Link Invariants, Holonomy Algebras and Functional Integration

Abstract

Given a principal G-bundle over a smooth manifold M, with G a compact Lie group, and given a finite-dimensional unitary representation of G, one may define an algebra of functions on the space of connections modulo gauge transformations, the ``holonomy Banach algebra'' H_b, by completing an algebra generated by regularized Wilson loops. Elements of the dual H_b* may be regarded as a substitute for measures on the space of connections modulo gauge transformations. There is a natural linear map from diffeomorphism- invariant elements of H_b* to the space of complex-valued ambient isotopy invariants of framed oriented links in M. Moreover, this map is one-to-one. Similar results hold for a C*-algebraic analog, the ``holonomy C*-algebra.'' These results clarify the relation between diffeomorphism-invariant gauge theories and link invariants, and the framing dependence of the expectation values of products of Wilson loops.