An analytic LTLT-equivariant index and noncommutative geometry

  • Doman Takata

    Kyoto University, Japan
An analytic $LT$-equivariant index and noncommutative geometry cover
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Let TT be a circle group and LTLT be its loop group. For infinite-dimensional Spinc^c-manifolds equipped with “almost free” LTLT-actions, we define “L2L^2-spaces consisting of sections of the Spinor bundles” without measures on manifolds, “LTLT-equivarinat Dirac operators”, and analytic indices valued in the representation group of LTLT. 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 CC^*-algebra which can be regarded as a crossed product of “the function algebra of the manifold by LTLT ”, without the algebra itself or the measure on LTLT. 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 LTLT in a new language.

Cite this article

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

DOI 10.4171/JNCG/330