Ambient space and integration of the trace anomaly

TL;DR

This paper employs ambient space techniques to derive and interpret the integrable components of the trace anomaly in even dimensions, revealing geometric origins of ambiguities in anomaly integration.

Contribution

It introduces a geometric ambient space framework to understand the trace anomaly and its ambiguities, especially in higher even dimensions.

Findings

Ambient space clarifies the geometric nature of anomaly ambiguities.

The topological anomaly is identified as the Q-curvature.

Ambiguities increase in dimensions d≥6 and are independent of renormalization schemes.

Abstract

We use the ambient space construction, in which spacetime is mapped into a special lightcone of a higher dimensional manifold, to derive the integrable terms of the trace anomaly in even dimensions. We argue that the natural topological anomaly is the so-called $Q$-curvature, which, when projected from the ambient space, always comes with a Weyl covariant operator that can naturally be adopted for the integration of the anomaly itself in the form of a nonlocal action. The use of the ambient space makes trasparent the fact that there are some new ambiguities in the integration of the anomaly, which we now understand geometrically from the ambient point of view. These ambiguities, which manifest themselves as undetermined parameters in the integrated nonlocal action, become more severe in dimensions $d\geq 6$ and do not seem to be related to a choice of the renormalization scheme.