5d Partition Functions with A Twist

We derive the partition function of 5d $\mathcal{N}=1$ gauge theories on the manifold $S_b^3\times {\mathrm{\Sigma }}_{\mathfrak{g}}$ with a partial topological twist along the Riemann surface, ${\mathrm{\Sigma }}_{\mathfrak{g}}$. 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 $\mathcal{N}=2$ Yang-Mills theory, in which case the partition function computes the 4d index of general class $\mathcal{S}$ theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to ${\cal M}3 \times \Sigma{\frak g}$withmoregeneralthree-manifolds${\cal M}_3$ and focus in particular on ${\mathcal{M}}_3={\mathrm{\Sigma }}_{{\mathfrak{g}}^{\mathrm{\prime }}}\times S^1$, in which case the partition function relates to the entropy of black holes in AdS${}_6$.