Counting orbifolds

Abstract

We present several methods of counting the orbifolds ${{{{\mathbb{C}^D}}} \left/ {\Gamma } \right.}$. A correspondence between counting orbifold actions on ${\mathbb{C}^D}$, brane tilings, and toric diagrams in D - 1 dimensions is drawn. Barycentric coordinates and scaling mechanisms are introduced to characterize lattice simplices as toric diagrams. We count orbifolds of ${\mathbb{C}^3}$, ${\mathbb{C}^4}$, ${\mathbb{C}^5}$, ${\mathbb{C}^6}$ and ${\mathbb{C}^7}$. Some remarks are made on closed form formulas for the partition function that counts distinct orbifold actions.