Fusion Potentials for G_k and Handle Squashing

TL;DR

This paper establishes a mathematical connection between the fusion ring of G_k conformal field theory and polynomial ideals, providing a residue formula for correlation functions and introducing a handle squashing operator.

Contribution

It introduces a novel isomorphism between the fusion ring and polynomial quotient rings, and derives a residue-like formula for correlation functions in Chern-Simons theory.

Findings

Fusion ring is isomorphic to P(u)/(\del V)

Derived a residue-like formula for correlation functions

Introduced a handle squashing operator with explicit formula

Abstract

Using Chern-Simons gauge theory, we show that the fusion ring of the conformal field theory G_k is isomorphic to P(u)/(\del V), where V is a polynomial in u and (\del V) is the ideal generated by the conditions \del V=0. We also derive a residue-like formula for the correlation functions in the Chern-Simons theory thus providing a RCFT version of the residue formula for the TLG models. An operator that acts like the measure in the residue formula has the ionterpretation of a handle squashing operator and an explicit formula for this operator is given.