Fermionic Quasi-Particle Representations for Characters of ${(G^{(1)})_1 \times (G^{(1)})_1 \o (G^{(1)})_2}$

TL;DR

This paper develops fermionic quasi-particle sum representations for characters of certain affine Lie algebra products, providing new insights into their structure and non-uniqueness in specific models.

Contribution

It introduces fermionic quasi-particle sum representations for characters of affine Lie algebra products for all simply-laced types, highlighting non-uniqueness in some cases.

Findings

Characters expressed as partition functions of massless quasi-particles.

Non-uniqueness of representations in the identity character of the critical Ising model.

Applicable to all simply-laced Lie algebras.

Abstract

We present fermionic quasi-particle sum representations for some of the characters (or branching functions) of ~${(G^{(1)})_1 \times (G^{(1)})_1 \o (G^{(1)})_2}$ ~for all simply-laced Lie algebras $G$. For given $G$ the characters are written as the partition function of a set of rank~$G$ types of massless quasi-particles in certain charge sectors, with nontrivial lower bounds on the one-particle momenta. We discuss the non-uniqueness of the representations for the identity character of the critical Ising model, which arises in both the $A_1$ and $E_8$ cases.