AGT 関係式 (3) 一般化に向け

て(String Advanced Lectures No.20)

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 23 日（水） 12:30-14:30

Contents

1. AGT relation for SU(2) quiver theory

2. Partition function of SU(N) quiver theory

3. Toda theory and W-algebra

4. Generalized AGT relation for SU(N) case

5. Towards AdS/CFT duality of AGT relation

AGT relation for SU(2) quiver

We now consider only the linear quiver gauge theories in AGT relation.We now consider only the linear quiver gauge theories in AGT relation.

Gaiotto’s discussion

An example : SW curve is a sphere with multiple punctures.An example : SW curve is a 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.

(↑ including 1-loop part)

U(1) part

[Alday-Gaiotto-Tachikawa ’09]AGT relation : SU(2) gauge theory AGT relation : SU(2) gauge theory Liouville theory Liouville theory !!

Gauge theory Liouville theory

coupling const. position of punctures

VEV of gauge fields internal momenta

mass of matter fields external momenta

1-loop part DOZZ factors

instanton part conformal blocks

deformation parameters Liouville parameters

4-dim theory : SU(2) quiver gauge theory

2-dim theory : Liouville (A1 Toda) field theory

In this case, the 4-dim theory’s partition function Z and the 2-dim

theory’s correlation function correspond to each other :

central charge :

Now we calculate Nekrasov’s partition function of 4-dim SU(N) quiver

gauge theory as the quantity of interest.

SU(2) case : We consider only SU(2)×…×SU(2) quiver gauge

theories.

SU(N) case : According to Gaiotto’s discussion, we consider, in

general, the

cases of SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2) x

SU(d’1) group,

where is non-negative.

SU(N) partition function

Nekrasov’s partition function of Nekrasov’s partition function of 4-dim gauge theory4-dim gauge theory

xx xxx

*

… …x

*

…

…

d’3 – d’2d’2 – d’1d’1

… ………

…

d3 – d2

d2 – d1

d1… ………

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 massmassflavor symm. of bifund. is U(1)

VEV# of d.o.f. depends on dk

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

Naive assumption is 2-dim AN-1 Toda theory, since Liouville theory is

nothing but A1 Toda theory. This means that the generalized AGT

relation seems

Difference from SU(2) case…

• VEV’s in 4-dim theory and momenta in 2-dim theory have more than

one d.o.f.

In fact, the latter corresponds to the fact that 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 .)

• In general, we don’t know how to calculate the conformal blocks of

Toda theory.

……

…

…

………

What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory?What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory?

Action :

Toda field with :

It parametrizes the Cartan subspace of AN-1 algebra.

simple root of AN-1 algebra :

Weyl vector of AN-1 algebra :

metric and Ricci scalar of 2-dim surface

interaction parameters : b (real) and

central charge :

Toda theory and W-algebra

What is AWhat is AN-1N-1 Toda theory? : some extension of Liouville theory Toda theory? : some extension of Liouville theory

• In this theory, there are energy-momentum tensor and higher spin fields

as Noether currents.

• The symmetry algebra of this theory is called W-algebra.

• For the simplest example, in the case of N=3, the generators are defined as

And, their commutation relation is as follows:

which can be regarded as the extension of Virasoro algebra, and where

　　　　　　 ,

What is AWhat is AN-1N-1 Toda field theory? Toda field theory? (continued)(continued)

We ignore Toda potential

(interaction) at this stage.

• The primary fields are defined as 　　　　　　　 ( is called

‘momentum’) .

• The descendant fields are composed by acting / 　 on the

primary fields as uppering / lowering operators.

• First, we define the highest weight state as usual :

Then we act lowering operators on this state, and obtain various

descendant fields as .

• However, some linear combinations of descendant fields accidentally

satisfy the highest weight condition. They are called null states. For

example, the null states in level-1 descendants are

• As we will see next, we found the fact that these null states in W-

algebra are closely related to the singular behavior of Seiberg-Witten

curve near the punctures. That is, Toda fields whose existence is

predicted by AGT relation may in fact describe the form (or behavior)

of Seiberg-Witten curve.

As usual, we compose the primary, descendant, and null fields.As usual, we compose the primary, descendant, and null fields.

• As we saw, Seiberg-Witten curve is generally represented as

and Laurent expansion near z=z0 of the coefficient function is

generally

• This form is similar to Laurent expansion of W-current (i.e. W-

generators)

• Also, the coefficients satisfy similar equations, except full-type

puncture’s case

This correspondence becomes exact, in some kind of ‘classical’ limit:(which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?)

• This fact strongly suggests that vertex operators corresponding non-

full-type punctures must be the primary fields which has null states in

their descendants.

