the qft notes i · approximation methods such as perturbation theory and the 1/n expansion remain...
TRANSCRIPT
THE QFT NOTES I
Badis Ydri
Institut fur Physik, Humboldt-Universitat zu Berlin, D-12489 Berlin-Germany
July 22, 2010
Abstract
In this article we review some aspects of noncommutative fuzzy field/matrix theory
with special emphasis on the phenomena of emergence of gauge theory, geometry and
supersymmetry from pure matrix models.
I DEDICATE THIS WORK TO THE MEMORY OF MY DAUGHTER
NOUR YDRI (24-NOVEMBER 06,28-JUNE 07)
1
Contents
1 Background 5
2 Noncommutativity In Quantum Mechanics 9
2.1 Spatial Noncommutativity In External Magnetic Fields . . . . . . . . . . . . . . 9
2.2 The Quantum Atom And Noncommutative Algebras . . . . . . . . . . . . . . . 10
2.3 Regularization By Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Few Elements of Noncommutative Geometry 15
3.1 The Noncommutative Geometry of Riemannian Manifolds . . . . . . . . . . . . 15
3.2 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Dixmier Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Spectral Triples or K-cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 The Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Example 6: Connes Trace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.7 Axioms of NCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.8 Cyclic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.8.1 Fredholm module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.8.2 The differential envelope Ω of A . . . . . . . . . . . . . . . . . . . . . . . 26
3.8.3 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.8.4 The cycle (Ω, d, T rs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.9 The Hochschild Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 The Fuzzy Sphere 34
4.1 The Continuum Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Continuum Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 The Complex Structure on TS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Quantization of S2 = CP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 K-cycles And Spectral Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.6 Coherent States And Star Products on Fuzzy CPN−1L . . . . . . . . . . . . . . . 45
4.7 Fuzzy Dirac Operators And The Absence of Fermion Doubling . . . . . . . . . . 49
4.8 The Ginsparg-Wilson Algebra on S2L . . . . . . . . . . . . . . . . . . . . . . . . 53
4.9 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.10 Fuzzy Monopoles:First Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Fuzzy Scalar Field Theory and Matrix Phase 62
5.1 Action and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 The effective action and The 2−Point Function . . . . . . . . . . . . . . . . . . 62
5.3 The 4−Point Function And Normal Ordering . . . . . . . . . . . . . . . . . . . 65
5.4 The Phase Structure and Effective Potential . . . . . . . . . . . . . . . . . . . . 67
5.5 Fuzzy S2 × S2 And Planar Limit:First Look . . . . . . . . . . . . . . . . . . . . 69
2
5.6 Real Quartic Pure Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Noncommutative Moyal-Weyl Spaces 76
6.1 The Weyl Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Noncommutative Gauge Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 Renormlaized Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3.1 The Effective Action and Feynman Rules . . . . . . . . . . . . . . . . . . 80
6.3.2 Vacuum Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.3 The UV-IR Mixing and The Beta Function . . . . . . . . . . . . . . . . . 85
6.4 Star Products on S2N And R2
θ And Planar Limit . . . . . . . . . . . . . . . . . 87
7 Fuzzy Gauge Field Theory And UV-IR Mixing 91
7.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Tangent Projective Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.4 Effective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5 The Normal Scalar Field Effective Action . . . . . . . . . . . . . . . . . . . . . . 97
7.6 The UV-IR mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.7 One-Matrix Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.8 The Partition Function in The Trivial Sector . . . . . . . . . . . . . . . . . . . . 103
7.9 Gauge Theory on Fuzzy S2 × S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8 Fuzzy Gauge Theory and Emergent Geometry 108
9 Fuzzy Fermions and Emergent Supersymmetry 115
10 Fuzzy CP2 116
10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.2 Quantizing CP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.3 The spectral triple CP2N = (MatN ,∆N ,HN) . . . . . . . . . . . . . . . . . . . . 119
10.4 Fuzzy gauge fields on CP2N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.5 Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.5.1 Fuzzy CP2 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.5.2 A Stable Fuzzy Sphere Phase . . . . . . . . . . . . . . . . . . . . . . . . 127
10.5.3 The Transition CP2−→S2 . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.6 Phase Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.7 Fermion on Fuzzy CP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.7.1 The Spinc Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.7.2 The Tangent Gammas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.7.3 The Dirac Operator on CP2 . . . . . . . . . . . . . . . . . . . . . . . . . 134
10.7.4 Fuzzification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3
11 Fuzzy OSP (2, 1) Gauge Supersymmetry 136
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.2 The continuum supersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.2.1 The Lie algebras osp(2, 1) and osp(2, 2) . . . . . . . . . . . . . . . . . . . 137
11.2.2 The supersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.2.3 Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
11.3 Scalar action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
11.3.1 The D−term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
11.3.2 The F−term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11.4 The fuzzy supersphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
11.5 Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
11.5.1 Klimcik differential complex . . . . . . . . . . . . . . . . . . . . . . . . . 148
11.5.2 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
11.5.3 The product ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
11.5.4 Gauge action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.6 The continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
11.7 A new fuzzy SUSY scalar action . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
12 The Noncommutative Torus 162
12.1 The Noncommutative Torus T dθ . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
12.2 NC U(N) Gauge Theory on T dθ . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
12.3 The Weyl-’t Hooft Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
12.4 SL(d, Z) Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.5 Morita Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4
1 Background
It can be argued using the principles of quantum mechanics and classical general relativity
that the picture of spacetime at the very large as being a smooth manifold must necessarily
break down at the Planck scale λp. At this scale localization looses its operational meaning and
one expects spacetime uncertainty relations which in turn strongly suggest that spacetime has
a quantum structure expressed by [xµ, xν ] = iλ2pQµν . The geometry of spacetime at the very
small is therefore noncommutative.
Noncommutative geometry allows for the description of the geometry of smooth differen-
tiable manifolds in terms of the underlying C∗−algebra of functions defined on these mani-
folds. Indeed given the following three data a) the algebra A = C∞(M) of complex valued
smooth and continous functions on a manifold M , b) the Hilbert space H = L2(M,S) of
square integrable sections of the irreducible spinor bundle over M and c) the Dirac operator
D = γµ(∂µ + 12ωµabγ
aγb) associated with the Levi-Civita connection ω one can reconstruct com-
pletely the differential geometry of the manifold M . These three data compose the so-called
spectral triple (A,H,D) corresponding to the Riemannian manifold M .
Noncommutative geometry is more general than ordinary differential geometry in that it
also enables us to describe algebraically the geometry of arbitrary spaces (which a priori do
not need to consist of points) in terms of spectral triples (A,H,D). The paradigm of NCG
adopted so often in physics is to generalize the ordinary commutative space M by replacing
the commutative algebra A by a noncommutative algebra Aθ. The result of this deformation
is in general a noncommutative space Mθ defined precisely by the spectral triple (Aθ,Hθ,Dθ)
where Hθ and Dθ are the corresponding deformations of the Hilbert space H and of the Dirac
operator D respectively [7, 6, 71].
Noncommutative geometry was also proposed (in fact earlier than renormalization) as a
possible way to eliminate ultraviolet divergences in quantum field theories [8]. The quantum
spacetime of [9] is Lorentz-covariant based on the commutation relations [xµ, xν ] = iλ2pQµν with
Qµν satisfying [xλ, Qµν ] = 0, QµνQµν = 0 and (1
2ǫµνλρQ
µνQλρ)2 = 1. As it turns out QFT on
this space is ultraviolet finite which is a remarkable (but not quite surprising) consequence of
spacetime quantization. Indeed this phenomena of “regularization by quantization” already
happens in quantum mechanics. For example while classical mechanics fail to explain the
blackbody radiation in the ultraviolet (the UV catastroph) quantum mechanics reproduces the
correct (finite) answer given by the famous experimentally verified Stefan-Boltzman law.
Noncommutative field theory is by definition a theory based on a noncommutative space-
time. The most studied examples in the literature are the Moyal-Weyl spaces Rdθ which corre-
spond to the case Qµν = θµν where θµν are rank 2 (or 1) antisymmetric constant tensors. This
clearly breakes Lorentz symmetry. The corresponding QFT is not UV finite and furthermore
it is plagued with the so-called UV-IR mixing phenomena [28]. There are also many open
questions about the renormalizability , unitarity and causality of these theories [29].
These theories attracted a lot of interest in recent years because of their appearance in
string theory [18]. It was discovered that the dynamics of open strings moving in a flat space
5
in the presence of a non-vanishing Neveu-Schwarz B-field and with Dp-branes is equivalent to
leading order in the string tension to a gauge theory on a Moyal-Weyl space Rdθ. Extension of
this result to curved spaces is also possible at least in one particular instance, namely the case
of open strings moving in a curved space with S3 metric. The resulting effective gauge theory
lives on a fuzzy sphere S2N [19] .
We can find very few fundamental physical models which are amenable to exact treatment.
Approximation methods such as perturbation theory and the 1/N expansion remain crucial
tools in analyzing different physical systems. Among the important approximation methods
for quantum field theories are strong coupling methods of lattice gauge theories which are
based on lattice discretisation of the underlying spacetime. They are among the rare effective
approaches for the study of confinement in QCD and for the non-perturbative regularization
of QFT. One feature of naive lattice discretisations however can be criticised. They do not
retain the symmetries of the exact theory except in some rough sense. A related feature is that
topology and differential geometry of the underlying manifolds are treated only indirectly by
limiting the couplings to “nearest neighbours”. Thus lattice points are generally manipulated
like a trivial topological set with a point being both open and closed. The upshot is that these
models have no rigorous representation of topological defects and lumps like vortices, solitons
and monopoles.
A new approach to discretisation inspired by non-commutative geometry is being developed
since a few years. The key remark here is that when the underlying spacetime or spatial cut
can be treated as a phase space and quantized with a parameter h assuming the role of h, the
emergent quantum space is fuzzy and the number of independent states per (“classical”) unit
volume becomes finite. “Fuzzified” compact manifolds are ultraviolet finite and support only
finitely many independent states. Their continuum limits are the semiclassical h → 0 limits.
This unconventional dicretization of classical topology is not at all equivalent to the naive one
and it does significantly overcome the previous criticisms.
There are other reasons also to pay attention to fuzzy spaces. There is much interest among
string theorists in matrix models and in describing D-branes using matrices. Fuzzy spaces
lead to matrix models too and their ability to reflect topology better than elsewhere should
be interesting to people. They let us devise new sorts of discrete models and are interesting
from that perspective. In addition it has now been discovered that when open strings end on
D-branes which are symplectic manifolds then the branes can become fuzzy. Fuzzy spaces are
also very relevant to the concepts of emergent geometry and emergent supersymmetry.
The central idea behind fuzzy spaces is discretisation by quantization. It relies on the
existence of a suitable Lagrangian and therefore it does not always work. An obvious limitation
is that the parent manifold has to be even dimensional. Successful use of fuzzy spaces for
QFT requires also good fuzzy versions of the Laplacian, Dirac equation, chirality operator. A
family of classical manifolds elegantly escaping these limitations are the co-adjoint orbits of Lie
groups. For semi-simple Lie groups they are the same as adjoint orbits. It is a theorem that
these orbits are symplectic. They can often be quantized when the symplectic forms satisfy the
Dirac quantization condition. The resultant fuzzy spaces are described by linear operators on
6
irreducible representations of the group. For compact orbits the latter are finite-dimensional.
In addition the elements of the Lie algebra define natural derivations and that helps to find
Laplacian and the Dirac operator. We can even define chirality with no fermion doubling and
represent monopoles and instantons. These orbits therefore are altogether well-adapted for
QFT. The most important of all co-adjoint orbits is the sphere CP1≃S2 = SU(2)/U(1) which
leads after quantization to the fuzzy sphere.
One of the principal goals of these notes is the construction of a new nonperturbative method
for gauge theories with and without fermions based on the fuzzy sphere S2N [20]. Nonperturba-
tive regularization of supersymmetry is also possible [36]. It was first proposed in [16] that S2N
can be used as a nonperturbative regularization of QFTs.
A gauge-covariant, chiral-invariant or SUSY-invariant regularization of gauge theories, QCD2,4
and SUSY2 in particular, can be achieved by quantizing the underlying spacetime manifold
thereby replacing it by a noncommutative matrix model or a “fuzzy manifold”. Indeed if the
underlying spacetime manifold can be treated as a phase space one can quantize it in the usual
way with a parameter θ assuming the role of h. Naturally the emergent quantum space is
fuzzy with noncommuting coordinates and a finite number of degrees of freedom and as a con-
sequence it is ultraviolet finite. The continuum limit is the semi-classical θ−→0 limit. These
are essentially matrix models [13].
Noncommutative Moyal-Weyl spaces are also matrix models not continuum manifolds. How-
ever they act on infinite dimensional Hilbert spaces and thus we can think of fuzzy spaces as
precisely finite dimensional regularizations of these spaces. In other words the whole problem
of noncommutative Moyal-Weyl gauge theories is also amenable to the methods and techniques
of random matrix theory [30].
A typical example of fuzzy manifolds is the fuzzy sphere S2N . In 4−dimensions we have
3 fuzzy manifolds which are obtained from co-adjoint orbits. CP1N×CP1
N , CP2N [31] and
squashed CP3N [32]. At this stage S2
N×S2N is the only serious candidate which incorporates
fermions and gauge fields without much complication .
The advantage of this regulator compared to ordinary lattice prescription is that discretiza-
tion by quantization is remarkably successful in preserving symmetries and topological features.
By construction the fuzzy spheres S2N and S2
N×S2N have the same rotational symmetries of the
corresponding continuum spheres. We can also show that we have exact U(k) “local” gauge
symmetries on these fuzzy spheres which are implemented by U(kN) and U(kN2) unitary
matrices respectively. Chiral fermions have a simple form in this setting and one can show
that chiral symmetry is maintained without any fermion doubling. For example we can write
Ginsparg-Wilson relations which resemble the Ginsparg-Wilson relations on the lattice (and
hence the analogy between the two regularizations). Yet on the fuzzy spheres S2N and S2
N×S2N
we can capture correctly most of the topology. We can show explicitly that nontrivial field
configurations such as monopoles and instantons are reproduced correctly using the language
of projective modules and K-theory[33, 34].
More importantly is the possiblity of having exact supersymmetry on S2N . In [35] we reviewed
Klimcık’s construction of noncommutative gauge theory on the fuzzy supersphere. This theory
7
has an exact osp(2, 1) supersymmetric U(1) gauge theory with a finite number of degrees of
freedom which can in principle treated numerically using Monte Carlo simulation.
The motivation for studying QFT’s on fuzzy models is 3-fold.
• 1) This is a novel way of simulating ordinary gauge theories with and without fermions
based on random matrix models. Simulating SUSY is also possible.
• 2) Fuzzy spaces because of their close connection to Moyal-Weyl NC spaces provide a
systematic way of regularizing QFT’s on Moyal-Weyl spaces.
• 3) They provide concrete models for emergent geometry. Indeed geometry in transition
is possible in certain matrix models ( those which correspond to fuzzy CPn ) without the
need to specify any Laplacians.
Thus any result obtained will be of relevance either to ordinary gauge theories especially with
regard to fermions with and without SUSY or to Moyal-Weyl field theories. Let us also note
that an alternative way of regularizing gauge theories on the Moyal-Weyl NC space is based on
the matrix model formulation of the twisted Eguchi-Kawai model [37, 38, 39, 40].
Among the main tools we propose to use in implementing the above fuzzy regularization is
numerical simulations. Monte Carlo simulation is conceptually and practically simple in this
case. Fuzzy actions look exactly like the continuum where integrals are replaced with finite
dimensional traces, derivatives with commutators and fields with finite dimensional matrices.
Updating procedures consist therefore in updating matrix elements of field configurations [41].
These configurations in pure gauge models are precisely the ”covariant” coordinates on the
fuzzy space. In a pure gauge model gauge invariant observables are directly constructed as
polynomials in these ”covariant” coordinates in contrast to what is done in lattice gauge theory.
Other nonperturbative approaches which could be exploited are the 1/N expansion, random
matrix theory and renormalization group. For example the scheme of [44] where flow equations
are obtained by integrating out a column and a row of N×N matrix thus reducing it to (N −1)×(N −1) matrix seems very suited. This technique was applied to the λφ4 theory in [43] and
to U(1) gauge theory in [42] to derive one-loop corrections of the effective action. Since these
are matrix models a more systematic application of this scheme should yield nonperturbative
information. The 1/N expansion and random matrix theory were applied to U(1) gauge theory
in [42, 45].
8
2 Noncommutativity In Quantum Mechanics
2.1 Spatial Noncommutativity In External Magnetic Fields
Spacetime noncommutativity is inspired by Quantum Mechanics. When classical phase is
quantized we replace the canonical positions and momenta xi, pj with Hermitian operators xi, pj
such that
[xi, pj] = ihδij . (1)
The quantum phase space is seen to be fuzzy, i.e points are replaced by Planck cells due to the
basic Heisenberg uncertainty principle
∆x∆p≥1
2h. (2)
Von Neumann called this “pointless geometry” meaning that there are no underlying points.
The so-called Von-Neumann algebras can be viwed as marking the birth of Noncommutative
geometry. The commutative limit is the quasiclassical limit h−→0. In general one can argue
using the principles of quantum mechanics and classical general relativity that the picture of
spacetime at the very large as being a smooth manifold must necessarily break down at the
Planck scale λp. Measuring for example the coordinate x of an event with an accuracy a will
cause by the Heisenberg principle an uncertainty in momentum of the order of 1/a. An energy
of the order of 1/a is transmitted to the system and concentrated at some time around x . This
in turn will generate a gravitational field by Einstein’s equations for the metric. The smaller
the uncertainty a the larger the gravitational field which can then trape any possible signal
from the event. At this scale localization looses thus its operational meaning, the manifold
picture breakes down and one expects spacetime uncertainty relations which in turn strongly
suggest that spacetime has a quantum structure expressed by
[xµ, xν ] = iλ2pQµν . (3)
The geometry of spacetime at the very small is therefore noncommutative. In above Qµν is
some tensor structure.
Noncommutative field theory is by definition a theory based on a noncommutative space-
time. The most studied examples in the literature are the Moyal-Weyl spaces Rdθ which corre-
spond to [xµ, xν ] = iQµν where Qµν = θµν is a rank 2 (or 1) antisymmetric constant tensor.
These theories attracted a lot of interest in recent years because of their appearance in
string theory. It was discovered that the dynamics of open strings moving in a flat space in the
presence of a non-vanishing Neveu-Schwarz B-field and with Dp-branes is equivalent to leading
order in the string tension to a gauge theory on a Moyal-Weyl space Rdθ. Extension of this
result to curved spaces is also possible at least in one particular instance, namely the case of
open strings moving in a curved space with S3 metric. The resulting effective gauge theory
lives on a fuzzy sphere S2L.
9
This phenomena happens already in Quantum Mechanics. Consider the following La-
grangian
Lm =m
2(dxi
dt)2 − dxi
dt.Ai , Ai = −B
2ǫijxj . (4)
After quantization the momentum space becomes noncommutative
[πi, πj ] = iBǫij , πi = mdxi
dt. (5)
Spatial noncommutativity arises in the limit m−→0, i.e from
L0 = −B2ǫijdxi
dtxj . (6)
In this case we have
[xi, xj] = iθǫij , θ =1
B. (7)
The limit m−→0 keeping B fixed is the projection onto the lowest Landau level (recall that
the mass gap is Bm
). This projection is also achieved in the limit B−→∞ keeping m fixed.
The is precisely what happens in string theory. We get noncommutative gauge theories
on Moyal-Weyl planes or fuzzy spheres depending on whether the strings in a Neveu-Schwarz
B-field are moving in a flat or curved (with S3 metric) backgrounds respectively. The corre-
sponding limit is α′−→0.
2.2 The Quantum Atom And Noncommutative Algebras
Quantum Mechanics is the very first example in which the space (the phase space in this
case) becomes noncommutative. This is obvious from the fact that the phase space coordinates
q and p will satisfy in the quantum theory the nontrivial commutation relation [q, p] = ih. Phase
space acquires a cell-like structure with minimum volume given roughly by h. In this section
we will rederive this result in an algebraic form in which the noncommutativity is established
at the level of the underlying algebra of functions.
It is a textbook result that the classical atom can be characterized by a set of positive real
numbers νi called the fundamental frequencies. The atom if viewed as a classical system will
radiate via its dipole moment interaction until it collapses. The intensity of this radiation is
given by
In ∝ | < ν, n > |4
< ν, n > =∑
i
niνi, ni∈Z. (8)
It is clear that all possible emitted frequencies < ν, n > form a group Γ under the addition
operation of real numbers
Γ = < n, ν >;ni∈Z. (9)
10
Indeed given two frequencies < ν, n >=∑
i niνi and < ν, n′>=
∑
i n′
iνi in Γ it is obvious that
< ν, n+ n′>=
∑
i(ni + n′
i)νi is also in Γ.
The algebra of classical observables of this atom can be obtained as the convolution algebra
of the the abelian group Γ. To see how this works exactly one first recalls that any function on
the phase space X of this atom can be expanded as (an almost) periodic series
f(q, p; t) =∑
n
f(q, p;n)e2πi<n,ν>t;n ≡ (n1, ..., nk). (10)
The Fourier coefficients f(q, p;n) are labelled by the elements n∈Γ. The convolution product
is defined by
f ∗ g(q, p; t;n) =∑
n1+n2=n
f(q, p; t;n1)g(q, p; t;n2) (11)
f(q, p; t;n) = f(q, p;n) exp(2πi < n, ν > t). (12)
This leads to the ordinary commutative pointwise multiplication of the corresponding functions
f(q, p; t) and g(q, p; t), namely
fg(q, p; t) ≡ f(q, p; t)g(q, p; t) =∑
n
f1 ∗ f2(q, p; t;n). (13)
The key property leading to this result is the fact that Γ is an abelian group.
If we take experimental facts in our account then we know that the atom must obey the
Ritz-Rydberg combination principle which says that a) rays in the spectrum are labeled with
two indices and b) frequencies of these rays obey the law of composition, viz
νij = νik + νkj. (14)
We write this as
(i, j) = (i, k) (k, j). (15)
The emitted frequencies νij are therefore not parametrized by the group Γ but rather by the
groupoid ∆ of all pairs (i, j). It is a groupoid since not all frequencies can be composed to give
another allowed frequency. Every element (i, j) has an inverse (j, i) and is associative.
The quantum algebra of observables is then the convolution algebra of the groupoid ∆ and
it turns out to be a noncommutative (matrix) algebra as one can see by rewriting (12) in the
form
F1F2(i,j) =∑
(i,k)(k,j)=(i,j)
F1(i,k)F2(k,j). (16)
One can easily check that F1F2 6=F2F1 so F ′s fail to commute.
To implement the element of the quantum algebra as matrices one should replace f(q, p; t;n) =
f(q, p;n)e2πi<n,ν>t by
F (Q,P ; t)(i,j) = F (Q,P )(i,j)e2πiνijt. (17)
From here the Heisenberg’s equation of motion, phase space canonical commutation relations
and Heisenberg’s uncertainty relations follow in the usual way.
11
2.3 Regularization By Quantization
Noncommutativity was proposed by Snyder (in fact earlier than renormalization) as a possi-
ble way to eliminate ultraviolet divergences in quantum field theories. The quantum spacetime
with [xµ, xν ] = iQµν with [xλ, Qµν ] = 0, QµνQµν = 0, (1
2ǫµνλρQ
µνQλρ)2 = 1 is Lorentz-covariant.
As it turns out QFT on this space is ultraviolet finite which is a remarkable (but not quite
surprising) consequence of spacetime quantization. Indeed this phenomena of “regularization
by quantization” already happens in quantum mechanics. For example while classical mechan-
ics fail to explain the blackbody radiation in the ultraviolet (the UV catastroph) quantum
mechanics reproduces the correct (finite) answer given by the famous experimentally verified
Stefan-Boltzman law.
This noncommutativity-induced regularization does not work in general. Examples: 1)
Moyal-Weyl spaces Rdθ , 2) noncommutative tori Td
θ , 3) fuzzy spaces and 4) Doplicher-
Frednahgen-Roberts space. Field theory on the first two is divergent while on the last two
is finite.
The phenomena of regularization by quantization is best understood by studying the exam-
ple of the balckbody radiation. We imagine the blackbody as a rectangular cavity of dimensions
L1, L2 and L3, with perfectly reflecting walls. Inside the cavity the electromagnetic field is as-
sumed to be in thermal equilibrium with the atoms of the walls. The electromagnetic field can
be pictured as a set of an infinite number of independent electromagnetic normal modes with
frequencies ω ranging from 0 to ∞ . The total energy E per unite volume of the cavity is given
by the general expression
E∆V =∫
∆E =∫ ∞
0< E >ω dNω. (18)
< E >ω is the average energy of a mode with frequency ω and dNω is the total number of all
normal modes with frequencies between ω and ω + dω.
Let us first consider one dimension i. It is a well known result that a given electromagnetic
normal mode is a standing wave which vanishes on the walls so that it must obviously stretch
an integer number of half-wavelenghts across the interval Li , i.e
kni=
2π
λni
=niπ
Li
. (19)
The number of modes which have wave-vectors knibetween ki and ki + dki is (with ∆ki the
difference between two successive modes kniand kni+1) given by
dki
∆ki
, ∆ki =π
Li
. (20)
The number of modes with wave-vectors between 0 and ki is given by the integral
∫ ki
0
Li
πdki =
∫ ki
−ki
Li
2πdki. (21)
12
Hence the number of modes with wave-vectors between ki and ki + dki is given by
Lidki
2π. (22)
In 3 dimensions this will take the form
dN~k = ∆Vd3~k
(2π)3, V = L1L2L3 (23)
Now we would like to find the number of normal modes with frequencies between ω and ω+dω.
This is easily achieved by using the identity k = ωc
in (23)
dNω
2= ∆V
ω2
c2dω
c
dΩ
8π3, (24)
Taking into account the fact that each mode with a wave-vector ~k corresponds to two different
photons with the two possible orthogonal polarizations , one needs therefore to double the
above answer
dNω =∆V ω2
π2c3dω. (25)
Each mode with frequency ω can be thought of as a classical one-dimensional harmonic oscillator
(HO),viz
H(q, p) =p2
2m+
1
2mω2q2. (26)
It is not difficult to prove that the average energy of this HO is given by
< E >ω=< H > =
∫∞−∞ dq
∫∞−∞ dpH(q, p)e−
H(q,p)kT
∫∞−∞ dq
∫∞−∞ dpe−
H(q,p)kT
= kT. (27)
Using this result in (18) together with (25) we obtain the Rayleigh-Jeans’s law for the density
of energy ∆E∆V
, namely
E =∫
∆E
∆V=∫ ∞
0
kT
π2c3ω2dω = ∞ (28)
This is the ultraviolet catastrophe which started all quantum physics.
The quantization of the harmonic oscillator (26) is a straightforward exercise in quantum
mechanics. The quantization of this classical system, i.e of the classical phase space (q, p)together with the basic Poisson bracket q, p = −1 leads to the quantum phase space (Q,P )with the basic commutation relation i
h[Q,P ] = −1. The quantum states |n > of the quantized
harmonic oscillator have energies En = hω(n+12). These energy levels are clearly non-degenerate
and therefore the partition function is given by
Z = Tre−HkT =
∞∑
0
e−hωkT
(n+ 12) =
e−hω2kT
1 − e−hωkT
. (29)
The average of the Hamiltonian H of the harmonic oscillator is then given by
13
< H > = kT 2 1
Z
dZ
dT=hω
2+
hω
ehωkT − 1
. (30)
Using this last result in (18) together with (25) we obatin the Planck’s law for the density of
energy ∆E∆V
, i.e
E =∫ ∆E
∆V=
h
π2c3
∫ ∞
0
ω3dω
ehωkT − 1
, (31)
In above we have also identified < E >ω=< H > − hω2
. The integral is now finite and the result
is exactly equal to the Stefan-Boltzmann law for blackbody radiation , namely
E =∫
∆E
∆V=
π2k4
15h3c3T 4. (32)
14
3 Few Elements of Noncommutative Geometry
3.1 The Noncommutative Geometry of Riemannian Manifolds
Noncommutative geometry (NCG) allows for the description of the geometry of smooth
differentiable manifolds in terms of the underlying C∗−algebra of functions defined on these
manifolds. Given the following three data a) the algebra A = C∞(M) of complex valued
smooth and continous functions on a manifold M , b) the Hilbert space H = L2(M,S) of square
integrable sections of the irreducible spinor bundle over M and c) the Dirac operator
D = γaeµa∇µ , ∇µ = ∂µ +
1
2ωµabγ
aγb. (33)
Then we can reconstruct completely the differential geometry of the manifold M .The Hilbert
Space H is endowed with the usual scalar product
(ψ, φ) =∫
dµ(g)ψ(x)φ(x) (34)
dµ(g) is the invariant volume form with the metric g. Elements f∈A acts as multiplicative
operators on H, i.e
(fψ)(x) = f(x)ψ(x), ψ∈H. (35)
∇µ in the formula (33) is the lift of the Levi-Civita connection to the bundle of spinors. The
γa are the flat gamma matrices, γa, γb = −2gµνeµae
νb . ea = eµ
a∂µ is an orthonormal basis of
vector fields on the manifold. The coefficients ωµab of the torsion free Levi-Civita connection
are the solutions of
∂µeaν − ∂νe
aµ − ωa
µbebν + ωa
νbebµ = 0. (36)
In the absence of spin it is enough to work with the Laplacian (instead with the above Dirac
operator)
∆ = D2 = −gµν(∇µ∇ν − Γρµν∇ρ) +
1
4R. (37)
R scalar curvature and Γρµν are the Christoffel symbols.
These three data A, H and D compose the so-called spectral triple (A,H,D) corresponding
to the Riemannian manifold M . There are several axioms which must be satisfied by this
spectral triple. Let us say a little more about this spectral triple. The metric on M is given in
terms of the Dirac opeartor D. For example we can show that the geodesic distance between
any two points on M is given by
d(p, q) = supf∈A|f(p) − f(q)| : ||[D, f ]||≤1. (38)
For example in one dimension D = ddx
and thus ||[D, f ]|| = sup| dfdx| and hence ||[D, f ]||≤1 leads
to sup| dfdx|≤1 . This is saturated by functions f(x) = x + constant and this gives the usual
15
distance in one dimension. The Riemannian measure on M (with dimension d) is given by the
so-called Dixmier trace∫
Mf ∝ trω(f |D|−d) , f∈A. (39)
Roughly speaking trωO picks the coefficient of the logarithmic divergence of the ordinary trace
trO . |D|−d is the analogue of the volume form. Finally the dimension of the spectral triple is
equal to the dimension of the space. The dimension is detremined from the behaviour of the
characteristic values of |D|−d, i.e
µn(|D|−d)−→constant×1
n. (40)
This is indeed true for a Riemannian manifold where the above formula coincidses with the
Weyl formula.
NCG is more general than ordinary differential geometry in that it also enables us to describe
algebraically the geometry of arbitrary spaces (which a priori do not even need to consist
of points) in terms of spectral triples (A,H,D). The paradigm of NCG adopted so often
in physics is to generalize the ordinary commutative space M by replacing the commutative
algebra A by a noncommutative algebra Aθ. The result of this deformation is in general a
noncommutative space Mθ defined precisely by the spectral triple (Aθ,Hθ,Dθ) where Hθ and
Dθ are the corresponding deformations of the Hilbert space H and of the Dirac operator Drespectively. The resulting noncommutative algebra can generically be characterized by a set
of parameters θ in analogy with the h of quantum mechanics .
For this prescription to actually work we need first to make sure that the algbera is indeed
enough to reconstruct the space, i.e we should be able to reconstruct M from A. The Gel’fand-
Naimark theorem gurantees this equivalence. Indeed any commutative C∗−algebra can be
realized as the algebra of complex valued functions over a compact Hausdorff space. It turns
out that the algebra alone can only capture topological aspects of the space. The metric aspects
are encoded in the absence of spin in the Laplacian whereas in the presence of spin they are
encoded in the Dirac operator.
Connes’ noncommutative geometry (NCG) is a very precise construction in which the above
basic theorem is explicitly implemented not only for smooth differentiable manifolds but also
for general spaces. One fruitful way of introducing NCG is by stating axioms of diferential
geometry in a form suitable for generalization. Differential geometry will appear therefore as
only a very special case of NCG.
Before we state the axioms we need first to write down the spectral or quantized calculus
which is a generalization of the usual calculus on manifolds. It can be summarized in the
following table
Complex variables −→ Operators on a Hilbert space
Real variables −→ Selfadjoint operators
Infinitesimal −→ Compact operators
Integral −→ Dixmier trace. (41)
16
The first two lines are essentially borrowed from QM whereas the third and last lines will be
explained in the next two sections.
3.2 Compact Operators
An operator T on a Hilbert space H is said to be compact if it can be approximated in
norm by finite rank operators. More precisely
∀ ǫ>0, ∃ a finite dimensional space E∈H : ||TE⊥|| < ǫ. (42)
With this definition it is clear that compact operators are in a sense small. One can alternatively
define compact operators as follows: They admit a uniformly convergent (in norm) expansion
of the form
T =∑
n≥0
µn(T )|ψn >< φn|. (43)
We have 0≤µi+1(T )≤µi(T ). The |ψn >, |φn > are orthonormal (not necessarily complete)
sets. One can now make the following remarks. 1) The size of the compact operator T (in-
finitesimal) is governed by the rate of decay of the sequence µn(T ) as n−→∞. 2) If we polar
decompose T = U |T | where |T | =√T ∗T and U is the phase of T then one can show that the
characteristic values µn(T ) of T are basically the eigenvalues of |T | with eigenvectors |φn >.
The characteristic values µn(T ) satisfy
µn+m(T1 + T2) ≤ µn(T1) + µm(T2)
µn+m(T1T2) ≤ µn(T1)µm(T2)
µn(TT1) ≤ ||T ||µn(T1)
µn(T1T ) ≤ ||T ||µn(T1). (44)
The T1 and T2 are compact operators and T is a bounded operator. For n = m = 0 the above
inequalities can be shown by using the fact that
µ0(T ) = sup||T |χ > || : |χ > ∈H, |||χ > ||≤1= ||T || ≡ the operator norm. (45)
This trivially satisfies all of (44). One can also show that µn(T ), n6=0 behaves as a norm as
follows. First let L(H) be the set of all bounded operators on H and Rn the set of all operators
with rank less than n, i.e Rn = S∈L(H) : dim(ImS)≤n. Then from the above first definition
of compact operators one can write
µn(T ) = dist(T,Rn), ∀n∈NLimµn(T ) −→ 0 n−→∞. (46)
17
This is another way of writing that the compact operator T is a norm limit of operators with
finite rank. From this definition and the obvious inclusions Rn + Rm ⊂ Rn+m the inequalities
(44) follow easily. In showing (44) we need also to use the fact that the set of compact operators
forms a two-sided ideal in L(H), i.e RnL(H) = L(H)Rn = Rn. This is obvious since compact
operators among bounded operators are like infinitesimal numbers among numbers.
A compact operator T is of order α∈R+ iff
∃ C <∞ : µn(T )≤Cn−α, ∀n≥1
⇔ µn = O(n−α), n−→∞. (47)
Example 1: Let us check that some of the intuitive rules of calculus of infinitesimals are still
valid for compact operators. For example if T1, T2 are of orders α, β then T1T2 is of order α+β.
Wwe start with
µn+m(T1T2)≤µn(T1)µm(T2). (48)
But µn(T1) = O(n−α), µm(T2) = O(m−β) and µp(T1T2) = O(p−γ). α , β and γ are the orders
of T1 , T2 and T1T2 respectively. Then
O((n+m)−γ) ≤ O(n−α)O(m−β)
O(nα)O(mβ) ≤ O((n+m)γ)
=⇒O(nα+β−γ) ≤ 1. (49)
Where we have assumed that n = m . One then concludes that γ = α + β.
Example 2: The volume ddx in d dimensions is an infinitesimal of order 1 and therefore the
differential dx is of order 1/d1.
Example 3: On a d-dimensional manifold M the Dirac operator D = D+ = |D| has the
eigenvalues (Weyl formula)
µj(D) ≃ 2π(d
ΩdvolM)1/dj1/d , for large j. (50)
So D−1 is infintesimal of order 1/d and therefore D−d is an infinitesimal of order 1 .
1This is because by definition the integral has as a domain the set of compact operators of order 1, in other
words ddxf(x) must be a compact operator of order 1 and therefore ddx is a compact operator of order 1.
Remember that a bounded operator f(x) times a compact operator ddx is still a compact operator of the same
order.
18
3.3 Dixmier Trace
One starts with the usual trace which has as a domain the space L1 of trace class operators.
Let T∈L1 be a positive and compact operator of order 1, then one can compute
σN (T ) ≡ TrT |N =N−1∑
n=0
µn(T )≤C lnN +C
′
N+ finite terms. (51)
In other words the ordinary trace is at most logarithmically divergent and should be replaced
by
TrT−→LimN−→∞γN(T )
γN(T ) =σN(T )
lnN=
1
lnN
N−1∑
n=0
µn(T ). (52)
The sequence γN(T ) satisfies
γN(T1 + T2) ≤ γN(T1) + γN(T2)≤γ2N(T1 + T2)(1 +ln 2
lnN). (53)
We can see immediately that γN is not linear and that linearity will be recovered if the sequence
γN converges. One needs then to replace (52) by something else, namely
Trω(T ) = LimωγN(T ). (54)
This is the Dixmier trace. Limω is a linear form on the space of bounded sequences γN. It is
positive, linear, scale invariant and it converges to the ordinary limit if the sequence on which
it is evaluated converges. Explicitly, it satisfies
Trω(T ) ≥ 0
Trω(λ1T1 + λ2T2) = λ1TrωT1 + λ2TrωT2
Trω(BT ) = Trω(TB), B is a bounded operator
Trω(T ) = 0, if T is of order higher than 1. (55)
The 4th equation means that infinitesimals of order 1 are in the domain of the Dixmier trace
while those of order higher than 1 have vanishing trace. The proof is pretty obvious from the
above construction.
The Dixmier trace can be extended to the whole space L1,∞, the space of trace class compact
operators of order 1 because of the linearity of the trace and the fact that L1,∞ is generated by
its positive part.
Example 4: The Laplacian on a d-dimensional torus T d = Rd/(2πZ)d (and its eigenvalues)
is (are)
∆ = −(∂
∂x1)2 − .....− (
∂
∂xd)2
~p2 = p21 + ...+ p2
d. (56)
19
One would like to compute Trω∆−d/2. The eigenvalues of ∆−d/2 are µp(∆−d/2) = |~p|−d. The
multiplicity of this eigenvalue is the number of points in Zd of length |~p| which is proportional
to the volume
Np+dp −Np = Ωdpd−1dp. (57)
The Ωd is a d− 1 dimensional sphere. Therefore
1
lnN
N−1∑
n=0
µn(T ) =1
lnNk
∑
p≤k
p−d ∼ 1
d ln k
∫ k
1p−d(Ωdp
d−1dp) ∼ Ωd
d=⇒
Trω(∆−d/2) =Ωd
d. (58)
Since on the torus |D|2 = ∆ this result can be written as
Trω(|D|−d) =Ωd
d. (59)
The Dirac operator here seems to play the role of the metric.
3.4 Spectral Triples or K-cycles
An arbitrary space X can be always defined by a set (A,H,D) where A is an involutive
algebra of bounded operators on the Hilbert space H and D = D+ is an operator acting on H
with the properties
D−1 is a compact operator on H⊥
[D, a] is a bounded operator for any a∈A. (60)
H⊥ is the orthogonal complement of the finite dimensional kernel of D.
The above space is compact in the sense that the spectrum of D is by construction discrete
with finite multiplicity. A noncompact space will be obtained if the algebra A has no unit.
More precisely we need to replace the first line in (60) by the following condition: For any a∈Aand λ not in R , a(D − λ)−1 is a compact operator. (A,H,D) is called the spectral triple or
K-cycle and contains everything that is to know about our space.
A point in the above K-cycle, or noncommutative space, is a state on the C∗ algebra A. In
other words a linear functional
ψ : A−→C
ψ(a∗a) ≥ 0, ∀a∈A||ψ|| = sup|ψ(a)| : ||a||≤1. (61)
One can check that ||ψ|| = ψ(1) = 1. The set S(A) of all states is a convex space. In other
words given any two states ψ1 and ψ2 and a real number 0≤λ≤1 then λψ1 +(1−λ)ψ2 is ∈S(A).
The boundary of S(A) is generated by pure states.
20
The spectral triple X = (A,H,D) is said to be even (otherwise it is said to be odd) if there
is a Z2 grading Γ of H satisfying
Γ2 = 1,
Γ+ = Γ
Γ, D = [Γ, a] = 0, ∀a∈A. (62)
The spectral triple X = (A,H,D) is said to be real (otherwise it is said to be complex) if
there is an antilinear isometry, J : H−→H which satisfies the following
J2 = ǫ(d)1,
JD = ǫ′
(d)DJ,
JΓ = (i)dΓJ
J+ = J−1 = ǫ(d)J. (63)
The mod 8 periodic functions ǫ(d) and ǫ′(d) are given by
ǫ(d) = (1, 1,−1,−1,−1,−1, 1, 1)
ǫ′
(d) = (1,−1, 1, 1, 1,−1, 1, 1). (64)
If the space X is a Riemannian spin manifold M then the real structure is exactly the CP
operation
Jψ = Cψ. (65)
C being the charge conjugation operator .
3.5 The Dirac Operator
The Dirac operator of the K-cycle X = (A,H,D) can be used to define a distance formula
on the space S(A): the space of states on the algebra A. Given two states (points) on A (of
X), ψ1 and ψ2 the distance between them is given by
d(ψ1, ψ2) = supa∈A|ψ1(a) − ψ2(a)| : ||[D, a]||≤1,(66)
D essentially contains all the metric informations of the space X.
Example 5: Let us check that the distance formula (66) will reduce to the ordinary distance
when the space X is an ordinary manifold M . In this case, A = C∞(M), D = γµ∂µ. The space
of states S(C∞(M)) is now the space of characters M(C∞) which can be identified with the
manifold itself as follows
x∈M −→ ψx∈M(C∞)
ψx(f) = f(x), ∀f∈C∞(M). (67)
21
The distance (66) takes then the form
d(x1, x2) = supf∈C∞(M)|f(x1) − f(x2)| : ||[D, f ]||≤1.(68)
Next since [D, f ] = γµ∂µf one has ||[D, f ]|| = supx∈M ||~∂f || and hence
|f(x2) − f(x1)| = |∫ x2
x1
~∂f.d~x|≤∫ x2
x1
|~∂f.d~x|≤∫ x2
x1
|~∂f |ds≤∫ x2
x1
||[D, f ]||ds≤∫ x2
x1
ds.
=⇒d(x1, x2) = Inf(
∫ x2
x1
ds). (69)
In the above proof we have assumed for simplicity that the functions f∈C∞(M) are real valued.
However the result (69) will also hold if f are complex valued functions on M . The only
difference is that one finds now that the norm of the bounded operator [D, f ] is equal to the
Lipschitz norm of f ,
||[D, f ]|| = ||f ||Lip = supx1 6=x2
|f(x1) − f(x2)|Inf(
∫ x2x1ds)
. (70)
3.6 Example 6: Connes Trace Theorem
Before we state the first axiom of NCG, one needs to do one more computation in which
one sees once more that the Dirac operator D = iγµ∂µ of a d dimensional spin manifold is
intimately related to the metric. More precisely, one would like to show that the Riemannian
measure on M is given by
Trωf |D|−d =∫
Mf(x)
√
detg(x)dx1∧dx2∧..∧dxd. (71)
The first step is to recognize that the Dirac operator D is a first order elliptic pseudodifferential
operator. The statement that D is a first order is trivial as one can see from its expression .
Being pseudodifferential operator means that it is an operator between two Hilbert spaces H1
and H2 of sections of Hermitian vector bundles over M which can be written in local coordinates
as
Dψ(x) =1
(2π)d
∫
eip(x−y)a(x, p)ψ(y)ddyddp , a(x, p) = −pµγµ. (72)
In this case H1 = H2 = L2(M,S): the Hilbert space of square integrable sections of the
irreducible spinor bundle over M . a(x, p) is called the principal symbol of the operator D and
since it is invertible for p 6=0 the Dirac operator is called elliptic .
It is not difficult to check that the principal symbol of the second order operator D2 will
be given by p21 = ηµνpµpν1 where we have assumed that we are in a locally flat metric (which
can always be done). One can then compute the principal symbol of D−2 as follows
D−2ψ(y) =1
(2π)d
∫
eip(y−x)a(x, p)ψ(x)ddxddp , a(x, p) =1
|p|21. (73)
22
This operator is of order −2. From here one can directly conclude that the operator |D|−d =
(D2)−d/2 is also a pseudodifferential operator of order −d and its principal symbol is given by
|p|−d1.
Now given any f∈C∞(M) it will act as a bounded multiplicative operator on the Hilbert
space and therefore the operator f |D|−d will be also a pseudodifferential operator of order −dwith a principal symbol given by f(x)|p|−d1.
The identity 1 which appears above is an N×N unit matrix which acts on the spinor bundle
so N = 2d/2 for even dimensional manifolds and N = 2(d−1)/2 for odd dimensional manifolds.
The next step is to use the famous Connes trace theorem which asserts that the Dixmier
trace of a pseudodifferential operator (of order −d ) over a d dimensional Riemannian manifold
is proportional to the Wodzicki residue of that operator. More precisely
TrωA =1
d(2π)dWresA
=1
d(2π)d
∫
S∗MTr[a(x, p)]σpdx
1∧dx2...∧dxd.
σp =d∑
j=1
(−1)j−1pjdp1∧..∧ ˆdpj∧...∧dpd.
S∗M = (x, p)∈T ∗M : |p| = 1. (74)
Therefore it is straight forward to see that
Trωf |D|−d =1
d(2π)dWresf |D|−d
=1
d(2π)d
∫
S∗MTr[f(x)|p|−d1]σpdx
1∧dx2...∧dxd.
=N
d(2π)d
∫
Sd−1σp
∫
Mf(x)dx1∧dx2...∧dxd.
=NΩd−1
d(2π)d
∫
Mf(x)dx1∧dx2...∧dxd. (75)
3.7 Axioms of NCG
Axiom 1: Dimension We are now ready to state the first axiom of NCG. It is written as
|D|−2 =d∑
µ,ν=1
[F,Xµ]∗ηµν [F,Xν ]. (76)
F is the sign of the Dirac operator, viz F = D|D| and it is assumed to satisfy [F, |D|−2] = 0.
Xµ are the generators of A while η = (ηµν) is in Md(A), i.e d×d matrices with entries in A.
The positive compact operator |D|−2 can be thought of as the square of the infinitesimal length
element over the space X = (A,H,D).
23
Usually this axiom is formulated as follows: D−1 is a compact operator of order 1/d where d
will be by definition the dimension of the above space (K-cycle) X ≡ (A,H,D). This is called
the Dimension Axiom.
Example 7: Let us compute the dimension of S2 from its Dirac operator. The Dirac operator
on S2 is known to have the form
D = ~σ.~L+ 1 (77)
Its square D2 has the spectrum
k2 = (j +1
2)2, j = l +
1
2or j = l − 1
2. (78)
k2 = (j + 12)2 is an eigenvalue of D2 with a total multiplicity equal to 4(j + 1
2) = 4k. One can
then compute
Trω|D|−2 = Limω1
lnNM
M∑
k=1
1
k2×4k
= Limω4
lnNMlnM. (79)
But NM =∑M
k=1 4k = 2M(M + 1) and therefore
Trω|D|−2 = 2. (80)
This equation means that |D|−2 is in the domain of the Dixmier trace and therefore it is a
compact operator of order 1. D−1 is then a compact operator of order 1/2 which leads to the
conclusion that the dimension of S2 is 2. (80) is exactly the Euler character of S2 .
Axiom 2: Reality One can use the real structure J introduced in equation (63) to define
the opposite algebra A0. Its elements a0 are defined by
a0 = Ja∗J+. (81)
We define the product
(ba)0 = a0b0. (82)
The two algebras A and A0 are also required to commute with each other , i.e
[a, b0] = 0, ∀a, b∈A. (83)
Equation (83) is the reality axiom of noncommutative geometry .
Axiom 3: First Order The real structure J and the Dirac operator D should also satisfy
one more condition known as the first order axiom of noncommutative geometry. For all a and
b∈A one must have
[[D, a], b0] = 0. (84)
24
Example 8: In the case of ordinary commutative manifold M with a Dirac operator D =
iγµ∂µ the opposite algebra A0 coincides with the algebra A itself since (ba)0 = a0b0 = b0a0 and
therefore the condition (84) will simply mean that the Dirac operator is a first order operator.
Axiom 4: Regularity For all a∈A the operator [D, a] is a bounded operator on the Hilbert
space H and both a and [D, a] are in the domain of δm for all integers m. δ being the derivation
defined by δ(a) = [|D|, a]. This is the algebraic formulation of the smoothness of the elements
of A.
There are three more axioms regarding orientability, finiteness of the K-cycle and Poincare
duality and K-theory whose discussions will take us out of the scope of these notes so we stop
here .
The subject of the next two sections will be the construction of noncommutative differential
calculus .
