Compact weak G_2-manifolds with conical singularities

TL;DR

This paper constructs 7-dimensional compact Einstein spaces with weak G_2-holonomy and conical singularities, providing explicit metrics and cohomology relations, advancing understanding of special holonomy manifolds in M-theory.

Contribution

It introduces explicit constructions of weak G_2-manifolds with conical singularities derived from non-compact G_2-holonomy spaces, including detailed metric and cohomology analysis.

Findings

Explicit metrics on weak G_2-manifolds with conical singularities

Cohomology of these manifolds expressed in terms of the base 6-manifold

Correspondence between non-compact G_2 spaces and weak G_2-manifolds with singularities

Abstract

We construct 7-dimensional compact Einstein spaces with conical singularities that preserve 1/8 of the supersymmetries of M-theory. Mathematically they have weak G_2-holonomy. We show that for every non-compact G_2-holonomy manifold which is asymptotic to a cone on a 6-manifold Y, there is a corresponding weak G_2-manifold with two conical singularities which, close to the singularities, looks like a cone on Y. Our construction provides explicit metrics on these weak G_2-manifolds. We completely determine the cohomology of these manifolds in terms of the cohomology of Y.