Explicit correspondences between gradient trees in $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$
Hidemasa Suzuki
TL;DR
The paper investigates an explicit FO-type correspondence between gradient trees on $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$ bounded by affine Lagrangian sections. By constructing explicit Schwarz–Christoffel disks $w_{\epsilon}$ and analyzing their $\epsilon\to0$ limits for $k=3,4$ using Schwarz–Christoffel theory, hypergeometric function expansions, and conformal moduli, it establishes convergence to the corresponding gradient trees, including edge-by-edge behavior and interior-edge lengths. The approach combines analytic and geometric techniques to realize the Fukaya–Oh correspondence in dimension one, providing concrete degeneration control and a detailed description of boundary phenomena via boundary collisions and moduli space degenerations. This clarifies the interplay between polygonal holomorphic disks and gradient-flow combinatorics, with potential implications for explicit computations in Fukaya categories and Floer-theoretic invariants in one-dimensional base cases.
Abstract
Fukaya and Oh studied the correspondence between pseudoholomorphic disks in $T^{*}M$ which are bounded by Lagrangian sections $\{L_{i}^ε\}$ and gradient trees in $M$ which consist of gradient curves of $\{f_{i}-f_{j}\}$. Here, $L_{i}^ε$ is defined by $L_{i}^ε=$\,graph$(εdf_{i})$. They constructed approximate pseudoholomorphic disks in the case $ε>0$ is sufficiently small. When $M=\mathbb{R}$ and Lagrangian sections are affine, pseudoholomorphic disks $w_ε$ can be constructed explicitly. In this paper, we show that pseudoholomorphic disks $w_ε$ converges to the gradient tree in the limit $ε\to+0$ when the number of Lagrangian sections is three and four.