Made available in DSpace on 2014-10-09T12:30:26Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:00:36Z (GMT). No. of bitstreams: 1 00956.pdf: 10327962 bytes, checksum: 71ac2457f781bcc0e415fc355cccc8fe (MD5) / Tese (Doutoramento) / IPEN/T / Massachusetts Institute of Technology - Cambridge, Mass - MIT

Made available in DSpace on 2014-10-09T12:30:26Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:00:36Z (GMT). No. of bitstreams: 1 00956.pdf: 10327962 bytes, checksum: 71ac2457f781bcc0e415fc355cccc8fe (MD5) / Tese (Doutoramento) / IPEN/T / Massachusetts Institute of Technology - Cambridge, Mass - MIT

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<div> We introduce the quantum toroidal superalgebra E_m|nassociated with the Lie superalgebra gl_m|n and initiate its study. For each choice of parity "s" of gl_m|n, a corresponding quantum toroidal superalgebra E_s is defined. </div><div> </div><div><br></div><div>To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. </div><div><br></div><div>The superalgebra E_s contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra U_q sl̂_m|n with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E_s, which exchanges the vertical and horizontal subalgebras.</div><div><br></div><div>If <i>m</i> and <i>n</i> are different and "s" is standard, we give a construction of level 1 E_m|n-modules through vertex operators. We also construct an evaluation map from E_m|n(q₁,q₂,q₃) to the quantum affine algebra U_q gl̂_m|n at level c=q₃^(m-n)/2.</div>

In this thesis we explore the geometric representation theory of shifted quantum affine algebras 𝒜^𝜇, using the critical K-theory of certain moduli spaces of infinite flags of quiver representations resembling the moduli of quasimaps to Nakajima quiver varieties. These critical K-theories become 𝒜^𝜇-modules via the so-called critical R-matrix 𝑅, which generalizes the geometric R-matrix of Maulik, Okounkov, and Smirnov. In the asymptotic limit corresponding to taking infinite instead of finite flags, singularities appear in 𝑅 and are responsible for the shift in 𝒜^𝜇. The result is a geometric construction of interesting infinite-dimensional modules in the category 𝒪 of 𝒜^𝜇, including e.g. the pre-fundamental modules previously introduced and studied algebraically by Hernandez and Jimbo. Following Nekrasov, we provide a very natural geometric definition of qq-characters for our asymptotic modules compatible with the pre-existing definition of q-characters. When 𝒜^𝜇 is the shifted quantum toroidal gl₁ algebra, we construct asymptotic modules DT_𝜇 and PT_𝜇 whose combinatorics match those of (1-legged) vertices in Donaldson--Thomas and Pandharipande--Thomas theories. Such vertices control enumerative invariants of curves in toric 3-folds, and finding relations between (equivariant, K-theoretic) DT and PT vertices with descendent insertions is a typical example of a wall-crossing problem. We prove a certain duality between our DT_𝜇 and PT_𝜇 modules which, upon taking q-/qq-characters, provides one such wall-crossing relation.

Designing photonic-plasmonic nanostructures with desirable electromagnetic properties is a central problem in modern photonics engineering. As limited by available materials, engineering geometry of optical materials at both element and array levels becomes the key to solve this problem. In this thesis, I present my work on the development of novel methods and design strategies for photonic-plasmonic structures and metamaterials, including novel Green’s matrix-based spectral methods for predicting the optical properties of large-scale nanostructures of arbitrary geometry. From engineering elements to arrays, I begin my thesis addressing toroidal electrodynamics as an emerging approach to enhance light absorption in designed nanodisks by geometrically creating anapole configurations using high-index dielectric materials. This work demonstrates enhanced absorption rates driven by multipolar decomposition of current distributions involving toroidal multipole moments for the first time. I also present my work on designing helical nano-antennas using the rigorous Surface Integral Equations method. The helical nano-antennas feature unprecedented beam-forming and polarization tunability controlled by their geometrical parameters, and can be understood from the array perspective. In these projects, optimization of optical performances are translated into systematic study of identifiable geometric parameters. However, while array-geometry engineering presents multiple advantages, including physical intuition, versatility in design, and ease of fabrication, there is currently no rigorous and efficient solution for designing complex resonances in large-scale systems from an available set of geometrical parameters. In order to achieve this important goal, I developed an efficient numerical code based on the Green’s matrix method for modeling scattering by arbitrary arrays of coupled electric and magnetic dipoles, and show its relevance to the design of light localization and scattering resonances in deterministic aperiodic geometries. I will show how universal properties driven by the aperiodic geometries of the scattering arrays can be obtained by studying the spectral statistics of the corresponding Green’s matrices and how this approach leads to novel metamaterials for the visible and near-infrared spectral ranges. Within the thesis, I also present my collaborative works as examples of direct and inverse designs of nanostructures for photonics applications, including plasmonic sensing, optical antennas, and radiation shaping.

