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The classification of topological phases of matter in the presence ofinteractions is an area of intense interest. One possible means ofclassification is via studying the partition function under modular transforms,as the presence of an anomalous phase arising in the edge theory of aD-dimensional system under modular transformation, or modular anomaly, signalsthe presence of a (D+1)-D non-trivial bulk. In this work, we discuss themodular 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 chiralcomplex free fermions in (2+1)-D and (3+1)-D are modular invariant. However, weshow in (3+1)-D that when the edge theory is coupled to a background U(1) gaugefield this results in the presence of a modular anomaly that is themanifestation of a quantum Hall effect in a (4+1)-D bulk. Using the modularanomaly, we find that the edge theory of (4+1)-D insulator with spacetimeinversion symmetry(P*T) and fermion number parity symmetry for each spinbecomes modular invariant when 8 copies of the edges exist.

Modular Anomalies in (2+1) and (3+1)-D Edge Theories