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In recent years symplectic geometry and symplectic topology have grown to large subbranches in mathematics and had a great impact on other areas in mathematics. When interested in geometry, a geometer always considers geometric structures that arise on immersed submanifolds. In symplectic geometry there is a distinguished class of immersions, known as Lagrangian submanifolds . In particular, minimal Lagrangian submanifolds, called special Lagrangians, are very important in mirror symmetry. Lagrangian mean curvature flow is an important example of Lagrangian deformation. From which we can get the special Lagrangian submanifolds. In recent years, there have been many papers about this subject and the result by K.Smoczyk and Mu-Tao Wang [WS] is very important and beautiful. Our main purpose in this article is to give a new proof for the main result in [WS] from the viewpoint of fully nonlinear partial differential equations.

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This thesis is devoted to the study of gauged linear sigma models (GLSMs) and mirror symmetry. The first chapter of this thesis aims to introduce some basics of GLSMs and mirror symmetry. The second chapter contains the author's contributions to new exact results for GLSMs obtained by applying supersymmetric localization. The first part of that chapter concerns supermanifolds. We use supersymmetric localization to show that A-twisted GLSM correlation functions for certain supermanifolds are equivalent to corresponding Atwisted GLSM correlation functions for hypersurfaces. The second part of that chapter defines associated Cartan theories for non-abelian GLSMs by studying partition functions as well as elliptic genera. The third part of that chapter focuses on N=(0,2) GLSMs. For those deformed from N=(2,2) GLSMs, we consider A/2-twisted theories and formulate the genuszero correlation functions in terms of Jeffrey-Kirwan-Grothendieck residues on Coulomb branches, which generalize the Jeffrey-Kirwan residue prescription relevant for the N=(2,2) locus. We reproduce known results for abelian GLSMs, and can systematically calculate more examples with new formulas that render the quantum sheaf cohomology relations and other properties manifest. We also include unpublished results for counting deformation parameters. The third chapter is about mirror symmetry. In the first part of the third chapter, we propose an extension of the Hori-Vafa mirrror construction [25] from abelian (2,2) GLSMs they considered to non-abelian (2,2) GLSMs with connected gauge groups, a potential solution to an old problem. We formally show that topological correlation functions of B-twisted mirror LGs match those of A-twisted gauge theories. In this thesis, we study two examples, Grassmannians and two-step flag manifolds, verifying in each case that the mirror correctly reproduces details ranging from the number of vacua and correlations functions to quantum cohomology relations. In the last part of the third chapter, we propose an extension of the Hori-Vafa construction [25] of (2,2) GLSM mirrors to (0,2) theories obtained from (2,2) theories by special tangent bundle deformations. Our ansatz can systematically produce the (0,2) mirrors of toric varieties and the results are consistent with existing examples which were produced by laborious guesswork. The last chapter briefly discusses some directions that the author would like to pursue in the future. / Doctor of Philosophy / In this thesis, I summarize my work on gauged linear sigma models (GLSMs) and mirror symmetry. We begin by using supersymmetric localization to show that A-twisted GLSM correlation functions for certain supermanifolds are equivalent to corresponding A-twisted GLSM correlation functions for hypersurfaces. We also define associated Cartan theories for non-abelian GLSMs. We then consider N =(0,2) GLSMs. For those deformed from N =(2,2) GLSMs, we consider A/2-twisted theories and formulate the genus-zero correlation functions on Coulomb branches. We reproduce known results for abelian GLSMs, and can systematically compute more examples with new formulas that render the quantum sheaf cohomology relations and other properties are manifest. We also include unpublished results for counting deformation parameters. We then turn to mirror symmetry, a duality between seemingly-different two-dimensional quantum field theories. We propose an extension of the Hori-Vafa mirror construction [25] from abelian (2,2) GLSMs to non-abelian (2,2) GLSMs with connected gauge groups, a potential solution to an old problem. In this thesis, we study two examples, Grassmannians and two-step flag manifolds, verifying in each case that the mirror correctly reproduces details ranging from the number of vacua and correlations functions to quantum cohomology relations. We then propose an extension of the HoriVafa construction [25] of (2,2) GLSM mirrors to (0,2) theories obtained from (2,2) theories by special tangent bundle deformations. Our ansatz can systematically produce the (0,2) mirrors of toric varieties and the results are consistent with existing examples. We conclude with a discussion of directions that we would like to pursue in the future.

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.

The construction introduced by Gross, Hacking and Keel in [28] allows one to construct a mirror family to (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. To do so one constructs a formal smoothing of a singular variety they call the n-vertex. By arguments of Gross, Hacking and Keel one knows that this construction can be lifted to an algebraic family if the intersection matrix for D is not negative semi-definite. In the case where the intersection matrix is negative definite the smoothing exists in a formal neighbourhood of a union of analytic strata. A proof of both of these is found in [GHK]. In our first project we use these ideas to find explicit formulae for the mirror families to low degree del Pezzo surfaces. In the second project we treat the remaining case of a negative semi-definite intersection matrix, corresponding to S being a rational elliptic surface and D a rational fibre. Using intuition from the first project we prove in the second project that in this case the formal family of their construction lifts to an analytic family.

Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted A and B) that are constructed from a nondegenerate quasihomogeneous polynomial W and a related group of symmetries G. In 2013, Tay proved that given two polynomials W1, W2 with the same quasihomogeneous weights and same group G, the corresponding A-models built with (W1, G) and (W2, G) are isomorphic. An analogous theorem for isomorphisms between orbifolded B-models remains to be found. This thesis investigates isomorphisms between B-models using polynomials in two variables in search of such a theorem. In particular, several examples are given showing the relationship between continuous deformation on the B-side and isomorphisms that stem as a corollary to Tay's theorem via mirror symmetry. Results on extending known isomorphisms between unorbifolded B-models to the orbifolded case are exhibited. A general pattern for B-model isomorphisms, relating mirror symmetry and continuous deformation together, is also observed.

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We consider the pair of a smooth complex projective variety together with an anti-canonical simple normal crossing divisor (we call it "log Calabi- Yau"). Standard examples are toric varieties together with their toric boundaries (we call them "toric pairs"). We provide a numerical criterion for a general log Calabi-Yau to be toric by an inequality between its dimension, Picard number and the number of boundary components. The problem originates in birational geometry and our proof is constructive, motivated by mirror symmetry. / text

New four dimensional N = 2 field theories can be engineered from compactifying six dimensional (2, 0) superconformal field theory on a punctured Riemann surface. Hitchin’s equation is defined on this Riemann surface and the fields in Hitchin’s equation are singular at the punctures. Four dimensional theory is entirely determined by the data at the punctures. Theory without lagrangian description can also be constructed in this way. We first construct new four dimensional generalized superconformal quiver gauge theory by putting regular singularity at the puncture. The algorithm of calculating weakly coupled gauge group in any duality frame is developed. The asymptotical free theory and Argyres-Douglas field theory can also be constructed using six dimensional method. This requires introducing irregular singularity of Hithcin’s equation. Compactify four dimensional theory down to three dimensions, the corresponding N = 4 theory has the interesting mirror symmetry. The mirror theory for the generalized superconformal quiver gauge theory can be derived using the data at the puncture too. Motivated by this construction, we study other three dimensional theories deformed from the above theory and find their mirrors. The surprising relation of above four dimensional gauge theory and two dimensional conformal field theory may have some deep implications. The S-duality of four dimensional theory and the crossing symmetry and modular invariance of two dimensional theory are naturally related.