5d Partition Functions with A Twist

TL;DR

This paper computes the partition function of 5d ${ m extbf{N}=1}$ gauge theories on specific manifolds with a topological twist, exploring its holographic implications, connections to 6d SCFTs, and black hole entropy in AdS$_6$.

Contribution

It derives a new expression for the 5d partition function on twisted manifolds, linking it to Bethe-Ansatz solutions, holography, 6d SCFTs, and black hole entropy.

Findings

Partition function expressed as a sum over Bethe-Ansatz solutions.

Large N limit reproduces holographic relations between free energies.

Partition function computes 4d index of class ${ m extbf{S}}$ theories and relates to black hole entropy.

Abstract

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