Cosets of Orbifold Compactification

The duality symmetry group of the cosets SU(n, 1) SU(n)⊗ U(1) , which describe the moduli space of a two-dimensional subspace of an orbifold model with (n − 1) complex Wilson lines moduli, is discussed. The full duality group and its explicit action on the moduli fields are derived. ∗ e-mail: [email protected]. 1 The derivation of a field theoretic low-energy effective action from string theory is a first step in attempting to relate string theory to the supersymmetric standard model or grand unified theories. In particular, such a field theory should have implications to fundamental questions in particle physics and cosmology, such as the study of gauge coupling unification scale, the masses of quarks and leptons and stringy inspired inflationary scenarios. Orbifold compactified heterotic string theories are of great phenomenological relevance. They constitute a large class of string vacua where the interactions can be computed explicitly using the underlying world-sheet conformal field theory [6]. The low-energy action is an N = 1 supergravity coupled to Yang-Mills and matter fields and their supersymmetric partners. With only terms with up to two derivatives in the bosonic fields, the theory is described in terms of three functionsthe Kähler potential K encoding the kinetic terms for the massless fields, the superpotential W containing the Yukawa couplings and the gauge f−function whose real part, at the tree level, determines the gauge couplings [4]. The functions K and W appear in the Lagrangian of the theory through the combination G = K + log|W |. (1) The orbifold models posses a set of continuous parameters, the toroidal moduli, parametrizing the size and shape of the orbifold. The vacuum expectation values of the moduli fields represent marginal deformations of the underlying conformal field theory of the orbifold [5]. The toroidal moduli fields belong to the untwisted sector of the orbifold and enter the space-time N = 1 supersymmetric four-dimensional Lagrangian as chiral fields with flat potentials to all orders in perturbation theory. In addition to toroidal moduli, the untwisted sector of the heterotic string theory compactified on orbifolds may also contain Wilson lines moduli [11]. These additional moduli exist in orbifold models where the twist defining the orbifold is realized on the E8 × E8 root lattice by a rotation [11]. Wilson line moduli are phenomenologicaly interesting because they lower the rank of the gauge group and thus leading to more realistic models. The moduli of the compactification on a d-dimensional torus T = R d Λ , where Λ is a d-dimensional lattice, are encoded in the metric Gij which is the lattice metric of Λ, an antisymmetric tensor Bij and Wilson lines Ai, where I is an E8 × E8 gauge lattice index and i is an internal lattice 2 index. The moduli space of toroidal compactification [10] is given (locally) by the coset space SO(d+ 16, d) SO(d+ 16, d)⊗ SO(d) . Toroidal compactifications lead to low-energy models with N = 4 supersymmetry and gauge groups of rank d + 16. Six-dimensional orbifolds [1,2] are obtained by identifying the points of the six-torus T6 under a cyclic group ZN = {θj , j = 0, · · · , N − 1}. In order to obtain consistent space-time supersymmetric theories, the twist should belong to SU(3) but not SU(2) [1, 3]. Furthermore, to reduce the rank of the gauge group, continuous Wilson line moduli must be introduced, this can be achieved by allowing the orbifold twist to act on the gauge sector of the theory as an automorphism of the E8 ×E8 root lattice. It can be demonstrated [8, 12] that the moduli spaces of orbifolds depend entirely on the eigenvalues of the twist and their multiplicities. The moduli space of the orbifold are parametrized by the T moduli corresponding to the Kähler deformations and the U moduli which correspond to the deformations of the complex structure. For each U modulus, the corresponding moduli space is described by the coset