O objetivo deste texto é apresentar uma classe de módulos para álgebra de Lie de campos vetoriais em um toro N -dimensional, Vect( T N ). O caso N = 1 nos dá a famosa álgebra de Witt (sua extensão central é álgebra de Virasoro). A álgebra Vect( T N ) apresenta um classe de módulos parametrizada por módulos de dimensão finita da álgebra gl N . Nosso objeto central de estudo são módulos induzidos dos módulos tensoriais de Vect( T N ) para Vect( T N +1 ). Estes módulos apresentam um quociente irredutível com espaços de peso de dimensão finita. A álgebra Vect( T N ) apresenta como subálgebra sl N +1 . Com a restrição da ação de Vect( T N ) a esta subálgebra obtemos o carácter deste quociente. Para obter um critério de irredutibilidade e construir sua realização de campo livre, consideramos uma classe de módulos para 1 (T N +1 )/ d 0 (T N +1 ) o Vect (T N ) , construída a partir de álgebras de vértice. Quando restritos a Vect (T N ) estes módulos continuam irredutíveis a menos que apareçam no chiral de De Rham. / The goal of this text is to present a class of modules for the Lie algebra of vector fields in a N -dimensional torus, Vect (T N ) . The case N = 1 give us the famous Witt algebra (its central extension is the Virasoro algebra). The algebra Vect( T N ) has a class of modules parametrized by finite dimensional gl N -modules. The central object of our study are modules induced from tensor modules for Vect( T N ) to Vect( T N +1 ). Those modules have an irreducible quotient such that every weight space has finite dimension. The algebra Vect( T N ) has as subalgebra sl N +1 . Restricting the action of Vect( T N ) to this subálgebra we have the character of this quotient. To obtain a irreducible critreria and construct a free field reazilation, we consider a class of modules for 1 (T N +1 )/ d 0 (T N +1 ) o Vect (T N ) , constructed from vertex algebras. When restricted to Vect (T N ) thesse modules remain irreducible, unless they belongs to the chiral De Rham complex.

Made available in DSpace on 2015-03-26T13:35:05Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1289687 bytes, checksum: 1760c5305e12b5504a52ff55f7e9035d (MD5) Previous issue date: 2008-02-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / We study Heisenberg model of classical spins lying on the toroidal support. The isotropic regime is characterised by a fractional soliton solution. Whenever the torus size is very large, R &#8594; &#8734;, its charge equals unity and the soliton effectively lies on a infinite cylinder. However, for R = 0 the spherical geometry is recovered and we obtain that configuration and energy of a soliton lying on a sphere. Vortex-like configurations are also supported: in a ring torus (R > r) such excitations present no core where energy could blow up. At the limit R &#8594; &#8734; we are effectively describing it on an infinite annulus (or cylinder, equivalently), where the spins appear to be practically parallel to each other, yielding no net energy. On the other hand, in a horn torus (R = r) a singular core takes place, while for R < r (self- intersecting spindle torus) two such singularities appear. If R is further diminished until vanish we recover vortex configuration on a sphere. Other formal solutions, without topological stability, are obtained and discussed with some details. / Estudamos o modelo de Heisenberg para spins clássicos no suporte toroidal. O regime isotrópico é caracterizado por uma solução solitônica fracionária. Quando o tamanho do toro é muito grande, R &#8594; &#8734;, sua carga se iguala à unidade e o sóliton efetivamente se comporta como no caso do cilindro infinito. Entretanto, para R = 0 a geometria esférica é recobrada e obtemos a configuração e a energia de um sóliton numa esfera. Configurações tipo vórtice também são suportadas: num ring torus (R > r) tais excitações não apresentam caroço onde a energia poderia divergir. No limite R &#8594; &#8734;estamos efetivamente descrevendo-o em um annulus infinito (ou cilindro, equivalentemente), onde os spins aparecem praticamente paralelos um ao outro, não tendo energia líquida. Por outro lado, em um horn torus (R = r) um caroço singular toma lugar, enquanto para R < r (self-intersectind spindle torus) duas singularidades deste tipo aparecem. Se R é diminuído até se anular, recuperamos a configuração de vórtice na esfera. Outras soluções formais, sem estabilidade topológica, são obtidas e discutidas com alguns detalhes.

This dissertation is an investigation into the classification of all hyperbolic manifolds which admit a reducible Dehn filling and a toroidal Dehn filling with distance 3. The first example was given by Boyer and Zhang. They used the Whitehead link. Eudave-Muñoz and Wu gave an infinite family of such hyperbolic manifolds using tangle arguments. I show in this dissertation that these are the only hyperbolic manifolds admitting a reducible Dehn filling and a toroidal Dehn filling with distance 3. The main tool to prove this is to use the intersection graphs on surfaces introduced and developed by Gordon and Luecke. / text

Approved for public release, distribution is unlimited / When a burst of air is produced in water, the result can be a toroidal bubble. This thesis is concerned with experimental investigations of three acoustical properties of toroidal bubbles: (i) propagation through high-intensity noise, (ii) emission, and (iii) scattering. In (i), an attempt to observe a recent prediction of the acoustic drag on a bubble is described, which is analogous to the Einstein-Hopf effect for an oscillating electric dipole in a fluctuating electromagnetic field. No effect was observed, which may be due to insufficient amplitude of the noise. In (ii), observations of acoustic emissions of volume oscillations of toroidal bubbles are reported. Surprisingly, the emission occurs primarily during the formation of a bubble, and is weak in the case of very smooth toroidal bubbles. In (iii), we describe an experiment to observe the effect of a toroidal bubble on an incident sound field. In addition to the acoustical investigations, we describe the construction of a large hallway apparatus for further investigations and for hands-on use by the public. The tank has cross section 2 feet by 2 feet and height 6 feet, and the parameters of reservoir pressure and time between air bursts are adjustable by the observer. / Lieutenant, United States Navy