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Observables and Microscopic Entropy of Higher Spin Black Holes

Geoffrey Compère, Juan I. Jottar, Wei Song

TL;DR

The paper develops a unified, gauge-invariant framework for higher spin black hole thermodynamics in AdS3, introducing tilde (tilded) conserved charges and conjugate sources, a canonical bulk entropy, and a bulk-boundary dictionary aligned with the asymptotic W-symmetries. By analyzing spin-2 and spin-3 cases and generalizing to sl(N,R) and hs[λ], it shows that canonical observables reproduce CFT counting when expressed in tilde variables, resolving discrepancies with holomorphic formalisms. The authors demonstrate that deformed CFTs retain their $ ext{W}$-symmetry structure and provide explicit holonomy-based conditions to define vacua and ensure thermodynamic consistency. The framework yields a consistent entropy and first-law relationship via a Legendre transform of the free energy and matches CFT partition functions across the bulk-boundary dictionary, with tilded variables providing a universal map across embeddings and algebras.

Abstract

In the context of recently proposed holographic dualities between higher spin theories in AdS3 and 1+1-dimensional CFTs with W-symmetry algebras, we revisit the definition of higher spin black hole thermodynamics and the dictionary between bulk fields and dual CFT operators. We build a canonical formalism based on three ingredients: a gauge-invariant definition of conserved charges and chemical potentials in the presence of higher spin black holes, a canonical definition of entropy in the bulk, and a bulk-to-boundary dictionary aligned with the asymptotic symmetry algebra. We show that our canonical formalism shares the same formal structure as the so-called holomorphic formalism, but differs in the definition of charges and chemical potentials and in the bulk-to-boundary dictionary. Most importantly, we show that it admits a consistent CFT interpretation. We discuss the spin-2 and spin-3 cases in detail and generalize our construction to theories based on the hs[λ] algebra, and on the sl(N,R) algebra for any choice of sl(2,R) embedding.

Observables and Microscopic Entropy of Higher Spin Black Holes

TL;DR

The paper develops a unified, gauge-invariant framework for higher spin black hole thermodynamics in AdS3, introducing tilde (tilded) conserved charges and conjugate sources, a canonical bulk entropy, and a bulk-boundary dictionary aligned with the asymptotic W-symmetries. By analyzing spin-2 and spin-3 cases and generalizing to sl(N,R) and hs[λ], it shows that canonical observables reproduce CFT counting when expressed in tilde variables, resolving discrepancies with holomorphic formalisms. The authors demonstrate that deformed CFTs retain their -symmetry structure and provide explicit holonomy-based conditions to define vacua and ensure thermodynamic consistency. The framework yields a consistent entropy and first-law relationship via a Legendre transform of the free energy and matches CFT partition functions across the bulk-boundary dictionary, with tilded variables providing a universal map across embeddings and algebras.

Abstract

In the context of recently proposed holographic dualities between higher spin theories in AdS3 and 1+1-dimensional CFTs with W-symmetry algebras, we revisit the definition of higher spin black hole thermodynamics and the dictionary between bulk fields and dual CFT operators. We build a canonical formalism based on three ingredients: a gauge-invariant definition of conserved charges and chemical potentials in the presence of higher spin black holes, a canonical definition of entropy in the bulk, and a bulk-to-boundary dictionary aligned with the asymptotic symmetry algebra. We show that our canonical formalism shares the same formal structure as the so-called holomorphic formalism, but differs in the definition of charges and chemical potentials and in the bulk-to-boundary dictionary. Most importantly, we show that it admits a consistent CFT interpretation. We discuss the spin-2 and spin-3 cases in detail and generalize our construction to theories based on the hs[λ] algebra, and on the sl(N,R) algebra for any choice of sl(2,R) embedding.

Paper Structure

This paper contains 18 sections, 181 equations, 2 figures.

Table of Contents

  1. Introduction and summary of results
  2. Warm-up: $sl(2,\mathds{R})$ with deformations
  3. The $sl(3,\mathds{R})$ case
  4. Summary of the holomorphic formalism
  5. Canonical charges and conjugate potentials
  6. Canonical entropy
  7. Matching to the (deformed) CFT
  8. AdS$_{3}$ vacua in the presence of higher spin deformations
  9. Generalization to $sl(N,\mathds{R})$ and $\hbox{hs}[\lambda]$
  10. Summary of the holomorphic formalism
  11. Canonical charges and conjugate potentials
  12. Boundary term and free energy
  13. Canonical entropy
  14. Matching to the (deformed) CFT
  15. Conventions
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Plot of $\mathcal{L}$ as a function of the deformation parameter $\mu$ for the vacuum solutions.
  • Figure 2: Plot of $\mathcal{W}$ as a function of the deformation parameter $\mu$ for the vacuum solutions.