On Affine Fusion and the Phase Model

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.