On the monoidal structure of matrix bi-factorisations
Nils Carqueville, Ingo Runkel
TL;DR
The paper develops a bimodule formulation of matrix bi-factorisations and proves that the associated category is monoidal, providing a rigorous framework for defects in two-dimensional Landau-Ginzburg models. It then extends to graded and multi-variable superpotentials, constructing explicit unit objects and associators and establishing monoidal structure for the finite-rank subcategories. By comparing with the bosonic subsector of $oldsymbol{\mathcal{N}=2}$ minimal models, the authors show a monoidal equivalence (in applicable subsectors) and compute fusing matrices on both sides, finding agreement that supports the LG/CFT correspondence. This work thus unifies matrix-factorisation defect descriptions with conformal field theory defect fusion, delivering concrete tools for calculating fusing matrices and testing dualities in defect theories.
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.