# Spin-harmonic structures and nilmanifolds

## TL;DR

This paper introduces spin-harmonic structures on low-dimensional Riemannian manifolds, linking them to well-known special holonomy groups and providing examples of balanced Spin(7) structures on compact 8-manifolds.

## Contribution

It defines spin-harmonic structures, relates them to classical geometric structures, and constructs new examples of balanced Spin(7) structures on compact 8-manifolds.

## Key findings

- Spin-harmonic structures relate to SU(2), SU(3), and G_2 structures.
- In dimension 8, they are equivalent to balanced Spin(7) structures.
- Examples of compact 8-manifolds with non-integrable balanced Spin(7) structures are provided.

## Abstract

We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type.