3.8 Cyclic Cohomology
3.8.1 Fredholm module
Given the K-cycle X = (A,H,D) we introduce a Fredholm module structure by defining on
H2 = H⊕H the operator
F =
(
0 D−1
D 0
)
. (85)
By construction (85) is such that F 2 = 1 . Let π be an involutive representation of the algebra
A on the Hilbert space H2 = H⊕H given by
∀f∈A : π(f) =
(
f 0
0 f
)
. (86)
The exterior derivative df of any element f of A is defined by df = i[F, π(f)] or
df = i
(
0 [D−1, f ]
[D, f ] 0
)
. (87)
We will assume that [F, π(f)] is a compact operator on H2 for any f in A. In other words df
is an infintesimal variable. we will also assume that π(f)(F − F+) is a compact operator. The
pair (H2, F ) define a Fredholm module, it is an even Fredholm module if the Hilbert space H
admits a Z/2 grading Γ. On H2 the chirality operator is therefore given by
Γ2 =
(
Γ 0
0 Γ
)
. (88)
By construction we have Γ22 = 1, Γ+
2 = Γ2,Γ2F = −FΓ2 and Γ2π(f) = π(f)Γ2 for any element
f of the algebra A.
25
The Schatten-Von Neumann ideal of compact operators Lp (where p is a real number ≥1)
is defined as the space of all bounded operators T on H2 such that the trace of |T |p is finite, in
other words
∞∑
n=0
(µn(T ))p <∞. (89)
The µn(T ) is the nth eigenvalue of |T |. The above condition simply means that the eigenvalues
of T must decrease fast enough at infinity. These classes are used to measure the size of the
differential [F, π(f)]. One other remark is that Lp⊂Lq if p≤q which can be written as
L1⊂L2⊂...⊂Lp⊂...⊂L∞. (90)
The L1 are trace-class operators, L2 are Hilbert-Schmidt operators and L∞ are the compact
operators.
A Fredholm module (H2, F ) is called p-summable if
[F, π(f)]∈Lp(H2), ∀f∈A. (91)
3.8.2 The differential envelope Ω of A
Let n≥0 be an even integer and let us assume that our Fredholm module (H2, F ) is (
n+1)-summable, in other words
[F, π(f)]∈Ln+1(H2), ∀f∈A. (92)
We can associate to the algebra A a bigger algebra Ω called the differential envelope of A in
the following way
Ω =⊕n
k=0Ωk. (93)
Ω0 = A and Ωk, k > 1 is the space generated by the operators
ω = π(f0)[F, π(f1)][F, π(f2)]....[F, π(fk)]. (94)
f0,f1....,fk are elements of A. In fact the operators ω define the space of k-forms over the algebra
A, in particular Ω1 is the space of one forms and Ω2 is the space of two-forms. The Holder
inequality is given by
Lp1Lp2.....Lpk⊂Lp, for1
p=
k∑
j=1
1
pj, (95)
One can immediately see that Ωk⊂Ln+1k . The product in Ω is the product of operators given
by
∀ψ∈Ωk, ∀φ∈Ωp : ψφ∈Ωk+p. (96)
26
3.8.3 The exterior derivative
The differential envelope Ω is a graded algebra in the following sense. In general one can
define the exterior derivative d as a map from Ω into Ω given by
dω = i(Fω − (−1)kωF ), ∀ω∈Ωk. (97)
For k = 0, (97) reduces precisely to (87). More precisely this exterior derivative dmaps k−forms
into (k + 1)−forms, in other words given a k−form ω one can compute
dω = i[
Fω − (−1)kωF]
= i[
iFπ(f0)[F, π(f1)][F, π(f2)]....[F, π(fk)]
− i(−1)kπ(f0)[F, π(f1)][F, π(f2)]....[F, π(fk)]F]
= i[F, π(f0)][F, π(f1)][F, π(f2)]....[F, π(fk)]. (98)
We have used the expression (94) of ω∈Ωk and the identity F [F, π(f)] = −[F, π(f)]F . From
(98) it is very clear that dω is in Ωk+1 which means that
d : Ωk−→Ωk+1. (99)
Now let us check that d is a graded derivation as follows
d(ω1ω2) = i[
Fω1ω2 − (−1)kω1ω2F]
= i[
[Fω1 − (−1)k1ω1F + (−1)k1ω1F ]ω2 − (−1)kω1ω2F]
= (dω1)ω2 + iω1
[
(−1)k1Fω2 − (−1)k1+k2ω2F]
= (dω1)ω2 + (−1)k1ω1(dω2). (100)
For k1 =even , (100) is exactly Leibnitz’s rule. One can write the definition (97) of the exterior
derivative as follows: dω = i(Fω − (−1)kωF ) = i(Fω − Γ2ωΓ2F ). The proof is simple and
consists in the observation that ω contains (k+1) elements of the algebra A which all commute
with Γ2 while the k operators F anticommute with Γ2 and therefore the extra sign (−1)k. So
dω = i(Fω − Γ2ωΓ2F ) = iΓ2(Γ2Fω − ωΓ2F ) = iΓ2[Γ2F, ω]. (101)
From (101) it is easily verified that d satisfies d2 = 0 , indeed
27
d2ω = iΓ2[Γ2F, dω] = iΓ2[Γ2F, iΓ2[Γ2F, ω]]
= −Γ2[Γ2F, Fω − Γ2ωΓ2F ] = −F (Fω − Γ2ωΓ2F ) + Γ2(Fω − Γ2ωΓ2F )Γ2F
= −F 2ω + FΓ2ωΓ2F − FΓ2ωΓ2F + ωF 2
= 0. (102)
The pair (Ω, d) defines a graded differential algebra .
3.8.4 The cycle (Ω, d, T rs)
One can define a closed graded trace of degree n , recalling that n is an even integer
introduced in equations (92) and (93), by
Trs : Ωn −→C
ω −→Trs(ω) = Tr′
(Γ2ω). (103)
Tr′(x) is defined only for those x’s which are such that the combination Fx+ xF is in L1(H2),
i.e it belongs to the space of trace class operators. It is given by
Tr′
(x) =1
2Tr(F (Fx+ xF )). (104)
To prove that the combination Fx + xF for x = Γ2ω is in fact in L1(H2) we simply compute
FΓ2ω + Γ2ωF = iΓ2dω where we have used the fact that ω∈Ωn and that n is even. dω is
clearly in Ωn+1 but by using Holder inequality we can check that Ωn+1⊂L1(H2). Hence dω and
therefore FΓ2ω + Γ2ωF are ∈L1(H2).
Since Trs(ω) depends only on dω and d2 = 0 it is trivial to see that Trs(dω) = 0. In other
words the trace (103) is closed. The trace (103) is also a graded trace because ∀ω1∈Ωk1 and
∀ω2∈Ωk2 such that k1 + k2 = n we find
Trs(ω1ω2) = − i
2TrΓ2Fd(ω1ω2)
= − i
2TrΓ2F
[
dω1ω2 + (−1)k1ω1dω2
]
= − i
2TrΓ2
[
(−1)k1+1dω1Fω2 + (−1)k1Fω1dω2
]
. (105)
In above we have used the identity Fdω1 + (−1)k1dω1F = 0. The triplet (Ω, d, T rs) defines a
cycle with dimension n over the the algebra A. It is a theorem that this cycle is essentially
determined by its character τ , i.e the (n+ 1)−linear function defined by
τn(π(f0), π(f1), π(f2)...., π(fn)) = Tr′
Γ2π(f0)[F, π(f1)][F, π(f2)].....[F, π(fn)]. (106)
τn is called the character or the cocycle of the cycle (Ω, d, T rs) .
28
3.9 The Hochschild Complex
Let A∗ be the algebraic dual of A , i.e the space of all linear functionals φ on A
φ : A −→C
f −→φ(f). (107)
A∗ is a bimodule in the sense that for any a and b in A and φ∈A∗ the object aφb is in A∗ defined
by
aφb(c) = φ(bca) (108)
Let now Cp = Cp(A,A∗) be the space of all p−linear maps from A to A∗. Any element T of
Cp can be viewed as a (p+ 1)−linear functional τ on A given by
τ(π(f0), π(f1), ...π(fp)) = [T (π(f1), ...π(fp))](π(f0))∈C. (109)
The Hochschild coboundary map b is defined as follows. To the boundary bT corresponds a
(p+ 2)−linear functional bτ given by
[bτ ](π(f0), ....π(fp+1)) = τ(π(f0)π(f1), ....π(fp+1))
+p∑
i=1
(−1)iτ(π(f0), ..., π(fi)π(fi+1), ..., π(fp))
+ (−1)p+1τ(π(fp+1)π(f0), ..., π(fp)). (110)
Hochschild cochains of degree p are defined to be those elements T∈Cp which are also linear
functionals τ on Ωp defined by
τ(ω) = τ(π(f0), π(f1), ..., π(fp)) , ω = π(f0)dπ(f1)....dπ(fp)∈Ωp. (111)
They must also vanish on the dΩp−1 part of Ωp, i.e
τ (dω′
) = 0. (112)
ω′∈Ωp−1. Now we define the Hochschild cocycles as all those Hochschild cochains which satisfy
the extra following condition
[bτ ](ω) = 0. (113)
[bτ ](ω) = [bτ ](π(f0), ..., π(fp+1)) and ω = π(f0)dπ(f1)...dπ(fp+1)∈Ωp+1. By definition the p−th
cohomology group of the algebra A with coefficients in A∗ is the cohomology Hp = Hp(A,A∗) of
the Hochschild complex (Cp(A,A∗), b). Finally we define the cyclic cocycles as those Hochschild
cocycles which satisfy
τγ = ǫ(γ)τ. (114)
29
The γ denotes any cyclic permutation of 0, 1, ...p2 and ǫ(γ) is the corresponding sign. For
even permutations it is plus whereas for odd permutations it is minus. Given an arbitrary
Hochschild cocycle τ we can associate to it a cyclic cocycle as follows
Aτ =∑
γ∈Γ
ǫ(γ)τγ . (115)
The Γ stands for the group of cyclic permutations of 0, 1, ..., p and A is a linear map from
Cp into Cp given by the above equation, i.e (115). Obviously the range of A is the subspace
Cpλ of Cp, namely the space of Hochschild cocycles which satisfy equation (114). Although
the Hochschild coboundary operator b does not commute with cyclic permutations , it can be
proven that it maps cyclic cocycles to cyclic cocycles. By definition the p−th cyclic cohomology
group of the algebra A with coefficients in A∗ is the cohomology HCp = HCp(A,A∗) of the
cyclic complex (Cpλ(A,A∗, b). Clearly (Cp
λ, b) is a subcomplex of the Hochschild complex.
In the previous section we associated an (n + 1)−summable Fredholm module structure
(H2, F ) to the even K-cycle X = (A,H,D). Then this Fredholm module was completely
characterized by the charcacter τn of its cycle (Ω, d, T rs). This character τn is explicitly given
by
τn(π(f0), π(f1), ..., π(fn)) =1
inTrΓ2π(f0)dπ(f1)...dπ(fn). (116)
Recall that n was taken to be even. This character is clearly an (n + 1)−linear map from the
algebra A into the complex numbers. It can be associated with an element Tn∈Cn(A,A∗) in
the following way
Tn[(π(f1), ..., π(fn))](π(f0)) = τn(π(f0), π(f1), ..., π(fn)). (117)
Tn is an n−linear map from A to A∗ . In the same way one can associate to τn a map τn from
Ωn into C by the equation
τn(ω) = τn(π(f0), π(f1), ..., π(fn)) =1
inTrΓ2ω
ω = π(f0)dπ(f1)....dπ(fn)∈Ωn.
(118)
Let us now check that this character τn is a cyclic cocycle. First one needs to check that it is
a Hochschild cochain, in other words for any ω′∈Ωn−1 we must have τn(dω
′) = 0. Indeed for
ω′∈Ωn−1 we have
dω′
= dπ(f1)dπ(f2)....dπ(fn). (119)
2τγ(π(f0), π(f1), ..., π(fp)) = τ(π(fn0), π(fn1
), ..., π(fnp)) where n0, n1, ..., np is the permutation γ∈Γ of
0, 1, ..., p.
30
Therefore
τn(dω′
) = τn(1, π(f1), ...π(fn)) = TrΓ2[F, π(f1)]....[F, π(fn)]
= TrΓ2Fπ(f1)[F, π(f2)]...[F, π(fn)]
− (−1)n−1TrFΓ2π(f1)[F, π(f2)]...[F, π(fn)]
= 0. (120)
We have also used the identity F [F, π(f)] = −[F, π(f)]F . Next one must check that τn is a
Hochschild cocycle, i.e bτn = 0 or more precisely
[bτn](π(f0), ..., π(fn+1)) = τn(π(f0)π(f1), ..., π(fn+1))
+n∑
i=1
(−1)iτn(π(f0), ..., π(fi)π(fi+1), ..., π(fn))
+ (−1)n+1τn(π(fn+1)π(f0), ..., π(fn)). (121)
To prove (121) let us simply compute the second term above
n∑
i=1
(−1)iτn(π(f0), ..., π(fi)π(fi+1), ..., π(fn)) = τ(π(fn+1)π(f0), ..., π(fn))
− τ(π(f0)π(f1), ..., π(fn+1)). (122)
From this last result one can easily see that bτn = 0 as desired. Finally one must check that τnis a cyclic cocycle, in other words
τn(π(f0), π(f1), ..., π(fn)) = τn(π(f1), ..., π(fn), π(f0)). (123)
Indeed
τn(π(f0), π(f1), ..., π(fn)) = TrΓ2π(f0)[F, π(f1)]...[F, π(fn)]
= TrΓ2π(f1)[F, π(f2)]...[F, π(fn)][F, π(f0)]
= τn(π(f1), ..., π(fn), π(f0)). (124)
With this result one concludes the proof that the character τn of the Fredholm module (H2, F ) is
a cyclic cocycle . The integer n in all the above equations is by construction the smallest integer
compatible with the (n+ 1)− summability of the Fredholm module (H2, F ) . For example it is
equal to 2 in the case of the sphere .
Example 9: The cyclic cocycle τ2 of the two dimensional cycle (Ω, d, T rs) is given by
τ2(π(f0), π(f1), π(f2)) = TrΓ2π(f0)[F, π(f1)][F, π(f2)]. (125)
31
By using equations (85), (86) and (88) we obtain
τ2(π(f0), π(f1), π(f2)) = TrΓf0[D−1, f1][D, f2] + TrΓ2f0[D, f1][D
−1, f2]. (126)
By using the identity [D−1, f ] = −D−1[D, f ]D−1 we obtain
τ2(π(f0), π(f1), π(f2)) = −TrΓD−1[D, f0]D−1[D, f1]D
−1[D, f2]. (127)
From this last result one sees that the cyclic cocycle τ2 can be interpreted as a fermionic one-
loop Feynman diagram with one insertion of the helicity operator Γ. For example if we consider
A to be the algebra of smooth functions on the sphere then (127) takes the form
τ2(π(f0), π(f1), π(f2)) = −∫
S2TrΓD−1[D, f0]D
−1[D, f1]D−1[D, f2]d
2x. (128)
The trace is now only over the spin indices. The element fi is a superposition of exponentials
exp(ikix) and therefore in the Fourier space [D, fi] will appear as an insertion of γµkµi at
the vertex i. D−1 appears as a propagator. Finally the overall conservation of momentum
δ(k0 + k1 + k2) reduces the number of variables to two.
Next one would like to extend the cyclic cocycle τ2 which is defined over the algebra A to a
cyclic cocycle τ e2 which is defined over the algebra M2(A), i.e the algebra of 2×2 matrices with
entries in the algebra A. The extended cyclic cocycle τ e2 should also satisfy the two conditions
satisfied by the original cyclic cocycle τ2, namely
τ e2 (π(f0), π(f1), π(f2)) = τ e
2 (π(f2), π(f0), π(f1))
bτ e2 = 0. (129)
This extension is given by ∀f0, f1, f2 inM2(A) we write (with f0 = σifi0, f1 = σif
i1 and f2 = σif
i2)
τ e2 (π(f0), π(f1), π(f2))≡Trσiσjσk.τ2(π(f i
0), π(f j1 ), π(fk
2 )), (130)
It is obvious that τ2(π(f i0), π(f j
1 ), π(fk2 )) is well defined since f i
0 , f j1 and fk
2 are all elements of
the algebra A. We can also (easily) check that the two properties given in equation (129) are
both satisfied. One of the central objects in these notes is
τ e2 (P ) ≡ τ e
2 (P, P, P ). (131)
P is an arbitrary idempotent ofM2(A) so it is a selfadjoint element of M2(A) which also satisfies
P 2 = P . As we will see (131) can be interpreted as the Chern character of some bundle. The
last thing one needs to do here is to check the stability of (131) under the deformation of P
among the idempotents ofM2(A). In other words under P−→P ‘ = UPU−1 where U is a unitary
transformation. One must have τ e2 (P ‘) = τ e
2 (P ). For infintesimal transformations U = 1 + T
we have P ‘ = P + δP where δP = [T, P ]. If we write P ‘ = σif′i and P = σif i then δP = σiδf
i
where δf i = f′i − f i. Hence
32
τ e2 (P ‘) = τ e
2 (P ‘, P ‘, P ‘)≡Trσiσjσk.τ2(π(f′i), π(f
′j), π(f′k))
= τ e2 (P, P, P ) + τ e
2 (P, P, δP ) + τ e2 (P, δP, P ) + τ e
2 (δP, P, P ). (132)
Hence
δτ e2 = 3τ e
2 (δP, P, P ). (133)
We have clearly used the fact that the extended cyclic cocycle τ e2 is symmetric under cyclic
permutations. Now from the fact that bτ e2 = 0 we have
τ e2 (TP 1, P 2, P 3) − τ e
2 (T, P 1P 2, P 3) + τ e2 (T, P 1, P 2P 3) − τ e
2 (P 3T, P 1, P 2) = 0. (134)
We get for P 1 = P 2 = P 3 the result
τ e2 (TP, P, P )− τ e
2 (PT, P, P ) = 0=⇒ τ e2 (δP, P, P ) = 0. (135)
Also we used δP = TP − PT . Hence δτ e2 = 0 and therefore the cyclic cocycle τ e
2 is stable.
33
4 The Fuzzy Sphere
4.1 The Continuum Sphere
The sphere is a two dimensional compact manifold defined by the set of all points (x1, x2, x3)
of R3 which satisfy :
x21 + x2
2 + x23 = R2. (136)
The algebra A of smooth , complex valued and square integrable functions on the sphere is
commutative with respect to the pointwise multiplication of functions. A basis for this algebra
can be chosen to be provided by the spherical harmonics Ylm, namely
f(x) = f(θ, φ) =∑
i1,...,ik
fi1...ikxi1 ...xik =∑
lm
clmYlm(θ, φ). (137)
A manifestly SU(2)−invariant description of A can be given as follows [64]. The algebra A is
the quotient of the algebra C∞(R3) of all smooth functions of R3 by its ideal I consisting of
all functions of the form : h(x)(xixi − R2). A scalar product on A is then given by: (f, g) =1
2πR
∫
d3xδ(xixi − ρ2)f ∗(x)g(x). Here f, g∈A and f(x),g(x) are their representatives in C∞(R3)
respectively. For example the norms of the generators xi of the algebra A are computed to be
‖xi‖2 = (xi, xi) = R2
3.
Now we would like to define the spinor bundle over the sphere S2. One starts first by
defining the Clifford algebra associated with the vector space R3. It is a complex algebra
generated by 3 self adjoint elements γα which satisfy the relations γαγβ + γβγα = 2δαβ, and
which are represented by 2×2 pauli matrices.
The spin group spin(3) is known to be equal to SU(2). It is the universal covering group
of SO(3). It consists of all the 2×2 transformations S(Λ) defined by Λαβγ
β = S−1(Λ)γαS(Λ)
and detS(Λ) = 1 where Λ is in SO(3). This map is clearly double valued because both S(Λ)
and −S(Λ) correspond to the same Λ in SO(3), in other words spin(3) covers SO(3) twice. For
transformations Λ near the identity, Λαβ ≃ δα
β + λαβ , the above map will have one solution given
by S(Λ) ≃ 1 + 14λαβγ
αγβ.
The spinor bundle over S2 will be defined in three steps. First one defines the principal fiber
bundle E = (spin(3), π,R3) over the base manifold R3 by the projection map
π : spin(3)−→SO(3) , S(Λ)−→Λ. (138)
The tangent space at each point p of the base manifold is R3 and therefore at each point p we
have a representation Λ of SO(3) which is acting naturally so that the above map (138) induces
essentially a projection of spin(3) onto R3. Thus the fiber of the above bundle E is spin(3)×Z2,
i.e π−1(p) = [spin(3)×Z2]p.
The second step is to associate with the above bundle E , a spinor bundle E3 over R3.
Following [65], this is done by first remarking that the structure group of E is spin(3). Then by
34
construction the associated bundle , E3 = (E3, π3,R3) , can have as a fiber any space on which
spin(3) is acting on the left. For obvious reasons we choose the Hilbert space of spinors
H3 = C∞(R3)⊗C2. (139)
If one now defines the right action of g∈spin(3) on the space spin(3)×H3 by
(h, ψ)g = (hg, g−1ψ); ∀h∈ spin(3), ψ∈H3. (140)
Then the associated spinor bundle E3 has the total space
E3 = [spin(3)×H3]/spin(3). (141)
In other words the two points (h, ψ) and (hg, g−1ψ) are identified. The projection map is defined
by
π3 : E3−→R3
[(h, ψ)]−→π3([(h, ψ)]) = π(h). (142)
[ ] denotes equivalence classes. The detail structure of the spinor bundle E3 is therefore given
by
H3−→E3−→R3. (143)
Its sections are by defintion the spinors ψ which are two components wave functions:
ψ =
(
ψ+
ψ−
)
. (144)
Both ψ+ and ψ− are in C∞(R3). ψ itself is in the Hilbert space H3.
Finally we can view R3 − 0 as a bundle over S2 where the fibers are half lines starting at
the center. Each point on the fiber is then given by its radial distance r. Therefore the spinor
bundle E2 over S2 can be thought of as the subbundle of E3 in which sections are independent
of the fiber coordinate r. In other words the E2 fiber is the subspace H2 of H3 in which wave
functions ψ do not depend on the radial coordinate r.
4.2 Continuum Dirac Operators
It is a known result that the Dirac operator in arbitrary coordinates on a manifold M is
given by [66]
D = iγµ(∂µ +1
8ωµab[γ
a, γb]). (145)
The γµ are the generators of the curved Clifford algebra, namely γµ, γν = 2gµν with γµ2 = 1
and γµ+ = γµ. The γa are the generators of the flat Clifford algebra which are defined as
follows. First one decomposes the metric gµν into tetrads, viz gµν = ηabeaµe
bν and ηab = gµνea
µebν
35
where ηab is the flat metric δab. The generators γa of the flat Clifford algebra are then defined
by γµ = γaEµa where Eµ
a is the inverse of eaµ given by Eµ
a = ηabgµνeb
ν . This Eµa satisfies therefore
the following equations Eµa e
bµ = δb
a and ηabEµaE
νb = gµν . Thus ea
µ is the matrix which transforms
the coordinate basis dxµ of the cotangent bundle T ∗x (M) to the orthonormal basis ea = ea
µdxµ
whereas Eµa is the matrix transforming the basis ∂/∂xµ of the tangent bundle Tx(M) to the
orthonormal basis Ea = Eµa
∂∂xµ . The above Dirac operator can then be rewritten as
D = iγaEµa (∂µ +
1
8ωµab[γ
a, γb]). (146)
The ωµab in the above equations is the affine spin connection one-form. All the differential
geometry of the manifold M is completely coded in the two following tensors. The curvature
two-form tensor Rab and the torsion two form-tensor T a. They are given by Cartan’s structure
equations
Rab = dωa
b + ωac∧ωc
b≡1
2Ra
bcdec∧ed
T a = dea + ωab∧eb≡1
2T a
bceb∧ec. (147)
The ωab means ωa
b = ωabµdx
µ. The Levi-Civita connection or Christoffel symbol Γµαβ on the
manifold M is determined by the two following conditions. First one must require that the
metric is covariantly constant, namely gµν;α = ∂αgµν − Γλαµgλν − Γλ
ανgµλ = 0. Secondly one
requires that there is no torsion , i.e T µαβ = 1
2(Γµ
αβ − Γµβα) = 0. The Levi-Civita connection
is then uniquely determined to be Γµαβ = 1
2gµν(∂αgνβ + ∂βgνα − ∂νgαβ). In the same way the
Levi-Civita spin connection is obtained by restricting the affine spin connection ωab to satisfy
the metricity and the no-torsion conditions respectively
ωab + ωba = 0 , dea + ωab∧eb = 0. (148)
The Levi-Civita spin connection on S2 with metric ds2 = ρ2dθ2 + ρ2sin2θdφ2 is given by
ω21 = cosθdφ. (149)
From the other hand the Levi-Civita spin connection on R3 with metric ds2 = dr2 + r2dθ2 +
r2sin2θdφ2 is given by
ω21 = cosθdφ , ω23 = sinθdφω13 = dθ. (150)
Now we are in the position to calculate the Dirac operators on the sphere S2 and on R3. On
the sphere we obtain
D2 = iγaEµa (∂µ +
1
4ωµabγ
aγb) = iγ1
R(∂θ +
1
2ctgθ) + i
γ2
Rsinθ∂φ. (151)
36
From the other hand we obtain on R3
D3 = iγaEµa (∂µ +
1
4ωµabγ
aγb) = iγ1
r(∂θ +
1
2ctgθ) + i
γ2
r sin θ∂φ + iγ3(∂r +
1
r). (152)
Thus D3 restricted on the sphere is related to D2 by the equation
D2 = D3|r=R − iγ3
R. (153)
This equation will always be our guiding rule for finding the Dirac operator on S2 starting from
the Dirac operator on R3 . However there is an infinite number of Dirac operators on S2 which
are all related by U(1) rotations and therefore they are all equivalent. The generator of these
rotations is given by the chirality operator γ on the sphere which is defined by
γ = ~σ.~n = γ+; γ2 = 1; γD2θ + D2θγ = 0, ~n =~x
R. (154)
D2θ is the Dirac operator on the sphere which is obtained from a reference Dirac operator D2g
by the transformation
D2θ = exp(iθγ)D2gexp(−iθγ)= (cos2θ)D2g + i(sin2θ)γD2g. (155)
Next we find algebraic global expressions of the Dirac operator D2 with no reference to any
local coordinates on the sphere S2. There are two different methods to do this which lead to
two Dirac operators on the sphere denoted in this section by D2g and D2w. The operator D2g
stands for the Dirac operator due to [64, 67, 68, 69] whereas D2w stands for the Dirac operator
due to [70]. On the continuum sphere these two Dirac operators are equivalent while on the
fuzzy sphere these operators become different. We start with the standard Dirac operator on
R3, viz
D3 = iσi∂i. (156)
The σi are the Pauli matrices. Now defining γ = ~σ.~xr
and the identity γ2 = 1 we can rewrite D3
as
D3 = γ2D3 = (~σ.~x
r)(~σ.~x
r)(iσi∂i) = i
γ
r(xi∂i + iǫkijσkxi∂j). (157)
Recalling that Lk = −iǫkijxi∂j one can finally find
D3 = iγ(∂r −~σ. ~Lr
). (158)
This operator is selfadjoint. On the sphere S2 the Dirac operator will be simply given by
37
D2 = D3|r=R − iγ3
ρ= −iγD2g. (159)
In above we have made the identification γ = γ3 and where D2g is the Dirac operator given by
D2g =1
R(~σ. ~L + 1). (160)
Another global expression for the Dirac operator D2 on the sphere can be found as follows
D3 = iσi∂i = i~σ[~n(~n.~∂) − ~n×(~n×~∂)] = iγ∂r +1
r2ǫijkσixjLk. (161)
Thus we get the operator
D2w =1
R2ǫijkσixjLk − i
γ
R. (162)
By using the identity iγR
= − 1R2 ǫijkσixj
σk
2one can rewrite equation (162) in the form
D2w =1
R2ǫijkσixj(Lk +
σk
2). (163)
From the above construction it is obvious that D2w = −iγD2g and therefore from equation
(155) one can make the following identification D2w = D2θ with θ = −π4. A more general Dirac
operator can be obtained from D2g by the general transformation (155).
The two Dirac operators (160) and (163) are clearly equivalent because one can show that
both operators have the same spectrum. This can be seen from the fact that D22g = D2
2w. The
spectrum of D2g can be derived from the identity
D2g =1
R[ ~J 2 − ~L2 +
1
4]. (164)
The eigenvalues of ~L2 are l(l+1) where l = 0, 1, 2, ... whereas the eigenvalues of ~J 2 are j(j+1)
where j = l±12. Hence we get the spectrum
D2g = ± 1
R(j +
1
2) , j =
1
2,3
2,5
2, .... (165)
The Laplacian on the sphere is defined by
∆ =1
R2~L2 = D2
2g −1
RD2g. (166)
38
4.3 The Complex Structure on TS2
This complex structure will provide essentially a volume form as well as a metric on the
tangent space TS2 of the sphere.
The complex structure J on the space TS2~n which is tangent to S2 at the point ~n is intro-
duced by the formula
Jij = ǫijknk. (167)
Next one can construct from the above complex structure a projector Pij defined by
Pij = −JikJkj = δij − ninj. (168)
It can also be rewritten in the form Pij = (niAdLi)2 where AdLi are the generators of the l = 1
adjoint representation of SU(2) defined by (AdLi)jk = iǫijk. A simple calculation leads to the
following basic identities among Jij and Pij
JijJjk = −Pik , PijPjk = Pik , PijJjk = Jik. (169)
The P is actually a projector on the tangent space TS2~n so the vector P~ξ is always in TS2
~n
where ~ξ is any vector in the vector space A3 = A⊗C3. This can be seen from the fact that
~n.P~ξ = Pijniξj = −(Jikni)Jkjξj = 0. The scalar product of any two tangent vectors P~ξ and
P~η will be given by P~ξ.P~η = (Pijξj)(Pikηk) = Pjkξjηk. This result illustrates the fact that Pij
plays the role of a metric on TS2~n. From this and from the identity 1
8π
∫
S2 ǫijknkdni∧dnj = 1 one
can see that Jij contains much more information on the metric aspects of TS2~n. More involved
calculations show that
[Li,Jjk] = iJilǫljk , [Li,Pjk] = iǫijlPlk + iǫiklPlj . (170)
The last commutation relations follow from the Jacobi identity [Lj, [Li,Jkl]] + [Jkl, [Lj,Li]] +
[Li, [Jkl,Lj]] = 0=⇒ǫilkJkj + ǫjikJkl + ǫljkJki = 0. It is proven by first rewriting it in the
form , [Li, Pjk] = −iJilJhkǫljh + iJjlJhiǫlkh. Then Jacobi identity gives , [Li, Pjk] = iǫijlPlk +
iJjlJhkǫilh + iJjlJhiǫlkh. Jacobi identity is used once more to recombine the last two terms in
this last equation and obtain finally the desired result .
From the complex structure Jij and the projector Pij one can construct the projectors P+ij
and P−ij , on the holomorphic and antiholomorphic parts respectively, of the tangent space TS2
~n.
They are given by
P±ij =
1
2(Pij±iJij). (171)
It is easy to check that P±ijP±
jk = P±ik, P±
ijP∓jk = 0, P±
ijPjk = P±ik and P±
ijJjk = ∓iP±k .
Finally the chirality operator (154) and the Dirac operators (160) and (163) can easily be
expressed in terms of the complex structure Jij and the projector Pij as
γ = − i
2!Jijσiσj , D2g =
1
RσiPij(Lj +
1
2σj) , D2w =
1
RJijσi(Lj +
σj
2). (172)
39
4.4 Quantization of S2 = CP1
The starting point is the Wess-Zumino term defined by
L = ΛiT r(σ3g−1g); g∈SU(2). (173)
The Λ is an undetermined real number. As it was shown in [72] this Lagrangian arises generally
when one tries to avoid the singularities of the phase space. In such cases a global Lagrangian
can not be found by a simple Legendre transformation of the Hamiltonian and therefore one
needs to enlarge the configuration space. A global Lagrangian over this new extended config-
uration space can then be shown to exist and it turns out to contain (173) as a very central
piece. Basically (173) reflects the constraints imposed on the system.
One example which was treated in [72] with great detail is the case of a particle with a
fixed spin. For a free particle one knows that the Lagrangian is given simply by the expression
L = m2~x
2. However if the particle is constrained to have a fixed spin given by ~S2 = Λ2 then
the phase space will be eight dimensional defined by ~x, ~p,Q, P where Q and P describes the
two independent spin degrees of freedom of the particle. Clearly Q and P can not be smooth
functions of ~S, they must show a singularity for at least one value of ~S. Indeed if they were
smooth functions of ~S there would have been no difference between this particle and the free
particle with arbitrary spin. To overcome this difficulty one can enlarge the configuration space
from R3 to R3×SU(2) over which the Lagrangian is now given by L = m2~x
2+ ΛiT r(σ3g
−1g).
The quantization of this sytem will fix the number Λ appropriately. In the Dirac quantization
scheme it is fixed to be of the form ±√
j(j + 1) where j = 0, 1/2, 1, ... In the Gupta-Bleuler
quantization approach Λ is fixed to be ±j. Another example for which the above term plays a
central role is the system of a charged particle in the field of a magnetic monopole.
We define the sphere S2 = ~x∈R3;∑3
i=1 x2i = R2 by the Hopf fibration
π : SU(2)−→S2 , g−→Rgσ3g−1 = ~x.~σ. (174)
Clearly the structure group U(1) of the principal fiber bundle U(1)−→SU(2)−→S2 leaves the
base point ~x invariant in the sense that all the elements geiσ3θ/2 of SU(2) are projected onto
the same point ~x on the base manifold S2. One can then identify the point ~x∈S2 with the
equivalence class
[geiσ3θ/2]∈SU(2)/U(1). (175)
Let us now turn to the quantization of the Lagrangian (173). First we parametrize the group
element g by the set of variables (ξ1, ξ2, ξ3). The conjugate momenta πi are given by the
equations πi = ∂L∂ξi
= ΛiT r(σ3g−1 ∂g
∂ξi ). ξi and πi will satisfy as usual the standard Poisson
brackets : ξi, ξj = πi, πj = 0 and ξi, πj = δij.
A change in the local coordinates ξ−→f(ǫ) which is defined by g(f(ǫ)) = exp(iǫiσi
2)g(ξ) will
lead to the identity ∂g(ξ)∂ξi
Nij(ξ) = iσj
2g(ξ) where Nij(ξ) = ∂fi(ǫ)
∂ǫj|ǫ=0. The modified conjugate
momenta ti are given by
ti = −πjNji =Λ
ρxi. (176)
40
They will then satify the interesting Poisson’s brackets
ti, g = iσi
2g
ti, g−1 = −ig−1σi
2ti, tj = ǫijktk. (177)
Putting equation (176) in the last of the equations (177) one can derive the following nice result
xi, xj = RΛǫijkxk which is the first indication that we are going to get a fuzzy sphere under
quantization. The classical sphere would correspond to Λ−→∞.
A more precise treatment would have to start by viewing equations (176) as a set of con-
straints rather than a set of identities on the phase space (ξi, ti). In other words the functions
Pi = ti − ΛRxi do not vanish identically on the phase space (ξi, ti). However their zeros will
define the physical phase space as a submanifold of (ξi, ti). To see that the Pi’s are not the
zero functions on the phase space one can simply compute the Poisson brackets Pi, Pj. The
answer turns out to be Pi, Pj = ǫijk(Pk − ΛRxk) which clearly does not vanish on the surface
Pi = 0. So the Pi’s should only be set to zero after the evaluation of all Poisson brackets. This
fact will be denoted by setting Pi to be weakly zero, i.e
Pi ≈ 0. (178)
Equations (178) provide the primary constraints of the system. The secondary constraints of the
system are obtained from the consistency conditions Pi, H ≈ 0 where H is the Hamiltonian
of the system. Since H is given by H = viPi where vi are Lagrange multipliers the requirement
Pi, H ≈ 0 will lead to no extra constraints on the system. It will only put conditions on the
v’s [72].
From equations (177) it is obvious that ti are generators of the left action of SU(2) on itself.
A right action can also be defined by the generators
tRj = −tiRij(g). (179)
The Rij(g) define the standard SU(2) adjoint representation: Rij(g)σi = gσjg−1. These right
generators satisfy the following Poisson brackets
tRi , g = −igσi
2
tRi , g−1 = iσi
2g−1
tRi , tRj = ǫijktRk . (180)
In terms of tRi the constraints (178) will then take the simpler form
tRi ≈ −Λδ3i. (181)
These constraints are divided into one independent first class constraint and two independent
second class constraints. tR3 ≈ −Λ is first class because on the surface defined by (181) one
41
have tR3 , tRi = 0 for all i. It corresponds to the fact that the Lagrangian (173) is weakly
invariant under the gauge transformations g−→gexp(iσ3θ2), namely L−→L − Λθ. The two
remaining constraints tR1 ≈ 0 and tR2 ≈ 0 are second class. They can be converted to a set of
first class constraints by taking the complex combinations tR± = tR1 ±itR2 ≈ 0. We would then
have tR3 , tR± = ∓itR± and therefore all the Poisson brackets tR3 , tR± vanish on the surface (181).
Let us now construct the physical wave functions of the system described by the Lagrangian
(173). One starts with the space F of complex valued functions on SU(2) with a scalar product
defined by (ψ1, ψ2) =∫
SU(2) dµ(g)ψ1(g)∗ψ2(g) where dµ stands for the Haar measure on SU(2).
The physical wave functions are elements of F which are also subjected to the constraints (181).
They span a subspace H of F. For Λ < 0 3 one must then have
tR3 ψ = −Λψ
tR+ψ = 0 (182)
In other words ψ transforms as the highest weight state of the spin s = L2
= |Λ| representation
of the SU(2) group. |Λ| is now being quantized to be either an integer or a half integer number.
The physical wave functions are then linear combinations of the form
ψ(g) =s∑
m=−s
Cm < lm|Ds(g)|ss >, |Λ| = s. (183)
The Ds(g) is the spin s = L2
representation of the element g of SU(2). Clearly the left action of
SU(2) on g will rotate the index m in such a way that < sm|Ds(g)|ss > transform as a basis
for the Hilbert space of the (2s+ 1)−dimensional irreducible representation s = |Λ| of SU(2).
Under the right action of SU(2) on g the matrix element < sm|Ds(g)|ss > will transform as
the heighest weight state s = |Λ| m = |Λ| of SU(2).
Observables of the system will be functions f(Li) ≡ f(L1, L2, L3) of the quantum operators
Li which are associated with ti. These functions are the only objects which will have by
construction4 weakly zero Poisson brackets with the constraints (181). These observables are
linear operators which act on the left of ψ(g) by left translation, namely
[iLiψ][g] =[ d
dtψ(e−i
σi2
tg)]
t=0(184)
The operators f(Li) can be represented by (L+ 1)×(L+ 1) matrices of the form
f(Li) =∑
i1,...,ik
αi1,...,ikLi1 ...Lik . (185)
The operators Li form by definition a complete set of SU(2) generators, namely they satisfy
[Li, Lj] = iǫijkLk and ~L2 = s(s + 1). The summation in (185) will clearly terminate because
3If Λ was positive the second equation of (182) should be replaced by tR−ψ = 0 and ψ would have then been
the lowest weight state of the spin s = L2
= Λ representation of the SU(2) group.4This is because by definition left and right actions of SU(2) commute.
42
the dimension of the space of all (L+1)×(L+1) matrices is finite equal to (L+1)2. In a sense
the Li’s provide the fuzzy coordinate functions on the fuzzy sphere S2L whereas fuzzy functions
are given by (185).
The last thing one would like to mention concerning the Wess-Zumino term (173) is its
relation to the symplectic structure on S2. The first claim is the fact that the symplectic
two-form ǫijknkdni∧dnj on S2 can be rewritten in the form
ω = Λid[
Trσ3g(σ, t)−1dg(σ, t)
]
= − Λ
2R3ǫijkxkdxi∧dxj = Λd cos θ∧dφ. (186)
The t in (186) is a time variable which goes from t1 to t2 while σ is a new extra parameter
which is chosen to be in the range [0, 1]. By definition g(1, t) = g(t) and ~n(1, t) = ~n(t). If one
defines the triangle ∆ in the plane (t, σ) by its boundaries ∂∆1 = (σ, t1), ∂∆2 = (σ, t2) and
∂∆3 = (1, t) then it is a trivial exercise to show that
SWZ =∫
∆ω =
∫ t2
t1Ldt+ Λi
∫ 1
0dσTrσ3
[
g(σ, t1)−1 ∂g
∂σ(σ, t1) − g(σ, t2)
−1 ∂g
∂σ(σ, t2)
]
. (187)
The equations of motion derived from this action are precisely those obtained from the Wess-
Zumino term (173). This is obvious from the fact that the second term of (187) will not
contribute to the equations of motion because it involves the fixed initial and final times where
g is not varied.
4.5 K-cycles And Spectral Triples
The ordinary sphere S2 is defined in global coordinates by the equation
~n∈R3 ,3∑
a=1
n2a = 1. (188)
A general function can be expanded in terms of spherical harmonics
f(~n) =∞∑
k=0
fkmYkm(~n). (189)
Global derivations are given by the generators of rotations in the adjoint representation of the
SU(2) group, namely
La = −iǫabcnb∂c , [La,Lb] = iǫabcLc. (190)
The Laplacian is given by
L2 = LaLa , eigenvalues l(l + 1) , l = 0, ...,∞. (191)
According to [71] all the geometry of the sphere is encoded in the K-cycle or spectral triple
(A,H,L2). A = C∞(S2) is the algebra of all functions f of the form (189) and H is the infinite
43
dimensional Hilbert space of square integrable functions on which the functions are represented.
In order to encode the geometry of the sphere in the presence of spin structure we use instead
the K-cycle (A,H,D, γ) [7]. The chirality operator γ and the Dirac operator D are given in
the case of S2 by equations (154) and (160) respectively.
As we have shown the fuzzy sphere is a particular deformation of the above triple which
is based on the fact that the sphere is the co-adjoint orbit SU(2)/U(1). The result of this
quantization is to replace the algebra C∞(S2) by the algebra of matrices MatL+1.
MatL+1 is the algebra of (L + 1)×(L+ 1) matrices which acts on an (L+ 1)−dimensional
Hilbert space HL with inner product (f, g) = 1L+1
Tr(f+g) where f, g∈MatL+1. The spin s = L2
IRR of SU(2) is found to act naturally on this Hilbert space. The generators are
[La, Lb] = iǫabcLc ,∑
a
L2a = c2≡
L
2(L
2+ 1). (192)
Generally the spherical harmonics Ykm(~n) become the canonical SU(2) polarization tensors Ykm
[53]. They are defined by
[La, [La, Ylm]] = l(l + 1)Ylm , [L±, Ylm] =√
(l∓m)(l±m+ 1)Ylm±1 , [L3, Ylm] = mYlm. (193)
They also satisfy
Y +lm = (−1)mYl−m ,
1
L+ 1TrYl1m1 Yl2m2 = (−1)m1δl1l2δm1,−m2. (194)
Matrix coordinates on S2L are defined by the k = 1 tensors as in the continuum, namely
x21 + x2
2 + x23 = 1 , [xa, xa] =
i√c2ǫabcxc, xa =
La√c2. (195)
“Fuzzy” functions on S2L are linear operators in the matrix algebra while derivations are inner
defined by the generators of the adjoint action of SU(2),i.e
La(φ) ≡ [La, φ]. (196)
A natural choice of the Laplacian operator ∆ on the fuzzy sphere is therefore given by the
following Casimir operator
∆L = L2a ≡ [La, [La, ..]]. (197)
Thus the algebra of matrices MatL+1 decomposes under the action of the group SU(2) as
L
2⊗L
2= 0⊕1⊕2⊕..⊕L. (198)
The first L2
stands for the left action of the group while the other L2
stands for the right action.
It is not difficult to convince ourselves that this Laplacain has a cut-off spectrum of the form
44
l(l + 1) where l = 0, 1, ..., L. As a consequence a general function on S2N can be expanded in
terms of polarization tensors as follows
f =L∑
l=0
l∑
m=−l
flmYlm. (199)
The fact that the summation over l involves only angular momenta which are ≤L originates of
course from the fact that the spectrum l(l + 1) of the Laplacian ∆ is cut-off at l = L.
The commutative continuum limit is given by L−→∞. This is the first limit of interest to
us in this article. Therefore the fuzzy sphere is a sequence of the following triples
(MatL+1, HL,∆L). (200)
Again in the presence of spin structure the fuzzy sphere S2L will be defined instead by the
K-cycle (A,H, D,Γ). The detailed construction of the Dirac operator D and of the chirality
operator Γ will be discussed shortly.
4.6 Coherent States And Star Products on Fuzzy CPN−1L
The sphere is the complex projective space CP1. The quantization of higher CPN−1 with
N > 2 is very similar to the quantization of CP1. This yields Fuzzy CPN−1L . In this section we
will explain this result by constructing the coherent states [74] and star products on all fuzzy
CPN−1L including the fuzzy sphere which corresponds to N = 2. We start with classical CPN−1
defined by the projectors5
P =1
N1 + αNn
ata , αN = −√
2(N − 1)
N. (201)
The requirement P 2 = P will lead to the defining equations of CPN−1 as embedded in RN2−1
given by
n2a = 1 , dabcnanb =
2
αN
N − 2
Nnc. (202)
The fundamental representation N of SU(N) is generated by the Lie algebra of Gell-Mann
matrices ta = λa
2, a = 1, ..., N2 − 1. These matrices satisfy
[ta, tb] = ifabctc
2tatb =1
Nδab1 + (dabc + ifabc)tc
Trtatbtc =1
4(dabc + ifabc) , T rtatb =
δab
2, T rta = 0. (203)
Let us specialize the projector (201) to the ”north” pole of CPN−1 given by the point ~n0 =
(0, 0, ..., 1). We have then the projector P0 = 1N1 + αN tN2−1 = diag(0, 0, ..., 1). So at the
5In this section we use N to denote the dimension of the space instead of the size of the matrix approximation.
45
”north” pole P projects down onto the state |ψ0 >= (0, 0, ..., 1) of the Hilbert space CN on
which the defining representation of SU(N) is acting .
A general point ~n∈ CPN−1 can be obtained from ~n0 by the action of an element g∈SU(N)
as ~n = g~n0. P will then project down onto the state |ψ >= g|ψ0 > of CN . One can show that
P = |ψ >< ψ| = g|ψ0 >< ψ0|g+ = gP0g+. (204)
gtN2−1g+ = nata. (205)
This last equation is the usual definition of CPN−1 . Under g−→gh where h∈U(N − 1) we
have htN2−1h+ = tN2−1 , i.e U(N − 1) is the stability group of tN2−1 and hence
CPN−1 = SU(N)/U(N − 1). (206)
Thus points ~n of CPN−1 are then equivalent classes [g] = [gh], h∈U(N − 1) .
The fuzzy sphere S2L is the algebra of operators acting on the Hilbert space H(2)
s which is the
(2s+1)−dimensional irreducible representation of SU(2). This representation can be obtained
from the symmetric product of L = 2s fundamental representations 2 of SU(2). Indeed given
any element g∈SU(2) its s−representation matrix U (s)(g) can be obtained as follows
U (s)(g) = U (2)(g)⊗s...⊗sU(2)(g), 2s− times. (207)
U (2)(g) is the spin 1/2 representation of g∈SU(2) .
Similarly fuzzy CPN−1L is the algebra of operators acting on the Hilbert space H(N)
s ; the
d(N)s −dimensional irreducible reprsentation of SU(N). This dimension is given explicitly by
d(N)s = (
(N − 1 + 2s)!
(N − 1)!(2s)!. (208)
Also this d(N)s −dimensional irreducible representation of SU(N) can be obtained from the
symmetric product of L = 2s fundamental representations N of SU(N).
Remark that for s = 1/2 we have d(N)1/2 = N and therefore H
(N)1/2 = CN is the fundamental
representation of SU(N). Clearly the states |ψ0 > and |ψ > of H(N)1/2 will correspond in H(N)
s
to the two states |~n0, s > and |~n, s > respectively so that |ψ0 >= |~n0,12> and |ψ >= |~n, 1
2>.
