On linear systems and higher dimensional gauge fields
type: Item
instance of: scholarly article
title: On linear systems and higher dimensional gauge fields
zbMATH Open document ID: 0698.53057
DOI: 10.1016/0034-4877(88)90048-1
published in: Reports on Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: The complete solvability of the linear system \(V_ a^{\gamma}(\partial_{\gamma}+Q_{\gamma})\psi =0\), \(a=1,...,k\), for the GL(N,\({\mathbb{C}})\) gauge potential \(Q_{\gamma}\), \(\gamma =1,...,d\), on \({\mathbb{C}}^ d\) was investigated by Ward (\(\psi\) is a column N-vector and \(Q_{\gamma}\) is an \(N\times N\) matrix). The reduction of GL(N,\({\mathbb{C}})\) to other Lie subgroups was obtained by imposing certain reality conditions. In the work a special case is considered, together with a reduction of SO(4n) to \(Sp(1)\times Sp(n)/{\mathbb{Z}}_ 2\). The geometry of the above linear system can be encoded into holomorphic data by using the generalized twistor correspondence (defined by a triple of manifolds: the flag manifold F of 1 and 2 dimensional subspaces in \({\mathbb{C}}^{2n+2}\), the projective space \({\mathbb{C}}P^{2n+1}\), the Grassmann manifold \(G_{2n}({\mathbb{C}}^{2n+2}))\). The Sp(n) gauge fields defined on an open set \(U(-{\mathbb{C}}^{4n})\) in \(G_{2n}({\mathbb{C}}^{2n+2})\) are investigated and it is proved that the corresponding bundles are classified by certain holomorphic bundles of rank 2n, defined on an open set \(U''\) in \({\mathbb{C}}P^{2n+1}\).
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 32L99
zbMATH DE Number: 4144903
zbMATH Keywords: holomorphic bundle
zbMATH Keywords: gauge potential
zbMATH Keywords: twistor correspondence
author: James F. Glazebrook
reviewed by: P. Prešnajder
MaRDI profile type: Publication
full work available at URL: 0034-4877(88)90048-1
OpenAlex ID: W2000060090
cites work: Q4721249
cites work: Q3735553
cites work: Quaternionic Kaehler manifolds
cites work: Q4725278
cites work: On self-dual gauge fields
cites work: Q3938902
description: scientific article; zbMATH DE number 4144903