Permutation branes and linear matrix factorisations

TL;DR

The work addresses how permutation branes in Gepner models can be described in the Landau-Ginzburg/ matrix-factorisation framework. It develops a dictionary between permutation branes and a class of linear matrix factorisations of W = sum i x_i^d, and uses Ext-group computations to match open-string spectra and topological data, including Witten indices, across tensor-product and orbifold settings. The authors provide explicit constructions for linear factorisations, map them to CFT permutation branes in both odd and even cycle-length cases, and verify the match with CFT results through detailed Ext computations and Macaulay2 checks, including quintic-like examples. The results strengthen the link between boundary CFT at the Gepner point and the derived-category/ LG description of D-branes, enabling practical brane-charge computations and suggesting avenues for extending the framework to more general boundary conditions.

Abstract

All the known rational boundary states for Gepner models can be regarded as permutation branes. On general grounds, one expects that topological branes in Gepner models can be encoded as matrix factorisations of the corresponding Landau-Ginzburg potentials. In this paper we identify the matrix factorisations associated to arbitrary B-type permutation branes.