# On Heterotic Vacua with Fermionic Expectation Values

## TL;DR

This paper investigates heterotic string backgrounds with fluxes and fermionic condensates, deriving conditions for supersymmetric solutions involving gaugino expectation values on G-structure manifolds.

## Contribution

It provides a comprehensive analysis of supersymmetric heterotic flux compactifications with fermionic condensates, extending previous Calabi-Yau results to more general G-structure manifolds.

## Key findings

- Derived generic conditions for supersymmetric solutions with gaugino condensates.
- Established integrability and Lichnerowicz constraints for such backgrounds.
- Identified how fluxes and condensates modify internal geometry requirements.

## Abstract

We study heterotic backgrounds with non-trivial $H$-flux and non-vanishing expectation values of fermionic bilinears, often referred to as gaugino condensates. The gaugini appear in the low energy action via the gauge-invariant three-form bilinear $\Sigma_{MNP}={\rm tr}\:\bar\chi\Gamma_{MNP}\chi$. For Calabi-Yau compactifications to four dimensions, the gaugino condensate corresponds to an internal three-form $\Sigma_{mnp}$ that must be a singlet of the holonomy group. This condition does not hold anymore when an internal $H$-flux is turned on and ${\cal O}(\alpha')$ effects are included. In this paper we study flux compactifications to three and four-dimensions on $G$-structure manifolds. We derive the generic conditions for supersymmetric solutions. We use integrability conditions and Lichnerowicz type arguments to derive a set of constraints whose solution, together with supersymmetry, is sufficient for finding backgrounds with gaugino condensate.