1. 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.

2. 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 ~ N 2 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

3.  M2-branes’ case : internal d.o.f. ~ N 3/2 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.  M5-branes’ case : internal d.o.f. ~ N 3 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

4. Intersecting M5-branes’ system makes 4-dim spacetime and 2-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. ?

5. Seiberg-Witten curve determines the field contents of 4-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(d 1 ) x SU(d 2 ) 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 brane D6-brane 4,5 6, 10 7,8,9 [Seiberg-Witten ’ 94] F1-string (from Hanany-Witten’s discussion) more generally… 5 increasing antifund.gaugebifund.fund.

6. 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. ~ direction of D4 ~ direction of NS5

7. 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

8. 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 2-dim conformal field theory. 0 ∞ deformation to 2-dim sphere multiple D4-branes NS5-branes 4,5 6 10 (compactified) For gauge group : SU(d 1 ) x SU(d 2 ) 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 – d2d2 – d1d1d3 – d2d2 – d1d1 … d’ 3 – d’ 2 d’ 2 – d’ 1 d’ 1 …… … … …… … … … … 0∞ Gaiotto’s discussion 8

9. 9 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(?) T N theory. … … …… S-dual … … SU(N) …………

10. T N 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. 10 What is the breakthrough provided by Gaiotto’s discussion? … … SU(N) U(1) … … … … … … SU(N) U(1) SU(N) U(1) SU(N) SU(N-1) U(1) SU(3) U(1) SU(2) U(1) … … interchangelengthen … … SU(N) U(1) SU(N) TNTN …

11. 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 link Nekrasov’s deformation parameters : background of graviphoton 1. The partition function of 4-dim gauge theory 11 (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.

12. 1-loop part of partition function of 4-dim quiver gauge theory We can obtain it of the analytic form : where 12 : 1-loop part can be written in terms of double Gamma function! mass VEV deformation parameters gauge antifund. bifund.fund.

13. We obtain it of the expansion form of instanton number : where : coupling const. and where Instanton part of partition function of 4-dim quiver gauge theory Young tableau 13 + (fractions of simple polynomials)

14. 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 14 2. The correlation function of 2-dim field theory descendants primaries

15. We obtain it of the factorization form of 3-point functions and propagators :  3-point function where  propagator (2-point function) : inverse Shapovalov matrix 15 Correlation function of 2-dim conformal field theory highest weight ~ simple punc.

16. [Alday-Gaiotto-Tachikawa ’ 09] AGT relation : SU(2) gauge theory  Liouville theory ! Gauge theoryLiouville theory coupling const. position of punctures VEV of gauge fields momentum of internal lines mass of matter fields momentum of external lines 1-loop partDOZZ factors instanton partconformal 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 :

17.  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) : (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  SU(N) Toda theory… !? … … … … … … … 17

18. 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 : 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 and coupling constant… ~ ‘quantization’ of Seiberg-Witten curve? 18 (or background physics)

19. 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 N 3 It is believed that this is an indication of some extra fields on M5-branes’ worldvolume : 19 Cancellation doesn’t work!! (T_T)

20. This story is related to AGT relation, if we compactify M5-branes’ worldvolume on 4-dim space X 4. We define 2-dim anomaly by integrating I 8 over X 4 : On the spacetime symmetry, we consider the following situation: TW NW We twist R 5 over X 4 so that N=2 supersymmetry on X 4 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. 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. 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

23. 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…return 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

24. 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 theory 24 For simplicity, we ignore Toda potential (interaction) at this present stage.

25. 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. 25

26. As we saw, Seiberg-Witten curve is generally represented as and Laurent expansion near z=z 0 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. [Kanno-Matsuo-SS-Tachikawa ’ 09] null condition ~ direction of D4 ~ direction of NS5 26

27. 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! 27 [Kanno-Matsuo-SS-Tachikawa ’ 09]

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

29.  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] Our plans of current and future research on generalized AGT relation 29

30. 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