Symmetries of abelian orbifolds

Abstract

Using the Polya Enumeration Theorem, we count with particular attention to ${{{{\mathbb{C}^3}}} \left/ {\Gamma } \right.}$ up to ${{{{\mathbb{C}^6}}} \left/ {\Gamma } \right.}$, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S _D . This produces a collection of multiplicative sequences, one for each cycle in the Cycle Index of the permutation group. A multiplicative sequence is controlled by its values on prime numbers and their pure powers. Therefore, we pay particular attention to orbifolds of the form ${{{{\mathbb{C}^D}}} \left/ {\Gamma } \right.}$ where the order of Γ is p ^α. We propose a generalization of these sequences for any D and any p.