Abstract
As is well known to physicists, the axial anomaly of the massless free fermion in Euclidean signature is given by the index of the corresponding Dirac operator. We use the Batalin–Vilkovisky (BV) formalism and the methods of equivariant quantization of Costello and Gwilliam to produce a new, mathematical derivation of this result. Using these methods, we formalize two conventional interpretations of the axial anomaly, the first as a violation of current conservation at the quantum level and the second as the obstruction to the existence of a well-defined fermionic partition function. Moreover, in the formalism of Costello and Gwilliam, anomalies are measured by cohomology classes in a certain obstruction–deformation complex. Our main result shows that—in the case of the axial symmetry—the relevant complex is quasi-isomorphic to the complex of de Rham forms of the space–time manifold and that the anomaly corresponds to a top-degree cohomology class which is trivial if and only if the index of the corresponding Dirac operator is zero.
Highlights
1.1 BackgroundThe axial anomaly is the failure of a certain classical symmetry of the massless free fermion to persist after quantization
It is well known to physicists that the axial anomaly is measured precisely by the index of the Dirac operator; the aim of this paper is to prove this fact in a mathematically rigorous context for perturbative quantum field theory (QFT)
The second way in which our work differs from existing treatments of anomalies is that we focus on a general approach to field theories and their quantization known as the Batalin–Vilkovisky (BV) formalism
Summary
The axial anomaly is the failure of a certain classical symmetry of the massless free fermion to persist after quantization (see [2] for an original reference on the topic). One can ask whether this symmetry persists after the massless free fermion is quantized This is in general not the case; the axial anomaly measures the obstruction to the promotion of this classical symmetry to a quantum one. There is a well-developed mathematical literature addressing many of the relevant issues in the case where one chooses B = B and restricts attention to a subspace of B which is an actual space (see [4,5,10]) These ideas make maneuvers of physicists (e.g., zeta-function regularization) precise in the special context of families of massless free fermionic field theories. We note that we work entirely in Euclidean signature; for a mathematical discussion of the axial anomaly in Lorentzian signature, see [1], which provides a historical survey of the topic
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