An open-closed Deligne-Mumford field theory associated to a Lagrangian submanifold

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.