Constructions of noncommutative instantons on T4 and K3

We generalize the spectral-curve construction of moduli spaces of instantons on T-4 and K-3 to noncommutative geometry. We argue that the spectral-curves should be constructed inside a twisted T-4 or K-3 that is an elliptic fibration without a section. rme demonstrate this explicitly for T-4 and to first order in the noncommutativity, for K-3. Physically, moduli spaces of noncommutative instantons appear as moduli spaces of theories with N = 4 supersymmetry in 2 +/- 1D. The spectral curves are related to Seiberg-Witten curves of theories with N = 2 in 3 + 1D. In particular, we argue that the moduli space of instantons of U(q) Yang-Mills theories on a noncommutative K-3 is equivalent to the Coulomb branch of certain 2 + 1D theories with N = 4 supersymmetry. The theories are obtained by compactifying the heterotic little-string theory on T-3 with global twists. This extends a previous result for noncommutative instantons on T-4. We also briefly discuss the instanton equation on generic curved spaces. (C) 2000 Elsevier Science B.V. All rights reserved.