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Asymptotic symmetries and thermodynamics of higher spin black holes in AdS3

Michael Ferlaino, Timothy J. Hollowood, S. Prem Kumar

TL;DR

This paper analyzes higher-spin black holes in SL(N,R) x SL(N,R) Chern-Simons theory, focusing on N=4 with a spin-4 chemical potential. It shows that the black hole solution drives an RG flow from an IR CFT with ${\cal W}_4$ symmetry to a UV fixed point governed by the non-principal ${\cal W}_4^{(2,1,1)}$ algebra, with conformal transformations generated by a twisted stress tensor $ ilde{T}$. Matching the Chern-Simons equations to UV Ward identities fixes the UV operator content and couplings, yielding a UV central charge $c_{UV}=20k/3$ and a ratio $c_{UV}/c_{IR}=1/9$, in agreement with the AdS radius ratio. The thermodynamics exhibits multiple BH branches, with the BTZ branch disappearing at high temperatures, and the UV thermodynamics is shown to be consistent with the UV fixed point description, linking holonomy constraints to phase structure and RG flow.

Abstract

We study black holes carrying higher spin charge in AdS3 within the framework of SL(N, R) x SL(N, R) Chern-Simons theory. Focussing attention on the N=4 case, we explicitly analyze the asymptotic symmetry algebra of black hole solutions with a chemical potential for spin-four charge. We demonstrate that the background describes an RG flow between an IR fixed point with W_4 symmetry and a UV fixed point with W-symmetry associated to a non-principal embedding of sl(2) in sl(4). Matching Chern-Simons equations with Ward identities of the deformed CFT, we show that the UV stress tensor is twisted by a certain U(1) current, and the flow is triggered by an operator with dimension 4/3 at the UV fixed point. We find independent confirmation of this picture via a consistent formulation of thermodynamics with respect to this UV fixed point. We further analyze the thermodynamics of multiple branches of black hole solutions for N=4,5 and find that the BTZ-branch, dominant at low temperatures, ceases to exist at higher temperatures following a merger with a thermodynamically unstable branch. We also point out an interesting connection between the RG flows and generalized KdV hierarchies.

Asymptotic symmetries and thermodynamics of higher spin black holes in AdS3

TL;DR

This paper analyzes higher-spin black holes in SL(N,R) x SL(N,R) Chern-Simons theory, focusing on N=4 with a spin-4 chemical potential. It shows that the black hole solution drives an RG flow from an IR CFT with symmetry to a UV fixed point governed by the non-principal algebra, with conformal transformations generated by a twisted stress tensor . Matching the Chern-Simons equations to UV Ward identities fixes the UV operator content and couplings, yielding a UV central charge and a ratio , in agreement with the AdS radius ratio. The thermodynamics exhibits multiple BH branches, with the BTZ branch disappearing at high temperatures, and the UV thermodynamics is shown to be consistent with the UV fixed point description, linking holonomy constraints to phase structure and RG flow.

Abstract

We study black holes carrying higher spin charge in AdS3 within the framework of SL(N, R) x SL(N, R) Chern-Simons theory. Focussing attention on the N=4 case, we explicitly analyze the asymptotic symmetry algebra of black hole solutions with a chemical potential for spin-four charge. We demonstrate that the background describes an RG flow between an IR fixed point with W_4 symmetry and a UV fixed point with W-symmetry associated to a non-principal embedding of sl(2) in sl(4). Matching Chern-Simons equations with Ward identities of the deformed CFT, we show that the UV stress tensor is twisted by a certain U(1) current, and the flow is triggered by an operator with dimension 4/3 at the UV fixed point. We find independent confirmation of this picture via a consistent formulation of thermodynamics with respect to this UV fixed point. We further analyze the thermodynamics of multiple branches of black hole solutions for N=4,5 and find that the BTZ-branch, dominant at low temperatures, ceases to exist at higher temperatures following a merger with a thermodynamically unstable branch. We also point out an interesting connection between the RG flows and generalized KdV hierarchies.

Paper Structure

This paper contains 20 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. SL$(\boldsymbol N,{\mathbb R})\times\text{SL}(\boldsymbol N,{\mathbb R})$ Chern-Simons Connections
  3. Higher spin chemical potentials and zero temperature flows
  4. Non-principal embeddings
  5. Asymptotic UV Symmetry for SL$(\boldsymbol 4,\mathbb R)$ Black Hole
  6. UV Ward identities
  7. Asymptotic algebra for the $(2,1,1)$ embedding
  8. Matching black hole parameters to ${\cal W}$-algebra currents
  9. Matching to ${\cal W}_4^{(2,1,1)}$ Ward identities
  10. Summary
  11. Thermodynamics
  12. Thermodynamic action:
  13. Holonomy conditions and phase structure
  14. Thermodynamics of the UV fixed point
  15. Discussions and Conclusions
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Energy and the grand potential for the six branches corresponding to spin-4 black holes in ${\rm SL}(4,{\mathbb R})$ Chern-Simons theory.
  • Figure 2: Energy versus temperature for real branches corresponding to spin-5 black holes from ${\rm SL}(5,{\mathbb R})$ Chern-Simons theory.