1. AGT 関係式 (3) 一般化に向け て (String Advanced Lectures No.20) 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 2010 年 6 月 23 日（水） 12:30-14:30

2. Contents 1. AGT relation for SU(2) quiver theory 2. Partition function of SU(N) quiver theory 3. Toda theory and W-algebra 4. Generalized AGT relation for SU(N) case 5. Towards AdS/CFT duality of AGT relation

3. AGT relation for SU(2) quiver We now consider only the linear quiver gauge theories in AGT relation. Gaiotto’s discussion

4. An example : SW curve is a sphere with multiple punctures. The Seiberg-Witten curve in this case corresponds to 4-dim N=2 linear quiver SU(2) gauge theory.  Nekrasov instanton partition function where equals to the conformal block of Virasoro algebra with for the vertex operators which are inserted at z=  Liouville correlation function (corresponding n+3-point function) where is Nekrasov’s full partition function. ( ↑ including 1-loop part) U(1) part

5. [Alday-Gaiotto-Tachikawa ’09] AGT relation : SU(2) gauge theory  Liouville theory ! Gauge theoryLiouville theory coupling const. position of punctures VEV of gauge fields internal momenta mass of matter fields external momenta 1-loop partDOZZ factors instanton part conformal blocks deformation parameters Liouville parameters  4-dim theory : SU(2) quiver gauge theory  2-dim theory : Liouville (A 1 Toda) field theory In this case, the 4-dim theory’s partition function Z and the 2-dim theory’s correlation function correspond to each other : central charge :

6. Now we calculate Nekrasov’s partition function of 4-dim SU(N) quiver gauge theory as the quantity of interest.  SU(2) case : We consider only SU(2)×…×SU(2) quiver gauge theories.  SU(N) case : According to Gaiotto’s discussion, we consider, in general, the cases of SU(d 1 ) x SU(d 2 ) x … x SU(N) x … x SU(N) x … x SU(d’ 2 ) x SU(d’ 1 ) group, where is non-negative. SU(N) partition function Nekrasov’s partition function of 4-dim gauge theory x xx x x * … … x * … … … d’ 3 – d’ 2 d’ 2 – d’ 1 d’ 1 …… … … … d3 – d2d2 – d1d1d3 – d2d2 – d1d1 …… … …

7. 1-loop part of partition function of 4-dim quiver gauge theory We can obtain it of the analytic form : where each factor is defined as : each factor is a product of double Gamma function!, gauge antifund. bifund.fund. mass flavor symm. of bifund. is U(1) VEV # of d.o.f. depends on d k deformation parameters

8. We obtain it of the expansion form of instanton number : where : coupling const. and and Instanton part of partition function of 4-dim quiver gauge theory Young tableau instanton # = # of boxes leg arm

9.  Naive assumption is 2-dim A N-1 Toda theory, since Liouville theory is nothing but A 1 Toda theory. This means that the generalized AGT relation seems  Difference from SU(2) case… VEV’s in 4-dim theory and momenta in 2-dim theory have more than one d.o.f. In fact, the latter corresponds to the fact that the punctures are classified with more than one kinds of N-box Young tableaux : (cf. In SU(2) case, all these Young tableaux become ones of the same type.) In general, we don’t know how to calculate the conformal blocks of Toda theory. … … … … … … … What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory?

10.  Action : Toda field with : It parametrizes the Cartan subspace of A N-1 algebra. simple root of A N-1 algebra : Weyl vector of A N-1 algebra : metric and Ricci scalar of 2-dim surface interaction parameters : b (real) and central charge : Toda theory and W-algebra What is A N-1 Toda theory? : some extension of Liouville theory

11. 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 A N-1 Toda field theory? (continued) We ignore Toda potential (interaction) at this stage.

12. The primary fields are defined as ( is called ‘momentum’). 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, some linear combinations of descendant fields accidentally satisfy the highest weight condition. They are called null states. For example, the null states in level-1 descendants are As we will see next, we found the fact that these null states in W-algebra are 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 in fact describe the form (or behavior) of Seiberg-Witten curve. As usual, we compose the primary, descendant, and null fields.

13. 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. W-generators) Also, the coefficients satisfy similar equations, except full-type puncture’s case This correspondence becomes exact, in some kind of ‘classical’ limit: (which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?) This fact strongly suggests that vertex operators corresponding non-full-type punctures must be the primary fields which has null states in their descendants. 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

14. 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 consider the screening operator (a special type of vertex operator) We can show that the state satisfies the highest weight condition, since the screening operator commutes with all the W-generators. (Note a strange form of a ket, since 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! [Kanno-Matsuo-SS-Tachikawa ’09]

15. Therefore, 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: Here, is the same form of β, is Weyl vector, and. The punctures on SW curve corresponds to the ‘degenerate’ fields!

16. Generalized AGT relation  Natural form : former’s partition function and latter’s correlation function  Problems and solutions for its proof correspondence between each kind of punctures and vertices: we can conjecture it, using level-1 null state condition. difficulty for calculation of conformal blocks: null state condition resolves it again! [Wyllard ’09] [Kanno-Matsuo-SS-Tachikawa ’09] … … … … … … … Correspondence : 4-dim SU(N) quiver gauge and 2-dim A N-1 Toda theory

17. We put the (primary) vertex operators at punctures, and consider the correlation functions of them: In general, the following expansion is valid: where and for level-1 descendants, : Shapovalov matrix It means that all correlation functions consist of 3-point functions and inverse Shapovalov matrices (propagator), where the intermediate states (descendants) can be classified by Young tableaux. On calculation of correlation functions of 2-dim A N-1 Toda theory descendants primaries

18. In fact, we can obtain it of the factorization form of 3-point functions and inverse Shapovalov matrices :  3-point function : We can obtain it, if one entry has a null state in level-1! where highest weight ~ simple punc. On calculation of correlation functions of 2-dim A N-1 Toda theory ’

19.  Case of SU(3) quiver gauge theory SU(3) : already checked successfully. [Wyllard ’09] [Mironov-Morozov ’09] SU(3) x … x SU(3) : We have checked successfully. [Kanno-Matsuo-SS ’10] SU(3) x SU(2) : We could check it, but only for restricted cases. [Kanno-Matsuo-SS ’10]  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] → subject of next talk… Our plans of current and future research on generalized AGT relation

20. Towards AdS/CFT of AGT  CFT side : 4-dim SU(N) quiver gauge theory and 2-dim A N-1 Toda theory 4-dim theory is conformal. The system preserves eight supersymmetries.  AdS side : the system with AdS 5 and S 2 factor and eight supersymmetries This is nothing but the analytic continuation of LLM’s system in M-theory. Moreover, when we concentrate on the neighborhood of punctures on Seiberg-Witten curve, the system gets the additional S 1 ~ U(1) symmetry. According to LLM’s discussion, such system can be analyzed using 3-dim electricity system: [Lin-Lunin-Maldacena ’04] [Gaiotto-Maldacena ’09]