New mirror pairs of Calabi-Yau orbifolds

TL;DR

This paper establishes a new representation-theoretic approach to mirror symmetry, constructing infinite pairs of orbifolds with mirror Hodge diamonds and analyzing their properties, especially in the context of Fermat quintic and symmetric group actions.

Contribution

It introduces a novel representation-theoretic framework for Borisov-Batyrev mirror symmetry and constructs new orbifold pairs with mirror Hodge structures, extending mirror symmetry to orbifolds.

Findings

Constructed infinitely many orbifold pairs with mirror Hodge diamonds.

Proved the conjecture that orbifold Hodge diamonds are also mirror.

Identified smooth Calabi-Yau threefolds via $ ext{Hilb}$ schemes with explicit mirror Hodge data.

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 $\P^4$, and $\tilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $\Gamma$ of the alternating group $A_5$, the $\Gamma$-Hilbert schemes $\Gamma$-$\Hilb(X)$ and $\Gamma$-$\Hilb(\tilde{X}^*)$ are smooth Calabi-Yau threefolds with (explicitly computed) mirror Hodge diamonds.