Gieseker-Nakajima moduli spaces Mk(n) parametrize the charge k noncommutative U(n) instantons on ℝ4 and framed rank n torsion free sheaves ε on ℂℙ;2 with ch2(ε) = k. They also serve as local models of the moduli spaces of instantons on general fourmanifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space X immersed into a Calabi-Yau fourfold Z. The local model Mk(n) of the corresponding instanton moduli space is the moduli space of charge k (noncommutative) instantons on origami spacetimes. There, X is modelled on a union of (up to six) coordinate complex planes ℂ2 intersecting in Z modelled on ℂ4. The instantons are shared by the collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions Mk γ(n) of Mk(n), motivated by the considerations of sewn gauge theories on orbifolds ℂ4/Γ. The geometry of the spaces Mk γ(n), more specifically the compactness of the set of torus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwinger identities recently found to be satisfied by the correlation functions of qq-characters viewed as local gauge invariant operators in the N = 2 quiver gauge theories. The cohomological and K-theoretic operations defined using Mk(n) and their quiver versions as correspondences provide the geometric counterpart of the qq-characters, line and surface defects.