Topological Field Theory and Quantum Holonomy Representations of Motion Groups

TL;DR

This paper explores the canonical quantization of abelian topological field theories on manifolds, revealing how wavefunctions form multi-dimensional representations of topological groups and deriving generalized linking numbers across dimensions.

Contribution

It introduces a topological extension of the action using sheaf cohomology and uncovers new global aspects of motion group presentations in any dimension.

Findings

Wavefunctions carry multi-dimensional topological group representations

Generalized linking numbers are derived in arbitrary dimensions

New insights into motion group presentations and quantum exchange statistics

Abstract

Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms of the Cech cohomology of the underlying spatial manifold, as required for topological invariance of the quantum field theory. The wavefunctions are found in the Hamiltonian formalism and are shown to carry multi-dimensional representations of various topological groups of the space. Expressions for generalized linking numbers in any dimension are thereby derived. In particular, new global aspects of motion group presentations are obtained in any dimension. Applications to quantum exchange statistics of objects in various dimensionalities are also discussed.