Superconnection character forms and the Cayley transform
type: Item
instance of: scholarly article
title: Superconnection character forms and the Cayley transform
zbMATH Open document ID: 0671.57013
DOI: 10.1016/0040-9383(88)90040-7
published in: Topology
publication date: 1988-01-01T00:00:00Z
review text: Let \(E=E^+\oplus E^-\) be a super vector bundle over a smooth manifold M equipped with an inner product, and let D be a connection on E compatible with the grading and the inner product. Let T: \(E^+\to E^- \) denote a vector bundle map and \(X=\left( \begin{matrix} 0\\ T\end{matrix} \begin{matrix} -T^*\\ 0\end{matrix} \right)\). In this situation the author introduced the superconnection character form \(\Phi\) (E,D,X) which is a closed form in \(\Omega\) (M) [Topology 24, 89-95 (1985; Zbl 0569.58030)]. In the present paper he describes the construction of this form in new notations, which allow to treat simultaneously the graded and ungraded cases. He discusses the Cayley transform sending a skew adjoint operator to a unitary operator. There is a correspondence between subbundles W of E and unitary automorphisms g of E which are inverted under conjugation by the involution \(\epsilon\) defined by the grading of E. For any automorphism g of E a form \(\Phi\) (E,D,g) is constructed which agrees with \(\Phi\) (E,D,X) in the case that g is the Cayley transform of X. It is shown that the form \(\Phi\) (E,D,g) is induced from the universal superconnection character form over a sufficiently large Grassmannian Gr(V) or unitary group U(V). Character forms \(\gamma_{2k}\) on Gr(V) and \(\gamma_{2k-1}\) on U(V) are introduced, which are closed forms and invariant under the action of U(V). The class of \(\gamma_{2k}\) in \(H^{2k}(Gr(V))\) is up to a factor equal to the Chern character class \(ch_ k\) of the canonical subbundle over Gr(V). These invariant classes are studied in detail and related to the universal superconnection character form. The last part deals with the problem of extending the results to infinite dimensions. For this reason the unitary group \(U^ p\) and Grassmannian \(Gr^ p\) of a Hilbert space H relative to the Schatten class \({\mathcal L}^ p(H)\) are defined. They are Banach manifolds for which the de Rham theorem holds, and they are classifying space…
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 4098308
zbMATH Keywords: de Rham cohomology
zbMATH Keywords: Grassmannian of a Hilbert space
zbMATH Keywords: super vector bundle
zbMATH Keywords: superconnection character form
zbMATH Keywords: Cayley transform
zbMATH Keywords: skew adjoint operator
zbMATH Keywords: unitary operator
zbMATH Keywords: universal superconnection character form
zbMATH Keywords: Chern character class
zbMATH Keywords: Schatten class
zbMATH Keywords: Banach manifolds
zbMATH Keywords: de Rham theorem
zbMATH Keywords: classifying spaces for K-theory
zbMATH Keywords: invariant character forms
reviewed by: K. H. Mayer
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90040-7
OpenAlex ID: W2023170243
author: Daniel Gray Quillen
description: scientific article; zbMATH DE number 4098308