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Asymptotic dynamics of AdS$_3$ gravity with two asymptotic regions

Marc Henneaux, Wout Merbis, Arash Ranjbar

TL;DR

This work addresses how AdS$_3$ gravity with two asymptotic AdS boundaries organizes its asymptotic degrees of freedom when holonomies around non-contractible cycles are dynamical. By formulating the problem in terms of Chern-Simons theory and performing Drinfeld–Sokolov reductions at each boundary, the authors show that the global zero modes (holonomies) couple the two boundaries via a radial Wilson line, while the boundary dynamics reduce to four chiral boson theories (one for each boundary and chirality) with explicit holonomy couplings. Depending on the type of $SL(2,\,\mathbb{R})$ holonomy—hyperbolic, elliptic, or parabolic—the reduced boundary actions map to geometric actions on Virasoro coadjoint orbits, with the central charge $c=\tfrac{3\ell}{2G}$ arising in the Virasoro algebra. The analysis clarifies how the gravitational phase space, including holonomies, Wilson lines, and boundary currents, is organized and how this structure extends to more general multi-boundary or higher-spin settings, providing a cohesive framework for understanding BTZ black holes and related geometries.

Abstract

The asymptotic dynamics of AdS$_3$ gravity with two asymptotically anti-de Sitter regions is investigated, paying due attention to the zero modes, i.e., holonomies along non-contractible circles and their canonically conjugates. This situation covers the eternal black hole solution. We derive how the holonomies around the non-contractible circles couple the fields on the two different boundaries and show that their canonically conjugate variables, needed for a consistent dynamical description of the holonomies, can be related to Wilson lines joining the boundaries. The action reduces to the sum of four free chiral actions, one for each boundary and each chirality, with additional non-trivial couplings to the zero modes which are explicitly written. While the Gauss decomposition of the $SL(2,\mathbb{R})$ group elements is useful in order to treat hyperbolic holonomies, the Iwasawa decomposition turns out to be more convenient in order to deal with elliptic and parabolic holonomies. The connection with the geometric action is also made explicit. Although our paper deals with the specific example of two asymptotically anti-de Sitter regions, most of our global considerations on holonomies and radial Wilson lines qualitatively apply whenever there are multiple boundaries, independently of the form that the boundary conditions explicitly take there.

Asymptotic dynamics of AdS$_3$ gravity with two asymptotic regions

TL;DR

This work addresses how AdS gravity with two asymptotic AdS boundaries organizes its asymptotic degrees of freedom when holonomies around non-contractible cycles are dynamical. By formulating the problem in terms of Chern-Simons theory and performing Drinfeld–Sokolov reductions at each boundary, the authors show that the global zero modes (holonomies) couple the two boundaries via a radial Wilson line, while the boundary dynamics reduce to four chiral boson theories (one for each boundary and chirality) with explicit holonomy couplings. Depending on the type of holonomy—hyperbolic, elliptic, or parabolic—the reduced boundary actions map to geometric actions on Virasoro coadjoint orbits, with the central charge arising in the Virasoro algebra. The analysis clarifies how the gravitational phase space, including holonomies, Wilson lines, and boundary currents, is organized and how this structure extends to more general multi-boundary or higher-spin settings, providing a cohesive framework for understanding BTZ black holes and related geometries.

Abstract

The asymptotic dynamics of AdS gravity with two asymptotically anti-de Sitter regions is investigated, paying due attention to the zero modes, i.e., holonomies along non-contractible circles and their canonically conjugates. This situation covers the eternal black hole solution. We derive how the holonomies around the non-contractible circles couple the fields on the two different boundaries and show that their canonically conjugate variables, needed for a consistent dynamical description of the holonomies, can be related to Wilson lines joining the boundaries. The action reduces to the sum of four free chiral actions, one for each boundary and each chirality, with additional non-trivial couplings to the zero modes which are explicitly written. While the Gauss decomposition of the group elements is useful in order to treat hyperbolic holonomies, the Iwasawa decomposition turns out to be more convenient in order to deal with elliptic and parabolic holonomies. The connection with the geometric action is also made explicit. Although our paper deals with the specific example of two asymptotically anti-de Sitter regions, most of our global considerations on holonomies and radial Wilson lines qualitatively apply whenever there are multiple boundaries, independently of the form that the boundary conditions explicitly take there.

Paper Structure

This paper contains 31 sections, 159 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $U(1)$ Chern-Simons theory
  3. Proper and improper gauge symmetries
  4. Zero modes for disk topology (a single boundary)
  5. Action
  6. Do the Kac-Moody currents form a complete set of observables?
  7. Connection with geometric action
  8. Zero modes for annulus topology (two boundaries)
  9. Action
  10. Connection with non-chiral boson
  11. Do the Kac-Moody currents form a complete set of observables?
  12. Periodicity of $\mu$
  13. Gauge fixing
  14. Extension to the non-abelian case
  15. Pure gravity
  16. ...and 16 more sections

Figures (2)

  • Figure 1: A fixed time slice of the eternal black hole solution is an infinite cylinder, which is topologically equivalent to the annulus. The two asymptotically AdS$_3$ boundaries are mapped on the two boundaries of the annulus.
  • Figure 2: A contractible contour composed of (i) the outer boundary, (ii) the radial ray $\rho$ joining the outer boundary to the inner boundary at $\varphi = 0$, (iii) the inner boundary (traveled in negative direction), (iv) the ray $\rho$ traveled in the other direction.