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

TL;DR

This paper develops a topological quantum field theory in (3+1) dimensions that generalizes anyonic statistics, showing how fractional statistics emerge from linking particles and strings, with explicit quantization and state representations.

Contribution

It introduces a canonical BF-type topological field theory with explicit solutions, revealing new quantum representations related to homology and linking in 3-manifolds.

Findings

Hilbert space is finite dimensional

Wavefunctions carry projective representations of gauge symmetries

Fractional statistics arise from linking of particles and strings

Abstract

We consider BF-type topological field theory coupled to non-dynamical particle and string sources on spacetime manifolds of the form $\IR^1\times\MT$, where $\MT$ is a 3-manifold without boundary. Canonical quantization of the theory is carried out in the Hamiltonian formalism and explicit solutions of the Schr\"odinger 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 $\MT$ is homologically non-trivial the wavefunctions in addition carry a multi-dimensional projective representation, in terms of the linking matrix of the homology cycles of $\MT$, 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 $\MT$. 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 $\MT$, and show that this canonical quantum field theory provides novel quantum representations of the cohomology of $\MT$ and its associated motion group.