On the relation between fractional charge and statistics

We revisit an argument, originally given by Kivelson and Roček, for why the existence of fractional charge necessarily implies fractional statistics. In doing so, we resolve a contradiction in the original argument, and in the case of a \nu = 1/m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> Laughlin holes, we also show that the standard relation between fractional charge and statistics is necessary by an argument based on a t’Hooft anomaly in a one-form global \mathcal{Z}_m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒵</mml:mi> </mml:mstyle> <mml:mi>m</mml:mi> </mml:msub> </mml:math> symmetry.