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 $M \to N$. 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 $\mathbb{RP}^2$ is the non-compact real form of the Fubini-Study metric on $M=\mathrm{SL}(3, \mathrm{R})/\mathrm{GL}(2, \mathrm{R})$. We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.