Symmetries of abelian Chern-Simons theories and arithmetic

We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including U(1)(k) Chern-Simons theory and (Z(k))(l) gauge theories. For example, we prove that U(1)(k) Chern-Simons theory is time-reversal invariant if and only if -1 is a quadratic residue modulo k, which happens if and only if all the prime factors of k are Pythagorean (i.e., of the form 4n + 1), or Pythagorean with a single additional factor of 2. Many distinct non-abelian finite symmetry groups are found.

Automated summary

This study determines the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern-Simons theories, revealing their sensitivity to the arithmetic properties of the Chern-Simons levels matrix and interesting connections with number theory. The research provides a complete characterization of the symmetries of abelian topological field theories and discovers many non-trivially time-reversal invariant theories through a quantum symmetry. Additionally, it identifies various non-abelian finite symmetry groups.