5d Partition Functions with A Twist

TL;DR

<3-5 sentence high-level summary> The paper derives and analyzes the exact partition function of 5d ${\cal N}=1$ gauge theories on $S^3_b \times \Sigma_\mathfrak g$ with a partial topological twist along $\Sigma_\mathfrak g$, expressing it as a Bethe-vacua sum that includes non-perturbative instanton effects via the Nekrasov-Shatashvili limit. It establishes a 2d A-model/TQFT structure for the effective theory on the twisted surface, and demonstrates universal large-$N$ relations connecting the 5d theory to its 6d UV completion and 4d indices, with holographic interpretations in AdS$_6$ and RG flows to AdS$_4$. The work provides explicit calculations for Seiberg-like theories, orbifolds, and E-string theories, including Schur-limit reductions and Casimir-energy analyses, and it extends the framework to more general manifolds and black-hole entropy studies in AdS$_6$. The results offer a powerful, exact approach to linking 5d gauge dynamics with higher-dimensional UV completions and lower-dimensional protected observables, with potential applications to holography and the microstate counting of AdS$_6$ black holes.

Abstract

We derive the partition function of 5d ${\cal N}=1$ gauge theories on the manifold $S^3_b \times Σ_{\frak g}$ with a partial topological twist along the Riemann surface, $Σ_{\frak g}$. 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 ${\cal M}_3 \times Σ_{\frak g}$ with more general three-manifolds ${\cal M}_3$ and focus in particular on ${\cal M}_3=Σ_{\frak g'}\times S^{1}$, in which case the partition function relates to the entropy of black holes in AdS$_6$.

Figures (4)

• Figure 1: The 5d Seiberg theory. The node represents a $USp(2N)$ gauge group, the solid line $N_{f}$ hypermultiplets in the fundamental representation and the dashed line an antisymmetric hypermultiplet.
• Figure 2: The 5d quiver gauge theories of discussed in Bergman:2012kr. Black nodes represent $SU(2N)$ gauge groups and white ones $USp(2N)$ gauge groups. Solid lines denote bifundamental hypermultiplets and dashed lines hypermultiplets in the antisymmetric representation. In addition, at any given node, $a$, one may have $N_{f}^{a}$ number of fundamental hypermultiplets, which we have not depicted.
• Figure 3: A simple class of 5d quiver gauge theories with free energy scaling as $N^3$. Black nodes represent $SU(2N)$ gauge groups, a line connecting two nodes denotes a bifundamental hypermultiplet, and the line ending on the same node denotes an adjoint hypermultiplet. No fundamental hypermultiplets are allowed in this case.
• Figure 4: The partition function of 4d $\mathcal{N}=1$ class $\mathcal{S}$ theories on $S^{3}_{b}\times S^{1}_{\beta}$ from the partition function of $\mathcal{N}=1$ gauge theories on $S^{3}_{b}\times \Sigma_{\mathfrak{g}}$. The precise relation is given in \ref{['rel4d5d']} with the mapping of parameters in \ref{['indrel2']}.