An open-closed Deligne-Mumford field theory associated to a Lagrangian submanifold
Amanda Hirschi, Kai Hugtenburg
TL;DR
This work develops a rigorous framework connecting open Gromov-Witten theory with the Fukaya category by constructing an open-closed Deligne--Mumford field theory (DMFT) in the presence of a compact embedded Lagrangian $L$ in a compact symplectic manifold $(X,\ abla)$. Central to the approach is a global Kuranishi chart for moduli spaces of open stable maps, extended to higher genus with boundary, together with coherent Thom forms, to realize a chain-level open-closed TCFT in a truncated setting. The authors prove the existence of a truncated open-closed DMFT whose open sector recovers the Fukaya $A_infty$-algebra of $L$ and whose closed sector recovers GW theory of $X$, with compatibility across boundary strata encoded by explicit orientation and Thom-system data. They further formulate a categorical DMFT in the Costello framework and outline applications to mirror symmetry and enumerative geometry, including genus-zero open invariants and their stability under deformation, and the pathway to higher-genus extensions. This unifies open and closed invariants within a universal perturbative setup and paves the way for deriving GW invariants from the Fukaya category and vice versa, via a principled chain-level formalism.
Abstract
Let $L \subset X$ be a compact embedded Lagrangian in a compact symplectic manifold. We present the moduli spaces of holomorphic maps of arbitrary genus with boundary on $L$ as a global Kuranishi chart, generalising the work of Abouzaid-McLean-Smith and Hirschi-Swaminathan. We use this to define an open-closed Deligne-Mumford theory whose open genus zero part is the Fukaya $A_\infty$ algebra associated to $L$, and whose closed part gives the Gromov--Witten theory of $X$. Combined with results of Costello, this has applications in obtaining Gromov--Witten invariants from the Fukaya category.