The Chiral Anomaly of the Free Fermion in Functorial Field Theory

TL;DR

This paper explores the chiral anomaly in free fermion theories within the functorial field theory framework, highlighting how the anomaly manifests as a twist involving complex lines and Clifford algebras.

Contribution

It provides a detailed construction of the anomaly theory as a functor mapping manifolds to Clifford algebras and bordisms to bimodules, clarifying the anomaly's mathematical structure.

Findings

The anomaly manifests as a twist in the functorial assignment.

The anomaly theory is constructed as a functor to Clifford algebras.

The construction clarifies the mathematical nature of the chiral anomaly.

Abstract

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.