Furthermore the equation |ψ >= g|ψ0 > becomes
|~n, s >= U (s)(g)|~n0, s > . (209)
U (s)(g) is the representation given by
U (s)(g) = U (N)(g)⊗s...⊗sU(N)(g), 2s− times. (210)
To any operator F on H(N)s (which can be thought of as a function on fuzzy CPN−1
L ) we
associate a ”classical” function Fs(~n) on a classical CPN−1 by
Fs(~n) =< ~n, s|F |~n, s > . (211)
46
The product of two such operators F and G is mapped to the star product of the corresponding
two functions
Fs ∗Gs(~n) =< ~n, s|F G|~n, s > . (212)
We will now compute this star product in a closed form. First we will use the result that
any operator F on the Hilbert space H(N)s admits the expansion
F =∫
SU(N)dµ(h)F (h)U (s)(h). (213)
U (s)(h) are assumed to satisfy the normalization
TrU (s)(h)U (s)(h′
) = d(N)s δ(h−1 − h
′
). (214)
Using the above two equations one can derive the value of the coefficient F (h) to be
F (h) =1
d(N)s
TrFU (s)(h−1). (215)
Using the expansion (213) in (211) we get
Fs(~n) =∫
SU(N)dµ(h)F (h)ω(s)(~n, h) , ω(s)(~n, h) = < ~n, s|U (s)(h)|~n, s > . (216)
On the other hand using the expansion (213) in (212) will give
Fs ∗Gs(~n) =∫ ∫
SU(N)dµ(h)dµ(h
′
)F (h)G(h′
)ω(s)(~n, hh′
). (217)
The computation of this star product boils down to the computation of ω(l)(~n, hh′). We have
ω(s)(~n, h) = < ~n, s|U (s)(h)|~n, s >=
[
< ~n,1
2|⊗s...⊗s < ~n,
1
2|][
U (N)(h)⊗s...⊗sU(N)(h)
][
|~n, 12> ⊗s...⊗s|~n,
1
2>]
= [ω( 12)(~n, h)]2s. (218)
ω( 12)(~n, h) = < ψ|U (N)(h)|ψ > . (219)
In the fundamental representation N of SU(N) we have U (N)(h) = exp(imata) = c(m)1 +
isa(m)ta and therefore
ω( 12)(~n, h) = < ψ|c(m)1 + isa(m)ta|ψ >= c(m) + isa(m) < ψ|ta|ψ > . (220)
ω( 12)(~n, hh
′
) = < ψ|U (N)(hh′
)|ψ >= < ψ|(c(m)1 + isa(m)ta)(c(m
′
)1 + isa(m′
)ta)|ψ >= c(m)c(m
′
) + i[c(m)sa(m′
) + c(m′
)sa(m)] < ψ|ta|ψ >− sa(m)sb(m
′
) < ψ|tatb|ψ > . (221)
47
Now it is not difficult to check that
< ψ|ta|ψ > = TrtaP =αN
2na
< ψ|tatb|ψ > = TrtatbP =1
2Nδab +
αN
4(dabc + ifabc)n
c. (222)
Hence we obtain
ω( 12)(~n, h) = c(m) + i
αN
2~s(m).~n. (223)
ω( 12)(~n, hh
′
) = c(m)c(m′
) − 1
2N~s(m).~s(m
′
) + iαN
2
[
c(m)sa(m′
)
+ c(m′
)sa(m)]
na − αN
4(dabc + ifabc)n
csa(m)sb(m′
). (224)
These two last equations can be combined to get the result
ω( 12)(~n, hh
′
) − ω( 12)(~n, h)ω( 1
2)(~n, h
′
) = − 1
2N~s(m).~s(m
′
) − αN
4(dabc + ifabc)n
csa(m)sb(m′
)
+α2
N
4nanbsa(m)sb(m
′
). (225)
We can remark that in this last equation we have got ride of all reference to c’s. We would like
also to get ride of all reference to s’s. This can be achieved by using the formula
sa(m) =2
iαN
∂
∂naω( 1
2)(~n, h). (226)
We get then
ω( 12)(~n, hh
′
) − ω( 12)(~n, h)ω( 1
2)(~n, h
′
) = Kab∂
∂naω( 1
2)(~n, h)
∂
∂nbω( 1
2)(~n, h
′
). (227)
Kab =2
Nα2N
δab − nanb +1
αN(dabc + ifabc)n
c. (228)
Therefore we obtain
Fs ∗Gs(~n) =2s∑
k=0
(2s)!
k!(2s− k)!Ka1b1 ....Kakbk
×∫
SU(N)dµ(h)F (h)[ω( 1
2)(~n, h)]2s−k ∂
∂na1ω( 1
2)(~n, h)...
∂
∂nakω( 1
2)(~n, h)
×∫
SU(N)dµ(h
′
)G(h′
)[ω( 12)(~n, h
′
)]2s−k ∂
∂nb1ω( 1
2)(~n, h
′
)...∂
∂nbkω( 1
2)(~n, h
′
). (229)
We have also the formula
(2s− k)!
(2s)!
∂
∂na1...
∂
∂nakFs(~n) =
∫
SU(N)dµ(h)F (h)[ω( 1
2)(~n, h)]2s−k ∂
∂na1ω( 1
2)(~n, h)...
∂
∂nakω( 1
2)(~n, h).
(230)
48
This allows us to obtain the final result [73]
Fs ∗Gs(~n) =2s∑
k=0
(2s− k)!
k!(2s)!Ka1b1 ....Kakbk
∂
∂na1...
∂
∂nakFj(~n)
∂
∂nb1...
∂
∂nbkGj(~n). (231)
Let us do some examples. Derivations on CPN−1 are generated by the vector fields La =
−ifabcnb∂/∂nc which satisfy [La,Lb] = ifabcLc. The corresponding action on the Hilbert space
H(N)s is generated by La and is given by
< ~n, s|U (s)(h−1)FU (s)(h)|~n, s >=< ~n0, s|U (s)(g−1h−1)FU (s)(hg)|~n0, s > . (232)
U (s)(h) is given by U (s)(h) = exp(iηaLa). Now if we take η to be small then one computes
< ~n, s|U (s)(h)|~n, s >= 1 + iηa < ~n, s|La|~n, s > . (233)
On the other hand we know that the representation U (s)(h) is obtained by taking the symmetric
product of 2s fundamental representations N of SU(N) and hence
< ~n, s|U (s)(h)|~n, s >= (< ~n,1
2|1 + iηata|~n,
1
2>)2s = 1 + i(2s)ηa
αN
2na. (234)
In above we have used the facts La = ta⊗s....⊗sta, |~n, s >= |~n, 12> ⊗s...⊗s|~n, 1
2> and the first
equation of (222). Finally we get the important result
< ~n, s|La|~n, s >= sαNna. (235)
We define the fuzzy derivative [La, F ] by
(LaF )s(~n) ≡ < ~n, s|[La, F ]|~n, s >= sαN
[
na ∗ Fs(~n) − Fs ∗ na(~n)]
= ifabcnc ∂
∂nbFs(~n). (236)
Finally we note the identity
1
d(N)s
TrF G =∫
CPN−1
Fs ∗Gs(~n). (237)
4.7 Fuzzy Dirac Operators And The Absence of Fermion Doubling
There is a major problem associated with conventional lattice approaches to the nonper-
turbative formulation of chiral gauge theories with roots in topological features. The Nielsen-
Ninomiya theorem [75] states that if we want to maintain chiral symmetry then one cannot
avoid the doubling of fermions in the usual lattice formulations. We will show that this problem
is absent on the fuzzy sphere and as consequence it will also be absent on fuzzy S2 × S2. It
49
does not arise on CP2 as well. We note here that there have also been important developments
[76, 77] in the theory of chiral fermions and anomalies in the usual lattice formulations.
The operators xi defined by xi = θLi, [xi, xj ] = iθǫijkxk and∑
a x2a = R2 are the fuzzy
coordinate functions on the fuzzy sphere S2L with θ = R/
√c2 and c2 = L2
a = L2(L
2+ 1). The
noncommutativity parameter characterizing the fuzziness of the sphere is θ. The commutators
[xi, xj] approaches zero when L−→∞ or equivalently θ−→0 which is the continuum limit.
We have already shown that in the continuum spinors ψ belong to the fiber H2 of the spinor
bundle E2 over the sphere. H2 is essentially a left A−module, in other words if f∈A and
ψ∈H2 then fψ∈H2. Recall that A = C∞(S2). H2 can also be thought of as the vector space
H2 = A⊗C2. The noncommutative analogue of the projective module H2 is the projective
module H2 = A⊗C2 where A = MatL+1. This is clearly an A−bimodule since there is a
left action as well as a right action on the space of spinors H2 by the elements of the algebra
A. The left action is generated by LLi = Li whereas the right action will be generated by LR
i
defined by LRi f = fLi for any f ∈ A. We also have [LL
i , LLj ] = iǫijkL
Lk and [LR
i , LRj ] = −iǫijkLR
k .
Derivations on the fuzzy sphere can be defined by Li = LLi − LR
i .
The fuzzy Dirac operators and the fuzzy chirality operators must be defined in such a way
that they act on the Hilbert space H2. The Dirac operators must anticommute with the fuzzy
chirality operators. They must be selfadjoint and reproduce the continuum operators in the
limit L−→∞. To get the discrete version of γ = σana one first simply replaces na by xa to get
σaxa. We can check that this operator does not square to 1. Indeed we can check that
1
(L2
+ 12)2
(
~σ.~L+1
2
)2
= 1. (238)
In other words the chirality operator in the discrete is given by
ΓL =1
L2
+ 12
(~σ.~L+1
2). (239)
By construction this operator has the correct continuum limit and it squares to one. However by
inspection ΓL does not commute with functions on S2L. The property that the chirality operator
must commute with the elements of the algebra is a fundamental requirement of the K-cycle
(A,H, D,Γ) desribing S2L. To overcome this problem one simply replace ~L by −~LR. Since these
generators act on the right of the algebra A , they will commute with anything which act on
the left and therefore the chirality operator will commute with the algebra elements as desired.
The fuzzy chirality operator is then given by
ΓR =1
L2
+ 12
(−~σ.~LR +1
2) (240)
The fuzzy version of Watamuras’s Dirac operator (163) is simply given by
D2w =1
R√c2ǫijkσiLj(Lk − LR
k +σk
2). (241)
50
By construction this Dirac operator has the correct continuum limit. It can also be rewritten
as
D2w = − 1
R2ǫijkσixjL
Rk . (242)
From this expression it is obvious that this Dirac operator is selfadjoint . Next we compute
D2wΓR = − 1
R(L2
+ 12)√c2
[ǫijkσlσiLjLRl L
Rk − iǫijkǫklmσlσiLjL
Rm − 2ǫijkLjL
Rk L
Ri +
1
2ǫijkσiLjL
Rk ]
ΓRD2w = − 1
R(L2
+ 12)√c2
[−ǫijkσlσiLjLRk L
Rl +
1
2ǫijkσiLjL
Rk ]. (243)
Taking the sum one gets
D2wΓR + ΓRD2w = 0. (244)
The fuzzy version of the Grosse-Klimcık-Presnajder Dirac operator defined by equation
(160) is simply given by
D2g =1
R(~σ.~L− ~σ.~LR + 1). (245)
This Dirac operator does not anticommute with the chirality operator (240) and therefore it is
no longer unitarily equivalent to D2w. Indeed the two operators D2g and D2w will not have the
same spectrum.
Let us start first with D2w. To find the spectrum of D2w one simply rewrites the square D22w
in terms of the two SU(2) Casimirs ~J2 and ~K2 where ~J and ~L are defined by
~J = ~L +~σ
2, ~L = ~L− ~LR. (246)
A straightforward computation leads to the result
D22w =
1
R2L2a
[
~L2(~LR)2 +1
2[~L2 + (~LR)2 − ~L2][1 − (
~σ
2)2 + ~J2 − 1
2~L2 − 1
2(~LR)2 − 1
2~L2]]
.
(247)
The eigenvalue j takes the two values j = l + 12
and j = l − 12
for each value of l where
l = 0, 1, ..., L. The eigenvalues of the above squared Dirac operator will then read
D22w(j) =
1
R2
[
(j +1
2)2 +
[l(l + 1)]2
4L2a
− l(l + 1)(j + 12)2
2L2a
]
. (248)
We get the spectrum
D2w(j = l ± 1
2) = ± 1
R(j +
1
2)
√
√
√
√[1 +1 − (j + 1
2)2
4L2a
]. (249)
51
The computation of the spectrum of the Dirac operator D2g is much easier. It turns out that
the spectrum of D2g is exactly equal to the spectrum of the continuum Dirac operator upto
the eigenvalue j = L− 12. Thus D2g is a better approximation to the continuum than D2w and
there is no fermion doubling. This can be seen from the equation
D2g =1
R[ ~J2 − ~L2 − 1
2(1
2+ 1) + 1]
=1
R[j(j + 1) − l(l + 1) +
1
4]. (250)
Again for each fixed value of l the quantum number j can take only the two values j = l + 12
and j = l − 12. For j = l + 1
2we get D2g(j) = 1
R(j + 1
2) with j = 1/2, 3/2, ..., L+ 1/2 whereas
for j = l − 12
we get D2g(j) = − 1R(j + 1
2) with j = 1/2, 3/2, ..., L− 1/2. The chirality operator
ΓR is equal +1 for D2g(j) = 1R(j + 1
2) with j = 1/2, 3/2, ..., L + 1/2 and it is equal −1 for
D2g(j) = − 1R(j + 1
2) with j = 1/2, 3/2, ..., L− 1/2. Thus the top modes with j = L+ 1/2 are
not paired. In summary we have the spectrum
D2g(j = l ± 1
2) = ± 1
R(j +
1
2) ,ΓR(j = l ± 1
2) = ±1, j =
1
2,3
2, ..., L− 1
2
D2g(j = l +1
2) =
1
R(j +
1
2) ,ΓR(j = l +
1
2) = +1, j = L+
1
2. (251)
By inspection one can immediately notice that there is a problem with the top modes j =
L + 12. The top eigenvalues j = L + 1
2in the spectrum of D2g are not paired to any other
eigenvalues which is the reason why D2g does not have a chirality operator. Indeed D2g does
not anticommute with ΓR. We find
D2gΓR + ΓRD2g =
2R
L+ 1D2
2g. (252)
This equation follows from the fact that
D2g =L+ 1
2R(ΓL + ΓR). (253)
The Dirac operator D2w vanishes on the top modes j = L + 12
and therefore the existence of
these modes spoils the invertibility of the Dirac operator D2w. The Dirac operator D2w has the
extra disadvantage of having a very different spectrum compared to the continuum. In other
words D2g is a much better Dirac operator than D2w if one can define for it a chirality operator.
Towards this end we note the following identity
[D2g,ΓR] = 2 i
√
1 − 1
(L+ 1)2D2w. (254)
This leads to the crucial observation that the two operators D2g and D2g anticommute, viz
D2gD2w + D2wD2g = 0. (255)
52
If we restrict ourselves to the subspace with j≤L − 12
then clearly D2g must have a chirality
operator. Let us then define the projector P by
P |L+1
2, j3 >= 0 , P |j, j3 >= |j, j3 > , for all j≤L− 1
2. (256)
Let us call V the space on which P projects down. The orthogonal space is W . Our aim is
to find the chirality operator of the Dirac operator PD2gP . To this end one starts by making
some observations concerning the continuum. From the basic continuum result D2w = −iγD2g
we can trivially prove the identity γ = iF2gF2w where F2g and F2w are the sign operators of
the Dirac operators D2g and D2w respectively defined by F2g = D2g
|D2g| and F2w = D2w
|D2w| . The
fuzzification of these expressions is only possible if one confine ourselves to the vector space V
since on the fuzzy sphere the operator F2w will not exist on the whole space V⊕W . Taking
all of these matters into considerations one ends up with the following chirality operator
ΓR′
= iF2gF2w. (257)
F2g =D2g
|D2g|, on V
= 0 , on W. (258)
F2w =D2w
|D2w|, on V
= 0 , on W. (259)
By construction (257) has the correct continuum limit. If it is going to assume the role of a
chirality operator on the fuzzy sphere it must also square to one on V , in other words one
must have on the whole space V⊕W : (ΓR′)2 = P . It should also be selfadjoint and should
anticommute with the Dirac operator PD2gP . The key requirement for all of these properties
to hold is the identity F2g,F2w = 0. This identity follows trivially from the result (255). It
is an interesting fact that the three operators F2g, F2w and ΓR′constitute a Clifford algebra on
V .
Thus we have established that fermions can be defined on S2L with no fermion doubling at
least in the absence of fuzzy monopoles. It is however easy to include them as well.
4.8 The Ginsparg-Wilson Algebra on S2L
The Ginsparg-Wilson chiral fermion has a Dirac operator D and a hermitian chirality op-
erator Γ squaring to unity. D and Γ fulfill the relations (where a is the lattice spacing)
D† = Γ D Γ, Γ, D = aDΓD. (260)
Now if Γ′ = Γ(a D) − Γ then
Γ′† = Γ
′
, Γ′2 = 1 aD = Γ(Γ + Γ
′
). (261)
53
Conversely given two idempotents Γ and Γ′ we have a Ginsparg-Wilson pair D = 1a
Γ(Γ + Γ′)
and Γ.
Our fermion on S2L admits such a formulation except that we choose Γ + Γ′ as the Dirac
operator. By using the chirality operators (239) and (240) we make the identifications
Γ = ΓR , Γ′
= ΓL. (262)
Then the Ginsparg-Wilson Dirac operator on the fuzzy sphere is
D = ΓRD2g ⇔ aD = ΓR(ΓR + ΓL) , a =2R
L+ 1. (263)
Let K be the algebra of the idempotents ΓR,ΓL. Introduce the hermitian operators
Γ1,2 =1
2(ΓR ± ΓL),Γ3 =
i
2[ΓR,ΓL],Γ0 =
1
2ΓR,ΓL. (264)
We can immediately compute
Γ21 =
1
2(1 + Γ0) =
(
aD2g
2
)2
, Γ22 =
1
2(1 − Γ0). (265)
Γ0 =2
(L+ 1)2[ ~J 2 − 2L2
a −1
4]. (266)
Γ3 =4R
√c2
(L+ 1)2D2w. (267)
Thus Γ0 and Γ2m, (m = 1, 2, 3) commute with all Γλ, (λ = 0, 1, 2, 3) and are in the center of K.
Furthermore we can compute
Γ21 + Γ2
2 = 1 = Γ23 + Γ2
0. (268)
In other words Γλ do not have eigenvalues exceeding 1 in modulus. Lastly it is trivial to check
that Γm (m = 1, 2, 3) mutually anticommute, viz
Γ1,Γ2 = Γ1,Γ3 = Γ2,Γ3 = 0. (269)
Since Γ0 depends only on ~J 2 the vector space of the algebra K can be split into the direct sum
⊕Vj of eigenspaces Vj of Γ0 with distinct eigenvalues cos 2θj . Furthermore since Γ0 commutes
with all Γλ it must commute with K and hence KVj are also eigenspaces of Γ0 with eigenvalues
cos 2θj . In other words KVj = Vj. On Vj the possible eigenvalues of Γ3 are of the form
± sin 2θj by (268) and those of Γ1,2 are ± cos θj ,± sin θj by (265). Both signs do occur on Vj
unless |Γ0| is 1 and hence Γ3 = 0. This can be seen as folows.
If Γ3 6=0 on Vj then |Γ0|6=1 and hence Γ1,2 have no zero eigenvalue and Γm/|Γm| generate a
Clifford algebra there. But if |Γ0| = 1 and Γ3 = 0 for j = j0 then Γ1 or Γ2 is also zero on Vj0
54
and we cannot infer that the nonzero Γm has eigenvalues of both signs there. We can only say
that its modulus |Γm| has its maximum value 1 there.
The spectrum and eigenstates of the Ginsparg-Wilson D = ΓRD2g can be found as follows.
We have
aD = 2(Γ21 + Γ2Γ1). (270)
It can be diagonalised since [Γ21,Γ2Γ1] = 0 and therefore [Γ2
1,D] = 0. On Vj we have Γ21 =
[a2D2g]
2 = cos2 θj1 and hence Γ22 = [− 1
L+1~σ(~LL + ~LR)]2 = sin2 θj . Similarly (Γ2Γ1)
2 = −Γ22Γ
21 =
− cos2 θj sin2 θj1 and thus the spectrum of aD on Vj is
aD(j) = 1 + exp(±2iθj). (271)
As for its eigenvectors we can proceed as follows. On Vj0 we have Γ1 or Γ2 = 0 so that aD is in
any case diagonal. On Vj (j 6=j0) we have |Γ2|6=0 and
aD = e−i
Γ2|Γ2|
π4 2[
Γ21 + i| sin θj|Γ1
]
ei
Γ2|Γ2|
π4 . (272)
So if Γ1ψ±j = ±(cos θj)ψ
±j then
aD[
e−i
Γ2|Γ2|
π4ψ±
j
]
= 2 cos θje±i
sin θj|sin θj |
θj[
e−i
Γ2|Γ2|
π4ψ±
j
]
. (273)
4.9 Monopoles
A point particle of electric charge q and mass m in the magnetic field of a monopole g is
described by the free Hamiltonian H = ~p2
2mtogether with the Poisson brackets xi, xj = 0,
pi, pj = qǫijkBk and xi, pj = δij . The ~B is the magnetic field of the monopole given
by ~B = − g4π
~xr3 where r is the radial distance between the monopole which is assumed to
be at rest at ~x = 0 and the point particle at ~x. The force acting on the electric charge
in the presence of the monopole is given by mdxi
dt= mxi, H = qǫijkxjBk. The canonical
angular momentum Li = ǫijkxj(mxk) of the point particle is not conserverd since we havedLi
dt= ǫijkxj(m
dxk
dt) = − qg
4πddt
(xi
r). Therefore Ji = Li + qg
4πni where ni = xi
rand dJi
dt= 0 should
be interpreted as the angular momentum of the point particle in the presence of a magnetic
monopole. In other words the point particle acquires an angular momentum in the direction of
the line joining the particle and the monopole.
It is a known result that in the case of an electric charge in the field of a monopole one can
not find a global system of canonical coordinates (~x, ~p) on the phase space T∗B and therefore
a global Lagrangian describing the above system can not be found by a simple Legendre trans-
formation of the Hamiltonian. To construct such a Lagrangian one enlarges the configuration
space B = ~x to a U(1) bundle E over B given by
E = R×SU(2) = (r, g) , τana = gτ3g−1. (274)
55
A global Lagrangian can then be written down as follows
L =1
2m∑
i
x2i + i
qg
4πTrτ3g
−1g
=1
2mr2 +
1
4mr2Tr[gg−1, gτ3g
−1]2 + iqg
4πTrτ3g
−1g. (275)
The above Lagrangian can be shown to be weakly invariant under the right U(1) action
g−→gei θ2τ3 where L−→L− qg
4πθ. Thus we have a fiber bundle structure U(1)−→SU(2)−→S2.
Following the same steps taken to quantize the Wess-Zumino term (173) one can start the
quantization of (275) by first parametrizing the group element g by a set of three real numbers
(ξ1, ξ2, ξ3). The conjugate momentum Πi associated to ξi is gievn by
Πi =∂L
∂ξi= i
qg
4πTrτ3g
−1 ∂g
∂ξi+
1
2mr2Tr[
∂g
∂ξig−1, gτ3g
−1][∂g
∂ξjg−1, gτ3g
−1]ξj. (276)
The modified conjugate momentum tk = −ΠiNik can also be computed along the same lines
which led to equation (176). The answer is tk = 12
qg4πTrτkgτ3g
−1−mr2
2Tr[i τk
2, gτ3g
−1][gg−1, gτ3g−1].
From this last equation the following constarint follows easily
P = nktk −qg
4π≈ 0 (277)
The Hamiltonian of the system can be computed in a standard fashion. It is given by
H =p2
r
2m+
1
4mr2Tr[gg−1, gτ3g
−1]2 + vP
=p2
r
2m+
1
2mr2[~t2 − (
qg
4π)2] + vP. (278)
The v is a Lagrange multiplier. We check that the first class constraint (277) have zero Poisson
bracket with the Hamiltonain (278) and therefore there is no secondary constraint. Observables
should of course have zero PB’s with P . These are r, pr, ti and ni or functions of them. Wave
functions of the system are ψ = ψ(r, g). The pr acts on ψ as the usual differential operator 1i
ddr
while ti acts by left multiplication, namely
[eiθitiψ](r, g) = ψ(r, e−iθiτi2 g). (279)
These wave functions should also satisfy the requirement
nktkψ =qg
4πψ. (280)
From the above two last equations one can easily find that
ψ(r, ge−iθτ32 ) = eiθ qg
4πψ(r, g). (281)
Any function of r and g admits the expansion
f(r, g) =∑
j
∑
m,n
cjm,n(r) < j,m|Dj(g)|j, n > . (282)
56
The g−→Dj(g) is the j IRR of SU(2). Then under the transformation g−→ge−iθτ32 each term in
the above expansion (282) will transform as < j,m|Dj(g)|j, n > −→e−iθn < j,m|Dj(g)|j, n >.
In other words wave functions ψ(r, g) should be functions of the form (282) with the Dirac
quantization condition qg4π
= −n where |n| can be either an integer or half-integer. Hence we
must haveqg
2π= ±N , N ∈ N. (283)
In the units where q = 2π we have g = ±N . The wave function of a particle of charge q in the
field of a monopole g which is at rest at r = 0 is thus given by the expansion
ψ(±N)(r, g) =∑
j,m
cjm(r) < j,m|D(j)(g)|j,∓N2> . (284)
The mathematical structure underlying this physical system is a U(1) principal fiber bundle
SU(2)−→S2. In other words for a fixed r = R the particle q moves on a sphere S2 and its
wave functions is a section of a U(1) bundle Γ(S2, SU(2)) over S2. They have the equivariance
property
ψ(N)(R, ge−iθτ32 ) = eiθ N
2 ψ(N)(R, g). (285)
They are not functions on S2 butfunctions on SU(2) because they depend on the specific point
on the U(1) fiber.
An alternative description of monopoles can be given in terms of K-theory and projective
modules. It is based on the Serre-Swan’s theorem [6, 7, 78] which states that there is a complete
equivalence between vector bundles over a compact manifoldM and projective modules over the
algebra C(M) of smooth functions on M . Projective modules are constructed from C(M)n =
C(M)⊗Cn where n is some integer through a certain projector p element in Mn(C(M)), i.e
the algebra of n×n matrices with entries in C(M) .
In our case M = S2 and C(M) = C∞(S2) ≡ A which is the algebra of smooth functions on
S2. For a monopole with winding number N = ±1 the appropriate projective module will be
constructed from A2 = A⊗C2. It is P(±1)A2 where P(±1) is the projector
P(±1) =1±~τ .~n
2. (286)
It is clearly an element of M2(A) and satisfies P(±1)2 = P(±1) and P(±1)+ = P(±1). P(±1)A2
describes a monopole with N = ±1 as one can directly check by computing its winding number
as follows
±1 =1
2πi
∫
TrP(±1)dP(±1)∧dP(±1). (287)
As opposed to the space of sections Γ(S2, SU(2)) elements of P(±1)A2 are by construction
invariant under the action g−→gexp(−iθ τ32).
Next we comment on the relation between the wave functions ψ(±1) given in equation
(284) and those belonging to P(±1)A2. The projector P(±1) can be rewritten as P(±1) =
D( 12) 1±τ3
2D( 1
2)+(g) where D( 1
2) : g−→D( 1
2)(g) = g is the 1
2IRR of SU(2). Hence we will have
57
P(±1)D( 12)(g)|± >= D( 1
2)(g)1±τ3
2|± >= D( 1
2)(g)|± > where |± > are defined by τ3|± >=
±|± >. In the same way one can show that P(±1)D( 12)(g)|∓ >= 0. This result means that
P(±1) = D( 12)(g)|± >< ±|D( 1
2)+(g) ⇔ (P(±1))ij = D
( 12)
i± (g)D( 12)+
±j (g). (288)
The < ±|D( 12)+(g) defines a map from P(±1)A2 into Γ(S2, SU(2)) as follows
< ±|D( 12)+(g) : P(±1)A2−→Γ(S2, SU(2))
|ψ > −→ < ±|D( 12)+(g)|ψ >= ψ(±1)(R, g). (289)
< ±|D( 12)+(g)|ψ > has the correct transformation law (285) under g−→gexp(−iθ τ3
2) as one can
check by using the basic equivariance property
D( 12)(ge−iθ
τ32 )|± >= e∓i θ
2D( 12)(g)|± > . (290)
In the same wayD( 12)(g)|± > defines a map Γ(S2, SU(2))−→P(±1)A2 which takes the wave func-
tions ψ(±1) to the two components elements ψ(±1)D( 12)(g)|± > of P(±1)A2. Under g−→gexp(−iθ τ3
2)
the two phases coming from ψ(±1) and D( 12)(g)|± > cancel exactly so that their product is a
function over S2.
For monopole charge ±N (N > 0) the corresponding projectors are
P(±N) =i=N∏
i=1
(1 ± ~τ (i).~n)
2. (291)
The ~τ (i) are commuting sets of Pauli matrices. They give the following sections of vector bundles
P(±N)A2N
where A2N
= A⊗C2N
consists of 2N -component vectors ξ = (ξ1, ξ2, ..., ξ2N ), ξi∈A.
We have ~τ (i).~n P(±N)ξ = ±P(±N)ξ where ~τ (i) is acting on the ith C2 factor. For the trivial
bundle we can use P0 = 1+τ32
(or 1−τ32
) or simply the identity. The Chern class of the projective
module P(±N)A2N
is trivially equal to ±N , viz
±N =1
2πi
∫
TrP(±N)dP(±N)∧dP(±N). (292)
For the case N = 1 we have computed explicitly the RHS of (292) and found it to be equal ±1.
For the case N > 1 the projector P(±N) is given by equation (291). It is the tensor product
of N projectors P(±1) and therefore by using the additivity of Chern classes the RHS of (292)
must be equal to ±N .
The Chern numbers (or the quantized fluxes) for the monopoles P(±N)A2N
are given by
±N = − 1
4π
∫
d(cos θ)∧dφ Tr γP(±N) [D2w,P(±N)] [D2w,P(±N)]. (293)
In above D2w = 1RǫijkσinjJk. To prove equation (293) one can first remark that [D2w,P(±N)] =
iσi∂i(P(±N)) where we have crucially used the fact ∂r(P(±N)) = 0. Next one shows by computing
the trace over the Pauli matrices σj that
TrγP(±N)[D2w,P(±N)]2 = −2iǫijknkTrP(±N)∂i(P(±N))∂j(P(±N)). (294)
58
The remaining trace is only over the Pauli matrices τ(i)j . Putting this last equation in (293)
gives (292). In arriving at (292) we have also used the identity −ǫijknkd(cosθ)∧dφ = dni∧dnj .
Let us note that this result remains valid if we use the Dirac operator D2g instead of D2w.
These numbers do not change under the deformation of P(±N) and therefore can be thought
of as soliton winding numbers which also proves the stability of the cyclic cocycle (293).
We rewrite the above winding numbers in the suggestive form
±N = −Trω
(
1
|D|2γ P(±N) [D,P(±N)] [D,P(±N)]
)
. (295)
The second line is Connes trace theorem [6]. |D| = positive square root of D†D while Trω is
the Dixmier trace. D in the above equation is either D2g or D2w. They both give the same
answer ±1.
4.10 Fuzzy Monopoles:First Look
The projector P(±1) will be replaced by a fuzzy projector p(±1) which we now find. We
proceed like before; we replace ~n by ~x and insist on the result to have the properties p(±1)2 =
p(±1) and p(±1)+ = p(±1). We also require this projector to commute with the chirality operator
ΓR. The answer for winding number N = +1 is
p(+1) =1
2+
1
L+ 1[~τ .~LL +
1
2]. (296)
This can be rewritten in the following useful form
p(+1) =~K(1)2 − (L
2− 1
2)(L
2+ 1
2)
(L2
+ 12)(L
2+ 3
2) − (L
2− 1
2)(L
2+ 1
2), ~K(1) = ~LL +
~τ
2. (297)
Thus p(+1) is the projector on the subspace with the maximum eigenvalue L2
+ 12. Similarly the
projector p(−1) will correspond to the subspace with minimum eigenvalue L2− l
2, namely
p(−1) =~K(1)2 − (L
2+ 1
2)(L
2+ 3
2)
(L2− 1
2)(L
2+ 1
2) − (L
2+ 1
2)(L
2+ 3
2). (298)
By construction (298) as well as (297) have the correct continuum limit (286) and they are in
the algebra M2(A) where A = MatL+1. Fuzzy monopoles with winding number ±1 are then
desribed by the projective modules p(±1)A2.
This last remark shows the way to fuzzify P(N). We substitute for ~K(1) the angular mo-
mentum operator
~K(N) = ~L+N∑
i=1
~τ (i)
2. (299)
Then we consider the subspace whereK(N)2≡ ~K(N). ~K(N) has the maximum eigenvalue kmax(kmax+
1), kmax = L/2+N/2 . On this space (~L+~τ (i)/2)2 has the maximum value (L/2+1/2)(L/2+3/2)
59
and ~τ (i).~L is hence L/2. The corresponding projector is
p(+N) =
∏
k 6=kmax[K(N)2 − k(k + 1)]
∏
k 6=kmax[kmax(kmax + 1) − k(k + 1)]
. (300)
p(−N) comes similarly from the least value kmin = L/2−N/2 of k. We assume that L ≥ N . We
remark that the limits as L→∞ of p(±N) are exactly P(±N) and not say P(±N) times another
projector. A proof starts by remarking that K(N)2 = L2(L
2+ 1) +
∑Ni=1
~L.~τ (i) +∑N
i,j=1~τ (i)
2~τ (j)
2
from which one concludes that ~K(N)2 is maximum when all the products ~L.~τ (i), i = 1, N and~τ (i)
2~τ (j)
2, i6=j, i, j = 1, N are maximum. But ~L.~τ (i) = (~L + ~τ (i)
2)2 − L
2(L
2+ 1) − 3
4can take only
the two different values L2
and −L2− 1 whereas ~τ (i)
2~τ (j)
2= 1
2(~τ (i)
2+ ~τ (j)
2)2 − 3
4for i6=j can take the
two values 14
and −34
so that K(N)2 is maximum for ~L.~τ (i) = L2
and ~τ (i).~τ (j) = 1 and it is given
by K(N)2 = L+N2
(L+N2
+ 1).
The analogue of P(±N)A2N
are the “projective modules”
p(±N)A2N
,A2N
= 〈(a1, a2, .., a2N ) : ai∈A〉 , A = MatL+1. (301)
These are the noncommutative substitutes for sections of vector bundles. If (a1, a2, ..., a2N ) is
regarded as a column then the column dimensions of p(±N)A2N
are 2(L2±N
2) + 1 as p(+N) and
p(−N) project down to the subspaces with kmax = L/2+N/2 and kmin = L/2−N/2 respectively
and its row dimension is L+ 1 as all the ai’s are in A = MatL+1. Their difference is ±N .
The fuzzy Dirac operators D2g and D2w are different and therefore one has to decide between
them. The fuzzy Dirac operator D2g does not admit a chirality operator so we will not work
with it here. The fuzzy Dirac operator D2w admits the fuzzy chirality operator ΓR but it has a
zero eigenvalue for j = L+ 12
so it must be regularized for its inverse to make sense. This will
be understood below but not done explicitly. The correct discrete version of (295) turns out to
be given by the formula
c(±N) = −TrǫP (±N)[F2w, P(±N)][F2w, P
(±N)]. (302)
F2w =
0 D2w
|D2w|D2w
|D2w| 0
, ǫ =
(
ΓR 0
0 ΓR
)
P (±N) =
(
1+ΓR
2p(±N) 0
0 1−ΓR
2p(±N)
)
. (303)
For p(+1) one finds that c(+1) = +1+[2(L+1)+1] while for p(−) we find c(−1) = −1+[2L+1].
They are both wrong if compared to (287). The correct answer is obtained by recognizing that
c(±1) is the index of the operator
f (±) =1 − ΓR
2p(±1) D2w
|D2w|p(±1) 1 + ΓR
2. (304)
60
This index counts the number of zero modes of f (±). By construction only the matrix elements
< p(±1)U−|f (±)|p(±1)U+ > where U± = 1±Γ2
A4 ( the extra A2 is due to spin) exist and therefore
f (±) is a mapping from V+ = p(±1)U+ to V− = p(±1)U−. Hence
Indexf (±) = dimV+ − dimV−. (305)
The chirality operator ΓR can be put in the form ΓR = 2L+1
[
l(l + 1) − (L+12
)2]
where l(l + 1)
is the eigenvalue of (−~LR + ~σ2)2. The values l = L
2± 1
2for which Γ|l= L
2± 1
2= ±1 defines the
subspace U± with dimension 2(L2± 1
2) + 1.
For p(+1) which projects down to the subspace with maximum eigenvalue kmax = L2
+ 12
of
the operator ~K(1) = ~L + ~τ2
the space V± has dimension [2(L2±1
2) + 1][2(L
2+ 1
2) + 1] and so the
index is
Indexf (+) = c(+1) = [2(L+ 1) + 1] + 1. (306)
For p(−1) which projects down to the subspace with minimum eigenvalue kmin = L2− 1
2of the
operator ~K(1) = ~L+ ~τ2
the space V± has dimension [2(L2±1
2) + 1][2(L
2− 1
2) + 1] and so the index
is
Indexf (−) = c(−1) = [2L+ 1] − 1. (307)
This result signals the existence of (extra unphysical) zero modes of the operator f (+) since we
know that c(+1) must be precisely equal to +1. Indeed for ΓR = +1 one must couple L2
+ 12
toL2
+ 12
and obtain j = L + 1, L, ..0 whereas for ΓR = −1 we couple L2
+ 12
to L2− 1
2and obtain
j = L, ..., 1. The integer j corresponds to the total angular momentum ~J = ~LL − ~LR + ~σ2
+ ~τ2.
The eigenvalues j(+1) = L+1 and 0 in V+ are not paired to anything. The extra piece in c(+1)
is therefore exactly equal to the number of the top zero modes namely 2j(+1) +1 = 2(L+1)+1.
These modes do not exist in the continuum and therefore they are of no physical relevance and
must be projected out. This can be achieved by replacing the projector p(+1) by a corrected
projector π(+1) = p(+1)[1− π(j(+1))] where π(j(+1)) projects out the top eigenvalue j(+1). Putting
π(+1) in (302) gives exactly c(+1) = +1 which is the correct answer.
The same analysis goes for p(−1). For ΓR = +1 one must couple L2− 1
2to L
2+ 1
2and obtain
j = L, .., 1 whereas for ΓR = −1 we couple L2− 1
2to L
2− 1
2and obtain j = L − 1, ..., 1, 0. The
eigenvalues j(−1) = L and 0 in V+ are not paired to anything. The extra piece in c(−1) is
therefore exactly equal to the number of the top zero modes namely 2j(−1) +1 = 2L+1. These
modes do not exist in the continuum and therefore they must also be projected out. This can
be achieved by replacing the projector p(−1) by a corrected projector π(−1) = p(−1)[1 − π(j(−1))]
where π(j(−1)) projects out the top eigenvalue j(−1). Again by putting π(−1) in (302) we get
exactly c(−1) = −1 which is the correct answer.
Generalization of this construction to monopoles wave functions with winding numbers ±Nis straightforward.
61
5 Fuzzy Scalar Field Theory and Matrix Phase
5.1 Action and Limits
A real scalar field φ on the fuzzy sphere is an N × N hermitian matrix where N = L + 1.
The action of a λφ4 model is given by
S =1
NTr[
Φ[La, [La,Φ]] +m2Φ2 + λΦ4]
. (308)
It has the correct continuum large N limit, viz
S =∫
dΩ
4π
[
ΦL2aΦ +m2Φ2 + λΦ4
]
. (309)
In perturbation theory of the matrix model (308) only the tadpole diagram can diverge in the
limit N −→ ∞ [54, 55]. On the fuzzy sphere the planar and non-planar tadpole graphs are
different and their difference is finite in the limit. This is the UV-IR mixing. This problem can
be removed by standard normal ordering of the interaction [56].
Non-perturbatively we find the phase diagram to contain an extra phase (the matrix phase or
the non-uniform ordered phase) between the two usual phases of the scalar model [41, 57, 84].
The usual phases are the disordered (rotationally invariant) phase (< Trφ >= 0) and the
uniform ordered phase ( < Trφ >= ±N√
−m2/2λ). In the novel ”matrix phase” we have
instead < Trφ >= ±(N−2k)√
−2m2/λ where k is some integer. The transition from disordered
to matrix is 3rd order with continous action and specific heat.
Another remarkable limit of the matrix action (308) is the limit of the noncommutative
Moyal-Weyl plane. This planar limit is defined byN −→ ∞, R −→ ∞ (the radius of the sphere)
keeping the ratio R2/√c2 = θ2 fixed. The parameter θ is the noncommutativity parameter and
c2 is the Casimir c2 = N2−14
. The coordinates on the fuzzy sphere are xa = RLa/√c2 with
commutation relations [xa, xb] = iθ2ǫabcxc/R. In the above planar limit restricting also to the
north pole on the sphere we have x3 = R and the commutation relations become [xi, xj ] = iθ2ǫij .
These are the commutation relations on the plane. We will give a more rigorous proof of this
statement using star products on the fuzzy sphere and on the Moyal-Weyl plane in the next
chapters. In this planr limit the matrix action becomes therefore
S =θ2
2Trθ
[
1
θ4Φ[xi, [xi,Φ]] +m2
θΦ2 + λθΦ
4]
. (310)
In above we have replaced R2
NTr = θ2
2Tr with θ2
2Trθ where Trθ is an infinite dimensional trace
and Φ is an infinite dimensional matrix (operator) on the corresponding Hilbert space. We
have also m2θ = m2/R2 and λθ = λ/R2.
5.2 The effective action and The 2−Point Function
We write the above action as
S =1
NTr(
Φ∆Φ +m2Φ2 + λΦ4)
, ∆ = L2a. (311)
62
To quantize this model we write Φ = Φ0 + Φ1 where Φ0 is a background field which satisfy the
classical equation of motion and Φ1 is a fluctuation. We compute
S[Φ] = S[Φ0] + TrΦ1
(
∆ +m2 + 4λΦ20
)
Φ1 + 2λTrΦ1Φ0Φ1Φ0 +O(Φ31). (312)
The linear term vanished by the classical equation of motion. Integration of Φ1 leads to the
effective action
Seff [Φ0] = S[Φ0] +1
2TR log Ω. (313)
The Laplacian Ω is given by
ΩBA,CD = (∆)BA,CD +m2δBCδAD + 4λ(Φ20)BCδAD + 2λ(φ0)BC(φ0)DA. (314)
Formally we write
Ω = ∆ +m2 + 4λΦ20 + 2λΦ0Φ
R0 . (315)
The matrix ΦR0 acts on the right. The 2−point function is deduced from
Squadeff =
1
NTrΦ0
(
∆ +m2)
Φ0 + λTR(
2
∆ +m2Φ2
0 +1
∆ +m2Φ0Φ
R0
)
. (316)
Let us insist that the trace TR is not the same as the trace Tr. To see this explicitly let us
introduce the propagator
(
1
∆ +m2
)AB,CD
=∑
k,k3
1
∆(k) +m2TAB
kk3(T+
kk3)DC . (317)
The eigenbasis Tkk3 is such ∆Tkk3 = ∆(k)Tkk3 where ∆(k) = k(k+ 1). The matrices Tkk3 are
the polarization tensors where k = 0, 1, 2, .., N − 1 and −k≤k3≤k, viz Tkk3 = Ykk3/√N . Thus
while the trace Tr is N dimensional the trace TR is actually N2 dimensional. We will also
need the resolution of the N2−dimensional identity matrix
δACδBD =∑
k,k3
TABkk3
(T+kk3
)DC . (318)
For any matrix ML acting on the left and any matrix MR acting on the right we have the
following matrix components
(ML)AB,CD = MACδBD , (MR)AB,CD = δACMDB. (319)
The planar contribution is thus given by
TR2
∆ +m2Φ2
0 = 2∑
k,k3
1
∆(k) +m2TrT+
kk3Φ2
0Tkk3
= 2∑
p,p3
∑
q,q3
φ(pp3)φ(qq3)∑
k,k3
1
∆(k) +m2TrT+
kk3Tpp3Tqq3Tkk3 . (320)
63
Similarly the non-planar contribution is given by
TR1
∆ +m2Φ0Φ
R0 =
∑
k,k3
1
∆(k) +m2TrT+
kk3Φ0Tkk3Φ0
=∑
p,p3
∑
q,q3
φ(pp3)φ(qq3)∑
k,k3
1
∆(k) +m2TrT+
kk3Tpp3Tkk3Tqq3. (321)
In above we have clearly expanded the matrix Φ0 as Φ0 =∑
kk3φ(kk3)Tkk3. Since Φ0 is a matrix
we can not move it across the polorization tensors and hence the contributions are different.
These contributions are finite by construction. Formally they become equal in the continuum
limit. However by doing the sums first then taking the limit we see immediately that they are
different even in the continuum limit. This is the source of the so-called UV-IR mixing. We
show this point next. We have the identities
∑
k3
TrT+kk3Tpp3Tqq3Tkk3 =
1
N(2k + 1)δp,qδp3,−q3(−1)p3 . (322)
∑
k3
TrT+kk3Tpp3Tkk3Tqq3 = (2k + 1)δp,qδp3,−q3(−1)p+p3+k+2s
p s s
k s s
. (323)
In above s is the spin of the SU(2) IRR, viz s = N−12
. Thus we obtain
TR2
∆ +m2Φ2
0 = 2∑
p,p3
|φ(pp3)|2ΠP , ΠP =1
N
∑
k
2k + 1
k(k + 1) +m2. (324)
TR1
∆ +m2Φ0Φ
R0 =
∑
p,p3
|φ(pp3)|2ΠN−P (p) , ΠN−P (p) =∑
k
2k + 1
k(k + 1) +m2(−1)p+k+2s
p s s
k s s
.
(325)
The UV-IR mixing is measured by the difference
ΠN−P − ΠP =1
N
∑
k
2k + 1
k(k + 1) +m
[
N(−1)p+k+2s
p s s
k s s
− 1]
. (326)
When the external momentum p is small compared to 2s = N − 1, one can use the following
approximation for the 6j symbols [54]
p s s
k s s
≈(−1)p+k+2s
NPp(1 − 2k2
N2), s→∞, p << 2s, 0≤k≤2s. (327)
Since Pp(1) = 1 for all p, only k >> 1 contribute in the above sum, and therefore it can be
approximated by an integral as follows
ΠN−P − ΠP =1
N
∑
k
2k + 1
k(k + 1) +m2
[
Pp(1 − 2k2
N2) − 1
]
=1
Nhp , hp =
∫ +1
−1
dx
1 − x+ 2m2
N2
[
Pp(x) − 1]
. (328)
64
Clearly we can drop the mass for large N . We have the generating function
∞∑
p=0
Pp(x)tp =
1√1 − 2tx+ t2
. (329)
We can immediately compute
∞∑
p=0
hptp =
2
1 − tln(1 − t). (330)
In other words h0 = 0 and hp>0 = −2∑p
n=11n. We obatin the following UV-IR mixing on the
sphere
ΠN−P − ΠP = − 2
N
p∑
n=1
1
n. (331)
This is non-zero in the continuum limit. It has also the correct planar limit on the Moyal-Weyl
plane. We will show this explicitly for S2 × S2. The planar contribution ΠP is given explicitly
by 1N
log N2
m2 (if we replace the sum in (324) by an integral). Thus the total quadratic effective
action is given by
Squadeff =
1
NTrΦ0
(
∆ +m2 + 3λ logN2
m2− 2λQ
)
Φ0. (332)
The operator Q = Q(L2) is defined by its eigenvalues Q(p) given by
QYpm = Q(p)Ypm , Q(p) =p∑
n=1
1
n. (333)
5.3 The 4−Point Function And Normal Ordering
The quartic effective action is obtained from (313) as follows. First we rewrite the Laplacian
Ω in the form
Ω = ∆ +m2 + 2λΦ20 + 2λ(φR
0 )2 + 2λΦ0ΦR0 . (334)
This is symmetric between left and right. The quartic effective action is then given by
Squarteff =
λ
NTrΦ4
0 − λ2TR[
2(
1
∆ +m2Φ2
0
)2
+ 2(
1
∆ +m2Φ2
0
)(
1
∆ +m2(ΦR
0 )2)
+(
1
∆ +m2Φ0Φ
R0
)2
+ 4(
1
∆ +m2Φ2
0
)(
1
∆ +m2Φ0Φ
R0
)]
=λ
NTrΦ4
0 − λ2∑
jj3
∑
ll3
∑
qq3
∑
tt3
φ(jj3)φ(ll3)φ(qq3)φ(tt3)[
2VP,P + 2VP,P + VN−P,N−P + 4VP,N−P
]
.
(335)
65
We use the notation ~k = (kk3) and introduce the interaction vertex v(~k,~j, ~p, ~q) = TrT+kk3Tjj3Tpp3Tqq3 .
The two planar-planar contributions are given by
VP,P (~j,~l, ~q,~t) =∑
kk3
∑
pp3
v(~k,~j,~l, ~p)
k(k + 1) +m2
v(~p, ~q,~t,~k)
p(p+ 1) +m2
VP,P (~j,~l, ~q,~t) =∑
kk3
∑
pp3
v(~k,~j,~l, ~p)
k(k + 1) +m2
v(~p,~k, ~q,~t)
p(p+ 1) +m2. (336)
The non-planar-non-planar contribution is given by
VN−P,N−P (~j,~l, ~q,~t) =∑
kk3
∑
pp3
v(~k,~j, ~p,~l)
k(k + 1) +m2
v(~p, ~q,~k,~t)
p(p+ 1) +m2. (337)
The planar-non-planar contribution is given by
VP,N−P (~j,~l, ~q, ~q,~t) =∑
kk3
∑
pp3
v(~k,~j,~l, ~p)
k(k + 1) +m2
v(~p, ~q,~k,~t)
p(p+ 1) +m2. (338)
In the continuum limit the planar-planar contribution VP,P remains finite and tends to the
commutative result. Thus with the continuum vertex w(~k,~j, ~p, ~q) =∫ dΩ
4πY +
kk3Yjj3Ypp3Yqq3 the
VP,P in the large N limit takes the form
VP,P (~j,~l, ~q,~t) =1
N
∑
kk3
∑
pp3
w(~k,~j,~l, ~p)
k(k + 1) +m2
w(~p, ~q,~t,~k)
p(p+ 1) +m2. (339)
We can check explicitly that this is indeed the commutative answer. It is finite. Furthermore
it is shown in [56] that all other contributions become equal in the continuum large N limit
to the above commutative result. Hence there is no difference between planar and non-planar
graphs and the UV-IR mixing is absent in this case.
Hence to remove the UV-IR mixing from this model a standard prescription of normal
ordering which amounts to the substraction of tadpoloe contributions will be sufficient. We
consider therefore the action
S =1
NTr[
Φ(
∆ +m2 − 3NλΠP + 2λQ)
Φ + λΦ4]
. (340)
In above Q = Q(L2a) is given for any N by the expression
QYpp3 = Q(p)Ypp3 , Q(p) = −1
2
∑
k
2k + 1
k(k + 1) +m
[
N(−1)p+k+2s
p s s
k s s
− 1]
. (341)
The first substraction is the usual tadpole substraction which renders the limiting commutative
theory finite. The second substraction is to cancel the UV-IR mixing. Although this action
does not have the correct continuum limit (due to the non-local substraction) the corresponding
quantum theory is standard Φ4 in 2 dimensions.
