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Thermodynamics of higher spin black holes in 3D

Justin R. David, Michael Ferlaino, S. Prem Kumar

TL;DR

This work analyzes the thermodynamics of spin-3 black holes in ${\rm SL}(3,\mathbb{R})\times{\rm SL}(3,\mathbb{R})$ CS theory with a spin-3 chemical potential, revealing four holonomy-allowed branches that interpolate between two AdS$_3$ backgrounds. It establishes that a UV fixed point with ${\cal W}_3^{(2)}\times{\cal W}_3^{(2)}$ symmetry governs the high-temperature regime and that Ward identities of the UV CFT precisely encode the UV/IR thermodynamic map. The authors derive a simple, consistent expression for the grand potential and show how UV variables provide a natural description of the high-$T$ phase, while detailing the geometry (wormhole vs black-hole gauge) and the puzzling features of Branch III. The results generalize to higher ${\rm SL}(N)$ theories and suggest deep connections between holonomy integrability, RG flows in AdS$_3$/CFT$_2$, and higher-spin holography, with potential checks via UV/IR Ward identities and free-field realizations.

Abstract

We examine the thermodynamic properties of recently constructed black hole solutions in SL(3,R) x SL(3,R) Chern-Simons theory in the presence of a chemical potential for spin-3 charge, which acts as an irrelevant deformation of the dual CFT with W_3 x W_3 symmetry. The smoothness or holonomy conditions admit four branches of solutions describing a flow between two AdS_3 backgrounds corresponding to two different CFTs. The dominant branch at low temperatures, connected to the BTZ black hole, merges smoothly with a thermodynamically unstable branch and disappears at higher temperatures. We confirm that the UV region of the flow satisfies the Ward identities of a CFT with W_3^(2) x W_3^(2) symmetry deformed by a spin-3/2 current. This allows to identify the precise map between UV and IR thermodynamic variables. We find that the high temperature regime is dominated by a black hole branch whose thermodynamics can only be consistently inferred with reference to this W_3^(2) x W_3^(2) CFT.

Thermodynamics of higher spin black holes in 3D

TL;DR

This work analyzes the thermodynamics of spin-3 black holes in CS theory with a spin-3 chemical potential, revealing four holonomy-allowed branches that interpolate between two AdS backgrounds. It establishes that a UV fixed point with symmetry governs the high-temperature regime and that Ward identities of the UV CFT precisely encode the UV/IR thermodynamic map. The authors derive a simple, consistent expression for the grand potential and show how UV variables provide a natural description of the high- phase, while detailing the geometry (wormhole vs black-hole gauge) and the puzzling features of Branch III. The results generalize to higher theories and suggest deep connections between holonomy integrability, RG flows in AdS/CFT, and higher-spin holography, with potential checks via UV/IR Ward identities and free-field realizations.

Abstract

We examine the thermodynamic properties of recently constructed black hole solutions in SL(3,R) x SL(3,R) Chern-Simons theory in the presence of a chemical potential for spin-3 charge, which acts as an irrelevant deformation of the dual CFT with W_3 x W_3 symmetry. The smoothness or holonomy conditions admit four branches of solutions describing a flow between two AdS_3 backgrounds corresponding to two different CFTs. The dominant branch at low temperatures, connected to the BTZ black hole, merges smoothly with a thermodynamically unstable branch and disappears at higher temperatures. We confirm that the UV region of the flow satisfies the Ward identities of a CFT with W_3^(2) x W_3^(2) symmetry deformed by a spin-3/2 current. This allows to identify the precise map between UV and IR thermodynamic variables. We find that the high temperature regime is dominated by a black hole branch whose thermodynamics can only be consistently inferred with reference to this W_3^(2) x W_3^(2) CFT.

Paper Structure

This paper contains 19 sections, 69 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and Summary
  2. ${\rm SL}(3,\mathbb R) \times {\rm SL}(3,\mathbb R)$ Chern-Simons theory
  3. Classical solutions
  4. Holonomy conditions and free energy
  5. On-shell action:
  6. Thermodynamical action:
  7. Multiple branches
  8. Low $T$ solutions:
  9. High $T$ solutions:
  10. Spacetime geometry
  11. Branch II and the extremal black hole
  12. Branches III and IV and spacetime metric
  13. The ${\cal W}_3^{(2)}$ CFT and branch III
  14. The ${\cal W}_3^{(2)}$ algebra:
  15. Ward identities for deformed ${\cal W}_3^{(2)}$ CFT
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Two different black holes dominate the ensemble at low and high temperatures. The BTZ-branch which reduces to the ordinary BTZ black hole at $\mu T=0$, merges with an unstable saddle at $\mu T_0\,=\,\tfrac{3}{16\pi}\sqrt{2\sqrt3 -3}$ and disappears. The high temperature partition function is dominated by the stable branch-III.
  • Figure 2: Thermodynamics of the four solutions to the holonomy/smoothness conditions, each carrying non-zero spin-three charge, plotted numerically for $\mu=1$.