A differential model for B-type Landau-Ginzburg theories

TL;DR

This paper develops a rigorous differential model for B-type open-closed topological Landau-Ginzburg theories on non-compact Kähler manifolds, generalizing previous models to cases with non-isolated critical points.

Contribution

It introduces a new mathematical framework for B-type Landau-Ginzburg theories that accommodates non-isolated critical loci and specializes to simpler models in Stein cases.

Findings

Constructs a differential model for non-compact Kähler manifolds with compact critical locus.

Shows specialization to Stein manifolds with finite critical set.

Provides a rigorous mathematical foundation for B-type Landau-Ginzburg theories.

Abstract

We describe a mathematically rigorous differential model for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is a non-compact K\"ahlerian manifold with holomorphically trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$ and whose critical locus is compact but need not consist of isolated points. We also show how this construction specializes to the case when $X$ is Stein and $W$ has finite critical set, in which case one recovers a simpler mathematical model.