# Relative quantum cohomology

### Jake P. Solomon

The Hebrew University of Jerusalem, Israel### Sara B. Tukachinsky

Tel Aviv University, Israel

## 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 $(X,L)=(CP_{n},RP_{n})$ 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.

## Cite this article

Jake P. Solomon, Sara B. Tukachinsky, Relative quantum cohomology. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1337