Kahler-Einstein and Kahler scalar flat supermanifolds

Abstract

Two results regarding Kähler supermanifolds with potential $K=A+C\theta \bar{\theta }$ are shown. First, if the supermanifold is Kähler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with Kähler potential $A$) has constant scalar curvature. As a corollary, every constant scalar curvature Kähler supermanifold has a unique superextension to a Kähler-Einstein supermanifold of one higher fermionic dimension. Second, if the supermanifold is itself scalar flat, then its base satisfies the equation $${\varphi }^{\bar{j}i}{\varphi }_{i\bar{j}}=2{\Delta }_0{S}_0+R_0^{\bar{j}i}{R}_{0i\bar{j}}-S_0^2,$$ where ${\Delta }_0$ is the Laplace operator, $S_0$ is the scalar curvature, and $R_{0i\bar{j}}$ is the Ricci tensor of the base, and $\varphi$ is some harmonic section on the base. Remarkably, precisely this equation arises in the construction of certain supergravity compactifications. Examples of bosonic manifolds satisfying the equation above are discussed.

9 pages--reference and examples added