Quantum geometry from the toroidal block

TL;DR

This paper explores the semi-classical expansion of toroidal conformal blocks, revealing how Seiberg-Witten geometry and epsilon-deformations naturally arise in conformal field theory, and proving key modularity and relations in N=2* theories.

Contribution

It demonstrates the emergence of Seiberg-Witten structures and epsilon-deformations from conformal blocks, providing new proofs and extending the understanding of modularity in 2d/4d correspondence.

Findings

Seiberg-Witten curve and differential emerge naturally from conformal blocks.

Derived epsilon1-deformations of prepotential relations.

Proved quasi-modularity of conformal block coefficients.

Abstract

We continue our study of the semi-classical (large central charge) expansion of the toroidal one-point conformal block in the context of the 2d/4d correspondence. We demonstrate that the Seiberg-Witten curve and (epsilon1-deformed) differential emerge naturally in conformal field theory when computing the block via null vector decoupling equations. This framework permits us to derive epsilon1-deformations of the conventional relations governing the prepotential. These enable us to complete the proof of the quasi-modularity of the coefficients of the conformal block in an expansion around large exchanged conformal dimension. We furthermore derive these relations from the semi-classics of exact conformal field theory quantities, such as braiding matrices and the S-move kernel. In the course of our study, we present a new proof of Matone's relation for N=2* theory.