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Dissertations / Theses on the topic 'Mirror symmetry'
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1
Branco, Lucas Castello. "Higgs bundles, Lagrangians and mirror symmetry." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:612325bd-6a7f-4d74-a85c-426b73ff7a14.
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Let Σ be a compact Riemann surface of genus g ≥ 2. This thesis is dedicated to the study of certain loci of the Higgs bundle moduli space. After recalling basic facts in the first chapter about G-Higgs bundles for a reductive group G, we begin the first part of the work, which deals with Higgs bundles for the real forms G<sub>0</sub> = SU* (2m), SO* (4m), and Sp(m, m) of G = SL(2m, C), SO(4m, C) and Sp(4m, C), respectively. The second part of the thesis deals with the Gaiotto Lagrangian. Motivated by mirror symmetry, we give a detailed description of the fibres of the G-Hitchin fibration containing generic G<sub>0</sub>-Higgs bundles, for the real groups G<sub>0</sub> = SU* (2m), SO* (4m) and Sp(m, m). The spectral curves associated to these fibres are examples of ribbons, i.e., non-reduced projective C-schemes of dimension one, whose reduced scheme are non-singular. Our description of these fibres is done in two different ways, each giving different and interesting insights about the fibre in question. One of the formulations is given in term of objects on the reduced curve, while the other in terms of the non-reduced spectral curve. A link is also provided between the two approaches. We use this description to give a proposal for the support of the dual BBB-brane inside the moduli space M(<sup>L</sup>G) of Higgs bundles for the Langlands dual group <sup>L</sup>G of G. In the second part of the thesis we discuss the Gaiotto Lagrangian, which is a Lagrangian subvariety of the moduli spaces of G-Higgs bundles, where G is a reductive group over C. This Lagrangian is obtained from a symplectic representation of G and we discuss some of its general properties. In Chapter 7 we focus our attention to the Gaiotto Lagrangian for the standard representation of the symplectic group. This is an irreducible component of the nilpotent cone for the symplectic Hitchin fibration. We describe this component by using the usual Morse function on the Higgs bundle moduli space, which is the norm squared of the Higgs field restricted to the Lagrangian in question. Lastly, we discuss natural questions and applications of the ideas developed in this thesis. In particular, we say a few words about the hyperholomorphic bundle, how to generalize the Gaiotto Lagrangian to vector bundles which admit many sections and give an analogue of the Gaiotto Lagrangian for the orthogonal group.
2
Mertens, Adrian. "Mirror Symmetry in the presence of Branes." Diss., lmu, 2011. http://nbn-resolving.de/urn:nbn:de:bvb:19-135464.
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Gu, Wei. "Gauged Linear Sigma Model and Mirror Symmetry." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/90892.
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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.<br>Doctor of Philosophy<br>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.
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Perevalov, Eugene V. "Type II/heterotic duality and mirror symmetry /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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Rossi, Paolo. "Symplectic Topology, Mirror Symmetry and Integrable Systems." Doctoral thesis, SISSA, 2008. http://hdl.handle.net/11577/3288900.
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Using Sympelctic Field Theory as a computational tool, we compute Gromov-Witten theory of target curves using gluing formulas and quantum integrable systems. In the smooth case this leads to a relation of the results of Okounkov and Pandharipande with the quantum dispersionless KdV hierarchy, while in the orbifold case we prove triple mirror symmetry between GW theory of target P^1 orbifolds of positive Euler characteristic, singularity theory of a class of polynomials in three variables and extended affine Weyl groups of type ADE.
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Rossi, Paolo. "Symplectic Topology, Mirror Symmetry and Integrable systems." Doctoral thesis, SISSA, 2008. http://hdl.handle.net/20.500.11767/4193.
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The plan of the work is the following:
² In Chapter 1 we recall, basically from [16] and [14], the ideas and methods of Symplectic
Field Theory. Our review will focus on the algebraic structure arising from topology,
more than on the geometry underlying it. In particular we de¯ne the SFT analogue of
the Gromov-Witten potential as an element in some graded Weyl algebra and consider
its properties (grading, master equations, semiclassical limit). We then stress (following
[18]) how this algebraic structure allows the appearence of a system of commuting
di®erential operators (on the homology of the Weyl algebra) which can be thought of as
a system of quantum Hamiltonian PDEs with symmetries. Sometimes this symmetries
are many enough to give rise to a complete integrable system (at least at the semiclassical
level) and we examine the main examples where this happens. Eventually we review
some results of [16] which turn out to be very useful in computations and which we
actually employ in the next chapters.
