Non-degeneracy of cohomological traces for general Landau-Ginzburg models

TL;DR

This paper proves the non-degeneracy of cohomological bulk and boundary traces in general Landau-Ginzburg models with non-isolated critical points, extending duality principles in mathematical physics.

Contribution

It establishes non-degeneracy results for cohomological traces in Landau-Ginzburg models with compact critical locus, generalizing Serre duality to broader settings.

Findings

Proves duality for hypercohomology of Koszul complex of W

Shows shift functor acts as Serre functor on D-branes category

Extends non-degeneracy results to models with non-isolated critical points

Abstract

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.