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

Let $B$ and $F=\frac 12F_{\mu \nu}dx^\mu \wedge dx^\nu$ be two forms, $F_{\mu \nu}$ 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 $\Sigma$ with boundaries $\partial \Sigma$, and the existence of edge states localised at $\partial \Sigma$ 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 $2\pi$ 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 vertexComment: Latex, 64 pages, SU-4240-516 (plus 1 uuencoded compressed tar file with the figures) Figures correcte

Edge States in 4D and their 3D Groups and Fields

It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral sector of a nonchiral field theory. We study the edge states of the abelian $BF$ system in four dimensions (4d) and show the following results in almost exact analogy: 1) The Lagrangian for these states is associated with a certain 2d generalization of the KM group. It describes a coadjoint orbit of this group as a Kirillov symplectic manifold and also the corresponding group representation. 2) It can be obtained from with a ``self-dual" or ``anti-self-dual" sector of a Lagrangian describing a massless scalar and a Maxwell field [ the phrase ``self-dual" here being used essentially in its sense in monopole theory]. There are similar results for the nonabelian $BF$ system as well. These shared features of edge states in 3d and 4d suggest that the edge Lagrangians for $BF$ systems are certain natural generalizations of field theory Lagrangians related to KM groups.Comment: 12 pages, SU-4240-42

Discretized Laplacians on an Interval and their Renormalization Group

The Laplace operator admits infinite self-adjoint extensions when considered on a segment of the real line. They have different domains of essential self-adjointness characterized by a suitable set of boundary conditions on the wave functions. In this paper we show how to recover these extensions by studying the continuum limit of certain discretized versions of the Laplace operator on a lattice. Associated to this limiting procedure, there is a renormalization flow in the finite dimensional parameter space describing the dicretized operators. This flow is shown to have infinite fixed points, corresponding to the self-adjoint extensions characterized by scale invariant boundary conditions. The other extensions are recovered by looking at the other trajectories of the flow.Comment: 23 pages, 2 figures, DSF-T-28/93,INFN-NA-IV-28/93, SU-4240-54

Quasi-Topological Quantum Field Theories and Z2Z_2 Lattice Gauge Theories

We consider a two parameter family of $Z_2$ gauge theories on a lattice discretization $T(M)$ of a 3-manifold $M$ and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space $\Gamma$. We show that there is a region $\Gamma_0$ of $\Gamma$ where the partition function and the expectation value of the Wilson loop for a curve $\gamma$ can be exactly computed. Depending on the point of $\Gamma_0$, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of $M$. The Wilson loop on the other hand, does not depend on the topology of $\gamma$. However, for a subset of $\Gamma_0$, depends on the size of $\gamma$ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.Comment: 19 pages, 13 figure

On two-dimensional quasitopological field theories

We study a class of lattice field theories in two dimensions that includes gauge theories. We show that in these theories it is possible to implement a broader notion of local symmetry, based on semi-simple Hopf algebras. A character expansion is developed for the quasitopological field theories, and partition functions are calculated with this tool. Expected values of generalized Wilson loops are defined and studied with the character expansion.Comment: 17 pages, 6 figure