Vector bundles on the noncommutative torus

from deformation quantization

Francesco D’Andrea

( joint work with G. Fiore & D. Franco )

15/07/2014

Frontiers of Fundamental Physics 14

Marseille, 15-18 July 2014

Line bundles: an example.

Mobius strip

• (Non-trivial) real line bundle on S1. A copy of R attached to any point of S1.

• Section = function associating to any point P of S1 a point on the line through P.

• In quantum mechanics: charged particle interacting with a magnetic monopole.

I wave function = section of a non-trivial vector bundle

I magnetic field = curvature of a connection on the bundle

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Complex line bundles on a torus.

ω2

ω1

⇒

• Torus algebra C(T2): universal C∗-algebra generated by two commuting unitaries.

• Complex structures on C/Λ ' T2 parametrized by τ = ω2/ω1 (Λ := ω1Z+ω2Z ).

• Take ω1 = 1 and τ = ω2 ∈ H :={z ∈ C : Im(z) > 0

}, call Eτ the complex manifold.

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Summary.

For 0 < θ < 1, let Aθ be the universal C∗-algebra generated by unitaries U and V with

UV = e2πiθVU .

The dense ∗-subalgebra (Frechet pre C∗-algebra with suitable seminorms. . . )

A∞θ :={ ∑

m,n∈Zam,nU

mVn : {am,n} ∈ S(Z2)}⊂ Aθ

is a strict deformation quantization of C(T2) associated to the action of R2 (Rieffel, 1993).

I finitely-generated projective Aθ-modules classified by Connes and Rieffel in the ‘80s.

I can they be obtained as deformations of vector bundles on the torus?

No action of R2 on line bundles: Rieffel’s tecnique cannot be used.

I they are deformations of line bundles on the elliptic curve Eτ = C/Λτ, provided

τ− pθ2i ∈ Z+ iZ ,

where p ∈ Z is the 1st Chern number of the bundle.

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Credits & motivations.

• Works on nc-tori and elliptic curves (quantum theta functions, etc.):

Y.I. Manin, M. Marcolli, F.P. Boca, J. Plazas, A. Polishchuk, A. Schwarz,

M. Vlasenko, I. Nikolaev, S. Mahanta, W.D. van Suijlekom, . . .

• Why? Hilbert’s 12th problem: description of the maximal abelian extension Kab of anumber field K in terms of special values of suitable meromorphic functions.

I K = Q : Kab generated by roots of unity, i.e. special values of eiz correspondingto torsion points of C∗ (Kronecker-Weber theorem).

I K = Q(τ) imaginary quadratic field: Kab generated by the j-invariant j(Eτ) andby the values of the Weierstrass’s elliptic function ℘(z; τ) at torsion points of Eτ.

I Manin’s idea: for a real quadratic field K = Q(θ), Aθ plays the role of Eτ.

• (Very ample) line bundles realize Eτ (and Aθ?) as complex projective variety.

• I. Nikolaev (arxiv:1404.4999) found generators of the Hilbert class field of K = Q(√D),

i.e. the analogue of the j-invariant for nc-tori.

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— PART I —

Nc-torus and Heisenberg modules

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The deformation point of view: twisted group algebras.

Let x,y ∈ R2. The C∗-algebra C(T2) is generated by the two functions:

u(x,y) := e2πix , v(x,y) := e2πiy .

By standard Fourier analysis,

C∞(T2) ≡{f =

∑m,n∈Z

am,numvn : {am,n} ∈ S(Z2)

}.

An associative product ∗θ on C∞(T2) is given on monomials by

(ujvk) ∗θ (umvn) = σ((j,k), (m,n)

)uj+mvk+n , ∀ j,k,m,n ∈ Z ,

where σ : Z2 × Z2 → C∗ is a 2-cocycle in the group cohomology complex of Z2, given by:

σ((j,k), (m,n)

):= eiπθ(jn−km) .

(C∞(T2), ∗θ) is a ∗-algebra (with undeformed involution), cocycle quantization of S(Z2)

with convolution product. The C∗-completion is the twisted group C∗-algebra C∗(Z2,σ).

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Quantization map & Moyal product.

An isomorphism Tθ : (C∞(T2), ∗θ)→ A∞θ is given on f =∑m,n∈Z am,nu

mvn by:

Tθ(f) :=∑m,n∈Z

am,ne−πimnθUmVn .

The phase factors are chosen to have Tθ(f)∗ = Tθ(f∗) for all f.

On the Schwartz space S(R2) one has the Moyal product (see e.g. Gayral et al., CMP 246,2004 & references therein):

(f ∗θ g)(z) =4

θ2

∫C×C

f(z+ ξ)g(z+ η)e4πiθ Im(ξη)dξdη ,

where z = x+ iy and dz = dxdy.

Extended by duality to tempered distributions, allows to define an associative product (byrestriction) on several interesting function spaces. E.g. B(R2), the set of smooth functionsthat are bounded together with all their derivatives, with ∗θ is a Frechet pre C∗-algebra.

