Gravity in 2+1 dimensions as a Riemann-Hilbert problem

TL;DR

This paper reformulates 2+1-dimensional gravity with point particles as a Riemann-Hilbert problem, providing explicit solutions for two particles and a conformal field theory representation for multiple particles.

Contribution

It introduces a gauge fixing that simplifies the problem to a Riemann-Hilbert formulation and connects gravity solutions to conformal field theory correlators.

Findings

Explicit hypergeometric solutions for two particles.

Representation of N-particle solutions via conformal field theory.

Reduction of gravity problem to a Riemann-Hilbert problem.

Abstract

In this paper we consider 2+1-dimensional gravity coupled to N point-particles. We introduce a gauge in which the $z$- and $\bar{z}$-components of the dreibein field become holomorphic and anti-holomorphic respectively. As a result we can restrict ourselves to the complex plane. Next we show that solving the dreibein-field: $e^a_z(z)$ is equivalent to solving the Riemann-Hilbert problem for the group $SO(2,1)$. We give the explicit solution for 2 particles in terms of hypergeometric functions. In the N-particle case we give a representation in terms of conformal field theory. The dreibeins are expressed as correlators of 2 free fermion fields and twistoperators at the position of the particles.