University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations Fall 2010 Balanced Metrics and Phenomenological Aspects of Heterotic Compactifications Tamaz Brelidze University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Elementary Particles and Fields and String Theory Commons Recommended Citation Brelidze, Tamaz, "Balanced Metrics and Phenomenological Aspects of Heterotic Compactifications" (2010). Publicly Accessible Penn Dissertations. 254. https://repository.upenn.edu/edissertations/254 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/254 For more information, please contact [email protected]. Balanced Metrics and Phenomenological Aspects of Heterotic Compactifications Abstract 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 ahler̈ 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 acuav 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. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Physics & Astronomy First Advisor Burt Ovrut Keywords String Theory, Balanced Metrics, Calabi-Yau, Heterotic Theory, Cosmic Strings, Donaldson Algorithm Subject Categories Elementary Particles and Fields and String Theory This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/254 BALANCED METRICS AND PHENOMENOLOGICAL ASPECTS OF HETEROTIC STRING COMPACTIFICATIONS Tamaz Brelidze A DISSERTATION in Physics and Astronomy Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2010 Supervisor of Dissertation Burt Ovrut Professor of Physics and Astronomy Graduate Group Chairperson Dissertation Committee Tony Pantev, Professor of Mathematics Masao Sako, Assistant Professor of Physics and Astronomy Mark Trodden, Professor of Physics and Astronomy Hugh Williams, Professor of Physics and Astronomy Acknowledgements First of all, I would like to thank my adviser for his everlasting guidance and support. Our long brainstorming sessions will always exemplify to me absolute dedication to science and scientific research. I am thankful for his providing a truly great opportunity to grow intellectually, and for making my Ph.D. journey a very rich experience. I gratefully acknowledge Volker Braun for mentoring me from the moment I joined the group. His insight, knowledge and patience were instrumental in com- pleting this work. I would like to thank Lara Anderson, for her endless enthusiasm frequent and insightful discussions. I am very grateful to my collaborators and people that I had an honor interacting closely during these years at Penn. Special appreciation goes to Michael Douglas, James Gray and Dan Wesley for very fruitful discussions. I am very grateful to Tony Pantev for being always available to shed light on the obscure world of mathematics with ease and extreme clarity. I would like to thank all current and previous members of the Penn String Theory Group, Vijay Balasubramanian, Mirjam Cvetiˇc,Paul Langacker, Gino Segre, Lara Anderson, Volker Braun, Inaki Garcia-Etxebarria, Yang-Hui He, Brent Nelson, Mike Schulz, Joan Simon, Timo Weigand, Mike Ambroso, Alexander Borisov, Bar- tolomiej Czech, Peng Gao, James Haverson, Klaus Larjo, Tommy Levi, Tao Lui, Robert Richter for providing a creative and enjoyable working environment. I am indebted to Alexander Borisov, David Chow and Robert Richter for useful comments on the manuscript. ii Last but not least, I would like to thank my committee members for their patience, advice and time. iii 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 iv on the K¨ahlerand 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 SU(3)C × SU(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. v Contents Acknowledgements ii Abstract iv List of Tables ix List of Figures xi 1 Introduction 1 1.0.1 String Theory . .1 1.1 Motivation . .4 1.1.1 The 10-dimensional effective heterotic action and supersymmetry . .4 1.1.2 Calabi-Yau three-folds . .5 1.1.3 Solutions of Hermitian Yang Mills Equations .