The sums of Rogers, Schur and Ramanujan and the Bose-Fermi correspondence in $1+1$-dimensional quantum field theory

TL;DR

This paper explores the connection between Rogers-Schur-Ramanujan sums and the Bose-Fermi correspondence in 1+1-dimensional quantum field theory, highlighting new fermionic sum representations of characters and branching functions.

Contribution

It introduces new fermionic sum representations for characters and branching functions related to affine Lie algebras and conformal field theory.

Findings

Fermionic sums provide alternative descriptions of spectra in quantum field theory.

Connections established between classical identities and modern quantum field theoretical models.

Recent results on fermionic sum representations of characters and branching functions.

Abstract

We discuss the relation of the two types of sums in the Rogers-Schur-Ramanujan identities with the Bose-Fermi correspondence of massless quantum field theory in $1+1$ dimensions. One type, which generalizes to sums which appear in the Weyl-Kac character formula for representations of affine Lie algebras and in expressions for their branching functions, is related to bosonic descriptions of the spectrum of the field theory (associated with the Feigin-Fuchs construction in conformal field theory). Fermionic descriptions of the same spectrum are obtained via generalizations of the other type of sums. We here summarize recent results for such fermionic sum representations of characters and branching functions. (To appear in C.N. Yang's 70th birthday Festschrift.)