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# Shiba check gauge

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```									    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.
Introduction
What is the multiple M-branes’ system like? (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.

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 N 3/2 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.

The near horizon geometry of M-branes is
Subject of today’s seminar                       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)
 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 !

3
Intersecting M5-branes’ system makes 4-dim spacetime and 2-dim surface.
• 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.)

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

• 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.

This is Seiberg-Witten system. [Seiberg-Witten ’94]      bundle of M5-branes

0,1,2,3
?
4,5

6,10                      4
Seiberg-Witten curve determines the field contents of 4-dim gauge theory.
• 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.

4,5
D4-brane (M5-brane)                                                               [Seiberg-Witten ’94]

flavor brane          color brane          flavor brane
6, 10
(length = infinite) (length ~ 1/coupling)
7,8,9
D6-brane

more generally…

antifund. gauge bifund.         fund.
F1-string
gluons / quarks
(from Hanany-Witten’s discussion)
NS5-brane (M5-brane)
increasing                       increasing

• 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                                                           .
5
A kind of ‘deformations’ makes clear the structure of Seiberg-Witten curve.
• 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

~ direction of D4   ~ direction of NS5

Note that
 The coefficient of y N is 1. : normalization which causes the divergence of              !
 The yN-1 term doesn’t exist. : suitable shift of coordinates
6
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
Gaiotto’s discussion
Seiberg-Witten curve may be described by 2-dim conformal field theory.
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]

NS5-branes
0                         ∞

0                                   ∞        deformation to
2-dim sphere

multiple D4-branes
…

…
4,5                                                      …                                  …

…
d3 – d2                            d’3 – d’2
…            d2 – d1               …            d’2 – d’1
10 (compactified)                                  …    …       d1                    … …          d’1

6                                               (All Young tableaux are composed by N boxes.)

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)                 8
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 :

S-dual

…       …

…       SU(N)     …                         …    SU(N)   SU(N)     …

• 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.

9
What is the breakthrough provided by Gaiotto’s discussion?
• TN theory is obtained as S-dual of SU(N) quiver gauge theory, as follows :
TN
interchange                                        lengthen   …         …
…                    …                         …
…                                              …

…                                                    …
…

In other words,                                                                                                  …

SU(N)                                                                 SU(N)
SU(N)          SU(N)      …          SU(N)          SU(N)
U(1)                                                                 U(1)
U(1)                                 U(1)

SU(N)                                                               U(1)
SU(N)          SU(N-1)     …   SU(3)          SU(2)
SU(N)                                                               U(1)
U(1)                           U(1)

• 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.
10
AGT relation
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. The partition function of 4-dim gauge theory

 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
(Sorry, they are different from Gaiotto’s ones!)
 Parameters
 coupling constants
 masses of fundamental / antifund. / bifund. fields and VEV’s of gauge fields     link

 Nekrasov’s deformation parameters : background of graviphoton
11
1-loop part of partition function of 4-dim quiver gauge theory

We can obtain it of the analytic form :

gauge         antifund.           bifund.               fund.

VEV             mass                    mass             mass

where                                 deformation parameters

< Case of SU(N) x SU(N’) >

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

12
Instanton part of partition function of 4-dim quiver gauge theory
We obtain it of the expansion form of instanton number :

where                           : coupling const. and       Young tableau

< Case of instanton # = 1 >

+

(fractions of simple polynomials)
where

13
2. The correlation function of 2-dim field theory
• We put the (primary) vertex operators                at punctures, and consider the
correlation functions of them:

• In general, the following expansion is valid:
primaries

descendants
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
14
Correlation function of 2-dim conformal field theory
We obtain it of the factorization form of 3-point functions and propagators :

 3-point function

where

highest weight
~ simple punc.

 propagator (2-point function) : inverse Shapovalov matrix
15
AGT relation : SU(2) gauge theory  Liouville theory !                  [Alday-Gaiotto-Tachikawa ’09]

 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 :

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
central charge :
16
Natural expectation : SU(N>2) gauge theory  SU(N) Toda theory… !?
 4-dim theory : SU(N) quiver gauge theory                   [Wyllard ’09]
[Kanno-Matsuo-SS-Tachikawa ’09]
 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            .)
17
Towards proof of AGT relation
(or background physics)

6-dim :             Multiple M5-branes’ worldvolume theory

compactification and             Correspondence of
coupling constant…               worldvolume anomaly
and central charge
4-dim :   SU(N) quiver gauge theory                         [Alday-Benini-Tachikawa ’09]

2-dim :                                           SU(N) Toda field theory
<concrete calculations>
Conformal blocks, Dotsenko-Fateev
integral, Selberg integral, …
[Mironov-Morozov-Shakirov-… ’09, ’10]

0-dim :                  Dijkgraaf-Vafa matrix model
~ ‘quantization’ of Seiberg-Witten curve?
18
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,

xN
Cancellation doesn’t work!! (T_T)
x   N3

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

19
Existence of Toda fields? : multiple M5-branes’ worldvolume anomaly
• This story is related to AGT relation, if we compactify M5-branes’ worldvolume on 4-dim
[Alday-Benini-Tachikawa ’09]
space X4. We define 2-dim anomaly by integrating I 8 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

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

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

,
(from AGT relation)
This is precisely the same as central charge of Toda theory!
20
Toda theory is quantum theory of SW curve? : Dijkgraaf-Vafa matrix model
[Dijkgraaf-Vafa ’09]
• 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
,

21
Toda theory is quantum theory of SW curve? : Dijkgraaf-Vafa matrix model
• 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 !

22
Towards generalized AGT relation
• 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

23
What is SU(N) Toda field theory? : some extension of Liouville field 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:                     For simplicity, we ignore
Toda potential (interaction)
at this present stage.

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

,

24
As usual, we compose the primary, descendant, and null fields.

• 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.                                                                           25
The singular behavior of SW curve is related to the null fields of W-algebra.
[Kanno-Matsuo-SS-Tachikawa ’09]
• As we saw, Seiberg-Witten curve is generally represented as

~ direction of D4   ~ direction of NS5

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

null condition
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.                                     26
The punctures on SW curve corresponds to the ‘degenerate’ fields!
• If we believe this suggestion, we can conjecture the form of             [Kanno-Matsuo-SS-Tachikawa ’09]

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          .
27
The punctures on SW curve corresponds to the ‘degenerate’ fields!
• 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                               .

28
Our plans of current and future research on generalized AGT relation
 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 W 4 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]

29
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!
30

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