
arXiv: 1906.04795
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.
High Energy Physics - Theory, bounding chains, open Gromov-Witten invariants, FOS: Physical sciences, 53D45, 53D37 (Primary), 14N35, 14N10, 53D12 (Secondary), Gromov-Witten axioms, Mathematics - Algebraic Geometry, Lagrangian submanifolds; Maslov index, relative quantum cohomology, Enumerative problems (combinatorial problems) in algebraic geometry, FOS: Mathematics, \(A_\infty\)-algebras, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, \(J\)-holomorphic curves, Algebraic Geometry (math.AG), open WDVV, Lagrangian submanifolds, High Energy Physics - Theory (hep-th), superpotentials, Mathematics - Symplectic Geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, stable maps, Symplectic Geometry (math.SG)
High Energy Physics - Theory, bounding chains, open Gromov-Witten invariants, FOS: Physical sciences, 53D45, 53D37 (Primary), 14N35, 14N10, 53D12 (Secondary), Gromov-Witten axioms, Mathematics - Algebraic Geometry, Lagrangian submanifolds; Maslov index, relative quantum cohomology, Enumerative problems (combinatorial problems) in algebraic geometry, FOS: Mathematics, \(A_\infty\)-algebras, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, \(J\)-holomorphic curves, Algebraic Geometry (math.AG), open WDVV, Lagrangian submanifolds, High Energy Physics - Theory (hep-th), superpotentials, Mathematics - Symplectic Geometry, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, stable maps, Symplectic Geometry (math.SG)
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