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Lambda<0 Quantum Gravity in 2+1 Dimensions I: Quantum States and Stringy S-Matrix

Kirill Krasnov

TL;DR

The work develops a quantum theory of pure gravity in AdS$_3$ with negative cosmological constant by encoding amplitudes for black-hole and point-particle processes as Liouville field theory correlators on Riemann surfaces, with point particles mapped to vertex operators and black-hole boundaries to surface boundaries. Hartle–Hawking states are defined via Euclidean path integrals and interpreted as Liouville/CFT partition functions on X or as chiral CS conformal blocks on the Schottky double $\tilde X$, linking the Lorentzian theory to a Teichmüller-space quantization. An inner product of HH states yields a naturally defined S-matrix for point-particle scattering, which takes the form of a sum over genera in a non-critical string theory whose worldsheet CFT is Liouville-like. The paper also discusses unitarity and black-hole entropy, arguing for a second-quantized Liouville theory to account for the entropy and emphasizing the holographic, stringy structure of Lambda$<0$ gravity in 2+1 dimensions. Overall, it provides a concrete framework where 3D gravity with topological sectors is recast as a string theory with Liouville dynamics on Riemann surfaces, offering insights into BH entropy and information questions in a controlled setting.

Abstract

We consider the theory of pure gravity in 2+1 dimensions, with negative cosmological constant. The theory contains simple matter in the form of point particles; the later are classically described as lines of conical singularities. We propose a formalism in which quantum amplitudes for process involving black holes and point particles are obtained as Liouville field theory (LFT) correlation functions on Riemann surfaces X. Point particles are described by LFT vertex operators, black holes (asymptotic regions) are in correspondence with boundaries of X. We analyze two examples: the amplitude for emission of a particle by the BTZ black hole, and the amplitude of black hole creation by two point particles. We then define an inner product between quantum states. The value of this inner product can be interpreted as the amplitude for one set of point particles to go into another set producing black holes. The full particle S-matrix is then given by the sum of all such amplitudes. This S-matrix is that of a non-critical string theory, with the world-sheet CFT being essentially the Liouville theory. Lambda<0 quantum gravity in 2+1 dimensions is thus a string theory.

Lambda<0 Quantum Gravity in 2+1 Dimensions I: Quantum States and Stringy S-Matrix

TL;DR

The work develops a quantum theory of pure gravity in AdS with negative cosmological constant by encoding amplitudes for black-hole and point-particle processes as Liouville field theory correlators on Riemann surfaces, with point particles mapped to vertex operators and black-hole boundaries to surface boundaries. Hartle–Hawking states are defined via Euclidean path integrals and interpreted as Liouville/CFT partition functions on X or as chiral CS conformal blocks on the Schottky double , linking the Lorentzian theory to a Teichmüller-space quantization. An inner product of HH states yields a naturally defined S-matrix for point-particle scattering, which takes the form of a sum over genera in a non-critical string theory whose worldsheet CFT is Liouville-like. The paper also discusses unitarity and black-hole entropy, arguing for a second-quantized Liouville theory to account for the entropy and emphasizing the holographic, stringy structure of Lambda gravity in 2+1 dimensions. Overall, it provides a concrete framework where 3D gravity with topological sectors is recast as a string theory with Liouville dynamics on Riemann surfaces, offering insights into BH entropy and information questions in a controlled setting.

Abstract

We consider the theory of pure gravity in 2+1 dimensions, with negative cosmological constant. The theory contains simple matter in the form of point particles; the later are classically described as lines of conical singularities. We propose a formalism in which quantum amplitudes for process involving black holes and point particles are obtained as Liouville field theory (LFT) correlation functions on Riemann surfaces X. Point particles are described by LFT vertex operators, black holes (asymptotic regions) are in correspondence with boundaries of X. We analyze two examples: the amplitude for emission of a particle by the BTZ black hole, and the amplitude of black hole creation by two point particles. We then define an inner product between quantum states. The value of this inner product can be interpreted as the amplitude for one set of point particles to go into another set producing black holes. The full particle S-matrix is then given by the sum of all such amplitudes. This S-matrix is that of a non-critical string theory, with the world-sheet CFT being essentially the Liouville theory. Lambda<0 quantum gravity in 2+1 dimensions is thus a string theory.

Paper Structure

This paper contains 10 sections, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Black hole and point particle spacetimes
  3. Analytic continuation
  4. Hartle-Hawking states
  5. Quantum amplitudes for processes involving point particles
  6. Emission of a point particle by the BTZ BH
  7. Black hole creation by two point particles
  8. Inner product and the S-matrix
  9. Remarks on unitarity and BH entropy
  10. Conclusions

Figures (4)

  • Figure 1: BTZ black hole: geometry of the time symmetry surface.
  • Figure 2: Time symmetry surface geometry of the single asymptotic region black hole with a torus wormhole inside the horizon
  • Figure 3: Symmetry surface geometry of the three asymptotic region black hole
  • Figure 4: The fundamental domain for the Schottky uniformization of a genus 2 surface.