Counting Orbifolds

TL;DR

This paper introduces methods for counting orbifolds of complex spaces by exploring their combinatorial and geometric representations, establishing correspondences with brane tilings and toric diagrams, and applying these to various dimensions.

Contribution

It develops new counting techniques for orbifolds using barycentric coordinates and scaling, linking orbifold actions to toric diagrams and brane tilings.

Findings

Counted orbifolds of C^3 to C^7.

Established correspondence between orbifold actions, brane tilings, and toric diagrams.

Discussed potential closed-form formulas for counting orbifold actions.

Abstract

We present several methods of counting the orbifolds C^D/Gamma. A correspondence between counting orbifold actions on 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 C^3, C^4, C^5, C^6 and C^7. Some remarks are made on closed form formulas for the partition function that counts distinct orbifold actions.