Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry

Abstract

Recently, a duality between Liouville theory and four dimensional N = 2 gauge theory has been uncovered by some of the authors. We consider the role of extended objects in gauge theory, surface operators and line operators, under this correspondence. We map such objects to specific operators in Liouville theory. We employ this connection to compute the expectation value of general supersymmetric 't Hooft-Wilson line operators in a variety of N = 2 gauge theories.

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

“…It means that outside a small disk D x ⊂ N there is a gauge transformation, which makes A 0 deformable to A 1 . One can loosely call such a modification adding a point-like instanton at x A −→ A + δ (1) x A (1.17) One can imagine a successive application of the modifications δ (1) x 1 δ (1) x 2 , which add point-like instantons at two distinct points x 1 = x 2 ∈ N, or adding two instantons at the same point, A −→ A + δ (2) x A, and so on. The specific realization of such modifications is possible in the string theory context, where the gauge theory instantons are the codimension four D-branes dissolved inside another brane [28].…”

“…in the AGT setup [1,2] to assign the X i (x)-observables to the non-intersecting loops on the curve C on which one compactifies the A 1 (0, 2)-theory, which define the α-coordinates in the system of Darboux coordinates on the moduli space of SL 2 local systems [87].…”

The stringy instanton partition function

“…Our approach should be compared with the results of [67]. Furthermore, we remark that the above constitutes a useful set-up to study the AGT correspondence [49,63,64].…”

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

“…Wilson and 't Hooft loops), and three-dimensional (domain wall) defects. Such defects have been considered in [11][12][13], and [14], respectively.…”