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Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type $A$

Hiraku Nakajima, Yuuya Takayama

TL;DR

The work establishes that Coulomb branches of framed quiver gauge theories of affine type $A$ are Cherkis bow varieties, linking the physics-driven Coulomb construction to a robust algebro-geometric model. It provides a detailed quiver description of bow varieties, proves a factorization property, and constructs a birational isomorphism to the Coulomb branch, yielding normal, stratified, and semismall structures. Under balanced dimensions, bow varieties coincide with Na-quiver varieties, and the authors develop deformation/resolution pictures, a collapsing morphism to chainsaw quiver varieties, and explicit local models. The paper further explores Hanany-Witten transitions and 3d mirror symmetry, situating bow/Coulomb structures within brane dualities and broadening the toolkit for studying moduli of instantons on Taub-NUT spaces and related hyper-Kähler geometries.

Abstract

We show that Coulomb branches of quiver gauge theories of affine type $A$ are Cherkis bow varieties, which have been introduced as ADHM type description of moduli space of instantons on the Taub-NUT space equivariant under a cyclic group action.

Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type $A$

TL;DR

The work establishes that Coulomb branches of framed quiver gauge theories of affine type are Cherkis bow varieties, linking the physics-driven Coulomb construction to a robust algebro-geometric model. It provides a detailed quiver description of bow varieties, proves a factorization property, and constructs a birational isomorphism to the Coulomb branch, yielding normal, stratified, and semismall structures. Under balanced dimensions, bow varieties coincide with Na-quiver varieties, and the authors develop deformation/resolution pictures, a collapsing morphism to chainsaw quiver varieties, and explicit local models. The paper further explores Hanany-Witten transitions and 3d mirror symmetry, situating bow/Coulomb structures within brane dualities and broadening the toolkit for studying moduli of instantons on Taub-NUT spaces and related hyper-Kähler geometries.

Abstract

We show that Coulomb branches of quiver gauge theories of affine type are Cherkis bow varieties, which have been introduced as ADHM type description of moduli space of instantons on the Taub-NUT space equivariant under a cyclic group action.

Paper Structure

This paper contains 65 sections, 63 theorems, 336 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bow varieties
  3. Original definition
  4. A quiver description of a bow variety
  5. Factorization map
  6. Stability conditions
  7. Characterization of the conditions (S1) and (S2)
  8. Numerical criterion
  9. Tangent space
  10. Balanced and cobalanced conditions
  11. Constituents of bow varieties
  12. Triangle part
  13. Definition
  14. Factorization
  15. Dimension estimate for -1
  16. ...and 50 more sections

Key Result

Theorem 2.1

There is a natural homeomorphism between the space ${\mathcal{M}}_{\mathrm{bow}}$ of the gauge equivalence classes of bow solutions and the space ${\mathcal{M}}_{\mathrm{quiver}}$ of the $S$-equivalence classes of quiver representations. Moreover it is an isomorphism of holomorphic symplectic manifo

Figures (3)

  • Figure 1: Quiver description of a bow variety
  • Figure 2: (a) A bow diagram of affine type $A_{\ell-1}$ and (b) its simplified version
  • Figure 3: Brane configuration

Theorems & Definitions (136)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Remark 2.6
  • ...and 126 more