Gieseker-Nakajima moduli spaces Mk(~n) parametrize the charge k noncommutative U(n) instantons on R~4 and framed rank n torsion free sheaves ε on CP~2 with ch_2(ε)= k. We define a generalization M_k({→ under letter n}) of M_k(n), the moduli space of charge k (noncommutative) instantons on origami spacetimes: a union X of (up to six) coordinate complex planes C~2 intersecting in C~4, the instantons of a collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions MJ~y_k({formula}) of Mk({formula}), motivated by the considerations of sewn gauge theories on orbifolds C~4/F. The geometry of the spaces O~y_k({formula}), 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({formula}) and their quiver versions as correspondences provide the geometric counterpart of the qq-characters, line and surface defects.
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