A study of time reversal symmetry of abelian anyons

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title> <jats:p>We perform a study of time reversal symmetry of abelian anyons <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{A} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> in 2+1 dimensions, in the spin structure independent cases. We will find the importance of the group <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{C} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> of <jats:italic>time-reversal-symmetric anyons</jats:italic> modulo <jats:italic>anyons composed from an anyon and its time reversal</jats:italic>. Possible choices of local Kramers degeneracy are given by quadratic refinements of the braiding phases of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{C} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula>, and the anomaly is then given by the Arf invariant of the chosen quadratic refinement. We also give a concrete study of the cases when |<jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{A} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula>| is odd or <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{A}={\left({\mathrm{\mathbb{Z}}}_2\right)}^N $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mfenced> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>N</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula>.</jats:p>