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

Abstract

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, £,V) which are not self-dual nor anti-self-dual.(1/ is an affine neighborhood in Qt = Gr(2, C4) of the stereographic compactification S4 of R4, E is a vector bundle on U and v is a connection on it whose curvature </> satisfies the inequalities *<f> * <j> and *<t> * -<f>.)In [4,5] Isenberg et al. and Manin have studied gauge fields whose curvature is neither self-dual nor anti-self-dual. We miss however actual examples of the theory and this we will give in this note. Moreover, we introduce in twistor theory the tool of reflexive sheaves (a kind of vector bundle with singularities) whose interest is very recent in geometry, and they seem to play a roll also in this context. We start by recalling the characterizations of such fields, which are well known by now. We use the language of coherent sheaves [1].The complexification of the stereographic compactification S4 of R4 is the complex 4-dimensional projective smooth quadric Q4. This is usually referred to by geometers as the Klein quadric. It can also be viewed as the Grassmann variety parametrizing all lines in projective space P3 = P(C4) and S4 as the subset of points corresponding to the lines L = PX(V) such that V = C2 is invariant under the antiholomorphic map C4 -» C4 consisting of multiplication by j g H (quaternions) when we decompose C4sH ffi^'H.Our construction depends on 14 complex parameters and produces an affine neighborhood U of S4 in <24 and a vector bundle on U together with with a connection on it whose curvature <j> is neither self-dual (*<¡> ¥= <f>) nor anti-self-dual