Partition functions on 3d circle bundles and their gravity duals

TL;DR

The work computes the large-${N}$ partition function of 3d ${ m N}=2$ theories on ${ m M}_{g,p}$ via supersymmetric localization and Bethe-vacua analysis, uncovering an ${N^{3/2}}$ scaling controlled by a dominant vacuum. It then constructs Euclidean minimal ${ m N}=2$ gauged supergravity backgrounds with ${ m M}_{g,p}$ boundaries (NUTs and Bolts) and uplifts them to eleven dimensions, computing renormalized on-shell actions. For ABJM, the Bolt solutions reproduce the field theory partition function in the appropriate large-${N}$ limit under uplift and flux-quantization constraints, and the analysis extends to general quivers with fractional R-charges and a proposed universal twist. The results illuminate a tight link between the extremal twisted superpotential, S^3 partition functions, and holographic fillings, and they point to a universal structure governing minimal supergravity truncations and holographic duals across a broad class of ${ m 3d}$ ${ m N}=2$ theories. The framework unifies topological twists, circle fibrations, and M-theory uplifts to yield tractable large-${N}$ checks of holography and new insights into dual gravity backgrounds.

Abstract

The partition function of a three-dimensional $\mathcal{N} =2$ theory on the manifold $\mathcal{M}_{g,p}$, an $S^1$ bundle of degree $p$ over a closed Riemann surface $Σ_g$, was recently computed via supersymmetric localization. In this paper, we compute these partition functions at large $N$ in a class of quiver gauge theories with holographic M-theory duals. We provide the supergravity bulk dual having as conformal boundary such three-dimensional circle bundles. These configurations are solutions to $\mathcal{N}=2$ minimal gauged supergravity and pertain to the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving $1/4$ of the supersymmetries. We discuss the conditions for the uplift of these solutions to M-theory, and compute the on-shell action via holographic renormalization. We show that the uplift condition and on-shell action for the Bolt solutions are correctly reproduced by the large $N$ limit of the partition function of the dual superconformal field theory. In particular, the $Σ_g \times S^1 \cong \mathcal{M}_{g,0}$ partition function, which was recently shown to match the entropy of $AdS_4$ black holes, and the $S^3 \cong \mathcal{M}_{0,1}$ free energy, occur as special cases of our formalism, and we comment on relations between them.

Figures (3)

• Figure 1: Value of the on-shell action in units of $\pi/ G_4$ for the branches of solutions with $g=0$ for the values $p=1,2,4,6$ respectively. In the picture the Bolt$_+$, Bolt$_-$ solutions and the mildly singular NUT$/\mathbb{Z}_p$ plus $\pm \frac{p}{2} -1$ unit of magnetic flux are depicted.
• Figure 2: Examples of moduli space for Bolts with $g>0$. The first row refers to the cases $g=1,p=1,4$, and the last two plots refer to $g=2, p=1$ and $g=3, p=4$ respectively.
• Figure 3: This plot is showing the behavior of the on-shell action (in units of $\pi/G_4$) for the Bolt$_+$ solution and the NUT$/\mathbb{Z}_p$ with the same units of flux for $1<p<12$, as a function of $p$. One can see that the Bolt$_+$ has always lower free energy, so dominates the ensemble.