The Geometry of the Loop Space and a Construction of a Dirac Operator

2005/05/04 by Andrew Stacey, Stacey, Andrew
Mathematics · Physics and Astronomy · #53C15 (Secondary) #58B20 (Primary) #58D15 #Advanced Differential Geometry Research #Algebraic Topology (math.AT) #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #math.DG #msc:53C15 #msc:58B20 #msc:58D15

paper · pdf · doi:10.48550/arxiv.math/0505077

52 pages, no figures

arxiv created 2005/05/04 · openalex publication_date 2005/05/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a construction of fibrewise inner products on the cotangent bundle of the smooth free loop space of a Riemannian manifold. Using this inner product, we construct an operator over the loop space of a string manifold which is directly analogous to the Dirac operator of a spin manifold.

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• The co-Riemannian Structure of Smooth Loop Spaces
• How to Construct a Dirac Operator in Infinite Dimensions
• The spinor bundle on loop space
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