66
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10 12 14
c/N
2
-b/N3/2
Disorder phase Non-Uniform Order phase
Uniform Order phase
Triple point (2.3 ± 0.2,0.52 ± 0.02)
N=2N=3N=4N=6N=8
N=10cN-2=(bN-3/2)2/4
Figure 1: The phase diagram of φ4 scalar field theory on the fuzzy sphere.
5.4 The Phase Structure and Effective Potential
The phase diagram of this model is shown on figure 1. We observe the ordinary disordered
and uniform ordered phases which exist also in the corresponding commutative theory. But
more importantly we observe an extra new phase between the disordered and uniform ordered
phase which we call matrix or non-uniform ordered phase. This structure can already be
understood by an analysis of the classical potential
V =1
NTr[
m2Φ2 + λΦ4]
. (342)
The minima (solutions of the equation of motion Φ(m2 +2λΦ2) = 0) are given by the following
configurations
Φ = 0 , disordered phase , Φ =
√
−m2
2λΓ , ordered/matrix phase. (343)
In above Γ is any Grassmanian element of the form Γ = (1, 1, 1...,−1,−1, ..,−1). The first k
elements of the diagonal matrix Γ is +1 and the remaining N − k elements are −1. The first
configuration in (343) is rotationally invariant (hence the name disordered) whereas the second
configuration in (343) is not rotationally invariant. We have spontaneous symmetry breaking
of rotational invariance in this model. In the commutative theory the matrix Γ can only be the
idenity function. In that case we will have an ordered phase characterized by
Φ =
√
−m2
2λ, ordered phase. (344)
This means in particular that the parameter m2 must be negative. In the noncommutative
theory, this phase exists and is renamed uniform ordered phase. In the noncommutative theory
there is also the possibility that Γ 6= 1 and hence we have a different phase from the usual
uniform ordered phase called the non-uniform ordered phase or matrix phase.
Let us define the following order parameters. The total power P and the power in the zero
modes P0 given by
67
P =<1
N1.5TrΦ2 > , P0 =<
1√N
(TrΦ)2 > . (345)
We also observe numerically that the powers P and P0 become straight lines for negative values
of m2 given essentially by the following theoretical prediction
P = 0 , for m2≥m2∗ , disordered phase.
P =−m2
2√Nλ
, for m2≤m2∗ , ordered/matrix phase. (346)
The situation with P0 is more involved since it will also depend on k. From the above potential
we have
P0 = 0 , for m2≥m2∗ , disordered phase.
P0 = − m2
2√Nλ
(2k −N)2 for m2≤m2∗ , ordered/matrix phase. (347)
We can use this type of reasoning to predict the following theoretical behaviour of the action
V = 0 , disordered phase.
V = −m4
4λ, ordered/matrix phase. (348)
Indeed it is found that the simulation results of < S > follow this behaviour to a very good
degree. In the above expressions m2∗ is a critical value. It agrees with the prediction of the
pure potential term (342) which has in the quantum theory a 3rd order phase transition which
occurs for negative m2 at m2∗ = −2N
√λ.
The quantum effective potential of this model (not to be confused with the above classical
potential) is derived from (313) with background Φ0 = ±φ1. We get the effective potential
Veff(φ) = m2φ2 + λφ4 +1
2TR log(∆ +m2 + 6λφ2). (349)
Since we want to take m2 very large we work instead with the rescaled couplings m2 = N2m2
and λ = N2m4λ. We get
Veff(φ)
N2= m2φ2 + m4λφ4 +
1
2N2TR log
(
φ2 +1
6m2λ+
∆
6N2m4λ
)
+1
2log(6N2m4λ).(350)
Thus we get the potential
Veff(φ)
N2= m2φ2 + m4λφ4 + log φ. (351)
The ground state is given by
φ2 =
√
1 − 4λ− 1
4m2λ. (352)
68
This makes sense for all λ such that
λ =N2λ
m4≤1
4. (353)
This gives the correct critical value
m4≥m4∗ = 4N2λ. (354)
The solution (873) corresponds to the two-cut solution where the cuts are centered around
Φ0 = ±φ1.
5.5 Fuzzy S2 × S2 And Planar Limit:First Look
By analogy with (308) the scalar theory with quartic self-interaction on fuzzy S2 × S2 is
S′
=1
N2TrTr
[
Φ∆Φ +m2Φ2 + λΦ4]
, ∆ = (L(1)a )2 + (L(2)
a )2. (355)
The Laplacians (L(1,2a )2 clearly correspond to the two different spheres. Φ is now an (N +1)2 ×
(N + 1)2 hermitian matrix. It can be expanded in terms of polarization operators as follows
Φ = NN−1∑
k=0
k∑
k3=−k
N−1∑
p=0
p∑
p3=−p
Φkk3pp3Tkk3Tpp3. (356)
The effective action of this model is still given by equation (313) with Laplacian ∆ = (L(1)a )2 +
(L(2)a )2, viz
S′
eff [Φ0] = S′
[Φ0] +1
2TR log Ω
Ω = ∆ +m2 + 4λΦ20 + 2λΦ0Φ
R0 . (357)
The 2−point function may now be deduced from
S′quadeff =
1
N2TrΦ0
(
∆ +m2)
Φ0 + λTR(
2
∆ +m2Φ2
0 +1
∆ +m2Φ0Φ
R0
)
. (358)
The Euclidean 4−momentum in this setting is given by (k, k3, p, p3) with square ∆(k, p) =
k(k + 1) + p(p+ 1). The propagator is given by
(
1
∆ +m2
)AB,CD
=∑
k,k3,p,p3
1
∆(k, p) +m2(Tkk3Tpp3)
AB((Tkk3Tpp3)+)DC . (359)
The one-loop correction to the 2−point function is
m2(k, p) = m2 + λ[
2ΠP + ΠN−P (k, p)]
. (360)
69
The planar contribution is given by
ΠP = 22s∑
a=0
2s∑
b=0
A(a, b) , A(a, b) =(2a+ 1)(2b+ 1)
a(a+ 1) + b(b+ 1) +m2. (361)
The non-planar contribution is given by
ΠN−P (k, p) = 22s∑
a=0
2s∑
b=0
A(a, b)(−1)k+p+a+bBkp(a, b) , Bab(c, d) = N2 a s s
c s s b s s
d s s.
(362)
As one can immediately see from these expressions both planar and non-planar graphs are finite
and well defined for all finite values of N . A measure for the fuzzy UV-IR mixing is again given
by the difference between planar and non-planar contributions which can be defined by the
equation
ΠN−P (k, p) − ΠP = 2s∑
a=0
s∑
b=0
A(a, b)[
(−1)k+p+a+bBkp(a, b) − 1]
. (363)
The fact that this difference is not zero in the continuum limit is what is meant by UV-IR
mixing on fuzzy S2 × S2. Equation (363) can also be taken as the regularized form of the
UV-IR mixing on R4. Removing the UV cut-off N−→∞ one can show that this difference
diverges as N2, viz
∆(k, p) −→ N2∫ 1
−1
∫ 1
−1
dtxdty2 − tx − ty
[
Pk(tx)Pp(ty) − 1]
. (364)
We have assumed that m2 << N . This is worse than what happens in the two-dimensional
case. In here not only that the difference survives the limit but also it diverges.
We can now state with some detail the continuum limit in which the fuzzy spheres approach
(in a precise sense) the noncommutative planes. We are interested in the canonical large
stereographic projection of the spheres onto planes. A planar limit can be defined as follows
θ2 =R2
√c2
= fixed as N,R→∞. (365)
We are now in a position to study what happens to the scalar field theory in this limit. First
we match the spectrum of the Laplacian operator on each sphere with the spectrum of the
Laplacian operator on the limiting noncommutative plane as follows
a(a+ 1) = R2~a2. (366)
The vector ~a is the two dimensional momentum on the noncommutative plane which corre-
sponds to the integer a. However since the range of a’s is from 0 to N − 1 the range of ~a2 will
be from 0 to 2NΛ2 where Λ = 1/θ. It is not difficult to show that the free action scales as
∑
a,b
∑
ma,mb
[
a(a+ 1) + b(b+ 1) +m2]
|φamabmb |2 =∫ 2NΛ2
0
d2~ad2~b
π2
[
~a2 +~b2 +M2]
|φ|~a|αa|~b|αb |2. (367)
70
The scalar field is assumed to have the scaling property φ|~a|φa|~b|φb = R3φamabmb which gives it
the correct mass dimension of −3. The αa and αb above are the angles of the two momenta
~a and ~b respectively. They are defined by αa = πma
aand αb = πmb
band hence they are in the
range [−π, π]. The mass parameter M of the planar theory is defined by M2 = m2/R2.
With these ingredients, it is not then difficult to see that the flattening limit of the planar
2−point function (361) is given by
ΠP
R2=
2
π2
∫ 2NΛ2
0
∫ 2NΛ2
0
d2~ad2~b
~a2 +~b2 +M2. (368)
This is the 2-point function on noncommutative R4 with the euclidean metric R2 × R2. By
rotational invariance it may be rewritten as
ΠP
R2=
2
π2
∫ 2NΛ2
0
d4k
k2 +M2. (369)
We do now the same exercise for the non-planar 2-point function (362). Since the external mo-
menta k and p are generally very small compared toN , one can use the following approximation
for the 6j-symbols
a s s
b s s
≈(−1)a+b
NPa(1 −
2b2
N2). N→∞, a << s, 0≤b≤2s. (370)
By using all the ingredients of the planar limit we obtain the result
ΠN−P (k, p)
R2=
2
π2(2π)2
∫ 2NΛ2
0
∫ 2NΛ2
0
(|~a|d|~a|)(|~b|d|~b|)~a2 +~b2 +M2
Pk(1 − θ4~a2
2R2)Pp(1 −
θ4~b2
2R2). (371)
Although the quantum numbers k and p in this limit are very small compared to s, they are large
themselves i.e. 1 << k, p << s. On the other hand, the angles νa defined by cos νa = 1 − θ4~a2
2R2
can be considered for all practical purposes small, i.e. νa = θ2|~a|R
because of the large R factor,
and hence we can use the formula (see for eg [58], page 72)
Pn(cos νa) = J0(η) + sin2 νa
2
[
J1(η)
2η− J2(η) +
η
6J3(η)
]
+O(sin4 νa
2), (372)
for n >> 1 and small angles νa, with η = (2n+ 1) sin νa
2. To leading order we then have
Pk(1 −θ4~a2
2R2) = J0(θ
2|~k||~a|) =1
2π
∫ 2π
0dαae
iθ2 cos αa|~k|~a|. (373)
This result becomes exact in the strict limit of N,R → ∞ where all fuzzy quantum numbers
diverge with R. We get then
ΠN−P (k, p)
R2=
2
π2
∫ 2NΛ2
0
∫ 2NΛ2
0
d2~ad2~b
~a2 +~b2 +M2eiθ2|~k|(|~a|cosαa)eiθ2|~p|(|~b| cos αb). (374)
71
By rotational invariance we can set θ2Bµνkµaν = θ2|~k|(|~a| cosαa), where B12 = −1. In other
words, we can always choose the two-dimensional momentum kµ to lie in the y-direction, thus
making αa the angle between aµ and the x-axis. The same is also true for the other exponential.
We thus obtain the canonical non-planar 2-point function on the noncommutative R4 (with
Euclidean metric R2 ×R2). Again by rotational invariance, this non-planar contribution to the
2-point function may be put in the compact form
ΠN−P (k, p)
R2=
2
π2
∫ 2NΛ2
0
d4k
k2 +M2eiθ2pBk. (375)
W can read immediately from the above calculation that the planar contribution is quadratically
divergent as it should be, i.e.
1
32π2
ΠP
R2=∫ 2NΛ2
0
d4k
(2π)4
1
k2 +M2=
1
8π2
N
θ2. (376)
The non-planar contribution remains finite in this limit, viz
1
32π2
ΠN−P (k, p)
R2=∫ 2NΛ2
0
d4k
(2π)4
1
k2 +M2eiθ2pBk =
1
8π2
[
2
E2θ4+M2 ln(θ2EM)
]
, Eν = Bµνpµ.
(377)
This is the answer of [28]. This is singular at p = 0 as well as at θ = 0.
5.6 Real Quartic Pure Matrix Models
Let us consider the matrix model (see [85] and references therein)
Z(µ, g) = e−N2F =∫
dφe−NTrV (φ) , V (φ) = µφ2 + gφ4. (378)
The φ is a hermitian N × N matrix. In order to compute this path integral one can use in
this case the saddle point method quite efficiently. The first step is to diagonalize φ as follows
φ = Uφ0U+ where φ0 is the diagonal matrix φ0 = (λ1, λ2, ..., λN). The unitary matrix U drops
from the action TrV (φ) since we have the U(N) symmetry
TrV (φ) = TrV (φ0) =N∑
i=1
V (λi) = µN∑
i=1
λ2i + g
N∑
i=1
λ4i . (379)
The measure becomes
dφ =N∏
i=1
dλi
∏
i<j
(λi − λj)2dUijdU
∗ij . (380)
Since the potential does not depend on U , the integral over the unitaries decouples and one
ends up with the path integral
Z(µ, g) = e−N2F =∫ N∏
i=1
dλie−N2S(λ) , S(λ) =
1
N
N∑
i=1
V (λi) −2
N2
∑
i<j
ln |λi − λj |. (381)
72
The second term in S is the so-called Vandermonde determinant.
In the large N limit the path integral is seen to be dominated by the configuration which is
the minimum of S. The saddle point configuration must therefore be a solution of the equation
of motion
∂S(λ)
∂λi=∂V (λi)
∂λi− 2
N
∑
j 6=i
1
λi − λj= 0. (382)
We introduce the resolvent (Green function)
W (z) =1
NTr
1
z −M=
1
N
N∑
i=1
1
z − λi
. (383)
From this definition we can immediately remark that W (z) is singular when z approaches the
spectrum of φ. In general the eigenvalues of φ are real numbers in some range [a, b]. In the
large N limit all statistical properties of the spectrum are encoded in the resolvent W (z). In
this limit we can also introduce a density of eigenvalues ρ(λ) which is positive definite and
normalized to one; ρ(λ)≥0 and∫ ba ρ(λ)dλ = 1. Thus the sum will be replaced by an integral
such that 1N
∑Ni=1 =
∫ ba ρ(λ)dλ and hence
W (z) =∫ b
aρ(λ)dλ
1
z − λ. (384)
We can immediately compute
∫ b
aρ(λ)λkdλ = − 1
2πi
∮
W (z)zkdz (385)
The contour is a large circle which encloses the interval [a, b]. In terms of the resolvent the
density of eigenvalues is therefore given by (obtained with a contour which is very close to [a, b])
ρ(λ) = − 1
2πi(W (λ+ i0) −W (λ− i0)). (386)
In other words knowing W (z) will give ρ(λ).
By using the Green function (383) we can rewrite (in the large N limit) the saddle point
equation (382) as
W 2(z) = V′
(z)W (z) − P (z) , P (z) =1
N
∑
i
V′(z) − V
′(λi)
z − λi. (387)
The solution of this quadratic equation is immediately given by
W (z) =1
2(V
′
(z) −√
V ′2(z) − 4P (z)). (388)
This is much simpler than the original saddle point equation. It remains only to determine the
coefficients of P (z) (which is a much smaller number of unkown) in order to determine W (z).
73
Knowing W (z) solves the whole problem since it will give ρ(λ). This W (z) can have many cuts
with endpoints located where the polynomial under the square-root vanishes. The number of
cuts is equal (at most) to the degree of V′so in our case it is equal (at most) to 3.
The classical minimum of the potential V is given by the condition V′(z) = 2z(µ+2gz2) = 0.
In other words V can have only one minimum at z = 0 for positive values of µ and g. Therefore
we can safely assume that the support of ρ(λ) will consist of one connected region [a, b] which
means that all eigenvalues of φ lie at the bottom of the well around φ = 0. In this case the
resolvent W (z) has one cut in the complex plane along [a, b] with branch points at z = a and
z = b. The polyonmial V′2(z)−4P (z) which is under the square-root must thus have two single
roots corresponding to the branch points while all other roots are double roots.
The above argument works when both µ and g are positive. For the more interesting case
when either µ or g is negative the potential can have two equivalent minima but the rest of the
analysis will still be valid in one of the phases of the model.
We get therefore
V′2(z) − 4P (z) = M2(z)(z − a)(z − b) (389)
Let us note that in the large z region the resolvent behaves as 1/z. Hence in this region
1
z∼ 1
2(V
′
(z) −M(z)√
(z − a)(z − b)). (390)
Or equivalently
M(z) = PolV
′(z)
√
(z − a)(z − b). (391)
Pol stands for the polynomial part of the ratio ( recall that M(z) must be a polynomial in any
case ).
Let us now compute
P (z) = 2µ+ 4gz2 + 4g∫
dλρ(λ)λ2 + 4gz∫
dλρ(λ)λ. (392)
In above we can use the fact that since the potential V is even we must have b = −a and
hence we must have the identity∫
dλρ(λ)λ = 0. Also let us remark that for z −→ ∞ we have
V′(z) = 4gz3 and hence M(z) must behave in this region as 4gz2 from (390). However M(z)
must be at most quadratic from the fact that P (z) is quadratic together with equation (389).
We must then have
M(z) = 4gz2 + c. (393)
Indeed we can compute directly from (391) that
M(z) = Pol2z(µ + 2gz2)√
z2 − a2= 4gz2 + 2(µ+ ga2). (394)
74
In other words c = 2(µ+ ga2). Putting all these things together we obtain
W (z) = µz + 2gz3 − (2gz2 + µ+ ga2)√z2 − a2. (395)
This function must still satisfy the condition W ∼ 1/z for large z. This gives an extra equation
which must be solved for a. We have then
W (z)√z2 − a2
=µz + 2gz3
√z2 − a2
− (2gz2 + µ+ ga2)
=µa2
2z2+
3ga4
4z2+O(
1
z4). (396)
In other words a must satisfy the quadratic equation
1 =µ
2a2 +
3g
4a4. (397)
The solution is
a2 =1
3g(−µ+
√
µ2 + 12g). (398)
Finally from (386) we derive the density of eigenvalues
ρ(λ) =1
π(2gλ2 + µ+ ga2)
√a2 − λ2. (399)
It is not difficult to check that the above density of eigenvalues is positive definite for positive
values of µ. For negative values of µ we must have the condition µ+ ga2 > 0 in order for ρ to
be positive definite on the interval [−a, a]. This leads to the requirement
µ2 < 4g. (400)
At µ2 = 4g we must have a first order phase transition from the phase with a density of
eigenvalues given by (399) with a support given by one cut which is the interval [−a, a] to a
different phase where the suport of the density of eigenvalues consists of two cuts. The original
cut [−a, a] splits therefore at λ = 0 when µ = 4g to two disconnected intervals. We will not
compute this new density of eigenvalues at this stage.
Another phase transition (a second order phase transition) which is more non-trivial occurs
at the value when g = −µ2/12. This is the pure gravity critical point. To see that there is
something higly non-trivial happening at this value we compute the density of eigenvalues ρ(λ).
We find
ρ(λ) =2g
π(λ2 − a2)
23 . (401)
The square-root singularities at the end-points are replaced by a different singular behaviour
given by the power 2/3.
The last remark is to find the density of eigenvalues for g = 0. We find the famous Wigner
semi-circle law, viz
ρ(λ) =1
πµ√a2 − λ2 , a2 =
2
µ. (402)
Everything can be obtained using ρ(λ). For example we can compute the free energy F and
specific heat Cv.
75
6 Noncommutative Moyal-Weyl Spaces
6.1 The Weyl Map
The noncommutative Moyal-Weyl space Rdθ is given by the following basic commutation
relations
[xi, xj ] = iθ2Bij . (403)
The hermitian operators xi, i = 1, ..., d are interpreted as the coordinates on the noncommuta-
tive space Rdθ and B is a dimensionless tensor assumed to be invertible. These operators act in
some infinite dimensional Hilbert space H. The noncommutativity parameter θ has dimension
of lenght so that the operators xi have dimension of (length)−1. Derivations on this Rdθ are
defined by
∂i = − i
θ2(B−1)ij xj = −∂+
i , [∂i, xj ] = δij , [∂i, ∂j ] =i
θ2(B−1)ij . (404)
We will require that the coordinate space representation Φ≡Φ(x) of any operator Φ≡Φ(x) to
be a Schwartz function of sufficiently rapid decrease at infinity, viz
Φ(x) =∫ ddk
(2π)dΦ(k)eikx. (405)
Then we may use the Weyl quantization to write
Φ(x) =∫ ddk
(2π)dΦ(k)eikx. (406)
This is precisely the Weyl symbol of the field Φ(x). We remark that Φ+ = Φ=⇒Φ∗(x) =
Φ(x) , and Φ∗(−k) = Φ(k). We introduce the Weyl map between operators and functions by
the relation
Φ(x) =∫
ddxΦ(x)∆(x, x). (407)
∆(x, x) =∫ ddk
(2π)deikxe−ikx. (408)
[∆+(x, x) = ∆(x, x)]. Remark also that in the commutative limit θ−→0 the operator ∆(x, x)
reduces to a delta function δd(x− x). We have the identity
[∂j , eikx] = ikje
ikx. (409)
Hence
[∂j , Φ(x)] =∫
ddx∂jΦ(x)∆(x, x) =∫
ddk
(2π)dikjΦ(k)eikx. (410)
76
We can also verify that
[∂i,∆(x, x)] = −∂i∆(x, x). (411)
eα∂∆(x, x)e−α∂ = ∆(x, x− α) , αi∈R. (412)
In above we have also used the identity
eα∂eikx = eiαkeikxeα∂ . (413)
Hence given any operator Φ(x) satifying all the above requirements on Rdθ , viz
Φ(x) =∫
ddxΦ(x)∆(x, x). (414)
[∂i, Φ(x)] =∫
ddx∂iΦ(x)∆(x, x). (415)
Φ(x) =∫
ddk
(2π)dΦ(k)eikx , Φ(x) =
∫
ddk
(2π)dΦ(k)eikx. (416)
We can immediately compute
TrΦ(x) =∫
d4xΦ(x)Tr∆(x, x). (417)
We can conclude from (412) that the trace Tr∆(x, x) is independent of x for any trace Tr on
H, i.e Tr∆(x, x−α) = Tr∆(x, x). In other words Tr∆(x, x) is simply an averall normalization
which can always be set equal to one, namely
TrΦ(x) =∫
d4xΦ(x) , T r∆(x, x) = 1. (418)
This last result written as TrΦ(x) =∫
ddxΦ(x) = Φ(0) yields immediately
Treikx = (2π)dδd(k) , T reikxeipx = (2π)dδd(k + p)
Treikxeipxeiqx = e−i θ2
2Bijkipj(2π)dδd(k + p+ q). (419)
Hence
Tr∆(x, x)∆(x, y) = δd(x− y). (420)
Using this last formula one can immediatey deduce
Φ(x) = TrΦ(x)∆(x, x). (421)
This means that the map ∆(x, x) provides indeed a one-to-one correspondence between fields
and operators.
77
As a final exercise we need to compute the maps φ12(x) of operators Φ12(x) which are of
the form Φ12(x) = Φ1(x).Φ2(x) , where Φ1(x) and Φ2(x) are arbitrary operators . We have
Φ12(x) = TrΦ12(x)∆(x, x)
= TrΦ1(x)Φ2(x)∆(x, x)
=∫
ddyddzΦ1(y)Φ2(z)Tr∆(x, y)∆(x, z)∆(x, x)
Tr∆(x, y)∆(x, z)∆(x, x) =∫
ddk
(2π)d
ddp
(2π)deik(x−y)eip(x−z)e−i θ2
2Bijkipj . (422)
It is not difficult to find therefore that
Φ12(x) =∫ ddk
(2π)d
ddp
(2π)dΦ1(k)Φ2(p)e
−i θ2
2Bijkipjei(k+p)x. (423)
We introduce the star product
f ∗ g(x) = ei2θ2Bij
∂∂ξi
∂∂ηj f(x+ ξ)g(x+ η)|ξ=η=0. (424)
By using this new product equation (423) can be rewritten as
TrΦ1(x)Φ2(x)∆(x, x) =∫
ddk
(2π)d
ddp
(2π)dΦ1(k)Φ2(p)e
ikx ∗ eipx = Φ1 ∗ Φ2(x). (425)
This result can also be put in the form
Φ1(x)Φ2(x) =∫
ddxΦ1 ∗ Φ2(x)∆(x, x). (426)
This leads to the identity
TrΦ1(x)Φ2(x) =∫
ddxΦ1 ∗ Φ2(x)≡∫
ddxΦ1(x)Φ2(x). (427)
It can also be generalized to the case of three fields , viz
TrΦ1(x)Φ2(x)Φ3(x)∆(x, x) = (Φ1 ∗ Φ2) ∗ Φ3(x) = Φ1 ∗ (Φ2 ∗ Φ3)(x). (428)
We can immediately conclude that ∗ is associative. In the presence of derivative we have on
the other hand the basic identities
Tr[∂i, Φ(x)] =∫
ddx∂iΦ(x)≡0. (429)
TrΦ1(x)[∂i, Φ2(x)] =∫
ddxΦ1(x) ∗ ∂iΦ2(x)≡∫
ddxΦ1(x)∂iΦ2(x). (430)
78
6.2 Noncommutative Gauge Actions
The noncommutative gauge theory actions of interest to us are matrix models of the form
[61]
Sθ =1
4g2Tr
∑
i,j
(
i[Di, Dj ] −1
θ2(B−1)ij
)2
. (431)
The coupling constant g is of dimension (mass)2− d2 . The trace is taken over some infinite dimen-
sional Hilbert space H and hence Tr[Di, Dj] is 6=0 in general. This trace carries also dimension
(length)d which is very obvious if one uses coherent states. The sector of this matrix theory
which corresponds to a noncommutative gauge field on Rdθ is defined by the configurations
Di = − 1
θ2(B−1)ij xj + Ai, A
+i = Ai. (432)
The components xi’s can be identified with those of a background noncommutative gauge
field whereas Ai’s are identified with the components of the dynamical U(N) noncommutative
gauge field. U(N) gauge transformations (with generators Ta, a = 1, ..., N2 in the fundamental
N−dimensional representation) are implemented by unitary matrices U = exp(iΛ) , U U+ =
U+U = 1 , Λ+ = Λ = ΛaTa acting in the Hilbert space H . The connection is
Di = −i∂i + AaiTa , Ai≡Aa
iTa. (433)
The curvature is given by
Fij = i[Di, Dj] −1
θ2(B−1)ij = [∂i, Aj] − [∂j , Ai] + i[Ai, Aj ] (434)
They transform as
Di−→UDiU+ , i.e Ai−→U AiU
+ − iU [∂i, U+]
Fij−→U FijU+. (435)
Small gauge transformations U = 1+ iΛ act as AΛi = Ai + i[Λ, Ai]− [∂i, Λ] , FΛ
ij = Fij + i[Λ, Fij ].
We can now write the following Weyl maps
Ai(x) =∫
ddxAi(x)∆(x, x) , [∂i, Aj](x) =∫
ddx∂iAj(x)∆(x, x). (436)
[Ai, Aj](x) =∫
ddxAi, Aj∗(x)∆(x, x)
Ai, Aj∗(x) = Ai ∗ Aj(x) −Aj ∗ Ai(x). (437)
Hence we obtain for the curvature
Fij(x) =∫
ddxFij(x)∆(x, x) , Fij(x)≡Fij = ∂iAj − ∂jAi + iAi, Aj∗. (438)
79
The actions (431) can therefore be rewritten as
Sθ =1
4g2
∑
ij
∫
ddxtr(Fij ∗ Fij). (439)
Gauge transformations similarly will have the map
U(x) = exp(
iΛ(x))
=∫
ddxU(x)∆(x, x) . (440)
U(x) = exp∗
(
iΛ(x))
≡∑
n=0
in
n!Λ ∗ Λ ∗ ... ∗ Λ(x) , n times. (441)
Λ(x) =∫
ddxΛ(x)∆(x, x). (442)
[Λ = ΛaTa , Λ = ΛaTa and Λa+ = Λa , Λa∗ = Λa]. These gauge transformations are such that
U ∗ U+(x) = U+ ∗ U(x) = 1 and they act as follows
Ai−→U ∗ Ai ∗ U+ − iU ∗ ∂iU+
Fij−→U ∗ Fij ∗ U+. (443)
6.3 Renormlaized Perturbation Theory
6.3.1 The Effective Action and Feynman Rules
The equations of motion are given by
δSθ = − i
g2
∫
ddxtr[δAν ∗ [Dµ, Fµν ]∗]=⇒[Dµ, Fµν ]∗ = 0. (444)
We recall that Dµ = −i∂µ + Aµ and [Dµ, f ]∗ = −i∂µf + [Aµ, f ]∗ . Let us now write
Aµ = A(0)µ + A(1)
µ . (445)
The background field A(0)µ satisfies the calssical equations of motion, viz [D(0)
µ , F (0)µν ]∗ = 0 and
A(1)µ is a quantum fluctuation. Using the fact that one can always translate back to the operator
formalism where∫
ddxtr behaves exactly like a trace we can compute∫
ddxtr[D(0)µ , A(1)
ν ]∗ ∗ [D(0)ν , A(1)
µ ]∗ =∫
ddxtr[
[D(0)µ , A(1)
µ ]∗ ∗ [D(0)ν , A(1)
ν ]∗ − [A(1)µ , A(1)
ν ]∗ ∗ [D(0)µ , D(0)
ν ]∗
]
=∫
ddxtr[
[D(0)µ , A(1)
µ ]∗ ∗ [D(0)ν , A(1)
ν ]∗ − iF (0)µν [A(1)
µ , A(1)ν ]∗
]
. (446)
Hence we compute upto quadratic terms in the fluctuation the action
Sθ[A] = Sθ[A(0)] +
1
2g2
∫
ddxtr[
[D(0)µ , A(1)
ν ]∗ ∗ [D(0)µ , A(1)
ν ]∗ − [D(0)µ , A(1)
µ ]∗ ∗ [D(0)ν , A(1)
ν ]∗
+ 2iF (0)µν ∗ [A(1)
µ , A(1)ν ]∗
]
. (447)
80
The linear term vanishes by the equations of motion. The gauge symmetry A′
µ = U ∗Aµ ∗U+ −iU ∗ ∂µU
+ reads in terms of the background and the fluctuation fields as follows
A(0)µ −→A(0)
µ
A(1)µ −→U ∗ A(1)
µ ∗ U+ + U ∗ [D(0)µ , U+]∗. (448)
This is in fact a symmetry of the full action Sθ[A] and not a symmetry of the truncated version
written above. This also means that we have to fix a gauge which must be covariant with
respect to the background gauge field. We choose the Feynamn-’t Hooft gauge given by the
actions
Sgf =1
2g2
∫
ddxtr[D(0)µ , A(1)
µ ]∗ ∗ [D(0)ν , A(1)
ν ]∗
Sgh = − 1
g2
∫
ddxtr[
c ∗D(0)µ D(0)
µ c+ c ∗ [A(1)µ , [D(0)
µ , c]∗]∗
]
. (449)
The partition function is therefore given by
Z[A(0)] = e−Sθ[A(0)]∫
DA(1)DcDc e−1
2g2 (Sdiam+Spara). (450)
In above the actions Sdiam and Spara are given by
Sdiam =∫
ddxtr[
[D(0)µ , A(1)
ν ]∗ ∗ [D(0)µ , A(1)
ν ]∗ − 2c ∗ (D(0)µ )2c
]
Spara = 2∫
ddxtr[F (0)µν ∗ A(1)
λ ∗ (Sµν)λρA(1)ρ ]. (451)
(Sµν)λρ = i(δµλδνρ − δµρδνλ) can be interpreted as the generators of the Lorentz group in the
spin one representation after Wick rotating back to Minkowski signature .
The one-loop effective action can be easily obtained from the above partition function. We
find the result
Γθ = Sθ[A(0)] − 1
2TrdTRLog
(
(D(0))2δij + 2iF (0)ij
)
+ TRLog(D(0))2. (452)
The operators (D(0))2 = D(0)i D(0)
i , D(0)i and F (0)
ij are defined through a star-commutator
and hence even in the U(1) case the action of these operators is not trivial. For example
D(0)i (A
(1)j )≡[D
(0)i , A
(1)j ]∗ = −i∂iA
(1)j + [A
(0)i , A
(1)j ]∗. The trace Trd is the trace associated with
the spacetime index i and TR corresponds to the trace of the different operators on the Hilbert
space H.
We find now Feynman rules for the noncommutative U(1) gauge theory. We start with the
diamagnetic part Sdiam of the action. This part of the action describes in a sense the motion
of the d − 2 physical degrees of freedom of the fluctuation field A(1)µ in the background field
A(0)µ which is very much like Landau diamagnetism. This can also be seen from the partition
function∫
DA(1)DcDc e−1
2g2 Sdiam = [det(D(0)µ )2]−
D−22 . (453)
81
The paramagnetic part Spara of the action describes the coupling of the spin one noncommuta-
tive current A(1)λ ∗ (Sµν)ρλA
(1)ρ to the background field A(0)
µ . This term is very much like Pauli
paramagnetism .
We write the diamagnetic action as follows
Sdiam =∫
ddx[
A(1)µ ∂2A(1)
µ − 2i∂µA(1)ν [A(0)
µ , A(1)ν ]∗ + [A(0)
µ , A(1)ν ]2∗ + 2c∂2c+ 2ic∂µ([A(0)
µ , c]∗)
+ 2ic[A(0)µ , ∂µc]∗ − 2c[A(0)
µ , [A(0)µ , c]∗]∗
]
. (454)
In momentum space we introduce the Fourier expansions
A(0)µ =
∫
kBµ(k)eikx , A(1)
µ =∫
kQµ(k)eikx , c =
∫
kC(k)eikx , c =
∫
kC(k)eikx ,
∫
k≡∫ ddk
(2π)d.
(455)
We also use the identities
eikx ∗ eipx = e−i2θ2k∧pei(k+p)x , [eikx, eipx]∗ = −2i sin(
θ2
2k∧p)ei(k+p)x , k∧p = ξµνkµpν . (456)
We compute now the following propagators
− 1
2g2
∫
ddxA(1)µ ∂2A(1)
µ = −1
2
∫
kQµ(k)( − 1
g2k2)Qµ(−k)=⇒− g2δµν
k2
− 1
2g2
∫
ddx 2c∂2c = −∫
kC(k)( − k2
g2)C(−k)=⇒− g2
k2. (457)
The vertex V (BQQ) is defined by
− 1
2g2
∫
ddx(
− 2i∂µA(1)ν [A(0)
µ , A(1)ν ]∗
)
=∫
k,p,qδk,p,qVνλµ(BQQ)
1
2Qν(k)Qλ(q)Bµ(p)
Vνλµ(BQQ) = −2i
g2(k − q)µδνλ sin(
θ2
2k∧q). (458)
We have used the notation δk1,k2,...,kn = (2π)dδd(k1 + k2 + ... + kn). The vertex V (QQBB) is
defined by
− 1
2g2
∫
ddx[A(0)µ , A(1)
ν ]2∗ =∫
k,p,q,lδk,p,q,lVµλνρ(QQBB)
1
4Bµ(k)Bλ(q)Qν(p)Qρ(l)
Vµλνρ(QQBB) =4
g2δµλδνρ
(
sin(θ2
2k∧p)sin(
θ2
2q∧l) + sin(
θ2
2q∧p)sin(
θ2
2k∧l)
)
. (459)
The vertex V (CCB) is given by
− 1
2g2
∫
ddx2ic[
∂µ[A(0)µ , c]∗ + [A(0)
µ , ∂µc]∗
]
=∫
k,p,lδk,p,lVµ(CCB)C(k)Bµ(p)C(l)
Vµ(CCB) = −2i
g2(l − k)µsin(
θ2
2p∧l). (460)
82
The vertex V (CCBB) is given by
1
2g2
∫
ddx 2c[A(0)µ , [A(0)
µ , c]∗]∗ =∫
k,p,q,lδk,p,q,lVµν(CCBB)
1
2Bµ(l)Bν(p)C(k)C(q)
Vµν(CCBB) =4
g2δµν
(
sin(θ2
2l∧k)sin(
θ2
2p∧q) + sin(
θ2
2p∧k)sin(
θ2
2l∧q)
)
. (461)
To calculate the paramagnetic vertex we write
F (0)µν =
∫
kFµν(k)e
ikx. (462)
Then
− 1
2g2Spara = − 1
g2
∫
ddx Fµν ∗ A(1)λ ∗ (Sµν)λρA
(1)ρ =
∫
k,p,qδk,p,qVµνλρ(FQQ)
1
2Fµν(k)Qλ(p)Qρ(q)
Vµνλρ(FQQ) = − 2
g2(δµρδνλ − δµλδνρ)sin(
θ2
2k∧q). (463)
6.3.2 Vacuum Polarization
The contribution of the diamagnetic vertices to the vacuum polarization tensor is given by
4 different diagrams. The graph with two BQQ vertices is equal to
Πµν(p)(BQQ) = (1
2)(4d)
∫
ksin2(
θ2
2k∧p)(2k − p)µ(2k − p)ν
k2(p− k)2. (464)
The graph with one BBQQ vertex is equal to
Πµν(p)(BBQQ) = (1
2)(−8dδµν)
∫
ksin2(
θ2
2k∧p) 1
k2. (465)
The graph with two BCC vertices is equal to
Πµν(p)(BCC) = (−1)(4)∫
ksin2(
θ2
2k∧p)(2k − p)µ(2k − p)ν
k2(p− k)2. (466)
The graph with one BBCC vertex is equal to
Πµν(p)(BBCC) = (−1)(−8δµν)∫
ksin2(
θ2
2k∧p) 1
k2. (467)
These contributions add to the diamagnetic polarization tensor
Πdiamµν (p) = 2(d− 2)
∫
ksin2(
θ2
2k∧p)
[
(p− 2k)µ(p− 2k)ν
k2(p− k)2− 2
k2δµν
]
. (468)
Using the identity 4sin2α = 2 − e2iα − e−2iα we can rewrite this result as a sum of planar and
non-planar contributions corresponding to planar and non-planar diagrams respectively. We
83
have then
Πdiamµν (p) = Πdiam,P
µν (p) + Πdiam,NPµν (p)
Πdiam,Pµν (p) = (d− 2)
∫
k
[
(p− 2k)µ(p− 2k)ν
k2(p− k)2− 2
k2δµν
]
Πdiam,NPµν (p) = −(d− 2)
∫
kcos(θ2k∧p)
[
(p− 2k)µ(p− 2k)ν
k2(p− k)2− 2
k2δµν
]
. (469)
We write now
1
k2(p− k)2=∫ 1
0dx
1
(P 2 − ∆)2, P = k − px , ∆ = x(x− 1)p2. (470)
Then we compute
Πdiam,Pµν (p) = −(d− 2)(p2δµν − pµpν)
∫ 1
0dx(1 − 2x)2
∫
P
1
(P 2 − ∆)2
+ (d− 2)∫ 1
0dx∫
P
1
(P 2 − ∆)2[4PµPν − 2(P 2 − ∆)δµν ]. (471)
Πdiam,NPµν (p) = (d− 2)(p2δµν − pµpν)
∫ 1
0dx(1 − 2x)2
∫
P
eiθ2P∧p
(P 2 − ∆)2
− (d− 2)∫ 1
0dx∫
P
eiθ2P∧p
(P 2 − ∆)2[4PµPν − 2(P 2 − ∆)δµν ]. (472)
In above we have used the fact that∫ 10 dx(−1 + 2x) 1
(P 2−∆)2= 0. Introducing also the Laplace
transforms
1
P 2 − ∆=∫ ∞
0e−P 2te∆tdt ,
1
(P 2 − ∆)2=∫ ∞
0e−P 2tte∆tdt. (473)
We get immediately that
∫
P
1
(P 2 − ∆)2=
1
(4π)d/2
∫ ∞
0dt t1−
d2 e−x(1−x)p2t. (474)
eiθ2P∧p
(P 2 − ∆)2=∫ ∞
0e−t(P− ip
2t)2te−x(1−x)p2te−
p2
4t dt
∫
P
eiθ2P∧p
(P 2 − ∆)2=
1
(4π)d/2
∫ ∞
0dtt1−
d2 e−x(1−x)p2te−
p2
4t , pµ = θ2ξµνpν . (475)
Hence
Πdiam,Pµν (p) = −(d− 2)
(4π)d/2(p2δµν − pµpν)
∫ 1
0dx(1 − 2x)2
∫ ∞
0dt t1−
d2 e−x(1−x)p2t. (476)
84
Πdiam,NPµν (p) =
(d− 2)
(4π)d2
(p2δµν − pµpν)∫ 1
0dx(1 − 2x)2
∫
dt t1−d2 e−x(1−x)p2te−
p2
4t
+(d− 2)
(4π)d2
pµpν
∫ 1
0dx∫
dt t−1− d2 e−x(1−x)p2te−
p2
4t . (477)
The contribution of the paramagnetic vertex to the vacuum polarization is given by one
graph with two FQQ vertices. This is equal to
< Fµν(p)Fλρ(−p) >= (1
2)(8)
∫
k(δµλδνρ − δµρδνλ)
sin2( θ2
2k∧p)
k2(p− k)2. (478)
The polarization tensor corresponding to this loop is given by the identity
1
2
∫
p< Fµν(p)Fλρ(−p) > Fµν(p)Fλρ(−p) =
1
2
∫
pΠpara
µν (p)Bµ(p)Bν(−p). (479)
Πparaµν (p) = 16(p2δµν − pµpν)
∫
k
sin2( θ2
2k∧p)
k2(p− k)2. (480)
In above we have clearly used the fact that Fµν(p) = ipµBν(p)− ipνBµ(p) + .... Going through
the same steps as before we rewrite this result as a sum of planar and non-planar contributions
as follows
Πparaµν (p) = Πpara,P
µν (p) + Πpara,NPµν . (481)
Πpara,Pµν (p) = 8(p2δµν − pµpν)
∫
k
1
k2(p− k)2
=8
(4π)d2
(p2δµν − pµpν)∫ 1
0dx∫ ∞
0dt t1−
d2 e−x(1−x)p2t. (482)
Πpara,NPµν (p) = −8(p2δµν − pµpν)
∫
k
cos(θ2k∧p)k2(p− k)2
= − 8
(4π)d2
(p2δµν − pµpν)∫ 1
0dx∫ ∞
0dt t1−
d2 e−x(1−x)p2te−
p2
4t . (483)
6.3.3 The UV-IR Mixing and The Beta Function
Let us first start by computing the tree level vacuum polarization tensor. we have
e−Sθ[A(0)] = e− 1
4g2
∫
ddxF(0)2µν = e
− 14g2
∫
pFµν(p)Fµν(−p)≡e−
12g2
∫
p(p2δµν−pµpν)Bµ(p)Bµ(−p)+...
. (484)
From this we conclude that
Πtree−levelµν =
1
g2(p2δµν − pµpν). (485)
85
As we have seen there are planar as well as non-planar corrections to the vacuum polarization
tensor at one-loop. Non-planar functions are generally UV finite because of the noncommuta-
tivity of spacetime whereas planar functions are UV divergent as in the commutative theory
and thus requires a renormalization. Indeed for t−→0 which corresponds to integrating over
arbitrarily high momenta in the internal loops we see that planar amplitudes diverge while non-
planar amplitudes are regularized by the exponential exp(− p2
4t) as long as the external momenta
p does not vanish.
Planar functions at one-loop are given from the above analysis by the expressions (also by
suppressing the tensor structure p2δµν − pµpν for simplicity and including an arbitrary mass
scale µ)
Πdiam,P(p) =1
(4π)d2
∫ 1
0
1
(µ2)2− d2
dx(1 − 2x)2
[x(1 − x) p2
µ2 ]2− d
2
(2 − d)Γ(2 − d
2)
Πpara,P(p) =8
(4π)d2
∫ 1
0
1
(µ2)2− d2
dx
[x(1 − x) p2
µ2 ]2− d
2
Γ(2 − d
2). (486)
In above we have also used the integrals (in Minkowski signature)
∫
P
P 2
(P 2 − ∆)2=d
2
(−1)i
(4π)d2
Γ(1 − d
2)
1
∆1− d2
,∫
P
1
(P 2 − ∆)2=
(−1)2i
(4π)d2
Γ(2 − d
2)
1
∆2− d2
∫
Pe−tP 2
=1
(4π)d2
1
td2
,∫
Pe−tP 2
P 2 =d
2
1
(4π)d2
1
td2+1
,∫
Pe−tP 2
PµPν =1
2δµν
1
(4π)d2
1
td2+1
Γ(2 − d
2) = −d− 2
2Γ(1 − d
2). (487)
In d = 4 + 2ǫ we obtain
Πdiam,P(p) =1
16π2
[
1
3(2
ǫ+ 2γ + 2) +
2
3lnp2
µ2+ 2
∫ 1
0dx (1 − 2x)2 lnx(1 − x)
]
Πpara,P =8
16π2
[
− 1
ǫ− ln
p2
µ2− γ −
∫ 1
0dx lnx(1 − x)
]
. (488)
Let us also define adiam = 116π2 (
23) and apara = 1
16π2 (−8). Obviously in the limit ǫ−→0 these
planar amplitudes diverge , i.e their singular high energy behaviour is logarithmically divergent.
These divergent contributions needs therefore a renormalization. Towards this end it is enough
as it turns out to add the following counter term to the bare action
δSθ = −1
4( − adiam + apara
ǫ)∫
ddxF (0)2µν . (489)
The claim of [59] is that this counter term will also substract the UV divergences in the 3−and 4−point functions of the theory at one-loop. The vacuum polarization tensor at one-loop
is therefore given by
Πone−loopµν = (p2δµν − pµpν)
1
g2r(µ)
+ Πdiam,NPµν + Πpara,NP
µν . (490)
86
1
g2r(µ)
= Πbar + Πcounter−term + Πdiam,P + Πpara,P
=1
g2+ (adiam + apara)ln
p2
µ2− 11
24π2γ +
1
24π2+
1
8π2
∫
dx[(1 − 2x)2 − 4]lnx(1 − x).
(491)
It is obvious that Πbar = 1g2 while Πcounter−term = −adiam+apara
ǫ. A starightforward calculation
gives then the beta function [59, 60]
β(gr) = µdgr(µ)
dµ= (adiam + apara)g3
r(µ)≡ =1
8π2(−11
3)g3
r(µ). (492)
This is equal to the beta function of ordinary pure SU(2) gauge theory. Non-planar functions
are finite in the UV because of the exponential e−p2
4t . However it is clear that this exponential
regularizes the behaviour at t−→0 only when the external momentum p is 6=0 . Indeed non-
planar functions are given by the following Hankel functions
I1 =∫
dt t1−d2 e−x(1−x)p2te−
p2
4t |d=4 =1
2[iπH
(1)0 (2i
√ab) + h.c]
I3 =∫
dt t−1− d2 e−x(1−x)p2te−
p2
4t |d=4 =π
4
a3/2
b1/2[H
(1)1 (2i
√ab) +H
(1)3 (2i
√ab) + h.c].
Where a = x(1− x)p2 and b = p2
4. These integrals are always finite when p 6=0 and diverge only
for θ−→0 and/or p−→0 as follows
I1 = −ln(
x(1 − x)p2p2)
I3 =16
(p)2
(
1 − x(1 − x)p2p2
8
)
. (493)
In the limit of small noncommutativity or small momenta we have therefore the infrared singular
behaviour
Πdiam,NPµν = −adiam(p2δµν − pµpν)ln p
2p2 +2
π2
pµpν
(p)2
Πpara,NPµν = −apara(p2δµν − pµpν)ln p
2p2. (494)
This also means that the renormalized vacuum polarization tensor diverges in the infrared limit
p−→0 which we take as the definition of the UV-IR mixing in this theory.
6.4 Star Products on S2N And R2
θ And Planar Limit
The commutation relations on the sphere read
[xa, xb] =iR√c2ǫabcxc. (495)
87
We define the stereographic projections A and A+ in terms of the operators xa as follows
A =1
2(x1 − ix2)B , A+ =
1
2B(x1 + ix2) , B =
2
R− x3
. (496)
The commutation relation [x1, x2] = iRx3/√c2 takes now the simpler form
[A,A+] = F (|A|2) , F (|A|2) = αB[
1 + |A|2 − αB
4|A|2 − RB
2
]
, α =θ2
R, |A|2 = AA+. (497)
The constraint∑
a x2a = R2 reads in terms of the new variables
α
4βB2 − (β +
α
2)B + 1 + |A|2 = 0 , β = R + α|A|2. (498)
This quadratic equation can be solved and one finds the solution
B≡B(|A|2) =2
α+
1
R + α|A|2[
1 −√
1 +4R2
α2+
4R
α|A|2
]
. (499)
For large N we have 1β
= 1R[1 − α |A|2
R] + O(α2) and hence α
2B = α
2R(1 + |A|2) or equivalently
[A,A+] = 12√
c2(1 + |A|2)2 + O(α2). From the formula |A|2 = L−(
√c2 − L3)
−2L+ it is easy to
find the spectrum of the operator F (|A|2). This is given by
F (|A|2)|s,m >= F (λs,m)|s,m > , s =N − 1
2. (500)
λs,m =c2 −m(m+ 1)
(√c2 −m− 1)2
=n(N − n)
(√c2 + s− n)2
= λn−1 , m = −s, ...,+s , n = s+m+ 1 = 1, ..., N.
