Construction of a family of non-self-dual gauge fields

Abstract:

Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [2] we construct a $15$-dimensional family of nontrivial complex gauge fields $(U,E,\nabla )$ which are not self-dual nor anti-self-dual. ($U$ is an affine neighborhood in ${Q_4} = \operatorname {Gr} (2,{{\mathbf {C}}^4})$ of the stereographic compactification ${S^4}$ of ${\mathbb {R}^4}$, $E$ is a vector bundle on $U$ and $\nabla$ is a connection on it whose curvature $\phi$ satisfies the inequalities ${}^{\ast }\phi \ne \phi$ and ${}^{\ast }\phi \ne - \phi$.)