Sublattice counting and orbifolds

Abstract

Abelian orbifolds of $\mathbb{C}^{3}$ are known to be encoded by hexagonal brane tilings. To date it is not known how to count all such orbifolds. We fill this gap by employing number theoretic techniques from crystallography, and by making use of Polya's Enumeration Theorem. The results turn out to be beautifully encoded in terms of partition functions and Dirichlet series. The same methods apply to counting orbifolds of any toric non-compact Calabi-Yau singularity. As additional examples, we count the orbifolds of the conifold, of the L ^aba theories, and of $\mathbb{C}^{4}$.