Trace Anomaly of Weyl Fermions via the Path Integral

We compute the trace, diffeomorphism and Lorentz anomalies of a free Weyl fermion in a gravitational background field by path integral methods. This is achieved by regularising the variation of the determinant of the Weyl operator building on earlier work by Leutwyler. The trace anomaly is found to be one half of the one of a Dirac fermion. Most importantly we establish that the potential parity-odd curvature term $R \tilde R$, corresponding to the Pontryagin density, vanishes. This is to the contrary of some recent findings in the literature which gave rise to a controversy. We verify, that the regularisation does not lead to (spurious) anomalies in the Lorentz and diffeomorphism symmetries. We argue that in $d = 2\;(\textrm{mod } 4)$ $P$- and $CP$-odd terms cannot appear and that for $d = 4\;(\textrm{mod } 4)$ they are absent at least at leading order.