Generalized Calabi-Yau metric and Generalized Monge-Ampere equation

TL;DR

This paper derives a generalized Monge-Ampère equation characterizing generalized Calabi-Yau manifolds within generalized Kahler geometry, linking geometric conditions to supergravity solutions.

Contribution

It introduces a non-linear PDE generalizing the Monge-Ampère equation for generalized Kahler potentials, connecting geometry with supergravity backgrounds.

Findings

Derived a generalized Monge-Ampère equation for generalized Calabi-Yau conditions

Connected solutions of the PDE to supergravity backgrounds with specific fields

Provided local conditions for generalized Kahler manifolds to be Calabi-Yau

Abstract

In the neighborhood of a regular point, generalized Kahler geometry admits a description in terms of a single real function, the generalized Kahler potential. We study the local conditions for a generalized Kahler manifold to be a generalized Calabi-Yau manifold and we derive a non-linear PDE that the generalized Kahler potential has to satisfy for this to be true. This non-linear PDE can be understood as a generalization of the complex Monge-Ampere equation and its solutions give supergravity solutions with metric, dilaton and H-field.