Holomorphic disks and GIT quotients

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.