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

Nikita Nekrasov

paper

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

arXiv:1608.07272

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 $E$ on $C P^{2}$ with $ch_{2} (E) = k$ . 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 $M_{k} (n)$ 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 $q q$ -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 $q q$ -characters, line and surface defects.

arXiv comment63 pages, 16 figures, paper 2 out of 5

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

Abstract

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