Open-closed Deligne-Mumford field theories: construction

TL;DR

This paper constructs open-closed Deligne-Mumford field theories from moduli spaces of stable curves with boundary, extending Fukaya's A-infinity algebra to higher genus and boundary components.

Contribution

It introduces a chain-level open-closed DMFT associated to a spin embedded Lagrangian, extending Fukaya's algebra and proving uniqueness up to homotopy.

Findings

Extends Fukaya A-infinity algebra to high genus and boundary curves.

Associates a unique open-closed DMFT to a given Lagrangian.

Supports Kontsevich's conjecture relating Fukaya categories to Gromov-Witten invariants.

Abstract

Open-closed Deligne--Mumford field theories are chain-level field theories based on moduli spaces of stable curves with boundary. We associate to a relatively spin embedded Lagrangian $L \subset (X,\omega)$ such an open-closed DMFT. It extends the Fukaya $A_\infty$ algebra to curves of arbitrarily high genus and with arbitrarily many boundary components and is unique up to homotopy. This is the first step in proving Kontsevich's conjecture that the Fukaya category determines the Gromov--Witten invariants of $X$, following a strategy delineated by Costello.