An analytic -equivariant index and noncommutative geometry

  • Doman Takata

    Kyoto University, Japan
An analytic $LT$-equivariant index and noncommutative geometry cover

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Let be a circle group and be its loop group. For infinite-dimensional Spin-manifolds equipped with “almost free” -actions, we define “-spaces consisting of sections of the Spinor bundles” without measures on manifolds, “-equivarinat Dirac operators”, and analytic indices valued in the representation group of . 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 -algebra which can be regarded as a crossed product of “the function algebra of the manifold by ”, without the algebra itself or the measure on . 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 in a new language.

Cite this article

Doman Takata, An analytic -equivariant index and noncommutative geometry. J. Noncommut. Geom. 13 (2019), no. 2, pp. 553–586

DOI 10.4171/JNCG/330