Connections on the path bundle of a principal fiber bundle

Abstract

Let ( P, M, π, G) be a principal fiber bundle. In the first section of this paper, we consider the Fréchet manifold PP of smooth paths in P along with important Fréchet submanifolds such as the space P P of paths in P which emanate from some fixed point u 0 ∈ P and the space π ∗ u0 ( L x 0 M) of paths at u 0 in P which project to the set L x 0 M of loops of M at x 0 = π( u 0). These Fréchet manifolds of paths of P are principal fiber bundles over corresponding path spaces of M with structure group a subgroup of the Fréchet Lie group of paths in G. The main result of Section 1 asserts that each connection of P induces a global section of the principal bundle P uoP →P xoM and also of the bundle ∗π u 0( L x 0 M) → L x 0 M. These sections may be used to show that both path bundles are trivial in the strong sense that they factor in the category of Fréchet manifolds. Section 2 considers connections on various principal bundles of paths of P such as PP, P uoP, π ∗ uo(P xo M) and π ∗ u 0( L x 0 M). Connections are defined as they usually are on any principal bundle except that the set of horizontal vectors þ γ at a path γ is required to be a C ∞( I, R )- submodule of the tangent space at γ for γ in PP, P uoP, π ∗ uo(P xo M) or π ∗ u 0( L x 0 M). We show that generally connections in this context define, along each path γ in P, a set of “horizontal” subspaces of P with the assignment of “horizontal” spaces along one path generally differing from the assignment of “horizontal” spaces along another path even when the two paths share common points of P. We characterize which connections on path space are induced from connections on P. We use the triviality of P uoP → P xoM to obtain an easy criterion for defining connections on P uoP and its submanifolds. In Section 3, we show how the idea of a “connection” as introduced by Polyakov fits into the present context and characterize which connections on P uoP may be obtained from a function from T (P uoP) into the Lie algebra of G as in the Polyakov case.