Genus Zero $\widehat{\mathfrak {su}}(n)_m$ Wess–Zumino–Witten Fusion Rules Via Macdonald Polynomials

Abstract

The Kac–Walton formula computes the fusion coefficients of genus zero $\widehat{\mathfrak {su}}(n)_m$ 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 $\widehat{\mathfrak {su}}(n)_m$ Wess–Zumino–Witten fusion coefficients can be retrieved directly from the (q, t)-Littlewood–Richardson coefficients through the parameter degeneration $(q,t)=\left( \exp \bigl ({\frac{2\pi i}{nc+m}}\bigr ), \exp \bigl ({\frac{2\pi ic}{nc+m}} \bigr )\right)$, $c\rightarrow 1$. The refinement thus establishes that at the level of the structure constants (q, t)-deformation provides a vehicle for performing the reduction modulo the fusion ideal via parameter specialization.