Open enumerative geometries for Landau-Ginzburg models

TL;DR

This paper reviews recent advances in defining open enumerative invariants for Landau-Ginzburg models, focusing on their mathematical foundations, computational methods, and connections to broader theories.

Contribution

It introduces a new framework for open enumerative invariants in Landau-Ginzburg models using multisection integrals over real orbifolds with boundary conditions.

Findings

Open invariants can be expressed as integrals over moduli spaces with boundary conditions.

Certain open invariants satisfy topological recursion and integrable hierarchies.

The paper identifies open problems and future directions in the field.

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.