On C∞(T2) ⊂ B(R2) one recovers the star product of the nc-torus.

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Heisenberg modules. [Connes 1980, Rieffel 1983]

Let p, s ∈ Z and p > 1. A right A∞θ -module structure on Ep,s := S(R)⊗ Cp is given by

ψ /U ={W( s

p+ θ, 0)⊗ (S∗)s

}ψ , ψ / V =

{W(0, 1)⊗ C

}ψ ,

where C,S ∈Mp(C) are the clock and shift operators, and W(a,b) unitaries on L2(R):{W(a,b)ψ

}(t) = e−πiabe2πibtψ(t− a) , ψ ∈ L2(R), a,b ∈ R.

An A∞θ -valued Hermitian structure on Ep,s is given by

〈ψ,ϕ〉 =∑m,n∈Z

UmVn∫∞−∞(ψ /UmVn|ϕ)t dt .

where for all ψ = (ψ1, . . . ,ψp) and ϕ = (ϕ1, . . . ,ϕp) ∈ Ep,s:

(ψ|ϕ)t :=

p∑r=1

ψr(t)ϕr(t) .

If θ ∈ RrQ, any finitely-gen. projective right Aθ-module is isomorphic either to (A∞θ )p orto a module Ep,s, with p and s coprime (p > 0 and s 6= 0) or p = 1 and s = 0.

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— PART II —

Elliptic curves and the WBZ transform

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Basics on elliptic curves.

Let us identify (x,y) ∈ R2 with z = x+ iy ∈ C. Fix τ ∈ H and let

Eτ = C/Λ , Λ := Z+ τZ ,

be the corresponding elliptic curve with modular parameter τ.

Let α : Λ× C→ C∗ be a smooth function and π : Λ→ EndC∞(C) given by

(?) π(λ)f(z) := α−1(λ, z)f(z+ λ) , ∀ λ ∈ Λ, z ∈ C .

Then π is a representation of the abelian group Λ if and only if

(‡) α(λ+ λ ′, z) = α(λ, z+ λ ′)α(λ ′, z) , ∀ z ∈ C, λ, λ ′ ∈ Λ.

An α satisfying (‡) is called a factor of automorphy for Eτ.

There is a corresponding line bundle Lα → Eτ with total space Lα = C× C/∼ , where

(z+ λ,w) ∼ (z,α(λ, z)w) , ∀ z,w ∈ C, λ ∈ Λ,

All line bundles on Eτ are of this form (Appell-Humbert theorem).10 / 20

Sections of line bundles.

Smooth sections of Lα ≡ subset Γα ⊂ C∞(C) of invariant functions under π(λ) in (?).

I if α = 1, Γα ≡ C∞(Eτ) are Λ-periodic functions (and C∞(C) is a C∞(Eτ)-module).

I if α holomorphic holomorphic elements of Γα are called theta functions: theyform a finite-dimensional vector space.

I if α unitary elements of Γα are quasi-periodic functions ( |f| is periodic ).

Note that in this case Γα ⊂ B(R2) is in the domain of Moyal product.

For any α, Γα is a finitely-generated projective C∞(Eτ)-submodule of C∞(C). Let

τ = ωx + iωy

with ωx,ωy ∈ R and ωy > 0. The smooth line bundle with degree p (unique for each p)can be obtained from the unitary factor of automorphy βp, where:

β(m+ nτ, x+ iy) = e−πiωxn2e−2πinx

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WBZ transform.

For fixed τ,p, let Fτ,p be the module with factor of automorphy βp. Let [n] := n mod p.

Proposition

Every f ∈ Fτ,p is of the form

f(z) =∑n∈Z

e2πinxeπin2ωxp f[n](y+ n

ωyp) ,

for some (unique) Schwartz functions f[1], . . . , f[p] ∈ S(R),

We denote by ϕτ,p : Fτ,p → S(R)⊗ Cp the bijection

f 7→ f = (f[1], . . . , f[p])t .

ϕτ,p is very similar to the Weil-Brezin-Zak transform of solid state physics (Folland, 1989).

Fτ,p is a pre Hilbert C∞(Eτ)-module with canonical Hermitian structure (f,g) 7→ f∗g.

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— PART III —

Vector bundles over the nc-torus

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Moyal deformation of bimodules.

C∞(R2) is a A∞θ -bimodule, with module structure:

(U . f)(x,y) = e2πixf(x,y+ 1

2θ), (f /U)(x,y) = e2πixf

(x,y− 1

2θ),

(V . f)(x,y) = e2πiyf(x− 12θ,y) , (f / V)(x,y) = e2πiyf(x+ 1

2θ,y) .

Let J be the antilinear involutive map:

Jf(x,y) = f(−x,−y) .

J( . )J sends A∞θ into its commutant, and transforms the left action into the right one.

