Modular Symmetry, Twisted Sectors and Flavour

Twisted sectors of heterotic orbifolds have long been of phenomenological interest, since their massless states are automatically localized at the fixed points, resulting in effective couplings exponentially suppressed by the distance between different points. In the field theory limit of large compactification radii, such nonlocal couplings vanish, but they acquire nonzero values through nonperturbative effects on the worldsheet involving stretched strings. Attempts have been made to explain the hierarchy of fermion masses (which generally requires some Yukawa couplings to be orders of magnitude smaller than unity) and the CKM charged current mixing of quarks, by using this suppression. The usual route to explaining small Yukawa couplings in a natural way is a (broken) flavour symmetry, so one might ask whether there is a symmetry associated with this suppression of nonlocal interactions. In fact, the targetspace modular invariance of heterotic strings might play such a role. This symmetry can be thought of as the extension of T-duality to a d-dimensional torus, where d ≥ 2. In the simplest case d = 2, the path integral for strings propagating on a background with modulus T ∼ V + iB, where V is the torus volume in string units and B is the antisymmetric tensor field (BMN ≡ BǫMN where M , N label the torus coordinates), is equivalent to that on a background T , where T ′ = (αT − iβ)/(iγT + δ), α, β, γ, δ ∈ Z, αδ − βγ = 1. This (P)SL(2,Z) symmetry is generated by S:T 7→ 1/T , T :T 7→ T + i. For 6-dimensional orbifolds, the full duality group may be