Partition functions on 3d circle bundles and their gravity duals

TL;DR

This paper computes the large N limit of partition functions of 3d $ ext{N}=2$ theories on circle bundle manifolds with holographic duals, matching supergravity solutions to field theory results and exploring their relations.

Contribution

It provides the supergravity duals for 3d circle bundle manifolds and matches their on-shell actions with large N field theory partition functions, extending previous results.

Findings

Supergravity solutions for circle bundle manifolds are constructed.

On-shell actions match large N field theory partition functions.

Special cases recover known black hole entropy and free energy results.

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 $\Sigma_g$, 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 $\Sigma_g \times S^1 \cong \mathcal{M}_{g,0}$ 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.