The exact Schur index of N = 4 N=4 SYM

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 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 N=2 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.