An Index Theorem for Loop Spaces
Copyright policy )An Index Theorem for Loop Spaces
We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of $KK$-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to find out an "appropriate form" of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact manifolds equipped with $S^1$-actions with compact fixed-point sets. In order to formulate it, we use a ring of formal power series.
41 pages
- Niigata University Japan
Mathematics - Differential Geometry, Index theory, Noncommutative geometry (à la Connes), fixed-point formula, Mathematics - Operator Algebras, FOS: Physical sciences, K-Theory and Homology (math.KT), Mathematical Physics (math-ph), index theorem, \(C^{\ast}\)-algebras of Hilbert manifolds, Equivariant \(K\)-theory, loop space, Witten genus, Differential Geometry (math.DG), \(\mathcal{R}\)\textit{KK}-theory, Mathematics - K-Theory and Homology, FOS: Mathematics, Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds, 19K56 (Primary), 19K35, 19L47, 58B20, 58B34 (Secondary), Operator Algebras (math.OA), Kasparov theory (\(KK\)-theory), Mathematical Physics
Mathematics - Differential Geometry, Index theory, Noncommutative geometry (à la Connes), fixed-point formula, Mathematics - Operator Algebras, FOS: Physical sciences, K-Theory and Homology (math.KT), Mathematical Physics (math-ph), index theorem, \(C^{\ast}\)-algebras of Hilbert manifolds, Equivariant \(K\)-theory, loop space, Witten genus, Differential Geometry (math.DG), \(\mathcal{R}\)\textit{KK}-theory, Mathematics - K-Theory and Homology, FOS: Mathematics, Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds, 19K56 (Primary), 19K35, 19L47, 58B20, 58B34 (Secondary), Operator Algebras (math.OA), Kasparov theory (\(KK\)-theory), Mathematical Physics
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