Symmetries of Abelian Chern-Simons Theories and Arithmetic

2019/04/29 by Delmastro, Diego, Gomis, Jaume · 2 citations
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Strongly Correlated Electrons (cond-mat.str-el)

paper · doi:10.48550/arxiv.1904.12884

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)_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.

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