Supersymmetric Gauge Theories, Quantization of $${\mathcal M}_\mathrm{flat}$$ , and Conformal Field Theory

Abstract

We will propose a derivation of the correspondence between certain gauge theories with N = 2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2, R)-connections represents a non-perturbative "skeleton" of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks. F.2 Projective local systems 126 F.3 Riemann-Hilbert type problems References 127 2. Riemann surfaces: Some basic definitions and results Let us introduce some basic definitions concerning Riemann surfaces that will be used throughout the paper.

Exact Results on $${\mathcal N}=2$$ Supersymmetric Gauge Theories

“…This resummation gets represented as an integral transformation with kernel F p,p ′ . Using (2.16), (2.21), (3.32) and certain identities satisfied by the kernel F p,p ′ α 3 α 4 α 2 α 1 established in [TV13], one may now verify explicitly that the S-duality relations (3.31) are indeed satisfied. 9 The IR duality conjectures can be used to describe the moduli space of vacua as manifold covered by charts with local coordinates a r , a D r .…”

q-Virasoro Modular Double and 3d Partition Functions

“…In particular, S-duality domain wall in 4d N = 2 theories of class S [155] are realized by 3d N = 2 U(N ) YM theories on S 3 b [156][157][158][159][160], and their partition functions are modular kernels of Liouville [161][162][163] or Toda theories [160]. Moreover, our construction of the W q,t modular double should easily lift to the elliptic case [164,165].…”

Isomonodromic Tau-Functions from Liouville Conformal Blocks

“…These operators generate a representation of the quantised algebra of algebraic functions on M flat (C) on the spaces of Virasoro conformal blocks [TV13]. The definition of the Verlinde loop operators given in [AGGTV, DGOT] can easily be rewritten as deformed traces over products of the operator-valued monodromy matrices defined in Section 3.4.…”

“…The fact that this sub-algebra becomes commutative for c = 1 leads to the existence of new representations for the quantised algebra of functions on M flat (C) related to the usual one by the transformation defined in Section 4.1. This representation is not unitarily equivalent to the one studied in [TV13] as the measure defining the scalar product for c > 25, the Liouville three-point function, can not be analytically continued to c = 1. It should be interesting to investigate this phenomenon and possible generalisations further.…”

“…We note that the quantum counterparts of the coordinates (σ, τ ) that can be defined away from c = 1 [TV13] remain non-commutative when c → 1. However, the algebra of all operators that can be constructed from the quantised coordinates (σ, τ ) contains the important sub-algebra generated by the Verlinde loop operators.…”