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3D Gravity, Point Particles and Liouville Theory

Kirill Krasnov

TL;DR

The paper proposes a quantum bulk/boundary correspondence between 3D gravity with negative cosmological constant and Liouville field theory (LFT), arguing that LFT non-normalizable states correspond to bulk point particles. It grounds this in the semiclassical limit by matching the gravity action for AdS with a line of conical singularities to the fixed-area Liouville action, with the identification $Q^2 = \frac{l}{4G\hbar}$ and the central charge $c_L = 1+6Q^2$. The authors demonstrate that the boundary vertex operators with $0<\alpha<Q/2$ implement bulk defects and that the mass–deficit relation $M l = \Delta+\bar{\Delta}$ emerges from the CFT data, while the bulk/boundary partition functions agree in the semiclassical regime. They discuss broader implications for holography, lattice quantum gravity models, and potential extensions to black hole physics and AdS$_3$/string theory, indicating a rich interplay between quantum LFT and 3D quantum gravity.

Abstract

This paper elaborates on the bulk/boundary relation between negative cosmological constant 3D gravity and Liouville field theory (LFT). We develop an interpretation of LFT non-normalizable states in terms of particles moving in the bulk. This interpretation is suggested by the fact that ``heavy'' vertex operators of LFT create conical singularities and thus should correspond to point particles moving inside AdS. We confirm this expectation by comparing the (semi-classical approximation to the) LFT two-point function with the (appropriately regularized) gravity action evaluated on the corresponding metric.

3D Gravity, Point Particles and Liouville Theory

TL;DR

The paper proposes a quantum bulk/boundary correspondence between 3D gravity with negative cosmological constant and Liouville field theory (LFT), arguing that LFT non-normalizable states correspond to bulk point particles. It grounds this in the semiclassical limit by matching the gravity action for AdS with a line of conical singularities to the fixed-area Liouville action, with the identification and the central charge . The authors demonstrate that the boundary vertex operators with implement bulk defects and that the mass–deficit relation emerges from the CFT data, while the bulk/boundary partition functions agree in the semiclassical regime. They discuss broader implications for holography, lattice quantum gravity models, and potential extensions to black hole physics and AdS/string theory, indicating a rich interplay between quantum LFT and 3D quantum gravity.

Abstract

This paper elaborates on the bulk/boundary relation between negative cosmological constant 3D gravity and Liouville field theory (LFT). We develop an interpretation of LFT non-normalizable states in terms of particles moving in the bulk. This interpretation is suggested by the fact that ``heavy'' vertex operators of LFT create conical singularities and thus should correspond to point particles moving inside AdS. We confirm this expectation by comparing the (semi-classical approximation to the) LFT two-point function with the (appropriately regularized) gravity action evaluated on the corresponding metric.

Paper Structure

This paper contains 4 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: quantum Liouville theory
  3. Particle interpretation
  4. Discussion

Figures (3)

  • Figure 1: The AdS space with a line of conical singularities can be described as AdS with a wedge removed.
  • Figure 2: Complex planes $z$ and $w$. The map from $w$ to $z$ is a covering, which is branched at fixed points of elliptic generators, in our case at $0,\infty$.
  • Figure 3: Surfaces of the upper-half space calculation.