Remarks on Topological Models and Fractional Statistics

One of the most intriguing aspects of Chern-Simons-type topological models is the fractional statis-tics of point particles which has been shown essential for our understanding of the fractional quan-tum Hall eects. Furthermore these ideas are applied to the study of high Tc superconductivity. We present here an recently proposed model for fractional spin with the Pauli term. On the other hand, in D=4 space-time, a Schwarz-type topological gauge theory with antisymmetric tensor gauge eld, namely B ^ F model, is reviewed. Antisymmetric tensor elds are conjectured as mediator of string interaction. A dimensional reduction of the previous model provides a (2+1) dimensional topological theory, which involves a 2-form B and a 0-form . Some recent results on this model are reported. Recently, there have been thoughts of generalizing unusual statistics to extended objects in others space-time dimensions, and in particular to the case of strings in four dimensions. In this context, discussions about fractional spin and antisymmetric tensor eld are presented. I B ^ F Models Schwarz-type theories are purely topological in the sense that their partition functions are independent of the metric and that the only observables in these theo-ries are topological invariants of the underlying space-time manifold M. Other observables describe linking and intersection number of manifolds of any dimension. Commonly called BF systems, they are character-ized by a BRST-gauge xed quantum action which dier from the classical action only by a BRST-commutator which contains the whole metric depen-dence of the quantum action. On the other hand, since the vaccum expectation value of a BRST-commutator vanishes, these eld theories may be obtained from the classical actions [1]. Furthemore, if we denote as Q the BRST-operator which is nilpotent, in these theories the energy-momentum tensor is Q trivial, i.e.