Relative quantum cohomology

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 $L\subset X$ 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 $(X,L)=(\mathbb{C}{P}^n,\mathbb{R}{P}^n)$ 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.

70 pages, 9 figures; corrected minor errors, updated bibliography

References in corpus (8)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Disk enumeration on the quintic 3-fold
• Kuranishi homology and Kuranishi cohomology
• The open Gromov-Witten-Welschinger theory of blowups of the projective plane
• WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6
• Involutions, obstructions and mirror symmetry
• Steenrod Pseudocycles, Lifted Cobordisms, and Solomon's Relations for Welschinger's Invariants
• Differential forms on orbifolds with corners

Cited by in corpus (11)

• WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6
• A construction of open descendant potentials in all genera
• The cyclic open-closed map, u-connections and R-matrices
• Differential forms on orbifolds with corners
• Solomon-Tukachinsky's vs. Welschinger's Open Gromov-Witten Invariants of Symplectic Sixfolds
• WDVV-based recursion for open Gromov-Witten invariants
• Relative quantum cohomology of the Chiang Lagrangian
• Open Gromov-Witten invariants from the Fukaya category
• A Geometric Depiction of Solomon-Tukachinsky's Construction of Open GW-Invariants
• Real Topological Recursions and WDVV Relations
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds