2016/08/25 by Nikita Nekrasov, Nekrasov, Nikita · 9 citations
Physics and Astronomy · Mathematics · #Black Holes and Theoretical Physics #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1608.07272
Gieseker-Nakajima moduli spaces Mk(n) parametrize the charge k\nnoncommutative U(n) instantons on bf R4 and framed rank n torsion\nfree sheaves \E on bf C bf P2 with rm\nch2(\E) = k. They also serve as local models of the moduli\nspaces of instantons on general four-manifolds. We study the generalization of\ngauge theory in which the four dimensional spacetime is a stratified space X\nimmersed into a Calabi-Yau fourfold Z. The local model bf Mk( nn) of the corresponding instanton moduli space is the moduli space of charge\nk (noncommutative) instantons on origami spacetimes. There, X is modelled\non a union of (up to six) coordinate complex planes bf C2 intersecting\nin Z modelled on bf C4. The instantons are shared by the collection\nof four dimensional gauge theories sewn along two dimensional defect surfaces\nand defect points. We also define several quiver versions bf M bf\nk\γ(\ bf n) of bf Mk( n), motivated by the\nconsiderations of sewn gauge theories on orbifolds bf C4/\Γ.\n The geometry of the spaces bf M bf k\γ(\ bf n), more\nspecifically the compactness of the set of torus-fixed points, for various\ntori, underlies the non-perturbative Dyson-Schwinger identities recently found\nto be satisfied by the correlation functions of qq-characters viewed as local\ngauge invariant operators in the \N=2 quiver gauge theories.\n The cohomological and K-theoretic operations defined using bf Mk( nn) and their quiver versions as correspondences provide the geometric\ncounterpart of the qq-characters, line and surface defects.\n