² In Chapter 2 we apply the methods of Symplectic Field Theory to the computation of
the Gromov-Witten invariants of target Riemann surfaces. Our computations reproduce
the results of [27], [28] which, in principle, solve the theory of target curves, but are fairly
more explicit and, above all, clarify the role of the KdV hierarchy in this topological
theory. More precisely we are able to describe the full descendant Gromov-Witten
potential as the solution to SchrÄodinger equation for a quantum dispersionless KdV
system. This quantization of KdV, already appearing in [31], can be easily dealt with
in the fermionic formalism to give extremely explicit results, like closed formulae for the
Gromov-Witten potential at all genera and given degree. These results where published
by the author in [32].
² In Chapter 3 we use basically the same techniques of Chapter 2 to compute the Gromov-
Witten theory of target curves with orbifold points (orbicurves). As in the smooth case,
the coe±cient for the Gromov-Witten potential are written in terms of Hurwitz numbers.
It turns out that we can even classify those target orbicurves whose potential involves
only a ¯nite number of these a priori unknown Hurwitz coe±cients, so that they can be
determined using WDVV equations. These polynomial P1-orbifolds are the object of our
study for the ¯nal part of this work. Moreover, we extend the theorem by Bourgeois
([4]) about Hamiltonian structures of ¯bration type to allow singular ¯bers (Seifert
¯brations), so that we can use our result on Gromov-Witten invariants of polynomial
P1-orbifolds to deduce the SFT-Hamiltonians of the ¯bration.
² In Chapter 4 we completely solve the rational Gromov-Witten problem for polynomial
P1-orbifolds. Namely we ¯nd a Landau-Ginzburg model which is mirror symmetric to
these spaces. This model consists in a Frobenius manifold structure on the space of
what we call for brevity tri-polynomials, i.e. polynomials of three variables of the form
¡xyz + P1(x) + P2(y) + P3(z). The main results here are the explicit construction of
the Frobenius manifold structure with closed expressions for °at coordinates and the
mirror theorem 4.0.3, i.e. the isomorphism of this Frobenius structure with the one on
the quantum cohomology of polynomial P1-orbifolds. From the polynomiality property
of the Frobenius potentials involved, one is able to show that there is also a third mirror symmetric partner in the picture, namely the Frobenius manifold associated to
extended a±ne Weyl groups of type A, D, E ([12]). The results of these last two chapters appeared in [33].
² In the Conclusions we summarize our results and analyze possible further developments
and directions to be explored.
7
Krefl, Daniel. "Real Mirror Symmetry and The Real Topological String." Diss., lmu, 2009. http://nbn-resolving.de/urn:nbn:de:bvb:19-102832.
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Williams, Matthew Michael. "Mirror Symmetry for Non-Abelian Landau-Ginzburg Models." BYU ScholarsArchive, 2019. https://scholarsarchive.byu.edu/etd/8560.
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We consider Landau-Ginzburg models stemming from non-abelian groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group G*, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors in general.
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Ueda, Kazushi. "Homological mirror symmetry for toric del Pezzo surfaces." 京都大学 (Kyoto University), 2006. http://hdl.handle.net/2433/144153.
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Kyoto University (京都大学)<br>0048<br>新制・課程博士<br>博士(理学)<br>甲第12069号<br>理博第2963号<br>新制||理||1443(附属図書館)<br>23905<br>UT51-2006-J64<br>京都大学大学院理学研究科数学・数理解析専攻<br>(主査)助教授 河合 俊哉, 教授 齋藤 恭司, 教授 柏原 正樹<br>学位規則第4条第1項該当
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Kadir, Shabnam Nargis. "The arithmetic of Calabi-Yau manifolds and mirror symmetry." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403756.
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Petracci, Andrea. "On Mirror Symmetry for Fano varieties and for singularities." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/55877.
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In this thesis we discuss some aspects of Mirror Symmetry for Fano varieties and toric singularities. We formulate a conjecture that relates the quantum cohomology of orbifold del Pezzo surfaces to a power series that comes from Fano polygons. We verify this conjecture in some cases, in joint work with A. Oneto. We generalise the Altmann–Mavlyutov construction of deformations of toric singularities: from Minkowski sums of polyhedra we construct deformations of affine toric pairs. Moreover, we propose an approach to the study of deformations of Gorenstein toric singularities of dimension 3 in the context of the Gross–Siebert program. We construct deformations of polarised projective toric varieties by deforming their affine cones. This method is explicit in terms of Cox coordinates and it allows us to give explicit equations for a construction, due to Ilten, which produces a deformation between two toric Fano varieties when their corresponding polytopes are mutation equivalent. We also provide examples of Gorenstein toric Fano 3-folds which are locally smoothable, but not globally smoothable.
