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

                                      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 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.

 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 !

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


                                                                   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.

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

   flavor brane          color brane          flavor brane
                                                                                      6, 10
  (length = infinite) (length ~ 1/coupling)

                                                             more generally…

              antifund. gauge bifund.         fund.
                                                             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                                                           .
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

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)

        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]

                                                                   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 :


                                                               …       …

             …       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.

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

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)

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:

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

 3-point function


        highest weight
        ~ simple punc.

 propagator (2-point function) : inverse Shapovalov matrix
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 :
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            .)
     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?
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,

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

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

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!
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 :


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 !

        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

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


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          .
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                               .

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]

   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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