Constructions of Non Commutative Instantons on $T^4$ and $K_3$

We generalize the spectral-curve construction of moduli spaces of instantons on $\text{MT}4$ and $K_3$ to noncommutative geometry. We argue that the spectral-curves should be constructed inside a twisted $\text{MT}4$ or $K_3$ that is an elliptic fibration without a section. We 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 $\text{SUSY}4$ supersymmetry in 2+1D. The spectral curves are related to Seiberg-Witten curves of theories with $\text{SUSY}2$ 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 $\mathcal{N}=4$ 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 $\text{MT}4$. We also briefly discuss the instanton equation on generic curved spaces.