| Summary: | Abstract We study 5d N $$ \mathcal{N} $$ = 1 supersymmetric field theories on closed five-manifolds M $$ \mathcal{M} $$ 5 which are principal circle bundles over simply-connected Kähler four-manifolds, M $$ \mathcal{M} $$ 4, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on M $$ \mathcal{M} $$ 5 through the insertion of a fibering operator, which introduces a non-trivial fibration over M $$ \mathcal{M} $$ 4, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such M $$ \mathcal{M} $$ 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 M $$ \mathcal{M} $$ 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.
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