Orbifold equivalent potentials
A polynomial \(V = V(x) \in k[x_1,\dots,x_m]\) is called a \textit{homogeneous potential} if is it homogeneous of degree \(2\) with respect to some given \(\mathbb Q_+\)-grading. Two such potentials \(V(x),W(y)\) (over the same base field \(k \subset \mathbb C\), but not necessarily the same number of variables \(m\)) are called \textit{orbifold equivalent} if there exists a finite-rank graded matrix factorization of \(W(y) - V(x) \in k[x_1,\dots, x_m,y_1,\dots y_n]\) whose left and right quantum dimensions are nonzero (concrete definitions, as well as a comparison with their more general counterparts in the context of pivotal categories, are given in the paper). The main result of the paper is a classification of orbifold equivalence classes of potentials over \(k = \mathbb C\) which define ADE singularities. Denoting by \(d\) the central charge, equivalence classes are \[ \begin{align*}{ &\{ A_{d-1} \}, \text{ for \(d\) odd},\cr &\{ A_{d-1}, D_{d/2+1} \} \text{ for \(d\) even, } d \not\in \{ 12,18,30 \},\cr &\{ A_{11}, D_7, E_6 \}, \{ A_{17}, D_{10}, E_7 \}, \{ A_{29}, D_{16}, E_8 \}.}\end{align*} \] The final section of the paper explains the relation of the above with the (conjectural) correspondence between \(\mathcal N = 2\) supersymmetric Landau--Ginzburg models and \(\mathcal N = 2\) superconformal field theories in two dimensions. The results of the paper also shed light on the relation between derived categories of matrix factorizations and Dynkin quiver representations.
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