Half-flat nilmanifolds

TL;DR

This paper introduces a double complex for Lie algebras that helps identify obstructions to half-flat SU(3)-structures and classifies 6-dimensional nilmanifolds with such structures.

Contribution

It presents a new double complex framework and provides a classification of nilmanifolds with invariant half-flat structures.

Findings

Cohomology of the double complex determines obstructions to half-flat structures.

Classification of 6-dimensional nilmanifolds with invariant half-flat SU(3)-structures.

Identification of conditions for existence of half-flat structures on nilmanifolds.

Abstract

We introduce a double complex that can be associated to certain Lie algebras, and show that its cohomology determines an obstruction to the existence of a half-flat SU(3)-structure. We obtain a classification of the 6-dimensional nilmanifolds carrying an invariant half-flat structure.