Deformed Schur Indices of BCD-type for N=4 Super Yang-Mills and Symmetric Functions

TL;DR

The paper addresses the calculation of deformed Schur indices for $ ext{N}=4$ SYM with $BCD_N$ gauge groups by formulating the indices as Koornwinder integrals and mapping them onto Macdonald polynomial normalization constants. It develops two complementary expansion schemes: (i) a Macdonald normalization constant expansion that generalizes Hatsuda’s $U(N)$ results to $BCD_N$, and (ii) a formal series in the fugacity $u$ that expresses the index as multi‑sum expressions over partitions with well‑defined coefficients. The work delivers explicit low‑rank results for $SO(2N)$, $SO(2N+1)$, and $Sp(2N)$, and investigates several limiting cases (e.g., $p=v=u=0$, $q=t$, and $q=u$) to test S‑duality and connect to known symmetric‑function theories such as Hall–Littlewood and inverse Kostka polynomials. Altogether, the approach provides exact, testable structures for dualities and giant graviton expansions in a broader class of gauge theories, with potential applications to defect setups and higher BPS indices.

Abstract

We investigate the deformed Schur index in four dimensional N=4 super Yang-Mills theories with $SO$ and $Sp$ gauge groups, generalizing Hatsuda's recent calculations. We express the deformed Schur index as integrals of Koornwinder polynomials and Macdonald polynomials, then perform the integrals in terms of the normalization constants of Macdonald polynomials. We provide explicit results for some low rank gauge groups and for expansion in a $u$ parameter. We discuss various special limits and the tests of S-duality.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2