1. AGT 関係式とその一般化に向け て (String Advanced Lectures No.22) 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 7 月 5 日（月） 14:00-15:40

2. Contents 1. Gaiotto’s discussion 2. AGT relation for SU(2) quiver theories 3. AGT-W relation for SU(N) quiver theories 4. AdS/CFT correspondence of AGT’s system

3. Gaiotto’s discussion

4. Seiberg-Witten curve  Low energy effective action (by Wilson’s renormalization : integration out of massive fields )  prepotential  potential for scalar field 4-dim N=2 SU(2) supersymmetric gauge theory [Seiberg-Witten ’94] classical 1-loop instanton : energy scale : Higgs potential (which breaks gauge symmetry) This breakdown is parametrized by

5. u (VEV) : shift of color brane mass : shift of flavor brane Singular points of prepotential, Seiberg-Witten curve and S-duality  The singular points of prepotential on u-plane By studying the monodromy of and, we can find that the prepotential has singular points. This can be described as These singular points means the emergence of new massless fields. This means that the prepotential must become a different form near a different singular point. ( S-duality)  M-theory interpretation : singular points are intersection points of M5-branes. (or D4/NS5-branes) [Witten ’97] : Seiberg-Witten curve in coupling

6. SU(2) generalized quivers [Gaiotto ’09] SU(2) gauge theory with 4 fundamental flavors (hypermultiplets)  S-duality group SL(2,Z) coupling const. : flavor sym. : SO(8) ⊃ SO(4)×SO(4) ～ [SU(2) a ×SU(2) b ]×[SU(2) c ×SU(2) d ] : (elementary) quark : monopole : dyon D4 NS5

7.  Subgroup of S-duality without permutation of masses In massive case, we especially consider this subgroup. mass : mass parameters can be associated to each SU(2) flavor. Then the mass eigenvalues of four hypermultiplets in 8 v is,. coupling : cross ratio (moduli) of the four punctures, i.e. z = Actually, this is equal to the exponential of the UV coupling → This is an aspect of correspondence between the 4-dim N=2 SU(2) gauge theory and the 2-dim Riemann surface with punctures. SU(2) gauge theory with massive fundamental hypermultiplets

8. SU(2) 1 ×SU(2) 2 gauge theory with fundamental and bifundamental flavors When each gauge group is coupled to 4 flavors, this theory is conformal. flavor symmetry ⊃ [SU(2) a ×SU(2) b ]×SU(2) e ×[SU(2) c ×SU(2) d ] flavor sym. of bifundamental hyper. : Sp(1) ～ SU(2) i.e. real representation S-duality subgroup without permutation of masses When the gauge coupling of SU(2) 2 vanishes or is very weak, we can discuss it in the same way as before for SU(2) 1. The similar discussion goes for (1 2). That is, this subgroup consists of the permutation of five SU(2)’s. cf. Note that two SL(2,Z) full S-duality groups do not commute! Here, we analyze only the boundary of the gauge coupling moduli space.

9. SU(2) 1 ×SU(2) 2 ×SU(2) 3 gauge theory with fund. and bifund. flavors (The similar discussion goes.) ■, ■ : weak : interchange

10. turn on/off a gauge coupling For more generalized SU(2) quivers : more gauge groups, loops…

11. Seiberg-Witten curve for quiver SU(2) gauge theories  massless SU(2) case In this case, the Seiberg-Witten curve is of the form If we change the variable as, this becomes  massless SU(2) n case or  mass deformation The number of mass parameters is n+3, because of the freedom. where are the solutions of VEV coupling polynomial of z of (n-1)-th order divergent at punctures

12. SU(3) generalized quivers SU(3) gauge theory with 6 fundamental flavors (hypermultiplets) This theory is also conformal. flavor symmetry U(6) : complex rep. of SU(3) gauge group kind of S-duality group : Argyres-Seiberg duality [Argyres-Seiberg ’07] coupling const. : flavor : U(6) ⊃ [SU(3)×U(1)]×[SU(3)×U(1)] : weak coupling U(6) ⊃ SU(6)×U(1) ～ [SU(3)×SU(3)×U(1)]×U(1) SU(6)×SU(2) ⊂ E 6 : infinite coupling of SU(3) theory Moreover, weakly coupled gauge group becomes SU(2) instead of SU(3) ! breakdown by VEV

