BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem

Gieseker-Nakajima moduli spaces $M_k(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\mathbf{R}}^4$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\mathbf{C}\mathbf{P}}^2$ with ${\rm ch}{2}({$\mathcal{E}$}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. 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 ${\bf M}{k}({\vec n})$ofthecorrespondinginstantonmodulispaceisthemodulispaceofcharge$k$(noncommutative)instantonsonorigamispacetimes\mathrm{.}There,$X$ is modelled on a union of (up to six) coordinate complex planes ${\mathbf{C}}^2$ intersecting in $Z$ modelled on ${\mathbf{C}}^4$. 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 ${\bf M}{\bf k}^{\gamma}({\vec{\bf n}})$of${\bf M}{k}({\vec n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\mathbf{C}}^4\mathrm{/}\mathrm{\Gamma }$. The geometry of the spaces ${\mathbf{M}}_{\mathbf{k}}^{\gamma }(\vec{\mathbf{n}})$, 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 $\mathcal{N}=2$ quiver gauge theories. The cohomological and K-theoretic operations defined using ${\bf M}_{k}({\vec n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects.