On exactness of the Abelian gauge field 1-form and on fractional statistics generated by Chern–Simons Lagrangian

The first part of the work deals with the model of minimal coupling between Abelian Chern–Simons field and some matter current. The exterior calculus tools are used to solve the gauge field equations in transversal, Weyl and Coulomb gauges. The second part contains a review of the model proposed by G. Semenoff (scalar matter field coupled with the topologically massive gauge field) from the viewpoint of the previous part's results. It is shown that $q$-deformed permutation algebra arises both in transversal and Weyl gauges. It is noted that the following alternative exists: the model admits only a Bose–Fermi transmutation in the non-simply connected region, or an anyonic-type field is admissible when the consideration is restricted by some contractible subset of the punctured plane.