5d partition functions with a twist

We derive the partition function of 5d N= 1 gauge theories on the manifold Sb3× Σg with a partial topological twist along the Riemann surface, Σg. 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 × Σg 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