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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