(501)
Now we introduce ordinary creation and annihilation operators a and a+ which are defined as
usual by [a, a+] = 1 with the canonical basis |n > of the number operator N = a+a , i.e N|n >=
n|n > , n = 0, 1, ... , and which also satisfy a|n >=√n|n− 1 > and a+|n >=
√n+ 1|n+ 1 >.
Next we embed the N−dimensional Hilbert space HN generated by the eigenstates |s,m > into
the infinite dimensional Hilbert space generated by the eigenstates |n > and then define the
maps on HN given by
A = fN (N + 1)a , A+ = a+fN (N + 1). (502)
It is an identity easy to check that (N+1)f 2N(N+1) = |A|2 and hence (N+1)f 2
N(N+1)|n−1 >=
nf 2N (n)|n−1 > and |A|2|s,m >= λn−1|s,m >. In other words we can identify the first N states
|n > in the infinite dimensional Hilbert space of the harmonic oscillator with the states of HN
via
|s,m >↔|n− 1 = s+m > . (503)
88
As a consequence we have the result fN(n) =√
λn−1
nwhich clealry indicates that the above map
is well define ( as it should be ) only for states n≤N . For example A|0 >= fL(N+1)a|0 >= 0 be-
cause a|0 >= 0 but also because A|0 >= 12(x1−ix2)B(|A|2)|s,−s >= B(λs,−s)
R2√
c2L−|s,−s >=
0 . The above map also vanishes identically on |s, s >= |N − 1 > as one might check. The
relation between F and fN is easily found to be given by
F (λn) = (1 + n)f 2N(n + 1) − nf 2
N (n). (504)
Now we define the corresponding coherent states by
|z;N >=1
√
MN (x)
N−1∑
n=0
zn
√n![fN(n)]!
|n > , MN(x) =N−1∑
n=0
(x)n
n!([fN(n)]!)2, x = |z|2
[fN (n)]! = fN (0)fN(1)...fN(n− 1)fN(n). (505)
By construction these states are normalized and they are such that
A|z;N >= z|z;N > − 1√
MN(|z|2)zN
√
(N − 1)![fN(N − 1)]!|N − 1 > . (506)
In the large N limit we can check that MN (x)−→(N − 1)(1 + x)N−2 and√
(N − 1)![fN(N −1)]!−→
√
π(N − 1) and hence A|z;N > −→z|z;N > which means that |z;N > becomes exactly
an A−eigenstate.
As it is the case with standard coherent states the above states |z;N > are not orthonormal
since < z1;N |z2;N >= MN(|z1|2)−12MN (|z2|2)−
12MN(z1z2). Using this result as well as the
completeness relation∫
dµN(z, z)|z;N >< z;N | = 1 where dµN(z, z) is the corresponding
measure we can deduce the identity
MN(1) =∫
dµN(z, z)MN(z)MN (z)
MN(|z|2) . (507)
This last equation allows us to determine that the measure dµ(z, z) is given by
dµN(z, z) = iMN (|z|2)XN(|z|2)dz∧dz = 2MN (|z|2)XN(|z|2)ρdρdθ. (508)
The function XN is defined by
∫ ∞
0dx xs−1XN(x) =
Γ(s)([fN(s− 1)]!)2
2π. (509)
The solution of this equation was found in [62] and it is given by
2πXN(x) = F1(γ +N, γ +N ;N + 1;−x) , γ =√c2 −
N − 1
2. (510)
For large N where |z|2 << N we can find that the behaviour of the measure dµN(z, z) coincides
with the ordinary measure on S2, viz
dµN(z, z)≃N − 1
2π
idz∧dz(1 + |z|2)2
. (511)
89
With the help of this coherent state |z;N > we can therefore associate to every operator O a
function ON(z, z) by setting < z;N |O|z;N >= ON(z, z). It is therefore clear that the trace of
the operator O is mapped to the inetgral of the function ON against the measure dµN(z, z), i.e
TrO =∫
dµN(z, z)ON(z, z). (512)
Given now two such operators O and P their product is associated to the star product of their
corresponding functions, namely
ON ∗ PN(z, z)≡ < z;N |OP |z;N >=∫
dµ(η, η)ON(η, z)MN(zη)MN(ηz)
MN(|z|2)MN(|η|2)PN(z, η)
ON(η, z) = (< z;N |η;N >)−1 < z;L|O|η;L > , PN(z, η) = (< η;N |z;N >)−1 < η;N |P |z;N > .
(513)
The large N limit of this star product is given by the Berezin star product on the sphere [63],
namely
ON ∗ PN (z, z) =N − 1
2π
∫ idη∧dη(1 + |η|2)2
ON(η, z)[
(1 + zη)(1 + ηz)
(1 + |z|2)(1 + |η|2)]N−2
PN(z, η). (514)
Finally we comment on the planar limit of the above star product which in fact is the central
point of our discussion here. In this limit we have x3 = −R where the minus sign is due to our
definition of the stereographic coordinate B in (496). The stereographic coordiantes B, A and
A+ are scaled in this limit as
B =1
R, A =
1
2RA , A = x1 − ix2 , A
+ =1
2RA+ , A+ = x1 + ix2 , xi = xi. (515)
This scaling means in particular that the coordinates z and z must scale as z = z/2R and
z = ¯z/2R. Since we already know that [A,A+] = 12√
c2(1 + |A|2)2 in the large N limit we can
immediately conclude that [A, A+] = 2θ2 in this limit or equivalently [x1, x2] = −iθ2. Next
from the result MN(x)−→(N − 1)(1 + x)N−2 when N−→∞ we can conlude that in this large
N limit we must have
MN (|z|2)−→N e1
2θ2 |z|2 . (516)
The measure dµN(z, z) behaves as
dµN(z, z) =i
4πθ2dz∧d¯z. (517)
Putting all these results together we obtain the Berezin star product on the plane [63], namely
O ∗ P (z, ¯z) =i
4πθ2
∫
dη∧d¯ηO(η, ¯z) e−1
2θ2 (z−η)(¯z−¯η)P (z, ¯η). (518)
Lastly it is easy to check that the trace behaves in the limit as follows
R2
NTr−→ i
8π
∫
dz∧¯z. (519)
90
7 Fuzzy Gauge Field Theory And UV-IR Mixing
7.1 Actions
From a string theory point of view the most natural gauge action on the fuzzy sphere is
the Alekseev-Recknagel-Schomerus action which is a particular combination of the Yang-Mills
action and the Chern-Simons term . It was shown in [19] that the dynamics of open strings
moving in a curved space with S3 metric in the presence of a non-vanishing Neveu-Schwarz B-
field and with Dp-branes is equivalent to leading order in the string tension to a gauge theory
on a noncommutative fuzzy sphere with a Chern-Simons term. The full U(n) action on the
fuzzy sphere they found is given by the combination
SARS[Da] = SY M [Da] + SCS[Da]. (520)
The 3 matrices Da’s are the covariant derivative with curvature Fab = i[Da, Db] + ǫabcDc and
the Yang-Mills and Chern-Simons-like actions are given respectively by
SY M [Da] =1
4g2(L+ 1)Trtr F 2
ab
SCS[Da] = − 1
6g2(L+ 1)Trtr
[
ǫabcFabDc + (D2a − L2
a)]
. (521)
This result is simply an extension of the original result of [18] in which strings moving in a flat
space in the presence of a constant N-S B-field are described in the limit α′−→0 by a Moyal-
Weyl noncommutative gauge theory. From string theory point of view the above combination
of Yang-Mills and Chern-Simons actions is therefore the most natural candidate for a gauge
action on the fuzzy sphere. In most of this notes we will thus work with the action
SARS[Da] =1
g2(L+ 1)Trtr
[
−1
4[Da, Db]
2 +i
3ǫabc[Da, Db]Dc
]
+n
6g2L2
a. (522)
This is also the action found in [51] in the context of IKKT matrix models [49, 50]. Gauge
transformations are implemented here by the unitary transformations U ∈ Un(L+1) as follows
Da→D′a = UCaU
−1. In above we have set R = 1 where R is the radius of the sphere . We will
also set N = n(L+ 1). The trace tr is the trace over the gauge group.
The first remark about this action is the fact that there is no quadratic term, i.e the term
D2a from the YM part cancels exactly the term D2
a from the CS. Furthermore we remark (as we
will show shortly) that in the Feynman gauge ξ = 1, the kinetic term reduces to L2: This is
simply the inverse propagator in the plane which can already be seen at the level of equations
of motion. Indeed varying the action yields the equations of motion
[Db, Fab] = 0 , Fab = [Da, Db] − iǫabcDc. (523)
91
As it was shown in [19] classical solutions in the presence of the Chern-Simons term which are
also absolute minima of the action are characterized by SU(2) IRR. (523) can also be solved
with general SU(2) representations as well as with diagonal matrices.
To state again the fact that the propagator is simply 1/L2 is our main reason why we prefer
the above action on the fuzzy sphere.
A final remark about the action (522) is to note that it has the extra symmetry Da−→Da +
αa1n(L+1) where αa are constant real numbers . This symmetry needs to be fixed by restricting
the covariant derivative Da to be traceless, i.e by removing the zero mode [47, 48]. This
symmetry manifests itself also in the form Aa−→Aa + αa1n(L+1) where Aa is the gauge field
defined by Da = La +Aa. Remark however that for Da = Ba1n(L+1) the action takes the value1
6g2n|L|2 whereas for Aa = Ba1n(L+1) the action is identically zero.
As it turns out the action (522) on its own does not describe in the continuum limit pure
gauge fields. Indeed we can show that in the continuum limit the gauge field Aa decomposes as
Aa = ATa +naΦ where AT
a is the field tangent to the sphere while Φna is the normal component
and as a consequence the gauge action becomes
SARS =1
4g2
∫
S2
dΩ
4πtr[
(F Tab)
2 − 4ǫabcFTabncΦ − 2[La + AT
a ,Φ]2 + 4Φ2]
. (524)
F Tab is clearly the curvature of the field AT
a , i.e F Tab = iLaA
Tb − iLbA
Ta + ǫabcA
Tc + i[AT
a , ATb ]. As
one can immediately see this theory consists of a 2−dimensional gauge field ATa with a Higgs
particle Φ.
7.2 Tangent Projective Module
Indeed the differential calculus on the fuzzy sphere is 3−dimensional and thus the field
content of this model consists necessarily of a 2-dimensional gauge field together with a scalar
fluctuation normal to the sphere. On the fuzzy sphere it is not possible to split the vector field~A in a gauge-covariant fashion into a tangent two-dimensional gauge field and a normal scalar
fluctuation. However although we do not have the analogue of ATa on the fuzzy sphere we can
still write a gauge-covariant expression for the normal scalar component in terms of Aa which
reads
Φ =1
2
(
xaAa + Aaxa +A2
a√c2
)
≡ D2a − L2
a
2√c2
. (525)
In above xa = La√c2
are the matrix coordinates on fuzzy. In the limit L → ∞ it is not difficult
to see that the matrix coordinates xa tend to the commutative coordinates na and the scalar
field Φ tends to ~n · ~A.
It is obvious that the 3−component gauge field is an element of the full projective mod-
ule Matn(L+1)⊗C3. Nevertheless strictly two-dimensional gauge fields can still be defined in
the fuzzy as elements of the tangent projective module P T (Matn(L+1)⊗C3) where P T is the
projector given by
P Tab = δab − xaxb. (526)
92
In the presence of a spin 1 field the algebra of matrices MatL+1 decomposes under the action
of the rotation group SU(2) as follows
ΓL2⊗ΓL
2⊗Γ1 = ΓL
2⊗(
ΓL2+1⊕ΓL
2⊕ΓL
2−1
)
. (527)
It is rather a trivial exercise to compute the projectors on the spaces ΓL2+1, ΓL
2−1 and verify
that P T = P+ + P−. In other words P+A and P−A are the components of the gauge field
tangent to the sphere whereas the normal component can be simply defined by P0A where
P0≡PN = 1 − P T which clearly projects on the space ΓL2
and reads in terms of components
PNab = xaxb. (528)
P T is the projector onto the fuzzy tangent bundle on the fuzzy sphere. We have (P T )2 = P T =
P T+, P Tabxb = 0, xaP
Tab = 0 and P T
abPTba = 2 which translates the fact that P T
abAb is indeed
a 2−dimensional gauge field. PN is the orthogonal projector which defines the fuzzy normal
bundle on the fuzzy sphere.
A more practical way of implementing the projection P T is to constrain in a gauge-covariant
way the gauge field Aa to satisfy an extra condition and as a consequence reduce the number
of its independent components from 3 to 2 . We adopt here the prescription of [52], viz
DaDa = c2. (529)
We incorporate the constraint (529) into the theory by adding to (522) the following scalar
action
Sm[Da] =2m2
g2(L+ 1)TrtrΦ2. (530)
This term in the continuum theory changes the mass term of the Higgs particle appearing in
(524) from√
2 to√
2(1 + 2m2) and hence in the large m limit the normal scalar field simply
decouples.
7.3 Quantization
The most general U(n) gauge action on the fuzzy sphere which is at most quartic in the
fields is therefore given by
S[Da] = SARS [Da] + Sm[Da] +ρ
g2(L+ 1)TrtrΦ − n
6g2L2
a. (531)
Or equivalently
S =1
g2(L+ 1)Trtr
[
−1
4[Da, Db]
2 +i
3ǫabc[Da, Db]Dc
]
+m2
2g2c2(L+ 1)Trtr(D2
a − c2)2
+ρ
2g2√c2(L+ 1)
Trtr(D2a − c2). (532)
93
We have also added a linear term in the scalar field with parameter ρ while the constant is
added so that the action vanishes for pure gauges Da = ULaU+.
The partition function of the theory is given by
ZL [Ja; g,m, ρ] =∫
∏3
a=1[dDa] e
−S[D]− 1g2(L+1)
TrJaDa. (533)
The equations of motion derived from the action (532) read in terms of the gauge-covariant
current Ja as follows
−i[Db, Fab] =m2
c2Da, D
2b − c2 + ρDa + Ja . (534)
Remark that the above action (532) for m6=0 does not enjoy the symmetry Ca−→Ca+αa1n(L+1)
and thus we can not simply remove the zero mode in this model.
We adopt the background field method to the problem of quantization. For simplicity we
consider the U(1) theory. In this case N = L+ 1. We make the change of variables Da −→ Ca
in the path integral. Then we separate the matrices as Ca = Da +Qa where Da are background
matrices and Qa are fluctuation matrices. We impose the covariant Lorentz gauge [Da, Qa] = 0.
The gauge fixing term and the Faddeev-Popov term are therefore given by
Sg.f + Sgh = − 1
2g2(L+ 1)Tr
[Da, Qa]2
ξ+
1
g2(L+ 1)Trb+[Da, [Da, b]]. (535)
We work in the gauge ξ−1 = 1 + m2
c2. This gauge becomes Feynman gauge ξ = 1 in the limit
N−→∞ and the Landau gauge ξ = 0 in the limit m−→∞. The partition function becomes
ZN [Ja; g,m, ρ] = e−S[Da]− 1
g2NTrDaJa det (D2)
∫
∏3
a=1[dQa] e
− 12g2N
TrQaΩabQb+.... (536)
The determinant Det(D2) comes from the integration over the ghost field whereas the Laplacian
Ωab is defined by
Ωab =ρ√c2δab + D2
cδab − 2iFab +2m2
c2(D2
c − c2)δab +4m2
c2DaDb. (537)
In above the notation Da and Fab means that the covariant derivative Da and the curvature
Fab act by commutators, i.e Da(M) = [Da,M ], Fab(M) = [Fab,M ] where M∈MatN . Similarly
D2(f) = [Da, [Da,M ]]. Performing the Gaussian path integral we obtain the one-loop effective
action
Γ[Da] = S[Da] +1
2Tr3TR log Ω − TR logD2. (538)
Note that the trace TR is the trace over 4 indices corresponding to the left action and right
action of operators on matrices . Tr3 means a trace associated with 3−dimensional rotations.
94
This result holds also for U(n) theories on the fuzzy sphere where only the meaning of the
symbols becomes different.
In the following we will concentrate on the theory with m = 0. We introduce the U(1)
gauge field by writing Da = La + Aa. The first term in (538) gives the full tree-level action of
the gauge field Aa. This reads
S[Da] =1
4g2NTrF 2
ab −1
2g2NǫabcTr[
1
2FabAc −
i
6[Aa, Ab]Ac]. (539)
In particular the quadratic action S2 reads
S2 = − 1
2g2TrL[La, Ab]
2 +1
2g2TrL[La, Aa]
2. (540)
The quadratic effective action is given by
Γ2 = S2 +1
2TR
1
L2(LA + AL) +
1
2TR
1
L2A2 − TR
1
L2Fab
1
LFab
− 1
4TR
1
L2(LA + AL)
1
L2(LA + AL). (541)
In the quadratic approximation the curvature is given by Fab = i[La, Ab] − i[Lb, Aa] + ǫabcAc.
7.4 Effective Action
The free propagator is defined by
(
1
L2
)AB,CD
=1
N
N−1∑
l=1
l∑
m=−l
1
l(l + 1)Y AB
lm (Y +lm)DC . (542)
The calculation of the first 3 terms in (541) is straightforward. By using extensively [53] we get
1
2TR
1
L2(LA + AL) =
1
2N
∑
pn
Tr[
La, Y†pn
] [
Aa, Ypn
]
p (p+ 1)= 2TrΦ − 2
NTrA2
a. (543)
1
2TR
1
L2A2 = − 1
2N
∑
l1m1
Tr[Aa, Y†l1m1
][Aa, Yl1m1 ]
l1(l1 + 1)=
1
NTrAaL2∆4Aa. (544)
−TR 1
L2Fab
1
LFab = − 1
N2
∑
k1m1
∑
k2m2
Tr[
Fab[Yk2m2 , Y†k1m1
]]
Tr[
Fab[Yk1m1 , Y†k2m2
]]
k1 (k1 + 1) k2 (k2 + 1)= − 1
NTrFab∆FFab.
(545)
The operators ∆4 ≡ ∆4(L2) and ∆F ≡ ∆F (L2) are defined in momentum space by their
eigenvalues ∆4(p1) and ∆F (p1) respectively which are given by the following loop sums
∆4(p1) =∑
l1,l2
2l1 + 1
l1(l1 + 1)
2l2 + 1
l2(l2 + 1)(1 − (−1)l1+l2+p1)(L+ 1)
p1 l1 l2L2
L2
L2
2l2(l2 + 1)
p1(p1 + 1). (546)
95
∆F (p1) = 2∑
l1,l2
2l1 + 1
l1(l1 + 1)
2l2 + 1
l2(l2 + 1)(1 − (−1)l1+l2+p1)(L+ 1)
l1 l2 p1L2
L2
L2
2
. (547)
The index p1 stands for external momentum whereas l1 and l2 are the internal momenta. The
1 in 1 − (−1)p1+l1+l2 comes from planar contribution whereas the −(−1)p1+l1+l2 comes from
non-planar contribution. The calculation of the last term of (541) is much more involved. We
get
−1
4TR
1
L2(LA + AL)
1
L2(LA + AL). = − 1
2N2
∑
l1m1
∑
l2m2
(
Tr[La, Yl1m1 ][Aa, Y+l2m2
]Tr[Lb, Yl2m2 ][Ab, Y+l1m1
]
l1(l1 + 1)l2(l2 + 1)
− Tr[La, Yl1m1 ][Aa, Y+l2m2
]Tr[Lb, Y+l1m1
][Ab, Yl2m2 ]
l1(l1 + 1)l2(l2 + 1)
)
= − 1
NTrAaLa∆3LbAb + ∆Γ2. (548)
Again the operator ∆3≡∆3(L2) is defined in momentum space by its eigenvalues ∆3(p1) given
by the loop sum
∆3(p1) =∑
l1,l2
2l1 + 1
l1(l1 + 1)
2l2 + 1
l2(l2 + 1)(1 − (−1)l1+l2+p1)(L+ 1)
p1 l1 l2L2
L2
L2
2
× l2(l2 + 1)
p21(p1 + 1)2
(l2(l2 + 1) − l1(l1 + 1)). (549)
By putting everything together (equations (543), (544), (545) and (548)) we obtain the full
U(1) effective action on S2N . This is given by
Γ2 = S2 + 2TrΦ +1
NTrAa(L2∆4 − 2)Aa −
1
NTrAaLa∆3LbAb
− 1
NTrFab∆FFab + ∆Γ2. (550)
This action must be gauge invariant in the limit N −→ ∞. To see this it is essential to use the
following identity
L2(∆3 − ∆4) + 2 = 0. (551)
In momentum space this reads as follows
∆3(p1) − ∆4(p1) = − 2
p1(p1 + 1). (552)
96
It is important to stress that this identity holds for any N . Thus the quadratic effective action
on the fuzzy sphere reads
Γ2 = S2 + 2TrΦ − 1
NTrFab(∆F − 1
2∆3)Fab −
1
2NǫabcTrFab∆3Ac + ∆Γ2. (553)
This establishes gauge invariance of the above quadratic effective action in the continuum large
N limit. As we will show shortly the action ∆Γ2 is also gauge invariant in the continuum limit.
However in order to show gauge invariance of the whole model for finite N we need also
to compute the quantum corrections to the cubic and quartic vertices. These corrections are
constrained by gauge invariance so that their effect is to replace Fab = i[La, Ab] − i[Lb, Aa] +
ǫabcAc in the above action by the full curvature Fab = i[La, Ab] − i[Lb, Aa] + ǫabcAc + i[Aa, Ab]
and to replace ǫabcFab(...)Ac by ǫabc(Fab − i3[Aa, Ab])(...)Ac (see equation (539) for the different
combinations of the gauge field which are gauge invariant). The full effective action for finite
N is therefore
Γ = S + 2TrΦ − 1
NTrFab(∆F − 1
2∆3)Fab −
1
2NǫabcTr(Fab −
i
3[Aa, Ab])∆3Ac + ∆Γ.
(554)
The action ∆Γ2 will also be modified when we include quantum corrections to the cubic and
quartic vertices.
7.5 The Normal Scalar Field Effective Action
The extra contribution ∆Γ2 in (548) is of the form
∆Γ2 =∑
p1n1
∑
p2n2
A−µ(p1n1)A−ν(p2n2)(−1)n1+ν[
Cp1−1mp1n11µC
p1−1−mp2n21ν (δp1,p2Λ
(−)(p1) + δp2,p1−2Σ(−)(p1))
+ Cp1+1mp1n11µC
p1+1−mp2n21ν (δp1,p2Λ
(+)(p1) + δp2,p1+2Σ(+)(p1))
]
. (555)
The functions Λ(±)(p1) and Σ(±)(p1) are loop sums of the form
∑
l1l2
2l1 + 1
l1(l1 + 1)
2l2 + 1
l2(l2 + 1)(1 − (−1)p1+l1+l2)(L+ 1)
p1 l1 l2L2
L2
L2
p2 l1 l2L2
L2
L2
X(l1, l2, p1).(556)
The explicit expressions of the X’s are not important for us here and thus we simply skip
writing them down. By inspection we can see that the Clebsch-Gordan coefficients appearing
in the action ∆Γ2 are exactly those which appear in the scalar mass term. Indeed we can
compute
1
4NTr(xaAa + Aaxa)
2 =∑
p1n1
∑
p2n2
A−µ(p1n1)A−ν(p2n2)(−1)n1+ν[
Cp1−1mp1n11µC
p1−1−mp2n21ν
× (δp1,p2λ(−)(p1) + δp2,p1−2σ
(−)(p1)) + Cp1+1mp1n11µC
p1+1−mp2n21ν (δp1,p2λ
(+)(p1) + δp2,p1+2σ(+)(p1))
]
.
(557)
97
The classical functions λ(±)(p1) and σ(±)(p1) (which we also do not write down here) are not
loops of the form (556) as it must be obvious. By comparing (555) and (557) we can immediately
deduce that the action ∆Γ2 in position space must involve anticommutators of xa and Aa instead
of commutators and hence it is a correction to TrΦ2 of the form
∆Γ2 = TrΦi∆ijΦj , Φi = xa(∇iAa) + (∇iAa)xa. (558)
The operators ∆ij ≡ ∆ij(L2) and ∇i ≡ ∇(L2) are determined from the requirement that (708)
agrees with (555). It is not difficult to show that this action is gauge invariant in the continuum
large N limit.
7.6 The UV-IR mixing
The criterion for the existence of a UV-IR mixing phenomena on the fuzzy sphere is defined
by the requirement that the fuzzy quantum effective action does not approach the quantum
effective action on the commutative sphere. For a U(1) theory on ordinary S2 the action (524)
is quadratic in the fields ATa and Φ and thus the quantum corrections are trivial, i.e the effective
action on S2 is essentially equal to the classical action (524). On the other hand the quantum
corrections on the fuzzy sphere yield the action Γ2. So to show the existence of a UV-IR mixing
phenomena in this model we need only to check that some (or all) of the operators in the above
effective action (553) do not vanish in the continuum limit.
In this section we concentrate only on the operators ∆3 and ∆4. We go back to equation
(546) and rewrite (actually doing one of the sums) the loop sum ∆4(p1) in the form
∆4(p1) =1
p1(p1 + 1)
∑
l
2l + 1
l(l + 1)
[
1 − (L+ 1)(−1)l1+p1+L
p1L2
L2
l1L2
L2
]
. (559)
As it turns out we can use in the large L limit the same approximation used in [54], namely
p1L2
L2
l1L2
L2
≃(−1)L+p1+l1
LPp1(1 −
2l21L2
). (560)
Pp are the Legendre polynomials which are defined by the generating function
1√1 − 2tx+ t2
=∞∑
p=0
Pp(x)tp. (561)
It is quite straightforward to conclude that in the limit we must have
∆4(p1) = − 1
p1(p1 + 1)
∫ 1
−1
dx
1 − x[Pp1(x) − 1]≡− h(p1)
p1(p1 + 1). (562)
By equation (552) we must also have
∆3(p1) = − h(p1) + 2
p1(p1 + 1). (563)
98
The number h(p1) can be evaluated using the following trick. We regularize the integral by
replacing the upper bound by 1 − δ. By using (561) one have the following result
∞∑
p=1
h(p)tp =2
1 − tln(1 − t). (564)
We can further write the above equation as follows
∞∑
p=1
h(p)tp = −2∞∑
p=0
∞∑
k=1
tp+k
k= −2
∞∑
p=1
p∑
k=1
tp
k. (565)
Thus
h(p1) = −2p1∑
l=1
1
l. (566)
Hence one obtains
∆4(p1) =2∑p1
l=11l
p1(p1 + 1), ∆3(p1) =
2∑p1
l=21l
p1(p1 + 1). (567)
Putting the above results together we obtain in the continuum the effective action
Γ2 = S2 + 2N∫
dΩ
4πΦ −
∫
dΩ
4πFab∆FFab +
1
2
∫
dΩ
4πFab∆3Fab −
1
2ǫabc
∫
dΩ
4πFab∆3Ac + ∆Γ2.
(568)
In above we can also use the result that the operator ∆F approaches zero in the continuum
limit. The 3rd and 4th terms give rise to a non-trivial quantum contribution to the U(1) action
on S2. In other words the U(1) theory on S2 obtained as a limit of a U(1) theory on S2N is not
a simple Gaussian theory. These two terms reflect the existence of a gauge invariant UV-IR
mixing on the fuzzy sphere which survives the continuum limit. The first and last corrections
are only relevant for the scalar sector of the theory and thus they do not lead to any UV-IR
mixing in the 2−dimensional gauge sector.
7.7 One-Matrix Model Formulation
Following [45] we show now that the 3−matrix action S[Da] given by equation (532) with a
particular value of the mass m2 can be derived from a much simpler 1−matrix model. Towards
this end we introduce Pauli matrices σa and define the operator
C = (1
2+ σaLa)⊗1n. (569)
It is a trivial exrecise to check that C = (j(j + 1) − (L+12
)2)⊗1n where j is the eigenvalue of
the operator ~J = ~L + ~σ2
which takes the two values L+12
and L−12
. The eigenvalues of C are
therefore L+12
with multiplicity n(L+ 2) and −L+12
with multiplicity nL . As it turns out this
99
matrix C can be obtained as a classical configuration of the following 2n(L + 1)−dimensional
1−matrix action
S[C] =1
4g2
1
L+ 1Trtr2
[
(L+ 1)4
16− (L+ 1)2
2C2 + C4
]
. (570)
Indeed the equations of motion derived from this action reads
C(C2 − (L+ 1)2
4) = 0. (571)
It is easy to see that C solves this equation of motion and that the value of the action in this
configuration is identically zero , i.e S[C = C] = 0. In general this equation of motion reads in
terms of the eigenvalues ci as ci(c2i − (L+1)2
4) = 0 which means that ci = 0,+L+1
2,−L+1
2. In other
words classical configurations are matrices of eigenvalues 0 , +L+12
and −L+12
with corresponding
multiplicities n0 , n+ and n− respectively which must clearly add up to n0 + n+ + n− =
2n(L+ 1). The action for each zero eigenvalue ci = 0 is given by S[ci = 0] = 14g2
(L+1)3
8which is
suppressed in the large L limit and thus these stationary points do not contribute in the large
L limit. Expanding around the vacuum C by writing C = 12+D0 +σaDa where D0 and Da are
n(L+ 1)×n(L+ 1) matrices and imposing the condition D0 = 0 we get
C2|D0=0 =1
4+D2
a +1
2ǫabcσcFab. (572)
Hence we obtain the action
S[C] =1
4g2
1
L+ 1Tr[
F 2ab + 2(D2
a − c2)2]
. (573)
In this context the solution C and all its unitary transformations UCU−1 are nothing but pure
gauges. We remark that the mass m2 is fixed in this action to be given by m2 = c2 and
the Chern-Simons term as well as the linear term in Φ are absent (compare with (532)). In
order to recover the Chern-Simons term we have to do more and consider instead the general
2n(L+ 1)−dimensional 1−matrix model with cubic interaction
S[C] =1
4g2
1
L+ 1Trtr2
[
α + βC2 + γC3 + δC4]
. (574)
The parameters α , β , γ and δ are real numbers which will be determined from the requirement
that this action S[C] must reduce somehow to (532) with some value of m2. A general 2n(L+
1)×2n(L+ 1) matrix can always be parametrized as follows C = 12+D0 + σaDa where D0 and
Da are n(L + 1) matrices. Substituting C = 12
+D0 + σaDa in the action S[C] we get after a
straightforward calculation
S[C] =1
4g2
1
L+ 1Tr[
δF 2ab + (γ + 4δD0)ǫabc
1
2Dc, Fab + (γ + 4δD0)D
2a + 2δ(D2
a)2
+ (δ + 2β + 6γD0 + 12D20δ)D
2a + 2δ[Dc, D0]
2 + 2α +γ
4+δ
8+β
2+ (δ +
3
2γ + 2β)D0 + (3δ
+ 3γ + 2β)D20 + (2γ + 4δ)D3
0 + 2δD40
]
. (575)
100
Clearly we must fix the field D0 to be a constant which means that D0 is proportional to the
identity matrix, viz
D0 = d01n(L+1). (576)
Then we fix δ and γ to be given by
δ = 1 , γ = −4
6− 4d0. (577)
We will also set ρ = 0 in (532) in which case we must choose β to be given by the equation
1 + 2β − 12d20 − 4d0 = −4c2. (578)
We get then the action
S[C] =1
4g2
1
L+ 1Tr[
F 2ab −
4
6(ǫabcDcFab +D2
a − c2) + 2(D2a − c2)
2 + S0
]
. (579)
The constant term is given by
S0 = −1
6+ 2α− β2
2+ 4βd0 + 4(2β − 1)d2
0 −64
3d3
0 − 24d40 −
4
6c2. (580)
We have therefore obtained the Yang-Mills action plus a Chern-Simons term with the correct
coefficients. However the mass term is fixed to be given by m2 = c2. The linear term in Φ is
absent because of (578). This term can be generated by choosing a different value of β given
by
1 + 2β − 12d20 − 4d0 = −4c2 +
2ρ√c2. (581)
The mass term will still be fixed to be given by m2 = c2.
The equation of motion in terms of C are given by
C(C2 +3γ
4C +
β
2) = 0. (582)
Again the matrix C = 0 solves this equation of motion but its action is still very large and hence
this configuration does not contribute in the large L limit. We notice also that this equation
of motion can be solved explicitly by the following matrix
C = [ − 3γ
8+ b(
1
2+ σaLa)]⊗1n , b =
2
L+ 1
√
9γ2
64− β
2. (583)
From this solution and from the eigenvalues of the matrix 12+σaLa we can immediately deduce
that ci = −3γ8
+bL+12
is an eigenvalue of C with multiplicity n+ = n(L+2) while ci = −3γ8−bL+1
2
101
is an eigenvalue with multiplicity n− = nL. In other words n+ +n− = 2n(L+1), n+−n− = 2n.
It is obvious that we have in this case
Trtr2C =(
− 3γ
8+
1
L+ 1
√
9γ2
64− β
2
)
2n(L+ 1). (584)
The general configuration C = 12
+D0 + Caσa (with D0 fixed as above) is however such that
Trtr2C = (1
2+ d0)2n(L+ 1). (585)
This means in particular that one must have for consistency
1
2+ d0 = −3γ
8+
1
L+ 1
√
9γ2
64− β
2. (586)
Solving this equation for β we obtain
2β = −c2 + [3 + (L+ 1)2]d0 + [9 − (L+ 1)2]d20. (587)
By comparing (578) and (587) we get a quadratic equation for d0 given by
(1 + c2)d20 − c2d0 −
3c2 + 1
4= 0. (588)
We get two solutions
d0 = −1
2,
3c2 + 1
2c2 + 2. (589)
Or equivalently
λ ≡ d0 +1
2= 0 ,
3c2 + 1
2c2 + 2+
1
2. (590)
We get the following values
δ(λ) = 1
γ(λ) =4
3(1 − 3λ)
β(λ) =1
2(1 − 3λ)2 − (L+ 1)2
2(1 − λ)2. (591)
The above solution C takes now the simpler form
C = [λ+ (1 − λ)σaLa]⊗1n. (592)
This solves the equation of motion C2+(1−3λ)C+ β(λ)2
= 0 and satisfies Trtr2C = 2Nλ. There
are n+ eigenevalues equal to −1−3λ2
+(1−λ)L+12
and n− eigenvalues equal to −1−3λ2
−(1−λ)L+12
.
The matrix action takes now the explicit form
Sλ[C] =1
4g2
1
L+ 1Trtr2[α(λ) + β(λ)C2 +
4
3(1 − 3λ)C3 + C4]. (593)
102
The constant α≡α(λ) is determined from the requirement that the action Sλ[C] vanishes for
C = C, i.e C is an absolute minimum . We have then Sλ[C = C] = 0 or equivalently
α(λ) =β2(λ)
4− β(λ)
6(1 − 3λ)2 − λ
3(L+ 1)2(1 − 3λ)(1 − λ)2. (594)
Finally by comparing equations (581) and (587) we get instead of (588) the equation
− ρ
2√c2
= (1 + c2)d20 − c2d0 −
3c2 + 1
4. (595)
Thus the coupling constant ρ is fixed in terms of d0 while d0 remains arbitrary. In this case we
have a whole class of one-matrix models labled by d0 (or equivalently λ) which gives the same
matrix action (532) with m2 fixed to be given by m2 = c2 and ρ fixed to be given by (595).
7.8 The Partition Function in The Trivial Sector
The partition function of the model (593) is defined by the following path integral (with
C0 = 12
+D0)
Z[g2, L, n;λ] =∫
DC δ(C0 − λ)e−Sλ[C]. (596)
The delta function can be represented by
δ(C0 − λ) =∫
DKe i2Trtr2(C−λ)J . (597)
J =
(
K 0
0 K
)
. (598)
It is clear that the matrix model Sλ[C] is symmetric under U(2N) symmetries where N = n(L+
1). Indeed all configurations of the form U−1CU , where C is some hermitian matrix C∈Mat2N
and U is an arbitrary transformation in U(2N) have the same action , i.e S[C] = S[U−1CU ]. In
other words for each matrix C we can always pick the unitary transformation U∈U(2N) which
will diagonalize C , i.e we write C as C = U−1CdU and think of U as angular variable while
we think of the diagonal matrix Cd (where the entries ci are given by the eigenvalues of C) as
a radial variable. This diagonalization is essentially a gauge fixing procedure in this setting.
As a consequence the measure DC decomposes as DC = DCdDU where DU is the normalized
Haar measure on the group U(2N) while DCd is given by
DCd =(2π)N(2N−1)
∏2Np=1p!
∏2N
i=1dci∆
2(Cd). (599)
The Vandermonde determinant ∆ is given by
∆2(Cd) =∏
i>j(ci − cj). (600)
103
The partition function reads in terms of the unitaries U∈U(2N) as follows
Z =∫
DK∫
DCdDU e−Sλ[Cd]−iT rλKei2Trtr2CdUJU−1
. (601)
Clearly the presence of the delta function in the form of the current J breakes the original
U(2N) symmetry of the action Sλ[C]. Indeed the constraint C0 = λ breakes the group U(2N)
down to SU(2)×U(N). The SU(2) factor corresponds to the rotational symmetries on the
sphere.
Similarly we diagonalize the current J by writing J = V JdV−1 where V ∈U(2N) and Jd is a
diagonal matrix with entries given by the eigenvalues of J . Since J is of the form J = Kσ0 we
can conclude that both V and Jd are of the same form , i.e V = Wσ0 and Jd = Kdσ0. Hence
K = WKdW−1 and the measure DK is now given by DK = DWDKd where DKd is of the
same form as (599), i.e
DKd =(2π)
N(N−1)2
∏Np=1p!
∏N
i=1dki∆
2(Kd). (602)
∆(k) is the Vandermonde determinant and ki are the eigenvalues of Kd. A straightforward
calculation gives now for the partition function
Z =∫
DKd
∫
DCdDU e−Sλ[Cd]−iT rλKdei2Trtr2CdUJdU−1
. (603)
We can use the identity
∫
DUe i2Trtr2CdUJdU−1
= (2
i)N(2N−1)
∏2N−1
p=1p!
det(ei2cijj)
∆(Cd)∆(Jd). (604)
7.9 Gauge Theory on Fuzzy S2 × S2
Fuzzy S2L × S2
L is the simplest 4 dimensional fuzzy space. It is a finite dimensional matrix
approximation of the cartesian product of two continuous spheres. This fuzzy space is defined
by a sequence of spectral triples
S2L × S2
L =
Mat(L+1)2 , HL,∆L
. (605)
The Mat(L+1)2 is the matrix algebra of dimension (L+ 1)2 and ∆L is a suitable Laplacian acting
on matrices which encodes the geometry of the space. It is defined by
∆L ≡ [LAB, [LAB, · ]] = L2AB. (606)
The LAB, with A,B = 1, 4, are the generators of the irreducible representation (L2, L
2) of SO (4).
The generators LAB (with LAB = −LBA) satisfy the commutation relations
[LAB, LCD] = ifABCDEFLEF
≡ i(
δBCLAD − δBDLAC + δADLBC − δACLBD
)
. (607)
104
The HL in (605) is the Hilbert space (with inner product < M,N >= 1(L+1)2
Tr(
M †N)
) which
is associated with the irreducible representation (L2, L
2) of SO(4).
Since SO(4) = [SU(2)×SU(2)]/Z2 we can introduce SU(2) (mutually commuting) gener-
ators L(1)a and L(2)
a by −2L(1)a = 1
2ǫabcLbc + La4 and −2L(2)
a = 12ǫabcLbc − La4 with a = 1, 2, 3
and ǫabc is the three dimensional Levi-Civita tensor. Then it can be easily shown that the two
SO(4) quadratic Casimir can be rewritten in the form (where ǫABCD is the four dimensional
Levi-Civita tensor)
L2AB = 4[(L(1)
a )2 + (L(2)a )2] = 2L(L+ 2)≡8c2
ǫABCDLABLCD = 8[(L(1)a )2 − (L(2)
a )2]≡0. (608)
Similarly the Laplacian L2AB reads in terms of the three dimensional indices as follows
L2AB = 4
[
(
L(1)a
)2+(
L(2)a
)2]
. (609)
Obviously L(1)a ≡ [L
(1)a , · ] and L(2)
a ≡ [L(2)a , · ]. For S2
L × S2L the algebra Mat(L+1)2 is generated
by the coordinate operators
x(1)a = R1
L(1)a√c2, x(2)
a = R2L(2)
a√c2
(610)
They satisfy3∑
a=1
(
x(i)a
)2= R2
i 1,[
x(i)a , x
(j)b
]
=i Ri√c2δijǫabcx
(i)c , i = 1, 2. (611)
In the limit L→ ∞ keeping R1 and R2 fixed we recover the commutative algebra of functions
on S2 ×S2. If we also choose to scale the radii R1 and R2 such as for example θ21 = R2
1/L1 and
θ22 = R2
2/L2 are kept fixed we obtain the non-commutative Moyal-Weyl space R2θ1× R2
θ2.
The algebra of matrices Mat(L+1)2 can be decomposed under the action of the two SU(2)
of SO(4) as Mat(L+1) ⊗Mat(L+1). As a consequence a general function on S2L × S2
L can be
expanded in terms of polarization tensors [53] as follows
φ =L∑
k1=0
k1∑
m1=−k1
L∑
k2=0
k2∑
m2=−k2
φk1m1k2m2 Yk1m1 ⊗ Yk2m2 . (612)
A U(n) gauge field on S2L×S2
L can be associated with a set of six hermitian matrices DAB ∈Matn(L+1)2 (DAB = −DBA) which transform homogeneously under the action of the group, i.e
DAB → UDABU−1, U ∈ U
(
n(L+ 1)2)
. (613)
We will mainly be interested in U(1) gauge theory in here. Extension to U(n) theories is
straightforward. The U(1) action is given by (with TrL = 1(L+1)2
Tr, g is the gauge coupling
constant and m is the mass of the normal components of the gauge field )
105
S =1
16g2
−1
4TrL[DAB, DCD]2 +
i
3fABCDEFTrL[DAB, DCD]DEF
+m2
8g2L2AB
TrL(D2AB − L2
AB)2 +m2
32g2L2AB
TrL(ǫABCDDABDCD)2. (614)
The equations of motion are given by
i[DCD, FAB,CD] +4m2
√c2DAB,Φ1 + Φ2 +
m2
√c2ǫABCDDCD,Φ1 − Φ2 = 0. (615)
As we will see shortly FAB,CD = i [DAB, DCD]+fABCDEFDEF can be interpreted as the curvature
of the gauge field on fuzzy S2L×S2
L whereas Φ1 and Φ2 (defined by D2AB −L2
AB = 8√c2(Φ1 +Φ2)
and ǫABCDDABDCD = 16√c2(Φ1 − Φ2)) can be interpreted as the normal components of the
gauge field on S2L × S2
L.
The most obvious non-trivial solution of the equations of motion (615) must satisfy FAB,CD =
0, D2AB = L2
AB and ǫABCDDABDCD = 0 (or equivalently FAB = 0, Φi = 0). This solution is
clearly given by the generators LAB of the irreducible representation (L2, L
2) of SO(4), viz
DAB = LAB. (616)
As it turns out this is also the absolute minimum of the model. By expanding DAB around
this vacuum as DAB = LAB + AAB and substituting back into the action (614) we obtain
a U(1) gauge field AAB on S2L×S2
L with the correct transformation law under the action of
the group, namely AAB−→UAABU−1 + ULABU
−1. The matrices DAB are thus the covariant
derivatives on S2L×S2
L. The curvature FAB,CD in terms of AAB takes the usual form FAB,CD =
iLABACD − iLCDAAB + fABCDEFAEF + i[AAB, ACD]. The normal scalar fields in terms of AAB
are on the other hand given by 8√c2(Φ1+Φ2) = LABAAB+AABLAB+A2
AB and 16√c2(Φ1−Φ2) =
ǫABCD(LABACD + AABLCD + AABACD).
We can verify this conclusion explicitly by introducing the matrices D(1)a = L(1)
a + A(1)a and
D(2)a = L(2)
a + A(2)a defined by
D(1)a ≡− 1
2
[
1
2ǫabcDbc +Da4
]
, D(2)a ≡− 1
2
[
1
2ǫabcDbc −Da4
]
. (617)
Clearly D(1)a (A(1)
a ) and D(2)a (A(2)
a ) are the components of DAB (AAB) on the two spheres
respectively. The curvature becomes F(i,j)ab = iL(i)
a A(j)b − iL(j)
b A(i)a + δijǫabcA
(i)c + i[A(i)
a , A(j)b ]
whereas the normal scalar fields become 2√c2Φi = (D(i)
a )2 − c2 = L(i)a A
(i)a + A(i)
a L(i)a + (A(i)
a )2.
In terms of this three dimensional notation the action (614) reads
S = S(1) + S(2) +1
2g2TrL
(
F(1,2)ab
)2. (618)
106
The S(1) and S(2) are the actions for the U(1) gauge fields A(1)a and A(2)
a on a single fuzzy sphere
S2L. They are given by
S(i) =1
4g2TrL
(
F(i,i)ab
)2 − 1
2g2ǫabcTrL
[
1
2F
(i,i)ab A(i)
c − i
6[A(i)
a , A(i)b ]A(i)
c
]
+2m2
g2TrLΦ2
i .
(619)
It is immediately clear that in the continuum limit L−→∞ the action (618) describes the
interaction of a genuine 4−d gauge field with the normal scalar fields Φi = n(i)a A
(i)a where n(i)
a
is the unit normal vector to the i-th sphere. The parameter m is precisely the mass of these
scalar fields. Let us also remark that in this limit the 3−dimensional fields A(i)a decompose as
A(i)a = (A(i)
a )T + n(i)a Φi where (A(i)
a )T are the tangent 2−dimensional gauge fields. Since the
differential calculus on S2L ×S2
L is intrinsically 6−dimensional we can not decompose the fuzzy
gauge field in a similar (gauge-covariant) fashion and as a consequence we can not write an
action on the fuzzy S2L × S2
L which will only involve the desired 4−dimensional gauge field.
107
8 Fuzzy Gauge Theory and Emergent Geometry
Our understanding of the fundamental laws of physics has evolved to a very geometrical
one. However, we still have very little understanding of the origins of geometry itself. This
understanding has been undergoing a significant evolution in recent years and it now seems
possible to understand classical geometry as an emergent concept. The notion of geometry as
an emergent concept is not new, see for example [3] for an inspiring discussion and [4, 5] for
some recent ideas. We examine such a phenomena in the context of noncommutative geometry
[7] emerging from matrix models, by studying a surprisingly rich three matrix model [10, 11, 12].
The matrix geometry that naturally emerges here has received attention as an alternative setting
for the regularization of field theories [13, 14, 15, 16] and as the configurations of D0 branes
in string theory [17, 19]. Here, however, the geometry emerges as the system cools, much as
a Bose condensate or superfluid emerges as a collective phenomenon at low temperatures, and
there is no background geometry in the high temperature phase.
We consider the most general single trace Euclidean action (or energy) functional for a
three matrix model invariant under global SO(3) transformations containing no higher than
the fourth power of the matrices. This model is surprisingly rich and in the infinite matrix
limit can exhibit many phases as the parameters are tuned. We find that generically the model
has two clear phases, one geometrical the other a matrix phase. Small fluctuations in the
geometrical phase are of a Yang-Mills and a scalar field around a ground state corresponding to
a round two-sphere. In the matrix phase there is no background spacetime geometry and the
fluctuations are those of the matrix entries around zero. In this note we focus on the subset of
parameter space where in the large matrix limit the gauge group is abelian.
For finite but large N we find that, in the low temperature phase, the model exhibits fluctu-
ation around a fuzzy sphere [20] and, in the infinite N limit, as the temperature is increased it
undergoes a transition where the entropy jumps, yet the model has critical fluctuations and a
divergent specific heat. As this critical coupling is approached the fuzzy sphere radius expands
to a critical radius and the sphere evaporates. The neighbourhood of the critical point exhibits
all the standard symptoms of a continuous 2nd order transition, such as large scale fluctu-
ations, critical slowing down (of the Monte Carlo routine) and is characterized by a specific
heat exponent which we argue is α = 1/2, a value consistent with our numerical simulations.
In the high temperature (strong coupling) phase the model is essentially a zero dimensional
Yang-Mills matrix model.
By studying the eigenvalues of operators in the theory we establish that, in the matrix
phase, the matrices Da are characterised by continuous eigenvalue distributions which undergo
a transition to a point spectrum characteristic of the fuzzy sphere phase. The point spectrum is
consistent with Da = La/R where La are su(2) angular momentum generators in the irreducible
representation given by the matrix size and R is the radius of the fuzzy sphere. The full model
received an initial study in [11] while a simpler version invariant under Da → Da + Λ1 arises
naturally as the configuration of D0 branes in the large k limit or boundary WZW model [19]
and has been studied numerically in [12]. The interpretation here is novel, as are the results on
108
the entropy and critical behaviour and the extension to the full model.
Let Da , a = 1, 2, 3, be three N×N hermitian matrices and let us consider the action
S = S0 + V , (620)
S0 =α4
N
[
− 1
4Tr[Da, Db]
2 +2i
3ǫabcTrDaDbDc
]
V =m2α4
N
[
− TrD2a +
1
2c2Tr(D2
a)2]
,
so that α4 = β plays the role of inverse temperature. The absolute minimum of the action
is given by Da = La. Expanding around this configuration, with Da = La + Aa, yields a
noncommutative Yang-Mills action with field strength
Fab = i[La, Ab] − i[Lb, Aa] + ǫabcAc + i[Aa, Ab] (621)
and gauge coupling g2 = 1/α4. As written the gauge field includes a scalar field,
Φ =1
2(xaAa + Aaxa +
A2a√c2
) , (622)
as the component of the gauge field normal to the sphere when viewed as embedded in R3 with
xa = La√c2
and c2 =∑
a L2a = (N2 − 1)/4.