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Bott, Christopher James. "Mirror Symmetry for K3 Surfaces with Non-symplectic Automorphism." BYU ScholarsArchive, 2018. https://scholarsarchive.byu.edu/etd/7456.
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Mirror symmetry is the phenomenon, originally discovered by physicists, that Calabi-Yau manifolds come in dual pairs, with each member of the pair producing the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. One important problem in mirror symmetry is that there may be several ways to construct a mirror dual for a Calabi-Yau manifold. Hence it is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?We specifically consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry. BHK mirror symmetry was inspired by the LandauGinzburg/Calabi-Yau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani Boissi`ere-Sarti originally showed that they agree when n = 2, and more recently Comparin-Lyon-Priddis-Suggs showed that they agree when n is prime. However, the n being composite case required more sophisticated methods. Whenever n is not divisible by four (or n = 16), this problem was solved by Comparin and Priddis by studying the associated lattice theory more carefully. In this thesis, we complete the remaining case of the problem when n is divisible by four by finding new isomorphisms and deformations of the K3 surfaces in question, develop new computational methods, and use these results to complete the investigation, thereby showing that the BHK and LPK3 mirror symmetry constructions also agree when n is composite.
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Matessi, Diego. "Constructions of Calabi Yau metrics and of special Lagrangian submanifolds." Thesis, University of Warwick, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.246770.
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Alim, Murad. "Mirror Symmetry, Toric Branes and Topological String Amplitudes as Polynomials." Diss., lmu, 2009. http://nbn-resolving.de/urn:nbn:de:bvb:19-103416.
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Hartmann, Heinrich [Verfasser]. "Mirror symmetry and stability conditions on K3 surfaces / Heinrich Hartmann." Bonn : Universitäts- und Landesbibliothek Bonn, 2011. http://d-nb.info/1016152949/34.
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Yang, Wenzhe. "The arithmetic geometry of mirror symmetry and the conifold transition." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:e55a7b22-a268-4c57-9d98-c0547ecdcef9.
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The central theme of this thesis is the application of mirror symmetry to the study of the arithmetic geometry of Calabi-Yau threefolds. It formulates a conjecture about the properties of the limit mixed Hodge structure at the large complex structure limit of an arbitrary mirror threefold, which is supported by a two-parameter example of a self-mirror Calabi-Yau threefold. It further studies the connections between this conjecture with Voevodsky's mixed motives. This thesis also studies the connections between the conifold transition and Beilinson's conjecture on the values of the L-functions at integral points. It carefully studies the arithmetic geometry of the conifold in the mirror family of the quintic Calabi-Yau threefold and its L-function, which is shown to provide a very interesting example to Beilinson's conjecture.
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Prince, Thomas. "Applications of mirror symmetry to the classification of Fano varieties." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43374.
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In this dissertation we discuss two new constructions of Fano varieties, each directly inspired by ideas in Mirror Symmetry. The first recasts the Fanosearch program (Coates--Corti--Kasprzyk et al.) for surfaces in terms of a construction related to the SYZ conjecture. In particular we construct Q-Gorenstein smoothings of toric varieties via an application of the Gross-Siebert algorithm to certain affine manifolds. We recover the theory of combinatorial mutation, which plays a central role in the Fanosearch program, from these affine manifolds. Combining this construction and the work of Gross--Hacking--Keel on log Calabi--Yau surfaces we produce a cluster structure on the mirror to a log del Pezzo surface proposed by Coates--Corti--et al. We exploit the cluster structure, and the connection to toric degenerations, to prove two classification results for Fano polygons. This cluster variety is equipped with a superpotential defined on each chart by a so-called maximally mutable Laurent polynomial. We study an enumerative interpretation of this superpotential in terms of tropical disc counting in the example of the projective plane (with a general choice of boundary divisor). In the second part we develop a new construction of Fano toric complete intersections in higher dimensions. We first consider the problem of finding torus charts on the Hori--Vafa/Givental model, adapting the approach taken by Przyjalkowski. We exploit this to identify 527 new families of four-dimensional Fano manifolds. We then develop an inverse algorithm, Laurent Inversion, which decorates a Fano polytope P with additional information used to construct a candidate ambient space for a complete intersection model of the toric variety defined by P. Moving in the linear system defining this complete intersection allows us to construct new models of known Fano manifolds, and also to construct new examples of Fano manifolds from conjectured mirror Laurent polynomials. We use this algorithm to produce families simultaneously realising certain collections of 'commuting' mutations, extending the connection between polytope mutation and deformations of toric varieties.
