Instanton Moduli as a Novel Map from Tori to K3-surfaces

A map is constructed from the moduli of hyper-KK ahler tori to hyper-KK ahler K3 surfaces which does not coincide with the Kummer map. The map takes a torus to the moduli space of SO(3) connections on a bundle with nontrivial rst Stiefel-Whitney class and rst Pontrjagin class equal to-4. This map is shown to intersect the Kummer moduli and also certain subvarieties of singular K3 surfaces. Our map is shown to satisfy the local Torelli theorem, and the K3-surfaces in its image are shown to carry a natural metric which is Calabi-Yau. x1 Introduction The aim of this paper is to study antiself-dual (ASD) instantons on a simple bundle P over a torus of real dimension four. In outline one nds that the moduli space of such instantons is a K3-surface and thus deenes a map assigning K3 surfaces to holomorphic tori (deenition 2.6.4). This map is not the map deened by Kummer, but it shares a number of properties with it. First of all it factorizes through singular K3-surfaces, which results in a non-singular surface containing many lines. Secondly it satisses the local Torelli theorem. The instanton moduli space, a singular K3 surface, comes with a natural metric, which is deened in terms of the L 2-norm of deformations of instantons. This metric is not at and is the celebrated Calabi-Yau or hyper-KK ahler metric (Theorem 2.6.1). The study of instantons on 4-dimensional tori was rst proposed as a suitable model to study the phenomena of quark connnement. Soon after mathematicians started to study the moduli spaces of instantons it became clear that they had a wealth of interesting geometrical properties, often related to quaternions, see, for example, Atiyah At] and Atiyah-Hitchin-Singer AHS]. Soon after that the moduli spaces on algebraic surfaces were shown to equal the moduli spaces of stable holomorphic structures on the bundle D3],,D5],UY], generalizing the (real) two dimensional case D1], AB]. In this paper we shall apply these geometric techniques to study the moduli space of SO(3)-instantons on a bundle P ! T with T a 4-dimensional torus. For P with w 2 (P) = 0; p 1 (P) = ?4 one can show that no instantons exist Muk1], BV], and the case which we study is the next simplest one, where w 2 (P) 6 = 0; p 1 (P) = ?4. The moduli space M(P) has real dimension 8, and T acts …