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AdS$_3$ black holes with primary Proca hair from a regularized Gauss-Bonnet coupling

Gokhan Alkac, Murat Mesta, Gonul Unal

TL;DR

This work introduces a consistent three-dimensional Einstein-Gauss-Bonnet framework within the generalized Proca class by regularizing the Gauss-Bonnet term through Weyl geometry. It derives an asymptotically $AdS_3$ black-hole solution with primary Proca hair, and analyzes how adding a scalar-tensor Gauss-Bonnet term and electric charge modifies the geometry, including regular and stealth BTZ branches. The key contributions are the explicit 3d vector-tensor Lagrangian with a closed-form reduced action, the Noether-charge–driven solution for a hair-bearing black hole, and the exploration of charged and nontrivially coupled generalizations. The findings provide a testbed for AdS$_3$/CFT$_2$ holography with hair, and they suggest directions for thermodynamics, rotations, and holographic entropy in these lower-dimensional higher-curvature theories.

Abstract

We construct a consistent three-dimensional Einstein-Gauss-Bonnet theory as a vector-tensor theory within the generalized Proca class by employing a regularization procedure based on the Weyl geometry, which was introduced recently in \link{2504.13084}. We then obtain an asymptotically AdS$_3$, static, and circularly symmetric black hole solution with primary Proca hair. Afterward, we investigate the effect of the scalar-tensor Gauss-Bonnet coupling constructed previously by different regularization schemes. We further generalize these solutions by incorporating an electric charge. As special cases, we find a regular black hole solution in addition to charged and uncharged stealth BTZ black hole solutions.

AdS$_3$ black holes with primary Proca hair from a regularized Gauss-Bonnet coupling

TL;DR

This work introduces a consistent three-dimensional Einstein-Gauss-Bonnet framework within the generalized Proca class by regularizing the Gauss-Bonnet term through Weyl geometry. It derives an asymptotically black-hole solution with primary Proca hair, and analyzes how adding a scalar-tensor Gauss-Bonnet term and electric charge modifies the geometry, including regular and stealth BTZ branches. The key contributions are the explicit 3d vector-tensor Lagrangian with a closed-form reduced action, the Noether-charge–driven solution for a hair-bearing black hole, and the exploration of charged and nontrivially coupled generalizations. The findings provide a testbed for AdS/CFT holography with hair, and they suggest directions for thermodynamics, rotations, and holographic entropy in these lower-dimensional higher-curvature theories.

Abstract

We construct a consistent three-dimensional Einstein-Gauss-Bonnet theory as a vector-tensor theory within the generalized Proca class by employing a regularization procedure based on the Weyl geometry, which was introduced recently in \link{2504.13084}. We then obtain an asymptotically AdS, static, and circularly symmetric black hole solution with primary Proca hair. Afterward, we investigate the effect of the scalar-tensor Gauss-Bonnet coupling constructed previously by different regularization schemes. We further generalize these solutions by incorporating an electric charge. As special cases, we find a regular black hole solution in addition to charged and uncharged stealth BTZ black hole solutions.

Paper Structure

This paper contains 6 sections, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. 3d regularized Gauss-Bonnet invariant
  3. Black hole solutions with primary hair
  4. Effect of the scalar-tensor Gauss-Bonnet term
  5. Addition of an electric charge
  6. Summary and outlook

Figures (2)

  • Figure 1: Figure shows behaviors of $f$, $w_0^2$, and $w_1$ for a determined set of parameters $\Lambda_0=-1.0$, $m=2.0$, and $\beta=0.1$. Each line is generated for a value of $q\in[0,10]$ as denoted in the figure.
  • Figure 2: The metric function in \ref{['metric2']} and the (minus) Ricci scalar as a function of $r$ are shown for a fixed set of parameters as noted at the top left.