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Open enumerative geometries for Landau-Ginzburg models

Mark Gross, Tyler L. Kelly, Ran J. Tessler

TL;DR

The paper surveys the development of open enumerative theories for Landau-Ginzburg models, focusing on open FJRW theory built from graded W-spin surfaces and their boundary conditions. It details moduli spaces as real orbifolds with corners, the construction of open Witten bundles and relative cotangent lines, and how boundary data yields well-defined open intersection numbers across PST, BCT, GKT, and TZ frameworks. It connects these invariants to mirror symmetry, LG/CY correspondence, and integrable hierarchies, and discusses wall-crossing phenomena that arise from varying boundary conditions. The work highlights key techniques, like multisection boundary conditions and compatibility constraints, and outlines major open problems including virtual fundamental chain constructions, higher-genus open theories, and a full open LG/CY program. Overall, the article synthesizes foundational methods, computational schemes, and ambitious conjectures guiding open enumerative geometry in LG models.

Abstract

We survey the recent progress in defining open enumerative theories for Landau-Ginzburg models. We illustrate the ideas required to develop these new foundations. In particular, we describe how to define the open enumerative invariants as integrals of multisections of certain vector bundles over a moduli space that is a real orbifold with corners, after prescribing boundary conditions for the multisections. We then explain the known situations where the open invariants satisfy certain forms of topological recursion relations, integrable hierarchies, or mirror symmetry. We end with a list of open questions and problems.

Open enumerative geometries for Landau-Ginzburg models

TL;DR

The paper surveys the development of open enumerative theories for Landau-Ginzburg models, focusing on open FJRW theory built from graded W-spin surfaces and their boundary conditions. It details moduli spaces as real orbifolds with corners, the construction of open Witten bundles and relative cotangent lines, and how boundary data yields well-defined open intersection numbers across PST, BCT, GKT, and TZ frameworks. It connects these invariants to mirror symmetry, LG/CY correspondence, and integrable hierarchies, and discusses wall-crossing phenomena that arise from varying boundary conditions. The work highlights key techniques, like multisection boundary conditions and compatibility constraints, and outlines major open problems including virtual fundamental chain constructions, higher-genus open theories, and a full open LG/CY program. Overall, the article synthesizes foundational methods, computational schemes, and ambitious conjectures guiding open enumerative geometry in LG models.

Abstract

We survey the recent progress in defining open enumerative theories for Landau-Ginzburg models. We illustrate the ideas required to develop these new foundations. In particular, we describe how to define the open enumerative invariants as integrals of multisections of certain vector bundles over a moduli space that is a real orbifold with corners, after prescribing boundary conditions for the multisections. We then explain the known situations where the open invariants satisfy certain forms of topological recursion relations, integrable hierarchies, or mirror symmetry. We end with a list of open questions and problems.
Paper Structure (38 sections, 17 theorems, 101 equations, 9 figures)

This paper contains 38 sections, 17 theorems, 101 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background on (closed) FJRW theory
  3. $W$-spin surfaces
  4. Moduli space and intersection theories
  5. Cohomological field theory and the FJRW class
  6. Relations with other theories and applications
  7. Mirror symmetry
  8. Integrable hierarchies
  9. The LG/CY correspondence
  10. $3$-spin structures and tautological relations on the moduli of curves
  11. Open FJRW theory
  12. Graded $W$-spin surfaces
  13. Lifting and alternations
  14. Point-insertion surfaces
  15. Different theories
  16. ...and 23 more sections

Key Result

Proposition 2.4

For any connected component $C_l$ of $|\widehat{C}|,$ with markings also denoted by $\{z_i\}_{i\in [l]}$ and half-nodes $\{p_h\}_{h\in N},$ it holds that $|\widehat{S}|$ is a line bundle and where $a_i,c_h$ are the twists.

Figures (9)

  • Figure 1: The partial normalization ${\nu}_q: C' \rightarrow C$ of $C$ at a node $q$.
  • Figure 2: A smooth connected marked genus 0 orbifold Riemann surface with boundary with two internal markings $z_1, z_2$ and a boundary marking $x_1$.
  • Figure 3: The five types of nodes on a nodal marked surface.
  • Figure 4: An example of two equivalent TZ $(x^r,\mathfrak{h})$-surfaces $\mathfrak{m}$-connected surfaces. Here $x_\zeta$ is alternating and can have twist $r-2m-2,$ for $0\leq m\leq\mathfrak{h},$ while $z_\xi$ has twist $m.$ Pointed arcs correspond to $\mathfrak{m}$-edges.
  • Figure 5: A graded $W$-spin disk and its corresponding dual graph $\Gamma$
  • ...and 4 more figures

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.6
  • Theorem 2.7
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • Definition 2.11
  • ...and 39 more