Simplified SFT moduli spaces for Legendrian links

TL;DR

This paper simplifies the computation of certain symplectic field theory invariants for Legendrian links by showing that, after isotopy, only simple holomorphic disks and annuli need to be considered, making the invariants more accessible.

Contribution

It introduces a method to make Legendrian links left-right-simple, reducing the complexity of moduli spaces to combinatorially manageable disks and annuli.

Findings

Holomorphic moduli spaces can be simplified after Legendrian isotopy.

Index 1 holomorphic maps are disks with 1 or 2 positive punctures.

Index 2 maps are disks or annuli without interior critical points.

Abstract

We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to $\mathbb{R}^{4}$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^{3}, \xi_{std})$. We allow our domains, $\Sigma$, to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^{1}(\Sigma)$. In general, $\mathcal{O}^{-1}(0)$ is not combinatorially computable. However after a Legendrian isotopy, $\Lambda$ can be made left-right-simple, implying that any $U$ of index $1$ is a disk with one or two positive punctures for which $\pi_{\mathbb{C}}\circ U$ is an embedding. Moreover, any $U$ of index $2$ is either a disk or an annulus with $\pi_{\mathbb{C}} \circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $\Lambda$ is combinatorially computable using only disks with $\leq 2$ positive punctures.