Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs

TL;DR

This paper develops a new method to compute 5d supersymmetric partition functions on complex five-manifolds using fibering operators, matching effective field theory with explicit calculations, and providing evidence for the Lockhart-Vafa formula.

Contribution

It introduces a novel approach to calculate Coulomb branch partition functions on five-manifolds via fibering operators and confirms the Lockhart-Vafa formula for the five-sphere.

Findings

Successfully compute Coulomb branch partition functions on complex five-manifolds.

Match low-energy effective theory with explicit one-loop calculations.

Provide evidence supporting the Lockhart-Vafa formula for five-sphere partition functions.

Abstract

We study 5d $\mathcal{N}=1$ supersymmetric field theories on closed five-manifolds $\mathcal{M}_5$ which are principal circle bundles over simply-connected K\"ahler four-manifolds, $\mathcal{M}_4$, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on $\mathcal{M}_5$ through the insertion of a fibering operator, which introduces a non-trivial fibration over $\mathcal{M}_4$, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such $\mathcal{M}_5$, which is conjectured to be the holomorphic `integrand' of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When $\mathcal{M}_4$ is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.