University, Diss., 2001--München.

University, Diss., 2004--Jena.

本論文給出了在鏡像對稱中對非拉格朗日A-膜的Fourier型的變換。 / SYZ構想斷言，鏡像對稱應該來自於一種在卡拉比-丘流形上逐纖維的Fourier-Mukai變換。在半平坦卡拉比-丘流形上的拉格朗日A-膜的情形下，這已經被實現。然而， Kapustin和Orlov指出，對於一些特殊的卡拉比-丘流形， A-膜範疇應該加上某些額外的物件。他們稱這些額外的物件為餘迷向A-膜。在半平坦卡拉比-丘流形的情況下，我們需要加入一些在每個纖維上是楊-米爾斯的A-膜以及B-膜。 / 我們首先推廣Nahm變換到環面上的楊-米爾斯叢。這也可以看作一種Fourier型的變換。然後我們在半平坦卡拉比-丘流形上實施逐纖維的這種Nahm變換。我們在一些半平坦卡拉比丘流形上構造了一些新的B-膜的例子。這些B-膜限制到每一個纖維環面上都是環面上的楊-米爾斯叢。並且我們驗證了在這種逐纖維的變換下，他們恰好就是Kapustin和Orlov所提出的餘迷向A 膜。 / This thesis gives the construction of Fourier type transformations in mirror symmetry for non-Lagrangian A-branes. / The SYZ proposal asserts that mirror symmetry should come from a fiberwise Fourier-Mukai transformation along torus fibrations on Calabi-Yau manifolds. This can be realized explicitly for Lagrangian A-branes in semi-flat case. However, Kapustin and Orlov pointed out that for certain Calabi-Yau manifolds some extra objects called coisotropic A-branes should be added into the category of A-branes. In semi-flat cases, we need to include A-and B-branes which are Yang-Mills along fibers. / We first generalize the Nahm transformation to Yang-Mills line bundles over tori which can also be regarded as a Fourier type transformation. Then we carry out a family version of this transformation for semi-flat Calabi-Yau manifolds. More precisely, we construct a new class of B-branes in semi-flat Calabi-Yau manifolds which are Yang-Mills line bundles when restricted to each fiber torus. And we show that this fiberwise transformation of these B-branes produce the coisotropic A-branes predicted by Kapustin and Orlov. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Yi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 61-62). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Fourier-Mukai Transformation --- p.10 / Chapter 2.1 --- A torus case --- p.10 / Chapter 2.1.1 --- Moduli space of flat U(1) bundles over T --- p.11 / Chapter 2.1.2 --- Poincare line bundle P --- p.12 / Chapter 2.1.3 --- Definition of the Fourier-Mukai Transformation for a torus --- p.13 / Chapter 2.1.4 --- Some concrete computations --- p.14 / Chapter 2.2 --- Semi-flat Calabi-Yau case --- p.15 / Chapter 2.2.1 --- Semi-flat Calabi-Yau manifolds and semi-flat branes --- p.15 / Chapter 2.2.2 --- Fourier-Mukai transformation for semi-flat branes --- p.18 / Chapter 3 --- Coisotropic A-branes --- p.23 / Chapter 3.1 --- Why Lagrangian branes are not enough? --- p.23 / Chapter 3.2 --- An example --- p.27 / Chapter 4 --- Nahm transformation --- p.29 / Chapter 4.1 --- Spinor bundle and the Dirac operator --- p.30 / Chapter 4.1.1 --- Clifford algebra and spin group --- p.30 / Chapter 4.1.2 --- Spinor bundle --- p.33 / Chapter 4.1.3 --- Dirac operator --- p.36 / Chapter 4.2 --- Nahm transformation for a torus (T, g) --- p.38 / Chapter 4.3 --- Fourier-Mukai transformation for coisotropic A-branes --- p.53

