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Journal of the European Mathematical Society
Article . 2023 . Peer-reviewed
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Article . 2024
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https://dx.doi.org/10.48550/ar...
Article . 2019
License: arXiv Non-Exclusive Distribution
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Relative quantum cohomology

Authors: Solomon, Jake P.; Tukachinsky, Sara B.;

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.

Keywords

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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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
8
Top 10%
Average
Top 10%
Green
gold