## Abstract

Gieseker-Nakajima moduli spaces M_{k}(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. 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 M_{k}(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 ℂ^{2} intersecting in Z modelled on 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 M_{k} ^{γ}(n) of M_{k}(n), motivated by the considerations of sewn gauge theories on orbifolds ℂ^{4}/Γ. The geometry of the spaces M_{k} ^{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 M_{k}(n) and their quiver versions as correspondences provide the geometric counterpart of the qq-characters, line and surface defects.

Original language | English |
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Pages (from-to) | 503-583 |

Number of pages | 81 |

Journal | Advances in Theoretical and Mathematical Physics |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |