Canonical Bf-type Topological Field Theory and Fractional Statistics of Strings

We consider BF-type topological field theory coupled to non-dynamical particle and string sources on spacetime manifolds of the form IR1 × M3, where M3 is a 3-manifold without boundary. Canonical quantization of the theory is carried out in the Hamiltonian formalism and explicit solutions of the Schrödinger equation are obtained. We show that the Hilbert space is finite dimensional and the physical states carry a one-dimensional projective representation of the local gauge symmetries. When M3 is homologically nontrivial the wavefunctions in addition carry a multi-dimensional projective representation, in terms of the linking matrix of the homology cycles of M3, of the discrete group of large gauge transformations. The wavefunctions also carry a one-dimensional representation of the non-trivial linking of the particle trajectories and string surfaces in M3. This topological field theory therefore provides a phenomenological generalization of anyons to (3 + 1) dimensions where the holonomies representing fractional statistics arise from the adiabatic transport of particles around strings. We also discuss a duality between large gauge transformations and these linking operations around the homology cycles of M3, and show that this canonical quantum field theory provides novel quantum representations of the cohomology of M3 and its associated motion group.