Rigidity and defect actions in Landau-Ginzburg models

TL;DR

This paper explores the categorical structure of topological defect lines in Landau-Ginzburg models, explicitly describing duality and fusion, and comparing defect actions to conformal field theories, revealing phase differences.

Contribution

It explicitly constructs the rigid and pivotal structure of defect categories in Landau-Ginzburg models with potential x^d using matrix factorizations.

Findings

Duality operation allows defect action computation on bulk fields.

Defect actions in Landau-Ginzburg models differ by phases from N=2 conformal field theories.

Provides a categorical framework for understanding defect fusion and duality.

Abstract

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential x^d, where defects are described by matrix factorisations of x^d-y^d. The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N=2 conformal field theories. We find that the two actions differ by phases.