The Atiyah Class and Complex Structure Stabilization in Heterotic Calabi-Yau Compactifications
Lara B. Anderson, James Gray, Andre Lukas, Burt Ovrut
TL;DR
This work develops and systematizes a perturbative mechanism to stabilize complex structure moduli in heterotic Calabi–Yau compactifications by leveraging holomorphic vector bundles and their Atiyah class. It presents three equivalent descriptions—the top-down Atiyah deformation analysis, a bottom-up jumping-Ext construction, and a 4D F-term effective theory—linking geometric constraints to explicit moduli stabilization outcomes. Through explicit SU(2) extension examples and further constructions, it demonstrates that large numbers of complex structure moduli can be fixed while preserving Calabi–Yau geometry, and that higher-order obstructions and hidden-sector setups can refine and complete stabilization. The approach offers a practical, algebraic-geometry-compatible route to moduli stabilization that complements traditional flux-based methods, with clear implications for realistic heterotic model building and phenomenology.
Abstract
Holomorphic gauge fields in N=1 supersymmetric heterotic compactifications can constrain the complex structure moduli of a Calabi-Yau manifold. In this paper, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed. We review the requisite deformation theory -- including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and, hence, non-supersymmetric. In addition, two equivalent approaches to this mechanism of moduli stabilization are presented. The first is an efficient computational algorithm for determining the supersymmetric moduli space, while the second is an F-term potential in the four-dimensional theory associated with vector bundle holomorphy. These three methods are proven to be rigorously equivalent. We present explicit examples in which large numbers of complex structure moduli are stabilized. Finally, higher-order corrections to the moduli space are discussed.