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

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.

arXiv commentv3: editing with referee's response

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• Classifying three-character RCFTs with Wronskian Index equalling $0$ or $2$
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• Hecke Relations, Cosets and the Classification of 2d RCFTs
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• Quasi-Characters in $\widehat{su(2)}$ Current Algebra at Fractional Levels
• A Passion for Theoretical Physics: a special issue in memory of Peter G. O. Freund