From 5d flat connections to 4d fluxes (the art of slicing the cone)

Abstract

We compute the Coulomb branch partition function of the 4d mathcal{N} = 2 vector multiplet on closed simply-connected quasi-toric manifolds B. This includes a large class of theories, localising to either instantons or anti-instantons at the torus fixed points (including Donaldson-Witten and Pestun-like theories as examples). The main difficulty is to obtain flux contributions from the localisation procedure. We achieve this by taking a detour via the 5d mathcal{N} = 1 vector multiplet on closed simply-connected toric Sasaki-manifolds M which are principal S1-bundles over B. The perturbative partition function can be expressed as a product over slices of the toric cone. By taking finite quotients M/ℤh along the S1, the locus picks up non-trivial flat connections which, in the limit h → ∞, provide the sought-after fluxes on B. We compute the one-loop partition functions around each topological sector on M/ℤh and B explicitly, and then factorise them into contributions from the torus fixed points. This enables us to also write down the conjectured instanton part of the partition function on B.