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

TL;DR

This paper computes boundary contributions to quantum anomalies for fermions in dimensions 3, 4, and 5, revealing new results in 5D and clarifying boundary effects on chiral anomalies.

Contribution

It provides explicit calculations of boundary anomalies in 3, 4, and 5 dimensions, including the first computation of the 5D anomaly with gravitational and gauge contributions.

Findings

Boundary terms dominate anomalies in odd dimensions.

New boundary contribution to chiral anomaly in 4D.

Explicit 5D anomaly including gravity and gauge fields.

Abstract

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