Orbifold equivalence: structure and new examples

Abstract

Orbifold equivalence is a notion of symmetry that does not rely on group actions. Among other applications, it leads to surprising connections between hitherto unrelated singularities. While the concept can be defined in a very general category-theoretic language, we focus on the most explicit setting in terms of matrix factorisations, where orbifold equivalences arise from defects with special properties. Examples are relatively difficult to construct, but we uncover some structural features that distinguish orbifold equivalencesmost notably a finite perturbation expansion. We use those properties to devise a search algorithm, then present some new examples including Arnold singularities.

McKay correspondence and orbifold equivalence

“…It was noticed there that several of the known examples of orbifold equivalence fit into this setting. Note that in the most of the known cases the orbifold equivalences are constructed by the computational methods ( [8], [15], [18], [13]). Our methods are geometric and allow to reproof some of the known results.…”

Dimensional & algorithmic reductions for Calogero-Ruijsenaars & Landau-Ginzburg models

“…In Chapter 6 we present a search algorithm for determining whether a given pair f and g are orbifold equivalent. This search algorithm follows parallel work from [38].…”

“…Theorem F (Theorem 6.3.2; [38]). Given two potentials f and g, there exists an algorithm that terminates if and only if f and g are orbifold equivalent.…”

“…The notion of orbifold equivalence was inspired by the study of (defects in) topological quantum field theories (see [8,12,19]) and it was first defined in the context of the study of equivariant and orbifold completions of the bicategory of Landau-Ginzburg models. Several examples have been explored in detail in the recent years [7,[38][39][40].…”

“…The present chapter is concerned with finding orbifold equivalences using computer search. The current state of the art is the algorithm presented in [38]. As recorded in Theorem 6.3.2, this algorithm terminates if and only if two potentials are orbifold equivalent.…”