Kahler-Einstein and Kahler scalar flat supermanifolds

TL;DR

This paper explores properties of K"ahler supermanifolds with specific potentials, establishing links between their geometric structures and scalar curvature, and deriving equations relevant to supergravity compactifications.

Contribution

It demonstrates that K"ahler-Einstein supermanifolds have bases with constant scalar curvature and characterizes scalar flat supermanifolds through a key differential equation.

Findings

K"ahler-Einstein supermanifolds' bases have constant scalar curvature.

Unique superextensions exist for constant scalar curvature K"ahler supermanifolds.

Derived equation relates base geometry to supergravity compactifications.

Abstract

Two results regarding K\"ahler supermanifolds with potential $K=A+C\theta\bar\theta$ are shown. First, if the supermanifold is K\"ahler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with K\"ahler potential $A$) has constant scalar curvature. As a corollary, every constant scalar curvature K\"ahler supermanifold has a unique superextension to a K\"ahler-Einstein supermanifold of one higher fermionic dimension. Second, if the supermanifold is itself scalar flat, then its base satisfies the equation $$ \phi^{\bar ji}\phi_{i\bar j}=2\Delta_0 S_0 + R_0^{\bar ji}R_{0i\bar j} - S_0^2, $$ 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 $\phi$ 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.