On linear systems and higher dimensional gauge fields
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 \({\mathbb{C}}P^{2n+1}\).
scientific article; zbMATH DE number 4144903