AGT 関係式とその一般化に向けて(Towards the generalization of AGT relation)

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 (Shotaro Shiba)

S. Kanno, Y. Matsuo, S.S. and Y. Tachikawa, Phys. Rev. D81 (2010) 046004. S. Kanno, Y. Matsuo and S.S., work in progress.

What is the What is the multiplemultiple M-branes’ system like? M-branes’ system like? (The largest motivation of my research)(The largest motivation of my research)

• The system of single M-brane in 11-dim spacetime is understood, at least

classically.

• However, at this time, we have too little information on the multiple M-

branes’ system.

• Now I hope to understand more on M-theory by studying the internal

degrees of freedom which the multiple branes’ systems must always have.

D-branes’ case : internal d.o.f ~ N2

• The superstrings ending on a D-brane compose the internal d.o.f.

• It is well known that this system is described by DBI action with gauge

symmetry of Lie algebra U(N), which is reduced to Yang-Mills theory in the

low-energy limit.

Introduction

2

M2-branes’ case : internal d.o.f. ~ N3/2

• The proposition of BLG model is the important breakthrough. [Bagger-Lambert ’07]

[Gustavsson ’07]

• We can derive the internal d.o.f. of order N3/2 naturally and successfully, using

the finite representation of Lie 3-algebra which is the gauge symmetry algebra

of BLG model. [Chu-Ho-Matsuo-SS ’08]

• However, at this moment, we don’t know at all what compose these d.o.f.

M5-branes’ case : internal d.o.f. ~ N3

Based on the recent research of AGT relation and its generalization, not a few

researchers now hope that [Alday-Gaiotto-

Tachikawa ’09] [Wyllard ’09] etc.

• Toda fields on 2-dim Riemann surface (or Seiberg-Witten curve [Seiberg-Witten ’94])

• W-algebra which is the symmetry algebra of Toda field theory

bring us some new understanding on the multiple M5-branes’ internal d.o.f !

The near horizon geometry of M-branes is AdS x S, so we can use AdS/CFT discussion.Then this internal d.o.f. corresponds to the entropy of AdS blackhole. (~

area of horizon)

3

Subject of today’s seminar

Intersecting M5-branes’ system makes Intersecting M5-branes’ system makes 4-dim spacetime4-dim spacetime and and 2-dim surface2-dim surface..

0,1,2,3

4,5

6,10

• From the condition of 11-dim supergravity (i.e. intersection rule), the

intersection surface of two bundles of M5-branes at right angles must be 3-

dim space.

• In this 3-dim space (i.e. 4-dim spacetime), N=2 gauge theory lives. (We see

this next.)

• The remaining part of M5-branes becomes 2-dim surface (complex 1-dim

curve).

• Since it is believed that M5-branes’ worldvolume theory is conformal (from

AdS/CFT), if 4-dim gauge theory is conformal, the theory on this 2-dim

surface (called as the Seiberg-Witten curve) must also be conformal field

theory.

bundle of M5-branes

4

This is Seiberg-Witten system. [Seiberg-Witten ’94]

In this time, M5-branes keep only ½ x ½ SUSYs.

?

Seiberg-Witten curve determines the field contents of Seiberg-Witten curve determines the field contents of 4-dim gauge theory4-dim gauge theory..

NS5-brane (M5-brane)

• Now we compactify 1-dim space out of 11-dim spacetime, and go to the D4-

NS5 system in superstring theory, since we have very little knowledge on M5-

brane.

• In string theory, (vibration modes) of F1-strings describe the gauge and

matter fields.

• The fields of this gauge theory are composed by F1-strings moving in 4-dim

spacetime.

• In general, gauge group is SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2)

x SU(d’1). This theory is conformal, when # of D6-branes is

.

gluons / quarks

D4-brane (M5-brane)

flavor brane(length = infinite)

color brane(length ~ 1/coupling)

flavor braneD6-brane

4,5

6, 107,8,9

[Seiberg-Witten ’94]

F1-string

(from Hanany-Witten’s discussion)

more generally…

5

increasing increasing

antifund.

gauge bifund. fund.

