An Index Theorem for Loop Spaces

2022/08/25 by Doman Takata, Takata, Doman
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2208.12129

Abstract

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 S¹-actions with compact fixed-point sets. In order to formulate it, we use a ring of formal power series.

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