University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2016 Abelian Gauge Symmetries in F-Theory and Dual Theories Peng Song University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons, and the Physics Commons Recommended Citation Song, Peng, "Abelian Gauge Symmetries in F-Theory and Dual Theories" (2016). Publicly Accessible Penn Dissertations. 2031. https://repository.upenn.edu/edissertations/2031 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2031 For more information, please contact [email protected]. Abelian Gauge Symmetries in F-Theory and Dual Theories Abstract In this dissertation, we focus on important physical and mathematical aspects, especially abelian gauge symmetries, of F-theory compactifications and its dual formulations within type IIB and heterotic string theory. F-theory is a non-perturbative formulation of type IIB string theory which enjoys important dualities with other string theories such as M-theory and E8 × E8 heterotic string theory. One of the main strengths of F-theory is its geometrization of many physical problems in the dual string theories. In particular, its study requires a lot of mathematical tools such as advanced techniques in algebraic geometry. Thus, it has also received a lot of interests among mathematicians, and is a vivid area of research within both the physics and the mathematics community. Although F-theory has been a long-standing theory, abelian gauge symmetry in Ftheory has been rarely studied, until recently. Within the mathematics community, in 2009, Grassi and Perduca first discovered the possibility of constructing elliptically fibered varieties with non-trivial toric Mordell-Weil group. In the physics community, in 2012, Morrison and Park first made a major advancement by constructing general F-theory compactifications with U(1) abelian gauge symmetry. They found that in such cases, the ellipticallyfibered Calabi-Yau manifold that F-theory needs to be compactified on has its fiber being a generic elliptic curve in the blow-up of the weighted projective space P(1;1;2) at one point. Subsequent developments have been made by Cvetiˇc, Klevers and Piragua extended the works of Morrison and Park and constructed general F-theory compactifications with U(1) U(1) abelian gauge symmetry. They found that in the U(1) × U(1) abelian gauge symmetry case, the elliptically-fibered Calabi-Yau manifold that F-theory needs to be compactified on has its fiber being a generic elliptic curve in the del Pezzo surface dP2. In chapter 2 of this dissertation, I bring this a step further by constructing general F-theory compactifications with U(1) × U(1) × U(1) abelian gauge symmetry. I showed that in the case with three U(1) factors, the general elliptic fiber is a complete intersection of two quadrics in P3, and the general elliptic fiber in the fully esolvr ed elliptic fibration is embedded as the generic Calabi-Yau complete intersection into Bl3P3, the blow-up of P3 at three generic points. This eventually leads to our analysis of representations of massless matter at codimension two singularities of these compactifications. Interestingly, we obtained a tri-fundamental representation which is unexpected from perturbative Type II compactifications, further illustrating the power of F-theory. In chapter 1 of this dissertation, I proved finiteness of a egionr of the string landscape in Type IIB compactifications. String compactifications give rise to a collection of effective low energy theories, known as the string landscape. However, it is not known whether the number of physical theories we can derive from the string landscape is finite. The vastness of the string landscape also poses a serious challenge to attempts of studying it. A breakthrough was made by Douglas and Taylor in 2007 when they studied the landscape of intersecting brane models in Type IIA compactifications on a particular Z2× Z2 orientifold. They found that two consistency conditions, namely the D6-brane tadpole cancellation condition, and the conditions on D6-branes that were required for N = 1 supersymmetry in four dimensions, only permitted a finite number of D6-brane configurations. These finite number of allowed D6-brane configurations thus result in only a finite number of gauge sectors in a 4D supergravity theory, allowing them to be studied explicitly. Douglas and Taylor also believed that the phenomenon of using tadpole cancellation and supersymmetry consistency conditions to restrict the possible number of allowed configurations to a finite one is not a mere coincidence unique to their construction; they conjectured that this phenomenon also holds for theories with magnetised D9- or D5-branes compactified on elliptically fibered Calabi-Yau threefolds. Indeed, this was what my collaborators and I also felt. To this end, I showed, using a mathematical proof, that their conjecture is indeed true for