Dirac equation on a G_2 manifold

TL;DR

This paper explores solutions to the Dirac equation on a G_2 holonomy manifold, analyzing their behavior near singularities and implications for chiral fermions in M-theory.

Contribution

It provides a large family of solutions on G_2 manifolds with asymptotic conical geometry, including all radial solutions, and studies their behavior near singularities.

Findings

No localized, square-integrable solutions at the origin.

Results align with the absence of chiral fermions in M-theory on the conifold.

Complementary approach to previous duality and anomaly analyses.

Abstract

We find a large family of solutions to the Dirac equation on a manifold of $G_2$ holonomy asymptotic to a cone over $S^3 \times S^3$, including all radial solutions. The behaviour of these solutions is studied as the manifold developes a conical singularity. None of the solutions found are both localised and square integrable at the origin. This result is consistent with the absence of chiral fermions in M-theory on the conifold over $S^3\times S^3$. The approach here is complementary to previous analyses using dualities and anomaly cancellation.