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Intersections of twisted cotangent bundles and symplectic duality

Naichung Conan Leung, Yunsong Wei

TL;DR

The paper develops a framework in which many symplectic resolutions are realized as intersections of twisted cotangent bundles $T^*G //_{\psi} H$, and studies their Langlands duals via intersections of dual twisted cotangent bundles, within the paradigm of 3d mirror symmetry and symplectic duality. It constructs and analyzes 4d mirror branes, fixed-point phenomena on Poisson slices, and a rich array of intersections including Bielawski slices, universal centralizers, quiver and bow varieties, and Slodowy-type varieties. Key technical contributions include a precise combinatorial description of fixed-point sets in terms of Weyl-group data, a closure criterion for families of closed root subsets, and explicit identifications of various intersections with known geometric objects (parabolic Slodowy varieties, quiver varieties, bow varieties). These results illuminate concrete 3d mirror dualities, provide tools to identify dual pairs, and connect physical dualities to rigorous algebro-geometric constructions with explicit fixed-point correspondences.

Abstract

We observe that numerous symplectic resolutions can be expressed as intersections of twisted cotangent bundles. Additionally, their dual symplectic resolutions can be derived from intersections of dual twisted cotangent bundles. We determine the collection of fixed points for certain intersections that are Poisson slices, extending the computations of fixed points for parabolic Slodowy varieties.

Intersections of twisted cotangent bundles and symplectic duality

TL;DR

The paper develops a framework in which many symplectic resolutions are realized as intersections of twisted cotangent bundles , and studies their Langlands duals via intersections of dual twisted cotangent bundles, within the paradigm of 3d mirror symmetry and symplectic duality. It constructs and analyzes 4d mirror branes, fixed-point phenomena on Poisson slices, and a rich array of intersections including Bielawski slices, universal centralizers, quiver and bow varieties, and Slodowy-type varieties. Key technical contributions include a precise combinatorial description of fixed-point sets in terms of Weyl-group data, a closure criterion for families of closed root subsets, and explicit identifications of various intersections with known geometric objects (parabolic Slodowy varieties, quiver varieties, bow varieties). These results illuminate concrete 3d mirror dualities, provide tools to identify dual pairs, and connect physical dualities to rigorous algebro-geometric constructions with explicit fixed-point correspondences.

Abstract

We observe that numerous symplectic resolutions can be expressed as intersections of twisted cotangent bundles. Additionally, their dual symplectic resolutions can be derived from intersections of dual twisted cotangent bundles. We determine the collection of fixed points for certain intersections that are Poisson slices, extending the computations of fixed points for parabolic Slodowy varieties.
Paper Structure (32 sections, 26 theorems, 139 equations)

This paper contains 32 sections, 26 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Fundamental Conventions
  3. Some Lie theoretic data
  4. Algebraic groups and actions
  5. Semisimple groups
  6. Hamiltonian action on the cotangent bundle of $G$
  7. Algebraic symplectic quotients
  8. Associated bundle
  9. Twisted Cotangent Bundles
  10. Affine Hamiltonian Lagrangian (AHL) $G$-bundles
  11. AHL $G$-bundles over homogeneous spaces
  12. Symplectic duality and S-duality
  13. Intersection of Hamiltonian $G$-varieties
  14. Intersection with Bielawski hyperkähler slices
  15. Universal centralizer and intersections of AHL $G$-bundles over $G/U$
  16. ...and 17 more sections

Key Result

Theorem 1

where and $\mathcal{X},\mathcal{Y}$ satisfying $\mathcal{X}\coprod \mathcal{Y}=\mathcal{I}\cup\mathcal{K}$ are subsets defined in Proposition general closed condition.

Theorems & Definitions (30)

  • Theorem 1: Theorem \ref{['fixed point set theorem']}
  • Lemma 1.1
  • Proposition 1.2: bisiecki1985holomorphic
  • Proposition 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Example 2.3
  • Lemma 2.4
  • Proposition 2.5: gan2002quantization
  • Proposition 2.6
  • ...and 20 more