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

Abstract

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→ 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 RP2 is the non-compact real form of the Fubini–Study metric on M= SL (3 , R) / GL (2 , R). We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.