Superconformal index for $\mathcal{N} = 4$ Super Yang-Mills and Elliptic Macdonald Polynomials

We establish a connection between the superconformal index of $\mathcal{N}=4$ $U(N)$ 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 $B_\lambda(p,q,t)$ and normalization constants $\mathcal{N}_\lambda(p,q,t)$. 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 the large $N$ limit, our formalism reduces to previously known results.