Relative scale separation in orbifolds of $S^2$ and $S^5$

TL;DR

This paper investigates the maximum achievable scale separation in orbifold compactifications involving spheres, using invariant theory to efficiently construct invariant spherical harmonics and quantify the relative eigenvalue ratios.

Contribution

It provides explicit calculations of the maximal scale separation ratios for specific orbifolds of $S^2$ and $S^5$, introducing efficient methods from invariant theory.

Findings

Maximum scale separation ratio for $S^2/ ext{finite subgroup of } SO(3)$ is 21.

Maximum scale separation ratio for $S^5/ ext{finite subgroup of } SU(3)$ is 12.

For smooth $S^5$, the maximal ratio is 4.2.

Abstract

In orbifold vacua containing an $S^q/\Gamma$ factor, we compute the relative order of scale separation, $r$, defined as the ratio of the eigenvalue of the lowest-lying $\Gamma$-invariant state of the scalar Laplacian on $S^q$, to the eigenvalue of the lowest-lying state. For $q=2$ and $\Gamma$ finite subgroup of $SO(3)$, or $q=5$ and $\Gamma$ finite subgroup of $SU(3)$, the maximal relative order of scale separation that can be achieved is $r=21$ or $r=12$, respectively. For smooth $S^5$ orbifolds, the maximal relative scale separation is $r=4.2$. Methods from invariant theory are very efficient in constructing $\Gamma$-invariant spherical harmonics, and can be readily generalized to other orbifolds.