Topological Landau-Ginzburg models on a world-sheet foam

TL;DR

The paper extends topological Landau-Ginzburg theory to world-sheet foams, where each 2D component carries its own $\mathcal{T}_i=(\boldsymbol{\phi}_i,W_i)$ and seams are encoded by matrix factorizations of the sum of adjacent potentials. It provides explicit constructions of boundary data via Lazaroiu's boundary connection, defines operator spaces through Jacobi algebras, and derives comprehensive correlator formulas for closed surfaces, boundaries, and foams, along with gluing rules that generalize standard 2D TQFT composition. A concrete Koszul-factorization example ties the formalism to Grassmannian quantum cohomology and to categorification programs for knot invariants, illustrating deep connections between matrix factorizations, boundary data, and 2-category structures. The framework yields a robust, local-to-global method for computing correlators on complex world-sheet topologies and clarifies how foam gluing translates into tensor-product and Ext-based operations, establishing a rich structure for 2D QFTs with boundaries and defects. Together, these results lay groundwork for systematic applications in categorification, open- and closed-string dualities, and higher-categorical formulations of 2D topological field theories with boundary and seam data.

Abstract

We define topological Landau-Ginzburg models on a world-sheet foam, that is, on a collection of 2-dimensional surfaces whose boundaries are sewn together along the edges of a graph. We use matrix factorizations in order to formulate the boundary conditions at these edges and produce a formula for the correlators. Finally, we present the gluing formulas, which correspond to various ways in which the pieces of a world-sheet foam can be joined together.