Index theory for short-ranged fields in higher dimensions
type: Item
instance of: scholarly article
title: Index theory for short-ranged fields in higher dimensions
zbMATH Open document ID: 0803.58050
published in: Journal of Functional Analysis
publication date: 1995-01-03T00:00:00Z
review text: Let \(M\) be a noncompact Riemannian spin manifold of even dimension with a warped end. This means that there exist a compact Riemannian manifold \((N,d\sigma)\) and a compact set \(K_ 0 \subset M\) such that \(M\setminus K_ 0\) is isometric to the product \((0,\infty) \times N\) which is equipped with the warped product metric \(dr^ 2 + f^ 2(r) d\sigma^ 2\), where \(f\) is a positive smooth function defined on \([0,\infty)\). Let \(E\) be a trivial Hermitian vector bundle over \(M\) and let \(\nabla_ 0\) be the connection on \(E\) which is defined by the ordinary exterior derivative \(d\), taken with respect to some fixed trivialization of \(E\). Any other metric connection on \(E\) is given by \(\nabla_ 0 + A\), for some \(A \in \Omega^ 1({\mathcal G}_ E)\), where \({\mathcal G}_ E = \{\gamma \in \text{Hom}_ \mathbb{C}(E)\mid \gamma +\gamma^* = 0\}\). \(F = dA + A \wedge A\) is called a short-ranged field, if \(F\) has a compact support. The author studies the twisted Dirac operator \(D_ A : C^ \infty(S \otimes E) \to C^ \infty(S \otimes E)\), where \(S\) is the spinor bundle and \(D_ A\) is constructed using a connection \(\nabla_ 0 + A\) on \(E\) such that the field \(F\) is short ranged. The author proves, that the restrictions \(D_ A^ \pm\) of \(D_ A\) to the spinors of positive resp. negative chirality have finite dimensional \(L^ 2\)-kernel, and he computes an index formula for the \(L^ 2\)-index of \(D^ +_ A\). In the proof the computation of the \(L^ 2\)-kernel is related to the boundary value problem studied by \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} [Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008)].
reviewed by: K. H. Mayer
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 599823
zbMATH Keywords: index theory
zbMATH Keywords: noncompact manifolds
zbMATH Keywords: kernel
zbMATH Keywords: twisted Dirac operator
zbMATH Keywords: boundary value problem
author: Nicolae Anghel
MaRDI profile type: Publication
full work available at URL: jfan.1994.1002
OpenAlex ID: W2079369985
DOI: 10.1006/JFAN.1994.1002
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description: scientific article; zbMATH DE number 599823