Mirror symmetry between orbifold curves and cusp singularities with group action

Abstract

We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-Hübsch transpose $(f^T,G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.

29 pages, Table 2 corrected, Assumption g=0 added to Theorem 25

References in corpus (8)

• A New Cohomology Theory for Orbifold
• On triangulated orbit categories
• A Generalized Construction of Mirror Manifolds
• FJRW rings and Landau-Ginzburg Mirror Symmetry
• The Mirror Map for Invertible LG Models
• K. Saito's Duality for Regular Weight Systems and Duality for Orbifoldized Poincare Polynomials
• Strange duality, mirror symmetry, and the Leech lattice
• A generalization of Gabriel's Galois covering functors and derived equivalences