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The Universal Phase Space of AdS3 Gravity

Carlos Scarinci, Kirill Krasnov

TL;DR

The paper presents a unified, explicit description of the AdS3 gravity phase space by embedding all fixed-topology moduli spaces into a universal space parametrized by two copies of the universal Teichmüller space ${\mathcal{T}}(1)$. It develops two complementary parametrizations—a maximal-surface cotangent-bundle picture and a generalized Mess map to ${\mathcal{T}}(1)\times{\mathcal{T}}(1)$—and demonstrates their equivalence as a symplectomorphism, grounded in the Chern-Simons formulation. The work connects holographic Fefferman-Graham data (stress tensor, Brown-Henneaux charges) to Bers-embedding quadratic differentials, giving explicit infinitesimal relations and formulas for charges, and clarifies how bulk moduli and asymptotic excitations coexist in a single, universal framework. It also clarifies how the bulk and boundary descriptions are related through harmonic decompositions of minimal Lagrangian maps and establishes a formal symplectic structure on the universal phase space, paving the way for potential quantum treatments and entropy investigations in AdS3 gravity.

Abstract

We describe what can be called the "universal" phase space of AdS3 gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the Universal Teichmüller space T(1) and is obtained from the correspondence between maximal surfaces in AdS3 and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each T(1) sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS3 spacetime) to the periods of the quadratic differentials arising via the Bers embedding of T(1)xT(1). Our construction also yields a symplectic map from T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial surface setting.

The Universal Phase Space of AdS3 Gravity

TL;DR

The paper presents a unified, explicit description of the AdS3 gravity phase space by embedding all fixed-topology moduli spaces into a universal space parametrized by two copies of the universal Teichmüller space . It develops two complementary parametrizations—a maximal-surface cotangent-bundle picture and a generalized Mess map to —and demonstrates their equivalence as a symplectomorphism, grounded in the Chern-Simons formulation. The work connects holographic Fefferman-Graham data (stress tensor, Brown-Henneaux charges) to Bers-embedding quadratic differentials, giving explicit infinitesimal relations and formulas for charges, and clarifies how bulk moduli and asymptotic excitations coexist in a single, universal framework. It also clarifies how the bulk and boundary descriptions are related through harmonic decompositions of minimal Lagrangian maps and establishes a formal symplectic structure on the universal phase space, paving the way for potential quantum treatments and entropy investigations in AdS3 gravity.

Abstract

We describe what can be called the "universal" phase space of AdS3 gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the Universal Teichmüller space T(1) and is obtained from the correspondence between maximal surfaces in AdS3 and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each T(1) sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS3 spacetime) to the periods of the quadratic differentials arising via the Bers embedding of T(1)xT(1). Our construction also yields a symplectic map from T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial surface setting.

Paper Structure

This paper contains 12 sections, 189 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Compact Spatial Topology
  3. Maximal Surfaces in AdS${}_3$ and Universal Teichmüller Space
  4. The Generalized Mess Parametrization of the AdS${}_3$ Phase Space
  5. The Generalized Mess Map
  6. Relation to the Fefferman-Graham Description: The Infinitesimal Case
  7. The Fefferman-Graham Stress Tensor and the Bers Embedding
  8. Charges
  9. The Phase Space Symplectic Structure
  10. Chern-Simons connections
  11. Discussion
  12. Appendix A: Universal Teichmüller theory

Figures (2)

  • Figure 1: Two deformation directions. One (left figure) corresponds to deforming the curve along which the maximal surface intersects the boundary. The other (middle figure) corresponds to deforming the complex structure on the maximal surface, or, geometrically, to deforming the constant "radial" coordinate foliations of the surface. A general point in the phase space deforms both the curve at infinity as well as the constant radial coordinate foliation of the maximal surface (right figure).
  • Figure 2: The diagram of maps