Galois Groups in Rational Conformal Field Theory

TL;DR

This paper explores the role of Galois groups in rational conformal field theory, showing they are abelian and linking them to the classification of RCFTs through polynomial relations.

Contribution

It introduces the use of Galois groups to classify RCFTs and proves their abelian nature, connecting algebraic structures to physical theories.

Findings

Galois groups in RCFT are all abelian.

Fusion rings can be described by polynomial relations.

Galois groups determine the classification of RCFTs.

Abstract

It was established before that fusion rings in a rational conformal field theory (RCFT) can be described as rings of polynomials, with integer coefficients, modulo some relations. We use the Galois group of these relations to obtain a local set of equation for the points of the fusion variety. These equations are sufficient to classify all the RCFT, Galois group by Galois group. It is shown that the Galois group is equivalent to the pseudo RCFT group. We prove that the Galois groups encountered in RCFT are all abelian, implying solvability by radicals of the modular matrix.