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Hamiltonian flow between standard module Lagrangians

Yujin Tong

TL;DR

The paper studies the Fukaya-category realization of standard modules in Aganagic's Coulomb-branch model for the $KLRW$ algebra, where two Lagrangian realizations, the $U$-shaped $T$-brane and the step $I$-brane $J$, realize the same standard module. It constructs a fiberwise-stops-preserving Hamiltonian flow whose long-time limit sends the mapping-cone $U$ to the $I$-fiber $J$, providing a geometric interpolation between these realizations and a generalized thimble. This yields a geometric explanation for the previously established categorical isomorphism obtained via holomorphic disc counting. By linking Hamiltonian dynamics on the Coulomb branch to the algebraic structure of the $KLRW$ standard modules, the work enhances the geometric intuition behind categorified knot-invariant constructions.

Abstract

In Aganagic's Fukaya category of the Coulomb branch of quiver gauge theory, the $T_θ$-brane algebra gives a symplectic realization of the Khovanov-Lauda-Rouquier-Webster (KLRW) algebra, where each standard module is known to admit two Lagrangian realizations: the 'U'-shaped $T$-brane and the step $I$-brane. We show that the latter arises as the infinite-time limit of the Hamiltonian evolution of the former, thus serving as a generalized thimble. This provides a geometric realization of the categorical isomorphism previously established through holomorphic disc counting.

Hamiltonian flow between standard module Lagrangians

TL;DR

The paper studies the Fukaya-category realization of standard modules in Aganagic's Coulomb-branch model for the algebra, where two Lagrangian realizations, the -shaped -brane and the step -brane , realize the same standard module. It constructs a fiberwise-stops-preserving Hamiltonian flow whose long-time limit sends the mapping-cone to the -fiber , providing a geometric interpolation between these realizations and a generalized thimble. This yields a geometric explanation for the previously established categorical isomorphism obtained via holomorphic disc counting. By linking Hamiltonian dynamics on the Coulomb branch to the algebraic structure of the standard modules, the work enhances the geometric intuition behind categorified knot-invariant constructions.

Abstract

In Aganagic's Fukaya category of the Coulomb branch of quiver gauge theory, the -brane algebra gives a symplectic realization of the Khovanov-Lauda-Rouquier-Webster (KLRW) algebra, where each standard module is known to admit two Lagrangian realizations: the 'U'-shaped -brane and the step -brane. We show that the latter arises as the infinite-time limit of the Hamiltonian evolution of the former, thus serving as a generalized thimble. This provides a geometric realization of the categorical isomorphism previously established through holomorphic disc counting.

Paper Structure

This paper contains 14 sections, 3 theorems, 28 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Acknowledgements.
  4. Background from the categorification of knot invariants
  5. Setup of both models
  6. The analogue between Aganagic's $\mathfrak{sl}_2$ and Seidel--Smith models
  7. Geometric proof in the Seidel--Smith analogue
  8. Main result: Geometric proof in Aganagic's model
  9. Heuristic description of the flow
  10. Structure and coordinates of the manifold
  11. Expected behavior of the flow
  12. Emergence of the $I$-fibers from the $T$-fibers
  13. Construction of the Hamiltonian function
  14. Verification of the Hamiltonian flow

Key Result

Theorem 1.1

In the Fukaya category of the Coulomb branch associated to the quiver gauge theory $(\Gamma,\vec{d})=(\bullet,1)$, the Hamiltonian evolution of the mapping-cone $T$-brane $U$, under flows preserving the fiberwise stops, asymptotically converges to the step $I$-brane $J$, which serves as the generali

Figures (13)

  • Figure 1: The $T_\theta$-branes representing the generators of the KLRW algebra.
  • Figure 2: Illustration of the standard module Lagrangians.
  • Figure 3: Illustration of the schemes. $\mathbb{C}^{\times}_y$ is depicted as a cylinder $\mathbb{R}_{\left|y\right|}\times S^1_{\mathrm{Arg}(y)}$ with the top $y\to \infty$ and the bottom $y\to 0$.
  • Figure 4: The Lagrangian $\tilde{U}$ (blue), $\tilde{J}$ (green, coinciding with the unstable manifold), and the stable manifold (orange).
  • Figure 5: Real coordinates for Aganagic's scheme.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Theorem 1.1: Main Theorem, informal
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • Example 4.1
  • Example 4.2
  • Remark 4.3