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Black Flower Microstates

Suvankar Dutta, Shruti Menon

TL;DR

The paper constructs stationary, non-axisymmetric black flower solutions in $AdS_3$ gravity within the Chern–Simons framework using collective field theory boundary conditions, enabling angularly modulated horizons controlled by an external potential $W(\theta)$. By mapping boundary degrees of freedom to relativistic fermions via bosonization, it organizes microstates in terms of Young diagrams and shows that microscopic counting reproduces the Bekenstein–Hawking entropy, including deformation-dependent corrections in the potential parameter $\lambda$. The approach yields explicit expressions for the bulk metric, horizon structure, temperature, and entropy, and demonstrates how nontrivial boundary data leads to a consistent microscopic interpretation of black hole entropy beyond axisymmetric BTZ. This work strengthens the holographic understanding of black hole thermodynamics in AdS$_3$ and opens avenues for nonperturbative studies and extensions to higher-spin or supersymmetric contexts.

Abstract

We investigate stationary, non-axisymmetric black hole solutions in AdS$_3$ gravity, known as black flower geometries, in the Chern-Simons formulation. Boundary conditions are specified by a collective field theory-inspired Hamiltonian with field-dependent chemical potentials and angularly inhomogeneous boundary data. We construct a tractable class of solutions and analyze their geometric and thermodynamic properties, obtaining an entropy with nontrivial dependence on the angular deformation. Upon quantization of the boundary theory via bosonization, the boundary degrees of freedom are mapped to relativistic free fermions. We explicitly construct and count the microstates associated with a given black flower geometry and find exact agreement with the Bekenstein-Hawking entropy.

Black Flower Microstates

TL;DR

The paper constructs stationary, non-axisymmetric black flower solutions in gravity within the Chern–Simons framework using collective field theory boundary conditions, enabling angularly modulated horizons controlled by an external potential . By mapping boundary degrees of freedom to relativistic fermions via bosonization, it organizes microstates in terms of Young diagrams and shows that microscopic counting reproduces the Bekenstein–Hawking entropy, including deformation-dependent corrections in the potential parameter . The approach yields explicit expressions for the bulk metric, horizon structure, temperature, and entropy, and demonstrates how nontrivial boundary data leads to a consistent microscopic interpretation of black hole entropy beyond axisymmetric BTZ. This work strengthens the holographic understanding of black hole thermodynamics in AdS and opens avenues for nonperturbative studies and extensions to higher-spin or supersymmetric contexts.

Abstract

We investigate stationary, non-axisymmetric black hole solutions in AdS gravity, known as black flower geometries, in the Chern-Simons formulation. Boundary conditions are specified by a collective field theory-inspired Hamiltonian with field-dependent chemical potentials and angularly inhomogeneous boundary data. We construct a tractable class of solutions and analyze their geometric and thermodynamic properties, obtaining an entropy with nontrivial dependence on the angular deformation. Upon quantization of the boundary theory via bosonization, the boundary degrees of freedom are mapped to relativistic free fermions. We explicitly construct and count the microstates associated with a given black flower geometry and find exact agreement with the Bekenstein-Hawking entropy.
Paper Structure (21 sections, 118 equations)

This paper contains 21 sections, 118 equations.

Table of Contents

  1. Introduction
  2. A Brief Review of AdS$_3$ Gravity, Chern--Simons Theory, and Collective Field Theory Boundary Conditions
  3. A special choice of boundary Hamiltonian
  4. Black flower solution
  5. Conditions on $\lambda$ and $W(\theta)$:
  6. Redefining the angular coordinate:
  7. Horizon radii:
  8. Euclidean temperature:
  9. Periodicity of $\varphi$:
  10. Triviality of holonomy of the gauge fields:
  11. A suitable choice of $c_\pm$:
  12. Entropy:
  13. Asymptotic charges
  14. Black flower microstates
  15. Construction of black flower microstates
  16. ...and 6 more sections