Genus Zero (su)over-cap(n)m Wess-Zumino-Witten Fusion Rules Via Macdonald Polynomials

The Kac-Walton formula computes the fusion coefficients of genus zero (su) over cap (n)(m) Wess-Zumino-Witten conformal field theories as the structure constants of the fusion algebra in the basis of Schur polynomials. Modulo a relation identifying the nth elementary symmetric polynomial with the unit polynomial, this fusion algebra is obtained from the algebra of symmetric polynomials in n variables by modding out a fusion ideal generated by the Schur polynomials of degree m + 1. The present work constructs a refinement of the fusion algebra associated with the Macdonald polynomials at q(m)t(n) = 1. The pertinent refined structure constants turn out to be given by the corresponding parameter specialization of Macdonald's (q, t)-Littlewood-Richardson coefficients that can be expressed alternatively in terms of the refined Verlinde formula. This reveals that the genus zero (su) over cap (n)(m) Wess-Zumino-Witten fusion coefficients can be retrieved directly from the (q, t)-Littlewood-Richardson coefficients through the parameter degeneration (q, t) = (exp(2 pi i/nc+m), exp(2 pi ic/nc+m)), c -> 1. The refinement thus establishes that at the level of the structure constants (q, 1)-deformation provides a vehicle for performing the reduction modulo the fusion ideal via parameter specialization.