13. Argyres-Seiberg duality for SU(3) gauge theory infinite coupling D4 NS5

14. SU(3) 1 ×SU(3) 2 gauge theory with fundamental and bifundamental flavors flavor symmetry of bifundamental Argyres-Seiberg duality

15. For more generalized SU(3) quivers : more gauge groups, loops… turn on/off a gauge coupling

16. Seiberg-Witten curve for SU(3) quiver gauge theories  massless SU(3) n case  massless SU(2)×SU(3) n-2 ×SU(2) case  mass deformation massless : massive : The number of mass parameters is n+3, because of the freedom. In both cases, SW curve can be rewritten as ( ), but the order of divergence of is different from each other.

17. SU(N) generalized quivers Seiberg-Witten curve in this case is of the form The variety of quiver gauge group where is reflected in the various order of divergence of at punctures. For example… Seiberg-Witten curve for massless SU(N) quiver gauge theories

18.  SU(2) quiver case order of divergence : mass parameters : flavor symmetry : SU(2)  SU(3) quiver case order of divergence : mass parameters : flavor symmetry : U(1) SU(3) Classification of punctures : divergence of massless SW curve at punctures

19.  SU(3) quiver case corresponding puncture :  SU(4) quiver case (and the natural analogy is valid for general SU(N) case) Classification of punctures : divergence of massless SW curve at punctures

20. AGT relation for SU(2) quivers

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

22. Nekrasov’s partition function of 4-dim gauge theory  Action classical part 1-loop correction : more than 1-loop is cancelled, because of N=2 SUSY. instanton correction : Nekrasov’s calculation with Young tableaux  Parameters coupling constants masses of fundamental / antifund. / bifund. fields and VEV’s of gauge fields deformation parameters : background of graviphoton or deformation (rotation) of extra dimensions (Note that they are different from Gaiotto’s ones!) Now we calculate Nekrasov’s partition function of 4-dim SU(2) quiver gauge theory as the quantity of interest. D4 NS5

23. 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 VEV deformation parameters

24. 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

25. The Nekrasov partition function for the simple case of SU(2) with four flavors is Since the mass dimension of is 1, so we fix the scale as,. (by definition)  Mass parameters : mass eigenvalues of four hypermultiplets : mass parameters of  VEV’s : we set --- decoupling of U(1) (i.e. trace) part. We must also eliminate the contribution from U(1) gauge multiplet. This makes the flavor symmetry SU(2) i ×U(1) i enhanced to SU(2) i ×SU(2) i. (next page…) SU(2) with four flavors : Calculation of Nekrasov function for U(2) U(2), actually Manifest flavor symmetry is now U(2) 0 ×U(2) 1, while actual symmetry is SO(8) ⊃ [SU(2)×SU(2)]×[SU(2)×SU(2)].

26. In this case, Nekrasov partition function can be written as where and  is invariant under the flip (complex conjugate representation) : which can be regarded as the action of Weyl group of SU(2) gauge symmetry.  is not invariant. This part can be regarded as U(1) contribution.  Surprising discovery by Alday-Gaiotto-Tachikawa In fact, is nothing but the conformal block of Virasoro algebra with for four operators of dimensions inserted at : SU(2) with four flavors : Identification of SU(2) part and U(1) part (intermediate state)

27. Correlation function of Liouville theory with. Thus, we naturally choose the primary vertex operator as the examples of such operators. Then the 4-point function on a sphere is 3-point function conformal block where  The point is that we can make it of the form of square of absolute value! … only if … using the properties : and Liouville correlation function

28. As a result, the 4-point correlation function can be rewritten as where and  It says that the 3-point function (DOZZ factor) part also can be written as the product of 1-loop part of 4-dim SU(2) partition function : under the natural identification of mass parameters : Example 1 : SU(2) with four flavors (Sphere with four punctures)

29. Example 2 : Torus with one puncture The SW curve in this case corresponds to 4-dim N=2* theory : N=4 SU(2) theory deformed by a mass for the adjoint hypermultiplet  Nekrasov instanton partition function This can be written as where equals to the conformal block of Virasoro algebra with  Liouville correlation function (corresponding 1-point function) where is Nekrasov’s partition function.

