Loop and Path Spaces and Four-Dimensional BF Theories: Connections, Holonomies and Observables

Abstract

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M.

TeX, 39 pages

References in corpus (1)

• Yang-Mills Theory as a Deformation of Topological Field Theory, Dimensional Reduction and Quark Confinement

Cited by in corpus (13)

• Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops
• Configuration spaces and Vassiliev classes in any dimension
• Wilson surfaces and higher dimensional knot invariants
• Parallel Transport over Path Spaces
• Non-Abelian Stokes theorem in action
• Non-abelian higher gauge theory and categorical bundle
• Topological Field Theory and Quantum Holonomy Representations of Motion Groups
• String Bracket and Flat Connections
• Negative forms and path space forms
• Strings with extended non-Abelian gauge interaction
• Holonomy and parallel transport in the differential geometry of the space of loops and the groupoid of generalized gauge transformations
• Thin homotopy and the holonomy approach to gauge theories
• Geometric prequantization on the path space of a prequantized manifold