A Note on a Standard Embedding on Half-Flat Manifolds

TL;DR

This paper explores the compactification of heterotic string theory on half-flat manifolds, showing how standard embedding with torsionful connections leads to gauge symmetry breaking and anomaly cancellation, aligning with recent findings.

Contribution

It demonstrates that standard embedding on half-flat manifolds involves torsionful connections, resulting in gauge symmetry breaking similar to Calabi-Yau cases, and confirms recent results using a different approach.

Findings

Compactification on half-flat manifolds involves torsionful connections.

Standard embedding leads to E8×E8 breaking to E6×E8.

Anomaly cancellation is achieved via torsionful connections.

Abstract

It is argued that the ten dimensional solution that corresponds to the compactification of $E_8 \times E_8$ heterotic string theory on a half-flat manifold is the product space-time $R^{1,2} \times Z_7$ where $Z_7$ is a generalized cylinder with $G_2$ riemannian holonomy. Standard embedding on $Z_7$ then implies an embedding on the half-flat manifold which involves the torsionful connection rather than the Levi-Civita connection. This leads to the breakdown of $E_8 \times E_8$ to $E_6 \times E_8$, as in the case of the standard embedding on Calabi-Yau manifolds, which agrees with the result derived recently by Gurrieri, Lukas and Micu (arXiv:0709.1932) using a different approach. Green-Schwarz anomaly cancellation is then implemented via the torsionful connection on half-flat manifolds.