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

Given a projective structure on a surface [Formula: see text] , 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 [Formula: see text] . 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 [Formula: see text] is the non-compact real form of the Fubini–Study metric on [Formula: see text] . We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.