Flat Connections for Characters in Irrational Conformal Field Theory

TL;DR

This paper extends the study of irrational conformal field theory (ICFT) to the torus, revealing that its characters satisfy heat-like equations with flat connections, and provides integral and geometric formulations for these characters.

Contribution

It introduces a geometric framework for ICFT characters on the torus, including integral representations and flat connection structures, advancing understanding of their mathematical properties.

Findings

Derived heat-like differential equations for ICFT characters

Obtained integral representations for coset characters

Presented a geometric formulation with flat connections as generalized Laplacians

Abstract

Following the paradigm on the sphere, we begin the study of irrational conformal field theory (ICFT) on the torus. In particular, we find that the affine-Virasoro characters of ICFT satisfy heat-like differential equations with flat connections. As a first example, we solve the system for the general $g/h$ coset construction, obtaining an integral representation for the general coset characters. In a second application, we solve for the high-level characters of the general ICFT on simple $g$, noting a simplification for the subspace of theories which possess a non-trivial symmetry group. Finally, we give a geometric formulation of the system in which the flat connections are generalized Laplacians on the centrally-extended loop group.