[Paper Review] Mirror symmetry between orbifold curves and cusp singularities with group action

Wolfgang Ebeling, Atsushi Takahashi|arXiv (Cornell University)|Mar 28, 2011

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper establishes a mirror symmetry correspondence between orbifold curves defined by invertible polynomials with finite symmetry groups and cusp singularities with group actions, showing that Dolgachev numbers of the orbifold curves match Gabrielov numbers of the mirror cusp singularities, and that the stringy Euler characteristic of the curve equals the equivariant Milnor number of the mirror singularity.

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.

Motivation & Objective

• To generalize homological mirror symmetry to the setting of orbifold curves and cusp singularities with group actions.
• To define and compute invariants—Dolgachev numbers for orbifold curves and Gabrielov numbers for cusp singularities with group actions—under mirror symmetry.
• To verify that the stringy Euler characteristic of the orbifold curve matches the G^T-equivariant Milnor number of the mirror cusp singularity.
• To recover known dualities such as Arnold’s strange duality and Seidel’s example as special cases of the proposed mirror symmetry.
• To show that Poincaré series of (f, G₀) can be expressed as a quotient of characteristic polynomials of dual pairs (f^T, G₀^T) and (F, G₀^T).

Proposed method

• Define an orbifold curve C_{(f,G)} as the quotient of the zero set of an invertible polynomial f by the extended group G̃ = ℂ^* ⋊ G.
• Construct the mirror cusp singularity F(x,y,z) = x^{γ′₁} + y^{γ′₂} + z^{γ′₃} - xyz, which is right equivalent to f^T - xyz.
• Use the Berglund-Hübsch transpose (f^T, G^T) to define the mirror dual pair.
• Define Dolgachev numbers as the isotropy orders of the orbifold curve’s singular points.
• Define Gabrielov numbers via the G^T-action on the Milnor fiber of F, capturing the equivariant topology of the singularity.
• Apply semi-orthogonal decompositions and orbit category constructions to relate derived categories of coherent sheaves and Fukaya categories.

Experimental results

Research questions

• RQ1 How do Dolgachev numbers of orbifold curves relate to Gabrielov numbers of mirror cusp singularities with group actions?
• RQ2 Does the stringy Euler characteristic of the orbifold curve equal the G^T-equivariant Milnor number of the mirror cusp singularity?
• RQ3 Can Arnold’s strange duality and Seidel’s example be recovered as special cases of this mirror symmetry framework?
• RQ4 Under what conditions can the Poincaré series of (f, G₀) be expressed as a quotient of characteristic polynomials of dual pairs?
• RQ5 What is the geometric and topological correspondence between the genus of the orbifold curve and the age-1 fixed-point structure of G^T?

Key findings

• The Dolgachev numbers of the orbifold curve C_{(f,G)} coincide exactly with the Gabrielov numbers of the mirror cusp singularity (F, G^T).
• The stringy Euler characteristic of C_{(f,G)} equals the G^T-equivariant Milnor number of the cusp singularity defined by F.
• The genus of the orbifold curve C_{(f,G)} equals the number of elements of age 1 in G^T that fix only the origin.
• For G = G₀, the Poincaré series of (f, G₀) is expressible as a quotient of the characteristic polynomials of (f^T, G₀^T) and (F, G₀^T).
• When G = G_f^{fin}, the maximal symmetry group, the mirror is the cusp singularity T_{2,3,6} with trivial group action, recovering the result of Ebeling-Takahashi.
• The example f(x,y,z) = x^5 + y^5 + z^5 with G₀ = ℤ/5ℤ yields a genus-2 curve with Dolgachev numbers (5,5,5), dual to the cusp singularity x^5 + y^5 + z^5 - xyz, recovering Seidel’s example.