Relative quantum cohomology

  • Jake P. Solomon

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

    Tel Aviv University, Israel
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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 with a bounding chain. Simultaneously, we define the quantum cohomology algebra of relative to 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 with 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