  We derive low-energy effective field theories for the quantum anomalous Hall
and topological superconducting phases. The quantum Hall phase is realized in
terms of free fermions with nonrelativistic dispersion relation, possessing a
global $U(1)$ symmetry. We couple this symmetry with a background gauge field
and compute the effective action by integrating out the gapped fermions. In
spite of the fact that the corresponding Dirac operator governing the dynamics
of the original fermions is nonrelativistic, the leading contribution in the
effective action is a usual Abelian $U(1)$ Chern-Simons term. The proximity to
a conventional superconductor induces a pairing potential in the quantum Hall
state, favoring the formation of Cooper pairs. When the pairing is strong
enough, it drives the system to a topological superconducting phase, hosting
Majorana fermions. Even though the continuum $U(1)$ symmetry is broken down to
a $\mathbb{Z}_2$ one, we can forge fictitious $U(1)$ symmetries that enable us
to derive the effective action for the topological superconducting phase, also
given by a Chern-Simons theory. To eliminate spurious states coming from the
artificial symmetry enlargement, we demand that the fields in the effective
action are $O(2)$ instead of $U(1)$ gauge fields. In the $O(2)$ case we have to
sum over the $\mathbb{Z}_2$ bundles in the partition function, which projects
out the states that are not $\mathbb{Z}_2$ invariants. The corresponding edge
theory is the $U(1)/\mathbb{Z}_2$ orbifold, which contains Majorana fermions in
its operator content.
