Transformations of Spherical Blocks

TL;DR

This paper investigates the relationship between supersymmetric gauge theories and conformal field theory, focusing on the epsilon-expansion of partition functions and their symmetry properties, including effects of surface operators and connections to quantum Painlevé VI.

Contribution

It introduces a novel analysis of the epsilon-expansion of partition functions using null vector decoupling equations and explores symmetry enhancements with surface operators in gauge theory.

Findings

Partition function expressed as a series in quasi-modular forms of Gamma(2)

Symmetry group extended to affine Weyl group with surface operators

Link established between null vector equations and quantum Painlevé VI

Abstract

We further explore the correspondence between N=2 supersymmetric SU(2) gauge theory with four flavors on epsilon-deformed backgrounds and conformal field theory, with an emphasis on the epsilon-expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Gamma(2), with the expected symmetry Weyl group of SO(8) semi-direct S_3. In the presence of an elementary surface operator, this symmetry is enhanced to an action of the affine Weyl group of SO(8) semi-direct S_4 on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painlev\'e VI equation.