• To see the structure of Seiberg-Witten curve, now we move each D4-brane

for longitudinal direction of NS5-branes to each distance.

• After this ‘deformation’, the gauge fields get VEV’s, and the matter fields get

masses. (This means, of course, that the gauge theory is no longer conformal.)

• In general cases, the Seiberg-Witten curve is described in terms of a

polynomial as

Note that

The coefficient of y

N is 1. : normalization which causes the divergence of

!

The y

N-1 term doesn’t exist. : suitable shift of coordinates

6

A kind of ‘deformations’ makes clear the structure of Seiberg-Witten curve.A kind of ‘deformations’ makes clear the structure of Seiberg-Witten curve.

~ direction of D4 　　 ~ direction of NS5

Contents

1. Introduction (pp.2-6)

2. Gaiotto’s discussion (pp.8-10)

3. AGT relation (pp.11-17)

4. Towards proof of AGT relation (pp.18-22)

5. Towards generalized AGT relation (pp.23-29)

6. Conclusion (p.30)

7

When we recognize the intersecting point of D4-branes and NS5-branes as

‘punctures’, 2-dim conformal field theory can be defined on Seiberg-Witten

curve. [Gaiotto ’09]

Seiberg-Witten curve may be described by Seiberg-Witten curve may be described by 2-dim conformal field theory2-dim conformal field theory..

0 ∞

deformation to2-dim sphere

multiple D4-branes

NS5-branes

4,5

6

10 (compactified)

For gauge group : SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2)

x SU(d’1)

(All Young tableaux are composed by N boxes.)

d3 – d2

d2 – d1

d1

…

d’3 –

d’2

d’2 –

d’1

d’1

… ……

…

… ……

…

… …

0 ∞

Gaiotto’s discussion

8

9

What is the breakthrough provided by Gaiotto’s discussion?What is the breakthrough provided by Gaiotto’s discussion?

• Therefore, 4-dim gauge theory relates to 2-dim theory at the following points :

gauge group 　　　　 type of punctures at z=0 and ∞ (which are classified with

Young tableaux)

coupling const. 　　　 length between neighboring punctures

• For example, when we infinitely lengthen a distance between punctures (i.e.

take a weak coupling limit), the following transformation occurs :

• Also, he strongly suggested that the larger class of 4-dim gauge theories than

those described by brane configurations in string theory can be recognized as

the 2-dim compactification of multiple M5-branes’ system. For example,

famous(?) TN theory.

… …… …

S-dual

……

SU(N) SU(N) SU(N)… … … …

• TN theory is obtained as S-dual of SU(N) quiver gauge theory, as follows

:

In other words,

• However, in the following, we concentrate on the systems of brane

configuration, i.e. the cases where 4-dim theory is a quiver gauge

theory.

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What is the breakthrough provided by Gaiotto’s discussion?What is the breakthrough provided by Gaiotto’s discussion?

…

…

SU(N)

SU(N)

SU(N)

U(1)

……

…

…

…

…

SU(N)

U(1) U(1)

SU(N)U(1)

SU(N)

SU(N-1)

U(1)

SU(3)

U(1)

SU(2)

U(1)

…

…

interchange lengthen

…

…

SU(N)

SU(N)

U(1)SU(N)

TN

…

AGT relation

Action (Besides the 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 link

Nekrasov’s deformation parameters : background of graviphoton

1. The partition function of 1. The partition function of 4-dim gauge theory4-dim gauge theory

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(Sorry, they are different from Gaiotto’s ones!)

AGT relation reveals the concrete correspondence between partition function of 4-dim

SU(2) quiver gauge theory and correlation function of 2-dim Liouville theory.

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

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: 1-loop part can be written in terms of double Gamma function!

< Case of SU(N) x SU(N’) >

mass massmassVEV

deformation parameters

gauge antifund. bifund. fund.