The relation with Moyal is as follows. The space B(R2) is an A∞θ sub-bimodule of C∞(R2).For a ∈ A∞θ and f ∈ B(R2), one can check that

a . f = σθ(a) ∗θ f and f / a = f ∗θ σθ(a) ,

where ∗θ is Moyal product, Tθ : C∞(T2)→ A∞θ the quantization map introduced before andσθ := T−1

θ the symbol map.

We will focus on right modules. . .14 / 20

Deformation of line bundles.

As one might expect, the subspaces Fi,p ⊂ C∞(R2) are not right A∞θ -submodules, but wecan try with Fτ,p for some non-trivial value of the modular parameter τ.

Recall that f ∈ C∞(C) belongs to Fτ,p iff:

(†) f(z+ 1) = f(z) and f(z+ τ) = e−πip(ωx+2x)f(z) ∀ z = x+ iy ∈ C.

Proposition

The vector space Fτ,p is a right A∞θ -module if and only if

τ− pθ2i ∈ Z+ iZ .

Proof. One has to prove that, for any f satisfying (†), f /U and f / V satisfy (†) too. . . �

We want to give a description in terms of Weyl operators, and compare Fτ,p withConnes-Rieffel modules.

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Line bundles and Connes-Rieffel modules.Fix r, s ∈ Z and let

τ = r+ i(s+ pθ2)

Using the WBZ transform ϕτ,p : Fτ,p → Hτ,p := S(R)⊗ Cp we transport the right modulestructure from Fτ,p to Hτ,p, and get a right action

fJa = ϕτ,p(ϕ−1τ,p(f) / a

)∀ a ∈ A∞θ , f ∈ Hτ,p.

Proposition

For all f ∈ Hτ,p,

fJU = eπirp{W( s

p+ θ, 0)⊗ (C∗)rS

}f , fJV =

{W(0, 1)⊗ (C∗)s

}f ,

where C,S ∈Mp(C) are the clock and shift operators.

Remarks.I For r = 0 and arbitrary s,p, there is a right A∞θ -module isomorphism Hτ,p ' Ep,s.

I If θ ∈ R r Q, every (finitely-gen. proj.) right module is the deformation of a line bundle(no vector bundle of higher rank is needed).

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Hermitian structures.

Let f,g ∈ Fτ,p.The canonical Hermitian structure (f,g) 7→ fg is naturally replaced by f ∗θ g .While fg is Λ-periodic and belongs to C∞(Eτ), if s = 1 the product f ∗θ g is Z2-periodicand belongs to C∞(T2).

Proposition

Let τ = i(1+ 12pθ). Then, for all f,g ∈ Fτ,p:

(†) Tθ(f ∗θ g) =∑

m,n∈ZVnUm

∫+∞−∞ (fJVnUm|g)t dt ,

where ( | )t and Tθ is as before, f = ϕτ,p(f) and g = ϕτ,p(g).

Equation (†) is similar to the Hermitian structure of Connes-Rieffel modules, but a differentright action is involved.

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— PART IV —

Some additional stuff

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A formal approach: Hopf cocycles & Drinfeld twist.

I If B is a bialgebra, A a left B-algebra module and M an Ao B-module, using acocycle twist based on B we get:

• a new bialgebra B? (with deformed coproduct),

• a new algebra A? (with deformed product),

• an A? o B?-module M?.

I If A = C∞(T2)[[ h]], with the cocycle twist

F = ei h∂x∧∂y

based on U(R2)[[ h]] we get a deformation quantization of A (the formal analogue ofthe algebra A∞θ of the noncommutative torus).

I We would like to deform the modules Fi,p ⊂ C∞(C) as well (here τ = i).

There is no obvious R2 action: if by contraddiction ∂x,∂y map Fi,p into itself, thereexists a flat connection

∇f = (∂xf)dx+ (∂yf)dy f ∈ Fi,p ,

meaning that the line bundle is trivial, i.e. the 1st Chern number is p = 0.

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Heisenberg twists.

I Two linear maps ∇1,∇2 : Fτ,p → Fτ,p are given by:

∇1f(z) = (∂x +2πipωy

y)f(z) , ∇2f(z) = ∂yf(z) .

(used e.g. by Connes-Rieffel to construct a connection with constant curvature)

I On O :=⊕p∈Z Fi,p consider the operators

a = 12(∇1 + i∇2) , a† = 1

2(−∇1 + i∇2) , q = ⊕p∈Zπp idFi,p

These operators satisfy the commutation relations of the Heisenberg Lie algebra h3:

[a, a†] = q , [q, . ] = 0 .

I O is a graded U(h3)-algebra module. Given any cocycle twist based on U(h3)[[ h]], wecan produce a new graded associative algebra O? =

⊕p∈Z(Fi,p[[ h]])? and each

(Fi,p[ h])? is automatically a module for the subalgebra (Fi,0[[ h]])? = C∞(T2)[[ h]]?.

I There are several known cocycle twists, but they give a commutative (Fi,0[[ h]])?.There are twists with non-trivial coassociator for which (Fi,0[[ h]])? is exactly thenc-torus algebra, but gives a non-associative deformation O?.

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Thank you for your attention.