From 5d Flat Connections to 4d Fluxes (the Art of Slicing the Cone)

Authors: Jim Lundin, Roman Mauch, Lorenzo Ruggeri Preprint number: UUITP-12/23 Abstract: We compute the Coulomb branch partition function of the 4d 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 obtaining flux contributions from the localisation procedure. We achieve this by taking a detour via the 5d 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 $h\to\infty$, 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.