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 S x R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary S, a compact, oriented Riemann surface of genus greater than one. Within this framework we can interpret the 2+1 dimensional vacuum Einstein equation as a decoupled ``dual'' version of the 2 dimensional SO(3) Hitchin equations. More precisely, we prove that if over S 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 S x R whose Levi--Civita connections are precisely the lifts of this induced flat connection and the conformal class induced by this cohomology class on S agrees with the fixed one. Conversely, given a singular, flat Lorentzian metric on S x R the restriction of its Levi--Civita connection gives rise to a real solution of the SO(3) Hitchin equations on S with respect to the conformal class induced by the corresponding cohomology class of the Lorentzian metric.