Matching the heterotic string on orbifolds and their resolutions

Abstract

We study the symmetry breaking mechanism under which a 6d orbifold compactification of the 10d heterotic string turns into a smooth Calabi--Yau compactification. This process is naturally required to preserve N=1 supersymmetry in 4d. The cause is the existence of a Fayet--Iliopoulos D--term generated by the anomalous U(1) gauge--symmetry on the orbifold theory. An orbifold is constructed by modding out a symmetry from a toroidal 6d lattice, resulting in an almost everywhere flat variety with the exception of fixed sets under the symmetry action. Those sets are fixed points and fixed tori, and the states localized at them are the so called twisted states. D--flatness leads to a vacuum with non--zero expectation values of twisted scalars. Those scalars play the role of blow--up modes: their vevs deform the local geometry and smooth out the singularities. We study Calabi--Yau manifolds obtained by blowing--up (resolving) the singularities using toric geometry. We analyze the massless spectrum and the anomaly cancellation on the deformed orbifold and in the resolution obtaining a perfect map. On the orbifold we can compute the full particle spectrum and the interactions using the CFT world--sheet description of the heterotic string. To compactify on the resolution, as the metric is unknown, we have to start with the 10d N=1 supergravity and super Yang--Mills effective theory and perform dimensional reduction. In the thesis we first review the 10d heterotic string, the heterotic supergravity, the orbifold and Calabi--Yau compactifications and the toric geometry techniques required for resolving orbifolds. We perform an study of potential orbifold 4d discrete symmetries in factorizable and non--factorizable orbifolds arising from the torus lattice automorphisms. We then come to our focus, which is the orbifold--resolution transition in two compact orbifold models with the Minimal Supersymmetric Standard Model Physics. First, we study the T6/Z7 orbifold and its resolution. This orbifold contains all the ingredients of realistic models. It is simpler because it is prime and has therefore only fixed points and no orbifold brother models. We find the field redefinitions that identify the orbifold and blow--up massless spectrum. A local index theorem is crucial in this process. We study then the Green--Schwarz anomaly cancellation mechanism after dimensional reduction on the resolution and from the massless spectrum and the field redefinitions on the orbifold. We find that both results perfectly agree. This determines the blow--up modes as non--universal axions on the Calabi--Yau manifold. After these encouraging results we study now a more involved case, this is the T6/Z6II orbifold and its resolution. As the orbifold is non--prime there are fixed tori. This makes the identification of the blow--up modes and the search of the field redefinitions more difficult. We overcome that difficulty exploring in a Mini--landscape of phenomenologically promising orbifold models to select a suitable one, and on it we are able to identify the blow--up modes. We find perfect agreement of the massless spectrum, including the orbifold generated mass terms. To culminate, we study in detail the anomaly cancellation mechanism. We find here as in the Z7 case that the blow--up modes play the role of the resolution non--universal axions. Our work interplays between string theory consistency, which expresses itself through 10d anomaly cancellation, and the physics and the geometry of supersymmetric space--time vacua.