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In the presence of boundaries, the quantum anomalies acquire additionalboundary terms. In odd dimensions the integrated conformal anomaly, for whichthe bulk contribution is known to be absent, is non-trivial due to the boundaryterms. These terms became a subject of active study in the recent years. In thepresent paper we continue our previous study [1], [2] and compute explicitlythe anomaly for fermions in dimensions $d=3, \ 4 \ $ and $5$. The calculationin dimension $5$ is new. It contains both contributions of the gravitationalfield and the gauge fields to the anomaly. In dimensions $d=3$ and $4$ wereproduce and clarify the derivation of the results available in theliterature. Imposing the conformal invariant mixed boundary conditions forfermions in odd dimension $d$ we particularly pay attention to the necessity ofchoosing the doubling representation for gamma matrices. In this representationthere exists a possibility to define chirality and thus address the question ofthe chiral anomaly. The anomaly is entirely due to terms defined on theboundary. They are calculated in the present paper in dimensions $d=3$ and $5$due to both gravitational and gauge fields. To complete the picture were-evaluate the chiral anomaly in $d=4$ dimensions and find a new boundary termthat is supplementary to the well-known Pontryagin term.

Fermions, boundaries and conformal and chiral anomalies in $d=3,\ 4$ and $5$ dimensions