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

Abstract

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.

Modular and duality properties of surface operators in N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ gauge theories

“…The equality between our results and those of [35] supports the proposal of a duality between the two types of surface operators in [52]. This also supports the conjecture of [59], based on [10,60,61], that in the presence of simple surface operators the instanton partition function is insensitive to whether they are realized as codimension-2 or codimension-4 operators. In section 7.1 we will comment on such relations in the case of higher rank gauge groups and will also make contact with the results for the twisted chiral rings when the surface defect is realized by coupling two-dimensional sigma-models to pure N = 2 SU(N) gauge theory.…”

“…Codimension-2 surface operators were systematically studied in [8] where, in the context of the of the 4d/2d correspondence [9], the instanton partition functions of N = 2 SU(2) super-conformal quiver theories with surface operators were mapped to the conformal blocks of a two-dimensional conformal field theory with an affine sl(2) symmetry. These studies were later extended to SU(N ) quiver theories whose instanton partition functions in the presence of surface operators were related to conformal field theories with an affine sl(N ) symmetry [10]. The study of codimension-4 surface operators was pioneered in [11] where the instanton partition function of the conformal SU(2) theory with a surface operator was mapped to the Virasoro blocks of the Liouville theory, augmented by the insertion of a degenerate primary field.…”

Orbifolds, defects and sphere partition function

“…It is particularly interesting to study such defects in the classes of supersymmetric theories in 2, 3 or 4 dimensions where various dualities or nontrivial relations among observables are known. For example, the inclusion of surface defects in 4D N = 2 supersymmetric gauge theories has been studied in the computation of instanton partition functions [3][4][5][6][7][8][9][10][11][12], superconformal indices [13][14][15][16][17] or sphere partition functions [18][19][20][21], and the results led to a more detailed understanding of the relation between 4D N = 2 SUSY theories and 2D conformal field theories [22][23][24], topological field theories or topological strings. The loop operators in 4D N = 2 theories were studied from a similar viewpoint in [25][26][27]; see [28] for a review.…”

Generalized matrix models and AGT correspondence at all genera

“…Extensions to higher rank gauge groups and Toda field theories were introduced and discussed in [19][20][21][22][23][24][25]. The refinement of the correspondence in presence of gauge theory observables has been presented and studied in [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. Moreover, some arguments for the derivation of the AGT correspondence were proposed in the M-theory context in [43,44] and via matrix models in .…”