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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