The Dirac spectrum
type: Item
instance of: scholarly article
title: The Dirac spectrum
zbMATH Open document ID: 1186.58020
DOI: 10.1007/978-3-642-01570-0
author: Nicolas Ginoux
published in: Lecture Notes in Mathematics
publication date: 2009-07-02T00:00:00Z
review text: This memoir is a survey on the spectral properties of the Dirac operator defined by a spin structure on a Riemannian manifold. I think that it can be used as a valuable guide to get introduced to this subject. The book is self-contained once some basic concepts of differential geometry are known, like vector bundles, Lie groups, principal bundles, connections, curvature, etc. All needed preliminaries on spin geometry are recalled in Chapter~1. The spin group \(\text{Spin}_n\) in dimension \(n\) is the non-trivial \(2\)-fold covering of \(\text{SO}_n\). A spin structure on an oriented Riemannian \(n\)-manifold \(M\) is a \(\text{Spin}_n\)-principal bundle over \(M\) which is a \(2\)-fold covering of the \(\text{SO}_n\)-principal bundle of oriented orthonormal frames of \(TM\) in an equivariant way with respect to the projection \(\text{Spin}_n\to\text{SO}_n\). It is proved that the family of spin structures on \(M\) is in one-to-one correspondence with \(H^1(M,\mathbb{Z}_2)\). It is recalled that \(\text{Spin}_n\) has one fundamental irreducible complex representation if \(n\) is odd, denoted by \(\Sigma_n\), and two if \(n\) is even, denoted by \(\Sigma_n^\pm\). The vector bundle \(\Sigma(M)\to M\) associated to the representation \(\Sigma_n\) (\(n\) odd) or \(\Sigma_n^+\oplus\Sigma_n^-\) (\(n\) even) is called the spinor bundle; in the even dimensional case, each \(\Sigma_n^\pm\) induces a vector bundle \(\Sigma^\pm M\), defining a splitting of \(\Sigma M\). The author also recalls the definitions of the Clifford multiplication \(\mu:TM^*\otimes\Sigma M\to\Sigma M\), the connection \(\nabla^\Sigma\) and the Hermitian inner product on \(\Sigma M\), which satisfy natural relations of compatibility. Then the Dirac operator \(D:\Gamma(\Sigma M)\to\Gamma(\Sigma M)\) is defined as the composition \[ \Gamma(\Sigma M) \overset{\nabla}{\longrightarrow} \Gamma(TM\otimes\Sigma M) \longrightarrow \Gamma(TM^*\otimes\Sigma M) \overset{\mu}{\longrightarrow} \Gamma(\Sigma M)\;, \]…
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 5574435
zbMATH Keywords: spin structure
zbMATH Keywords: spinors
zbMATH Keywords: Dirac operator
zbMATH Keywords: spectrum
zbMATH Keywords: eigenvalues
zbMATH Keywords: boundary conditions
zbMATH Keywords: lower eigenvalue estimates
zbMATH Keywords: upper eigenvalue estimates
zbMATH Keywords: isospectrality
zbMATH Keywords: harmonic spinors
zbMATH Keywords: spectral gap
zbMATH Keywords: pinching Dirac eigenvalues
zbMATH Keywords: conformal spectral invariants
zbMATH Keywords: convergence of eigenvalues
zbMATH Keywords: eta-invariants
zbMATH Keywords: positive mass theorems
zbMATH Keywords: twistor spinors
zbMATH Keywords: Killing spinors
reviewed by: Jesús A. Álvarez López
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-01570-0
OpenAlex ID: W4213330757
description: scientific article; zbMATH DE number 5574435