  We formulate and prove an index theorem for loop spaces of compact manifolds
in the framework of $KK$-theory. It is a strong candidate for the
noncommutative geometrical definition (or the analytic counterpart) of the
Witten genus. In order to find out an "appropriate form" of the index theorem
to formulate a loop space version, we formulate and prove an equivariant index
theorem for non-compact manifolds equipped with $S^1$-actions with compact
fixed-point sets. In order to formulate it, we use a ring of formal power
series.