We obtain it of the expansion form of instanton number :

where : coupling const. and

where

Instanton part Instanton part of partition function of 4-dim quiver gauge theoryof partition function of 4-dim quiver gauge theory

Young tableau

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+

< Case of instanton # = 1 >

(fractions of simple polynomials)

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

consider the correlation functions of them:

• In general, the following expansion is valid:

For the case of Virasoro algebra, , and e.g. for level-

2,

: Shapovalov matrix

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

propagator , and the intermediate states (i.e. descendant fields) can be

classified by Young tableaux.

Parameters (They correspond to parameters of 4-dim gauge theory!)

position of punctures

momentum of vertex operators for internal / external lines

central charge of the field theory

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2. The correlation function of 2. The correlation function of 2-dim field theory2-dim field theory

descendants

primaries

We obtain it of the factorization form of 3-point functions and propagators :

3-point function

where

propagator (2-point function) : inverse Shapovalov matrix15

Correlation functionCorrelation function of 2-dim conformal field theory of 2-dim conformal field theory

highest weight~ simple punc.

[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 momentum of internal lines

mass of matter fields momentum of external lines

1-loop part DOZZ factors

instanton part conformal blocks

deformation parameters 　 Liouville parameters

16

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

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

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

correlation function correspond each other :

central charge :

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

2-dim theory : SU(N) Toda field theory

• Similarly, we want to study on correspondence between partition function of

4-dim theory and correlation function of 2-dim theory :

• This discussion is somewhat complicated, since in these cases, punctures

are classified with more than one kinds of Young tableaux (which composed by

N boxes) :

< 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]

Natural expectation : SU(N>2) gauge theory Natural expectation : SU(N>2) gauge theory SU(N) Toda theory… !? SU(N) Toda theory… !?

……

… ……

…

…

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Multiple M5-branes’ worldvolume theory

SU(N) quiver gauge theory

SU(N) Toda field theory

Dijkgraaf-Vafa matrix model

Towards proof of AGT relation

6-dim :

4-dim :

2-dim :

0-dim :

<concrete calculations>Conformal blocks, Dotsenko-Fateev integral, Selberg integral, …

[Mironov-Morozov-Shakirov-… ’09, ’10]

Correspondence of worldvolume anomaly and central charge[Alday-Benini-Tachikawa ’09]

Contradiction? of compactification andcoupling constant…

~ ‘quantization’ of Seiberg-Witten curve?18

(or background physics)

Existence of Toda fields? : multiple M5-branes’ worldvolume anomaly

First, we remember how the anomaly is cancelled in the single M5-brane’s

case.For example, [Berman ’07] for a review.

worldvolume fields : bosons (5 d.o.f.) / fermions (8 d.o.f.) / self-dual 2-form field

(3 d.o.f.)

inflow mechanism (interaction term in the 11-dim supergravity action at 1-loop level

in l p) :

Chern-Simons interaction (which needs careful treatment because of presence of

M5-branes) :

Therefore, when we naively consider, in the case of (multiple) N M5-branes’

case,

x N

x N3

It is believed that this is an indication of some extra fields on M5-branes’

worldvolume :

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Cancellation doesn’t work!! (T_T)

• This story is related to AGT relation, if we compactify M5-branes’

worldvolume on 4-dim space X4. We define 2-dim anomaly by integrating I8

over X4:

• On the spacetime symmetry, we consider the following situation:

TW NW

• We twist R5 over X4 so that N=2 supersymmetry on X4 remains. In this case,

N=(0,2) supersymmetry with U(1) R-symmetry remains on . The general

form of anomaly is

• Especially, in the case of with Nekrasov’s deformation

,

This is precisely the same as central charge of Toda theory!

[Alday-Benini-Tachikawa ’09]

20

Existence of Toda fields? : multiple M5-branes’ worldvolume anomaly

F : external U(1) bundle coupling to U(1)R symmetry

(from AGT relation)

21

• We consider 4-dim and 2-dim system in type IIB string theory.

