Parallel transport in the determinant line bundle: The zero index case
type: Item
instance of: scholarly article
title: Parallel transport in the determinant line bundle: The zero index case
zbMATH Open document ID: 0648.58036
DOI: 10.1007/BF01205549
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: Let X be a compact spin manifold of even dimension, \(S=S_+\oplus S_-\) the spinor bundle of X and E a hermitian vector bundle over X. Let \(\bar S\) and \(\bar E\) denote the pullback of S resp. E to \({\mathbb{R}}\times X\), and let \({\bar \nabla}^ E\) be a connection on \(\bar E.\) For any \(y\in {\mathbb{R}}\) the connection \(\nabla^ E_ y\) on E is induced by \({\bar \nabla}^ E\) and \(\partial_ y: L^ 2(X,S_+\otimes E)\to L^ 2(X,S_-\otimes E)\) is the Dirac-operator also called Weyl operator coupled to the connection \(\nabla^ E_ y\) on E and the metric on X. A general construction of \textit{J.-M. Bismut} and \textit{D. S. Freed} [Commun. Math. Phys. 106, 159-176 (1986) and 107, 103-163 (1986)] applied to this special case yields Hilbert bundles \(H_+\) and \(H_-\) over \({\mathbb{R}}\) and a bundle map \(\partial: H_+\to H_-\) given by the Weyl operators \(\partial_ y\), and a determinant line bundle L over \({\mathbb{R}}\) with an inner product and a compatible connection. The authors assume that \(\partial_ y\) is invertible for all \(y\in {\mathbb{R}}\) and study the parallel displacement in L from -\(\infty\) to \(+\infty\). This parallel transport is described in terms of the determinants det \(\partial_{+\infty}\partial^*_{\infty}\) and det \(\partial_{- \infty}\partial^*_{-\infty}\), the dimension of the kernel and the spectral asymmetry \(\eta\) (H) of a self adjoint Dirac operator H on \(\bar S\otimes \bar E\). Since \({\mathbb{R}}\times X\) is not compact \(\eta\) (H) cannot be defined in the usual way. The result is in a special case an extension of a formula of Bismut and Freed (loc. cit.) for the holonomy in the determinant line bundle over a loop.
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 4058489
zbMATH Keywords: index theory
zbMATH Keywords: Dirac-operator
zbMATH Keywords: Weyl operator
zbMATH Keywords: determinant line bundle
zbMATH Keywords: connection
author: Stephen DellaPietra
author: Vincent Della Pietra
reviewed by: K. H. Mayer
MaRDI profile type: Publication
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Wikidata QID: Q125361054
full work available at URL: https://doi.org/10.1007/bf01205549
OpenAlex ID: W1479997822
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description: scientific article; zbMATH DE number 4058489