Galois groups in rational conformal field theory II. The discriminant

TL;DR

This paper derives a formula expressing the discriminant of fusion ring relations in conformal field theories as a product involving the modular matrix, revealing its integer nature and prime divisors related to the level.

Contribution

It provides a new explicit formula for the discriminant of fusion ring relations in conformal field theories, linking it to the modular matrix and prime divisors.

Findings

Discriminant expressed as product of modular matrix row powers

Discriminant is always an integer

Discriminant's prime divisors divide the level

Abstract

We express the discriminant of the polynomial relations of the fusion ring, in any conformal field theory, as the product of the rows of the modular matrix to the power -2. The discriminant is shown to be an integer, always, which is a product of primes which divide the level. Detailed formulas for the discriminant are given for all WZW conformal field theories.