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Patch-wise localization with Chern-Simons forms in five dimensional supergravity

Edoardo Colombo, Vasil Dimitrov, Dario Martelli, Alberto Zaffaroni

TL;DR

This work develops a patch-wise equivariant localization framework for five-dimensional gauged supergravity with vector multiplets to compute on-shell actions of general supersymmetric solutions, including nontrivial topology and gauge-field fluxes. The on-shell Lagrangian is recast as a topological term plus a total derivative, enabling the action to be determined from topological data via localization on toric bases. The authors derive a universal localized action formula that captures bulk and boundary contributions, and verify it with explicit examples such as Kerr–Newman black holes, black rings, lenses, and solitons, both in AdS and asymptotically flat settings. They connect the topological data to thermodynamic potentials, magnetic flux quantization, and flat connections through UV–IR relations, providing a robust tool to predict entropy and thermodynamics of yet-unknown AdS solutions. The results bridge holographic extremization and the equivariant volume viewpoint, offering a powerful approach to explore a broad landscape of supersymmetric gravitational configurations without solving the full field equations explicitly.

Abstract

In this paper, using equivariant localization for foliations, we compute the on-shell action of a general class of supersymmetric solutions of five-dimensional gauged supergravity with vector multiplets. Unlike previous literature, we also allow for a non-trivial topology for the space-time as well as for the gauge fields. In practice, we achieve this by covering the spacetime manifold with patches and localizing in each patch. We derive a general formula for the on-shell action that depends on topological data only and can be used without a detailed knowledge of the solution. Our final result is relevant for the physically interesting examples of multi-center black holes, black rings and black lenses, topological solitons and Euclidean black saddles. We also show how to connect the topological data with the thermodynamic data: electrostatic potential, the magnetic fluxes and the possible flat connections of the solutions. Our formula for the on-shell action is derived in the context of gauged supergravity, but it is straightforward to take the ungauged limit. Thus, we reproduce known results for a large class of explicit asymptotically flat supersymmetric solutions, and we provide predictions for AdS solutions still to be found.

Patch-wise localization with Chern-Simons forms in five dimensional supergravity

TL;DR

This work develops a patch-wise equivariant localization framework for five-dimensional gauged supergravity with vector multiplets to compute on-shell actions of general supersymmetric solutions, including nontrivial topology and gauge-field fluxes. The on-shell Lagrangian is recast as a topological term plus a total derivative, enabling the action to be determined from topological data via localization on toric bases. The authors derive a universal localized action formula that captures bulk and boundary contributions, and verify it with explicit examples such as Kerr–Newman black holes, black rings, lenses, and solitons, both in AdS and asymptotically flat settings. They connect the topological data to thermodynamic potentials, magnetic flux quantization, and flat connections through UV–IR relations, providing a robust tool to predict entropy and thermodynamics of yet-unknown AdS solutions. The results bridge holographic extremization and the equivariant volume viewpoint, offering a powerful approach to explore a broad landscape of supersymmetric gravitational configurations without solving the full field equations explicitly.

Abstract

In this paper, using equivariant localization for foliations, we compute the on-shell action of a general class of supersymmetric solutions of five-dimensional gauged supergravity with vector multiplets. Unlike previous literature, we also allow for a non-trivial topology for the space-time as well as for the gauge fields. In practice, we achieve this by covering the spacetime manifold with patches and localizing in each patch. We derive a general formula for the on-shell action that depends on topological data only and can be used without a detailed knowledge of the solution. Our final result is relevant for the physically interesting examples of multi-center black holes, black rings and black lenses, topological solitons and Euclidean black saddles. We also show how to connect the topological data with the thermodynamic data: electrostatic potential, the magnetic fluxes and the possible flat connections of the solutions. Our formula for the on-shell action is derived in the context of gauged supergravity, but it is straightforward to take the ungauged limit. Thus, we reproduce known results for a large class of explicit asymptotically flat supersymmetric solutions, and we provide predictions for AdS solutions still to be found.

Paper Structure

This paper contains 59 sections, 407 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Supersymmetric solutions of 5D gauged supergravity
  3. Symmetric scalar cosets.
  4. STU model.
  5. Minimal supergravity.
  6. Supersymmetric solutions in the time-like class
  7. The on-shell Lagrangian
  8. Geometry of solutions with U(1)$^3$ symmetry
  9. Cohomology of ${\rm d}\eta^I$
  10. Toric geometry on the base and basic cohomology.
  11. A linear relation between the $\alpha^I_a$.
  12. The limit to ungauged supergravity
  13. Patch-wise localization of the on-shell action
  14. The on-shell action with patches and interfaces
  15. The localization formula
  16. ...and 44 more sections

Figures (2)

  • Figure 1: The disposition of the patches $U_{(a)}$, interfaces $U_{(ab)}$ and intersection of interfaces $U_{(0,\ldots,d)}$ in the toric diagram for the four-dimensional base of a generic solution with $d=4$.
  • Figure 2: The triangulation of the fan.