The exact Schur index of $\mathcal{N}=4$ SYM

TL;DR

This paper derives an exact, explicit expression for the Schur index of four-dimensional $ ext{N}=4$ supersymmetric Yang-Mills theory with gauge group $U(N)$, using a fermionic gas representation to facilitate calculations.

Contribution

It reformulates the Schur index for $ ext{N}=4$ SYM as a fermionic gas partition function, enabling exact finite $N$ and large $N$ computations.

Findings

Explicit formulas for fixed $N$ Schur index.

All-orders large $N$ expansion derived.

Simplified integral expressions for the index.

Abstract

The Witten index counts the difference in the number of bosonic and fermionic states of a quantum mechanical system. The Schur index, which can be defined for theories with at least $\mathcal{N}=2$ supersymmetry in four dimensions is a particular refinement of the index, dependent on one parameter $q$ serving as the fugacity for a particular set of charges which commute with the hamiltonian and some supersymmetry generators. This index has a known expression for all Lagrangian and some non-Lagrangian theories as a finite dimensional integral or a complicated infinite sum. In the case of $\mathcal{N}=4$ SYM with gauge group $U(N)$ we rewrite this as the partition function of a gas of $N$ non interacting and translationally invariant fermions on a circle. This allows us to perform the integrals and write down explicit expressions for fixed $N$ as well as the exact all orders large $N$ expansion.