Gravitational interpretation of the Hitchin equations

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFTtype 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 one. 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. 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. AMS Classification: Primary: 53C50; Secondary: 58J10, 83E99