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Borokhov, Vadim Aleksandrovich Kapustin Anton N. "Monopole operators and mirror symmetry in three-dimensional gauge theories /." Diss., Pasadena, Calif. : California Institute of Technology, 2004. http://resolver.caltech.edu/CaltechETD:etd-04062004-015855.
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Sigrist, Norbert. "First-order design of mirror systems with no axial symmetry." Diss., The University of Arizona, 1999. http://hdl.handle.net/10150/284660.
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All-reflective imaging systems that are asymmetrical and eccentric have the advantage of providing more degrees of freedom to improve image quality. A disadvantage of these asymmetrical imaging systems is that they suffer from asymmetric mapping. This asymmetric mapping manifests itself mainly in the presence of keystone distortion and anamorphism. Due to the increase in degrees of freedom, the complexity of such systems escalates; thus, the designer is confronted with the difficult task of determining optimal starting points. This work addresses several first-order aspects of the design and characterisation of asymmetrical, all-reflective, aspherical, eccentric imaging systems. In contrast to the work of Stone and Forbes, which is based upon the theory of Hamiltonian optics and includes both the first- and second-order considerations, this work is based upon the theory of collineation. Because of the inherent simplicity of the collinear mapping, which is a projective transformation, we are able to present a simple but certainly not naive way of designing and characterising such asymmetrical all-reflective imaging systems. The simplicity of this proposition has the advantage that we can gain insights into asymmetrical mapping behaviour. Specifically, we apply the collinear mapping model on all-reflective asymmetrical imaging systems resulting in the description of how the mapping between conjugate planes may be described. First we will define keystone distortion and anamorphism. Then we will introduce and investigate the significance of the Cardinal points and planes, the Scheimpflug condition and the horizon planes and show how they are applied in the designing of imaging systems that are free of both keystone distortion and anamorphism. Having established a first-order layout of the optical system, we will then develop a process for converting the first-order layouts into imaging systems consisting of real aspheric surfaces.
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Johnson, Jared Drew. "An Algebra Isomorphism for the Landau-Ginzburg Mirror Symmetry Conjecture." BYU ScholarsArchive, 2011. https://scholarsarchive.byu.edu/etd/2793.
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Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. We verify this conjecture for a wide class of singularities on the level of Frobenius algebras, generalizing work of Krawitz [10]. We also review the relevant parts of the constructions.
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Zhang, Xiangwen 1984. "Mean curvature flow for Lagrangian submanifolds with convex potentials." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=111593.
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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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Mukai, Daichi. "Mirror symmetry of nonabelian Landau-Ginzburg orbifolds with loop type potentials." Kyoto University, 2020. http://hdl.handle.net/2433/253068.
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Pumperla, Max [Verfasser], and Bernd [Akademischer Betreuer] Siebert. "Unifying Constructions in Toric Mirror Symmetry / Max Pumperla. Betreuer: Bernd Siebert." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2012. http://d-nb.info/1026332966/34.
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Böhm, Janko [Verfasser], and Frank-Olaf [Akademischer Betreuer] Schreyer. "Mirror symmetry and tropical geometry / Janko Böhm. Betreuer: Frank-Olaf Schreyer." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2011. http://d-nb.info/1051094968/34.
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Lu, Wenxuan. "Instanton correction, wall crossing and mirror symmetry of Hitchin's moduli spaces." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/67809.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 205-210).<br>We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We will show that the two instanton correction problems are equivalent in an appropriate sense via the identification of the wall crossing formulas in the metric problem with consistency conditions in the complex structure problem. This is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which till now has been mostly studied in the framework of geometric Langlands duality. This result provides examples of Calabi-Yau varieties where the instanton correction (in the sense of mirror symmetry) of metrics and complex structures can be determined. This equivalence also relates certain enumerative problems in foliations to some gluing constructions of affine varieties.<br>by Wenxuan Lu.<br>Ph.D.
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Vaintrob, Dmitry. "Mirror symmetry and the K theory of a p-adic group." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/104578.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 59-61).<br>Let G be a split, semisimple p-adic group. We construct a derived localization functor Loc : ... from the compactified category of [BK2] associated to G to the category of equivariant sheaves on the Bruhat-Tits building whose stalks have finite-multiplicity isotypic components as representations of the stabilizer. Our construction is motivated by the "coherent-constructible correspondence" functor in toric mirror symmetry and a construction of [CCC]. We show that Loc has a number of useful properties, including the fact that the sections ... compactifying the finitely-generated representation V. We also construct a depth </= e "truncated" analogue Loc(e) which has finite-dimensional stalks, and satisfies the property RIP ... V of depth </= e. We deduce that every finitely-generated representation of G has a bounded resolution by representations induced from finite-dimensional representations of compact open subgroups, and use this to write down a set of generators for the K-theory of G in terms of K-theory of its parahoric subgroups.<br>by Dmitry A. Vaintrob.<br>Ph. D.
