Instantons and Bows for the Classical Groups

Abstract

The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds ${\mathbb{R}}^4/\Gamma$ by a finite subgroup $\Gamma \subset SU(2).$ We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm's equations [Che09] via the Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak03], Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory [COS11]. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.

arXiv commentTo appear in memorial volume dedicated to Michael Atiyah. 53 pages, 1 figure

References in corpus (22)

• S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory
• Mirror Symmetry in Three Dimensional Gauge Theories
• Aspects of N=2 Supersymmetric Gauge Theories in Three Dimensions
• Singular Monopoles and Supersymmetric Gauge Theories in Three Dimensions
• Branes, Instantons, And Taub-NUT Spaces
• Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
• Instantons on Gravitons
• Moduli Spaces of Instantons on the Taub-NUT Space
• A survey on Nahm transform
• On Yang--Mills instantons over multi-centered gravitational instantons
• On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics
• Super Yang-Mills Theory with Impurity Walls and Instanton Moduli Spaces
• Geometric construction of new Yang-Mills instantons over Taub-NUT space
• Calorons, Nahm's equations on S^1 and bundles over P^1xP^1
• Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem
• Towards geometric Satake correspondence for Kac-Moody algebras -- Cherkis bow varieties and affine Lie algebras of type $A$
• Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d N=2 supersymmetric theories
• Spatially periodic instantons: Nahm transform and moduli
• On the energy spectrum of Yang--Mills instantons over asymptotically locally flat spaces
• Harmonic functions and instanton moduli spaces on the multi-Taub--NUT space
• Instantons on ALE spaces for classical groups
• Instantons on multi-Taub-NUT Spaces II: Bow Construction

Cited by in corpus (1)

• $G_2$-instantons on the ALC members of the ${\mathbb{B}}_7$ family