Parity anomaly in four dimensions

@article{Kurkov2017ParityAI,
  title={Parity anomaly in four dimensions},
  author={Maxim Kurkov and Dmitri Vassilevich},
  journal={Physical Review D},
  year={2017},
  volume={96},
  pages={025011}
}
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 a 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. 

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