Dirac Charge Quantization and Generalized Differential Cohomology

TL;DR

This paper explores the cancellation of anomalies in Type I superstring theory using KO-theory and generalized differential cohomology, providing a geometric interpretation of the Green-Schwarz mechanism and charge quantization.

Contribution

It introduces a refined geometric framework for anomaly cancellation in superstring theory via KO-theory and generalized differential cohomology, extending previous approaches.

Findings

Anomaly cancellation depends on properties of a quadratic form in KO-theory.

Charge quantization is interpreted through generalized differential cohomology.

The paper refines the equation Tr R^2 = Tr F^2 within KO-theory.

Abstract

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.