Hamiltonian flow between standard module Lagrangians

TL;DR

This paper demonstrates that a specific Lagrangian realization in the Fukaya category can be obtained as an infinite-time Hamiltonian flow limit, linking geometric and categorical perspectives in symplectic geometry.

Contribution

It establishes a geometric interpretation of the step Lagrangian as an infinite-time limit of Hamiltonian evolution of the U-shaped Lagrangian in the context of Coulomb branch Fukaya categories.

Findings

The step Lagrangian arises as a Hamiltonian flow limit of the U-shaped Lagrangian.

Provides a geometric realization of a categorical isomorphism via Hamiltonian dynamics.

Connects holomorphic disc counting with Hamiltonian flow limits in symplectic geometry.

Abstract

In Aganagic's Fukaya category of the Coulomb branch of quiver gauge theory, the $T_\theta$-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.