Computations, Transition Formulae, and the Local Index Formula
In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the literature and which have acted as guide posts to the development of trace and determinant methods in geometric analysis over the past few decades. There are basic equivalences, indispensable in both theoretical developments and exact computations, between coefficients in the asymptotic expansions of zeta traces, heat traces and resolvent traces. The middle part of the chapter presents and proves these identifications, and provides an application to the computation of relative zeta determinants. The final part of the chapter turns to residue trace and residue determinant computations. Based on a residue determinant formula for the spectral zeta function at zero, an elementary proof of the local Atiyah Singer index theorem is given.
Keywords: Regularized trace, regularized determinants, explicit computations, resolvent trace, heat trace, complex powers, pole structure, asymptotic expansion, Atiyah‐Singer, index theorem, Riemann Roch Hirzebruch
Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.
If you think you should have access to this title, please contact your librarian.