Sublattice Counting and Orbifolds

TL;DR

This paper develops a number theoretic approach using Polya's Enumeration Theorem to count abelian orbifolds of C^3 and other toric Calabi-Yau singularities, providing explicit enumeration methods.

Contribution

It introduces a novel application of crystallography and number theory to systematically count orbifolds of complex geometries, extending previous partial results.

Findings

Count of abelian orbifolds of C^3 using partition functions.

Extension of counting methods to other toric Calabi-Yau singularities.

Explicit enumeration of orbifolds of the conifold, L^{aba} theories, and C^4.

Abstract

Abelian orbifolds of 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 C^4.