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Sheridan, Nicholas (Nicholas James). "Homological mirror symmetry for a Calabi-Yau hypersurface in projective space." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/73374.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 365-369).<br>This thesis is concerned with Kontsevich's Homological Mirror Symmetry conjecture. In Chapter 1, which is based on [1], we consider the n-dimensional pair of pants, which is defined to be the complement of n + 2 generic hyperplanes in CPn. The pair of pants is conjectured to be mirror to the Landau-Ginzburg model (Cn+2 , W), where W = z1...zn+2 We construct an immersed Lagrangian sphere in the pair of pants, and show that its endomorphism A.. algebra in the Fukaya category is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror,.giving some evidence for the Homological Mirror Symmetry conjecture in this case. In Chapter 2, which is based on [2], we build on these results to prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d =/> 3.<br>by Nicholas Sheridan.<br>Ph.D.
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MAGID, DIANE ALEXIS. "FRAMES OF REFERENCE, THE PERCEPTION OF SYMMETRY AND THE MIRROR ILLUSION (ENANTIOMORPHS)." Diss., The University of Arizona, 1986. http://hdl.handle.net/10150/188154.
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The relationship between symmetry and apparent reversals of enantiomorphic (mirror-reflected) objects was investigated. Subjects were presented with a series of standard and enantiomorphic books with various structural symmetries. The object directions (top-front-right) assigned to standard books were compared with the directions assigned to their enantiomorphs and the axes of apparent reversal determined. The primary finding was that apparent reversals were not limited to the left-right dimension. Reversals of top-bottom and front-back were also obtained. In most cases, apparent reversals occurred along the axis of structural (geometric) symmetry. However, symmetry defined in structural terms did not always predict apparent reversals. In certain cases, subjects perceived reversals most often along the left-right axis, even though (depending on the book) reversals of top-bottom or front-back were equally possible. The concept of perceived symmetry, which includes but is not limited to structural symmetry, is developed. Also, the influence of perceived symmetry on frames of reference is discussed.
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Barrott, Lawrence Jack. "Convergence of the mirror to a rational elliptic surface." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/285007.
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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.
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Rainville, Stéphane Jean Michel. "The spatial mechanisms mediating the perception of mirror symmetry in human vision /." Thesis, McGill University, 1999. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=36688.
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The present thesis reports psychophysical and modeling studies on the spatial properties of visual mechanisms mediating the perception of mirror symmetry in human vision. In a first set of experiments, patterns were filtered for power spectra that decayed with spatial frequency according to variable slopes. Results revealed that symmetry detection is optimal if contrast energy is roughly equated across log-frequency bands (i.e. 1/f2) and that, under such conditions, spatial scales contribute equally and independently to symmetry perception. In a second study, random-noise patterns were filtered for various orientation bands. Results showed that symmetry perception is possible at all orientations, is mediated by oriented mechanisms, and is computed independently in different orientation channels. Data also revealed that the dimensions of the spatial integration region (IR) for symmetry vary with orientation in a way that approximately matches the spatial distribution of information in the stimulus. Finally, symmetry detection was measured for bandpass textures of variable spatial density and variable contrast polarity. For such patterns, it was found that symmetry is computed at a spatial scale proportional to stimulus density and that mechanisms insensitive to contrast polarity (i.e. second-order) are involved in the scale-selection process.<br>Overall, results from empirical and modeling work revealed an intimate link between symmetry perception and the properties of spatial filters. In particular, I argue that the size of the IR tends to vary such that a fixed amount of information is integrated irrespective of the spatial properties of the stimulus. Implications for the functional architecture of symmetry perception are discussed, and a paradigm for future research in symmetry perception is proposed in which spatial filtering is extended to higher orders of spatial complexity.
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Basalaev, Alexey [Verfasser]. "Mirror symmetry for simple elliptic singularities with a group action / Alexey Basalaev." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2015. http://d-nb.info/1073604454/34.
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Mertens, Adrian [Verfasser], and Christian [Akademischer Betreuer] Römelsberger. "Mirror Symmetry in the presence of Branes / Adrian Mertens. Betreuer: Christian Römelsberger." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2011. http://d-nb.info/1016172877/34.
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Rainville, Stéphane Jean Michel. "The spatial mechanisms mediating the perception of mirror symmetry in human vision." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/NQ64650.pdf.
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Buchholz, Arne Verfasser], and Hannah [Akademischer Betreuer] [Markwig. "Tropical covers, moduli spaces & mirror symmetry / Arne Buchholz. Betreuer: Hannah Markwig." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2014. http://d-nb.info/1060715953/34.
