Thesis (Ph.D.) - University of Queensland, 2005. / Includes bibliography.

This thesis presents a detailed study of phenomena related to topological solitons (in $2$-dimensions). Topological solitons are smooth, localised, finite energy solutions in non-linear field theories. The problems are about the moduli spaces of lumps in the projective plane and vortices on compact Riemann surfaces. Harmonic maps that minimize the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions in real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the behaviour of lumps and their symmetries. An interesting feature is that the moduli space of charge $3$ lumps is a $7$- dimensional manifold of cohomogeneity one. In this thesis, we discuss the charge $3$ moduli space, calculate its metric and find explicit formula for various geometric quantities. We discuss the moment of inertia (or angular integral) of moduli spaces of charge $3$ lumps. We also discuss the implications for lump decay. We discuss interesting families of moduli spaces of charge $5$ lumps using the symmetry property and Riemann-Hurwitz formula. We discuss the K\"ahler potential for lumps and find an explicit formula on the $1$-dimensional charge $3$ lumps. The metric on the moduli spaces of vortices on compact Riemann surfaces where the fields have zeros of positive multiplicity is evaluated. We calculate the metric, K\"{a}hler potential and scalar curvature on the moduli spaces of hyperbolic $3$- and some submanifolds of $4$-vortices. We construct collinear hyperbolic $3$- and $4$-vortices and derive explicit formula of their corresponding metrics. We find interesting subspaces in both $3$- and $4$-vortices on the hyperbolic plane and find an explicit formula for their respective metrics and scalar curvatures. We first investigate the metric on the totally geodesic submanifold $\Sigma_{n,m},\, n+m=N$ of the moduli space $M_N$ of hyperbolic $N$-vortices. In this thesis, we discuss the K\"{a}hler potential on $\Sigma_{n,m}$ and an explicit formula shall be derived in three different approaches. The first is using the direct definition of K\"ahler potential. The second is based on the regularized action in Liouville theory. The third method is applying a scaling argument. All the three methods give the same result. We discuss the geometry of $\Sigma_{n,m}$, in particular when $n=m=2$ and $m=n-1$. We evaluate the vortex scattering angle-impact parameter relation and discuss the $\frac{\pi}{2}$ vortex scattering of the space $\Sigma_{2,2}$. Moreover, we study the $\frac{\pi}{n}$ vortex scattering of the space $\Sigma_{n,n-1}$. We also compute the scalar curvature of $\Sigma_{n,m}$. Finally, we discuss vortices with impurities and calculate explicit metrics in the presence of impurities.

Made available in DSpace on 2015-09-17T15:26:53Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-07-27. Added 1 bitstream(s) on 2015-09-17T15:45:05Z : No. of bitstreams: 1 000848282.pdf: 636772 bytes, checksum: 5cce337d62dc9c9febd4fbe947aab2c6 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A teoria de defeitos integráveis em teoria de campos em 1+1 dimensões, foi introduzida pela escola de York [16, 17, 22], utilizando transformações de Bäcklund para descrever o defeito. Nesta dissertação estudamos o modelo de Schrödinger não linear na presença de um defeito integrável. Estudamos tanto o modelo discreto [29] como o modelo contínuo dentro dos formalismos lagrangiano [23] e da matrizr [7]. Construímos também o formalismo hamiltoniano para o modelo de Schrödinger não linear na presença de um defeito integrável. Discutimos e relacionamos os formalismos lagrangiano, hamiltoniano e da matriz r / The theory of integrable defects in 1+1 field theory, was introduced by the school of York [16, 17, 22], employing B¨acklund transformation in order to describe the defect. In this dissertation we have studied the nonlinear Schrödinger model in the presence of an integrable defect. We study both, the discrete model [29] as the continuous model within the lagrangian [23] andr matrix [7] formalisms. Also we built the hamiltonian formalism for nonlinear Schrödinger model in the presence of an integrable defect. We discuss and relate the lagrangian, hamiltonian and r matrix formalisms

Tratamos vários modelos que descrevem a interação entre dois subsistemas quânticos. Mostramos que dois deles são exatamente soluveis, achamos as soluções correspondentes e mostramos a relação das propriedades espectrais do banho com a decoerência. Fazemos uma análise crítica do uso de liouvillianos com forma de Lindblad e de liouvillianos obtidos através da aproximação de Born-Markov, empregando entre outras coisas os resultados dos modelos exatos. Propomos novos esquemas de aproximação que descrevem com exatidão o regime de tempos curtos e o acoplamento de dois sistemas \"pequenos\", e os testamos com os modelos exatamente solúveis antes discutidos. Fazemos aplicações aos sais orgânicos de transferência de carga, a um experimento de produção de gatos e da observação de sua decoerência subsequente e às zonas de Ramsey empregadas perifericamente nesta experiência. / Several models describing the interaction between two quantum systems are examined. The exact solution of two of them is found and the relation between the bath\'s spectral properties and decoherence is shown. Using these results and several others, we analyze the use of both Lindbland-form and Born-Markov liouvillians. New approximation schemes, appropriate for short times and small subsystems, are proposed and tested (using the above mentioned exact models). Applications are made to the treatment of organic salts, to the analysis of a recently reported experiment of production of cat states and observation of its subsequent decoherence, and to the understanding of the Ramsey zones, peripherically used in this experience.

Orientador: Adolfo Maia Junior / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T09:39:54Z (GMT). No. of bitstreams: 1 Gama_MarceloCristino_M.pdf: 1393843 bytes, checksum: e0929d5eaf3b27295beb403045153a88 (MD5) Previous issue date: 2006 / Resumo: O objetivo deste trabalho é estudar a aplicação das Equações de Hill na Cosmologia Inflacionária, buscando informações a respeito das bandas de ressonância dessas equações e sobre a produção de partículas no Universo pós-inflacionário. Para isto estudamos as Equações de Hill e dois de seus casos particulares, as Equações de Mathieu e Lamé, suas bandas de estabilidade e ressonância. Analisamos o coeficiente de Floquet, que regula a determinação dessas bandas e apresentamos alguns métodos de cálculo desse coeficiente. Estudamos alguns modelos cosmológicos: o Modelo Cosmológico Padrão, o Modelo Inflacionário e o Modelo Caótico. Apresentamos alguns métodos de reconstrução do potencial do Inflaton a partir de informações sobre a configuração atual do Universo. Por fim, apresentamos nossa contribuição original com alguns teoremas sobre as Equações de Hill e mostramos como eles podem ser usados para a reconstrução do Potencial Inflacionário / Abstract: The objective of this work is to study Hill?s Equation in the Inflationary Cosmology context, searching for information about their resonance bands and about the particle production in the Universe after inflation. For this, we study the Hill?s Equation and two of their particular cases, namely the Mathieu?s and Lamé?s Equations, their stability and resonance bands. We analyse the Floquet?s coefficient, which controls the determination of these bands and we show some methods to calculate this coefficient. We also study some cosmological models: the Standard Cosmological Model, the Inflationary Model and the Chaotic Model. We show some methods of Inflaton Potential Reconstruction from information about the actual configuration for the Universe. Finally we present our original contribution with some theorems about Hill?s Equation and show how they can be used in order to reconstruct the Inflationary Potential / Mestrado / Fisica-Matematica / Mestre em Matemática Aplicada

No description available.

No description available.

No description available.

<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>

No description available.