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Symmetries of Abelian Chern-Simons Theories and Arithmetic

Diego Delmastro, Jaume Gomis

TL;DR

The paper delivers a complete characterization of unitary and anti-unitary symmetries for abelian Chern-Simons theories, revealing that symmetry structures are governed by deep arithmetic of the level data. For $U(1)_k$, time-reversal invariance occurs precisely when $-1$ is a quadratic residue modulo $k$ (i.e., $k\in\mathbb{T}$), with Lagrangian invariance further restricted to Pell-type values in $\mathbb{P}$; unitary symmetries form a finite $\mathbb{Z}_2$-factor group whose size is tied to the number of distinct primes dividing $k$. The authors extend these results to general abelian theories described by a $K$-matrix, deriving concrete anti-unitary and unitary symmetry conditions via $Q$ and $P$ matrices and presenting duality criteria that relate self-dual and dual pairs through $Q^tK_1^{-1}Q-K_2^{-1}=P$. A rich set of examples and connections to number theory (including Pell equations and Hardy-Littlewood conjectures) underscore the intricate interplay between topology and arithmetic in abelian CS theories.

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.

Symmetries of Abelian Chern-Simons Theories and Arithmetic

TL;DR

The paper delivers a complete characterization of unitary and anti-unitary symmetries for abelian Chern-Simons theories, revealing that symmetry structures are governed by deep arithmetic of the level data. For , time-reversal invariance occurs precisely when is a quadratic residue modulo (i.e., ), with Lagrangian invariance further restricted to Pell-type values in ; unitary symmetries form a finite -factor group whose size is tied to the number of distinct primes dividing . The authors extend these results to general abelian theories described by a -matrix, deriving concrete anti-unitary and unitary symmetry conditions via and matrices and presenting duality criteria that relate self-dual and dual pairs through . A rich set of examples and connections to number theory (including Pell equations and Hardy-Littlewood conjectures) underscore the intricate interplay between topology and arithmetic in abelian CS theories.

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 Chern-Simons theory and gauge theories. For example, we prove that Chern-Simons theory is time-reversal invariant if and only if is a quadratic residue modulo , which happens if and only if all the prime factors of are Pythagorean (i.e., of the form ), or Pythagorean with a single additional factor of . Many distinct non-abelian finite symmetry groups are found.

Paper Structure

This paper contains 22 sections, 18 theorems, 198 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction and Summary
  2. TQFTs and Symmetry
  3. $U(1)_k$ Chern-Simons
  4. Symmetries of $U(1)_k$
  5. Anti-unitary Symmetries
  6. Unitary Symmetries
  7. Minimal abelian TQFT
  8. $U(1)^n$ Chern-Simons theory
  9. Symmetries of $U(1)^n$
  10. Anti-unitary symmetries
  11. Unitary Symmetries.
  12. Dualities of $U(1)^n$
  13. Examples
  14. Notation and definitions.
  15. Definition 2.1.3
  16. ...and 7 more sections

Key Result

Proposition 3.1

The bosonic theory $U(1)_k$, with $k$ even, is never time-reversal invariant.

Figures (1)

  • Figure 3: Braiding of anyons: if at least one of $a,b$ is abelian, then the two lines may be unbraided, a process that generates a phase $B(a,b)\in U(1)$.

Theorems & Definitions (63)

  • Definition 2.1
  • Proposition 3.1
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.6
  • ...and 53 more