In the large N limit, taken with α and m held fixed the action for small fluctuations becomes
that of a U(1) gauge field coupled to a scalar field defined on a background commutative two-
sphere [10]. For large m the scalar field is not excited. The model with m = 0 appears as a low
energy limit of string theory [19].
Thus the limit N−→∞ is the limit of the continuum sphere. The other limit of interest is a
double scaling noncommutative planar limit of large R and large L taken together (restricting
the theory around the north pole for example) keeping 2R2/√c2 fixed equal to θ2. The above
action (620) is seen to tend for large m to the action
Sθ =θ2
4g2TrF 2
ij =θ2
4g2Tr
∑
i,j
(
i[Di, Dj] −1
θ2(ξ−1)ij
)2
. (623)
Here Di = − 1θ2 (ξ
−1)ij xj + Ai and [xi, xj ] = iθ2ξij where ξ−1 is an invertible tensor. In two
dimensions this action is the infinite dimensional matrix model describing U(1) gauge theory
on the noncommutative Moyal-Weyl plane (Tr here is an infinite dimensional trace ).
One can see the background geometry as that of a fuzzy sphere [20] by noting that the xa
satisfy
x21 + x2
2 + x23 = 1 , [xa, xb] =
i√c2ǫabcxc, (624)
and the algebra generated by products of the xa is the algebra of all N × N matrices with
complex coefficients. The geometry enters through the Laplacian [14]
L2 = [La, [La, ..]]. (625)
109
which has the same spectrum as the round Laplacian on the commutative sphere but cut off at
a maximum angular momentum L = N − 1. The fluctuations of the scalar have this Laplacian
as kinetic term.
The ground state is found by considering the configuration Da = φLa where φ plays the
role of the inverse radius of the sphere. The effective potential [10, 21] obtained by integrating
out fluctuations around this background is given, in the large N limit, by
Veff
2c2= α4
[
φ4
4− φ3
3+m2(
φ4
4− φ2
2)]
+ log φ2 (626)
It is unbounded from below at the origin, but, for sufficiently large α, exhibits a local minimum
near φ = 1 which disappears for α < α∗. The critical curve α∗ is determined from the point at
which the real roots of ∂Veff/∂φ = 0 merge and dissapear. This occurs at
φ∗ =φ∗
1 +m2=
3
8(1 +m2)
1 +
√
1 +32t
9
(627)
and gives the critical curve
2(1 +m2)α4∗(0)
α4∗
= 1 +16t
3(1 +
8t
9) + (1 +
32t
9)3/2 (628)
where t = m2(1+m2) and α∗(0) = (8/3)3/4. This expression interpolates between α∗ = (8/3)3/4
with φ∗ = 34
for m2 = 0 and α∗ = ( 8m2 )
1/4, with φ∗ = 1√
2for large m2. Thus, as the system is
heated, the radius, R = φ−1, expands form R = 1, at large α to R−1∗ = φ∗ at α∗. When α < α∗
the fuzzy sphere solution no longer exists, and, in effect, the fuzzy sphere evaporates as the
radius approaches R∗.
Furthermore, the entropy, S =< S > /N2, is given by
S =3
4+
iα4
12N2ǫabcTr(DaDbDc) −
m2α4
2N2Tr(DaDa) (629)
and for the fuzzy sphere phase becomes
S =3
4− α4
(
1
24φ3 − m2
8φ2
)
(630)
with φ given by the local minimum of Veff . Expanding in the neighbourhood of the critical
point gives
S = S∗ −1
4
(φ2∗ + 2tφ)
√
3φ∗ + 4t(β − βc)
1/2 + . . . (631)
and we can see that this predicts that the transition has a divergent specific heat with exponent
α = 1/2. For the special case m = 0 we find S∗ = 512
while in the matrix phase the entropy
takes the constant value S = 34
and ∆S = 13, i.e. the entropy increases in a discrete jump of 1
9
per degree of freedom (there are 3 matrices).
110
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0 1 2 3 4 5 6
Cv
α∼
N=16N=24N=32N=48
theory
Figure 2: The specific heat for the model S0, where the parameter m is set to zero.
An initial numerical study of the model (620), with m 6= 0, was reported in [11], whereas
the model with m2 = 0 was studied in [12]. Our recent systematic study is reported in [1, 2].
In Monte Carlo simulations we use the Metropolis algorithm and the action (620). The errors
were estimated using the jackknife method.
We observe that the entropy S collapses well for different N and defining αs as the value of
α at which curves of the average value of the action, < S >, for different N cross, gives a good
estimate of the location of the transition.
For m2 = 0 we observe (figure 2) a divergence in the specific heat, Cv :=< (S− < S >)2 >
/N2, as the critical value αs is approached from above, i.e. from the low temperature phase.
For α > αs the model is in the fuzzy sphere phase and Cv rapidly approaches Cv = 1 as α is
increased. For α < αs the model is in a matrix phase and Cv = 0.75, retaining this constant
value right from the transition at αs to α = 0. For large N we observe that the location αmax
of the peak in Cv and the minimum αmin coincide to the numerical accuracy we have explored
and agree well with the coupling, αs, at which < S > intersect for different values of N . The
critical coupling determined either as αmax or αmin from Cv, or as αs gives good agreement
with the theoretical value (628) and the numerical value found in [12].
The behaviour when m 6= 0 is qualitatively similar to m = 0. We observe again that, for
large α, the model is in the fuzzy sphere phase where Cv = 1.
As α is decreased for fixed N , the specific heat goes through a peak at αmax, with the peak
value decreasing as m2 is increased. As α is further decreased from αmax, Cv goes through
a minimum at αmin and then increases again, while asymptoting to the value 0.75 at α = 0.
As the value of m2 is increased, our numerical study confirms that the fuzzy sphere-matrix
model transition is shifted to lower values of α. Extrapolating m2 → ∞ we infer that the
critical coupling goes to zero. This limiting model should be related to the fuzzy Yang-Mills
model without scalar field [22]. By extrapolating the measured values of αmax and αmin to
N = ∞ we obtain the critical value αc which agrees with αs and hence we infer is detecting
the matrix-to-sphere phase transition. The location, αc of this transititon, however, deviates
measurably from (628) and for example for m2 = 200 we find αc = 0.403 ± 0.014 while (628)
111
-1
-0.5
0
0.5
1
1.5
-8 -6 -4 -2 0 2 4 6
Ln α∼ s
, α∼ c
Ln m2
Matrix Phase
Fuzzy Sphere Phase
Ln α∼ sLn α∼ ctheory
Figure 3: The phase diagram.
gives α∗ = 0.446982. Thus, though (628) gives a good indicative value for the transition it is
not precise. Our results are summarised in a phase diagram in figure 3.
We further observe that the jump in entropy decreases as m is increased and the nature of
the transition seems to changed from one with a jump in the entropy to one with continuous
entropy. Our theoretical analysis (631) indicates a divergent specific heat with exponent α = 1/2
but with a narrowing critical regime and with the critical point (628) shifted to smaller values
of α. As m is increased we therefore still expect critical fluctuations with a divergent specific
heat though the region where the specific heat is divergent becomes very narrow and our data
is not precise enough to access the critical regime. We have not been able to determine with
any precision where the transition becomes continuous, however the entropy jump appears to
dissapear at m2 ∼ 20. It is also possible that beyond this point the transition is in fact 3rd
order for large m, a behaviour typical of many matrix models, however, the persistence of the
critical line as determined by αs the crossing point of the entropy curves indicates the transition
is probably 2nd order continuous. We know of no other model that exhibits transitions of the
type presented here and further numerical and theoretical study is warranted.
More detail on the structure of the phases can be obtained from the distribution of eigen-
values of observables. Here we focus on D3 (by rotational symmetry all matrices have the same
eigenvalue distribution). The characteristic behaviour of the distributions of eigenvalues in the
fuzzy sphere and matrix phases is illustrated in figures 4 and 5 respectively. Thus we see that,
as one crosses the critical curve αs in figure 3, the eigenvalue distribution of D3 undergoes a
transition from a point spectrum to a continuous distribution symmetric around zero which is
fit by the one-cut distribution [26] for single matrix in a quartic potential.
We have thus found clear evidence for two distinct phases in the model. They can be cleanly
separated by the curve αs, figure 3, obtained as the value of the coupling at which the entropy
curves for different matrix sizes cross. The high temperature phase is described by Yang-
Mills in zero dimensions so there is no background spacetime geometry. As the system cools
a geometrical phase condenses and at sufficiently low temperatures the system is described
by small fluctuations of a U(1) gauge field coupled to a massive scalar field. The critical
112
0
0.2
0.4
0.6
0.8
1
1.2
-15 -10 -5 0 5 10 15
eige
nval
ues
dist
ribut
ion
of D
3
eigenvalue
N=24
Figure 4: The eigenvalues distribution for D3, with N = 24, in the fuzzy sphere for α = 5.00
and m2 = 200. It corresponds to D3 = L3.
0
0.005
0.01
0.015
0.02
0.025
-60 -40 -20 0 20 40 60
eige
nval
ues
dist
ribut
ion
of D
3
eigenvalues
N=24N=32N=48
Figure 5: The eigenvalues distribution for D3 in the matrix phase for N = 24, 32, 48, α = 0.20
and m2 = 200.
113
temperature is pushed to higher temperatures as the scalar field mass is increased. We also
find evidence that the UV/IR mixing [28] typical of many non-commutative field theories is
linked to the geometrical instability and dissapears as the scalar mass is increased. Once the
geometrical phase is well established the specific heat takes the value 1 with the gauge and
scalar fields each contributing 1/2.
When the system enters the matrix phase the fuzzy sphere configuration collapses under
fluctuations. In this phase, there is no underlying spacetime geometry and the model is de-
scribed by a pure matrix model. The system in this phase is well approximated by 3 decoupled
matrices each with the same quartic potential. The value Cv = 0.75 which coincides with the
the specific heat of 3 independent matrix models with quartic potential in high temperature
limit is therefore consistent with this interpretation.
The model of emergent geometry described here, though reminiscent of the random matrix
approach to two dimensional gravity [5] is in fact very different. The manner in which spacetime
emerges is also different from that envisaged in string pictures where continuous eigenvalue
distributions [4] or a Liouville mode [27] give rise to extra dimensions. It is closely connected
to the D0 brane scenario described in [17] and the m = 0 version is a dimensionally reduced
version of a boundary WZN models in the large k limit [19]. It is not difficult to invent higher
dimensional models with essentially similar phenomenology to that presented here (see [23] and
[25]). The model can be extended rather trivially to a dynamical one [24], the phase structure
is essentially the same though the detailed properties will differ and only space is emergent.
We believe the scenario gives an appealing picture of how a geometrical phase might emerge
as the system cools and suggests a very novel scenario for the emergence of geometry in the
early universe. In such a scenario the temperature can be viewed as an effect of other degrees
of freedom present in a more realistic model but not directly participating in the transition we
describe.
114
9 Fuzzy Fermions and Emergent Supersymmetry
115
10 Fuzzy CP2
10.1 Definition
CP2 is a Kahler manifold which is the orbit of SU(3) through the hypercharge operator
Y . The group SU(3) has eight generators ti which satisfy [ti, tj] = ifijktk; the hypercharge is
Y = 2√3t8. In the 3 representation the generators are 1
2λi where the λi are the eight Gell-Mann
matrices. As the stability group of Y is U(2) we have
CP2 = SU(3)/U(2). (632)
CP2 is also a projective complex space or the space of C1 subspaces in C3. If ξ ∈ C3 − 0 a
point of CP2 is the equivalence class 〈ξ〉 = 〈λξ〉 for all λ ∈ C1 −0. Choosing λ = (∑ |ξi|2)−
12
we see that CP2 = 〈ξ〉 = 〈ξeiθ〉: (∑ |ξi|2) = 1. Hence
CP2 = S5/U(1). (633)
The eight Gell-Mann matrices form a basis for the real vector space of traceless hermitian
matrices ∑ ξiλi, ξ = (ξ1, ..., ξ8) ∈ R8. So CP2 is a submanifold of R8. Let dijk be the totally
symmetric SU(3)-invariant tensor defined by
λiλj =2
3δij + ( dijk + ifijk)λk (634)
Then
ξ ∈ CP2 ⇐⇒ dijkξiξj = constant × ξk. (635)
10.2 Quantizing CP2
The Lagrangian giving fuzzy CP2 is
L = i NTrY g(t)−1 g(t). (636)
In above g(t)∈SU(3) and N is an arbitrary constant which is yet to be determined. A point
ξ(t) ∈ CP2 is related to g(t) by ξ(t)iλi = g(t)Y g−1(t) while the symplectic form on CP2 is
iNd[
TrY g−1dg]
= −i NTrY [g−1dg ∧ g−1dg].
Let us parametrize the group element g(t) by a set of eight real numbers θi, i = 1, ..., 8
and write g(t) = exp(iθiλi/2). The conjugate momenta πi associated with θi are given by
πi = ∂L∂θi
= iNT rY g−1 ∂g∂θi . These equations are essentially providing a set of constraints and
therefore one should rewrite them as πi − iNT rY g(t)−1 ∂g∂θi ≈ 0. Now if we change the local
parametrization θi−→fi(ǫ) such that g(f(ǫ)) = exp(iǫiλi/2)g(θ) where f(0) = θ. Then one can
show that the new conjugate momenta Λi = −πjNji are given by
Λi ≈N
2Tr[gY g−1λi]. (637)
116
In above we have used the very useful identity iλi
2g(θ) = Nji(θ)
∂g∂θj with Nji(θ) =
∂fj(ǫ)
∂ǫi|ǫ=0. Here
≈ denotes weak equality in the sense of Dirac. Using πi, πj = θi, θj = 0 and θi, πj = δijone can also prove the following identities
Λi, g = iλi
2g , Λi,Λj = fijkΛk. (638)
In other words Λi are the generators of SU(3) transformations which act naturally on the left
of g(t). They generate symmetries of the Lagrangian (636) as L does not change under the
transformations g−→hg where h is any constant element in SU(3).
SU(3) can also act on the right of g(t). The generators of this action can be given in terms
of Λi by ΛRj = −ΛiU
(1,1)ij (g) where U(g)(1,1) is the adjoint representation of the element g of
SU(3) defined by U(1,1)ij (g)λi = gλjg
−1. Similarly these new generators ΛRi satisfy the identities
ΛRi , g = −igλi
2, ΛR
i ,ΛRj = fijkΛ
Rk . (639)
In terms of these right generators the constraints (637) take the simpler form
ΛRi ≈ − N√
3δi8, (640)
These are primary constraints. As in the case of the sphere there are no secondary constraints
since the Hamiltonian commutes with (637) .
From (639) it is obvious that the constarints ΛRi ≈ − N√
3δi8 for i = 1, 2, 3, 8 are first class
constraints whereas for i = 4, 5, 6, 7 they are second class constraints6. We can make a set of
first class constraints which is classically equivalent to all the constraints by taking appropriate
complex combinations of the above second class constraints. For N≥0 we take
Π8 = Y R =2√3ΛR
8 ≈ −2
3N
Πi = IRi ≈ 0
Π45 = ΛR4 − iΛR
5 ≈ 0
Π67 = ΛR6 − iΛR
7 ≈ 0. (641)
For N≤0 the two above last equations are replaced with ΛR4 + iΛR
5 ≈ 0 and ΛR6 + iΛR
7 ≈ 0. Now
one can check that the Poisson brackets Πi,Πj vanish weakly on the surface Πj ≈ 0 for all i.
These constraints can be realized on functions on SU(3). As all isospin singlets (for example
the s-quark or the Ω−) have hypercharge in integral multiples of 23
we find that N ∈ Z. With
N fixed accordingly the constraints together mean that for right action we have highest weight
isospin singlet states of hypercharge −23N .
An IRR of SU(3) is labeled by (n1, n2) where ni ∈ N. It comes from the symmetric
product of n1 3’s and n2 3’s. A tensor Ti1...in1j1...jn2
for (n1, n2) has n1 upper indices, n2 lower
6Indeed one can easily check that the Poisson brackets ΛRi ,Λ
Rj do weakly vanish on the surface ΛR
j ≈− N
√
3δj8 only for i = 1, 2, 3, 8.
117
indices and is traceless,viz Ti1i2...in1i1j2...jn2
= 0 . Within an IRR the orthonormal basis can be written
as |(n1, n2), I2, I3, Y 〉 where I2, I3 and Y are square of isospin, its third component and the
hypercharge.
Let g → U (n1,n2)(g) define the representation (n1, n2) of SU(3). Then the functions given
by < (n1, n2), I2, I3, Y |U (n1,n2)(g)|(n1, n2), 0, 0,−2
3N > fulfill the constraints . By the Peter-
Weyl theorem, their linear span∑
ξ(n1,n2)I2,I3,Y 〈(n1, n2), I
2, I3, Y |U (n1,n2)(g)|(n1, n2), 0, 0,−23N〉 gives
all the functions of interest. If N = N ≥ 0, that requires that (n1, n2) = (N, 0). These are just
the symmetric products of N 3’s. If N = −N ≤ 0, (n1, n2) = (0, N) and we get the symmetric
product of N 3’s. The representations that we get by quantizing the Lagrangian (636) are thus
(N, 0) or (0, N).
The coordinates ξi on CP2 are such that∑
ξi ξi is a constant function which we can take to
be 1. On quantization, ξi become the operators ΛLi which we denote by xi. Since
∑
ΛLi ΛL
i = c2I
and c2 = 13N2 +N in (N, 0) or (0, N) we have
xi =ΛL
i√
13N2 +N
,∑
xi xi = I. (642)
So
[xi, xj] =i
√
13N2 +N
fijk xk (643)
These operators commute in the large N limit. It is a remarkable fact that xi fulfill (635) for
any N if xi’s belong to (N, 0) or (0, N), viz
dijkxixj =N3
+ 12
√
13N2 +N
xk. (644)
The algebra A generated by xi is what substitutes for the algebra of functions A = C∞(CP2).
It is the full matrix algebra in the IRR. Any f ∈ A has the partial-wave expansion
f(ξ) =∑
fnI2,I3,Y 〈(n1, n2), I
2, I3, Y |U (n1,n2)(g)|(n1, n2), 0, 0, 0〉. (645)
ξα λα = g λ8 g−1. (646)
The ket |(n1, n2), 0, 0, 0〉 exists only if n1 = n2 so that the sum in (645) can be restricted to
n1 = n2. If F ∈ A then F too has an expansion like (645) where the series is cut-off at
n = N . That is because of the following. The SU(3) Lie algebra has two actions on F . The
left action F → LLα F = Λα F and the right action F → −LR
α F = −F Λα. The derivation
F → adLα F = LLα F −LR
α F = [Λα, F ] is the action which annihilates I and corresponds to the
su(3) action on CP2. As F transforms as (N, 0) (for N ≥ 0 say) for ΛLα and as (0, N) for −ΛR
α
A decomposes into direct sum of IRR’s, viz (N, 0) ⊗ (0, N) = ⊕Nn=0(n, n). If 〈(n, n), I2, I3, Y 〉
furnishes a basis for (n, n) then F =∑N
0 FnI2,I3,Y |(n, n), I2, I3, Y 〉. Identifying this basis with
the one in (645) for n ≤ N we see that F transforms like a function on CP2 with a terminating
partial wave expansion.
118
10.3 The spectral triple CP2N = (MatN ,∆N ,HN)
In this section we will summarize the main results of this chapter so far with a slight change
of notation. Let Ta, a = 1, ..., 8 be the generators of SU(3) in the symmetric irreducible
representation (n, 0) of dimension N = 12(n+ 1)(n+ 2). They satisfy
[Ta, Tb] = ifabcTc . (647)
T 2a =
1
3n(n + 3) ≡ |n|2 , dabcTaTb =
2n+ 3
6Tc. (648)
Let ta = λa
2(where λa are the usual Gell-Mann matrices) be the generators of SU(3) in the
fundamental representation (1, 0) of dimension N = 3. They also satisfy
2tatb =1
3δab + (dabc + ifabc)tc
tr3tatb =1
2δab , tr3tatbtc =
1
4(dabc + ifabc). (649)
The N−dimensonal generator Ta can be obtained by taking the symmetric product of n copies
of the fundamental 3−dimensional generator ta, viz
Ta = (ta⊗1⊗...⊗1 + 1⊗ta⊗...⊗1 + ... + 1⊗1⊗...⊗ta)symmetric. (650)
In the continuum CP2 is the space of all unit vectors |ψ > in C3 modulo the phase. Thus
eiθ|ψ >, for all θ∈[0, 2π[ define the same point on CP2. It is obvious that all these vectors
eiθ|ψ > correspond to the same projector P = |ψ >< ψ|. Hence CP2 is the space of all
projection operators of rank one on C3. Let HN and H3 be the Hilbert spaces of the SU(3)
representations (n, 0) and (1, 0) respectively. We will define fuzzy CP2 through the canonical
SU(3) coherent states as follows. Let ~n be a vector in R8, then we define the projector
P3 =1
31 + nata (651)
The requirement P 23 = P3 leads to the condition that ~n is a point on CP2 satisfying the
equations
[na, nb] = 0 , n2a =
4
3, dabcnanb =
2
3nc. (652)
We can write
P3 = |~n, 3 >< 3, ~n|. (653)
We think of |~n, 3 > as the coherent state in H3 (level 3 × 3 matrices) which is localized at the
point ~n of CP2. Therefore the coherent state |~n,N > in HN (level N ×N matrices) which is
localized around the point ~n of CP2 is defined by the projector
PN = |~n,N >< N,~n| = (P3⊗P3⊗...⊗P3)symmetric. (654)
119
We compute that
tr3taP3 =< ~n, 3|ta|~n, 3 >=1
2na , trNTaPN =< ~n,N |Ta|~n,N >=
n
2na. (655)
Hence it is natural to identify fuzzy CP2 at level N = 12(n + 1)(n + 2) (or CP2
N) by the
coordinates operators
xa =2
nTa. (656)
They satisfy
[xa, xb] =2i
nfabcxc , x
2a =
4
3(1 +
3
n) , dabcxaxb =
2
3(1 +
3
2n)xc. (657)
Therefore in the large N limit we can see that the algebra of xa reduces to the continuum
algebra of na. Hence xa−→na in the continuum limit N−→∞.
The algebra of functions on fuzzy CP2N is identified with the algebra of N×N matrices
MatN generated by all polynomials in the coordinates operators xa. Recall that N = 12(n +
1)(n + 2). The left action of SU(3) on this algebra is generated by (n, 0) whereas the right
action is generated by (0, n). Thus the algebra MatN decomposes under the action of SU(3)
as
(n, 0)⊗(0, n) = ⊗np=0(p, p). (658)
A general function on fuzzy CP2N is therefore written as
F =n∑
p=0
F(p)I2,I3,Y T
(p,p)I2,I3,Y . (659)
The T(p,p)I2,I3,Y are SU(3) matrix polarization tensors in the irreducible representation (p, p). I2, I3
and Y are the square of the isospin, the third component of the isospin and the hypercharge
quantum numbers which characterize SU(3) representations.
The derivations on fuzzy CP2N are defined by the commutators [Ta, ..]. The Laplacian is then
obviously given by ∆N = [Ta, [Ta, ...]]. Fuzzy CP2N is completely determined by the spectral
triple CP2N = (MatN ,∆N ,HN). Now we can compute
trNFPN =< ~n,N |F |~n,N >= FN (~n) =n∑
p=0
F(p)I2,I3,Y Y
(p,p)I2,I3,Y (~n). (660)
The Y(p,p)I2,I3,Y (~n) are the SU(3) polarization tensors defined by
120
Y(p,p)I2,I3,Y (~n) =< ~n,N |T (p,p)
I2,I3,Y |~n,N > . (661)
Furthermore we can compute
trN [Ta, F ]PN =< ~n,N |[Ta, F ]|~n,N >= (LaFN)(~n) , La = −ifabcnb∂c. (662)
And
trNFGPN =< ~n,N |FG|~n,N >= FN ∗GN(~n). (663)
The star product on fuzzy CP2N is found to be given by [73]
FN ∗GN(~n) =n∑
p=0
(n− p)!
p!n!Ka1b1 ...Kapbp∂a1 ...∂apFN(~n)∂b1 ...∂bpGN(~n)
Kab =2
3δab − nanb + (dabc + ifabc)nc. (664)
10.4 Fuzzy gauge fields on CP2N
We will introduce fuzzy gauge fields Aa, a = 1, ..., 8, through the covariant derivatives Da,
a = 1, ..., 8, as follows
Da = Ta + Aa. (665)
Da are N×N matrices which transform under the action of U(N) as follows Da−→UDaU+
where U∈U(N). Hence Aa are N×N matrices which transform as Aa−→UAaU+ + U [Ta, U
+].
In order that the field ~A be a U(1) gauge field on fuzzy CP2N it must satisfies some additional
constraints so that only four of its components are non-zero. These are the tangent components
to CP2N . The other four components of ~A are normal to CP2
N and in general they will be
projected out from the model.
Let us go back to the continuum CP2 and let us consider a gauge field Aa, a = 1, ..., 8,
which is strictly tangent to CP2 . By construction this gauge field must satisfy
Aa = P TabAb , P
T = (naAdta)2. (666)
P T is the projector which defines the tangent bundle over CP2. The normal bundle over CP2
will be defined by the projector PN = 1 − P T . Explicitly these are given by
P Tab = ncnd(Adtc)ae(Adtd)eb = ncndfcaefdbe , P
Nab = δab − ncndfcaefdbe. (667)
In above we have used the fact that the generators in the adjoint representation (1, 1) satisfy
(Adta)bc = −ifabc. Remark that we have the identities naPTab = nbP
Tab = 0. Hence the condition
(666) takes the natural form
naAa = 0. (668)
121
This is one condition which allows us to reduce the number of independent components of Aa
by one. We know that there must be three more independent constraints which the tangent
field Aa must satisfy since it has only 4 independent components. To find them we start from
the identity [79]
dabkdcdk =1
3
[
δacδbd + δbcδad − δabδcd + fcakfdbk + fdakfcbk
]
. (669)
Thus
ncnddabkdcdk =2
3
[
nanb −2
3δab + ncndfcakfdbk
]
. (670)
By using the fact that dcdkncnd = 23nk we obtain
dabknk = nanb −2
3δab + ncndfcakfdbk. (671)
Hence it is a straightforward calculation to find that the gauge field Aa must also satisfy the
conditions
dabknkAb =1
3Aa. (672)
In the case of S2 the projector P T takes the simpler form P Tab = δab−nanb and hence PN
ab = nanb.
From equation (671) we have on CP2
P Tab = dabcnc − nanb +
2
3δab , P
Nab = −dabcnc + nanb +
1
3δab. (673)
If we choose to sit on the “north pole” of CP2, i.e ~n = (0, 0, 0, 0, 0, 0, 0,− 2√3) then we can find
that P T = diag(0, 0, 0, 1, 1, 1, 1, 0) and as a consequence PN = (1, 1, 1, 0, 0, 0, 0, 1) . So Adta,
a = 1, 2, 3, 8 correspond to the normal directions while Adta, a = 4, 5, 6, 7 correspond to the
tangent directions. Indeed by substituting ~n = (0, 0, 0, 0, 0, 0, 0,− 2√3) in equation (672) and
using d8ij = 1√3δij where i, j = 1, 2, 3 and d8αα = − 1
2√
3where α = 4, 5, 6, 7 and d888 = − 1√
3
we get A1 = A2 = A3 = A8 = 0 which is what we want. In fact (672) already contains (668).
In other words it contains exactly the correct number of equations needed to project out the
gauge field Aa onto the tangent bundle of CP2.
Let us finally say that given any continuum gauge field Aa which does not satisfy the
constraints (668) and (672) we can always make it tangent by applying the projector P T . Thus
we will have the tangent gauge field
ATa = P T
abAb = dabcncAb − na(nbAb) +2
3Aa. (674)
Similarly the fuzzy gauge field must satisfy some conditions which should reduce to (668) and
(672). Implementing (668) and (672) in the fuzzy setting is quite easy since we will only need
122
to return to the covariant derivatives Da and require them to satisfy the SU(3) identities (648),
viz
D2a =
1
3n(n + 3) , dabcDaDb =
2n+ 3
6Dc. (675)
So Da are almost the SU(3) generators except that they fail to satisfy the fundamental com-
mutation relations of SU(3) given by equation (647). This failure is precisely measured by the
curvature of the gauge field Aa, namely
Fab = i[Da, Db] + fabcDc = i[Ta, Ab] − i[Tb, Aa] + fabcAc + i[Aa, Ab]. (676)
The continuum limit of this object is clearly given by the usual curvature on CP2, viz Fab =
iLaAb − iLbAa + fabcAc + i[Aa, Ab].
Next we need to write down actions on fuzzy CP2N . The first piece is the usual Yang-Mills
action
SYM =1
4g2NTrF 2
ab. (677)
The second piece in the action is a potential term which has to implement the constraints (675)
in some limit. Towards the end of writing this potential we will introduce the four normal
scalar fields on fuzzy CP2N by the equations (see equations (675))
Φ =1
n(D2
a −1
3n(n + 3)) =
1
2xaAa +
1
2Aaxa +
1
nA2
a−→naAa. (678)
Φc =1
n(dabcDaDb −
2n+ 3
6Dc) =
1
2dabcxaAb +
1
2dabcAaxb −
2n+ 3
6nAc +
1
ndabcAaAb
−→ dabcnaAb −1
3Ac. (679)
We add to the Yang-Mills action the potential term
V0 =β
NTrΦ2 +
M2
NTrΦ2
a. (680)
In the limit where the parameters β and M2 are taken to be very large positive numbers we
can see that only configurations Aa (or equivalently Da) such that Φ = 0 and Φc = 0 dominate
the path integral which is precisely what we want.
The total action is then given by
S1 =1
2g2NTrF 2
ab +β
NTrΦ2 +
M2
NTrΦ2
c
=1
g2NTr[
− 1
4[Da, Db]
2 + ifabcDaDbDc
]
+3n
4g2NTrΦ +
β
NTrΦ2 +
M2
NTrΦ2
c . (681)
123
This is essentially the same action considered in [79]. However this action is different from the
action considered in [82] which is of the form
S0 =1
g2NTr[
− 1
4[Da, Db]
2 +2i
3fabcDaDbDc
]
. (682)
The first difference is between the cubic terms which come with different coefficients. The
second more crucial difference is the presence of the potential term in our case. The linear term
in Φ is actually a part of the Yang-Mills action.
The equations of motion derived from the action S0 are
[Da, Fab] = 0. (683)
These are solved by the fuzzy CP2N configurations
Da = Ta (684)
and also by the diagonal matrices
Da = diag(d1a, d
2a, ..., d
Na ). (685)
We think of these diagonal matrices (including the zero matrix) as describing a single point in
accordance with the IKKT model [49].
More interestingly is the fact that these equations of motion are also solved by the fuzzy
S2N configurations
Di = Li , i = 1, 2, 3 and Dα = 0 α = 4, 5, 6, 7, 8. (686)
Indeed in this case [Da, Fab] = [Di, Fib]. For b = j this is equal to 0 because fijk = ǫijk whereas
for b = α this is equal to zero because fijα = 0. In above Li are the generators of SU(2) in the
irreducible representation N−12
.
The equations of motion derived from the action S1 are on the other hand given by
i
g2[Db, Fab] +
1
2g2fabcFbc + 2βΦ, Da +M2
(
2dabcΦc, Db −2n+ 3
3Φa
)
= 0. (687)
Now the only solutions of these equations of motion are the CP2N configurations (684). Thus the
potential term has eliminated the diagonal matrices (685) as possible solutions. This classical
observation will not hold in the quantum theory for all values of the parameter M2 since there
will always be a region in the phase space of the theory where the vacuum solution is not
Da = Ta but Da = 0. However when we take M2 to be very large positive number then we
can see that Da = Ta becomes quantum mechanically more stable. Hence by neglecting the
potential term we can not at all speak of the space CP2N since it will collapse rather quickly
under quantum fluctuations to a single point. The potential term has also eliminated the fuzzy
S2N configurations (686) as possible solutions. In fact even if we set M = β = 0 in the above
equations of motion the fuzzy S2N configurations (686) are not solutions.
124
The other major difference between S1 and S0 is that if we expand around the fuzzy CP2N
solution Da = Ta by writing Da = Ta + Aa and then substitute back in S1 and S0 we find that
S0 does not yield in the continuum limit the usual pure gauge theory on CP2. It contains an
extra piece which resembles the Chern-Simons action ( although it is strictly real ). S1 will
yield on the other hand the desired pure gauge theory on CP2N in the limit M2−→∞ and hence
it has the correct continuum limit. If we do not take the limit M2−→∞ then S1 will describe
a gauge theory coupled to 4 adjoint scalar fields which are the normal components of Aa.
The only motivation for S0-as far as we can see- is its similarity to the fuzzy S2 action which
looks precisely like S0 with the replacement fabc−→ǫabc. Also it is found that perturbation
theory with S0 is simpler than perturbation theory with S1. More importantly it is found
that S0 allows for some new topology change which does not occur with S1. In particular the
transitions CP2N−→S2
N and S2N−→CP2
N are possible in the quantum theory of S0.
Using the background field method we find that the one-loop effective action in the gauge
ξ−1 = 1 + 2g2βi
n2 is given by
Γi[D] = Si[D] +1
2Tr8TR log Ωi
ab − TR logD2a. (688)
Si[D] =1
g2TrN
(
− 1
4[Da, Db]
2 + iαifabcDaDbDc
)
+ ρiTrNΦ + βiTrNΦ2 +M2i TrNΦ2
c .
(689)
Ωiab = D2
cδab − 2iFab + 2i(1 − 3αi
2)fabcDc +
2g2ρi
nδab +
2g2βi
n2(4DaDb + 2nδabΦ) +
2g2M2i
n2
(
− daa′cdbb′cDa′Db′ + 4daa′cdbb′cDa′Db′ − 2(2n+ 3
3)dabcDc + 2ndabcΦc + (
2n+ 3
6)2δab
)
.
(690)
The trace TR corresponds to the left and right actions of operators on matrices whereas Tr8 is
the trace associated with 8−dimensional rotations. Given a matrix O the operator O is given
by O(..) = [O, ..], for example Da(Qa) = [Da, Qa]. For S0 we have α0 = 23, ρ0 = β0 = M0 = 0
while for S1 we have α1 = 1, ρ1 = 3n4g2 , β1 = β and M1 = M .
10.5 Quantum Theory
10.5.1 Fuzzy CP2 Phase
Let us first neglect the potential term in Si, i.e we will set βi = Mi = 0 or equivalently
V0 = 0 in S1. The effective potential is given by the formula (688) where the background field
is chosen such that
Da = φTa. (691)
125
We want to study the stability of the fuzzy CP2N vacuumDa = Ta against quantum fluctuations.
The φ is an order parameter which measures in a well defined obvious sense the radius of CP2N .
The effective potential in this configuration is given by
Si[D] =3|n|2g2
[
1
4φ4 − αi
2φ3 +
g2ρi
3n(φ2 − 1)
]
+ 6N2 log φ. (692)
Case 1 For S0 we have α0 = 23
and ρ0 = 0. Thus the quantum effective potential is
Veff =Γ0[D]
6N2=
2
3n2g2
[
1
4φ4 − 1
3φ3]
+ logφ+ constant. (693)
The quantum minimum of the model is given by the value of φ which solves the equation
V′
eff = 0. It is not difficult to convince ourselves that this equation of motion will admit a
solution only up to an upper critical value g∗ of the gauge coupling constant g beyond which
the configuration Da = φTa collapses. At this value g∗ the potential Veff becomes unbounded
from below. The conditions which will yield the critical value g∗ are therefore V′
eff = V′′
eff = 0.
We find immediately
φ∗ =3
4, n2g2
∗ =2
9(3
4)4 = 0.0703. (694)
Above the value g∗ we do not have a fuzzy CP2N , in other words the space CP2
N evaporates at
this point. This critical point separates two distinct phases of the model, in the region above g∗we have a “matrix phase” while in the region below g∗ we have a “fuzzy CP2
N” phase in which
the model admits the interpretation of being a U(1) gauge theory on CP2.
By going from small values of g (g≤g∗ corresponding to the “fuzzy CP2N phase”) towards
large values of g we get through the value g∗ where the space CP2N decays. Looking at this
process the other way around we can see that starting from large values of g (g>g∗ corresponding
to the “matrix phase”) and going through g∗ we generate the space CP2N dynamically. It seems
therefore that we have generated quantum mechanically the spectral triple which defines the
space CP2N .
For the purpose of comparing with the numerical results of [82] we define the coupling
constant α such that α4N = 1g2 . Then the critical value (694) is seen to occur at
α∗ = 2.309. (695)
This is precisely the result of the Monte Carlo simulation reported in equation (3.2) of [82].
Case 2 For S1 we have α1 = 1 and ρ1 = 3n4g2 and hence the effective potential is given by
Veff =Γ1[D]
6N2=
2
3n2g2
[
1
4φ4 − 1
2φ3 +
1
4φ2]
+ logφ+ constant. (696)
A direct calculation yields the critical values
126
φ∗ =9 +
√17
16, n2g2
∗ =φ2∗
4(φ∗ −
2
3) = 0.02552. (697)
This g∗ is smaller than the g∗ obtained in (694) and hence the fuzzy CP2N is more stable in the
model S0 than it is in the model S1 which is largely due to the linear term proportional to Φ
in S1. In other words attempting to put true gauge theory on fuzzy CP2N causes the space to
decay more rapidly. However for S0 the true vacuum is the fuzzy sphere S2N and not the fuzzy
CP2N as we will now discuss
10.5.2 A Stable Fuzzy Sphere Phase
We know that there is also a fuzzy sphere solution (686) for the model S0. We consider then
the background field
Di = φTi , i = 1, 2, 3 , Dα = 0 , α = 4, 5, 6, 7, 8. (698)
We want now to study the stability of this vacuum against quantum fluctuations. The φ is now
an order parameter which measures the radius of the fuzzy sphere S2N . The classical potential
in this configuration is
S0[D] =2c2g2
[
1
4φ4 − 1
3φ3]
. (699)
In above c2 = N2−14
is the Casimir of SU(2) in the irreducible representation N−12
( N =12(n + 1)(n + 2) ). It is clear that 2c2 >> 3|n|2 and |n|2
c2<< 1 in the large n limit. Hence the
action (699) around the classical minimum φ = 1 is much smaller than the classical part of the
action (692). In other words the fuzzy sphere is more stable than the fuzzy CP2N in this case.
The quantum corrections are given in this case by
1
2Tr3TR log Ωij +
1
2Tr5TR log Ωαβ − TR log φ2L2
i . (700)
In above
Ωij =(
φ2L2k +
2g2ρ
n
)
δij + 2iφ(φ− 3α
2)ǫijkLk , Ωαβ =
(
φ2L2k +
2g2ρ
n
)
δαβ − 3iαφfαβiLi.
(701)
Following the same arguments of the previous section (only now it is SU(2) representation
theory which is involved) we have in the large n limit
1
2Tr3TR log Ωij − TR logφ2L2
i =1
2Tr3TR log φ2δij − TR logφ2 + .. = N2 logφ+ ... (702)
We can also argue that we have
1
2Tr5TR log Ωαβ + .. = 5N2 log φ+ ... (703)
127
In other words the configurations (698) although they are fuzzy sphere configurations they know
(through their quantum interactions) about the other SU(3) structure present in the model.
Classically this SU(3) structure is not detected at all by these configurations in the classical
potential (699). The effective potential becomes in this case
Veff =Γ0[D]
6N2=
1
12g2
[
1
4φ4 − 1
3φ3]
+ log φ+ constant. (704)
A direct calculation yields the critical value
φ∗ =3
4, g2
∗ =1
36(3
4)4 = 0.0087875. (705)
In terms of the coupling α define by α4 = 1g2 the critical value g∗ reads
α∗ = 3.26. (706)
This is again what is measured in the Monte Carlo simulation of the model S0 as it is reported
in equation (4.2) of [82]. Therefore we have a fuzzy sphere phase above α∗ and a matrix phase
below α∗. The model S0 can also be in a fuzzy CP2N phase for n2g2
∗ below the second value
of (694) which for large enough n is much smaller than the value n2g2∗ with g2
∗ given by the
second equation of (705). However we have seen in the previous paragraph that this CP2N
will decay rather quickly to a single point which (by the discussion of the present section) can
only happen by going first across a fuzzy sphere phase. We have then the transition pattern
CP2N−→S2
N−→0. In the limit where α4 = 2/n2g2 is kept fixed we can see that the above
critical value (705) is infinitely large which means that the model S0 is mostly in the fuzzy
sphere phase. The matrix phase shrinks to zero and the fuzzy sphere is completely stable
in this limit since the fuzzy CP2N phase can occur only at very small values of the coupling
constant n2g2.
10.5.3 The Transition CP2−→S2
Now we include the effect of the potential term V0. The relevant model is given by the action
S1. Naturally the calculation becomes more complicated in this case. After a long calculation
we get in the configuration Da = φTa the quantum effective potential
VM−→∞ =2
3n2g2
[
1
4φ4 − 1
2φ3 +
1
4φ2 +
g2β
9(φ2 − 1)2 +
g2M2
27(φ2 − φ)2
]
+1
3log φ. (707)
The calculation of the critical values in terms of the mass parameters M2 = g2M2 and γ is
done in the same way as before and it yields the following equations. The critical radius occurs
at the solutions of the equation
[1 +4M2
9(γ +
1
3)]φ2
∗ −9
8[1 +
4M2
27]φ∗ +
1
4− M2
9(2γ − 1
3) = 0. (708)
128
In the limit M−→∞ we get the solution
φ∗−→9+√
81 + 64(1 + 3γ)(1 − 6γ)
16(1 + 3γ), M−→∞. (709)
The choice of the plus sign instead of the minus sign is so that when γ goes to zero (in other
words β−→0) this solution will reduce to the first equation of (697). This agreement is due
to the fact that the limit γ−→0 is formally equivalent to the limit M−→0 ( since γ = βM2 ).
Indeed for very small values of M we get the potential
VM−→0 =2
3n2g2
[
1
4φ4 − 1
2φ3 +
1
4φ2 +
g2β
9(φ2 − 1)2 +
g2M2
27(φ2 − φ)2
]
+ log φ. (710)
This will also lead to the equation (708) which for M−→0 admits the solution given by the
first equation of (697).
The critical value of the coupling constant g∗ (or equivalently α∗) is given on the other hand
by the equation
n2g2∗
2=
1
α4∗
=1
2φ2∗
[
3
4(1 +
4M2
27)φ∗ −
1
2+
2M2
9(2γ − 1
3)]
. (711)
Hence in the limit M−→∞ we get the behavior
α4∗ =
18
M2φ2∗(φ∗ + 4γ − 2
3) + 9
4φ2∗(3φ∗ − 2)
−→ 18
M2φ2∗(φ∗ + 4γ − 2
3). (712)
The equation of motion ∂VM−→∞
∂φ= 0 could admit in general four real solutions where the one
with the least energy can be identified with the radius of fuzzy CP2N . This solution is found
to be very close to 1. However this is only true up to an upper value of the gauge coupling
constant g (or equivalently a lower bound of α) for every fixed value of M beyond which the
equation of motion ceases to have any real solutions. At this value the fuzzy CP2N collapses
under the effect of quantum fluctuations and we cross to a pure matrix phase. As the mass M
is sent to infinity it is more difficult to reach the matrix phase and hence the presence of the
mass makes the fuzzy CP2N solution Da = φTa more stable. In fact when M2−→∞ the critical
value α∗ approaches zero.
We repeat the large mass analysis for the model S0. In other words we add the potential V0
to the action S0 and study the effective potential when M,β−→∞. The interest in this action
lies in the fact that it admits ( at least for M = β = 0 ) a fuzzy sphere solution and hence
we can contemplate a transition (at the level of the phase diagram) between fuzzy CP2N and
fuzzy S2N when we take the limit M,β−→0. As before we consider fuzzy CP2
N configurations
Da = φTa, a = 1, ..., 8. For S0 + V0 (in other words non-zero values of M and β) these
configurations are in fact the true vacuum as we have discussed previously. When V0 = 0 the
fuzzy S2N configurations become the true minimum. The calculation of the quantum corrections
with non-zero V0 is exactly identical to what we have done previously and we end up with the
effective potential
129
VM−→∞ =2
3n2g2
[
1
4φ4 − 1
3φ3 +
g2β
9(φ2 − 1)2 +
g2M2
27(φ2 − φ)2
]
+1
3log φ. (713)
In the large M limit we get the same critical value (709). The critical value of g ( or equivalently
α ) is found on the other hand to be given by
α4∗ =
2
n2g2∗
=18
M2φ2∗(φ∗ + 4γ − 2
3) + 9
2φ3∗. (714)
So again in the large mass limit the fuzzy CP2N phase is stable even for the model S0.
However we know from our previous discussion that in the limit M−→0 the minimum of
the model S0 should tend to the fuzzy sphere solutions. Thus it is important to consider also
the fuzzy sphere configurations Di = φTi , i = 1, 2, 3 , Dα = 0 , α = 4, 5, 6, 7, 8. The effective
potential in these configurations is
VM−→0 =Γ0[D]
6N2=
1
12g2
[
1
4φ4 − 1
3φ3 +
M2γc22n2
(φ2 − |n|2c2
)2 +M2
2
((
2n+ 3
6n2
)2
φ2 +c23n4
φ4)]
+ log φ. (715)
The critical values are
φ∗ =3
4− 4M2
3
[
9γc28n2
− γ
3(1 +
3
n) +
3c28n4
+1
2
(
2n+ 3
6n2
)2]
+O(M4). (716)
α4∗ =
2
n2g2∗
=96
n2φ3∗
1
1 + 4M2
3φ∗[γ(1 + 3
n) − 3
2(2n+3
6n2 )2]−→256γ
3M2 +O(M4). (717)
So when M−→0 this α∗ goes to zero which is consistent with the result (705). This equation
tell us how we actually approach this critical value α∗ = 0.
The intersection of this equation with (714) gives a one-loop estimation of the value MT at
which the vacuum of the model S0 goes from a fuzzy sphere S2N to a fuzzy CP2
N as we increase
the mass parameter M . Equivalently the intersection point occurs at the value MT at which
the vacuum of the model S0 goes from a fuzzy CP2N to a fuzzy sphere S2
N as we decrease M .
10.6 Phase Structure
The model S1 . This is the correct model which describes U(1) gauge theory in the continuum
limit at least classically. The minimum of the model (for non-zero potential) can only be fuzzy
CP2N . There are two phases. In the fuzzy CP2
N phase we have a U(1) gauge theory on fuzzy
CP2N whereas in the matrix phase the fuzzy CP2
N configurations Da = φTa collapse and we
end up with a U(N) gauge theory on a single point. We have the qualitative behavior of a first
130
order phase transition which occurs between these two regions of the phase space. However it is
obvious from the critical line (714) that when the mass M of the four normal scalar components
of the 8−dimensional gauge field on fuzzy CP2N goes to infinity it is more difficult to reach the
transition line. In this limit the fuzzy CP2N phase dominates while the matrix phase shrinks to
zero. Therefore we can say that we have a nonperturbative regularization of U(1) gauge theory
on fuzzy CP2N .
The model S0 + V0 . This is a string-theory-inspired gauge model which does not go in the
continuum limit to the usual U(1) gauge theory on CP2 even classically. Indeed it can be shown
that it contains in the continuum limit (in addition to the usual Yang-Mills term) a Chern-
Simons-like term. However this model has a more interesting phase structure since it allows for
the (quantum) transitions between fuzzy S2N and fuzzy CP2
N . The main reason behind this
remarkable feature lies in the fact that when M−→∞ the absolute minimum of the model is
the fuzzy CP2N configurations Da = φTa whereas in the limit M−→0 the absolute minimum of
the model is the fuzzy sphere configurations Di = φTi, i = 1, 2, 3 and Dα = 0, α = 4, 5, 6, 7, 8.
The phase diagram of this model with the particular value γ = 112
is plotted in figure (6) for
illustration . The phase diagram consists of 3 phases.
1) The fuzzy CP2N phase : This is the region with M≥MT and above the line (714) where
the absolute minimum of the model is the fuzzy CP2N configurations Da = φTa and where
the field theory is some U(1) gauge theory on fuzzy CP2N . Recall that MT is the value of the
mass parameter M at which the two curves (714) and (717) intersect. The fuzzy CP2N phase
dominates the phase diagram when M−→∞.
2) The matrix phase : This phase shrinks to zero when M−→∞. This occurs at the points
of the phase diagram which are below the two lines (714) and (717).
3) The fuzzy S2N phase : These are the points which have M≤MT and which are above
the line (717) where the absolute minimum of the model is the fuzzy S2N configurations Di =
φTi, i = 1, 2, 3, Dα = 0, α = 4, 5, 6, 7, 8 and where the field theoy is a U(1) gauge theory on
fuzzy S2N with complicated coupling to 6 adjoint scalars.
For other approaches to topology change using finite dimensional matrix algebra and fuzzy
physics see [80, 81].
10.7 Fermion on Fuzzy CP2
10.7.1 The Spinc Structure
CP2 is not a spin manifold but a spinc [83]. The obstruction to the CP2 spin structure
comes from noncontractibile two-spheres in CP2. The SU(2)×SU(2) IRR’s of the non-existent
spinors are as follows, 1) Left-handed spinors : (1/2, 0) with quantum numbers (I, Y ) = (1/2, 0)
131
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7
2(n
g *)-2
m
The phase diagram for the model S0 with gamma =1/12
matrix phase
fuzzy CP2n phase
fuzzy S2N
phase
equation (5.23) equation (5.27)
Figure 6: The phase diagram for the model S0 with γ = 112
. In this case φ∗ = 1 and hence
the large mass expansion (714) becomes α4∗ = 108
4m2+27while the small mass expansion (717) is
α4∗ = 64
9m2 with m = M . The intersection point occurs at MT . This looks like a triple point.
under the action of the two-fold cover SU(2)×U(1) of U(2), 2) Right-handed spinors : (0, 1/2)
with quantum numbers (I, Y ) = (0, 1) and (0,−1) .
