[Paper Review] New mirror pairs of Calabi-Yau orbifolds

Alan Stapledon|arXiv (Cornell University)|Nov 23, 2010

Algebraic Geometry and Number Theory31 references5 citations

TL;DR

This paper establishes a representation-theoretic version of Borisov-Batyrev mirror symmetry for Calabi-Yau orbifolds using equivariant stringy invariants and combinatorial methods. It constructs infinitely many new mirror pairs of orbifolds with mirror Hodge diamonds, proving that for any subgroup Γ ⊆ A₅ acting on the Fermat quintic threefold, the Γ-Hilbert schemes of X and its mirror are smooth Calabi-Yau threefolds with explicitly computed mirror Hodge diamonds.

ABSTRACT

We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When $X$ is the Fermat quintic in $¶^4$, and $ ilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $Γ$ of the alternating group $A_5$, the $Γ$-Hilbert schemes $Γ$-$\Hilb(X)$ and $Γ$-$\Hilb( ilde{X}^*)$ are smooth Calabi-Yau threefolds with (explicitly computed) mirror Hodge diamonds.

Motivation & Objective

• To prove a representation-theoretic version of Borisov-Batyrev mirror symmetry for Calabi-Yau orbifolds.
• To construct infinitely many new mirror pairs of orbifolds with mirror Hodge diamonds using group actions on reflexive polytopes.
• To establish a McKay-type correspondence between orbifold Hodge numbers of quotient spaces X/Γ and X*/Γ when Γ ⊆ SL(M).
• To verify that Γ-Hilbert schemes of mirror hypersurfaces yield smooth Calabi-Yau threefolds with mirror Hodge diamonds.
• To provide a purely combinatorial proof of mirror symmetry for equivariant stringy invariants, avoiding deep intersection cohomology results.

Proposed method

• Introduce the equivariant stringy invariant $ E_{ ext{st}, ho}(Z;u,v) $ with values in the complex representation ring $ R( ho) $ of a finite group Γ acting on a Gorenstein variety Z.
• Derive a general formula for the equivariant Hodge-Deligne polynomial of a non-degenerate hypersurface in a torus (Theorem 4.10), using combinatorial tools on cones and representations.
• Use the equivariant stringy invariant formula (Proposition 5.5) and its simplification for reflexive polytopes (Corollary 5.7) to relate $ E_{ ext{st}, ho}(X;u,v) $ and $ E_{ ext{st}, ho}(X^*;u,v) $ via $ (-u)^{d-1} ext{det}( ho) $.
• Apply the main identity $ E_{ ext{st}, ho}(X;u,v) = (-u)^{d-1} ext{det}( ho) imes E_{ ext{st}, ho}(X^*;u^{-1},v) $ to deduce representation-theoretic mirror symmetry in cohomology.
• Construct crepant resolutions via Γ-Hilbert schemes for Γ ⊆ A₅ acting on the Fermat quintic and its mirror, ensuring smoothness and Calabi-Yau structure.
• Verify the Hodge diamond mirror symmetry for all non-trivial proper subgroups of A₅, including explicit Hodge diamond tables for Γ = ℤ₂, ℤ₂×ℤ₂, ℤ₃, ℤ₅, A₄, S₃, D₅.

Experimental results

Research questions

• RQ1 Can a representation-theoretic version of mirror symmetry be established for Calabi-Yau orbifolds using equivariant stringy invariants?
• RQ2 Do the orbifolds $ ilde{X}/ ho $ and $ ilde{X}^*/ ho $ have mirror Hodge diamonds when $ ho o ext{SL}(M) $, even if singular?
• RQ3 Are the Γ-Hilbert schemes of mirror hypersurfaces smooth Calabi-Yau threefolds with mirror Hodge diamonds for subgroups Γ ⊆ A₅?
• RQ4 Can the mirror symmetry relation for stringy invariants be proven combinatorially without relying on intersection cohomology?
• RQ5 Do the orbifold Hodge numbers $ h^{p,q}_{ ext{orb}} $ of $ ilde{X}/ ho $ and $ ilde{X}^*/ ho $ satisfy a McKay-type duality when $ ho o ext{SL}(M) $?

Key findings

• The paper proves a representation-theoretic mirror symmetry identity: $ E_{ ext{st}, ho}(X;u,v) = (-u)^{d-1} ext{det}( ho) imes E_{ ext{st}, ho}(X^*;u^{-1},v) $, valid for any finite group Γ acting on a reflexive polytope.
• For the Fermat quintic threefold X and its mirror $ ilde{X}^* $, the Γ-Hilbert schemes $ ext{Hilb}_ ho(X) $ and $ ext{Hilb}_ ho( ilde{X}^*) $ are smooth Calabi-Yau threefolds with explicitly computed mirror Hodge diamonds for all subgroups Γ ⊆ A₅.
• When Γ acts freely (e.g., Γ = ℤ₅), the quotients $ X/ ho $ and $ ilde{X}^*/ ho $ are smooth Calabi-Yau manifolds with mirror Hodge diamonds, providing new mirror pairs not in Batyrev-Borisov’s construction.
• For Γ ⊆ A₅, the orbifold Hodge numbers satisfy $ h^{p,q}_{ ext{orb}}( ilde{X}/ ho) = h^{d-1-p,q}_{ ext{orb}}( ilde{X}^*/ ho) $, confirming a McKay-type correspondence for non-abelian orbifold singularities.
• The Hodge diamonds for Γ = ℤ₂, ℤ₂×ℤ₂, ℤ₃, ℤ₅, A₄, S₃, D₅ are computed explicitly, showing $ h^{1,1} = 59, 41, 49, 21, 29, 33, 19 $ respectively, and $ h^{2,0} = h^{1,0} = 1, 0 $, confirming mirror symmetry.
• The results extend to the cubic surface case (d=3), where $ X/ ho $ and $ ilde{X}^*/ ho $ have orbifold Hodge diamonds with $ h^{1,0}_{ ext{orb}} = 0 $, $ h^{2,0}_{ ext{orb}} = 1 $, $ h^{1,1}_{ ext{orb}} = 20 $, and the Γ-Hilbert schemes are crepant resolutions.