# Parity anomaly in four dimensions

## TL;DR

This paper defines and computes the parity anomaly in four-dimensional manifolds with boundary, linking spectral asymmetry of the Dirac operator to boundary Chern-Simons currents, ensuring gauge invariance and consistency.

## Contribution

It introduces a novel definition of the parity anomaly in four dimensions and computes it explicitly, establishing its relation to boundary Chern-Simons terms.

## Key findings

- The parity anomaly in 4D is associated with spectral asymmetry.
- The anomaly can be computed directly on manifolds with boundary.
- The resulting action remains gauge invariant despite unusual Chern-Simons levels.

## Abstract

In an analogy to the odd-dimensional case we define the parity anomaly as the part of the one-loop effective action for fermions associated with spectral asymmetry of the Dirac operator. This quantity is computed directly on four-dimensional manifolds with boundary and related to the Chern-Simons current on the boundary. Despite a quite unusual Chern-Simons level obtained, the action is gauge invariant and passes all consistency checks.