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

We study 5d $\mathcal{N}=1$ supersymmetric field theories on closed five-manifolds ${\mathcal{M}}_5$ which are principal circle bundles over simply-connected Kähler 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.