Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry

We study aspects of Heterotic/F-theory duality for compactifications with Abelian discrete gauge symmetries. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with n-sections, associated with the Tate-Shafarevich group Z_n. Such models are obtained by studying first a specific toric set-up whose associated Heterotic vector bundle has structure group Z_n. By employing a conjectured Heterotic/F-theory mirror symmetry we construct dual geometries of these original toric models, where in the stable degeneration limit we obtain a discrete gauge symmetry of order two and three, for compactifications to six dimensions. We provide explicit constructions of mirror-pairs for symmetric examples with Z_2 and Z_3, in six dimensions. The Heterotic models with symmetric discrete symmetries are related in field theory to a Higgsing of Heterotic models with two symmetric abelian U(1) gauge factors, where due to the Stuckelberg mechanism only a diagonal U(1) factor remains massless, and thus after Higgsing only a diagonal discrete symmetry of order n is present in the Heterotic models and detected via Heterotic/F-theory duality. These constructions also provide further evidence for the conjectured mirror symmetry in Heterotic/F-theory at the level of fibrations with torsional sections and those with multi-sections.Comment: 25 pages, 4 figure

European Journal of Combinatorics Index, Volume 27

BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p

Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly by generic or tuned Weierstrass/Tate models for elliptic fibrations over complex base surfaces. This includes a relatively small number of somewhat exotic constructions, including elliptic fibrations over non-toric bases, models with new Tate tunings that can give rise to exotic matter in the 6D F-theory picture, tunings of gauge groups over non-toric curves, tunings with very large Hodge number shifts and associated nonabelian gauge groups, and tuned Mordell-Weil sections associated with U(1) factors in the corresponding 6D theory.Comment: 92 pages, 7 figures; v6: cleaned up errors in reference

Abelian Gauge Symmetries in F-Theory and Dual Theories

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 resolved 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 region 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

Toric Geometry and String Theory

In this thesis we probe various interactions between toric geometry and string theory. First, the notion of a top was introduced by Candelas and Font as a useful tool to investigate string dualities. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We classify all tops and give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group. Secondly, we compute all loop closed and open topological string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular, we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds. We determine the BPS structure of the amplitudes, and illustrate our general results in various examples with and without D-branes. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots. This thesis is based on hep-th/0303218 (with H. Skarke), hep-th/0405083 and hep-th/0411227 (with B. Florea and M. Marino).Comment: Oxford University DPhil Thesis (Advisor: Philip Candelas), accepted October 2005, 152 pp., 43 figure