Journal of Noncommutative Geometry

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Volume 13, Issue 2, 2019, pp. 553–586
DOI: 10.4171/JNCG/330

Published online: 2019-07-17

An analytic $LT$-equivariant index and noncommutative geometry

Doman Takata[1]

(1) Kyoto University, Japan

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 $C^*$-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.

Keywords: Infinite-dimensional manifolds, loop groups, Dirac operators, spectral triples, crossed product algebras

Takata Doman: An analytic $LT$-equivariant index and noncommutative geometry. J. Noncommut. Geom. 13 (2019), 553-586. doi: 10.4171/JNCG/330