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

Abstract

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 ${\mathrm{ch}}_2(\mathcal{E})=k$. 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 ${\mathbf{M}}_k(\vec{n})$ 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 ${\mathbf{C}}^2$ 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 ${\mathbf{M}}_{\mathbf{k}}^{\gamma }(\vec{\mathbf{n}})$ of ${\mathbf{M}}_k(\vec{n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\mathbf{C}}^4/\Gamma$. 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 ${\mathbf{M}}_k(\vec{n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects.

63 pages, 16 figures, paper 2 out of 5

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