  The classification of topological phases of matter in the presence of
interactions is an area of intense interest. One possible means of
classification is via studying the partition function under modular transforms,
as the presence of an anomalous phase arising in the edge theory of a
D-dimensional system under modular transformation, or modular anomaly, signals
the presence of a (D+1)-D non-trivial bulk. In this work, we discuss the
modular transformations of conformal field theories along a (2+1)-D and a
(3+1)-D edge. Using both analytical and numerical methods, we show that chiral
complex free fermions in (2+1)-D and (3+1)-D are modular invariant. However, we
show in (3+1)-D that when the edge theory is coupled to a background U(1) gauge
field this results in the presence of a modular anomaly that is the
manifestation of a quantum Hall effect in a (4+1)-D bulk. Using the modular
anomaly, we find that the edge theory of (4+1)-D insulator with spacetime
inversion symmetry(P*T) and fermion number parity symmetry for each spin
becomes modular invariant when 8 copies of the edges exist.