elliptically fibered Calabi-Yau threefolds p X B whose base B satisfy a few easily-checked conditions (summarized in chapter 1 of this dissertation). In particular, these conditions are satisfied by, although not limited to, the almost Fano twofold bases B given by the toric varieties associated to all 16 reflexive two-dimensional polytopes and the del Pezzo surfaces dPn for n = 0;1; :::; 8. This list, in particular, also includes the Hirzebruch surfaces F0 = P1 ×P1;F1 = dP1;F2. My proof also allowed us to derive the explicit and computable bounds on all flux quanta and on the number of D5-branes. These bounds only depends on the topology of the base B and are independent on the continuous moduli of the compactification, in particular the Kahler moduli, as long as the supergravity approximation is valid. Physically, my proof showed that these compactifications only give rise to a finite number of four-dimensional N = 1 supergravity theories, and that these theories only have finitely many gauge sectors with finitely many chiral spectra. Such finiteness properties are not observed in generic quantum field theories, further fortifying superstring theory as a more promising theory. In chapter 3 of this dissertation, I study abelian gauge symmetries in the duality between F-theory and E8 × E8 heterotic string theory. It is conjectured that F-theory, when compactified on an elliptic K3-fibered (n + 1)-dimensional Calabi-Yau manifold X B, and heterotic string theory when compactified on an elliptically fibered n-dimensional Calabi-Yau manifold Z B with the same base B, are dual to each other. Thus under such duality, in particular, if the F-theory compactification admits abelian gauge symmetries, the dual heterotic string theory must admit the same abelian gauge symmetry as well. However, how abelian gauge symmetries can arise in the dual heterotic string theory has never been studied. The main goal of this chapter is to study exactly this. We start with F-theory compactifications with abelian gauge symmetry. With the help of a mathematical lemma as well as a computer code that I came up with, I was able to construct a rich list of specialized examples with specific abelian and nonabelian gauge groups on the F-theory side. The computer code also directly computes spectral cover data for each example constructed, allowing us to further analyze how abelian gauge symmetries arise on heterotic side. Eventually, we found that in general, there are three ways in which U(1)-s can arise on the heterotic side: the case where the heterotic theory admits vector bundles with S(U(1) ×U(m)) structure group, the case where the heterotic theory admits vector bundles with SU(m)×Zn structure group, as well as the case where the heterotic theory admits vector bundles with structure groups having a centralizer in E8 which contains a U(1) factor. Another important achievement was my discovery of the non-commutativity of the semi-stable degeneration map which splits a K3 surface into two half K3 surfaces, and the map to Weierstrass form, which was not previously known in the literature. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Physics & Astronomy First Advisor Mirjam Cvetic Keywords abelian gauge theory, algebraic geometry, F-theory, string compactification, string landscape Subject Categories Mathematics | Physics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2031 ABELIAN GAUGE SYMMETRIES IN F-THEORY AND DUAL THEORIES Peng Song 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 2016 Mirjam Cvetic,ˇ Professor of Physics and Astronomy Supervisor of Dissertation Philip Nelson, Professor of Physics and Astronomy Graduate Group Chairperson Dissertation Committee Mirjam Cvetic,ˇ Professor of Physics and Astronomy Antonella Grassi, Professor of Mathematics Elliot Lipeles, Professor of Physics and Astronomy Philip Nelson, Professor of Physics and Astronomy Evelyn Thomson, Professor of Physics and Astronomy ABELIAN GAUGE SYMMETRIES IN F-THEORY AND DUAL THEORIES COPYRIGHT 2016 Peng Song ACKNOWLEDGMENTS This dissertation would not have been possible without the support of my family, my advi- sor, my research group and my good friends. First and foremost, I want to thank my family for being so supportive and understand- ing. It is their morale support that fuelled me to move on throughout these years. Next, I would like to thank my advisor Mirjam Cvetic,ˇ for her constant guidance in my pursuit of physics and mathematics. Her keen insights and intuitions have always shed light on the obscurities I encountered during my research works. There were many occasions in which Professor Cveticˇ have successfully solved problems using innovative and creative approaches that no one else in my group thought of, creating important breakthroughs. I am extremely grateful to the great mathematician Ron Donagi, from whom I learnt a great deal of mathematics and physics.