A bstract We derive the partition function of 5d $$ \mathcal{N}=1 $$ N = 1 gauge theories on the manifold $$ {S}_b^3 \times {\varSigma}_{\mathfrak{g}} $$ S b 3 × Σ g with a partial topological twist along the Riemann surface, $$ {\varSigma}_{\mathfrak{g}} $$ 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 AdS 6 and AdS 4 spacetimes, reproducing known holographic relations between the corresponding free energies on S 5 and S 3 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 $$ \mathcal{N}=2 $$ 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 $$ {\mathrm{\mathcal{M}}}_3 \times {\varSigma}_{\mathfrak{g}} $$ ℳ 3 × Σ g with more general three-manifolds ℳ 3 and focus in particular on $$ {\mathrm{\mathcal{M}}}_3 = {\varSigma}_{\mathfrak{g}}^{\prime}\times {S}^1 $$ ℳ 3 = Σ g ′ × S 1 , in which case the partition function relates to the entropy of black holes in AdS 6 .

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