Modular flavour symmetry and orbifolds

We develop a bottom-up approach to flavour models which combine modular symmetry with orbifold constructions. We first consider a 6d orbifold $\mathbb{T}^2/\mathbb{Z}_N$, with a single torus defined by one complex coordinate $z$ and a single modulus field $\tau$, playing the role of a flavon transforming under a finite modular symmetry. We then consider 10d orbifolds with three factorizable tori, each defined by one complex coordinate $z_i$ and involving the three moduli fields $\tau_1, \tau_2, \tau_3$ transforming under three finite modular groups. Assuming supersymmetry, consistent with the holomorphicity requirement, we consider all 10d orbifolds of the form $(\mathbb{T}^2)^3/(\mathbb{Z}_N\times\mathbb{Z}_M)$, and list those which have fixed values of the moduli fields (up to an integer). The key advantage of such 10d orbifold models over 4d models is that the values of the moduli are not completely free but are constrained by geometry and symmetry. To illustrate the approach we discuss a 10d modular seesaw model with $S_4^3$ modular symmetry based on $(\mathbb{T}^2)^3/(\mathbb{Z}_4\times\mathbb{Z}_2)$ where $\tau_1=i,\ \tau_2=i+2$ are constrained by the orbifold, while $\tau_3=\omega$ is determined by imposing a further remnant $S_4$ flavour symmetry, leading to a highly predictive example in the class CSD$(n)$ with $n=1-\sqrt{6}$.