Calabi-Yau Orbifolds and Torus Coverings

TL;DR

This paper develops a new analytical method to count Abelian orbifolds of toric Calabi-Yau singularities using torus coverings, verifying previous algorithms and linking orbifold counting to polynomial equations modulo prime numbers.

Contribution

It introduces a novel analytical approach for counting Abelian orbifolds of Calabi-Yau singularities, connecting torus coverings with polynomial equations modulo primes.

Findings

Verified previous algorithm results for orbifold counting.

Established a correspondence between torus covers and orbifolds C^D/Z_p.

Linked orbifold invariance to discrete symmetries in quiver gauge theories.

Abstract

The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to the counting of orbifolds of toric Calabi-Yau singularities, with particular attention to Abelian orbifolds of C^D. By doing so, the work introduces a novel analytical method for counting Abelian orbifolds, verifying previous algorithm results. One identifies a p-fold cover of the torus T^{D-1} with an Abelian orbifold of the form C^D/Z_p, for any dimension D and a prime number p. The counting problem leads to polynomial equations modulo p for a given Abelian subgroup of S_D, the group of discrete symmetries of the toric diagram for C^D. The roots of the polynomial equations correspond to orbifolds of the form C^D/Z_p, which are invariant under the corresponding subgroup of S_Ds. In turn, invariance under this subgroup implies a discrete symmetry for the corresponding quiver gauge theory, as is clearly seen by its brane tiling formulation.