Gravity duals of supersymmetric gauge theories on three-manifolds

TL;DR

The authors construct a broad class of gravity duals for ${ m N}=2$ gauge theories on curved 3-manifolds by using Euclidean self-dual solutions of four-dimensional minimal gauged supergravity with ball topology, establishing a universal holographic free-energy formula that depends only on the supersymmetric Killing vector data. They show that, for toric self-dual fillings on the four-ball, the renormalized on-shell action matches the large-$N$ localization predictions, providing an exact gauge/gravity check for theories defined on general backgrounds. The approach connects local geometric data (Toda/Kähler structures and Killing spinors) to global holographic quantities, and the paper provides explicit examples (AdS$_4$, Taub-NUT-AdS$_4$, and Plebanski-Demianski) illustrating the general construction, as well as an infinite-parameter extension via $m$-pole toric metrics. The results also relate holographic invariants to boundary conformal data and bulk topological invariants through index theorems, offering a versatile framework for computing holographic observables beyond round-sphere backgrounds. Overall, the work strengthens the gauge/gravity correspondence in curved-background settings and opens directions for extending to broader observables and higher dimensions.

Abstract

We study gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector, finding a simple closed formula when the solution has U(1) x U(1) symmetry. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localization. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories with a large N limit, defined on a general class of background three-manifold geometries.

Figures (4)

• Figure 1: Overview of the metrics discussed in the main part of the paper and in the present appendix. The arrows point from a metric to a special case of the metric, except the wavy arrow which corresponds to a conformal transformation, i.e. equation \ref{['Kahler']}.
• Figure 2: Contour plots in the $(\rho,\eta)$ quadrant.
• Figure 3: Overview of the metrics and coordinate transformations in this subsection. Straight arrows denote changes of coordinates, while wavy arrows denote conformal transformations (with the indicated conformal factor). The bottom arrow points only one way, to represent the fact that the Plebanski-Demianski metric, with a choice of Killing vector $K$, is a special case of the general self-dual Einstein metric.
• Figure 4: An illustration of the map between $(\rho,\eta)$-coordinates and $(R,S)$-coordinates: $\rho = \sqrt{R^2 - 1} \sqrt{1-S^2}$, $\eta = R S$. The conformal boundary, defined by $y^\mathrm{can}$=0, is simply a segment in the $(R,S)$ plane. This is mapped to an arc intersecting the $\rho=0$ axis at two points ($A$ and $B$). The three marked points on this axis $\eta=-1,0,1$ correspond to the location of the three monopoles in (\ref{['3-pole F']}), with $\eta=1$ corresponding to the NUT. The parameters $a,b,c$ are choosen to correspond to region $C$ in Figure 3 of CP.