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

TL;DR

This work uncovers a number-theoretic bridge between rational conformal field theories via Hecke operators, showing that the conductor N and quadratic residues govern Galois permutations of fusion data. By introducing an effective central charge, the authors connect Hecke images to their progenitors, preserving fusion rules and often yielding isomorphic modular tensor categories. They extend the analysis beyond modular data to duality (F- and B-matrices), establishing that Galois symmetry permeates the entire RCFT data under Hecke transformations. Through detailed examples (notably M(3,5), M(2,k), and simple-current reductions), the paper demonstrates unitary/non-unitary outcomes and provides a coherent framework for predicting when Hecke images correspond to physically meaningful theories. The results suggest deep arithmetic structure in RCFT/MTC and motivate extending Hecke-theoretic methods to broader classes of theories and their VOA realizations.

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.

Figures (5)

• Figure 1: A geometric illustration of fusion, as the composition of two 4-point functions.
• Figure 2: Fusion matrix between blocks. The labels of the matrix entries, i.e. $p$ and $q$, take the positions of the "propagator".
• Figure 3: Braiding matrix between blocks. The labeling of the matrix entries, i.e. $p$ and $q$, take the positions of the "propagator".
• Figure 4: A simple loop transformation of conformal block.
• Figure 5: Braiding eigenvalues.