Vertex Operators for the BF System and its Spin-Statistics Theorems

Abstract

Let $B$ and $F=\frac 12F_{μν}dx^μ\wedge dx^ν$ be two forms, $F_{μν}$ being the field strength of an abelian connection $A$. The topological $BF$ system is given by the integral of $B\wedge F$. 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 manifold $Σ$ with boundaries $\partial Σ$, and the existence of edge states localised at $\partial Σ$ 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 creatin 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 this 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