Adjunctions and defects in Landau-Ginzburg models

TL;DR

This work shows that the bicategory $\mathcal{LG}_k$ of Landau-Ginzburg models has well-behaved left and right adjoints for every defect (1-morphism), with evaluation and coevaluation maps described explicitly by Atiyah classes and homological perturbation. It provides a concrete, computable framework to evaluate arbitrary string diagrams in the defect TFT, recovering the Kapustin-Li disk correlator and proving the Cardy condition in full generality (over arbitrary rings $k$). The approach unifies bulk, boundary, and defect sectors into a single bicategorical picture, enabling transparent derivations of open/closed TFT data and their compatibility via residues, Bar/Koszul models, and perturbative inverses. The results extend to a graded pivotal structure and a robust defect action on bulk fields, clarifying the CFT/LG correspondence in the topological setting and offering tools for constructing orbifolds and computing correlators in LG-based TFTs.

Abstract

We study the bicategory of Landau-Ginzburg models, which has potentials as objects and matrix factorisations as 1-morphisms. Our main result is the existence of adjoints in this bicategory and a description of evaluation and coevaluation maps in terms of Atiyah classes and homological perturbation. The bicategorical perspective offers a unified approach to Landau-Ginzburg models: we show how to compute arbitrary correlators and recover the full structure of open/closed TFT, including the Kapustin-Li disk correlator and a simple proof of the Cardy condition, in terms of defect operators which in turn are directly computable from the adjunctions.

Figures (1)

• Figure 1.1: Part of a worldsheet with defect lines and field insertions

Theorems & Definitions (133)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Lemma 2.6
• Proposition 2.7
• proof
• Remark 2.8
• Lemma 2.9