FUNCTIONAL INTEGRATION ON SPACES OF CONNECTIONS

摘要

Suppose that G is a compact connected Lie group and P --> M is a smooth principal G-bundle. We define a ''cylinder function'' on the space A of smooth connections on P to be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M. Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(A). Let a ''generalized measure'' on A be a bounded linear functional on Fun(A). We construct a generalized measure mu(0) on A that is invariant under all automorphisms of the bundle P (not necessarily fixing the base M). This result extends previous work which assumed M was real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph with n edges embedded in M determines a C*-subalgebra of Fun(A) isomorphic to C(G(n)), and the generalized measure mu(0):Fun(A) --> C restricts to the linear functional on C(G(n)) given by integration against normalized Haar measure on G(n). Our result implies that the group G of gauge transformations acts as unitary operators on L-2(A), the Hilbert space completion of Fun(A) in the norm F(2)=mu(0)(F(2))(1/2). Using ''spin networks,'' we construct explicit functions spanning the subspace L-2(A/G)subset of or equal to L-2(A) consisting of vectors invariant under the action of G. (C) 1997 Academic Press. [References: 24]