Relative scale separation in orbifolds of S2 and S5

Abstract

In orbifold vacua containing an Sq/Γ factor, we compute the relative order of scale separation, r, defined as the ratio of the eigenvalue of the lowest-lying Γ-invariant state of the scalar Laplacian on Sq, to the eigenvalue of the lowest-lying state. For q = 2 and Γ finite subgroup of SO(3), or q = 5 and Γ 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 S5 orbifolds, the maximal relative scale separation is r = 4.2. Methods from invariant theory are very efficient in constructing Γ-invariant spherical harmonics, and can be readily generalized to other orbifolds.