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

Abstract

Abstract We compute the Coulomb branch partition function of the 4d N $$ \mathcal{N} $$ = 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 N $$ \mathcal{N} $$ = 1 vector multiplet on closed simply-connected toric Sasaki-manifolds M which are principal S 1-bundles over B. The perturbative partition function can be expressed as a product over slices of the toric cone. By taking finite quotients M/ℤ h along the S 1, 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/ℤ h 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.

Keywords

• Nonperturbative Effects
• Supersymmetric Gauge Theory
• Extended Supersymmetry
• Field Theories in Higher Dimensions