Fusion potentials I

TL;DR

This paper investigates Gepner's conjecture that fusion rings in rational conformal field theories can be represented as polynomial rings modulo an ideal derived from a potential, confirming this in several cases with single-variable polynomials.

Contribution

It demonstrates that in multiple instances, the fusion ring is isomorphic to a polynomial ring in one variable constrained by a potential, supporting Gepner's conjecture.

Findings

Fusion rings can be represented by single-variable polynomial quotients.

Gepner's conjecture holds in various cases for rational conformal field theories.

The approach simplifies understanding the algebraic structure of fusion rings.

Abstract

We reconsider the conjecture by Gepner that the fusion ring of a rational conformal field theory is isomorphic to a ring of polynomials in $n$ variables quotiented by an ideal of constraints that derive from a potential. We show that in a variety of cases, this is indeed true with {\it one-variable} polynomials.