On the monoidal structure of matrix bi-factorisations

TL;DR

This paper studies the tensor product structure of matrix bi-factorisations using bimodules, establishing a monoidal category with applications in physics and conformal field theory, and compares different categorical descriptions of defect lines.

Contribution

It formulates matrix factorisations as bimodules to show they form a monoidal category and compares this with a conformal field theory perspective, including explicit computations of 6j-symbols.

Findings

Bimodule matrix factorisations form a monoidal category.

The monoidal category has a physical interpretation in Landau-Ginzburg models.

Explicit comparison of categories via 6j-symbols.

Abstract

We investigate tensor products of matrix factorisations. This is most naturally done by formulating matrix factorisations in terms of bimodules instead of modules. If the underlying ring is C[x_1,...,x_N] we show that bimodule matrix factorisations form a monoidal category. This monoidal category has a physical interpretation in terms of defect lines in a two-dimensional Landau-Ginzburg model. There is a dual description via conformal field theory, which in the special case of W=x^d is an N=2 minimal model, and which also gives rise to a monoidal category describing defect lines. We carry out a comparison of these two categories in certain subsectors by explicitly computing 6j-symbols.