Chern Forms on Mapping Spaces

Abstract

We state a Chern–Weil type theorem which is a generalization of a Chern–Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps \(Map_b(M,N)\) following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1, 19, 20]. We specialize to the case \(N=G\) to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group \(H^{1/2}_b(S^{1},G)\). In this last example, we precise Freed's construction [5] on the loop group: The cohomology of the first Chern form of any holomorphic connection in the class considered is given by the Kahler form.