Hecke Relations in Rational Conformal Field Theory

TL;DR

This paper introduces Hecke operators acting on vector-valued modular forms in rational conformal field theories, revealing new relations between characters and enabling construction of characters for theories with increasing central charge.

Contribution

It extends Galois symmetry to relations between RCFT characters and develops Hecke operators as tools for generating and relating RCFT characters with different central charges.

Findings

Hecke operators relate RCFT characters with different central charges.

Constructed infinite sets of RCFT characters with positive integer coefficients.

Established connections between Hecke operators and solutions to modular linear differential equations.

Abstract

We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their $q$ expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.