标题： 异质紧化注释 显示英文标题

标题： Notes on Heterotic Compactifications

摘要： 本文技术札记描述了关于杂化紧化的一对结果。首先，我们给出一个例子，表明通常用于摄动杂化紧化的 anomaly-freedom 约束（即匹配第二陈特征类）对于涉及无挠层的紧化是不正确的。其次，我们纠正了文献中关于杂化紧化中无质量粒子计数的错误。 显示英文摘要

摘要： In this technical note we describe a pair of results on heterotic compactifications. First, we give an example demonstrating that the usual statement of the anomaly-freedom constraint for perturbative heterotic compactifications (meaning, matching second Chern characters) is incorrect for compactifications involving torsion-free sheaves. Secondly, we correct errors in the literature regarding the counting of massless particles in heterotic compactifications.