Non-conformal limit of AGT relation from the 1-point torus conformal block

115

Recently Alday and Tachikawa [1] proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal N = 2 SU(2) quiver gauge theories in the presence of a certain surface operator. In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N ) symmetry and instanton partition functions in conformal N = 2 SU(N ) quiver gauge theories in the presence of a surface operator. We also discuss the extension to non-conformal N = 2 SU(N ) theories.

Section: Jhep01(2011)045mentioning

confidence: 83%

Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories

Kozçaz

Pasquetti

Passerini

et al. 2011

115

“…This relation for a torus was further studied in [27,53]. The identification of the parameters of the two is as follows: the momenta of the vertex operators, the internal momenta and the complex structure moduli correspond, respectively, to the mass parameters of the hypermultiplets, the Coulomb moduli and the gauge coupling constants in the gauge theory.…”

Section: From Liouville Theory To Generalized Matrix Modelmentioning

confidence: 99%

Seiberg-Witten curve via generalized matrix model

Maruyoshi

Yagi

2011

Abstract:We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N = 2 SU(2) n gauge theory with circular quiver diagram by the AGT relation. We first verify that the generalized matrix model is obtained from the perturbative calculation of the Liouville correlation function. We then derive the Seiberg-Witten curve for N = 2 gauge theory as a spectral curve of the generalized matrix model.

“…Recently a renewed interest has developed due to a conjecture [10,11] that Liouville theory on a Riemann surface of genus g is related to a certain class of N = 2, 4-dimensional gauge theories and the conjecture has been supported by extensive tests on genera 0 and 1 [10,12,13] and proven in a class of cases [14,15]. At the classical level the key point in solving the theory is the determination of the accessory parameters which on the sphere are related to the semiclassical limit of the operator product expansion via the Polyakov relation.…”

Section: Introductionmentioning

confidence: 99%

Accessory parameters for Liouville theory on the torus

Menotti

2012

We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.