The Chiral Anomaly of the Free Fermion in Functorial Field Theory
Matthias Ludewig, Saskia Roos
TL;DR
This work analyzes the chiral anomaly of the free fermion within the language of twisted, functorial field theory. It constructs an anomaly theory T that assigns to a (d−1)-dimensional spin manifold Y the Clifford algebra $ ext{Cl}(W_Y)$ and to a d-dimensional bordism X a bimodule $T(X)=\Lambda L_X \otimes \Lambda^{ ext{top}}(\mathcal{H}_{X^{\mathrm{cl}}})_J$, with Lagrangian boundary data coming from harmonic spinors. Functoriality is established up to controlled isomorphisms via a Gluing Theorem for Lagrangian relations and a coherence theorem, and the finite-dimensional obstruction K explains the algebraic origin of the anomaly as a failure of strict second quantization to be functorial. The anomaly theory connects to broader index-theoretic structures and provides a pathway to extended TFT formulations, including connections to the Dai–Freed framework and Connes fusion in operator-algebraic settings. Overall, the paper reframes the fermionic determinant as a determinant-line-valued object that encodes index-theoretic information through a geometrically and algebraically coherent twist.
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.