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# Advances in Theoretical and Mathematical Physics

## Volume 21 (2017)

### Number 2

### BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem

Pages: 503 – 583

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n2.a4

#### Author

#### Abstract

Gieseker–Nakajima moduli spaces $\mathcal{M}_k (n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on $\mathbb{R}^4$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on $\mathbb{CP}^2$ 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 $\mathfrak{M}_k (\vec{n})$ 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 $\mathbb{C}^2$ intersecting in $Z$ modelled on $\mathbb{C}^4$. 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 $\mathfrak{M}^{\gamma}_{\mathrm{k}} (\underline{\vec{\mathrm{n}}})$ of $\mathfrak{M}_k (\vec{n})$, motivated by the considerations of sewn gauge theories on orbifolds $\mathbb{C}^4 / \Gamma$.

The geometry of the spaces $\mathfrak{M}^{\gamma}_{\mathrm{k}} (\underline{\vec{\mathrm{n}}})$, 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 $\mathrm{K}$-theoretic operations defined using $\mathfrak{M}_k (\vec{n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects.