1. AGT 関係式 (2) AGT 関係式 (String Advanced Lectures No.19) 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 9 日（水） 12:30-14:30

2. Contents 1. Gaiotto’s discussion for SU(2) 2. SU(2) partition function 3. Liouville correlation function 4. Seiberg-Witten curve and AGT relation 5. Towards generalized AGT relation

3. Gaiotto’s discussion for SU(2) [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

4.  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

5. SU(2) partition function  Action 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 deformation parameters : background of graviphoton or deformation of extra dimensions Nekrasov’s partition function of 4-dim gauge theory (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.

6. 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

7. 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

8. 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)].

9. 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)

10. 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

11. 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)

12. 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.

13. 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.

14. 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

15. 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??

16. Towards generalized AGT  Natural generalization of AGT relation seems the correspondence between partition function of 4-dim SU(N) quiver gauge theory and correlation function of 2-dim A N-1 Toda theory : This discussion is somewhat complicated, since in SU(N>2) case, 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.) [Wyllard ’ 09] [Kanno-Matsuo-SS-Tachikawa ’ 09] … … … … … … …