Rigidity and defect actions in Landau-Ginzburg models
Nils Carqueville, Ingo Runkel
TL;DR
The paper constructs and analyzes the rigid and pivotal monoidal structure of the defect category $MF_{bi}(W)$ for one-variable Landau-Ginzburg models with potential $W$, explicitly giving right duals, evaluation/coevaluation maps, and a pivotal structure. It then uses this framework to derive explicit defect actions on bulk fields, comparing the LG results with $ ext{N}=2$ minimal model CFT data and showing phases arise in the correspondence, though certain phase-insensitive quantities agree. The study clarifies how defect fusion, duality, and defect-induced maps on bulk fields interact with the triangulated and Grothendieck structures, highlighting both compatibilities and essential discrepancies between LG and CFT descriptions. The discussion points toward higher-categorical extensions (bicategories of LG theories) and outlines directions for extending rigidity and pivotality analyses to multi-variable potentials, with an eye toward a refined understanding of dualities in the defect landscape.
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.