The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and fractionally quantized, generalizing previous results for Wilson lines.

中文翻译：

---

Chern-Simons 不变量和杂种优势超势

四维异质优势理论中的超势包含由与紧化几何的规范丛和切丛相关的全纯陈-西蒙斯不变量产生的项。这些效应对于该理论的许多关键特征至关重要，包括真空稳定性和模量稳定性。尽管它们很重要，但文献中很少有工具可以计算给定杂种优势真空中的这种影响。在这项工作中，我们提出了新的技术来明确确定异性串紧缩中的全纯 Chern-Simons 不变量。我们计算中的关键技术成分是规范丛和切丛之间的实丛态射。我们发现，除了标准嵌入之外，还有大量示例，其中陈-西蒙斯超势消失了。