An L²-Index Theorem for Dirac Operators on S¹ * R³

2000/09/14 by Tom M. W. Nye, Nye, Tom M. W., Michael A. Singer +1
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Spectral Theory in Mathematical Physics #hep-th #math.DG

paper · pdf · doi:10.48550/arxiv.math/0009144

14 pages, Latex, to appear in the Journal of Functional Analysis

arxiv created 2000/09/14 · openalex publication_date 2000/09/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An expression is found for the L²-index of a Dirac operator coupled to a connection on a U_n vector bundle over S¹×\mathbb R³. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on S¹. An excision theorem due to Gromov, Lawson, and Anghel reduces the index theorem to this special case. The index formula can be expressed using the adiabatic limit of the η-invariant of a Dirac operator canonically associated to the boundary. An application of the theorem is to count the zero modes of the Dirac operator in the background of a caloron (periodic instanton).

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