Orbifold equivalent potentials

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 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.

Orbifold equivalence: structure and new examples

“…If one tries to generate examples of orbifold equivalences truly beyond simple singularities, one soon realises that the approach taken in [13] is neither general nor systematic enough. In that work, the method employed to find expressions like (2.4) was to set one of the variables x i , y j occurring in W (x, y) = V 1 (x) − V 2 (y) to zero, to pick some simple matrix factorisation Q of the resulting potentialW and to complete Q to a graded matrix factorisation Q(x, y) of the full W (x, y) using quasi-homogeneous entries that contain the missing variable -under additional simplifying constraints such as J = −adjugate(E).…”

“…We summarise some abstract properties of the notions of orbifold equivalence and quantum dimensions in a theorem; all statements were proven before, see [11,14,13] and references therein:…”

“…The most interesting orbifold equivalences so far have been found for simple singularities of ADE type [14,13]. The potentials are…”

“…The A-D orbifold equivalences are related to (simple current) orbifolds in the CFT context, but the A-E orbifold equivalences do not arise from any group action [13]; they are examples of "symmetries" beyond groups. For the purposes of elucidating some general observations to be made later, and also of conveying an idea of the typical complexity of the matrix factorisations involved, we reproduce a concrete example of an orbifold equivalence from [13], namely that between V A 11 = x 12 1 + x 2 2 and V E 6 = y 3 1 + y 4 2 . In this case, the smallest possible (see subsection 3.2) orbifold equivalence is of rank 2.…”

Realizing IR theories by projections in the UV

“…Thus, left IR boundary conditions are in one-to-one correspondence with right P -modules in the UV. 5 Analogously one finds that right IR boundary conditions B IR lift to left P -modules B UV = R † ⊗ B IR in the UV, and defects D IR of the IR theory lift to P -bimodules D U V = R † ⊗ D IR ⊗ R. Importantly, P itself is the UV lift of the IR identity defect: R I = A straight-forward generalization of the discussion of IR bulk fields shows that IR defect fields are lifted to bimodule morphisms of the respective UV lifted defects, which again due to the special properties of P are nothing but the defect fields of the UV lifts.…”