Relative quantum cohomology

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