Multinomials and Polynomial Bosonic Forms for the Branching Functions of the $\widehat{su}(2)_{M}\times \widehat{su}(2)_{N}/\widehat{su}(2)_{M+N}$ Conformal Coset Models

TL;DR

This paper introduces explicit q-multinomial formulas that express the branching functions of certain conformal coset models, generalizing previous binomial and trinomial approaches and connecting statistical models with algebraic structures.

Contribution

It provides new explicit q-multinomial expressions for the branching functions of specific coset models, extending prior binomial/trinomial methods to multinomials.

Findings

Explicit q-multinomial formulas for configuration sums

Connection between RSOS models and coset branching functions

Representation of branching functions via multinomials and double sums

Abstract

We give explicit expressions for the q-multinomial generalizations of the q-binomials and Andrews' and Baxter's q-trinomials. We show that the configuration sums for the generalized RSOS models in regime III studied by Date et al. can be expressed in terms of these multinomials. This generalizes the work of ABF and AB where configuration sums of statistical mechanical models have been expressed in terms of binomial and trinomial coefficients. These RSOS configuration sums yield the branching functions for the $\widehat{su}(2)_{M}\times \widehat{su}(2)_{N}/\widehat{su}(2)_{M+N}$ coset models. The representation in terms of multinomials gives Rocha-Caridi like formulas whereas the representation of Date et al. gives a double sum representation for the branching functions.