Constructions of noncommutative instantons on T4 and

We generalize the spectral-curve construction of moduli spaces of instantons on T4 and K3 to noncommutative geometry. We argue that the spectral-curves should be constructed inside a twisted T4 or K3 that is an elliptic fibration without a section. We demonstrate this explicitly for T4 and to first order in the noncommutativity, for K3. 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 K3 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 T3 with global twists. This extends a previous result for noncommutative instantons on T4. We also briefly discuss the instanton equation on generic curved spaces.