WZW Partition Functions from Supersymmetric Localization

TL;DR

This work proves that diagonal modular invariant WZW partition functions for \\widehat{SU(N)}_k can be realized as lattice sums derived from supersymmetric localization, clarifying when this lattice representation matches the CFT numerator. The authors implement a two-step approach: (1) Weyl folding of theta-function sums to reexpress integrable-weight sums as a Weyl-orbit sum over \\Lambda_W/(k+N)\\ Lambda_R, and (2) a partial Poisson resummation of the Z_{susy} lattice sum to produce the same theta-product structure. They extend Murthy and Witten's result to general N>=2 with N+k even, provide counterexamples for odd N+k, and show that suitable Weyl-invariant lattices between the root and weight lattices can capture non-diagonal invariants (including SU(2)_k) via an integer-matrix decomposition. Altogether, the paper builds a rigorous bridge between localization-inspired lattice constructions and modular-invariant WZW partition functions, enabling explicit lattice realizations of both diagonal and certain non-diagonal invariants.

Abstract

We prove a conjecture of Murthy and Witten which expresses diagonal modular invariant WZW partition functions as lattice sums.