Global Anomalies in Canonical Gravity

TL;DR

This paper investigates the presence of global diffeomorphism anomalies in 3+1 canonical gravity with chiral fermions, showing that under certain conditions, such anomalies do not occur on specific spatial manifolds.

Contribution

It demonstrates the absence of SU(2)-induced diffeomorphism anomalies in canonical gravity on S^3 and Lens spaces, assuming the Hatcher conjecture.

Findings

No SU(2) anomalies on S^3 or Lens spaces under the conjecture

Global anomalies depend on the topology of the spatial manifold

Diffeomorphism anomalies are linked to the homotopy groups of the diffeomorphism group

Abstract

In this note we study the structure of diffeomorphism anomalies in 3+1 canonical gravity coupled to a chiral massless fermion. We find that when the spatial manifold is S^3 or a Lens space L(p,q), the first homotopy group of the related diffeomorphism group can be nontrivial and hence the question of global anomalies becomes relevant. Here we show that for gravity coupled to SU(2) chiral fermions, assuming the strong form of the Hatcher conjecture, SU(2)-induced diffeomorphism anomalies do not occur whenever the spatial manifold is S^3 or a Lens space.