Can gapped systems have gravitational anomalies?

Question

I'm interested in gravitational anomalies both which arise when coupling to curvature fluctuations or when placing the system on a manifold of non-trivial topology.

Intuitively, we can compare the curvature radius to 1/gap if we have some characteristic velocity. However, an anomaly is some discrete behavior, so even if the curvature radius is much larger than 1/gap, the effect may be hard to measure but it could still be there, so this does not rule out a gravitational anomaly of the first kind.

Answer 1

The answer to my question is Yes.

Embarrassingly, one of the simplest examples is given by the fermionic quasistring topological order I described in my paper http://arxiv.org/abs/1404.4385 . The magic is that the 5th oriented bordism group is generated by the mapping torus of complex conjugation on CP^2. Thus, if we consider the fermionic quasistring top order on CP^2, the action changes by a sign when we perform the large diffeomorphism of complex conjugation on CP^2.