The spinc structure on CP2 is achieved by introducing an additional U(1) connection for
spinors which amounts to adding a hypercharge of magnitude |Y | = 1. Thus the quantum
numbers in the spinc case follows by adding an additional hypercharge which we can take to be
−1, viz 1) Left-handed spinc: (I, Y ) = (1/2,−1) and 2) Right-handed spinc : (I, Y ) = (0, 0)
and (0,−2). These are precisely the U(2) quantum numbers of the representation space of
tangent γ′s which will be found in the next section. The SU(3) IRR’s have to contain these
U(2) IRR’s . They are not symmetric between left- and right-handed spinors.
10.7.2 The Tangent Gammas
We start from the Clifford algebra on R8, i.e from the 16×16 matrices γi (i = 1, 2, ...8) with
the relations
γi, γj = 2δij, γ†i = γi (718)
The γ− matrices which will occur in the Dirac operator act by left multiplication on the algebra
Mat16 generated by γi. The CP2 γ′s are the tangent projections γiPij. Clearly there are only
four of them at each ξ which are linearly independent. P is the projector onto the tangent
bundle defined by
P =1
3(ξiadλi)
2 , (adλi)jk = −2ifijk. (719)
132
The goal now is to find a four-dimension subspace of Mat16 at each ξ on which the CP2 γ′s
can act. Let us find this subspace at ξ0. Define the fermionic creation-annihilation operators
a†1 =1
2(γ4 + iγ5), a1 =
1
2(γ4 − iγ5),
a†2 =1
2(γ6 + iγ7), a2 =
1
2(γ6 − iγ7). (720)
Let
|0〉 = a1a2, |1〉 = a†1|0〉 |2〉 = a†2|0〉 |3〉 = a†1a†2|0〉. (721)
These states span a 4−dimensional space. The matrices γ4,5,6,7 act irreducibly on this space.
For an appropriate subspace at other points of CP2 we use the fact that SU(3) acts transitively
on CP2 or more precisely the relation gλ8g−1 = λiξ. The algebra of SU(3) can be realised
using γi, viz
tci =1
4ifijkγjγk (722)
Their 16−dimensional representation can be split into 8⊕8 using the projectors P± = 1±γ9
2,
γ9 = γ1γ2...γ8. The γi transform as an 8 under the action
[tcj , γi] := ad tjγi = ifjilγl. (723)
Let T (g) be the image of g in this SU(3) representation. Thus T (g) acts on Mat16 according
to AdT (g)M = T (g)MT (g)−1. The above 4−dimensional vector space and its basis at e can
be labelled as V (e) and |ν, e〉 = |ν〉 , ν = 0, 1, 2, 3. The vector space and its basis at g are
then
V (g) = AdT (g)V (e) = T (g)V (e)T (g)−1,
|ν, g〉 = AdT (g)|ν; e〉 = T (g)|ν, e〉T (g)−1. (724)
It is on this vector space that γiPij acts by left-multiplication.
a+α transform as (K+, K0) while aα transform as (K−, K0) (α = 1, 2). The vector space
V (e) decomposes into the direct sum
(I, Y ) = (0,−2)⊕(1
2,−1)⊕(0, 0). (725)
The state |0, e〉 ≡ |0〉 has I = 0, Y = −2 and such a vector occurs only in 10, namely Ω−, thus
|0, g〉∈10. Also |α, e〉 (α = 1, 2) has I = 12,Y = −1 which is a vector which occurs only in 8,
10 and 27. Thus |α, g〉 (α = 1, 2) transforms as the direct sum 8⊕10⊕27. There remains |3, e〉with I = Y = 0. U(2) singlets with I = Y = 0 are contained only in SU(3) singlet and in 8
and 27 so that |3, g〉 transforms as 1⊕8⊕27.
133
10.7.3 The Dirac Operator on CP2
We require of the CP2 Dirac operator D that it is linear in derivatives and anticommutes
with the chirality operator
Γ = − 1
4!ǫijklγiγjγkγl. (726)
At ξ = ξ0, Γ = −γ4γ5γ6γ7 and is +1 on |0, e〉 and |3, e〉, and −1 on |α, e〉(α = 1, 2). Hence
Γ = +1 on |0, g〉, |3, g〉 and −1 on |α, g〉 for all g. The former have even chirality and the latter
have odd chirality. Now γiPij anticommutes with Γ. We introduce the SU(3) generators
Ji = Li + ad tci (727)
Li = −ifijkξj∂
∂ξk(728)
They commute with Γ. Hence the Dirac operator must be defined by
D = γiPijJj. (729)
This Dirac operator anticommutes with Γ by construction and it acts on A⊗Mat16. But there
are only four tangent gammas at each ξ so we have to reduce A⊗Mat16 to V (g). A⊗Mat16consists of sections of the trivial U(2)−bundle over CP2. But that is not the case for |0, g〉 and
|α, g〉 (α = 1, 2). Under g→g u, |0, g〉 transforms as an SU(2) singlet with Y = −2 and |α, g〉transforms as an SU(2) doublet with hypercharge Y = −1. We regard elements of A⊗Mat16as functions of g ∈ SU(3) which are invariant under the substitution g → g u, u ∈ U(2). Now
consider in the case of |0, g〉 the wave functions D(N1,N2)(II3Y )(0,0,2). They exist only if N2 = N1 + 3.
Let us consider the following combination
D(N,N+3)(I,I3,Y )(0,0,2)|0, g〉. (730)
This is invariant under g→gu and can form constituents of a basis for the expansion of functions
in A⊗Mat16. The remaining elements of a basis can be found in the same way. They are
1√2D
(N,N)
(I,I3,Y )( 12,− 1
2,1)|1, g〉 +
1√2D
(N,N)
(I,I3,Y )( 12, 12,1)|2, g〉
1√2D
(N,N+3)
(I,I3,Y )( 12,− 1
2,1)|1, g〉 +
1√2D
(N,N+3)
(I,I3,Y )( 12, 12,1)|2, g〉
D(N,N)(I,I3,Y )(0,0,0)|3, g〉 (731)
”Orbital” SU(3) momentum Lα does not act on the individual factors in (730,731) which are
functions on SU(3) and not just CP2. We need thus to lift them to operators J Li which act on
g in such a manner that ξ transform under SU(3) as ξ→hξ. In other words J Li are generators
of SU(3)L, the left-regular representation, and the Dirac equation is to be reinterpreted with
Jj replaced by Jj = J Lj + ad tcj. The action of D on typical basis vectors like (730) can be
computed explicitly. One finds
134
DD(N,N+3)(I,I3,Y )(0,0,2)|0, g〉 = −4
3
J Ri D
(N,N+3)(I,I3,Y )(0,0,2)
AdT [tc8, [tc8, γi]]|0, e〉. (732)
The J Ri are the generators of SU(3) acting on the right of g. The general wave function for
even and odd chiralities are
ξ(N,N+3)(I,I3,Y ) D
(N,N+3)(I,I3,Y )(0,0,2)|0, g〉 , ξ
(N,N)(I,I3,Y )D
(N,N)(I,I3,Y )(0,0,0)|3, g〉. (733)
η(N,N)(I,I3,Y )D
(N,N)
(I,I3,Y )( 12,− 3−2α
2,1)|α, g〉 , η(N,N+3)
(I,I3,Y ) D(N,N+3)
(I,I3,Y )( 12,− 3−2α
2,1)|α, g〉. (734)
The ξ(N1,N2)(I,I3,Y ) and η
(N1,N2)(I,I3,Y )∈C and repeated indices are summed. Since γjPji anticommutes with
Γ we can represent the effect of D on wave functions in terms of the off-diagonal block matrix
(
0 d
d+ 0
)
(735)
This acts on
ξ(N1,N2)(I,I3,Y )D
(N1,N2)(I,I3,Y )a
η(N1,N2)(I,I3,Y )D
(N1,N2)(I,I3,Y )b
. (736)
10.7.4 Fuzzification
In summary D acts on a subspace of A⊗Mat16 and thus the fuzzy Dirac operator D acts
on a subspace of A⊗Mat16 where A is obtained from A by restricting “orbital” SU(3) IRR’s to
(n, n), n≤N . The fuzzy Dirac operator D is then obtained from the continuum D by projection
to this subspace. The operator D commutes only with the total SU(3) Casimir J2i and not with
orbital SU(3) Casimir L2i . This causes edge effects distorting the spectrum of D for those states
having (n, n) near (N,N) which D mixes with (n′, n
′), n
′≥N . This particular edge phenomenon
does not occur for S2 = CP1 where orbital angular momentum L2α commutes with the Dirac
operator. Thus we choose to introduce the cut-off not on the orbital Casimir, but on the total
Casimir, retaining all states upto the cut-off. This will give a fuzzy Dirac operator D with
a spectrum exactly that of the continuum operator D upto the cut-off point and which has
chirality (chirality Γ of D commutes with J2i ) and no fermion doubling.
Since the fuzzy D is just a restriction of the continuum D we can continue to use (729) in
calculation just remembering the truncation of the spectrum. In the final expressions like (736)
the (N1, N2) labels the IRR and the Dirac operator acts in subspace with fixed (N1, N2). So
the cut-off can be introduced on (N1, N2).
135
11 Fuzzy OSP (2, 1) Gauge Supersymmetry
11.1 Introduction
As we have seen the differential calculus on the fuzzy sphere is 3−dimensional and as a
consequence a spin 1 vector field ~C is intrinsically 3−dimensional. Each component Ci, i =
1, 2, 3, is an element of some matrix algebra MatN . Thus U(1) symmetry will be implemented
by U(N) transformations. On the fuzzy sphere S2N it is not possible to split the vector field
~C in a gauge-covariant fashion into a tangent 2-dimensional gauge field and a normal scalar
fluctuation. Thus in order to reduce the number of independent components from 3 to 2 we
impose the gauge-covariant condition
1
2(xiCi + Cixi) +
C2i√
N2 − 1= 0. (737)
xi = Li/√
L2i ( where Li are the generators of SU(2) in the irreducible representation N−1
2of
the group ) are the matrix coordinates on fuzzy S2N . The action on the fuzzy sphere S2
N is given
by
SN [C] =1
4Ng2TrF 2
ij −1
2Ng2ǫijkTrL
[
1
2FijCk −
i
6[Ci, Cj]Ck
]
. (738)
Fij is the curvature on the fuzzy sphere, viz Fij = i[Yi, Yj]+ǫijkYk where the covariant derivatives
Yi are defined by Yi = Li+Ci. The action (738) with the constraint (737) was studied extensively
in these notes.
Following [36, 64] we will derive in this section the supersymmetric analogue of the action
(738). Let us summarize here the main results. Instead of the SU(2) vector ~C = (Ci) we
will have an OSP (2, 2) supervector A = (A±,W,Ci, B±). The 5 superfields Ci, B± transform
as a superspin 1 multiplet under OSP (2, 1) while the remaining 3 superfields A± and W will
transform as a superspin 1/2 under OSP (2, 1). All these superfields are elements of the algebra
Mat(2L+ 1, 2L). We define the supersymmetric curvature by F = δA+A ∗A = (F±, f, ci, b±)
where the exterior derivative δ and the associative product ∗ are defined appropriately on forms.
The action by analogy with (738) reads
SL[A] = αStr ⊳ F ∗ F + βSTr(A ∗ F − 1
3A ∗ A ∗ A). (739)
The first term is similar to the usual Yang-Mills action whereas the second term is a ( real-
valued ) Chern-Simons-like contribution. α and β are two real parameters. The Hodge triangle
⊳ is defined as the identity map between one-forms and two-forms and thus ⊳F should be
considered as a one-form. This action will have the correct continuum limit provided we impose
the following supersymmetric- and gauge-covariant conditions on the supergauge field A. The
first condition is the supersymmetric anlogue of (737) defined by
136
[D+, A−] − [D−, A+] +1
4D0,W + [A+, A−] +
1
4W 2 = 0. (740)
We will also impose the following supersymmetric constraints
b+ = b− = c+ = c− = c3 = 0. (741)
These constraints will reduce the number of independent components of A and F from 8 to
2. D±,0 are the generators of OSP (2, 2) in the complement of OSP (2, 1). The generators of
OSP (2, 1) are denoted by Ri, V±. The expressions of the curvatures F±, f and ci, b± in terms
of the gauge fields A±,W and Ci, B± are given by
F± = 2[X∓, Y±]±2[X±, Y3] + [X0, Z±] + 2X± , f = 4Z+, X− − 4Z−, X+ + 2X0
c± = ∓2X±, X± − Y± , c3 = 2X+, X− − Y3 , b± = [X0, X±] − Z±. (742)
The supercovariant derivatives X±, X0, Yi and Z± are defined by X± = D± + A± , X0 =
D0 +W ,Yi = Ri + Ci and Z± = V± +B±.
For other constructions of fuzzy supersymmetric gauge models see [87]. The work [88] is a
numerical study of the type of supersymmetry which is involved in IKKT models [49, 50] so it
is not the same as fuzzy SUSY.
11.2 The continuum supersphere
11.2.1 The Lie algebras osp(2, 1) and osp(2, 2)
We start with the osp(2, 1) Lie algebra. It consists of three even generators R± and R3 and
two odd generators V± with commutation and anticommutation relations
[R+, R−] = 2R3 , [R3, R±] = ±R± (743)
and
[R3, V±] = ±1
2V± , [R±, V±] = 0 , [R±, V∓] = V±. (744)
V±, V± = ±1
2R± , V±, V∓ = −1
2R3. (745)
The following notation is also useful R± = Λ1 ± iΛ2, R3 = Λ3, V+ = Λ4 and V− = Λ5. The
above commutation and anticommutation relations take now the following forms respectively
[Λi,Λj] = iǫijkΛk , [Λi,Λα] =1
2(σi)βαΛβ , Λα,Λβ =
1
2(Cσi)αβΛi. (746)
i, j, k = 1, 2, 3 and α, β = 4, 5. The charge conjugation C is such that C45 = −C54 = 1 and
C44 = C55 = 0. The most important point is that V± transform as an SU(2) spinor.
137
Let us also introduce osp(2, 2). We add two more odd generators D± and one even generator
D0 with the commutation and anticommutation relations
[R3, D±] = ±1
2D± , [R±, D±] = 0 , [R±, D∓] = D±. (747)
D±, D± = ∓1
2R± , D±, D∓ =
1
2R3. (748)
D±, V± = 0 , D±, V∓ = ±1
4D0. (749)
and
[D0, Ri] = 0 , [D0, V±] = D± , [D0, D±] = V±. (750)
Again we denote D+ = Λ6, D− = Λ7 and D0 = Λ8. Then the commutation and anticommuta-
tion relations (747) and (748) take the forms respectively
[Λi,Λα] =1
2(σi)βαΛβ , Λα,Λβ = −1
2(Cσi)αβΛi. (751)
Here α, β = 6, 7. Note that D± transform also as an SU(2) spinor. Equation (744) and (747)
can be put in the form
[Λi,Λα] =1
2(σi)βαΛβ (752)
Equation (745), (748) and (749) can be put together in the form
Λα,Λβ =1
2(Cσi)αβΛi +
1
4(ǫC)αβΛ8. (753)
Equation (750) takes the form
[Λ8,Λi] = 0 , [Λ8,Λα] = ǫαβΛβ. (754)
Here ( in the last three equations ) α, β = 4, 5, 6, 7 and
σi =(
σi 0
0 σi
)
, C =(
C 0
0 −C)
, ǫ =(
0 12
12 0
)
. (755)
Also
σ1 =(
0 1
1 0
)
, σ2 =(
0 −ii 0
)
, σ3 =(
1 0
0 −1
)
. (756)
138
11.2.2 The supersphere
The supersphere S(3,2) ( with ordinary S3 as its even part ) is given by the points ψ∈ C(2,1)
which satisfy
|ψ|2 = 1. (757)
We have ψ = (z, θ) = (z1, z2, θ) and ψ = (z, θ) = (z+1 , z
+2 , θ) and the norm is given by
|ψ|2 ≡ ψψ = zz + θθ = |z1|2 + |z2|2 + θθ. (758)
In above z1, z2 are complex variables and θ, θ are Grassmann numbers. The group manifold
of osp(2, 1) is S(3,2) in the same way that the group manifold of su(2) is S3. Furthermore the
supersphere S(2,2) is an adjoint orbit of OSP (2, 1) in the same way that the sphere S2 is an
adjoint orbit of SU(2). In other words we must consider the supersymmetric Hopf fibration
S1−→S(3,2)−→S(2,2) by analogy with the Hopf fibration S1−→S3−→S2. We define thus the
coordinates functions on S(2,2) by the following functions on S3,2
ωa(ψ) = ψΛ( 12)
a ψ , a = 1, ..., 5. (759)
A point on S(2,2) is given by the supervector ω = (ω1, ..., ω5). Λ( 12)
a are the generators of
OSP (2, 1) in the 3−dimensional fundamental representation characterized by the superspin
j = 12. It consists of the SU(2) irreducible representations 1
2and 0. The generators are given
explicitly by
Λ( 12)
i =1
2
(
σi 0
0 0
)
, Λ( 12)
4 =1
2
( 0 0 −1
0 0 0
0 −1 0
)
, Λ( 12)
5 =1
2
( 0 0 0
0 0 −1
1 0 0
)
. (760)
Remark that under ψ−→ψh, h = exp(iγ) with γ real numbers we have ωa−→ωa. Thus the
points ψh on S(3,2) correspond to the same point ω on S(2,2). This shows that OSP (2, 1) is a
principal U(1) bundle over the coset space OSP (2, 1)/U(1) which is diffeomorphic to the sphere
S(2,2). We can compute the explicit expressions
ωi =1
2zσiz , ω4 = −1
2(z+
1 θ + z2θ) , ω5 =1
2(−z+
2 θ + z1θ). (761)
By using these equations we can immediately compute
ω2i + Cαβωαωβ =
1
4. (762)
This is the defining equation of the supersphere S(2,2). We define the grade adjoint ++ by
z++i = z+
i , θ++ = θ and θ++ = −θ and by the requirement that (AB)++ = (−1)dAdBB++A++
where dA and dB are the degrees of A and B respectively. For an even object the degree is
equal 0 while for an odd object the degree is equal to 1. Hence we have the reality conditions
ω++i ≡ ω+
i = ωi , ω++α = −Cαβωβ , α, β = 4, 5. (763)
139
The action of the group OSP (2, 2) on S(2,2) preserves (762) and (763) but it is not the same
as the adjoint action of the group OSP (2, 1). This is because the Lie algebra osp(2, 1) is not
invariant under the action of the generators Λ6, Λ7 and Λ8 of OSP (2, 2). Let us define the
OSP (2, 2) coordinates functions
Ωa(ψ) = ψΛ( 12)
a ψ , a = 1, ..., 8. (764)
They define an OSP (2, 2) vector. We will have the following extra generators ( in addition to
(760) ) in the 3−dimensional fundamental representation j = 12
of OSP (2, 2)
Λ( 12)
8 =
(
12 0
0 2
)
, Λ( 12)
6 =1
2
( 0 0 1
0 0 0
0 −1 0
)
, Λ( 12)
7 =1
2
( 0 0 0
0 0 1
1 0 0
)
. (765)
Explicitly we have
Ω8 = 2 − zz , Ω6 =1
2(z+
1 θ − z2θ) , Ω7 =1
2(z+
2 θ + z1θ) (766)
Ω++8 ≡ Ω+
8 = Ω8 , Ω++α = CαβΩβ , α, β = 6, 7. (767)
We can immediately compute
−1
4Ω2
8 + CαβΩαΩβ = −1
4, α, β = 6, 7. (768)
By adding (762) ( with the substitutions ωi−→Ωi and ωα−→Ωα , α = 4, 5 ) and (768) we get
the OSP (2, 2) Casimir
Ω2i −
1
4Ω2
8 + CαβΩαΩβ = 0. (769)
Let us also compute the following
ω3ω4 = (1
2|z1|2 −
1
2|z2|2)ω4 = −1
4(|z1|2z+
1 θ + |z1|2z2θ − |z2|2z+1 θ − |z2|2z2θ) (770)
(ω1 + iω2)ω5 = (z+1 z2)ω5 =
1
2(|z1|2z2θ − |z2|2z+
1 θ). (771)
Hence by using also√
ω2i = 1
2zz we obtain
Ω6 = − 1√
ω2i
(ω3ω4 + (ω1 + iω2)ω5). (772)
By using Ω++6 = Ω7, ω
++4 = −ω5, ω
++5 = ω4, ω
++i = ωi and (i)++ = −i we obatin
Ω7 =1
√
ω2i
(ω3ω5 − (ω1 − iω2)ω4). (773)
Finally by using again√
ω2i = 1
2zz we obtain
Ω8 = 2 − 2√
ω2i . (774)
140
11.2.3 Laplacians
Define ni = 2Rωi, nα = 2Rωα. Then
n2i + Cαβnαnβ = R2. (775)
The delta function δ(n2i + Cαβnαnβ − R2) will have an expansion of the general form δ(n2
i +
Cαβnαnβ −R2) = δ(n2i − R2) + CαβnαnβX where X is given by
X =1
2[
d
dn5dn4
δ(n2i + Cαβnαnβ −R2)]n4=n5=0 =
dδ(n2i −R2)
dn2i
. (776)
Thus
δ(n2i + Cαβnαnβ −R2) =
1
2Rδ(r −R) +
n4n5
2rR
dδ(r − R)
dr, n2
i = r2. (777)
In above we have assumed that r≥0. A scalar superfield Φ on S(2,2) has an expansion of the
form ( with α, β = 4, 5 )
Φ = φ0 + Cαβψαnβ + φ1Cαβnαnβ . (778)
φ0 ≡ φ0(ni) and φ1 ≡ φ1(ni) are scalar functions while ψ is a Majorana spinor field with
two components Grassman functions ψα ≡ ψα(ni). In the terminology of supesymmetry in
4−dimensional Minkowski spacetime the field φ0 is the D−term of the superfield Φ while the
field φ1 is the F−term of the superfield. The integral of this superfield over the supersphere is
defined by ( with dΩ denoting the solid angle )
I(Φ) =∫
r2drdΩdn4dn5δ(n2i + Cαβnαnβ − R2)Φ. (779)
Since the volum form and the delta function are invariant under the OSP (2, 1) action the
integral should be invariant under the susy action on Φ. A straightforward calculation ( using
also∫
dn4 =∫
dn5 = 0 and∫
dn4dn5n4n5 = −1 ) we obtain
I(Φ) =∫
dΩ[
d
dr(r
2Rφ0) − Rφ1
]
r=R. (780)
Thus as in the case of supersymmetry in 4 dimensions the integral depends only on the D−and F−terms of the superfield. The OSP (2, 1) and OSP (2, 2) Laplacians ( by inspection of
equations (762), (768) and (769) ) are given respectively by the equations
K2,1 = Λ2i + CαβΛαΛβ
K2,2 = Λ2i −
1
4Λ2
8 + CαβΛαΛβ. (781)
The Laplacian on S(2,2) is given by
∆ = K2,1 −K2,2 =1
4Λ2
8 + Λ6Λ7 − Λ7Λ6. (782)
The action for the scalar superfield Φ is given by
S = I(Φ++∆Φ) =∫
r2drdΩdn4dn5δ(n2i + Cαβnαnβ − R2)Φ++∆Φ. (783)
141
11.3 Scalar action
In the calculation of the above action we need the D− and F−terms of the superfield
Φ++∆Φ. Because of the constraint the superfield Φ can be rewritten in the form Φ = φ2 +
Cαβψαnβ where φ2 = φ0 + φ1(R2 − r2). Hence the D−term of the superfield Φ++∆Φ is
[Φ++∆Φ]0 = φ++2 ∆φ2 = φ2∆φ2. (784)
The F−term will be extracted from
[Φ++∆Φ]1 = (Cαβψαnβ)++∆(Cαβψαnβ) = (Cαβψαnβ)∆(Cαβψαnβ). (785)
In above we have assumed that the superfield Φ is real and hence Φ++ = Φ or equivalently
φ++0 = φ+
0 = φ0, φ++1 = φ+
1 = φ1 and ψ++α = −Cαβψβ . We have also assumed that cross terms
are linear in nα which we will show.
11.3.1 The D−term
First we calculate the D−component. The action of the generators Λ6 = D+ and Λ7 = D−on φ0 is defined by 7
Λ6φ0 ≡ (D+ni)∂iφ0 = −1
2
[
(σi)66n6 + (σi)76n7
]
∂iφ0 = −1
2[n6∂3 + n7∂+]φ0
Λ7φ0 ≡ (D−ni)∂iφ0 = −1
2
[
(σi)67n6 + (σi)77n7
]
∂iφ0 = −1
2[n6∂− − n7∂3]φ0. (786)
These equations are consistent with the commutation relations [Λα,Λi] = −12(σi)βαΛβ where
α, β = 6, 7. Let us also say that the operators D± correspond to the generatorsD± in the adjoint
representation of OSP (2, 2). Furthermore ∂± = ∂1 ± i∂2 and n6, n7 are given by n6 = 2RΩ6,
n7 = 2RΩ7 and hence we must have from (772) and (773) the expressions
n6 = −1
r(n3n4 + (n1 + in2)n5) , n7 =
1
r(n3n5 − (n1 − in2)n4). (787)
Remark that Λ6φ0 and Λ7φ0 are odd and hence a second action of Λ6 and Λ7 will involve
anticommutation relations instead of commutation relations. We have
Λ7Λ6φ0 = −1
2
[
Λ7(n6∂3φ0) + Λ7(n7∂+φ0)]
= −1
2
[
(Λ7n6)∂3φ0 − n6(Λ7∂3φ0) + (Λ7n7)∂+φ0 − n7(Λ7∂+φ0)]
Λ6Λ7φ0 = −1
2
[
Λ6(n6∂−φ0) − Λ6(n7∂3φ0)]
= −1
2
[
(Λ6n6)∂−φ0 − n6(Λ6∂−φ0) − (Λ6n7)∂3φ0 + n7(Λ6∂3φ0)]
(788)
7Take the case of the ordinary generators of SU(2) denoted here by Λi = Ri. We know that Λiφ0 ≡(Riφ0)(~n) = −iǫijknj∂kφ0. This can be put in the form Λiφ0 = (Rinj)∂jφ0.
142
The quantities (Λα∂3φ0) and (Λα∂±φ0) will be given by similar expressions to (786). From the
anticommutation relations D±, D± = ∓12R± and D±, D∓ = 1
2R3 we have
Λ6n6 = D+n6 = −1
2n+ , Λ6n7 = D+n7 =
1
2n3 , n+ = n1 + in2
Λ7n6 = D−n6 =1
2n3 , Λ7n7 = D−n7 = +
1
2n− , n− = n1 − in2. (789)
Note here that the odd coordinates associated with Λ6,7 will always be denoted by n6,7 although
we will denote sometimes the operators Λ6 and Λ7 by D+ and D− respectively. So n+ and n−are always bosonic coordinates associated with Λ+ = Λ1 + iΛ2 and Λ− = Λ1− iΛ2. We compute
( with L3 = i(n1∂2 − n2∂1) )
Λ7Λ6φ0 = −1
4[(+ni∂i + L3)φ0 − n6n7∂
2φ0]
Λ6Λ7φ0 = −1
4[(−ni∂i + L3)φ0 + n6n7∂
2φ0]. (790)
Thus
(Λ6Λ7 − Λ7Λ6)(φ0) =1
2[ni∂iφ0 − n6n7∂
2φ0]. (791)
Similarly Λ8(φ0) = D0(ni)∂iφ0 = 0 since [Λ8, Ri] = [D0, Ri] = 0. Hence
φ0∆(φ0) =1
2φ0[ni∂iφ0 − n6n7∂
2φ0] =1
2φ0
[
r∂rφ0 +R2 − r2
2∂2φ0
]
=1
2φ0
[
R2 − r2
2∂2
r +R2
r∂r +
1
2(R2
r2− 1)L2
a
]
φ0.(792)
In above we have used the results n6n7 = −n4n5 = −R2−r2
2and ∂2 = ∂2
r + 2r∂r + L2
a
r2 . Finally we
get
d
dr
[
r
2Rφ0∆φ0
]
r=R=R
4
[(
dφ0
dr
)2]
r=R− 1
4Rφ0L2
aφ0. (793)
The corresponding action is
I0 =∫
dΩd
dr
[
r
2Rφ0∆φ0
]
r=R=R
4
∫
dΩ(
dφ0
dr
)2
+1
4R
∫
dΩ(Laφ0)2. (794)
The full action coming from the D−term is obtained from above by replacing φ0 with φ2. We
get
ID =∫
dΩd
dr
[
r
2Rφ2∆φ2
]
r=R=R
4
∫
dΩ(
dφ2
dr
)2
+1
4R
∫
dΩ(Laφ2)2
=∫
dΩd
dr
[
r
2Rφ2∆φ2
]
r=R=R
4
∫
dΩ(
dφ0
dr− 2Rφ1
)2
+1
4R
∫
dΩ(Laφ0)2. (795)
143
11.3.2 The F−term
Now we have ( with D− = Λ7, D+ = Λ6 and α = 4, 5)
Λ6(ψα) = D+(ni)∂iψα =1
2(D+n+)∂−ψα +
1
2(D+n−)∂+ψα + D+(n3)∂3ψα
= −1
2n7∂+ψα − 1
2n6∂3ψα
Λ7(ψα) = D−(ni)∂iψα =1
2(D−n+)∂−ψα +
1
2(D−n−)∂+ψα + D−(n3)∂3ψα
= −1
2n6∂−ψα +
1
2n7∂3ψα. (796)
In above we have also used the fact ( which we can check from the commutation relations
[D∓, R±] = −D±, [D±, R±] = 0 and [D±, R3] = ∓12D± ) that
D−(n+) = −n6 ,D+(n−) = −n7 ,D+(n+) = D−(n−) = 0 ,D+(n3) = −1
2n6 , D−(n3) =
1
2n7(797)
We will also need ( from the anticommutation relations D±, V± = 0 and D±, V∓ = ±14D0
with D0 = Λ8 ) the actions
D+(n5) =1
4n8 ,D+(n4) = 0 ,D−(n5) = 0 ,D−(n4) = −1
4n8. (798)
The even coordinate n8 is defined by n8 = 2RΩ8 = 2(2R− r). Next
Λ6(Cαβψαnβ) = Λ6(ψ4n5 − ψ5n4) = Λ6(ψ4).n5 − ψ4D+(n5) − Λ6(ψ5).n4 + ψ5D+(n4)
= Λ6(ψ4).n5 −1
4ψ4n8 − Λ6(ψ5).n4
= −1
4ψ4n8 +
1
2n7n5∂+ψ4 +
1
2n6n5∂3ψ4 −
1
2n7n4∂+ψ5 −
1
2n6n4∂3ψ5
Λ7(Cαβψαnβ) = Λ7(ψ4n5 − ψ5n4) = Λ7(ψ4).n5 − ψ4D−(n5) − Λ7(ψ5).n4 + ψ5D−(n4)
= Λ7(ψ4).n5 − Λ7(ψ5).n4 −1
4ψ5n8
= −1
4ψ5n8 +
1
2n6n5∂−ψ4 −
1
2n7n5∂3ψ4 −
1
2n6n4∂−ψ5 +
1
2n7n4∂3ψ5.
(799)
By using now n7n5 = −1rn−n4n5, n7n4 = n6n5 = −1
rn3n4n5 and n6n4 = 1
rn+n4n5 we obtain
Λ6(Cαβψαnβ) = −1
4ψ4n8 −
1
2r(n−∂+ψ4 + n3∂3ψ4 − n3∂+ψ5 + n+∂3ψ5)n4n5
Λ7(Cαβψαnβ) = −1
4ψ5n8 −
1
2r(n3∂−ψ4 − n−∂3ψ4 + n+∂−ψ5 + n3∂3ψ5)n4n5. (800)
We need now to compute the following ( using also D±(n8) = −n4,5 )
Λ6(ψαn8) =1
2D+(n+)∂−ψα.n8 +
1
2D+(n−)∂+ψα.n8 + D+(n3)∂3ψα.n8 − ψαD+(n8)
144
= −1
2n7∂+ψα.n8 −
1
2n6∂3ψα.n8 + ψαn4
Λ7(ψαn8) =1
2D−(n+)∂−ψα.n8 +
1
2D−(n−)∂+ψα.n8 + D−(n3)∂3ψα.n8 − ψαD−(n8)
= −1
2n6∂−ψα.n8 +
1
2n7∂3ψα.n8 + ψαn5. (801)
Next step is to compute the following action
Λ7
(
1
2r(n−∂+ψ4 + n3∂3ψ4 − n3∂+ψ5 + n+∂3ψ5)n4n5
)
= − 1
8rn5n8[n−∂+ψ4 + n3∂3ψ4 − n3∂+ψ5
+ n+∂3ψ5]
Λ6
(
1
2r(n3∂−ψ4 − n−∂3ψ4 + n+∂−ψ5 + n3∂3ψ5)n4n5
)
= − 1
8rn4n8(n3∂−ψ4 − n−∂3ψ4 + n+∂−ψ5
+ n3∂3ψ5). (802)
This action corresponds to Λ6,7 acting on the factor n4n5. The action of Λ6,7 on the other terms
leads to products which involve n4n5 and n6,7 and hence they are zero by (787).
We now compute in a straightforward way
n5Λ6Λ7(Cαβψαnβ) =n4n5n8
8r
[
(n−∂+ − n+∂−)ψ5 − (n3∂− − n−∂3)ψ4 −2r
n8ψ5
]
n4Λ6Λ7(Cαβψαnβ) =n4n5n8
8r(n3∂+ − n+∂3)ψ5. (803)
n4Λ7Λ6(Cαβψαnβ) =n4n5n8
8r
[
(−n+∂− + n−∂+)ψ4 − (n3∂+ − n+∂3)ψ5 +2r
n8ψ4
]
n5Λ7Λ6(Cαβψαnβ) =n4n5n8
8r(n3∂− − n−∂3)ψ4. (804)
So ( with L3 = 12(n+∂− − n−∂+), L± = L1 ± iL2 = ∓(n±∂3 − n3∂±) and ψ = (ψ1, ψ2))
(Cαβψαnβ)(Λ6Λ7 − Λ7Λ6)(Cαβψαnβ) =n4n5n8
4r
[
ψ4L−ψ4 − ψ4L3ψ5 − ψ5L3ψ4 − ψ5L+ψ5
− r
n8
ψ4ψ5 +r
n8
ψ5ψ4
]
= −Cαβnαnβn8
8rψT (σaLa +
r
n8
)(Cψ). (805)
The full result is then
(Cαβψαnβ)∆(Cαβψαnβ) = −Cαβnαnβn8
8rψT (σaLa + 2
r
n8
)(Cψ). (806)
The contribution of this F−term to the action is given by
IF = −R∫
dΩ[
− n8
8rψT (σaLa + 2
r
n8)(Cψ)]|r=R =
R
4
∫
dΩψT (σaLa + 1)(Cψ). (807)
The total action
I =R
4
∫
dΩ(
dφ0
dr− 2Rφ1
)2
+1
4R
∫
dΩ(Laφ0)2 +
R
4
∫
dΩψT (σaLa + 1)(Cψ). (808)
145
11.4 The fuzzy supersphere
We consider the irreducible representation with OSP (2, 1) superspin equal L. This repre-
sentation consists of the direct sum of the SU(2) representations with spins L and L− 12. Let
L(L,L) be the space of linear operators from the corresponding representation space into itself.
The action of the superalgebra OSP (2, 2) on L(L,L) is described by the operators
Ri =( R
(L)i 0
0 R(L− 1
2)
i
)
, Vα =(
0 V(L,L− 1
2)
α
V(L− 1
2,L)
α 0
)
D0 =(
2L 0
0 2L+ 1
)
, Dα =(
0 −V (L,L− 12)
α
V(L− 1
2,L)
α 0
)
. (809)
The dimension of the first block of Ri and D0 is (2L+ 1)× (2L+ 1) while the dimension of the
second block is (2L)× (2L). The upper and lower off-diagonal blocks are therefore rectangular
matrices with dimensions (2L+1)×(2L) and (2L)×(2L+1) respectively. In the above equation
the definitions of R(l)i are the usual ones, i.e ( with R
(l)± = R
(l)1 ± iR
(l)2 and l = L,L− 1
2)
(R(l)± )ll3±1,ll3 =
√
(l ∓ l3)(l ± l3 + 1) , (R(l)3 )ll3,ll3 = l3, (810)
whereas V(L,L− 1
2)
α and V(L− 1
2,L)
α are given by
(V(L,L− 1
2)
± )Ll3± 12,L− 1
2l3
= −1
2
√
L± l3 +1
2, (V
(L− 12,L)
± )L− 12l3± 1
2,Ll3
= ∓1
2
√
L∓ l3. (811)
We will also denote the operators given in (809) by Λ(L)i ≡ Ri, i = 1, 2, 3, Λ(L)
α ( ≡ V±, D± ),
α = 4, 5, 6, 7 and Λ(L)8 ≡ D0. For L = 1
2we get the 3−dimensional fundamental representation
of OSP (2, 2) given in (760) and (765).
The above irreducible representation with superspin L is characterized by the value of the
OSP (2, 1) Casimir operatorK2,1 = R2i +CαβVαVβ which is equal L(L+ 1
2) in this representation,
viz
K2,1 = R2i + CαβVαVβ = L(L+
1
2). (812)
The above operators (809) give also a non-typical irreducible representation of OSP (2, 2) char-
acterized by the value of the OSP (2, 2) Casimir operator
K2,2 = R2i + CαβVαVβ − CαβDαDβ − 1
4D2
0 = 0. (813)
This means in particular two things, 1) this representation ( as opposed to typical ones of
OSP (2, 2) ) is irreducible with respect to the OSP (2, 1) subgroup and 2) the OSP (2, 2) gen-
erators Dα and D0 can be realized nonlinearly in terms of the OSP (2, 1) generators.
146
The space L(L,L) is isomorphic to the algebra of supermatrices Mat(2L + 1, 2L). The
dimension of the Hilbert space on which this algebra acts is N = (2L+1)+(2L) = 4L+1. The
coordinates operators on the fuzzy supersphere are defined by ni = 2RΩi, n4,5 = 2RΩ4,5 where
Ωi =Ri
2√
L(L+ 12), Ω4 =
V+
2√
L(L+ 12), Ω5 =
V−√
2L(L+ 12). (814)
The remaining coordinates operators n6,7,8 = 2RΩ6,7,8 are similarly defined by
Ω8 =D0
2√
L(L+ 12), Ω6 =
D+
2√
L(L+ 12), Ω7 =
D−
2√
L(L+ 12). (815)
These coordinates operators satisfy the commutation and anticommutation relations ( with
a, b, c = 1, ..., 8 )
[na, nb = nanb − (−1)dnadnb nbna =iR
√
L(L+ 12)fabcnc. (816)
The definition of the structure constants fabc is obvious from (746) and (751). These coordinates
operators must also satisfy the constraints
n2i + Cαβnαnβ = R2. (817)
n2i + Cαβnαnβ − 1
4n2
8 = 0. (818)
The continuum limit is defined by L−→∞ in which na−→na and Ωa−→Ωa. To see this more
explicitly we notice that under the adjoint action of OSP (2, 1) the algebra Mat(2L + 1, 2L)
decomposes as
Mat(2L+ 1, 2L) ≡ L⊗ L = 0 ⊕ 1
2⊕ 1 ⊕ ...⊕ 2L− 1
2⊕ 2L. (819)
The dimension of this space is N2 and a generic element is a polynomial in ni,4,5. Recall that
n6,7,8 can be realized nonlinearly in terms of the ni,4,5. Among these polynomials we can define
the matrix superspherical harmonics. A given N ×N supermatrix can be expanded in terms of
these superspherical harmonics. In the continuum limit Mat(2L+1, L) approaches the algebra
of superfunctions on the supersphere. In partiuclar the matrix superspherical harmonics go
to the ordinary superspherical harmonics which are the eigensuperfunctions of the Casimir
operator R2i + CαβVαVβ and R3.
A very important remark is to note that elements of Mat(2L + 1, 2L) ( in other words
superfields ) can be even or odd if Mat(2L + 1, 2L) is defined over a graded commutative
algebra P instead of the field of complex numbers. In this case we will denote this algebra by
Mat(2L + 1, 2L;P ). In the fuzzy case we have the definitions RiΦ = [Ri,Φ] and VαΦodd =
147
Vα,Φodd, VαΦeven = [Vα,Φeven] where Φ is any element of Mat(2L+ 1, 2L;P ), Φodd is an odd
element of Mat(2L + 1, 2L;P ) and Φeven is an even element of Mat(2L + 1, 2L;P ). Strictly
speaking the fuzzy supersphere is identified with the even elements of Mat(2L + 1, 2L;P )
while the odd elements will be crucial in constructing gauge theories. The inner product on
Mat(2L+ 1, 2L;P ) is defined by
(Φ1,Φ2) ≡ STrΦ++1 Φ2. (820)
This satisfies STr(1N) = 1 and STr[X, Y = 0.
A general supermatrix Φ∈Mat(2L+ 1, 2L;P ) and its graded involution Φ++ are given by
Φ =(
φR ψR
ψL φL
)
, Φ++ =(
φ++R ∓ψ++
L
±ψ++R φ++
L
)
. (821)
φR and φL are (2L+ 1)× (2L+ 1) and (2L) × (2L) matrices while ψR and ψL are respectively
(2L+ 1) × (2L) and (2L) × (2L+ 1) matrices. In Φ++ the upper signs refer to the case when
Φ is an even superfield ( in which case the off-diagonal blocks are fermionic and the diagonal
blocks are bosonic), while the lower signs refer to the case when Φ is an odd ( in which case
the off-diagonal blocks are bosonic and the diagonal blocks are fermionic ). We remark that
STrΦ = Tr2L+1φR − (−1)|Φ|Tr2LφL.
The Laplacian on the fuzzy supersphere is given by
∆ = K2,1 −K2,2 =1
4D2
0 + D6D7 −D7D6. (822)
The definition of D0,6,7 are obvious by analogy with Ri and V4,5 given above. The fuzzy
supersphere of size N = 4L+ 1 is by definition the spectral triple consisting of 1) the algebra
of supermatrices Mat(2L+ 1, 2L) together with 2) the representation space of the superspin L
of OSP (2, 1) with inner product given by the supertrace STr and graded involution given by
++ and 3) the Laplacian ∆ which is the most important ingredient. The Laplacian fixes the
metric aspects of the space uniquely while the algebra alone will only give topology.
11.5 Gauge theory
11.5.1 Klimcik differential complex
The Laplacian on the fuzzy supersphere depends only on the OSP (2, 2) generators D±,0
in the adjoint representation. This means in particular that the OSP (2, 2) generators in the
directions D± are the supersymmetric covariant derivatives on the fuzzy supersphere while the
OSP (2, 2) generators V± are the supersymmetry generators. A gauge field on the supersphere
is a superspin 1/2 multiplet composed of 3 superfields A± and W in the directions D± and
D0 resepctively. These superfields A± and W transform under OSP (2, 1) in the same way as
D± and D0. The supercovariant derivatives ( as opposed to the covariant derivatives in the
non-supersymmetric case ) are thus
X± = D± + A± , X0 = D0 +W. (823)
148
In order to construct gauge theory on the fuzzy supersphere we must in fact start from an
OSP (2, 2) supervector. Thus we need to add a superspin 1 multiplet composed of 5 more
superfields Ci and B± ( which transform under OSP (2, 1) in the same way as V± and Ri ) with
supercovariant derivatives
Yi = Ri + Ci , Z± = V± +B±. (824)
In the following we will construct explicitly the action principle of the OSP (2, 2) vector gauge
superfield (A±,W,Ci, B±). In the case of the fuzzy supersphere this action principle will be
a supermatrix model. We will also need to write down constraints which must be satisfied
by these superfields in order to have the correct number of degrees of freedom on the fuzzy
supersphere.
The differential complex over the fuzzy supersphere is defined by
ΨN = ⊕3j=0Ψ
jN (825)
The elements of ΨjN are the j−forms. We have the following identifications
Ψ0N = Ψ3
N = Mat(2L + 1, 2L) , Ψ1N = Ψ2
N = ⊗8i=1Mat(2L + 1, 2L)i. (826)
We must clearly have Mat(2L+ 1, 2L)i = Mat(2L + 1, 2L). A zero-form is thus an element Φ
of the algebra Ψ0N = Mat(2L+ 1, 2L) while a one-form is an element of Ψ1
N of the form
A = (A±,W,Ci, B±). (827)
The 5 superfields Ci, B± transform as a superspin 1 multiplet under OSP (2, 1) while the
remaining 3 superfields A± and W will transform as a superspin 1/2 under OSP (2, 1). All
these superfields are elements of the algebra Mat(2L + 1, 2L). We can also write zero-forms
and one-forms as 3N × 3N supermatrices of the form
M1 = r+ ⊗ C− + r− ⊗ C+ + 2r3 ⊗ C3 + 2v+ ⊗ B− − 2v− ⊗B+ − 2d+ ⊗A− + 2d− ⊗ A+
− 1
2d0 ⊗W.
M0 = 13 ⊗ Φ. (828)
In above Λ( 12)
i ≡ ri, i = 1, 2, 3(±, 3), Λ( 12)
α = vα, α = 4(+), 5(−), Λ( 12)
α = dα, α = 6(+), 7(−) and
Λ( 12)
8 ≡ d0 are the supermatrices of the 3−dimensional superspin 1/2 fundamental representation
of OSP (2, 2) corresponding to the generators Ri, V±, D± and D0 respectively. We will also use
the notation Λi ≡ Ri, i = 1, 2, 3(±, 3), Λα = Vα, α = 4(+), 5(−), Λα = Dα, α = 6(+), 7(−) and
Λ8 ≡ D0.
Similarly two-forms and three-forms are given by a = (a±, w, ci, b±)∈Ψ2N = Ψ1
N and φ∈Ψ3N =
Ψ0N . We write the corresponding 3N × 3N supermatrices as
M2 = r+ ⊗ c− + r− ⊗ c+ + 2r3 ⊗ c3 + 2v+ ⊗ b− − 2v− ⊗ b+ − 2d+ ⊗ a− + 2d− ⊗ a+ − 1
2d0 ⊗ w.
M3 = 13 ⊗ φ. (829)
149
Let us introduce the quadratic Casimirs
CG = r+ ⊗R− + r− ⊗R+ + 2r3 ⊗ R3 + 2v+ ⊗ V− − 2v− ⊗ V+ − 2d+ ⊗D− + 2d− ⊗D+
− 1
2d0 ⊗D0
CH = r+ ⊗R− + r− ⊗R+ + 2r3 ⊗ R3 + 2v+ ⊗ V− − 2v− ⊗ V+
C = CG − CH = −2d+ ⊗D− + 2d− ⊗D+ − 1
2d0 ⊗D0. (830)
11.5.2 The exterior derivative
We introduce a coboundary operator δ : ΨiN−→Ψi+1
N defined on 0−forms by
δΦ =(
r+ ⊗R− + r− ⊗R+ + 2r3 ⊗R3 + 2v+ ⊗ V− − 2v− ⊗ V+
− 2d+ ⊗D− + 2d− ⊗D+ − 1
2d0 ⊗D0
)
Φ. (831)
The exterior derivative on one-forms is on the other hand given by
δM1 = δGM1 − δHMH1 . (832)
MH1 is the orthogonal projection of M1 from G⊗Mat(2L + 1, 2L) into H ⊗Mat(2L + 1, 2L)
where G = osp(2, 2) and H = osp(2, 1). In other words
MH1 = r+ ⊗ C− + r− ⊗ C+ + 2r3 ⊗ C3 + 2v+ ⊗ B− − 2v− ⊗B+. (833)
The exterior derivatives δG and δH are defined by ( with the notation M1 ≡ hA ⊗ CA and
adO ≡ O = [O, .])
δGM1 = 2(−1)hΛηabadΛ( 12)
a hA ⊗ adΛbCA +1
2dGM1
δHM1 = 2(−1)hΛηabadΛ( 12)
a hA ⊗ adΛbCA +1
2dHM1. (834)
η, η stand for the block diagonal matrices η = 2(13, C,−1/4), η = 2(13, C) and a, b = 1, ..., 8
in δGM1 and a, b = 1, ..., 5 in δHM1. The Dynkin numbers dG, dH are defined by
STrXY =4L2
dGSTrG(X1Y1) , STrXY =
4L2
dHSTrH(X1Y1). (835)
where STrG, STrH are the supertraces in the adjoint representations of G and H respectively.
We choose X = Y = Ri so XY = R2i = diag((R
(L)i )2, (R
(L− 12)
i )2) for H and XY = R2i =
diag((R(L)i )2, (R
(L− 12)
i )2; (R(L− 1
2)
i )2, (R(L−1)i )2) for G. The supermatrices X1 and Y1 correspond
to the representation L = 1. Using the property STrΦ = Tr2L+1φR−Tr2LφL of STr we compute
for G that STrR2i = 6L2 and hence dG = 6 while for H we compute STrR2
i = 3L(L + 12) and
hence dH = 6 + 3/L.
