In section 13.6 of Nakahara, the parity anomaly is in odd dimensional spacetime.

From the paper Fermionic Path Integral And Topological Phases by Witten, the problem appears as one cannot define the sign of the path integral,

$$S[\bar{\psi},\psi;A]=\int d^{2n+1}x\bar{\psi}iD \!\!\!\!/\,\psi,$$ $$\mathcal{Z}=\det(iD \!\!\!\!/\,)=\prod_{\lambda\in\mathrm{spec}}\lambda,$$

because there are infinite number of positive and negative eigenvalues $\lambda$.

The number of eigenvalues flowing through $\lambda=0$ is related with the index theorem in $2n+2$ dimenions.

Does the partiy anomaly appear in even dimensions?

From Nakahara's derivation, I don't see anything related with the dimension of spacetime. If this anomaly exists in odd dimensions, then why doesn't it appear in even dimensions?

  • $\begingroup$ Related: physics.stackexchange.com/q/43317/2451 $\endgroup$
    – Qmechanic
    Oct 25, 2018 at 20:58
  • $\begingroup$ @Qmechanic Is there parity anomaly in even dimensions? $\endgroup$
    – Valac
    Oct 25, 2018 at 21:02
  • $\begingroup$ Mass terms are parity invariant in even dimensions only. Thus, Pauli-Villars (which consists of introducing very massive ghost fields) may only cause an anomaly in odd dimensions. $\endgroup$ Oct 25, 2018 at 21:15
  • $\begingroup$ @AccidentalFourierTransform Thank you for your comment. Would you please elaborate the first sentence "Mass terms are parity invariant in even dimensions only." in the answer. Thanks a lot. $\endgroup$
    – Valac
    Oct 25, 2018 at 23:40


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