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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

                           Contradiction? of
                           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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posted:5/29/2011
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