The \(\zeta\)-determinant and Quillen determinant for a Dirac operator on a manifold with boundary
type: Item
instance of: scholarly article
title: The \(\zeta\)-determinant and Quillen determinant for a Dirac operator on a manifold with boundary
zbMATH Open document ID: 0973.58017
DOI: 10.1007/PL00001651
published in: Geometric and Functional Analysis. GAFA
publication date: 2001-12-03T00:00:00Z
review text: The proof of the coincidence of the \(\zeta\)-determinant and the Quillen determinant of a Dirac operator on an odd dimensional manifold with boundary, which was announced in [\textit{S. G. Scott} and \textit{K. P. Wojciechowski}, C. R. Acad. Sci., Paris, Sér. I, Math. 328, No.~2, 139-144 (1999; Zbl 0957.58026)] is presented. Precisely, the operator \({\mathcal D} : C^\infty(M,S)\to C^\infty(M, S)\) is defined on a compact connected manifold \(M\) with boundary \(Y\), where \(S\) is a bundle of Clifford modules. It is assumed \(Y\) has a color neighborhood \(N = [0, 1]\times Y\) and \(\mathcal D\) has the form \({\mathcal D}_{|N}= G(\partial_u + B)\), where \(G : S|Y \to S|Y\) is a unitary bundle isomorphism, \(B\) the corresponding Dirac operator on \(Y\). Let \(\Pi_\geq\) be the projection from \(L^2(Y; S|Y)\) to the space spanned by the eigenvectors corresponding to the nonnegative eigenvalues of \(B\). The Grassmannian \(\text{Gr}({\mathcal D})\) is defined to be the space of all pseudodifferential projections which differ from \(\Pi_\geq\) by an operator of order \(-1\). The elliptic boundary value problem \({\mathcal D}_P\) is the operator \(\mathcal D\) with domain determined by a boundary condition \(P\), which is extensively studied in [\textit{B. Booss-Bavnbek} and \textit{K. P. Wojciechowski}, `Elliptic boundary problems for Dirac operators', Boston (1993; Zbl 0797.58004), hereafter referred to as [1]]. In that book, the selfadjoint Grassmannian \(\text{Gr}^*_\infty({\mathcal D})\) is introduced. In [\textit{B. Booss-Bavnbek} and \textit{K. P. Wojciechowski}, Commun. Math. Phys. 121, 1-9 (1989; Zbl 0683.58044)], \(H^2(\text{Gr}^*_\infty({\mathcal D});\mathbb{Z}) = 0\) is shown. Existence of \(\zeta\)-determinants for \({\mathcal D}_P\) is proved in [\textit{K. P. Wojciechowski}, Commun. Math. Phys. 201, No.~2, 423-444 (1999; Zbl 0948.58022)]. Strictly saying, the Quillen determinant is a section of the determinant bundle of \(\text{Gr}({\mathcal D})\). If…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 1562879
zbMATH Keywords: Dirac operator
zbMATH Keywords: elliptic boundary condition
zbMATH Keywords: \(\zeta\)-determinant
zbMATH Keywords: Quillen determinant
zbMATH Keywords: Grassmannian
author: Simon Scott
author: Krzysztof P. Wojciechowski
MaRDI profile type: Publication
full work available at URL: pl00001651
OpenAlex ID: W2164527666
description: scientific article; zbMATH DE number 1562879