[Paper Review] Mapping pure gravity to strings in three-dimensional anti-de Sitter geometry

Bo Sundborg|arXiv (Cornell University)|May 31, 2013

Black Holes and Theoretical Physics3 citations

TL;DR

This paper establishes a classical map between three-dimensional pure gravity in anti-de Sitter space (AdS₃) and string theory in the same background by showing they arise as different gauge fixings of the same underlying SL(2,R) × SL(2,R) Chern-Simons gauge theory. The key result is a local, invertible correspondence between gravity solutions and string solutions via constrained Wess-Zumino-Witten (WZW) models, with winding states in string theory mapping to singular Liouville stress tensors in gravity.

ABSTRACT

Strings propagating in three-dimensional anti-de Sitter space with a background antisymmetric tensor field are well understood, even at the quantum level. Pure three-dimensional gravity with a negative cosmological constant is potentially important because of the existence of black hole solutions and an asymptotic conformal symmetry, but it is mysterious and surprisingly resistant to analysis. In this letter, the two theories are related by a map on the classical level. The map is obtained by gauge fixing the string completely, like in a light cone gauge, and comparing the resulting constrained theory with the boundary theory obtained from gravity by imposing the appropriate asymptotic boundary conditions. The two theories are formally related as different gauge fixings of the same gauge theory.

Motivation & Objective

• To resolve the long-standing ambiguity in defining a consistent quantum spectrum for 3D pure gravity with a negative cosmological constant.
• To bridge the gap between the poorly understood classical and quantum behavior of 3D AdS₃ gravity and the well-controlled string theory in the same background.
• To establish a formal, local map between AdS₃ gravity and string theory by identifying them as different gauge fixings of the same gauge-theoretic framework.
• To clarify the role of boundary conditions and constraints—particularly the Brown-Henneaux and Virasoro conditions—in connecting gravity and string solutions.
• To investigate the physical interpretation of winding modes in string theory from the gravity side, especially their potential singularities in spacetime coordinates.

Proposed method

• Formalize 3D AdS₃ gravity using the Chern-Simons formulation with SL(2,R) × SL(2,R) gauge connections, where the vielbein is expressed as $ e = A - \bar{A} $, and the equations of motion reduce to flatness conditions $ F = \bar{F} = 0 $.
• Implement asymptotic boundary conditions inspired by Brown and Henneaux, leading to a non-chiral WZW model on the boundary, which is then reduced to Liouville theory via Gauss decomposition.
• Use the Gauss decomposition $ G = ABC $ with $ A, B, C $ parametrizing SL(2,R) to express the WZW field $ G(\xi, \bar{\xi}) = g(\xi)\bar{g}(\bar{\xi}) $, and derive the Liouville field $ \Phi $ in terms of chiral fields $ \phi, \bar{\phi}, x, y $.
• Apply constraints $ J[E_+] = \kappa $, $ \bar{J}[E_-] = -\kappa $, and $ J[H] = \bar{J}[H] = 0 $ to fix the gauge freedom and relate WZW solutions to gravity solutions via the Liouville equation.
• Compare the gauge-fixing procedure in gravity (using $ J[H] = \bar{J}[H] = 0 $) with that in string theory (using Virasoro constraints $ T = \bar{T} = 0 $), showing both are consistent with the same underlying gauge symmetry.
• Invert the perspective: treat $ J[E_\pm] = \pm\kappa $ as fundamental and view $ J[H] = \bar{J}[H] = 0 $ and $ T = \bar{T} = 0 $ as alternative gauge fixings, enabling a local map between gravity and string solutions.

Experimental results

Research questions

• RQ1 Can a direct classical map be established between three-dimensional pure gravity in AdS₃ and string theory in the same background?
• RQ2 How do the different constraint structures—specifically $ J[H] = \bar{J}[H] = 0 $ in gravity versus $ T = \bar{T} = 0 $ in string theory—relate to each other within a unified gauge-theoretic framework?
• RQ3 What is the spacetime interpretation of string theory winding modes when mapped back to gravity solutions?
• RQ4 Do the Virasoro constraints in string theory and the $ J[H] $ constraints in gravity lead to equivalent solution spaces, and if not, what are the differences in their solution structure?
• RQ5 Can the classical duality between gravity and string theory in AdS₃ be extended to the quantum regime, given the well-understood quantum structure of the string side?

Key findings

• The paper establishes a local, invertible map between classical solutions of 3D AdS₃ gravity and string theory by showing both arise as different gauge fixings of the same SL(2,R) × SL(2,R) Chern-Simons gauge theory.
• The constraint $ J[E_+] = \kappa $, $ \bar{J}[E_-] = -\kappa $, which fixes the light-cone gauge, is common to both gravity and string theory, serving as a unifying anchor in the map.
• Solutions with non-zero winding numbers in string theory correspond to Liouville stress tensors with localized singularities in $ \tau - \sigma $ and $ \tau + \sigma $ coordinates, which translate to singularities in $ t - \varphi $ and $ t + \varphi $ spacetime coordinates in gravity.
• Despite these singularities in the Banados gauge, the asymptotic structure of such solutions remains globally compatible with Brown-Henneaux boundary conditions, suggesting they are physically acceptable.
• The gauge-fixing procedure in gravity (via $ J[H] = \bar{J}[H] = 0 $) algebraically fixes the WZW-to-Liouville map, while the Virasoro constraints in string theory fix only derivatives, potentially allowing for a larger solution space depending on boundary conditions.
• The existence of this classical map suggests that 3D AdS₃ gravity can be interpreted as a theory of a (hyper-)surface in a symmetric space, where the shape of the surface encodes the spacetime geometry, offering a new geometric picture of gravity.