Necessary conditions for the existence of Morita Contexts in the bicategory of Landau-Ginzburg Models

TL;DR

This paper investigates the conditions under which Morita contexts exist in the bicategory of Landau-Ginzburg models, establishing necessary conditions and demonstrating their insufficiency through matrix determinant properties.

Contribution

It introduces necessary conditions for Morita contexts in Landau-Ginzburg bicategories and shows these are not sufficient, providing a clearer understanding of their structural limitations.

Findings

Necessary conditions for Morita contexts are identified.

Necessary conditions are proven not to be sufficient.

A trivial sufficient condition is stated.

Abstract

We use a matrix approach to study the concept of Morita context in the bicategory $\mathcal{LG}_{K}$ of Landau-Ginzburg models on a particular class of objects. In fact, we first use properties of matrix factorizations to state and prove two necessary conditions to obtain a Morita context between two objects of $\mathcal{LG}_{K}$. Next, we use a celebrated result (due to Schur) on determinants of block matrices to show that these necessary conditions are not sufficient. Finally, we state a trivial sufficient condition.