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

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.

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

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  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. 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 freedomwhich the multiple branes’ systems must always have. • D-branes’ case : internal d.o.f ~ N2 • The superstringsending 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.

  3. M2-branes’ case : internal d.o.f. ~ N3/2 • The proposition of BLG model is the important breakthrough. [Bagger-Lambert ’07] [Gustavsson ’07] • We can derive the internal d.o.f. of order N3/2 naturally and successfully, using the finite representation of Lie 3-algebra which is the gauge symmetry algebra of BLG model. • [Chu-Ho-Matsuo-SS ’08] • However, at this moment, we don’t know at all what compose these d.o.f. • M5-branes’ case : internal d.o.f.~ N3 • Based on the recent research of AGT relation and its generalization, not a few researchers now hope that[Alday-Gaiotto-Tachikawa ’09] [Wyllard ’09] etc. • Toda fields on 2-dim Riemann surface (or Seiberg-Witten curve[Seiberg-Witten ’94]) • W-algebra which is the symmetry algebra of Toda field theory • bring us some new understanding on the multiple M5-branes’ internal d.o.f ! The near horizon geometry of M-branes is AdSx S, so we can use AdS/CFT discussion. Then this internal d.o.f. corresponds to the entropy of AdS blackhole. (~ area of horizon) Subject of today’s seminar

  4. 0,1,2,3 4,5 6,10 ? 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.) • 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. In this time, M5-branes keep only ½ x ½ SUSYs. This isSeiberg-Witten system.[Seiberg-Witten ’94] bundle of M5-branes

  5. 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. • 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 . 4,5 D4-brane (M5-brane) [Seiberg-Witten ’94] flavor brane (length = infinite) color brane (length ~ 1/coupling) flavor brane 6, 10 D6-brane 7,8,9 more generally… antifund. gauge bifund. fund. F1-string gluons / quarks (from Hanany-Witten’s discussion) NS5-brane (M5-brane) increasing increasing 5

  6. 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 • Note that • The coefficient of yN is 1. : normalization which causes the divergence of ! • The yN-1 term doesn’t exist. : suitable shift of coordinates ~ 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)

  8. 4,5 10 (compactified) … … 6 … … … 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 … … … d’3–d’2 d’2–d’1 d’1 d3–d2 d2–d1 d1 … … … (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)

  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 compactificationof multiple M5-branes’ system. For example, famous(?) TN theory. … … S-dual … … … … … … … … SU(N) SU(N) SU(N)

  10. SU(N) SU(N) SU(N) U(1) What is the breakthrough provided by Gaiotto’s discussion? • TN theory is obtained as S-dual of SU(N) quiver gauge theory, as follows : • In other words, • However, in the following, we concentrate on the systems of brane configuration, i.e. the cases where 4-dim theory is a quiver gauge theory. TN interchange lengthen … … … … … … … … … … SU(N) SU(N) … SU(N) SU(N) SU(N) SU(N) U(1) U(1) U(1) U(1) … U(1) … SU(N-1) SU(3) SU(2) U(1) U(1)

  11. 1. The partition function of 4-dim 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. • Action (Besides the classical part…) • 1-loop correction : more than 1-loop is cancelled, because of N=2 supersymmetry. • instanton correction : Nekrasov’s calculation with Young tableaux • Parameters • coupling constants • masses of fundamental/antifund./bifund. fields and VEV’s of gauge fields link • Nekrasov’s deformation parameters : background of graviphoton (Sorry, they are different from Gaiotto’s ones!)

  12. gauge bifund. fund. antifund. VEV mass mass mass deformation parameters 1-loop part of partition function of 4-dim quiver gauge theory • We can obtain it of the analytic form : • where < Case of SU(N)x SU(N’)> : 1-loop part can be written in terms of double Gamma function! 12

  13. < Case of instanton #=1 > + 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 where Young tableau (fractions of simple polynomials) 13

  14. primaries descendants 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

  15. 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 • propagator (2-point function) : inverse Shapovalov matrix highest weight ~ simple punc. 15

  16. 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 : central charge :

  17. Natural expectation : SU(N>2) gauge theory  SU(N) Toda theory… !? • 4-dim theory : SU(N) quiver gauge theory • 2-dim theory : SU(N) Toda field theory • Similarly, we want to study on correspondence between partition function of 4-dim theory and correlation function of 2-dim theory : • This discussion is somewhat complicated, since in these cases, punctures are classified with more than one kinds of Young tableaux (which composed by N boxes) : < full-type > < simple-type > < other types > (cf. In SU(2) case, all these Young tableaux become ones of the same type .) [Wyllard ’09] [Kanno-Matsuo-SS-Tachikawa ’09] … … … … … … …

  18. Towards proof of AGT relation (or background physics) 6-dim : Multiple M5-branes’ worldvolume theory Contradiction? of compactification and coupling constant… Correspondence of worldvolume anomaly and central charge [Alday-Benini-Tachikawa ’09] 4-dim : SU(N) quiver gauge theory 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

  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 lp) : • Chern-Simons interaction (which needs careful treatment because of presence of M5-branes) : • Therefore, when we naively consider, in the case of (multiple) N M5-branes’ case, • x N • x N3 • It is believed that this is an indication of some extra fields on M5-branes’ worldvolume : Cancellation doesn’t work!! (T_T) 19

  20. 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 space X4. We define 2-dim anomaly by integrating I8 over X4: • On the spacetime symmetry, we consider the following situation: • TW NW • We twist R5 over X4 so that N=2 supersymmetry on X4 remains. In this case, N=(0,2) supersymmetry with U(1) R-symmetry remains on . The general form of anomaly is • Especially, in the case of with Nekrasov’s deformation • , • This is precisely the same as central charge of Toda theory! [Alday-Benini-Tachikawa ’09] F : external U(1) bundle coupling to U(1)R symmetry (from AGT relation) 20

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

  22. 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

  23. 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

  24. 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: • which can be regarded as the extension of Virasoro algebra, and where • , For simplicity, we ignore Toda potential (interaction) at this present stage.

  25. 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.

  26. The singular behavior of SW curve is related to the null fields of W-algebra. • As we saw, Seiberg-Witten curve is generally represented as • and Laurent expansion near z=z0 of the coefficient function is generally • This form is similar to Laurent expansion of W-current (i.e. definition of W-generators) • Also, the coefficients satisfy the similar equation, except the full-type puncture’s case • This correspondence becomes exact, when we take some ‘classical’ limit : • (which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?) • This fact strongly suggest that vertex operators corresponding non-full-type punctures must be the primary fields which has null states in their descendant fields. [Kanno-Matsuo-SS-Tachikawa ’09] ~ direction of D4~ direction of NS5 null condition 26

  27. The punctures on SW curve corresponds to the ‘degenerate’ fields! • 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 . [Kanno-Matsuo-SS-Tachikawa ’09]

  28. 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 .

  29. 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 W4 algebra, but is mostly streightforward. • Case of SU(∞) quiver gauge theory • In this case, we consider the system of infinitely many M5-branes, which may relate to AdS dual system of 11-dim supergravity. • AdS dual system is already discussed using LLM’s droplet ansatz, which is also governed by Toda equation. [Gaiotto-Maldacena ’09] 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 requiresthat 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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