Systems of correlation functions, coinvariants and the Verlinde algebra

TL;DR

This paper establishes an isomorphism between correlation function spaces and coinvariant spaces in affine Kac-Moody algebras, generalizing Frenkel-Zhu's theorem and describing these spaces via fusion coefficients.

Contribution

It proves the isomorphism of Gaberdiel-Goddard spaces with coinvariants and extends Frenkel-Zhu's theorem to a broader context.

Findings

Gaberdiel-Goddard spaces are isomorphic to coinvariant spaces

These spaces decompose into tensor products with fusion coefficient multiplicities

Generalization of Frenkel-Zhu's theorem

Abstract

We study the Gaberdiel-Goddard spaces of systems of correlation functions attached to an affine Kac-Moody Lie algebra $\gh$. We prove that these spaces are isomorphic to the spaces of coinvariants with respect to certain subalgebras of $\gh$. This allows to describe the Gaberdiel-Goddard spaces as direct sums of tensor products of irreducible $\g$-modules with multiplicities given by fusion coefficients. We thus reprove and generalize Frenkel-Zhu's theorem.