Geometry of Anomalies

Abstract

The purpose of the chapter is to present an interpretation of the BRST transformations, the BRST cohomology and its cocycles in the framework of the principal fiber bundle geometry. The basic idea is to use the evaluation map $ev: \textsf{P}\times \mathcal{G}\rightarrow \textsf{P}$, where $\textsf{P}$ is the principal fiber bundle where the gauge theory is defined and $\mathcal G$ is the relevant group of gauge transformations. It turns out that this simple setup contains all the information concerning the BRST transformations. Exploiting this new tool, the formulas of Chap. 4 are rewritten in a geometrical language. This treatment is extended to diffeomorphisms and local Lorentz transformations. This language is essential to understand the origin of anomalies, which lies in the geometry of the classifying space and classifying bundle. More precisely, local anomalies of field theory are rooted in the cohomology of the classifying space. It is argued that locality in quantum field theory is linked to universality. This is the main result of this chapter and a fundamental result for the comprehension of the anomaly problem. It is also a crucial preliminary for the application of the family’s index theorem in the next chapter.