Holomorphic disks and GIT quotients

TL;DR

This paper establishes a correspondence between moduli spaces of holomorphic disks in a G-invariant Lagrangian and its GIT quotient, providing a formula to compute disk potentials in the quotient from the original space.

Contribution

It introduces a novel correspondence and computational formula relating holomorphic disks and disk potentials between a space and its GIT quotient.

Findings

Established a correspondence between moduli spaces of holomorphic disks

Derived a formula for the disk potential of the quotient from the original space

Provided computational tools under positivity and topological assumptions

Abstract

Let $G$ be a connected compact Lie group and let $\mathbb{G}$ be its complexification. In this paper, we establish a correspondence between the moduli spaces of holomorphic disks bounded by a $G$-invariant Lagrangian submanifold $L \subseteq X$ and those bounded by its quotient $L/G$ in the GIT quotient $X \mathbin{/\mkern-6mu/} \mathbb{G}$. Under suitable positivity and topological assumptions, we derive a computationally effective formula for the disk potential of $L/G$ from that of $L$ via the {semistable disk potential}, which reflects the choice of a level set of a value of the moment map.