Fusion Algebras of Fermionic Rational Conformal Field Theories via a Generalized Verlinde Formula

TL;DR

This paper generalizes the Verlinde formula to fermionic rational conformal field theories, revealing how their fusion coefficients relate to bosonic projections and providing explicit calculations for specific fermionic models.

Contribution

It introduces a generalized Verlinde formula for fermionic theories and explores the structure of their fusion algebras, including cases where axioms are weakened.

Findings

Fusion coefficients relate to bosonic projections.

Fusion coefficients for conjugate Ramond fields are 1 or 2.

Explicit fusion algebra calculations for fermionic W(2,d)-models.

Abstract

We prove a generalization of the Verlinde formula to fermionic rational conformal field theories. The fusion coefficients of the fermionic theory are equal to sums of fusion coefficients of its bosonic projection. In particular, fusion coefficients of the fermionic theory connecting two conjugate Ramond fields with the identity are either one or two. Therefore, one is forced to weaken the axioms of fusion algebras for fermionic theories. We show that in the special case of fermionic W(2,d)-algebras these coefficients are given by the dimensions of the irreducible representations of the horizontal subalgebra on the highest weight. As concrete examples we discuss fusion algebras of rational models of fermionic W(2,d)-algebras including minimal models of the $N=1$ super Virasoro algebra as well as $N=1$ super W-algebras SW(3/2,d).