Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

TL;DR

This paper classifies 6-dimensional homogeneous spaces with trivial canonical bundle and explores solutions to heterotic string equations on these spaces, identifying unique non-Kähler examples and constructing explicit solutions.

Contribution

It provides a classification of 6D unimodular Lie algebras with complex structures and trivial canonical bundle, and characterizes the spaces admitting invariant solutions to heterotic equations.

Findings

Classification of 6D unimodular Lie algebras with complex structures and trivial canonical bundle.

Identification of specific non-Kähler homogeneous spaces admitting invariant solutions.

Explicit construction of heterotic solutions on the Nakamura manifold.

Abstract

We classify all the $6$-dimensional unimodular Lie algebras $\mathfrak{g}$ admitting a complex structure with non-zero closed $(3,0)$-form. This gives rise to $6$-dimensional compact homogeneous spaces $M=\Gamma\backslash G$, where $\Gamma$ is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection $\nabla^{\varepsilon,\rho}$ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being Hermitian-Yang-Mills, we prove that if a compact non-K\"ahler homogeneous space $M=\Gamma\backslash G$ admits an invariant solution with respect to some non-flat connection $\nabla$ in the family $\nabla^{\varepsilon,\rho}$, then $M$ is a nilmanifold with underlying Lie algebra $\mathfrak{h}_3$, a solvmanifold with underlying algebra $\mathfrak{g}_7$, or a quotient of the semisimple group SL(2,$\mathbb{C}$). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-K\"ahler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.