## Abstract

We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov–Witten invariants of a Lagrangian submanifold L ⊂ X with a bounding chain. Simultaneously, we define the quantum cohomology algebra of X relative to L and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov–Witten invariants. Another result is a vanishing theorem for open Gromov–Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov–Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov–Witten invariants of .X; L/ D .CP ^{n}; RP ^{n}/ with n odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya–Oh–Ohta–Ono theory of bounding chains.

Original language | English |
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Pages (from-to) | 3497-3573 |

Number of pages | 77 |

Journal | Journal of the European Mathematical Society |

Volume | 26 |

Issue number | 9 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© 2023 European Mathematical Society.

## Keywords

- A algebra
- Gromov–Witten axiom
- J -holomorphic
- Lagrangian submanifold
- Open WDVV
- bounding chain
- open Gromov–Witten invariant
- relative quantum cohomology
- stable map
- superpotential