Higher Self-Duality

Abstract

We use the totally trace-free component of Thorpe's middle curvature operator $R_{2n}:{\Lambda }^{2n}\to {\Lambda }^{2n}$ to define higher self-duality on oriented Riemannian $4n$-manifolds. We show that higher self-dual metrics are absolute minimizers of a natural conformal energy, whose Bach tensor is symmetric, trace-free, and divergence-free, resolving a question posed by Kastor. We further show that the self- and anti-self-dual blocks give a signed-square formula for the top Pontryagin density. Compact self-dual examples include the Fubini-Study metric, though Einstein metrics need not be higher Bach-flat in general.

19 pages; v2: substantially rewritten around trace-free Thorpe-Kulkarni curvature; proves higher Bach symmetry and a top Pontryagin identity