Dirac Charge Quantization and Generalized Differential Cohomology
Dirac Charge Quantization and Generalized Differential Cohomology
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狄拉克电荷量化和广义微分上同调
DOI：
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发表时间：
2000
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通讯作者：
D. Freed
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文献类型：
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作者：
D. Freed
这里的主要新结果是消除了I型超弦中的全局反常，无论有没有D-膜。我们在这里的论证依赖于用KO理论对2形式阿贝尔规范场的精确解释；反常消除是由Dirac算子族的完整Atiyah-Singer指数定理的几何形式推导出来的。这是格林-施瓦茨机制的改进版。似乎对这一机制的几何解释--在阿贝尔规范场的电耦合中抵消局部和全局费米子异常与局部和全局异常--总是以类似的方式进行。例如，之前与M.Hopkins的一篇论文(HEP-TH/0002027)解释了用这些术语消除带有D-膜的类型II中的异常。本文的重点是关于阿贝尔规范场和狄拉克电荷量子化的一般性讨论。也就是说，在广义微分上同调理论中，我们通过将阿贝尔规范场解释为余链，在泛函积分中实现了电荷的量子化。我们的论述包括基本例子和超弦理论的例子。微分上同调的数学基础目前正在开发中；我们在这里只给出一个草图。类型I中的异常抵消依赖于KO理论中某一二次型的性质，我们在与M.Hopkins合著的附录中对此进行了分析。特别地，通常的方程`‘Tr R^2=Tr F^2’被精化为KO时空理论中的方程。
The main new result here is the cancellation of global anomalies in the Type I superstring, with and without D-branes. Our argument here depends on a precise interpretation of the 2-form abelian gauge field using KO-theory; then the anomaly cancellation follows from a geometric form of the full Atiyah-Singer index theorem for families of Dirac operators. This is a refined version of the Green-Schwarz mechanism. It seems that a geometric interpretation of this mechanism-the cancellation of local and global fermion anomalies against local and global anomalies in the electric coupling of an abelian gauge field-always proceeds in a similar manner. For example, a previous paper with M. Hopkins (hep-th/0002027) explains the cancellation of anomalies in Type II with D-branes in these terms. The focal point of this paper is a general discussion about abelian gauge fields and Dirac charge quantization. Namely, we argue that quantization of charge is implemented in the functional integral by interpreting abelian gauge fields as cochains in a generalized differential cohomology theory. Our exposition includes elementary examples as well as examples from superstring theory. The mathematical underpinnings of differential cohomology are currently under development; we only give a sketch here. The anomaly cancellation in Type I depends on properties of a certain quadratic form in KO-theory, which we analyze in an appendix written jointly with M. Hopkins. In particular, the usual equation ``Tr R^2 = Tr F^2' is refined to an equation in the KO-theory of spacetime.