LT-equivariant index from the viewpoint of KK-theory: A global analysis on the infinite-dimensional Heisenberg group

Abstract

Let T be a circle group, and LT be its loop group. We hope to establish an LT-equivariant index theory for Hamiltonian LT-spaces, from the viewpoint of KK-theory. We have already constructed several objects in the previous paper (Takata, arXiv:1701.06055), including a Hilbert space H which can be regarded as a “Hilbert space consisting of L2-sections of a Spinor bundle on the infinite-dimensional manifold”, an “LT-equivariant Dirac operator D” acting on H, and the “twisted group C∗-algebra of LT”, in an ad hoc way.In this paper, we will construct more sophisticated objects for the index problem, in terms of KK-theory. Concretely, we will define analytic index of D as a K-theory element of the “twisted group C∗-algebra of LT” mentioned above. Then, we will also define the “latter half” of the assembly map at the “index element” (H,D). Finally, we will prove that the value of the “latter half” of the assembly map at the “index element”, coincides with the above analytic index of D. This is an infinite-dimensional version of a part of the index theorem for complete Riemannian manifolds with proper cocompact actions of Kasparov.We are mainly interested in the LT-equivariant index problem of infinite-dimensional manifolds, and so we use the language of representation of LT. However, our theory can be easily generalized to the setting of infinite-dimensional abstract Heisenberg groups equipped with some additional structures. This is potentially useful for researchers of other areas.