Deformed Schur indices and Macdonald polynomials

TL;DR

The paper derives an exact, finite-$N$ expression for the two-parameter deformed Schur index $I_N(t,u;q)$ in $\mathcal{N}=4$ $U(N)$ SYM by exploiting Macdonald polynomials, yielding a compact sum over partitions: $I_N(t,u;q)=\frac{(q;q)_\infty}{(t;t)_N (t^N q;q)_\infty}\sum_{\ell(\lambda)\le N} u^{|\lambda|}\prod_{i=1}^{\ell(\lambda)}\frac{(t^{N-i+1};q)_{\lambda_i}}{(t^{N-i}q;q)_{\lambda_i}}$. The authors also extend to line operators, analyze the large-$N$ limit to obtain $I_\infty(t,u;q)=\frac{(q;q)_\infty}{(t;t)_\infty (u;u)_\infty}$, and formulate giant-graviton expansions for finite-$N$ corrections, including explicit, though intricate, recursion structures. The work connects gauge-theory indices with Macdonald symmetry structures, enabling efficient finite-$N analyses and providing a path toward generalizations to full indices, elliptic deformations, and other gauge groups via Macdonald-Koornwinder polynomials. This approach offers precise, analytic control over BPS spectra and giant graviton regimes in AdS/CFT contexts, with potential applications to dualities and line-operator dynamics.

Abstract

The Schur index in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with $U(N)$ gauge group has a natural two-parameter deformation. We find that a matrix integral in such a deformed Schur index can be exactly evaluated by using Macdonald polynomials. The resulting expression is a simple combinatorial summation over partitions. An extension to line operator indices is straightforward. In particular, for an anti-symmetric representation, the line operator index has a relatively simple form. We further discuss infinite $N$ analysis and finite $N$ giant graviton expansions.