Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. The problem of classification of superintegrable systems can be approached by considering associated geometric structures. To this end, we invoke the invariant theory of Killing tensors (ITKT), and the recursive version of the Cartan method of moving frames to derive joint invariants. We are able to intrinsically characterise and interpret the arbitrary parameters appearing in the general form of the Smorodinsky-Winternitz superintegrable potential, where we determine that the more general the geometric structure associated with the SW potential is, the fewer arbitrary parameters it admits. Additionally, we classify the multi-separability of the Tremblay-Turbiner-Winternitz (TTW) system. We provide a proof that only for the case k = +/- 1 does the general TTW system admit orthogonal separation of variables with respect to both Cartesian and polar coordinates. / A study towards the classification of superintegrable systems defined on the Euclidean plane.

No description available.

The discovery of integrability on both sides of the duality between planar N=4 super Yang-Mills theory and free type IIB string theory in AdS5 × S5 has lead to great progress in our understanding of the AdS/CFT correspondence. Similar integrable structures also appear in the more recent three-dimensional superconformal N=6 Chern-Simons-matter theory constructed by Aharony, Bergman, Jafferis and Maldacena (ABJM), as well as in its gravity dual, type IIA string theory on AdS4 × CP3. However, new interesting complications arise in the AdS4/CFT3 duality. In the conjectured all-loop Bethe equations by Gromov and Vieira the dispersion relation of the magnons has a non-trivial coupling dependence which is parametrized by a function that is only known to the leading order at weak and strong coupling. In the first part of this thesis I discuss our calculations of the next-to-leading correction to this function at weak coupling. We compute this function from four-loop Feynman diagrams in the SU(2) × SU(2) sector of the ABJM model. As a consistency check we have performed the calculation both in a component formalism and using superspace techniques. At strong coupling the fundamental excitations of the integrable model are the giant magnons. The topic of the second part of this thesis is the spectrum of these giant magnons in CP3. Furthermore, I discuss our analyses of the finite-size corrections beyond the asymptotic Bethe ansatz. At weak coupling we have computed the leading four-loop wrapping diagrams in the ABJM model. At the strong coupling side of the duality I discuss our results for the exponentially suppressed finite-size corrections to the energy of giant magnons.

Thesis (Ph.D.) - University of Glasgow, 2008. / Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, Department of Mathematics, 2008. Includes bibliographical references. Print version also available.

Thesis (Ph.D.)--Victoria University of Wellington, 2009. / Includes bibliographical references.

Thesis (Doctoral)--Universität Bonn, 1990. / Includes bibliographical references.

In this thesis we study gauge/gravity duals in the 5d/6d AdS/CFT correspondence. We start with field theories defined on squashed five-spheres with SU(3) × U(1) symmetry. These five-sphere backgrounds are continuously connected to the round sphere. We find a one-parameter family of 3/4 BPS deformations and a two-parameter family of (generically) 1/4 BPS deformations. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplift to massive type IIA supergravity. We holographically renormalize the Romans theory, and use our general result to compute the renormalized on-shell actions for the solutions. The results agree perfectly with the large N limit of the dual gauge theory partition function, which we compute using large N matrix model techniques. In addition we compute BPS Wilson loops in these backgrounds, both in supergravity and in the large N matrix model, again finding precise agreement. We conjecture a general formula for the partition function on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then proceed to study Euclidean Romans supergravity in six dimensions with a non-trivial Abelian R-symmetry gauge field. We show that supersymmetric solutions are in one-to-one correspondence with solutions to a set of differential constraints on an SU(2) structure. As an application of our results we (i) show that this structure reduces at a conformal boundary to the five-dimensional rigid supersymmetric geometry previously studied, (ii) find a general expression for the holographic dual of the VEV of a BPS Wilson loop, matching an exact field theory computation, (iii) construct holographic duals to squashed Sasaki-Einstein backgrounds, again matching to a field theory computation, and (iv) find new analytic solutions to the squashed five-sphere background. We also analyse the classification of gravity duals with zero B-field.

No description available.

