Balanced Metrics and Phenomenological Aspects of Heterotic Compactifications

ABSTRACT BALANCED METRICS AND PHENOMENOLOGICAL ASPECTS OF HETEROTIC STRING COMPACTIFICATIONS Tamaz Brelidze Burt Ovrut, Advisor This thesis mainly focuses on numerical methods for studying Calabi-Yau manifolds. Such methods are instrumental in linking models inspired by the mi- croscopic physics of string theory and the observable four dimensional world. In particular, it is desirable to compute Yukawa and gauge couplings. However, only for a relatively small class of geometries can those be computed exactly using the rather involved tools of algebraic geometry and topological string theory. Numerical methods provide one of the alternatives to go beyond these limitations. In this work we describe numerical procedures for computing Calabi-Yau metrics on complete intersections and free quotients of complete intersections. This is accomplished us- ing the balanced metrics approach and enhancing its previous implementations with tools from Invariant Theory. In particular, we construct these metrics on generic quintics, four-generation quotients of the quintic, Schoen Calabi-Yau complete inter- sections and the quotient of a Schoen manifold with the Z3 × Z3 fundamental group that was previously used to construct a heterotic standard model. We also investi- gate the dependence of Donaldson’s algorithm on the integration scheme, as well as on the K ̈ahler and complex moduli. We then construct a numerical algorithm for explicitly computing the spectrum of the Laplace-Beltrami operator on Calabi-Yau threefolds. One of the inputs of this algorithm is the Calabi-Yau metric. To illustrate our algorithm, the eigenvalues and eigenfunctions of the Laplacian are computed nu- merically on two different quintic hypersurfaces, some Z5 × Z5 quotients of quintics, and the Calabi-Yau threefold with the Z3 × Z3 fundamental group of the heterotic standard model. We then explain the degeneracies of the eigenvalues in terms of the irreducible representations of the finite symmetry groups of the threefolds. We also study the cosmic string solutions in softly broken N = 1 supersym- metric theories that arise from heterotic string compactifications with the MSSM spectrum. These vacua have the S U (3)C × S U (2)L × U (1)Y gauge group of the stan- dard model augmented by additional an U (1)B −L. The B-L symmetry is sponta- neously broken by a vacuum expectation value of one of the right-handed sneutrinos, which leads to U (1)B −L cosmic string solutions. We present a numerical analysis that demonstrates that boson condensates can, in principle, form for theories of this type. However, the weak Yukawa and gauge couplings of the right-handed sneu- trino suggests that bosonic superconductivity will not occur in the simplest vacua in this context. Fermion superconductivity is also disallowed by the electroweak phase transition, although bound state fermion currents can exist