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Jost, Jan Niklas [Verfasser], and Thomas [Akademischer Betreuer] Reichelt. "Mirror Symmetry for Del Pezzo Surfaces / Jan Niklas Jost ; Betreuer: Thomas Reichelt." Heidelberg : Universitätsbibliothek Heidelberg, 2021. http://d-nb.info/1227711492/34.
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Jost, Jan [Verfasser], and Thomas [Akademischer Betreuer] Reichelt. "Mirror Symmetry for Del Pezzo Surfaces / Jan Niklas Jost ; Betreuer: Thomas Reichelt." Heidelberg : Universitätsbibliothek Heidelberg, 2021. http://d-nb.info/1227711492/34.
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Smorenburg, Ana Renée Pascale. "Mirror (a)symmetry? : visuo-proprioceptive interactions in individuals with spastic hemiparetic cerebral palsy." Thesis, Manchester Metropolitan University, 2012. http://e-space.mmu.ac.uk/314042/.
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The work presented in this thesis aimed to get more insight into the previously reported positive effects of mirror visual feedback in children with spastic hemiparetic cerebral palsy (SHCP) and into visuo-proprioceptive interactions in children and adolescents with SHCP during goal-directed matching tasks. Individuals with SHCP have unilateral motor impairments that hamper them in accurate movement performance. In conjunction with the motor problems, these individuals experience sensory problems. The first study in this thesis (chapter two) found that mirror visual feedback of the impaired arm in SHCP led to significantly higher levels of neuromuscular activity than mirror visual feedback of the less-impaired arm. This indicates that the mirror-effect was not just caused by the illusory perception of symmetry between two limbs, and confirmed that the beneficial effect is dependent on mirror visual feedback of the less-impaired arm. In chapter three and four it was demonstrated that the ability of children with SCHP to match one (matching) hand with the position of the other (reference) hand, without visual information, is deteriorated when compared to typically developing children. However, if visual information of the static reference arm was available to the participants, the matching accuracy of the matching hand was significantly higher. Mirror visual feedback of the reference arm, generated by placing a mirror in between the arms in the sagittal plane, created the illusion that both hands were already at the endpoint. However, this did not impact upon the matching accuracy of the matching arm and resulted in similar error scores as regular feedback of the reference arm. Chapter five showed that moving the less-impaired arm in synchrony with the impaired arm resulted in higher matching accuracy than moving the impaired arm alone. Moreover, mirror visual feedback of the less-impaired arm improved matching accuracy for a subset of the participants. The effects of a short practice of a bimanual matching task with (mirror) visual feedback of the less-impaired arm on matching accuracy of the impaired arm was studied in chapter six. The results showed a higher matching accuracy of the impaired arm after the practice period. However, the role of the mirror is still inconclusive in this respect. From this it can be concluded that for individuals with SHCP practice of a matching movement can induce a transfer from visual to proprioceptive control of movement. Taken together, the work in this thesis showed that the deficit in position sense of the impaired arm in individuals with SHCP can be modified by visual feedback of the less-impaired arm. Although the role of mirror visual feedback is still inconclusive, it seems that motor learning can induce a transfer from visual to proprioceptive control of movement, which can have implications for therapy.
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Huang, Yu-Chien Ph D. Massachusetts Institute of Technology. "Elliptic fibrations among toric hypersurface Calabi-Yau manifolds and mirror symmetry of fibrations." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/124593.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2019<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 245-255).<br>In this thesis, we investigate the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau three folds by (1) constructing explicitly elliptically fibered Calabi-Yau threefolds with large Hodge numbers using Weierstrass model techniques motivated by F-theory, and comparing the Tate-tuned Wierstrass model set with the set of Calabi-Yau threefolds constructed using toric hypersurface methods, and (2) systematically analyzing directly the fibration structure of 4D reflexive polytopes by classifying all the 2D subpolytopes of the 4D polytopes in the Kreuzer and Skarke database of toric Calabi-Yau hypersurfaces. With the classification of the 2D fibers, we then study the mirror symmetry structure of elliptic toric hypersurface Calabi-Yau threefolds. We show that the mirror symmetry of Calabi-Yau manifolds factorizes between the toric fiber and the base: if there exist 2D mirror fibers of a pair of mirror reflexive polytopes, the base and fibration structure of one hypersurface Calabi-Yau determine the base of the other.<br>by Yu-Chien Huang.<br>Ph. D.<br>Ph.D. Massachusetts Institute of Technology, Department of Physics
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Francis, Amanda. "New Computational Techniques in FJRW Theory with Applications to Landau Ginzburg Mirror Symmetry." BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3265.
