On Affine Fusion and the Phase Model

TL;DR

This paper reviews the integrable phase model realization of affine fusion in $su(n)$ WZNW conformal field theories, highlighting its algebraic structure, connection to the modular S matrix, and implications for fusion multiplicities and higher-genus fusion.

Contribution

It introduces a new realization of affine fusion via the phase model, clarifying the role of noncommutative Schur polynomials and extending understanding of fusion at higher genus.

Findings

Demonstrates the phase model's role in affine fusion

Shows how threshold levels and higher-genus fusion are accommodated

Connects noncommutative Schur polynomials to fusion multiplicities

Abstract

A brief review is given of the integrable realization of affine fusion discovered recently by Korff and Stroppel. They showed that the affine fusion of the $su(n)$ Wess-Zumino-Novikov-Witten (WZNW) conformal field theories appears in a simple integrable system known as the phase model. The Yang-Baxter equation leads to the construction of commuting operators as Schur polynomials, with noncommuting hopping operators as arguments. The algebraic Bethe ansatz diagonalizes them, revealing a connection to the modular S matrix and fusion of the $su(n)$ WZNW model. The noncommutative Schur polynomials play roles similar to those of the primary field operators in the corresponding WZNW model. In particular, their 3-point functions are the $su(n)$ fusion multiplicities. We show here how the new phase model realization of affine fusion makes obvious the existence of threshold levels, and how it accommodates higher-genus fusion.