An analytic $LT$-equivariant index and noncommutative geometry

Abstract

Let $T$ be a circle group and $LT$ be its loop group. For infinite-dimensional Spin${}^c$-manifolds equipped with “almost free” $LT$-actions, we define “$L^2$-spaces consisting of sections of the Spinor bundles” without measures on manifolds, “$LT$-equivarinat Dirac operators”, and analytic indices valued in the representation group of $LT$. They have been studied already in the context of geometric quantization of Hamiltonian loop group spaces. However, we introduce a new perspective, noncommutative geometry, to the study of index theory for infinite-dimensional manifolds, in this paper. More precisely, we construct a noncommutative $C^{\ast }$-algebra which can be regarded as a crossed product of “the function algebra of the manifold by $LT$ ”, without the algebra itself or the measure on $LT$. Moreover, we combine all of them in terms of spectral triples. As expected, the triple is not finitely summable. Lastly, we add some applications including the Borel–Weil theory for $LT$ in a new language.