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

J Geom Anal. 2018;28(3):2780-2811. doi: 10.1007/s12220-017-9934-9. Epub 2017 Oct 12.


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 RP 2 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.

Keywords: Anti-self-dual metrics; Cartan geometry; Einstein metrics; Gauge theory; Projective structures.