Orbifold equivalent potentials
Nils Carqueville, Ana Ros Camacho, Ingo Runkel
TL;DR
The paper introduces an orbifold equivalence relation among potentials defined via finite-rank graded matrix factorisations with nonzero left and right quantum dimensions, proving this is an equivalence relation in a pivotal bicategory framework. It specializes to ADE simple singularities over C, classifying the equivalence classes and providing explicit matrix factorisations to realise the ADE pairings, including D-to-A and E-to-A type correspondences. Beyond objects, it shows these equivalences induce derived-category level correspondences and links to Dynkin quiver representations through equivariant completion. The work then compares these mathematical structures to two-dimensional N=2 conformal field theories, arguing that the ADE orbifold classes reflect CFT Morita equivalences and that defect spectra and module decompositions align with CFT predictions, thereby supporting the LG/CFT correspondence. Collectively, the results yield new, structurally rich connections between matrix factorisations, category theory, and rational CFT in the ADE landscape.
Abstract
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 type {A_{d-1}} for d odd, {A_{d-1},D_{d/2+1}} for d even 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.