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Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A and B models, which are created based on a choice of an affine singularity with a group of symmetries. Both models are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras), and they are also Frobenius manifolds. We give a result relating stabilization of singularities in classical singularity to its counterpart in the Landau-Ginzburg setting. The A model comes from so-called FJRW theory and can be de fined up to a full cohomological field theory. The structure of this model is determined by a generating function which requires the calculation of certain numbers, which we call correlators. In some cases the their values can be computed using known techniques. Often, there is no known method for finding their values. We give new computational methods for computing concave correlators, including a formula for concave genus-zero, four-point correlators and show how to extend these results to find other correlator values. In many cases these new methods give enough information to compute the A model structure up to the level of Frobenius manifold. We give the FJRW Frobenius manifold structure for various choices of singularities and groups.
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Jones, Daniel M. "A convergent beam electron diffraction study of some rare-earth perovskite oxides /." Connect to this title, 2007. http://theses.library.uwa.edu.au/adt-WU2008.0057.
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Brown, Matthew Robert. "Construction and Isomorphism of Landau-Ginzburg B-Model Frobenius Algebras." BYU ScholarsArchive, 2016. https://scholarsarchive.byu.edu/etd/5652.
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Landau-Ginzburg Mirror Symmetry provides for the construction of two algebraic objects, called the A- and B-models. Special cases of these models–constructed using invertible polynomials and abelian symmetry groups–are well understood. In this thesis, we consider generalizations of the B-model, and specifically address the associativity of the multiplication in these models. We also prove an explicit B-model isomorphism for a class of polynomials in three variables.
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Cordner, Nathan James. "Isomorphisms of Landau-Ginzburg B-Models." BYU ScholarsArchive, 2016. https://scholarsarchive.byu.edu/etd/5882.
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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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Argüz, Nurömür Hülya [Verfasser], and Bernd [Akademischer Betreuer] Siebert. "Log geometric techniques for open invariants in mirror symmetry / Nurömür Hülya Argüz ; Betreuer: Bernd Siebert." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2017. http://d-nb.info/1123729506/34.
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Argüz, Nurömür Hülya Verfasser], and Bernd [Akademischer Betreuer] [Siebert. "Log geometric techniques for open invariants in mirror symmetry / Nurömür Hülya Argüz ; Betreuer: Bernd Siebert." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2017. http://nbn-resolving.de/urn:nbn:de:gbv:18-82774.
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Jones, Daniel M. "A convergent beam electron diffraction study of some rare-earth perovskite oxides." University of Western Australia. School of Physics, 2008. http://theses.library.uwa.edu.au/adt-WU2008.0057.
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This work describes detailed convergent beam electron diffraction (CBED) studies of GdAlO3 and LaAlO3 perovskites. CBED patterns tilted away from major zone axes have been found to have high sensitivity to the presence of mirror or glide mirror symmetry. Such patterns confirm to high accuracy that the space group of GdAlO3 is orthorhombic, Pnma. Tilted patterns from this well characterised structure also serve as benchmarks against which similar patterns may be compared. In the case of LaAlO3, tilted patterns enable the space group to be confirmed as rhombohedral R3c, previously claimed to be cubic (Fm3c) by CBED. Furthermore, no evidence for the low symmetry (I2/a or F1) phases proposed for LaAlO3 has been observed. The LaAlO3 study also gives a careful assessment of the influence of tilted specimen surfaces on the CBED data. Within the qualitative scope of these experiments, no symmetry degrading effects could be observed. Some preliminary Quantitative CBED (QCBED) data from LaAlO3 is also presented. This shows it will be possible to make a detailed study of the bonding charge density (Δρ) in this material when combined with X-ray diffraction data. Also included is a brief CBED study of LaFeO3, a material that is isostructural with GdAlO3. Although this is restricted to exact zone axis patterns, it is noted that tilted patterns have significant potential to improve the quality of the symmetry determination.
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Wu, Ruoxu. "Notes on Some (0,2) Supersymmetric Theories in Two Dimensions." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/77921.
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This thesis is devoted to a discussion of two-dimensional theories with (0,2) supersymmetry. Examples of two-dimensional (0,2) gauged linear sigma models (GLSMs) are constructed for various spaces including Grassmannians, complete intersections in Grassmannians, and non-complete intersections such as Pfaffians. Generalizations of (2,2) Toda dual theories to (0,2) Toda-like theories are also discussed and some examples are given, including products of projective spaces and del Pezzo surfaces. Correlation functions are computed to show the examples are the correct mirror models.<br>Ph. D.
