# Symmetries of Abelian Chern-Simons Theories and Arithmetic

## TL;DR

This paper characterizes the symmetries of abelian Chern-Simons theories, revealing deep connections with number theory and identifying conditions for time-reversal invariance based on prime factorization.

## Contribution

It provides a complete classification of unitary and anti-unitary symmetries of abelian topological field theories, linking symmetry properties to arithmetic characteristics of the level matrix.

## Key findings

- Identifies conditions for time-reversal invariance in $U(1)_k$ theories based on quadratic residues.
- Classifies non-trivial quantum symmetries including various finite groups.
- Connects symmetry properties with prime factorization and number theory concepts.

## Abstract

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 $(\mathbb Z_k)_\ell$ 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.