Almost Calabi-Yau with torsion 6-manifolds and the instanton condition

TL;DR

This paper investigates conditions under which the curvature of torsion connections on almost complex Calabi-Yau with torsion 6-manifolds becomes an SU(3)-instanton, revealing a link to the parallelism of torsion.

Contribution

It establishes a precise equivalence between the SU(3)-instanton condition and torsion parallelism on ACYT 6-manifolds, extending to non-compact and balanced cases.

Findings

Curvature of torsion connection is an SU(3)-instanton iff torsion is parallel.

Results apply to both compact and non-compact ACYT 6-manifolds.

Conditions identified for Strominger-Bismut connection in CYT 6-manifolds.

Abstract

It is observed that on a compact almost complex Calabi-Yau with torsion (ACYT) 6-manifold with co-closed Lee form the curvature of the torsion connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the torsion connection. The same conclusion holds for any (non necessarily compact) balanced ACYT 6-manifold. In particular, on a CYT 6-manifold the Strominger-Bismut connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the Strominger-Bismut connection provided either the CYT 6-manifold is compact with co-closed Lee form or it is a balanced CYT 6-manifold.