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Pavanelli, Simone. "Mirror symmetry for a two parameter family of Calabi-Yau three-folds with Euler characteristic 0." Thesis, University of Warwick, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396954.
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Goldner, Christoph Jan [Verfasser]. "Enumerative aspects of Tropical Geometry : Curves with cross-ratio constraints and Mirror Symmetry / Christoph Jan Goldner." Tübingen : Universitätsbibliothek Tübingen, 2021. http://d-nb.info/1233678248/34.
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Merrell, Evan D. "A Maple Program for Computing Landau-Ginzburg A- and B-Models and an Exploration of Mirror Symmetry." BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3322.
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Mirror symmetry has been a significant area of research for geometry and physics for over two decades. Berglund and Hubsch proposed that for a certain family of singularities W, the so called "transposed" singularity WT should be the mirror partner of W. cite{BH} The techniques for constructing the orbifold LG models to test this conjecture were developed by FJR in cite{FJR} with a cohomological field theory generalized from the study of r-spin curves. The duality of LG A- and B-models became more elaborate when Krawitz cite{Krawitz} generalized the Intriligator-Vafa orbifold B-model to include contributions from more than one sector.This thesis presents a program written in Maple for explicitly computing bases for both LG A- and B-model rings, as well as the correlators for A-models to the extent of current knowledge. Included is a list of observations and conjectures drawn from computations done in the program.
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Comparin, Paola. "Symétrie miroir et fibrations elliptiques spéciales sur les surfaces K3." Thesis, Poitiers, 2014. http://www.theses.fr/2014POIT2281/document.
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Une surface K3 est une surface X complexe compacte projective lisse qui a fibré canonique trivial et h0;1(X) = 0. Dans cette thèse on s'intéresse à deux problèmes pour ces surfaces. D'abord on considère des surfaces K3 obtenues comme recouvrement double de P2 ramifié le long d'une sextique. On classifie les fibrations elliptiques sur ces surfaces et leur groupe de Mordell-Weil, c'est-à-dire le groupe des sections. Vu que une section de 2-torsion définit une involution de la surface (dite involution de van Geemen-Sarti), alors en classifiant les fibrations et les section de 2-torsion on obtient une classification complète des involutions de van Geemen-Sarti sur ce type de surfaces K3. On montre aussi comment calculer l'équation de la fibration et on étudie le quotient par l'involution de van Geemen-Sarti. Ensuite on montre la construction de Berglund-Hübsch-Chiodo-Ruan (BHCR): il s'agit d'une construction miroir qui part d'un polynôme dans un espace projectif à poids et d'un groupe d'automorphismes (avec certaines propriétés) et qui donne, en toute dimension, des paires de variétés Calabi-Yau. Ces deux variétés sont l'une miroir de l'autre en sense classique. On classifie toutes les paires de surfaces K3 obtenues avec cette construction qui aient en plus un automorphisme non{symplectique d'ordre premier p > 3. Pour les surfaces K3 une autre notion de symétrie miroir a été introduite par Dolgachev et Nikulin : la symétrie pour K3 polarisées (LPK3). On montre dans la thèse comment polariser les surfaces obtenues avec la construction BHCR et on preuve que deux surfaces miroir au sense BHCR, dûment polarisées, appartiennent à deux familles miroir LPK3<br>A K3 surface is a complex compact projective surface X which is smooth and such that its canonical bundle is trivial and h0;1(X) = 0. In this thesis we study two different topics about K3 surfaces. First we consider K3 surfaces obtained as double covering of P2 branched on a sextic curve. For these surfaces we classify elliptic fibrations and their Mordell-Weil group, i.e. the group of sections. A 2-torsion section induces a symplectic involution of the surface, called van Geemen-Sarti involution. The classification of elliptic fibrations and 2-torsion sections allows us to classify all van Geemen-Sarti involutions on the class of K3 surfaces we are considering. Moreover, we give details in order to obtain equations for the elliptic fibrations and their quotient by the van Geemen-Sarti involutions. Then we focus on the mirror construction of Berglund-Hübsch-Chiodo-Ruan (BHCR). This construction starts from a polynomial in a weighted projective space together with a group of diagonal automorphisms (with some properties) and gives a pair of Calabi-Yau varieties which are mirror in the classical sense. The construction works for any dimension. We use this construction to obtain pairs of K3 surfaces which carry a non-symplectic automorphism of prime order p > 3. Dolgachev and Nikulin proposed another notion of mirror symmetry for K3 surfaces: the mirror symmetry for lattice polarized K3 surfaces (LPK3). In this thesis we show how to polarize the K3 surfaces obtained from the BHCR construction and we prove that these surfaces belong to LPK3 mirror families