Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories

TL;DR

This paper explores how Hecke operators induce Galois symmetries in rational conformal field theories, affecting their modular data, fusion, and braiding structures, and proposes extending these operators to modular tensor categories.

Contribution

It establishes a connection between Hecke operators, Galois symmetries, and the structure of RCFTs and modular tensor categories, providing new tools for classification.

Findings

Hecke operators relate RCFT characters with different central charges.

Galois symmetries influence fusion and braiding matrices.

Effective central charge links different theories via Galois automorphisms.

Abstract

Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion algebras. We show that the conductor $N$ of a RCFT and the quadratic residues mod $N$ play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Hecke operators to the full data of modular tensor categories. The Galois symmetry encountered in the modular data appears in the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators.