Hopping in the Phase Model to a Non-Commutative Verlinde Formula for Affine Fusion

M A Walton

doi:10.1088/1742-6596/474/1/012033

Abstract

Korff and Stroppel discovered a realization of su(n) affine fusion, the fusion of the su(n) Wess-Zumino-Novikov-Witten (WZNW) conformal field theory, in the phase model, a limit of the q-boson hopping model. This integrable-model realization provides a new perspective on affine fusion, explored in a recent paper by the author. The role of WZNW primary fields is played in it by non-commutative Schur polynomials, and fusion coefficients are thus given by a non-commutative version of the Verlinde formula. We present the extension to all Verlinde dimensions, of arbitrary genus and any number N of points. The level-dependence of affine fusion is also discussed, using the concept of threshold level, and its generalization to threshold weight.

Highlights

Affine fusion is found in several mathematical and physical contexts. It is a natural generalization of the tensor product of representations of simple Lie algebras; a simple truncation thereof controlled by a non-negative integer, the level
The Korff-Stroppel integrable realization of su(n) affine fusion [9] leads to the noncommutative Verlinde formula (14) for affine fusion coefficients, that can be extended to a non-commutative formula (15) for arbitrary Verlinde dimensions [14]
The realization offers a new perspective on affine fusion, that should deepen our understanding

Summary

Introduction

Affine fusion is found in several mathematical and physical contexts. It is a natural generalization of the tensor product of representations of simple Lie algebras; a simple truncation thereof controlled by a non-negative integer, the level. To see the relation to Young tableaux, note that the hopping operator ai is associated with the weight Λi − Λi−1 The set of these affine weights have horizontal parts equal to the weights of the basic su(n) irreducible representation L(Λ1), that can be labelled by Young tableau i. Korff and Stroppel showed that the hopping operators of the phase model realize su(n) affine fusion, a truncation of the su(n) tensor product. The non-commutative Schur polynomial sΛ1+Λ2(A) for the adjoint representation of su(3) equals the sum of: a2a3a2 Comparing this last result with (9), we already see a couple of important differences. It becomes clear that the noncommutative Schur polynomial sλ(A) plays the role in the phase model of the primary field φλ in the WZNW conformal field theory It is the integrability of the phase model that gives rise to duality in affine fusion.

Higher-Genus and Level-Dependence

Conclusion