4-dim : Topological strings on Calabi-Yau 3-fold

2-dim : Seiberg-Witten curve embedded in Calabi-Yau 3-fold

• Dijkgraaf-Vafa matrix model may provide a bridge between them.

matrix model is powerful tool of description of topological B-model strings.

matrix model is also related to Liouville and Toda systems (, as we will see

concisely).

• Concretely, the partition function of 4-dim theory and the correlation function

of 2-dim theory may be connected via the partition function of matrix model :

where

,

[Dijkgraaf-Vafa ’09]

Toda theory is quantum theory of SW curve? : Dijkgraaf-Vafa matrix model

22

• It is known that the free fermion system ( ) can describe the system of

creation and annihilation of D-branes which are extended, for example, as

• To define this system, we ‘quantize’ Seiberg-Witten curve as , so the

following chiral path integral must be given naturally :

• On the other hand, it is known that x classically act on fermions as

• To sum up, in ‘quantum’ theory, x may be represented as

• This means that an additional term is given in chiral path integral :

When we bosonize the fermions, this additional term is nothing but the Toda

potential !

Toda theory is quantum theory of SW curve? : Dijkgraaf-Vafa matrix model

• In the previous section, we saw some evidence(?) that Toda fields live on

Seiberg-Witten curve or multiple M5-branes’ worldvolume.

• Now let us return the discussion on generalization of AGT relation. To do

this, we need to consider…

momentum of Toda fields in vertex operators :

Again, in SU(N>2) case, we need to determine the form of vertex operators which

corresponds to each kind of punctures (classified with Young tableaux).

how to calculate the conformal blocks of W-algebra: 3pt functions and

propagators

correspondence between parameters of SU(N) quiver gauge theory and

those of SU(N) Toda field theory

Towards generalized AGT relation

23

• 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 SU(N) Toda field theory? : some extension of Liouville field theoryWhat is SU(N) Toda field theory? : some extension of Liouville field theory

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For simplicity, we ignore

Toda potential (interaction)

at this present stage.

• The primary fields are defined as 　　　　　　　 , so 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, (special) linear combinations of descendant fields accidentally

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

example, the null states in level-1 descendant fields are

• As we will see next, we found the fact that this null state in W-algebra is

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 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.

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• 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. definition of W-

generators)

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

puncture’s case

This correspondence becomes exact, when we take some ‘classical’ limit :

(which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?)

• This fact strongly suggest that vertex operators corresponding non-full-type

punctures must be the primary fields which has null states in their descendant

fields.

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

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• 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 define the screening operator (a special type of vertex operator)

• We can easily show that the state satisfies the highest

weight condition, since the screening operator commutes with all the W-

generators.

(Note that 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!

27

[Kanno-Matsuo-SS-Tachikawa ’09]

• Therefore, when we write the simple root as (as

usual), 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 of Liouville case

.

Here, is the same form of β,

is Weyl vector, and .

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

Case of SU(3) quiver gauge theory

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

SU(3) x … x SU(3) : We checked 1-loop part, and now calculate instanton

part.

SU(3) x SU(2) : We check it now, but correspondence seems very

complicated!

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]

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

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Conclusion

It is well known that Seiberg-Witten system can be regarded as the multiple

M5-branes’ system. This system is composed by intersecting M5-branes, and

can be described by (direct sum? of) 4-dim quiver gauge theory and 2-dim

conformal field theory on Seiberg-Witten curve.

Recently, it was strongly suggested that the partition function of 4-dim theory

and the correlation function of 2-dim theory closely correspond to each other.

In particular, this correspondence requires that Toda (or Liouville) field should

live in 2-dim theory on Seiberg-Witten curve.

We showed that the singular behavior of SW curve near punctures corresponds

to the composition of null states in W-algebra. Also, we conjectured the

momentum of vertex operators corresponding each kind of punctures.

Again, we expect that this subject brings us new understanding on M5-branes!

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