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Call number: LD2668 .T4 1978 J69 / Master of Science

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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

We study mirror symmetry via Fourier-Mukai-type transformations, which we call SYZ mirror transformations, in view of the ground-breaking Strominger-Yau-Zaslow Mirror Conjecture which asserted that the mirror symmetry for Calabi-Yau manifolds could be understood geometrically as a T-duality modified by suitable quantum corrections. We apply these transformations to investigate a case of mirror symmetry with quantum corrections, namely the mirror symmetry between the A-model of a toric Fano manifold X¯ and the B-model of a Landau-Ginzburg model (Y, W). Here Y is a noncompact Kahler manifold and W : Y → C is a holomorphic function. We construct an explicit SYZ mirror transformation which realizes canonically the isomorphism QH*X&d1; ≅Ja cW between the quantum cohomology ring of X¯ and the Jacobian ring of the function W. We also show that the symplectic structure oX¯ of X¯ is transformed to the holomorphic volume form eWOY of ( Y, W). Concerning the Homological Mirror Symmetry Conjecture, we exhibit certain correspondences between A-branes on X¯ and B-branes on (Y, W) by applying the SYZ philosophy. / Chan, Kwok Wai. / Adviser: Nai Chung Conan Leung. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3536. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 52-56). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis (M.S.)--University of Delaware, 2007. / Principal faculty advisor: Guang R. Gao, Dept. of Electrical and Computer Engineering. Includes bibliographical references.

R.M. Redheffer. / Includes bibliographical references. / Army Signal Corps Contract no. W-36-039 sc-32037.

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (March 1, 2007) Vita. Includes bibliographical references.

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (March 1, 2007) Vita. Includes bibliographical references.