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

TL;DR

This paper investigates the modular properties of edge theories in higher-dimensional topological phases, revealing anomalies linked to quantum Hall effects and conditions for modular invariance in complex systems.

Contribution

It demonstrates the presence of modular anomalies in (3+1)-D edge theories coupled to gauge fields and identifies conditions for modular invariance involving multiple copies of the system.

Findings

(3+1)-D chiral fermions are modular invariant without gauge coupling

Coupling to U(1) gauge fields introduces modular anomalies in (3+1)-D

Eight copies of the edge theory restore modular invariance in (4+1)-D with symmetries

Abstract

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.