Comments on Heterotic Flux Compactifications

TL;DR

This paper analyzes heterotic flux compactifications with torsion, revealing conditions for smooth solutions, the role of higher-order terms, and the implications of flux and torsion on the geometry and gauge sector.

Contribution

It clarifies the geometric and physical constraints of heterotic flux compactifications, especially regarding torsion, flux, and higher-order corrections, and establishes conditions for smooth, consistent solutions.

Findings

No-go theorem for compactification with non-zero H and zero divergence of H.

Smooth solutions require non-zero divergence of H and comparable H^2 and dH.

Inclusion of α' R^2 terms is necessary for consistent truncation and solution stability.

Abstract

In heterotic flux compactification with supersymmetry, three different connections with torsion appear naturally, all in the form $\omega+a H$. Supersymmetry condition carries $a=-1$, the Dirac operator has $a=-1/3$, and higher order term in the effective action involves $a=1$. With a view toward the gauge sector, we explore the geometry with such torsions. After reviewing the supersymmetry constraints and finding a relation between the scalar curvature and the flux, we derive the squared form of the zero mode equations for gauge fermions. With $\d H=0$, the operator has a positive potential term, and the mass of the unbroken gauge sector appears formally positive definite. However, this apparent contradiction is avoided by a no-go theorem that the compactification with $H\neq 0$ and $\d H=0$ is necessarily singular, and the formal positivity is invalid. With $\d H\neq 0$, smooth compactification becomes possible. We show that, at least near smooth supersymmetric solution, the size of $H^2$ should be comparable to that of $\d H$ and the consistent truncation of action has to keep $\alpha'R^2$ term. A warp factor equation of motion is rewritten with $\alpha' R^2$ contribution included precisely, and some limits are considered.