The singular behavior of SW curve is related to the null fields of W-algebra.The singular behavior of SW curve is related to the null fields of W-algebra.[Kanno-Matsuo-SS-Tachikawa ’09]

null condition

~ direction of D4 　 ~ direction of NS5

• If we believe this suggestion, we can conjecture the form of

momentum of Toda field in vertex operators

, which corresponds to each kind of punctures.

• To find the form of vertex operators which have the level-1 null state,

it is useful to consider the screening operator (a special type of vertex

operator)

• We can show that the state satisfies the highest

weight condition, since the screening operator commutes with all the

W-generators.(Note a strange form of a ket, since the screening operator itself has non-zero

momentum.)

• This state doesn’t vanish, if the momentum satisfies

for some j. In this case, the vertex operator has a null state at level

.

The punctures on SW curve corresponds to the ‘degenerate’ fields!The punctures on SW curve corresponds to the ‘degenerate’ fields![Kanno-Matsuo-SS-Tachikawa ’09]

• Therefore, the condition of level-1 null state becomes for

some j.

• It means that the general form of mometum of Toda fields

satisfying this null state condition is

.

Note that this form naturally corresponds to Young tableaux

.

• More generally, the null state condition can be written as

(The factors are abbreviated, since they are only the images under Weyl

transformation.)

• Moreover, from physical state condition (i.e. energy-momentum is

real), we need to choose , instead of naive

generalization:

Here, is the same form of β,

is Weyl vector,

and .

The punctures on SW curve corresponds to the ‘degenerate’ fields!The punctures on SW curve corresponds to the ‘degenerate’ fields!

Generalized AGT relation

Natural form : former’s partition function and latter’s correlation

function

Problems and solutions for its proof

• correspondence between each kind of punctures and vertices:

we can conjecture it, using level-1 null state condition.

< full-type > < simple-type > < other types >

• difficulty for calculation of conformal blocks: null state condition

resolves it again!

[Wyllard ’09][Kanno-Matsuo-SS-Tachikawa ’09]

……

…

…

………

Correspondence : 4-dim SU(N) quiver gauge and 2-dim ACorrespondence : 4-dim SU(N) quiver gauge and 2-dim AN-1N-1 Toda theoryToda theory

• We put the (primary) vertex operators at punctures, and

consider the correlation functions of them:

• In general, the following expansion is valid:

where

and for level-1 descendants,

: Shapovalov matrix

• It means that all correlation functions consist of 3-point functions and

inverse Shapovalov matrices (propagator), where the intermediate

states (descendants) can be classified by Young tableaux.

On calculation of correlation functions of 2-dim AOn calculation of correlation functions of 2-dim AN-1N-1 Toda theory Toda theory

descendants

primaries

In fact, we can obtain it of the factorization form of 3-point functions

and inverse Shapovalov matrices :

3-point function : We can obtain it, if one entry has a null state in

level-1!

wherehighest weight~ simple punc.

On calculation of correlation functions of 2-dim AOn calculation of correlation functions of 2-dim AN-1N-1 Toda theory Toda theory

’

Case of SU(3) quiver gauge theory

SU(3) : already checked successfully. [Wyllard ’09] [Mironov-Morozov ’09]

SU(3) x … x SU(3) : We have checked successfully. [Kanno-Matsuo-SS ’10]

SU(3) x SU(2) : We could check it, but only for restricted cases. [Kanno-Matsuo-

SS ’10]

Case of SU(4) quiver gauge theory

• In this case, there are punctures which are not full-type nor simple-type.

• So we must discuss in order to check our conjucture (of the simplest

example).

• The calculation is complicated because of W4 algebra, but is mostly

streightforward.

Case of SU(∞) quiver gauge theory

• In this case, we consider the system of infinitely many M5-branes, which may

relate to AdS dual system of 11-dim supergravity.

• AdS dual system is already discussed using LLM’s droplet ansatz, which is

also governed by Toda equation. [Gaiotto-Maldacena ’09] →

subject of next talk…

Our plans of current and future research on generalized AGT relationOur plans of current and future research on generalized AGT relation

Towards AdS/CFT of AGT

CFT side : 4-dim SU(N) quiver gauge theory and 2-dim AN-1Toda

theory

• 4-dim theory is conformal.

• The system preserves eight supersymmetries.

AdS side : the system with AdS5 and S2 factor and eight

supersymmetries

• This is nothing but the analytic continuation of LLM’s system in M-

theory.

• Moreover, when we concentrate on the neighborhood of punctures

on Seiberg-Witten curve, the system gets the

additional S1 ~ U(1) symmetry.

• According to LLM’s discussion, such system can

be analyzed using 3-dim electricity system:

[Lin-Lunin-Maldacena ’04]

[Gaiotto-Maldacena ’09]