Link Invariants, Holonomy Algebras and Functional Integration

Given a principal G-bundle over a smooth manifold M , with G a compact Lie group, and given a finite-dimensional unitary representation ρ of G, one may define an algebra of functions on A/G, the “holonomy Banach algebra” Hb, by completing an algebra generated by regularized Wilson loops. Elements of the dual H∗b may be regarded as a substitute for measures on A/G. There is a natural linear map from Diff0(M)-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.”