Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four

Given a projective structure on a surface $N$ , we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space M of a certain rank 2 affine bundle $M\to N$ . The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on ${\mathrm{RP}}^2$ is the non-compact real form of the Fubini-Study metric on $M=\mathrm{SL}(3,R)/\mathrm{GL}(2,R)$ . We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.