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NUTs, Bolts, and Spindles

Matteo Kevin Crisafio, Alessio Fontanarossa, Dario Martelli

TL;DR

The paper constructs and analyzes new Euclidean supersymmetric solutions of minimal gauged supergravity with spindle-bolt topology, arising from the Plebański–Demiański family. By combining local PD geometry, ambitoric/toric structure, and global Seifert-bundle regularity, the authors derive topological and flux quantization data that determine the bulk filling and boundary theory in holography. They show that the holographically renormalized on-shell action is captured by equivariant localization, with twist and anti-twist yielding markedly different extremization behavior and leading to precise large-$N$ predictions for 3d ${\cal N}=2$ SCFTs on Seifert orbifolds. The results unify and extend known supersymmetric AdS$_4$ solutions, establish a robust boundary/bulk dictionary via flat connections and spindle data, and illuminate how boundary geometry encodes bulk filling, with potential extensions to higher dimensions and broader holographic contexts.

Abstract

We construct new infinite classes of Euclidean supersymmetric solutions of four dimensional minimal gauged supergravity comprising a $U (1) \times U (1)$-invariant asymptotically locally hyperbolic metric on the total space of orbifold line bundles over a spindle (bolt). The conformal boundary is generically a squashed, branched, lens space and the graviphoton gauge field can have either twist or anti-twist through the spindle bolt. Correspondingly, the boundary geometry inherits two types of rigid Killing spinors, that we refer to as twist and anti-twist for the three-dimensional Seifert orbifolds, as well as some specific flat connections for the background gauge field, determined by the data of the spindle bolt. For all our solutions we compute the holographically renormalized on-shell action and compare it to the expression obtained via equivariant localization, uncovering a markedly distinct behaviour in the cases of twist and anti twist. Our results provide precise predictions for the large $N$ limit of the corresponding localized partition functions of three-dimensional $\mathcal{N}=2$ superconformal field theories placed on Seifert orbifolds.

NUTs, Bolts, and Spindles

TL;DR

The paper constructs and analyzes new Euclidean supersymmetric solutions of minimal gauged supergravity with spindle-bolt topology, arising from the Plebański–Demiański family. By combining local PD geometry, ambitoric/toric structure, and global Seifert-bundle regularity, the authors derive topological and flux quantization data that determine the bulk filling and boundary theory in holography. They show that the holographically renormalized on-shell action is captured by equivariant localization, with twist and anti-twist yielding markedly different extremization behavior and leading to precise large- predictions for 3d SCFTs on Seifert orbifolds. The results unify and extend known supersymmetric AdS solutions, establish a robust boundary/bulk dictionary via flat connections and spindle data, and illuminate how boundary geometry encodes bulk filling, with potential extensions to higher dimensions and broader holographic contexts.

Abstract

We construct new infinite classes of Euclidean supersymmetric solutions of four dimensional minimal gauged supergravity comprising a -invariant asymptotically locally hyperbolic metric on the total space of orbifold line bundles over a spindle (bolt). The conformal boundary is generically a squashed, branched, lens space and the graviphoton gauge field can have either twist or anti-twist through the spindle bolt. Correspondingly, the boundary geometry inherits two types of rigid Killing spinors, that we refer to as twist and anti-twist for the three-dimensional Seifert orbifolds, as well as some specific flat connections for the background gauge field, determined by the data of the spindle bolt. For all our solutions we compute the holographically renormalized on-shell action and compare it to the expression obtained via equivariant localization, uncovering a markedly distinct behaviour in the cases of twist and anti twist. Our results provide precise predictions for the large limit of the corresponding localized partition functions of three-dimensional superconformal field theories placed on Seifert orbifolds.

Paper Structure

This paper contains 34 sections, 352 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Plebański-Demianski solutions
  3. Non-accelerating vs accelerating solutions
  4. Ambitoric structure
  5. Notable old solutions
  6. Non-accelerating solutions
  7. Local Killing spinors
  8. Global analysis
  9. Metric at the boundary
  10. Metric in the bulk
  11. Gauge field and Killing spinors in the bulk
  12. Gauge field and Killing spinors at the boundary
  13. Toric data
  14. Seifert orbifolds with twist and anti-twist
  15. Quantization conditions
  16. ...and 19 more sections

Figures (2)

  • Figure 1: Summary of supersymmetric euclidean $\mathrm{AdS}_4$ solutions to $d=4$, $\mathcal{N}=2$ minimal gauged supergravity containing nuts and bolts. In the central column the symmetry and the bulk content are sketched, while the lateral columns show when a certain supersymmetric solution is contained in another one by some limiting procedure which will be explained in detail in section \ref{['subsect:Limits to the old solutions']}.
  • Figure 2: Labelled polytope associated to normal vectors \ref{['toric-fan']}. The normal singularity $\mathbb{C}/\mathbb{Z}_{v}$ shows up as a label in the polytope. figure \ref{['fig:non-compact polytope non-zerot']} is valid for $t\neq 0$, whilst figure \ref{['fig:non-compact polytope zerot']} is for $t=0$.