Landau-Ginzburg models -- old and new

TL;DR

This paper explores the development of Stability Hodge Structure (SHS) in the context of Landau-Ginzburg models, aiming to advance mathematical understanding and solve longstanding problems in algebraic geometry.

Contribution

It introduces the perspective of Landau-Ginzburg models to the study of SHS and discusses conjectures and applications through simple examples.

Findings

Identification of SHS within Landau-Ginzburg models

Proposed conjectures based on simple examples

Potential applications to algebraic geometry problems

Abstract

In the last three years a new concept -- the concept of wall crossing has emerged. The current situation with wall crossing phenomena, after papers of Seiberg-Witten, Gaiotto-Moore-Neitzke, Vafa-Cecoti and seminal works by Donaldson-Thomas, Joyce-Song, Maulik-Nekrasov-Okounkov-Pandharipande, Douglas, Bridgeland, and Kontsevich-Soibelman, is very similar to the situation with Higgs Bundles after the works of Higgs and Hitchin -- it is clear that a general "Hodge type" of theory exists and needs to be developed. Nonabelian Hodge theory did lead to strong mathematical applications -- uniformization, Langlands program to mention a few. In the wall crossing is is also clear that some "Hodge type" of theory exists -- Stability Hodge Structure (SHS). This theory needs to be developed in order to reap some mathematical benefits --- solve long standing problems in algebraic geometry. In this paper we look at SHS from the perspective of Landau--Ginzburg models and we look at some applications. We consider simple examples and explain some conjectures these examples suggest.