Mak, Kit Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 78-81). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Preliminaries on Number Theory --- p.10 / Chapter 2.1 --- Finite Fields --- p.10 / Chapter 2.2 --- p-adic Numbers --- p.13 / Chapter 2.3 --- The Teichmuller Representatives --- p.16 / Chapter 2.4 --- Character Theory --- p.18 / Chapter 3 --- Basic Calabi-Yau Geometry --- p.26 / Chapter 3.1 --- Definition and Basic Properties of Calabi-Yau Manifolds --- p.26 / Chapter 3.2 --- Calabi-Yau Manifolds of Low Dimensions --- p.29 / Chapter 3.3 --- Constructions of Calabi-Yau Manifolds --- p.32 / Chapter 3.4 --- Importance of Calabi-Yau Manifolds in Physics --- p.35 / Chapter 4 --- Number of Points on Calabi-Yau Manifolds over Finite Fields --- p.39 / Chapter 4.1 --- The General Method --- p.39 / Chapter 4.2 --- The Number of Points on a Family of Calabi-Yau Varieties over Finite Fields --- p.45 / Chapter 4.2.1 --- The Case ψ = 0 --- p.45 / Chapter 4.2.2 --- The Case ψ ß 0 --- p.50 / Chapter 4.3 --- The Number of Points on the Affine Mirrors over Finite Fields --- p.55 / Chapter 4.3.1 --- The Case ψ = 0 --- p.55 / Chapter 4.3.2 --- The Case ψ ß 0 --- p.56 / Chapter 4.4 --- The Number of points on the Projective Mirror over Finite Fields --- p.59 / Chapter 4.5 --- Summary of the Results and Related Conjectures --- p.61 / Chapter 5 --- The Relation Between Periods and the Number of Points over Finite Fields modulo q --- p.67 / Chapter 5.1 --- Periods of Calabi-Yau Manifolds --- p.67 / Chapter 5.2 --- The Case for Elliptic Curves --- p.69 / Chapter 5.2.1 --- The Periods of Elliptic Curves --- p.69 / Chapter 5.2.2 --- The Number of Fg-points on Elliptic Curves Modulo q --- p.70 / Chapter 5.3 --- The Case for a Family of Quintic Threefolds --- p.73 / Chapter 5.3.1 --- The Periods of Xψ --- p.73 / Chapter 5.3.2 --- The Number of F9-points on Quintic Three- folds Modulo q --- p.75 / Bibliography --- p.78

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Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Jacques Distler. Vita. Includes bibliographical references. Also available from UMI.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--Rutgers University, 2008. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 79-86).

In this dissertation we study several topics related to the geometry and physics of heterotic string compactification. After an introduction to some of the basic ideas of this field, we review the heterotic line bundle standard model construction and a complex structure mod- uli stabilization mechanism associated to certain hidden sector gauge bundles. Once this foundational material has been presented, we move on to the original research of this disser- tation. We present a scan over all known heterotic line bundle standard models to examine the frequency with which the particle spectrum is forced to change, or "jump," by the hidden sector moduli stabilization mechanism just mentioned. We find a significant percentage of forced spectrum jumping in those models where such a change of particle content is possible. This result suggests that one should consider moduli stabilization concurrently with model building, and that failing to do so could lead to misleading results. We also use state of the art techniques to study Yukawa couplings in these models. We find that a large portion of Yukawa couplings which naively would be expected to be non-zero actually vanish due to certain topological selection rules. There is no known symmetry which is responsible for this vanishing. In the final part of this dissertation, we study the Chern-Simons contribution to the superpotential of heterotic theories. This quantity is very important in determining the vacuum stability of these models. By explicitly building real bundle morphisms between vec- tor bundles over Calabi-Yau manifolds, we show that this contribution to the superpotential vanishes in many cases. However, by working with more complicated, and realistic geome- tries, we also present examples where the Chern-Simons contribution to the superpotential is non-zero, and indeed fractional. / Doctor of Philosophy / String theory is a candidate for a unified theory of all of the known interactions of nature. To be consistent, the theory needs to be formulated in 9 spatial dimensions, rather than the 3 of everyday experience. To connect string theory with reality, we need to reproduce the known physics of 3 dimensions from the 9 dimensional theory by hiding, or "compactifying," 6 directions on a compact internal space. The most common choice for such an internal space is called a Calabi-Yau manifold. In this dissertation, we study how the geometry of the Calabi-Yau manifold determines physical quantities seen in 3 dimensions such as the number of particle families, particle interactions and potential energy. The first project in this dissertation studies to what extent the process of making the Calabi-Yau manifold rigid, something which is required observationally, affects the particle spectrum seen in 3 dimensions. By scanning over a large model set, we conclude that computation of the particle spectrum and such "moduli stabilization" issues should be considered in concert, and not in isolation. We also showed that a large portion of the interactions that one would naively expect between the particles in such string models are actually absent. There is no known symmetry of the theory that accounts for this structure, which is linked to the topology of the extra spatial dimensions. In the final part of the dissertation, we show how to calculate previously unknown contributions to the potential energy of these string theory models. By linking to results from the mathematics literature, we show that these contributions vanish in many cases. However, we present examples where it is non-zero, a fact of crucial importance in understanding the vacua of heterotic string theories.