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

TL;DR

This paper develops a formalism for 2+1 dimensional quantum gravity with negative cosmological constant, showing it can be described as a non-critical string theory with Liouville field theory correlators representing quantum amplitudes involving black holes and point particles.

Contribution

It introduces a Liouville field theory-based formalism for quantum amplitudes in 2+1D gravity, connecting black hole processes to string theory.

Findings

Quantum amplitudes are expressed as Liouville CFT correlators.

Black holes and particles are represented by boundary conditions and vertex operators.

The resulting S-matrix corresponds to a non-critical string theory.

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.