Symmetries of Abelian Orbifolds

TL;DR

This paper uses the Polya Enumeration Theorem to count abelian orbifolds of complex spaces, focusing on their symmetry properties and developing generalized multiplicative sequences for various dimensions and group orders.

Contribution

It introduces a method to count and analyze abelian orbifolds using cycle index and multiplicative sequences, extending to general dimensions and prime power orders.

Findings

Counted orbifolds up to C^6/Gamma

Developed multiplicative sequences controlled by prime powers

Proposed generalization for any dimension and prime

Abstract

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, 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 C^D/Gamma where the order of Gamma is p^alpha. We propose a generalization of these sequences for any D and any p.