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) fourdimensional quantum BF -theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M .

Topological Field Theory and Quantum Holonomy Representations of Motion Groups

“…From the source continuity equation 16) and from the definitions we find 20) and which are orthonormal,…”

String holonomy and extrinsic geometry in four-dimensional topological gauge theory

“…The present formalism, which involves non-dynamical point particles, naturally incorporates the particle-string Aharonov-Bohm phases [17,19], extrinsic curvature terms [18,19], and similar long-range string intersection interactions [19] that have been discussed extensively in Higgs models. The emergence of smooth surface invariants in this topological field theory is intriguing in light of recent work [40] on observables in nonabelian BF theories which suggests that surface observables yield possibly new invariants of immersed surfaces in 4-manifolds. In the case of non-topological deformations of BF theory, these observables may be relevant to the quark confinement problem [14].…”

Strings with extended non-Abelian gauge interaction

“…The horizontal paths with respect to a connection A on a given principal bundle P create the principal G-bundle P A that is an example of principal G-bundle on the path space. A connectionĀ with a nonabelian 2-form B define the special connection on the G-bundle P A [4]. Let R be an irreducible representation R of the compact gauge group G. The product of R and its contragradient representationR can be expanded to the direct sum of irreducible representations R i :…”

“…We begin with putting together an embedded cylinder Cyl ∈ LP Σ, the bundles (3.16) valued tensors (G, B), and a pair of connections (A,Ā) in the representation R. As a extension of the holonomy construction for the special connection [4], the gauge invariant which comprises all these objects can be constructed:…”