Liouville Correlation Functions from Four-dimensional Gauge Theories

TL;DR

The paper advances the AGT-type program by matching Nekrasov's instanton sums for four-dimensional N=2 SCFTs associated with punctured Riemann surfaces to Virasoro conformal blocks of Liouville theory, and showing that Liouville correlators arise from integrating the squared full Nekrasov partition function over Coulomb moduli. It provides explicit demonstrations at genus 0 and 1 (sphere and torus cases) and extends the construction to general g,n via linear and necklace quivers, proposing a universal dictionary between gauge theory data (UV couplings, Coulomb parameters, masses) and Liouville data (conformal blocks, three-point functions). A key insight is the separation into an instanton (conformal block) part and a one-loop (DOZZ) part, with the full Liouville correlator recovered by an S-duality-invariant integral over moduli. The work lays the groundwork for broad generalizations to higher rank, W-algebras, and ADE Toda theories, and highlights numerous open problems and potential connections to geometric and topological structures in quantum field theory.

Abstract

We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.

Figures (8)

• Figure 1: Three distinct ways to decompose a four-punctured sphere into two pairs of pants. They correspond to three distinct weakly-coupled frames of $SU(2)$ gauge theory with four flavors.
• Figure 2: Two examples to sew a six-punctured sphere from four pairs of pants. Left: a standard linear quiver theory. Right: a generalized quiver theory, where three $SU(2)$ gauge groups couple to four hypermultiplets, denoted by the three-punctured sphere at the center.
• Figure 3: Placement of labels of the conformal blocks we use.
• Figure 4: Pictorial representation of the propagator $K^{-1}$, shown in (a), the vertex with two external legs $R$, shown in (b) and the vertex with one external leg $S$, shown in (c). External legs are represented with solid lines, while internal legs are represented with dashed lines.
• Figure 5: Sewing of building blocks into sphere (a) and and torus (b) conformal blocks. In the figure we see a five-point conformal block in the sphere and two points on the torus.