The toroidal block and the genus expansion

TL;DR

This paper explores the connection between 4D N=2* supersymmetric gauge theories and 2D conformal field theories, focusing on the genus expansion and modular properties of conformal blocks on a torus, revealing new insights into topological string theory.

Contribution

It uncovers the modular structure of conformal blocks in the gauge/CFT correspondence and links the genus expansion to quasi-modular forms in topological string theory.

Findings

Conformal blocks exhibit quasi-modular behavior at large intermediate weights.

The genus expansion provides a new perspective on the topological string partition function.

Analytic structure of the topological string is clarified in the field theory limit.

Abstract

We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of N=2* gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasi-modular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.