Differential models for B-type open-closed topological Landau-Ginzburg theories

TL;DR

This paper develops differential models for B-type open-closed topological Landau-Ginzburg theories on non-compact Calabi-Yau manifolds, providing explicit cochain level constructions and verifying key axioms.

Contribution

It introduces a new family of cochain level models for Landau-Ginzburg theories with explicit trace and map constructions, advancing the mathematical understanding of these theories.

Findings

Models satisfy most open-closed topological field theory axioms on cohomology

Explicit cochain level formulas for traces and maps are provided

Conjecture that all axioms are satisfied at the cochain level

Abstract

We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is any non-compact Calabi-Yau manifold and $W$ is any holomorphic complex-valued function defined on $X$ whose critical set is compact. The models are constructed at cochain level using smooth data, including the twisted Dolbeault algebra of polyvector valued forms and a twisted Dolbeault category of holomorphic factorizations of $W$. We give explicit proposals for cochain level versions of the bulk and boundary traces and for the bulk-boundary and boundary-bulk maps of the Landau-Ginzburg theory. We prove that most of the axioms of an open-closed topological field theory are satisfied on cohomology and conjecture that the remaining axioms are also satisfied.