Balanced Hermitian geometry on 6-dimensional nilmanifolds

TL;DR

This paper characterizes the invariant balanced Hermitian structures on 6-dimensional nilmanifolds, showing the holonomy reduction of the Bismut connection occurs only for abelian complex structures, and links these structures to solutions of the Strominger system.

Contribution

It provides a complete description of invariant balanced Hermitian structures on 6-dimensional nilmanifolds and establishes a connection to Strominger system solutions for abelian complex structures.

Findings

Holonomy of Bismut connection reduces to a proper subgroup of SU(3) iff the complex structure is abelian.

Abelian complex structures guarantee invariant balanced Hermitian structures solve the Strominger system.

Classification of invariant balanced Hermitian structures on 6-dimensional nilmanifolds.

Abstract

The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the holonomy group of the associated Bismut connection reduces to a proper subgroup of SU(3) if and only if the complex structure is abelian. As an application we show that if J is abelian then any invariant balanced J-Hermitian structure provides solutions of the Strominger system.