Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families

We continue our study of gauge equivariant K-theory. We thus study the analysis of complexes endowed with the action of a family of compact Lie groups and their index in gauge equivariant K-theory. We introduce various index functions, including an axiomatic one, and show that all index functions coincide. As an application, we prove a topological index theorem for a family D = (D_b)_b∈B of gauge-invariant elliptic operators on a G-bundle X → B, where G → B is a locally trivial bundle of compact groups, with typical fiber G. More precisely, one of our main results states that a-ind(D) = t-ind(D) ∈ K_G⁰ (X), that is, the equality of the analytic index and of the topological index of the family D in the gauge-equivariant K-theory groups of X. The analytic index ind_a(D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel K_G-classes of D. The topological index is defined purely in terms of the principal symbol of D.