Self-dual Yang–Mills equations in split signature

Abstract

We study the self-dual Yang-Mills equations in split signature. We give a special solution, called the basic split instanton, and describe the ADHM construction in the split signature. Moreover a split version of t'Hooft ansatz is described.

The Penrose transform in split signature

“…where ∂ i := ∂ ∂x i , see [1]. If A = i A i dx i is the connection form of ∇ then V , in these local coordinates, is given by…”

“…where P is the space of complex lines in T, M is the Grassmannian of complex 2-planes in T, F is the space of pairs of nested 1-and 2-dimensional subspaces of T, and where µ and ν are the natural holomorphic maps. Both maps µ and ν are fiber bundle maps where the fibers of µ are isomorphic to CP 2 and the fibers of ν are isomorphic to CP 1 . Given any open subset U of M we set…”

“…The space of 2-forms on M has a natural decomposition into two subspaces of self-dual and anti-self-dual 2-forms, see [14]. For a given Lie group G, a G-SDYM field on M is a vector bundle V with structure group G and a connection ∇ on V compatible with G whose curvature is self-dual, see [1,20,21]. By a split instanton, we mean a U (n)-SDYM field on M .…”

“…In the introduction, for simplicity, we only consider the Penrose transform for O(−2) which is a very important case. There are three real forms: (1) Euclidean: This corresponds to the real form SO 0 (5, 1) of SL(4, C) which gives the totally real sub-manifold S 4 of M. In this case we have the fibration π : P → S 4 , see [2]. Using this fibration, we have the following Penrose transform in the Euclidean case H 1 (π −1 (U ), O(−2)) → ker | U which is an isomorphism.…”