150
We have explicitly
δM1 = δGM⊥1 + (δG − δH)MH
1 , M⊥1 = M1 −MH
1 . (836)
We can immediately compute
2δG(d± ⊗ A∓) = 4d∓ ⊗R±A∓±4d± ⊗R3A∓ + 2d0⊗V±A∓ − 2v± ⊗D0A∓
− 4r± ⊗D∓A∓∓4r3⊗D±A∓ + dGd± ⊗ A∓, (837)
and
1
2δG(d0 ⊗W ) = −2d+ ⊗ V−W + 2d− ⊗ V+W + 2v+ ⊗D−W − 2v− ⊗D+W
+1
4dGd0 ⊗W. (838)
Hence
δGM⊥1 = −4r− ⊗D+A+ + 4r+ ⊗D−A− + 4r3 ⊗ (D+A− + D−A+)
− 2v− ⊗ (D0A+ −D+W ) + 2v+ ⊗ (D0A− −D−W )
− d− ⊗ (4R+A− + 4R3A+ + 2V+W − dGA+)
+ d+ ⊗ (4R−A+ − 4R3A− + 2V−W − dGA−)
+ d0 ⊗ (−2V+A− + 2V−A+ − 1
4dGW ). (839)
Furthermore ( with the notation MH1 = hA ⊗ CA )
(δG − δH)MH1 = (−1)hA(−4add+ ⊗D− + 4add− ⊗D+ − add0 ⊗D0)(hA ⊗ CA)
+1
2(dG − dH)MH
1 . (840)
We compute
2(−4add+ ⊗D− + 4add− ⊗D+ − add0 ⊗D0)(ri ⊗ Ci) = 4d+ ⊗ (D−C3 −D+C−)
+4d− ⊗ (D+C3 + D−C+), (841)
and
−2(−4add+ ⊗D− + 4add− ⊗D+ + add0 ⊗D0)(v+ ⊗ B− − v− ⊗ B+) =
−2d+ ⊗D0B− + 2d− ⊗D0B+ + 2d0 ⊗ (D+B− −D−B+). (842)
Thus
(δG − δH)MH1 =
1
2(dG − dH)(r+ ⊗ C− + r− ⊗ C+ + 2r3 ⊗ C3 + 2v+ ⊗B− − 2v− ⊗ B+)
+ 2d+ ⊗ (−D0B− + 2D−C3 − 2D+C−) − 2d− ⊗ (−D0B+ − 2D+C3 − 2D−C+)
+ 2d0 ⊗ (D+B− −D−B+). (843)
151
The final result is ( with dG = 4, dH = 6 )
δA± = 2A± + 2D∓C± − 2R±A∓±2D±C3∓2R3A± + D0B± − V±W
δW = 2W + 4V+A− − 4V−A+ + 4D−B+ − 4D+B−
δB± = −B± + D0A± −D±W
δC3 = −C3 + 2D+A− + 2D−A+
δC± = −C±∓4D±A±. (844)
Lastly the action of the coboundary operator on two-forms a three-forms is given by the obvious
definitions ( with M2 = hA ⊗ cA )
δM2 = −1
213 ⊗HAcA
= D+a− −D−a+ +1
4D0w − V+b− + V−b+ − 1
2R+c− − 1
2R−c+ −R3c3. (845)
δM3 = δ(13 ⊗ φ) = 0. (846)
11.5.3 The product ∗
The associative product ∗ between the forms is a map ∗ : ΨiN ⊗Ψj
N−→Ψi+jN defined for i = 1
by ( with hA ⊗XA standing for one-forms, two-forms and three-forms )
(13 ⊗ Φ) ∗ (hA ⊗XA) = hA ⊗ ΦXA. (847)
For i = 2 we have
(hA ⊗ CA) ∗ (13 ⊗ Φ) = hA ⊗ CAΦ , (hA ⊗ CA) ∗ (13 ⊗ φ) = 0, (848)
and
(hA ⊗ CA) ∗ (h′
A ⊗ C′
A) = (hA ⊗ CA) ∗G (h′
A ⊗ C′
A) − (hA ⊗ CA) ∗H (h′
A ⊗ C′
A), (849)
where
(hA ⊗ CA) ∗G,H (h′
A ⊗ C′
A) = 2(−1)Ch′
ad(hA)h′
B ⊗ CAC′
B
(850)
The indices A and B run over the superalgebra H for ∗H whereas for ∗G they run over the
superalgebra G. Explicitly we have
(hA ⊗ CA) ∗ (h′
A ⊗ C′
A) = 2(ri ⊗ Ci + v+ ⊗ B− − v− ⊗B+) ∗G ( − 2d+ ⊗ A′
− + 2d− ⊗A′
+
− 1
2d0 ⊗W
′
) + ( − 2d+ ⊗ A− + 2d− ⊗A+ − 1
2d0 ⊗W ) ∗G (h
′
A ⊗ C′
A).
(851)
152
The first line is computed to be given by
First line = 2d+ ⊗ (B−W′
+ 2C−A′
+ − 2C3A′
−) − 2d− ⊗ (B+W′
+ 2C+A′
− + 2C3A′
+)
+ 2d0 ⊗ (B−A′
+ − B+A′
−). (852)
The second line is computed to be given by
Second line = 2d+ ⊗ (−WB′
− + 2A−C′
3 − 2A+C′
−) + 2d− ⊗ (WB′
+ + 2A+C′
3 + 2A−C′
+)
− 2d0 ⊗ (A−B′
+ − A+B′
−) + 4r+ ⊗A−A′
− − 4r− ⊗ A+A′
+ + 4r3 ⊗ (A−A′
+ + A+A′
−)
− 2v+ ⊗ (A−W′ −WA
′
−) + 2v− ⊗ (A+W′ −WA
′
+). (853)
Thus we obtain
A± ∗ A′
± = ±2A±C′
3∓2C3A′
± +WB′
± −B±W′
+ 2A∓C′
± − 2C±A′
∓
W ∗W ′
= 4A−B′
+ + 4B+A′
− − 4A+B′
− − 4B−A′
+
C± ∗ C ′
± = ∓4A±A′
±
C3 ∗ C′
3 = 2A−A′
+ + 2A+A′
−
B± ∗B± = WA′
± − A±W′
. (854)
Also for i = 2 we have
(hA ⊗ CA) ∗ (h′
A ⊗ c′
A) = −1
2(−1)h
′CSTr(hAh
′
B) ⊗ CAc′
B. (855)
By using the identities STrrirj = 12δij , STrv±v∓ = ∓1
2, STrd±d∓ = ±1
2and STrd2
0 = −2 ( all
other supertraces are zero ) we obtain immdediately the results
(hA ⊗ CA) ∗ (h′
A ⊗ c′
A) = −1
2C+c
′
− − 1
2C−c
′
+ − C3c′
3 + A+a′
− −A−a′
+ +1
4Ww
′
+ B−b′
+ − B+b′
−. (856)
For i = 3 we have the two non-vanishing products (hA⊗cA)∗ (13⊗Φ) and (hA⊗cA)∗ (h′
A⊗C ′
A)
with obvious definitions by analogy with the products (13⊗Φ)∗(hA⊗cA) and (h′
A⊗C′
A)∗(hA⊗cA). In particular the product of two-forms with one-forms is given as above with reversed order
of small and capital letters. For i = 4 we have one non-zero product given by (13⊗φ)∗ (13⊗Φ)
while the rest are zero.
This coboundary operator is nilpotent, i.e it satisfies δ2 = 0. The product ∗ is compatible
with δ so that the Leibniz rule is respected. Thus we must have δ(X i ∗ Y j) = δX i ∗ Y j +
(−1)iX i ∗ δY j.
11.5.4 Gauge action
We consider one-forms A = (A±,W,Ci, B±) satisfying the reality condition A++ = A. Thus
we must have A++± = ±A∓, W++ = W , C++
i = Ci and B++± = ∓B∓. We define the curvature
by
F = δA+ A ∗ A = (F±, f, ci, b±). (857)
153
We can immediately compute
F± = 2[X∓, Y±]±2[X±, Y3] + [X0, Z±] + 2X±
f = 4Z+, X− − 4Z−, X+ + 2X0
c± = ∓2X±, X± − Y±
c3 = 2X+, X− − Y3
b± = [X0, X±] − Z±. (858)
Recall that X± = D± + A±, X0 = D0 + W , Yi = Ri + Ci and Z± = V± + B±. We define the
supersymmetric noncommutative U(1) gauge action by
SL[A] = αStr ⊳ F ∗ F + βSTr(A ∗ δA +2
3A ∗ A ∗ A)
= αStr ⊳ F ∗ F + βSTr(A ∗ F − 1
3A ∗ A ∗ A). (859)
The first term is similar to the usual Yang-Mills action whereas the second term is a ( real-
valued ) Chern-Simons-like contribution. α and β are two real parameters. The Hodge triangle
⊳ is defined as the identity map between Ψ1N and Ψ2
N and thus ⊳F should be considered as a
one-form. Explicitly we have
⊳F ∗ F = b−b+ − b+b− − c2i + F+F− − F−F+ +1
4f 2
A ∗ F = B−b+ − B+b− − Cici + A+F− − A−F+ +1
4Wf, (860)
and
A ∗ A ∗ A = B−[W,A+] − B+[W,A−] − 2C+A−A− + 2C−A+A+ − 2C3A+, A−+ A+(2[C3, A−] − 2[C−, A+] + [W,B−]) −A−( − 2[C3, A+] − 2[C+, A−] + [W,B+])
+ WA−, B+ −WA+, B−. (861)
We need first to show gauge invariance of the above action. The invariance of the Yang-Mills
term is obvious whereas the invariance of the Chern-Simons-like term requires some work in
order to be established. Towards this end we will need to rewrite the Chern-Simons-like term
in a completely covariant fashion.
First by thinking about CG as a one-form we can show after a long calculation that we must
have
STrCG ∗ F = STr(
− 2D+A− + 2D−A+ + V−B+ − V+B− −RiCi −1
2D0W
)
+1
2STrW
(
D−, B+ − D+, B− − V−, A+ + V+, A−)
− STrA+
(
[D−, C3] − [D+, C−] − 1
2[D0, B−] + [R−, A+] − [R3, A−] − 1
2[W,V−]
)
154
− STrA−
(
[D−, C+] + [D+, C3] +1
2[D0, B+] − [R+, A−] − [R3, A+] +
1
2[W,V+]
)
− STrC+D−, A− + STrC−D+, A+ −1
2STrB+([D0, A−] − [D−,W ])
+1
2STrB−([D0, A+] − [D+,W ]) − STrC3
(
D−, A+ + D+, A−)
. (862)
In above we have used the results
STrD∓F± = STr(
±R∓C± + 2D±A∓±R3C3 +D∓A± − V∓B±±1
4D0W
+ 2A∓[D∓, C±]±2A±[D∓, C3] −WD∓, B±)
STrV∓b± = STr(
−D∓A±∓1
4D0W − V∓B± −WV∓, A±
)
1
4STrD0f = STr
(
D+A− −D−A+ + V−B+ − V+B− +1
2D0W − A−[D0, B+] + A+[D0, B−]
)
STrR∓c± = STr(
∓ 4D∓A± −R∓C±±2A±[R∓, A±])
STrR3c3 = STr(
D+A− −D−A+ − R3C3 −A−[R3, A+] − A+[R3, A−])
. (863)
More calculation yields
STrCG ∗ F = STr(
−D+A− +D−A+ − V−B+ + V+B− −RiCi +1
2D0W
)
+1
2STrW
(
1
4f − 1
2W − A−, B+ + B−, A+
)
− STrA+
(
− 1
2F− + A− − [A−, C3] + [A+, C−] +
1
2[W,B−]
)
− STrA−
(
1
2F+ − A+ − [A−, C+] − [A+, C3] −
1
2[W,B+]
)
− STrC+(1
4c− +
1
4C− − 1
2A−, A−) − STrC−(
1
4c+ +
1
4C+ +
1
2A+, A+)
− 1
2STrB+(b− +B− − [W,A−]) +
1
2STrB−(b+ +B+ − [W,A+])
− STrC3
(
1
2c3 +
1
2C3 − A−, A+
)
. (864)
Equivalently
STrCG ∗ F =1
2STrA ∗ F − 1
2STrA ∗ A ∗ A− 1
2STr
(
(CH + AH) ∗ ⊳(CH + AH) − CH ∗ ⊳CH
)
− STr(
(C⊥ + A⊥) ∗ ⊳(C⊥ + A⊥) − C⊥ ∗ ⊳C⊥
)
. (865)
In above C⊥ = C = CG − CH , AH is the projection of A in the directions along the generators
of H = osp(2, 1) and A⊥ is the corresponding orthogonal part. Explicitly we have
STr(CH + AH) ∗ ⊳(CH + AH) = STr( − Y 2i − Z+Z− + Z−Z+)
STr(C⊥ + A⊥) ∗ ⊳(C⊥ + A⊥) = STr(1
4X2
0 +X+X− −X−X+). (866)
155
The supersymmetric noncommutative U(1) gauge action becomes
SL[A] = αStr ⊳ F ∗ F +β
3STr
(
2(CH + AH) ∗ F + 2(C⊥ + A⊥) ∗ F + (CH + AH) ∗ ⊳(CH + AH)
+ 2(C⊥ + A⊥) ∗ ⊳(C⊥ + A⊥) − CH ∗ ⊳CH − 2C⊥ ∗ ⊳C⊥
)
. (867)
This establishes gauge invariance of the system under
CG + A−→U ∗ (CG + A) ∗ U++. (868)
U is a zero-form with U++ = U+ and hence this transformation law means
X±,0−→UX±,0U+ , Yi−→UYiU
+ , Z±−→UZ±U+. (869)
The next step is to notice that the system as it stands contains too many degrees of freedom
and hence we must impose some extra constraints in order to reduce the number of independent
components of A and F from 8 to 2 since we are in two dimensions. We impose
(δA + A ∗ A)H = 0 , ⇔ b+ = b− = c+ = c− = c3 = 0, (870)
and
(C⊥ + A⊥) ∗ ⊳(C⊥ + A⊥) − C⊥ ∗ ⊳C⊥ = 0. (871)
Both constraints are obviously gauge covariant.
These two constraints as well as the action (867) are invariant under all OSP (2, 1) super-
symmetry transformations. Indeed the two quantities (CH +AH)∗⊳(CH +AH) and (C⊥+A⊥)∗⊳(C⊥ +A⊥) are separately invariant under OSP (2, 1) which is the reason behind the invariance
of the second constraint and the 4th and 5th terms of the action under OSP (2, 1). Furthermore
a generic one-form and a generic two-form will always decompose under OSP (2, 1) into a direcrt
sum of a superpin 1/2 multiplet and a superspin 1 multiplet. For example for the one-form A
and for the two-form F the components A+, A−,W and F+, F−, f form OSP (2, 1) multiplets
with superspin 1/2 while the other five components B+, B−, Ci of A and b+, b−, ci of F form
multiplets with superspin 1. This is the reason why the first constarint is OSP (2, 1) invariant.
The invariance of the rest of the action under OSP (2, 1) is ovbious since it is covariant under
full OSP (2, 2).
11.6 The continuum limit
The constraint (871) reads explicitly
[D+, A−] − [D−, A+] +1
4D0,W + [A+, A−] +
1
4W 2 = 0. (872)
In the continuum limit this becomes
n6A− − n7A+ +1
4n8W = 0. (873)
156
After some calculation we get the solution ( by using ω6 = 12(z1θ − z2θ), ω7 = 1
2(z1θ + z2θ),
ω8 = 2 − zz and zz + θθ = 1 )
W = zz(Aθ
z+2
−Aθ
z2) = A
θ
z+2
− Aθ
z2, (874)
where
A+ =1
2(A− z+
1
z+2
A) , A− = −1
2(A +
z1z2A). (875)
The constraint (870) leads to the equations
B± = [D0, A±] − [D±,W ]
C± = ∓4D±, A±C3 = 2D+, A− + 2D−, A+. (876)
We need now to compute the action ( with L−→∞ )
SL[A] = αSTr(F+F− − F−F+ +1
4f 2) + βSTr(A+F− − A−F+ +
1
4Wf). (877)
Explicitly we have
F± = [D0, [D0, A±]] + 2A± − [D0, [D±,W ]] − [V±,W ]∓12[D∓, D±, A±]±12[D2±, A∓]
f = 2W + 4D+, [D−,W ] − 4D−, [D+,W ] + 4V+, A− − 4D+, [D0, A−]− 4V−, A+ + 4D−, [D0, A+]. (878)
Because of the constraint (873) we have only two independent superfields. We will work in
the local coordinates t = z1/z2, t = z+1 /z
+2 , b = −θ/z2 and b = −θ/z+
2 . We introduce the
parametrization
A+ =1
2(A− tA) , A− = −1
2(A+ tA) , W = bA− bA. (879)
In terms of t, t and b,b the supersymmetryic covariant derivatives D,D and the supersymmetric
charges Q,Q are given respectively by
D = ∂b + b∂t , D = ∂b + b∂t, (880)
and
Q = ∂b − b∂t , Q = ∂b − b∂t. (881)
In terms of t, t and b,b the OSP (2, 2) generators are given by
R+ = −∂t − t2∂t − tb∂b , R− = ∂t + t2∂t + tb∂b , R3 = t∂t − t∂t +1
2b∂b −
1
2b∂b
V+ =1
2(Q+ tQ) , V− =
1
2(Q− tQ), (882)
157
and
D0 = b∂b − b∂b , D+ =1
2(D − tD) , D− = −1
2(D + tD). (883)
Let us compute
(D20 + 2)A+ =
1
2bDA+
1
2bDA+ bbDDA− tbbDDA− 1
2btDA− 1
2btDA+ A− tA.(884)
(D0D+ + V+)(W ) = −1
2bb(tDDA + DDA) +
3
2bb(D2A+ tD2A) +
1
2(tA−A) + (tb− 1
2b)DA
+ (−b+1
2tb)DA+
1
2(bDA− tbDA). (885)
In above we have used the identities D2 = ∂t, D2 = ∂t, DD = ∂b∂b, DD = ∂b∂b and D, D = 0.
We can also compute
−12D−(D+A+) = 3DDA+ − 3ttDDA+ + 3tD2A+ − 3tD2A+ − 3bDA+
= −3
2tDDA− 3
2ttDDA− 3
2tD2A +
3
2t2D2A+
3
2tD2A +
3
2DDA− 3
2ttD2A
− 3
2bDA+
3
2t2tDDA +
3
2tbDA+
3
2tA− 3
2bDA− 3
2ttbDA. (886)
Also
12D2+A− = 3D2A− + 3t2D2A− + 3tbDA−
= −3
2D2A− 3
2tD2A− 3
2A− 3
2t2D2A− 3
2t2tD2A− 3
2tbDA− 3
2ttbDA. (887)
We get immediately ( with y = 1 + tt+ bb, ω = DA + DA and 2ω6 = −tb+ b )
F+ = −3
2y(
D2A+ tD2A + tDDA− DDA+ bω)
+ (b− tb)ω
= −3
2
(
D(yω) + tD(yω))
− 4ω6ω. (888)
Since F+++ = F− we must have ( by using also D++ = −D, D++ = D, A++ = A, A++ = −A
and ω++ = ω, 2ω++6 = 2ω7 = −tb − b )
F− = −3
2
(
D(yω)− tD(yω))
− 4ω7ω. (889)
Also
4D+D−W − 4D−D+W = 2(1 + tt)DDW − bDW − bDW= 2y(bDDA− bDDA) − 2(1 + tt)ω −W − bbω. (890)
4V+A− − 4V−A+ = 2y(bD2A + bD2A) − (1 + tt)ω −W. (891)
158
−4D+D0A− + 4D−D0A+ = (1 + tt)(DD0A− DD0A) + bD0A + bD0A
= y(bDDA− bDDA+ bD2A+ bD2A− ω). (892)
Hence
f = 3y(bDDA− bDDA+ bD2A+ bD2A− 2ω) + 2(y + bb)ω
= 3(bD(yω) + bD(yω)) − 4(y + bb)ω. (893)
We can immediately compute the Yang-Mills action
SY M [A] = αSTr(F+F− − F−F+ +1
4f 2)
= αSTr(
9
2yD(yω)D(yω) + 4y2ω2
)
=α
2πi
∫
dtdtdbdb
y
(
9
2yD(yω)D(yω) + 4y2ω2
)
. (894)
In the last line we have also converted the supertrace into a superintegral. Similarly the Chern-
Simons action becomes
SCS[A] = βSTr(A+F− − A−F+ +1
4Wf)
= βSTr(
− 3
4y(
AD(yω) + AD(yω)))
=β
2πi
∫
dtdtdbdb
y
(
− 3
4y(
AD(yω) + AD(yω)))
. (895)
The last step is to rewrite the above actions in terms of components of the superfields A and
A. Introduce
iA = ζ + bv +1
2bw + iu
1 + tt+ bb(
η
1 + tt+ ∂tζ)
iA = −ζ + bv − 1
2bw − iu
1 + tt+ bb(
η
1 + tt+ ∂tζ). (896)
w and u are real bosonic fields while v is a complex bosonic field. Clearly w++ = w, u++ =
u, v++ = v. The fermionic fields ζ and η are such that ζ++ = −ζ , ζ++ = ζ , η++ = η, η++ = −η.We can immediately compute
iyω = iu+ bη − bη + bb(
(1 + tt)(∂tv − ∂tv) +i
1 + ttu)
. (897)
The Kahler term can now be put in the form ( with∫
dbdbbb = −1,∫
db = 0,∫
db = 0 )
1
2πi
∫
dtdtdbdb
y
(
9α
2yD(yω)D(yω)
)
=1
2πi
∫
dtdt(9
2α)(
− (1 + tt)2(∂tv − ∂tv)2 + ∂tu∂tu
+u2
(1 + tt)2+ η∂tη − ∂tη.η − 2iu(∂tv − ∂tv)
)
. (898)
159
The superpotential takes the form
1
2πi
∫ dtdtdbdb
y(4αy2ω2) =
1
2πi
∫
dtdt(4α)(
2ηη
1 + tt− u2
(1 + tt)2+ 2iu(∂tv − ∂tv)
)
. (899)
Simmilarly
1
2πi
∫
dtdtdbdb
y(−3
4βy)
(
AD(yω) + AD(yω))
=1
2πi
∫
dtdt(−3
4β)(
2ηη
1 + tt− u2
(1 + tt)2
+ 2iu(∂tv − ∂tv))
. (900)
In order to cancel the coupling between the fields u and v we choose β = −23α. This will also
cancel the mass term of the u field. We obtain finally ( with SL[A] = SY M [A] + SCS[A] and
L−→∞ )
SL[A] =1
2πi
∫
dtdt(9
2α)(
− (1 + tt)2(∂tv − ∂tv)2 + ∂tu∂tu+ η∂tη + η∂tη +
2ηη
1 + tt
)
. (901)
11.7 A new fuzzy SUSY scalar action
The next natural step is to take the action (739) with the corresponding constraints (740)
and (741) and write the whole thing in terms of the components of X±,0, Yi, Z± thus reducing
the supertrace STr to an ordinary trace Tr. The fermionic fields should then be integrated
out before we can attempt any numerical investigation. This complicated exercise will not be
pursuited here.
A possibly much simpler supersymmetric action than the above pure gauge action is given
by the following fuzzy supersymmetric scalar action. We introduce the superscalar fields ΦH
and Φ⊥ defined by the expressions
ΦH = (CH + AH) ∗ ⊳(CH + AH) = −Y 2i − Z+Z− + Z−Z+
Φ⊥ = (C⊥ + A⊥) ∗ ⊳(C⊥ + A⊥) =1
4X2
0 +X+X− −X−X+. (902)
The action we write ( without any extra constraints and with full supersymmetry ) is
SL[Φ] = STr(a⊥Φ⊥ + b⊥Φ2⊥ + c⊥Φ3
⊥ + d⊥Φ4⊥ + ...)
+ STr(aHΦH + bHΦ2H + cHΦ3
H + dHΦ4H + ...). (903)
This action is supersymmetric for the same reason that (739) is supersymmetric. It is gauge
covariant since ΦH and Φ⊥ are gauge covariant fields. The parameters a, b, c, d.. are the coupling
constants of the model. The partition function thus reads
ZL[a, b, c, d, ...] =∫
dX±dX0dZ±dYie−SL[Φ]. (904)
160
We conclude this article by introducing the matrix components of X±,0, Yi, Z± in the follow-
ing way. The superfields X0 = D0 +W and Yi = Ri +Ci are real scalar superfields so that they
are even elements of the superalgebra Mat(2L+1, 2L) while X± = D±+A± and Z± = V±+B±are odd elements of Mat(2L+1, 2L) ( we think of them as real spinor superfields ). This means
that instead of considering the algebra Mat(2L + 1, 2L) over the field of complex numbers C
we consider it over a graded commutative algebra P. Then the one-forms are actually elements
of the space
Ψ1N(P ) = G0 ⊗Mat(2L + 1, 2L;P )0 ⊕G1 ⊗Mat(2L+ 1, 2L;P )1. (905)
G0 and G1 are the even and odd parts of the super-Lie algebra G = osp(2, 2) while Mat(2L+
1, 2L;P )0,1 are the subspaces of Mat(2L + 1, 2L;P ) with even and odd grading respectively
with respect to the gradings of Mat(2L + 1, 2L) and P. Ψ1N(P ) is isomorphic to the space of
one-forms Ψ1N we had constructed previously.
Let us introduce the (2L+1)× (2L+1) fermionic matrices ψ±R, χ±R, the 2L×2L fermionic
matrices ψ±L, χ±L, the (2L+1)×2L bosonic matrices X±R, Z±R and the 2L× (2L+1) bosonic
matrices X±L, Z±L as follows
X± =(
ψ±R X±R
X±L ψ±L
)
, X− = X+++ =
(
ψ+++R +X++
+L
−X+++R ψ++
+L
)
. (906)
Z± =(
χ±R Z±R
Z±L χ±L
)
, Z− = Z+++ =
(
χ+++R +Z++
+L
−Z+++R χ++
+L
)
. (907)
Furthermore we introduce the (2L + 1) × (2L + 1) bosonic matrices YiR = Y ++iR , X0R = X++
0R ,
the 2L × 2L bosonic matrices YiL = Y ++iL , X0L = X++
0L , the (2L + 1) × 2L fermionic matrices
φiR = −φ++iL , φ0R = −φ++
0L and the 2L × (2L+ 1) fermionic matrices φiL = φ++iR , φ0L = φ++
0R as
follows
Yi =(
YiR φiR
φiL YiL
)
, X0 =(
X0R φ0R
φ0L X0L
)
. (908)
161
12 The Noncommutative Torus
12.1 The Noncommutative Torus T dθ
The noncommutative (NC) plane Rdθ is obtained by replacing the commutative coordinates
xi by hermitian unbounded operators xi which satisfy the commutation relations
[xi, xj ] = iθij . (909)
Thus Rdθ is the algebra of functions which is generated by the operators xi. The algebra
of functions on the NC torus T dθ is the proper subalgebra of Rd
θ which is generated by the
operators
za = exp(2πi(Σ−1)ai xi). (910)
In terms of za the commutation relations (909) read
zbza = zazbexp(2πiΘab) , Θab = 2π(Σ−1)ai θij(Σ
−1)bj. (911)
Σia is the d× d period matrix of T d
θ which satisfies ΣiaΣ
ja = δij . The indices i, j = 1, ..., d denote
spacetime directions whereas a, b = 1, ..., d denote directions of the frame bundle of T dθ . The
two points x and x+ Σai i on the noncommutative torus are identified ( where the summation
over i is understood and the index a is fixed ). For the square torus Σai is proportional to δa
i .
Let us recall that a general function on the commutative torus is given by
f(x) =∑
~m∈Zd
f~me2πi(Σ−1)a
i maxi
. (912)
The corresponding operator on the noncommutative torus is given by
f =∑
~m∈Zd
e2πi(Σ−1)ai maxi
f~m (913)
or equivalently
f =∑
~m∈Zd
∏d
a=1(za)maeπi
∑
a<bmaΘabmbf~m. (914)
It is not difficult to show that
f =∫
ddxf(x) ⊗ ∆(x). (915)
The product ⊗ is the tensor product between the coordinate and operator representations. The
operator ∆(x) is periodic in x given by
∆(x) =1
|detΣ|∑
~m∈Zd
∏d
a=1(za)
ma∏
a<beiπmaΘabmbe−2πi(Σ−1)a
i maxi
(916)
162
or
∆(x) =1
|detΣ|∑
~m∈Zd
e2πi(Σ−1)ai maxi
e−2πi(Σ−1)ai maxi
. (917)
The star product ∗ on the noncommutative torus can be introduced by means of the map ∆(x).
Indeed it is the the star product f1∗f2(x) of the two functions f1 and f2 ( and not their ordinary
product f1(x)f2(x) ) which corresponds to the Weyl operator f1f2 given by
f1f2 =∫
ddxf1 ∗ f2(x) ⊗ ∆(x). (918)
Equivalently we have
Trf1f2∆(x) = f1 ∗ f2(x). (919)
Let us also recall that derivations on the noncommutative torus are anti-hermitian linear oper-
ators ∂i defined by the commutation relations [∂i, xj ] = δij or equivalently
[∂i, za] = 2πi(Σ−1)ai za (920)
and [∂i, ∂j ] = icij where cij are some real-valued c-numbers. In particular we have the result
[∂i, f ] =∫
ddx∂if(x) ⊗ ∆(x). (921)
12.2 NC U(N) Gauge Theory on T dθ
The basic NC action we will study is given by [38]
SY M = − 1
4g2
∫
ddxtrN (Fij − fij)2∗. (922)
The curvature Fij is defined by Fij = ∂iAj − ∂jAi + i[Ai, Aj]∗ where ∗ is the canonical star
product on the NC plane Rdθ . Ai is a U(N) gauge field on the NC plane Rd
θ while fij is some
given constant curvature and g is the gauge coupling constant. Local gauge transformations
are defined as usual by
AUi = U ∗ Ai ∗ U+(x) − iU ∗ ∂iU
+ , FUij = U ∗ Fij ∗ U+(x). (923)
U(x) are N×N star-unitary matrices, in other words U(x) is an element of U(N) which satisfies
U ∗ U+(x) = U+ ∗ U(x) = 1N .
Classically the action (922) is minimized by gauge fields of non-zero topological charge which
on compact spaces are given by multi-valued functions. We need therefore to define these gauge
configurations of non-zero topological charge on the corresponding covering spaces. Gauge field
on the NC torus T dθ is thus the gauge field Ai on the NC space Rd
θ with the twisted boundary
conditions
163
Ai(x+ Σja j) = Ωa(x) ∗ Ai(x) ∗ Ω+
a (x) − iΩa(x) ∗ ∂iΩa(x)+. (924)
If we try writing the N×N star-unitary transition functions Ωa, a = 1, ..., d, in the infinitesimal
form Ωa(x) = 1+ iΛa(x) we can show that −i[Ai,Λa]∗−∂iΛa +O(Λ2) = 0 ( since the two points
x and x+ Σja j are identified we have Ai(x+ Σj
a j) = Ai(x) ) . We can immediately conclude
that the functions Λa do not exist and hence (924) are global large gauge transformations.
Furthermore by computing Ai(x+ Σja j + Σj
b j) in the following two different ways
Ai(x+ Σja j + Σj
b j) = Ωb(x+ Σja j) ∗ Ai(x+ Σj
a j) ∗ Ω+b (x+ Σj
a j)
− iΩb(x+ Σja j) ∗ ∂iΩb(x+ Σj
a j)+ (925)
and
Ai(x+ Σja j + Σj
b j) = Ωa(x+ Σjb j) ∗ Ai(x+ Σj
b j) ∗ Ω+a (x+ Σj
b j)
− iΩa(x+ Σjb j) ∗ ∂iΩa(x+ Σj
b j)+; (926)
we get the consistency conditions
Ωb(x+ Σja j) ∗ Ωa(x) = Ωa(x+ Σj
b j) ∗ Ωb(x). (927)
12.3 The Weyl-’t Hooft Solution
We will choose the gauge in which the N ×N star-unitary transition functions Ωa take the
form
Ωa(x) = eiαaixi ⊗ Γa (928)
where Γa are constant SU(N) matrices while αai is a d × d real matrix which represents the
U(1) factor of the U(N) group. We will also assume that αai is chosen such that (αΣ)T = −αΣ.
The corresponding background gauge field is introduced by
ai = −1
2Fijx
j ⊗ 1N . (929)
By using (928) and (929) it is a trivial exercise to show that (924) takes the form
1
2Fij(x
j + Σja) ⊗ 1N =
1
2Fij
(
eiαaixi ∗ xj ∗ e−iαaixi)
⊗ ΓaΓ+a + αai ⊗ 1N . (930)
By using the identity eiαaixi ∗xj ∗e−iαaix
i
= xj +θjkαak and the star-unitary condition ΓaΓ+a = 1
we reach the equation
1
2FijΣ
ja =
1
2Fijθ
jkαak + αai. (931)
164
The two solutions for α in terms of F and for F in terms of α are given respectively by
α = −ΣTF1
θF + 2, F = 2αT 1
Σ − θαT. (932)
Now by putting (928) in the consistency conditions (927) we obtain the d−dimensional Weyl-’t
Hooft algebra
ΓaΓb = e2πN
iQabΓbΓa (933)
where Qab = N2π
(αaiθijαbj + αbiΣ
ia − αaiΣ
ib) are the components of the antisymmetric matrix Q
of the non-abelian SU(N) ’t Hooft fluxes across the different non-contractible 2−cycles of the
noncommutative torus. Equivalently Q is given by
Q =N
2π(αθαT − 2αΣ). (934)
By construction Qab, for a fixed a and b, is quantized, i.e Qab ∈ Z. This can be seen for
example by taking the determinant of the two sides of the Weyl-’t Hooft algebra (933). This
quantization condition is a generic property of fluxes on compact spaces with non-contractible
2−cycles.
Now let us write the full gauge field Ai as the sum of the non-trivial gauge solution ai and
a fluctuation gauge field Ai, viz Ai = ai +Ai. It is a straightforward exercise to check that the
fluctuation field Ai transforms in the adjoint representation of the gauge group. In particular
under global large gauge transformations we have
Ai(x+ Σja j) = Ωa(x) ∗ Ai(x) ∗ Ω+
a (x). (935)
We can then compute Fij = Fij + f ∗ij where the curvature of the fluctuation field Ai is given by
Fij = DiAj−DjAi+i[Ai, Aj ]∗ with the covariant derivative defined by DiAj = ∂iAj +i[ai,Aj]∗.
The curvature of the background gauge field ai is given by f ∗ij = ∂iaj − ∂jai + i[ai, aj ]∗ =
Fij + 14Fikθ
klFlj . By requiring that the curvature f ∗ij of the background gauge field ai to be
equal to the constant curvature fij so that we have
Fij +1
4Fikθ
klFlj = fij (936)
we can immediately see that the action (922) becomes
SY M = − 1
4g2
∫
ddxtrN (Fij(x))2∗. (937)
This means in particular that the classical solutions of the model in terms of the fluctuation
field Ai are given by the condition of vanishing curvature, i.e Fij = 0. Hence the requirement
f ∗ij = fij is equivalent to the statement that the vacuum solution of the action is given by
165
Ai = 0. The fluctuation gauge field Ai has vanishing flux and as a consequence is a single-
valued function on the torus.
Finally let us note that the identity (931) can be put in the matrix form 12F (Σ− θαT ) = αT
or equivalently
1
1 − θαT Σ−1= 1 +
1
2θF. (938)
By squaring we can derive the identity
(
1
1 − θαT Σ−1
)2
= 1 + θf ∗. (939)
Furthermore by using the two identities f = (1 + 14Fθ)F and F (Σ− θαT ) = 2αT together with
the two facts ΣT = Σ−1 and (αΣ)T = −αΣ we can show that the antisymmetric matrix Q of
the non-abelian SU(N) ’t Hooft fluxes given by (934) can be rewritten as
Q =N
2πΣ−1f(1 − θαT Σ−1)2Σ. (940)
By using the identity (946) and Θ = 2πΣ−1θ Σ it is a straightforward matter to derive the
relationship between the curvature fij of the vacuum gauge configuration ai on T dθ and the
SU(N) ’t Hooft magnetic fluxes Qab. This is given by
Σ−1fΣ = 2π1
N −QΘQ. (941)
12.4 SL(d, Z) Symmetry
We assume that d is an even number. We may use the modular group SL(d, Z) of the
torus T d to transform the flux matrix Q into Q0 where Q = ΛTQ0Λ. Λ is an arbitrary discrete
SL(d, Z) symmetry which can be chosen such that Q0 is skew-diagonal, i.e
Q0 =
0 q1−q1 0
0 q d2
−q d2
0
. (942)
Under this SL(d, Z) transformation the d−dimensional Weyl-’t Hooft algebra (933) becomes
Γ0aΓ
0b = e
2πN
iQ0abΓ0
bΓ0a. (943)
The transformed twist eating solutions Γ0a are given in terms of the old twist eaters Γa by the
formula
Γa =∏d
b=1(Γ0
b)Λba . (944)
166
In order to verify these relations explicitly it is enough to restrict ourselves to two dimensions,
i.e d = 2. Extension to higher dimensions is straightforward. In two dimensions we have
Γ1 = (Γ01)
Λ11(Γ02)
Λ21 , Γ2 = (Γ01)
Λ12(Γ02)
Λ22 . (945)
We note ( from (943) ) the identity
Γ0Ja
a Γ0Jb
b = e2πN
iJaQ0ab
JbΓ0Jb
b Γ0Ja
a . (946)
We can immediately show that
Γ1Γ2 = e2πN
i
(
Λ21Q021Λ12+Λ11Q0
12Λ22
)
Γ2Γ1
= e2πN
i(ΛT Q0Λ)12Γ2Γ1. (947)
But ΛTQ0Λ = Q which is precisely what we want.
Let us introduce, given the rank N of the SU(N) gauge group and the fluxes qi ∈ Z
(i = 1, ..., d2), the following integers
xi = gcd(qi, N) , li =N
xi, mi =
qixi. (948)
Since li and mi, for every fixed value of i, are relatively prime there exists two integers ai and
bi such that aili + bimi = 1. Let us introduce the following 4 matrices
L0 =
l1l1
l d2
l d2
, M0 =
0 m1
−m1 0
0 m d2
−m d2
0
A0 =
a1
a1
a d2
a d2
, B0 =
0 −b1b1 0
0 −bd2
bd2
0
. (949)
We can then easily verify that Q0 = NM0L0−1 and A0L0 + B0M0 = 1. If we rotate back
to a general basis where Q = ΛTQ0Λ, L = Λ−1L0Λ′, M = ΛTM0Λ
′, A = Λ
′−1A0Λ and
B = Λ′−1B0(ΛT )−1 then we obtain
Q = NML−1 , AL+BM = 1. (950)
167
Let us recall that Λ is the SL(d, Z) transformation which represents the automorphism sym-
metry group of the NC torus T dθ . As it turns out the extra SL(d, Z) transformation Λ
′will
represent the automorphism symmetry group of the dual NC torus T dθ′
.
It is a known result that a necessary and sufficient condition for the existence of d inde-
pendent matrices Γ0a which solve the Weyl-’t Hooft algebra (943) is the requirement that the
product l1...l d2
divides the rank N of the gauge group, viz
N = N0l1...l d2. (951)
The integer N/N0 is identified as the dimension of the irreducible representation of the Weyl-’t
Hooft algebra. As we will see shortly the integer N0 is the rank of the group of matrices which
commute with the twist eating solutions Γ0a. More explicitly the matrices Γ0
a can be taken in
the subgroup SU(N0) ⊗ SU(l1) ⊗ ...⊗ SU(l d2) of SU(N) as follows ( i = 1, ..., d
2)
Γ02i−1 = 1N0 ⊗ 1l1 ⊗ ...⊗ Vli ⊗ ...⊗ 1l d
2
Γ02i = 1N0 ⊗ 1l1 ⊗ ...⊗ (Wli)
mi ⊗ ...⊗ 1l d2
. (952)
Vli and Wli are the usual SU(li) clock and shift matrices which satisfy VliWli = exp(2πili
)WliVli .
They are given respectively by the explicit expressions
Vli =
0 1
0 0 1
0 0 0 1
0 0 0 0 .
0 1
1 0
, Wli =
1
e2πili
e4πli
.
e2π(li−1)
li
.
(953)
Let us remark that (Wli)mili = 1li and V li
li= 1li and hence (Γ0
2i−1)li = (Γ0
2i)li = 1N . In general
we have ( for each b = 1, ..., d)
(Γ1)L1b(Γ2)
L2b ...(Γd)Ldb = 1N . (954)
12.5 Morita Equivalence
The fluctuation gauge field Ai corresponds to a Weyl operator Ai given by the map (915), viz
Ai =∫
ddxAi(x)⊗∆(x). Similarly the global large gauge transformation Ωa corresponds to the
Weyl operator Ωa =∫
ddxΩa(x)⊗ ∆(x). Hence by using the identity evi∂i∆(x)e−vi∂i = ∆(x−v)for v ∈ Rd we can rewrite the constraints (935) as follows
eΣia∂iAie
−Σia∂i = ΩaAiΩ
+a . (955)
168
To be more precise the operator eΣia∂i means here 1 ⊗ eΣ
ia∂i where ⊗ stands for the tensor
product between the coordinate and operator representations. The Weyl operator Ai can be
expanded in an SU(N0)⊗SU(l1...l d2) invariant way. Recall that N0 is the rank of the group of
matrices which commute with the twist eating solutions Γ0a. Thus we may write
Ai =∫
ddk eikixi⊗∑
~j mod L
d∏
a=1
(Γa)ja⊗ai(k,~j). (956)
The matrices Γa are given in terms of the twist eaters Γ0a by the formula (944). ai(k,~j) is an
N0 ×N0 matrix-valued function which is periodic in ~j so that we have ai(k, ja) = ai(k, ja +Lab)
for each b = 1, ..., d. Therefore we have (j1, j2, ..., jd) ∼ (j1 + L1b, j2 + L2b, ..., jd + Ldb) for each
b = 1, ..., d. For example in two dimensions we can see ( by using (954) ) that we have the
result (Γ1)j1+L1b(Γ2)
j2+L2b = (Γ1)j1(Γ2)
j2 and hence (j1, j2) ∼ (j1 + L1b, j2 + L2b).
By putting (928) and (956) in the constraint (955) we obtain
∫
ddk eiki(xi+Σia)⊗
∑
~j mod L
d∏
a=1
(Γa)ja⊗ai(k,~j) =
∫
ddk eikixi
e−iαaiθijkj⊗∑
~j mod L
Γa
d∏
b=1
(Γb)jbΓ+
a ⊗ai(k,~j). (957)
We work in the special basis where Q = Q0 and Γa = Γ0a and then use covariance of the torus
under SL(d, Z) symmetry to extend the result to a general basis. In this special basis where
Q = Q0 and Γa = Γ0a and for a given value of the index a ( say a = 1 ) the matrix Γ0
a will
commute with all factors in the product∏d
b=1 (Γb)jb except one which we will call Γ0
b ( for
example for a = 1 we will have b = 2 ). It is then trivial to verify from the identity (946)
that Γ0a
∏db=1 (Γ0
b)jbΓ0+
a = e2πiN
Q0ab
jb∏d
b=1 (Γ0b)
jb. By rotating back to a general basis we obtain
the formula
Γa
d∏
b=1
(Γb)jbΓ+
a = e2πiN
Qabjb
d∏
b=1
(Γb)jb. (958)
The constraint (957) becomes
∫
ddk eiki(xi+Σia)(
1 − e2πi(ξa+ 1N
Qabjb))
⊗∑
~j mod L
d∏
a=1
(Γa)ja⊗ai(k,~j) = 0.
(959)
The vector ξa is defined by
ξa = − ki
2π(Σi
a − θijαaj) = − ki
2π(Σ − θαT )ia. (960)
The above equation is solved by ai(k,~j) = 0 and if ai(k,~j) does not vanish we must have instead
ξa +1
NQabjb = na ∈ Z. (961)
169
By using Q = NML−1 and QT = −Q we can rewrite this constraint as
ξa = mcL−1ca where mc = nbLbc + jbMbc. (962)
Recalling the identity AL+BM = 1 we can immediately see that this last equation is solved by
the integers nb = maAab and jb = maBab. In terms of the momentum ~k the solution ξa = mcL−1ca
reads ki = 2πmaβai with
β = − 1
(Σ − θαT )L. (963)
Hence the solution of equation (959)-or equivalently of the constraint(955)- when ai(k,~j) does
not vanish is given by the Weyl operator (956) such that
ki = 2πmaβai , ja = mbBba ∀ ma∈Z. (964)
For every fixed set of d integers ma the solution for ki and ja is unique modulo L and thus the
Weyl operator Ai becomes ( with ai(~m) ≡ ai(2πmaβai, mbBba))
Ai =∑
~m∈Zd
e2πimaβaixi
d∏
a=1
(Γa)mbBba⊗ai(~m). (965)
In the special basis (942) we can show the following
d∏
a=1
( d∏
b=1
(Γ0b)
B0ab
)ma
=d∏
a=1
(
(Γ01)
B0a1(Γ2)
B0a2 ...
)ma
= (Γ02)
m1B012(Γ0
1)m2B0
21 ...
= e2πiN
m1(B012Q0
21B021)m2(Γ0
1)m2B0
21(Γ02)
m1B012 ... (966)
and
d∏
a=1
(Γ0a)
mbBba = (Γ01)
m2B021(Γ0
2)m1B0
12 ... (967)
Thus in general we must have the identity
d∏
a=1
(Γa)mbBba =
d∏
a=1
( d∏
b=1
(Γb)Bab
)ma∏
a<b
e−2πiN
ma(BQBT )abmb . (968)
Next it is straightforward to show the identity
e2πi∑
a,imaβaixi =
d∏
a=1
(
e2πi∑
iβaixi
)ma∏
a<b
eπimaΘ1ab
mb
Θ1ab = 2πβaiθijβbj . (969)
170
Thus the gauge field becomes
Ai =∑
~m∈Zd
d∏
a=1
(z′
a)ma eπi
∑
a<bmaΘ
′
abmb⊗ai(~m) (970)
where
z′
a = e2πiβaixi ⊗d∏
b=1
(Γb)Bab
Θ′
ab = Θ1ab −
2
N(BQBT )ab. (971)
By using Q = NML−1 and AL + BM = 1 we obtain Θ′ = 2πβθβT − 2L−1BT + 2ABT .
Next by using β = −12L−1Σ−1(θF + 2), Σ−1 = ΣT , Θ = 2πΣ−1θΣ and 1 + θf = (1 + 1
2θF )2
we can compute that 2πβθβT = L−1Σ−1(1 + θf)ΣΘ(L−1)T . Furthermore from the identity
Q = NML−1 = N2π
Σ−1f(1 + θf)−1Σ we can show that 1 + θf = ΣL(L − ΘM)−1Σ−1 and
hence 2πβθβT = −(L − ΘM)−1(L−1Θ)T . Finally by using AL + BM = 1 or equivalently
AΘ +BML−1Θ = L−1Θ and the fact that (BML−1Θ)T = BT + (ΘM −L)L−1BT we conclude
that
Θ1 = 2πβθβT = − 1
L− ΘM(AΘ +B)T + L−1BT . (972)
Hence
Θ′
= − 1
L− ΘM(AΘ +B)T − L−1BT + 2ABT (973)
Since ABT is an integral matrix we have immediately e2iπma(ABT )abmb = 1. Similarly we can
show that e−iπma(L−1BT )abmb = e−iπξaja = e−iπnaja = ±1, thus
Θ′ = − 1
L− ΘM(AΘ +B)T . (974)
The commutation relations satisfied by the operators z′
a can be computed ( first in the special
basis (942) then rotating back to a general basis ) to be given by
z′
bz′
a = z′
az′
be2πi(2πβθβT − 1
NBQBT )
ab (975)
and thus
z′
bz′
a = z′
az′
bexp(2πiΘ′
ab) , Θ′
ab = 2π(Σ′−1)a
i θ′
ij(Σ′−1)b
j . (976)
The covariant derivative in the Weyl-t’Hooft solution was found to be given by Di = ∂i− i2Fij x
j .
We compute
[Di, z′
a] = 2πi(Σ′−1)a
i z′
a (977)
171
where
(Σ′−1)a
i = βak(1 +1
2θF )ki (978)
or equivalently
Σ′
= Σ(ΘM − L). (979)
By comparing the expansion (970) to the expansion (914) and the commutation relations (976)
and (977) to the commutation relations (911) and (920) we can immediately conclude that the
original NC torus T dθ is replaced with a dual NC torus T d
θ′where θ
′= (Σ
′Θ
′Σ
′−1)/2π. Indeed
we have obtained the replacements za−→z′
a, ∂i−→Di, Θ−→Θ′and Σ−→Σ
′. By analogy with
(916) we can therefore define on T dθ′
a mapping ∆′(x
′) of fields into operators as follows
∆′
(x′
) =1
|detΣ′ |∑
~m∈Zd
∏d
a=1(z
′
a)ma∏
a<beiπmaΘ
′
abmbe−2πi(Σ
′−1)ai max
′i
. (980)
The expansion (970) thus becomes
Ai =∫
ddx′
∆′
(x′
) ⊗A′
i(x′
) (981)
where
A′
i(x′
) =∑
~m∈Zd
e2πi(Σ′−1)a
i max′i
ai(~m). (982)
This is a single-valued U(N0) gauge field on the NC torus T dθ′
of volume |detΣ′ |. The new
operator trace Tr′is related to Tr by
Tr′
trN0 =N0
N
|detΣ′ ||detΣ| Tr trN . (983)
Finally it is a trivial exercise to check that the action (937) becomes on the dual torus T dθ′ given
by
SY M = − 1
4g′2
∫
ddx′
trN0(F′
ij(x′
))2∗ (984)
where
g′2 = g2N0
N
|detΣ′ ||detΣ| =
g2
N0Nd−1|det(ΘQ−N)|. (985)
172
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