5d partition functions with a twist

We derive the partition function of 5d N= 1 gauge theories on the manifold S_b³× Σ_g with a partial topological twist along the Riemann surface, Σ_g. This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large N limit, where it is related to holographic RG flows between asymptotically locally AdS₆ and AdS₄ spacetimes, reproducing known holographic relations between the corresponding free energies on S⁵ and S³ and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric N= 2 Yang-Mills theory, in which case the partition function computes the 4d index of general class S theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to ℳ₃ × Σ_g with more general three-manifolds ℳ₃ and focus in particular on ℳ3=Σ'_g × S¹, in which case the partition function relates to the entropy of black holes in AdS₆.