$G_2$ holonomy metrics and wrapped D6-branes

TL;DR

This paper derives torsion class constraints for supergravity backgrounds from wrapped D6-branes on special Lagrangian cycles, and constructs G2-holonomy metrics via lifting to 11 dimensions, revealing geometric structures and calibrations.

Contribution

It introduces a method to determine torsion class constraints for G2-holonomy metrics from wrapped D6-branes and constructs explicit calibrating forms in these backgrounds.

Findings

Derived torsion class constraints for G2-holonomy backgrounds.

Constructed closed 3- and 4-forms calibrating cycles.

Lifted 10D backgrounds to 11D G2-holonomy manifolds.

Abstract

We determine torsion class constraints for the supergravity background produced by D6-branes wrapping special Lagrangian cycles in a Calabi-Yau 3-fold. We employ a recently introduced method which involves probing the putative background by all possible supersymmetric brane configurations. We then lift this background to 11-dimensions to a product of 4-d Minkowski space and a 7-fold of G2-holonomy. The latter is a particular U(1) bundle over an almost complex manifold of SU(3) structure with specific torsion class constraints. We construct the closed 3- and 4-forms which calibrate the 3- and 4-cycles in the G2-holonomy manifold.