Gravitational interpretation of the Hitchin equations

Abstract

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1-dimensional vacuum general relativity theory on Σ × R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ , a compact, oriented Riemann surface of genus greater than 1. Within this framework we can interpret the 2+1-dimensional vacuum Einstein equation as a decoupled “dual” version of the two-dimensional SO(3) Hitchin equations. More precisely, we prove that if over Σ with a fixed conformal class a real solution of the SO(3) Hitchin equations with induced flat SO(2, 1) connection is given, then there exists a certain cohomology class of non-isometric, singular, flat Lorentzian metrics on Σ × R whose Levi-Civita connections are precisely the lifts of this induced flat connection and the conformal class induced by this cohomology class on Σ agrees with the fixed one. Conversely, given a singular, flat Lorentzian metric on Σ × R the restriction of its Levi-Civita connection gives rise to a real solution of the SO(3) Hitchin equations on Σ with respect to the conformal class induced by the corresponding cohomology class of the Lorentzian metric.