# Chern-Dirac bundles on non-K\"ahler Hermitian manifolds

## TL;DR

This paper introduces Chern-Dirac bundles and operators on Hermitian manifolds, establishing their properties and connections to various cohomology theories, thus extending classical Dirac theory to non-Kähler settings.

## Contribution

It defines Chern-Dirac bundles and operators on Hermitian manifolds and links harmonic spinors to Dolbeault, Bott-Chern, and Aeppli cohomologies, extending Dirac theory.

## Key findings

- Chern-Dirac bundles generalize classical Dirac bundles to Hermitian manifolds.
- Harmonic spinors correspond to Dolbeault, Bott-Chern, and Aeppli cohomology classes.
- The tensor product of canonical and anticanonical bundles forms a bigraded Chern-Dirac bundle.

## Abstract

We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.