The Chern Character of θ-summable Fredholm Modules over dg Algebras and Localization on Loop Space

2019/01/15 by Batu Güneysu, Güneysu, Batu, Matthias Ludewig +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1901.04721

openalex publication_date 2019/01/15 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We introduce the notion of a ϑ-summable Fredholm module over a locally convex dg algebra Ω and construct its Chern character as a cocycle on the entire cyclic complex of Ω, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack in the context of loop spaces. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type localization formula on loop space.

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