The supersymmetric NUTs and bolts of holography

TL;DR

This work maps the landscape of supersymmetric fillings of biaxially squashed Lens space boundaries in Euclidean AdS4, revealing that a given boundary can admit multiple fillings with different topologies, notably Taub-NUT-AdS (topology $\mathbb{R}^4/\mathbb{Z}_p$) and Taub-Bolt-AdS (topology $\mathcal{M}_p$). It develops the general $SU(2)\times U(1)$-invariant solutions of four‑dimensional gauged supergravity, solves the Killing spinor equations for 1/2 and 1/4 BPS sectors, and analyzes the regular self-dual Einstein geometries (Taub-NUT-AdS and Quaternionic-Eguchi-Hanson) along with their instanton fields. The paper then studies non-self-dual bolt solutions, constructs their moduli spaces, and computes holographic free energies, establishing precise matches with large-$N$ localization results in the 1/2 and 1/4 BPS Taub-NUT-AdS cases. A crucial part of the work is the uplift to eleven dimensions, which restricts which bolt solutions admit M-theory realizations depending on the internal Sasaki-Einstein manifold $Y_7$, and it discusses the implications for dual 3d field theories, including ABJM-like theories and R-symmetry twists. Overall, the study exposes a rich phase structure of supersymmetric AdS4 fillings and clarifies how bulk topology and boundary data encode dual field-theory dynamics on curved backgrounds.

Abstract

We show that a given conformal boundary can have a rich and intricate space of supersymmetric supergravity solutions filling it, focusing on the case where this conformal boundary is a biaxially squashed Lens space. Generically we find that the biaxially squashed Lens space S^3/Z_p admits Taub-NUT-AdS fillings, with topology R^4/Z_p, as well as smooth Taub-Bolt-AdS fillings with non-trivial topology. We show that the Taub-NUT-AdS solutions always lift to solutions of M-theory, and correspondingly that the gravitational free energy then agrees with the large N limit of the dual field theory free energy, obtained from the localized partition function of a class of N=2 Chern-Simons-matter theories. However, the solutions of Taub-Bolt-AdS type only lift to M-theory for appropriate classes of internal manifold, meaning that these solutions exist only for corresponding classes of three-dimensional N=2 field theories.

Figures (3)

• Figure 1: The moduli space of 1/2 BPS solutions with biaxially squashed $S^3/\mathbb{Z}_p$ boundary, with squashing parameter $s$. The arrows denote identification of solutions on different branches. Notice that these moduli spaces are connected for each $p$, but that for $p\geq 2$ the space multiply covers the $s$-axis. The self-dual Quaternionic-Eguchi-Hanson solution QEH$_p$ appears as a special point on the positive branch for $p\geq 3$.
• Figure 2: Plots of the free energies $I(s)$ of the different branches for $p=1,2,5,12$, respectively. The first plot is the free energy of the 1/2 BPS Taub-NUT-AdS solution. In the other plots the green curve is the free energy $\tfrac{1}{p}I_\mathrm{NUT}$ of the Taub-NUT-AdS$/\mathbb{Z}_p$ solution, while the dotted line in magenta is the free energy $I_\mathrm{NUT+flux}^\mathrm{orb}$, including the contribution of $\pm\frac{p}{2}$ units of flux at the orbifold singularity. The red curve is the free energy $I_{\mathrm{Bolt_-}}$ of the negative branch. The blue curve is the free energy $I_{\mathrm{Bolt_+}}$ of the positive branch. The free energies of the special solutions are marked with points.
• Figure 4: Plots of the free energies $I(s)$ of the different branches for $p=1,2,5,12$, respectively. The dotted lines in magenta are the free energies $I_\mathrm{NUT+flux_\pm}^\mathrm{orb}$, including the contribution of $\pm\frac{p}{2}-1$ units of flux at the orbifold singularity. The red lines are the free energies $I_{\mathrm{Bolt_-}}$ of the negative branches. The blue lines are the free energies $I_{\mathrm{Bolt_+}}$ of the positive branches. The special solutions are marked with points.