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D1-D5 black hole microstate counting from supergravity

Vyacheslav. S. Rychkov

TL;DR

The authors directly quantize the moduli space of regular D1-D5 microstate geometries in Type IIB supergravity, modeling the moduli as a closed curve F(s) in four dimensions. Using the CWZ covariant symplectic form and a new consistency condition, they derive that the curve components obey chiral boson commutation relations with a prefactor fixed to $\alpha=\pi\mu^2$, enabling a microstate counting argument that yields a finite fraction of the D1-D5 entropy via a 4-component chiral boson system. They implement a reduction to a plane-wave background to compute the symplectic form, and show that the resulting degeneracy scales as $\Gamma\sim\exp\left(2\pi\sqrt{(c/6)N_1N_5}\right)$ with $c=4$, consistent with Cardy-like expectations for the two-charge system. The approach provides a principled SUGRA route to microstate counting and lays groundwork for extending to more complex (e.g., 3-charge) moduli spaces and horizon-entropy connections.

Abstract

We quantize the moduli space of regular D1-D5 microstates, directly from Type IIB SUGRA. The moduli space is parametrized by a smooth closed non-selfintersecting curve in four dimensions, and we derive that the components of the curve satisfy chiral boson commutation relations, with the correct value of the effective Planck constant previously conjectured using U-duality. We use the Crnkovic-Witten-Zuckerman covariant quantization method, previously used to quantize the `bubbling AdS' geometries, combined with a certain new `consistency condition' which allows us to reduce the computation to quantizing perturbations around the plane wave.

D1-D5 black hole microstate counting from supergravity

TL;DR

The authors directly quantize the moduli space of regular D1-D5 microstate geometries in Type IIB supergravity, modeling the moduli as a closed curve F(s) in four dimensions. Using the CWZ covariant symplectic form and a new consistency condition, they derive that the curve components obey chiral boson commutation relations with a prefactor fixed to , enabling a microstate counting argument that yields a finite fraction of the D1-D5 entropy via a 4-component chiral boson system. They implement a reduction to a plane-wave background to compute the symplectic form, and show that the resulting degeneracy scales as with , consistent with Cardy-like expectations for the two-charge system. The approach provides a principled SUGRA route to microstate counting and lays groundwork for extending to more complex (e.g., 3-charge) moduli spaces and horizon-entropy connections.

Abstract

We quantize the moduli space of regular D1-D5 microstates, directly from Type IIB SUGRA. The moduli space is parametrized by a smooth closed non-selfintersecting curve in four dimensions, and we derive that the components of the curve satisfy chiral boson commutation relations, with the correct value of the effective Planck constant previously conjectured using U-duality. We use the Crnkovic-Witten-Zuckerman covariant quantization method, previously used to quantize the `bubbling AdS' geometries, combined with a certain new `consistency condition' which allows us to reduce the computation to quantizing perturbations around the plane wave.

Paper Structure

This paper contains 14 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. D1-D5 geometries
  3. Fields and charges
  4. Counting D1-D5 microstates
  5. Quantization from SUGRA
  6. Symplectic form quantization
  7. Consistency condition
  8. Fixing the prefactor
  9. Simplifying assumptions
  10. Field perturbations
  11. A coordinate transformation
  12. Reduction to the plane wave
  13. Plane wave symplectic form
  14. Discussion

Figures (2)

  • Figure 1: The class of closed curves containing a unit interval $\mathcal{I}$ along $x_{1}$, and otherwise having an arbitrary profile. We evaluate the symplectic form around any curve from this class, and for the perturbations supported on $\mathcal{I}$ and directed along $x_{2}$. This symplectic form is then extended to the full moduli space by uniqueness.
  • Figure 2: The 'cutoff' $C^{\infty}$ function $\chi(r)$ interpolates smoothly between 1 at $r=0$ and 0 at $r=\infty$.