30. Example 3 : 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

31. [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 :

32. According to Gaiotto’s discussion, SW curve for SU(2) case is. In massive cases, has double poles. Then the mass parameters can be obtained as, where is a small circle around the a-th puncture. The other moduli can be fixed by the special coordinates, where is the i-th cycle (i.e. long tube at weak coupling). Note that the number of these moduli is 3g-3+n. (g : # of genus, n : # of punctures) SW curve and AGT relation Seiberg-Witten curve and its moduli

33. The Seiberg-Witten curve is supposed to emerge from Nekrasov partition function in the “semiclassical limit”, so in this limit, we expect that. In fact, is satisfied on a sphere, then has double poles at z i. For mass parameters, we have, where we use and. For special coordinate moduli, we have, which can be checked by order by order calculation in concrete examples. Therefore, it is natural to speculate that Seiberg-Witten curve is ‘quantized’ to at finite. 2-dim CFT in AGT relation : ‘quantization’ of Seiberg-Witten curve??

34. AGT-W relation for SU(N)

35. 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 …… … …

36. 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

37. 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

38.  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?

39.  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

40. 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.

41. 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.

42. 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

43. 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]

44. 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!

45. 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

46. 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 ’

47.  Now we are interested in the Nekrasov’s partition function of 4-dim SU(N) quiver gauge theory.  It seems natural that generalized AGT relation (or AGT-W relation) clarifies the correspondence between Nekrasov’s function and some correlation function of 2-dim A N-1 Toda theory:  Main difference from SU(2) case: Not all flavor symmetries are SU(N), e.g. bifundamental flavor symmetry. Therefore, we need the condition which restricts the d.o.f. of momentum β in Toda vertex which corresponds to each (kind of) puncture. → level-1 null state condition [Wyllard ’09] [Kanno-Matsuo-SS-Tachikawa ’09] N-1 Cartans SU(N) U(1) SU(N) U(1) SU(N) U(1) SU(N) … N-1 d.o.f. AGT relation : 4-dim SU(N) quiver gauge and 2-dim A N-1 Toda theory

48.  Correspondence between each kind of punctures and vertices : we conjectured it, using level-1 null state condition for non-full-type punctures. full-type : correponds to SU(N) flavor symmetry (N-1 d.o.f.) simple-type : corresponds to U(1) flavor symmetry (1 d.o.f.) other types : corresponds to other flavor symmetry The corresponding momentum is of the form which naturally corresponds to Young tableaux. More precisely, the momentum is, where [Kanno-Matsuo-SS-Tachikawa ’09] … … … … … … … Level-1 null state condition resolves the problems of AGT-W relation.

49.  Difficulty for calculation of conformal blocks : Here we consider the case of A 2 Toda theory and W 3 -algebra. In usual, the conformal blocks are written as the linear combination of which cannot be determined by recursion formula. However, in this case, thanks to the level-1 null state condition we can completely determine all the conformal blocks. Also, thanks to the level-1 null state condition, the 3-point function of primary vertex fields can be determined completely: Level-1 null state condition resolves the problems of AGT-W relation.

50.  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 section… Our plans of current and future research on generalized AGT relation

51. AdS/CFT of AGT’s system

52.  CFT side : 4-dim SU(N ≫ 1) quiver gauge theory and 2-dim A N-1 Toda theory 4-dim theory is conformal. The system preserves eight (1/2×1/2) supersymmetries.  AdS side : the system with AdS 5 and S 2 factor and 1/2 BPS state of AdS 7 ×S 4 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: [Gaiotto-Maldacena ’09] [Lin-Lunin-Maldacena ’04]

53. On the near horizon (dual) spacetime and its symmetry The near horizon region of M5-branes is AdS 7 ×S 4 spacetime. Then, what is the near horizon of intersecting M5-branes like?  0,1,2,3-direction : 4-dim quiver gauge theory lives here. All M5-branes must be extended.  7-direction : compactification direction of M → IIA Only M5(D4)-branes must be extended.  8,9,10-direction and 5-direction : corresponding to SU(2)×U(1) R-symmetry No M5-branes are extended to the former, and only M5(NS5)-branes are to the latter. Then the result is … (original AdS 7 × S 4 ) r

