Symmetries, Anomalies and Duality in Chern-Simons Matter Theories

Abstract

This thesis investigates the properties of quantum field theory with Chern-Simons interaction in three spacetime dimension. We focus on their symmetries and anomalies. We find many theories exhibit the phenomenon of duality - different field theories describe the same long-distance physics, and we will explore its consequence.

We start by discussing Chern-Simons matter dualities with unitary gauge groups. The theories can couple to background gauge field for the global U(1) symmetry, and we produce new dualities by promoting the fields to be dynamical. We then continue to discuss theories with orthogonal and symplectic gauge groups and their dualities. For the orthogonal gauge algebra there can be discrete levels in addition to the ordinary Chern-Simons term, and the dualities require specific discrete levels as well as precise global forms of the gauge groups. We present several consistency tests for the dualities, such as consistency under deformation by the mass terms on both sides of the duality. When the matter fields are heavy the dualities reduce at long distance to the level-rank dualities between Chern-Simons theories, which we prove rigorously. We clarify the global form of the gauge groups, and we show the level-rank dualities hold generally only between spin topological quantum field theories.

Next, we apply the dualities to describe the symmetry that only emerges in the infrared. For example, we argue that quantum electrodynamics (QED3) with two fermions has an emergent unitary O(4) symmetry. Similarly, we argue U(1)2 coupled to one scalar with the Wilson-Fisher interaction has an emergent SO(3) symmetry. We also investigate the microscopic symmetry and match its `t Hooft anomaly across the duality, thus providing another evidence for the duality.

Finally, we comment on the time-reversal symmetry T in three spacetime dimension. We find examples where the square of T does not equal the fermion parity, but instead it is modified by the Z2 magnetic symmetry. This occurs in QED3 with two fermions. We also clarify the dynamics of QED3 with fermions of higher charges using the conjectured dualities. The results are generalized to theories with SO(N) gauge group.