An analytic \(LT\)-equivariant index and noncommutative geometry (Q2327771)

The main result of this work is the construction of a spectral triple for an infinite dimensional manifold which admits an action of the loop group \(LT\) (here \(T\) is the circle). The analytic index of the Dirac operator involved takes values in the Grothendieck group \(R^{\tau}LT\) associated with positive energy representations of \(LT\). Moreover, applications of this construction are given: An infinite dimensional version of Bott's quantization, and a re-statement of the Borel-Weil lemma for \(LT\). This paper is inspired by work of \textit{N. Higson} et al. [Adv. Math. 135, No. 1, 1--40 (1998; Zbl 0911.46040)].