Gravitational anomaly and topological phases

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Since the quantum Hall effect, the notion of topological
phases of matter has been extended to those that are well-defined (or:

``protected'') in the presence of a certain set of
symmetries, and that exist in dimensions higher than two. In the (fractional)
quantum Hall effects (and in ``chiral'' topological phases in general),
Laughlin's thought experiment provides a key insight into their topological
characterization; it shows a close connection between topological phases and
quantum anomalies.

By taking various examples, I will demonstrate that
quantum anomalies serve as a useful tool to diagnose (and even define)
topological properties of the systems.

For chiral topological phases in (2+1) dimensions and
(3+1) dimensional topological superconductors, I will discuss topological
responses of the system which involve a cross correlation between thermal
transport, angular momentum, and entropy. We also argue that gravitational
anomaly is useful to study symmetry protected topological phases in (2+1)