Parallel transport in the determinant line bundle: The non-zero index case
type: Item
instance of: scholarly article
title: Parallel transport in the determinant line bundle: The non-zero index case
zbMATH Open document ID: 0629.53029
DOI: 10.1007/BF01239012
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: The authors consider a compact spin manifold X of even dimension with spin bundle \(S=S_+\oplus S_- \to X.\) For a Hermitian vector bundle \(E\to X\) over X, the Weyl operators are defined by \(\partial_ y:L^ 2(X,S_+\otimes E)\mapsto L^ 2(X,S_-\otimes E)\) coupled to the connection \(\nabla^ E_ y\) and the (y-independent) metric on X. With these structures a smooth determinant line bundle \({\mathcal L}\) over \({\mathbb{R}}\) is constructed, and a natural Hermitian metric and a compatible connection are introduced. The parallel transport in \({\mathcal L}\) is investigated in the case that index \(\partial_ y\) is not necessarily zero. It is expressed in terms of the spectral asymmetry of a Dirac operator on \({\mathbb{R}}\times X\). The pull-backs of the inner product and connection on \({\mathcal L}\) are calculated. These results are used then to obtain the formulas for curvature in \({\mathcal L}\) and parallel transport in \({\mathcal L}\). The calculus of the resolvent of the Dirac operator and the notational conventions are given in two appendices.
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 4024213
zbMATH Keywords: spin manifold
zbMATH Keywords: Hermitian vector bundle
zbMATH Keywords: Weyl operators
zbMATH Keywords: determinant line bundle
zbMATH Keywords: Dirac operator
zbMATH Keywords: parallel transport
author: Stephen DellaPietra
author: Vincent Della Pietra
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
cites work: The chiral determinant and the eta invariant
cites work: On the heat equation and the index theorem
cites work: Spectral asymmetry and Riemannian Geometry. I
cites work: The analysis of elliptic families. II: Dirac operators, êta invariants, and the holonomy theorem
cites work: Parallel transport in the determinant line bundle: The zero index case
cites work: The eta invariant for a class of elliptic boundary value problems
cites work: Q3874765
cites work: Determinants of Cauchy-Riemann operators on a Riemann surface
cites work: Q4185363
cites work: Q5542418
full work available at URL: https://doi.org/10.1007/bf01239012
OpenAlex ID: W4232581549
description: scientific article; zbMATH DE number 4024213