Superconformal index for $$ \mathcal{N} $$ = 4 super Yang-Mills and elliptic Macdonald polynomials

A bstract We establish a connection between the superconformal index of $$ \mathcal{N} $$ N = 4 U ( N ) SYM and the elliptic Ruijsenaars-Schneider integrable system. The index admits an expression in terms of elliptic Macdonald polynomials, which leads to a compact summation over generalized partitions involving the structure constants B λ ( p , q , t ) and normalization constants $$ {\mathcal{N}}_{\lambda}\left(p,q,t\right) $$ N λ p q t . By solving the elliptic Ruijsenaars-Schneider model perturbatively in the elliptic parameter p , a systematic expansion of the index in powers of p is obtained. We check that in various limits, namely a deformed 1/2 BPS limit and especially the large N limit, our formalism reduces to previously known results.