Topological Field Theory and Quantum Holonomy Representations of Motion Groups

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 C 8 ech 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 projective 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. 2000Academic Press

Gauge-invariant factorization and canonical quantization of topologically massive gauge theories in any dimension

“…As demonstrated later, this degeneracy is restricted by gauge invariance under large gauge transformations (LGT). Since the physical wave functional Ψ P [A] in the TFT sector coincides with that of a pure topological quantum field theory, one recovers the results of R. J. Szabo [10] who solved in the Schrödinger picture the pure topological B ∧ F theory (as well as in the presence of sources) in any dimension.…”

“…As is well-known, functions obeying such a double periodicity condition are nothing other than the generalised Riemann theta functions defined in any dimension on the complex N p -torus [10], with the compact reduced phase space resulting from the requirement of invariance under LGT,…”

“…Third, these types of quantum field theories display profound connections between mathematics and physics for what concerns topological properties related, say, to the motion group, the Ray-Singer torsion and link theory. These connections appear within the canonical quantisation 2 of these systems [10,11]. Furthermore, within the context of dynamical relativistic (quantum) field theories in any spacetime dimension, which is a general framework of potential relevance to fundamental particle physics as well as mathematical investigations for their own sake, such topological B ∧ F terms may be considered to define couplings between two independent tensor fields whose dynamics is characterised by the following action, provided the spacetime manifold M is endowed now with a Lorentzian metric structure (of mostly negative signature) allowing for the introduction of the associated Hodge * operator,…”

Exotic statistics for strings in 4d BF theory

“…As pointed out by Balanchandran and others [1,4,29,37,38], exotic statistics are possible for closed strings in 3d space, since there are topologically non-trivial ways to exchange unknotted unlinked circles in R 3 . The statistics of such theories are governed not by the braid group B n , but by a larger group: the "loop braid group" LB n .…”

“…In related work, Szabo [38] has obtained a different class of representations using BF theory with abelian gauge group. Surya [37] has also studied representations of the loop braid group.…”