VERTEX OPERATORS FOR THE BF SYSTEM AND ITS SPIN–STATISTICS THEOREMS

Abstract

Let B and [Formula: see text] be two-forms, Fµν being the field strength of an Abelian connection A. The topological BF system is given by the integral of B ∧ F. With "kinetic energy" terms added for B and A, it generates a mass for A, thereby suggesting an alternative to the Higgs mechanism, and also gives the London equations. The BF action, being the large length and time scale limit of this augmented action, is thus of physical interest. In earlier work, it has been studied on spatial manifolds Σ with boundaries ∂Σ, and the existence of edge states localized at ∂Σ has been established. They are analogous to the conformal family of edge states to be found in a Chern–Simons theory in a disc. Here we introduce charges and vortices (thin flux tubes) as sources in the BF system and show that they acquire an infinite number of spin excitations due to renormalization, just as a charge coupled to a Chern–Simons potential acquires a conformal family of spin excitations. For a vortex, these spins are transverse and attached to each of its points, so that it resembles a ribbon. Vertex operators for the creation of these sources are constructed and interpreted in terms of a Wilson integral involving A and a similar integral involving B. The standard spin–statistics theorem is proved for these sources. A new spin–statistics theorem, showing the equality of the "interchange" of two identical vortex loops and 2π rotation of the transverse spins of a constituent vortex, is established. Aharonov–Bohm interactions of charges and vortices are studied. The existence of topologically nontrivial vortex spins is pointed out and their vertex operators are also discussed.