54. The most general gravity solution with such symmetry is Note that the spacetime solution is constructed from a single function which obeys 3-dim Toda equation (In the following, we consider the cases where the source term is non-zero.) cf. coordinates of 11-dim spacetime: LLM ansatz : 11-dim SUGRA solution with AdS 5 x S 2 factor and SUSYs [Lin-Lunin-Maldacena ’04]

55. The neighborhood of punctures : Toda equation with source term We consider the system of N M5(D4)-branes and K M5(NS5)-branes (N ≫ K ≫ 1), and locally analyze the neighborhood of punctures (intersecting points). M5(NS5)-branes wrap AdS 5 ×S 1, which is conformal to R 1,5. So, including the effect of M5(D4)-branes, the near horizon geometry is also AdS 7 ×S 4 : When we set the angles and (i.e. U(1) symm. for β-direction), we can determine the correspondence to LLM ansatz coordinates as where. Note that D → ∞ along the segment r=0 and 0 ≦ y ≦ 1. This means that Toda equation must have the source term, whose charge density is constant along the segment: S1S1 S1S1

56. In this simplified situation, 11-dim spacetime has an additional U(1) symmetry. Moreover, the analysis become much easier, if we change the variables: Note that this transformation mixes the free and bound variables: (r, y, D) → (ρ, η, V)… Then LLM ansatz and Toda equation becomes ( ) and i.e. This is nothing but the 3-dim cylindrically symmetric Laplace equation. For simplicity, we concentrate on the neighborhood of the punctures. ρ η

57. From the U(1) symmetry of β-direction, the source must exist at ρ=0. Near, LLM ansatz becomes more simple form (using ) Note that at (i.e. at the puncture), The circle is shrinking The circle is not shrinking. This makes sense, only when the constant slope is integer. In fact, this integer slopes correspond to the size of quiver gauge groups. ( → the next page…) For more simplicity, we concentrate on the neighborhood of the punctures.

58. The neighborhood of punctures : Laplace equation with source term We consider the such distribution of source charge:  When the slope is 1, we get smooth geometry.  When the slope is k, which corresponds to the rescale and, we get A k-1 singularity at and, since the period of β becomes. In general, if the slope changes by k units, we get A k-1 singularity there. This can be regard the flavor symmetry of additional k fundamental hypermultiplets. This means the source charge corresponds to nothing but the size of quiver gauge group. N

59. Near, the potential can be written as (since, ) Then we obtain, So the boundary condition (~ source at r=0) is On the source term : AdS/CFT correspondence for AGT relation ! integer x * x

60. 3-dim Toda equation, 2-dim Toda equation and their correspondence  3-dim Toda equation :  2-dim Toda equation (after rescaling of μ) : 2-dim Toda  Correspondence : or [proof] The 2-dim equation (without curvature term, for simplicity) says Therefore, under the correspondence, this 2-dim equation exactly becomes the 3-dim equation: differential of differential elementcoordinate

61. To obtain the source term, we consider OPE of kinetic term of 2-dim equation and the vertex operator : ( ) Then using the correspondence, we obtain In massless case, (since we consider AdS/CFT correspondence). According to our ansatz, this is of the form where : N elements (Weyl vector) : k elements Source term from 2-dim Toda equation source??

62. Towards the correspondence of “source” in AdS/CFT context…? For full [1,…,1]-type puncture: For simple [N-1,1]-type puncture : For [l 1,l 2,…]-type puncture :

63. Conclusion  AGT relation reveals the interesting correspondence between 4-dim N=2 linear or necklace SU(2) quiver gauge theory and 2-dim Liouville theory.  We show (in part) that AGT-W relation for 4-dim linear SU(3) quiver gauge theory and 2-dim A 2 Toda theory is satisfied, by checking 1-loop factor and some lower levels of instanton factor. Here we use effectively the level-1 null state condition for vertices in Toda theory.  As one way to study AGT-W relation for SU(N ≫ 1) quiver gauge theory, it can be useful to discuss AdS/CFT correspondence. Our conjecture for general vertices in Toda theory enables us to study this correspondence. This will be an important future work.