Differential Models for B-Type Open–Closed Topological Landau–Ginzburg Theories

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 TFT (topological field theory) are satisfied on cohomology and conjecture that the remaining two axioms (namely non-degeneracy of bulk and boundary traces and the topological Cardy constraint) are also satisfied.