AGT 関係式 (2) AGT 関係式(String Advanced Lectures No.19)
高エネルギー加速器研究機構 (KEK)
素粒子原子核研究所 (IPNS)
柴 正太郎
2010 年 6 月 9 日(水) 12:30-14:30
Contents
1. Gaiotto’s discussion for SU(2)
2. SU(2) partition function
3. Liouville correlation function
4. Seiberg-Witten curve and AGT relation
5. Towards generalized AGT relation
Gaiotto’s discussion for SU(2)
[Gaiotto ’09]
SU(2) gauge theory with 4 fundamental flavors (hypermultiplets)
S-duality group SL(2,Z)
coupling const. :
flavor sym. : SO(8) ⊃ SO(4)×SO(4) ~
[SU(2)a×SU(2)b]×[SU(2)c×SU(2)d]
: (elementary) quark
: monopole
: dyon
Subgroup of S-duality without permutation of masses
In massive case, we especially consider this subgroup.
• mass : mass parameters can be associated to each SU(2) flavor.
Then the mass eigenvalues of four hypermultiplets in 8v is ,
.
• coupling : cross ratio (moduli) of the four punctures, i.e. z =
Actually, this is equal to the exponential of the UV coupling
→ This is an aspect of correspondence between the 4-dim N=2 SU(2)
gauge theory and the 2-dim Riemann surface with punctures.
SU(2) gauge theory with massive fundamental hypermultiplets
SU(2) partition function
Action
classical part
1-loop correction : more than 1-loop is cancelled, because of N=2
supersymmetry.
instanton correction : Nekrasov’s calculation with Young tableaux
Parameters
coupling constants
masses of fundamental / antifund. / bifund. fields and VEV’s of gauge fields
deformation parameters : background of graviphoton or deformation of extra
dimensions
Nekrasov’s partition function of Nekrasov’s partition function of 4-dim gauge theory4-dim gauge theory
(Note that they are different from Gaiotto’s ones!)
Now we calculate Nekrasov’s partition function of 4-dim SU(2) quiver
gauge theory as the quantity of interest.
1-loop part 1-loop part of partition function of 4-dim quiver gauge theoryof partition function of 4-dim quiver gauge theory
We can obtain it of the analytic form :
where each factor is defined as
: each factor is a product of double Gamma function!
,
gauge antifund. bifund. fund.
mass massmassVEV
deformation parameters
We obtain it of the expansion form of instanton
number :
where : coupling const. and
and
Instanton part Instanton part of partition function of 4-dim quiver gauge theoryof partition function of 4-dim quiver gauge theory
Young tableau
< Young tableau >
instanton # = # of boxes
leg
arm
SU(2) with four flavors : Calculation of Nekrasov function for U(2)SU(2) with four flavors : Calculation of Nekrasov function for U(2)
U(2), actually
Manifest flavor symmetry is now
U(2)0×U(2)1 , while actual symmetry is
SO(8)⊃[SU(2)×SU(2)]×[SU(2)×SU(2)].
In this case, Nekrasov partition function can be written as
where and
is invariant under the flip (complex conjugate representation) :
which can be regarded as the action of Weyl group of SU(2) gauge
symmetry.
is not invariant. This part can be regarded as U(1)
contribution.
Surprising discovery by Alday-Gaiotto-Tachikawa
In fact, is nothing but the conformal block of Virasoro
algebra with
for four operators of dimensions inserted at :
SU(2) with four flavors : Identification of SU(2) part and U(1) partSU(2) with four flavors : Identification of SU(2) part and U(1) part
(intermediate state)
Correlation function of Liouville theory with .
Thus, we naturally choose the primary vertex operator
as the examples of such operators. Then the 4-point function on a
sphere is
3-point function conformal block
where
The point is that we can make it of the form of square of absolute
value!
… only if
… using the properties : and
Liouville correlation function
As a result, the 4-point correlation function can be rewritten as
where and
It says that the 3-point function (DOZZ factor) part also can be
written as the product of 1-loop part of 4-dim SU(2) partition function
:
under the natural identification of mass parameters :
Example 1 : SU(2) with four flavors (Sphere with four punctures)
Example 2 : Torus with one puncture
The SW curve in this case corresponds to 4-dim N=2* theory :
N=4 SU(2) theory deformed by a mass for the adjoint hypermultiplet
Nekrasov instanton partition function
This can be written as
where equals to the conformal block of Virasoro algebra with
Liouville correlation function (corresponding 1-point function)
where is Nekrasov’s partition
function.
Example 3 : Sphere with multiple punctures
The Seiberg-Witten curve in this case corresponds to 4-dim N=2
linear quiver SU(2) gauge theory.
Nekrasov instanton partition function
where equals to the conformal block of
Virasoro algebra with for the vertex operators which are
inserted at z=
Liouville correlation function (corresponding n+3-point function)
where is Nekrasov’s full partition function.
• According to Gaiotto’s discussion, SW curve for SU(2) case is
.
• In massive cases, has double poles.
• Then the mass parameters can be obtained as ,
where is a small circle around the a-th puncture.
• The other moduli can be fixed by the special coordinates
,
where is the i-th cycle (i.e. long tube at weak coupling).
Note that the number of these moduli is 3g-3+n. (g : # of genus, n : # of punctures)
SW curve and AGT relation
Seiberg-Witten curve and its moduli
• The Seiberg-Witten curve is supposed to emerge from Nekrasov
partition function in the “semiclassical limit” , so in this
limit, we expect that .
• In fact, is satisfied on a
sphere,
then has double poles at zi .
• For mass parameters, we have ,
where we use and .
• For special coordinate moduli, we have ,
which can be checked by order by order calculation in concrete
examples.
• Therefore, it is natural to speculate that Seiberg-Witten curve is
‘quantized’ to at finite .
2-dim CFT in AGT relation : ‘quantization’ of Seiberg-Witten curve??
Towards generalized AGT Natural generalization of AGT relation seems the correspondence
between partition function of 4-dim SU(N) quiver gauge theory and
correlation function of 2-dim AN-1 Toda theory :
• This discussion is somewhat complicated, since in SU(N>2) case,
the punctures are classified with more than one kinds of N-box
Young tableaux :
< full-type > < simple-type > < other types >
(cf. In SU(2) case, all these Young tableaux become ones of the same
type .)
[Wyllard ’09][Kanno-Matsuo-SS-Tachikawa ’09]
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