To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numbers, the left and right quantum dimension. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories.Restricted to ADE singularities, the resulting equivalence classes of potentials are those of typeeven but not in {12, 18, 30}, and {A 11 , D 7 , E 6 }, {A 17 , D 10 , E 7 } and {A 29 , D 16 , E 8 }. This is the result expected from two-dimensional rational conformal field theory, and it directly leads to new descriptions of and relations between the associated (derived) categories of matrix factorisations and Dynkin quiver representations.

We prove a tensor equivalence between full subcategories of a) graded matrix factorisations of the potential x d − y d and b) representations of the N = 2 minimal super vertex operator algebra at central charge 3 − 6/d, where d is odd. The subcategories are given by a) permutation-type matrix factorisations with consecutive index sets, and b) Neveu-Schwarztype representations. The physical motivation for this result is the Landau-Ginzburg / conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [BR], where an isomorphism of fusion rules was established.

In this paper we study the computational feasibility of an algorithm to prove orbifold equivalence between potentials describing Landau-Ginzburg models. Through a comparison with state-of-the-art results of Gröbner basis computations in cryptology, we infer that the algorithm produces systems of equations that are beyond the limits of current technical capabilities. As such the algorithm needs to be augmented by 'inspired guesswork', and we provide two new examples of applying this approach.

No abstract

In this brief note we prove orbifold equivalence between two potentials described by strangely dual exceptional unimodular singularities of type $K_{14}$ and $Q_{10}$ in two different ways. The matrix factorizations proving the orbifold equivalence give rise to equations whose solutions are permuted by Galois groups which differ for different expressions of the same singularity.Comment: Appendix by the second author and Federico Zerbin