A first look at Weyl anomalies in shape dynamics

TL;DR

This paper investigates potential Weyl anomalies in shape dynamics by adapting gauge cohomology tools, concluding that under locality assumptions, no local anomalies arise in 3+1 dimensions, supporting the consistency of shape dynamics.

Contribution

It develops a Hamiltonian gauge cohomology framework to analyze Weyl anomalies in shape dynamics, providing a complete classification in 3+1 dimensions.

Findings

Spatial Weyl anomalies are always local under assumptions.

Temporal anomalies depend on the shape dynamics Hamiltonian.

No local Weyl anomalies are found in 3+1 dimensions if extra terms are local.

Abstract

One of the more popular objections towards shape dynamics is the suspicion that anomalies in the spatial Weyl symmetry will arise upon quantization. The purpose of this short paper is to establish the tools required for an investigation of the sort of anomalies that can possibly arise. The first step is to adapt to our setting Barnich and Henneaux's formulation of gauge cohomology in the Hamiltonian setting, which serve to decompose the anomaly into a spatial component and time component. The spatial part of the anomaly, i.e. the anomaly in the symmetry algebra itself ($[\Omega, \Omega]\propto \hbar$ instead of vanishing) is given by a projection of the second ghost cohomology of the Hamiltonian BRST differential associated to $\Omega$, modulo spatial derivatives. The temporal part, $[\Omega, H]\propto\hbar$ is given by a different projection of the first ghost cohomology and an extra piece arising from a solution to a functional differential equation. Assuming locality of the gauge cohomology groups involved, this part is always local. Assuming locality for the gauge cohomology groups, using Barnich and Henneaux's results, the classification of Weyl cohomology for higher ghost numbers performed by Boulanger, and following the descent equations, we find a complete characterizations of anomalies in 3+1 dimensions. The spatial part of the anomaly and the first component of the temporal anomaly are always local given these assumptions even in shape dynamics. The part emerging from the solution of the functional differential equations explicitly involves the shape dynamics Hamiltonian, and thus might be non-local. If one restricts this extra piece of the temporal anomaly to be also local, then overall no \emph{local} Weyl anomalies, either temporal or spatial, emerge in the 3+1 case.