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-07Bitstream added on 2014-06-13T19:53:20Z : No. of bitstreams: 1 000759972.pdf: 3491220 bytes, checksum: 2c6f0f812b13ff3ed611e4f1161ac822 (MD5) / PROPG/Reitoria / Neste trabalho investigamos algumas propriedades dinâmicas de dois modelos descritos por mapeamentos discretos: (i) mapa quadrático com perturbação paramétrica e; (ii) modelo do acelerador de Fermi concentrando particularmente na dinâmica dissipativa. No caso (i) e com a introdução de uma perturbação paramétrica o espaço de parâmetros é bidimensional permitindo assim um estudo de suas estruturas periódicas. Por outro lado o modelo do acelerador de Fermi descrito em (ii) consiste de uma particula clássica confinada entre duas paredes rígidas sendo uma delas fixa e outra movendo-se periodicamente no tempo. A partícula sofre colisões com ambas paredes, que assumiremos serem inelásticas. Isso implica em uma perda fracional de energia a cada choque. O são observadas nela. Mostramos que as estruturas periódicas presentes no espaço de parâmetros é bidimensional e estruturas periódicas também modelo do acelerador de Fermi obedecem a uma regra de organização descrita por uma equação diofantina / Some dynamical properties are investigated in this work considering two models described by discret mappings: (i) a quadratic map under a parametric perturbation and; (ii) a Fermi accelerator model focusing particularly in the dissipative dynamics. In case (i) and with the introduction of a parametric perturbation the parameter space becomes two-dimensional allowing us to study periodic structures present in such space. On the other hand, the Fermi accelerator model described in case (ii), consists of a classical particle confined to bounce between two rigid walls. One of them is fixed and the other one is assumed to move periodically in time. Inelastic collisions are considered leading the particle to suffer a fractional loss of energy upon collision. The parameter space is also two-dimensional and periodic structures are observed. We show that the organization of such structures is described by a diophantine equation

Nesta tese, através de métodos perturbativos adequados, resultados rigorosos são obtidos para dois sistemas dinâmicos específicos. Primeiro, apresentamos uma investigação matemática do fenômeno de localização dinâmica em uma classe de sistemas de dois níveis periodicamente e quase-periodicamente dependente do tempo. Nossos resultados são baseados em um procedimento de eliminação iterativa de termos polinominais da série de Lindstedt, a qual é proposta como solução de uma certa equação de Riccati associada. Tal procedimento é desenvolvido aqui de uma forma sistemática para adequá-lo ao efeito de localização em qualquer ordem de perturbação. No caso quase-periódico esse procedimento nos leva apenas a uma série de Lindstedt formal bem definida. No caso periódico, uma solução perturbativa convergente é obtida e, em particular, uma expansão perturbativa convergente para a frequência secular é apresentada. O caso particular do campo monocromático é discutido em detalhes onde cômputos numéricos das soluções são apresentadas e os resultados são exibidos em termos de certas probabilidades de transição entre os dois auto-estados do sistema. Segundo, consideramos em uma equação de Hill perturbada da forma + (p IND.0(t) + p IND.1(t)) = 0 onde p IND.0 é real analítica e periódica, p IND.1 é real analítica quase-periódica e R é pequeno. Assumindo condições Diophantinas nas frequências do sistema desacoplado, i.e., as frequências dos potenciais externos p IND.0 e p IND.1 e a frequência própria da equação de Hill não-perturbado (=0), e assumindo apenas uma condição de não-degenerescência específica sobre o potencial perturbador p IND.1, provamos que soluções quase-periódicas da equação não-pertrubada são estáveis se estiver em um conjunto de Cantor de medida relativamente grande em [- IND.0. IND.0] C R, onde IND.0 é pequeno o suficiente. Nosso método é baseado em um procedimento de resoma da série de Lindstedt formal obtida como solução de uma equação de Riccati associada ao problema de Hill. Finalmente, salientamos que os sistemas acima são matematicamente aparentados. De fato, ambos passam pela solução de certas equações de Riccati bastante parecidas. Tais soluções são procuradas em termos de séries de Lindstedt expandidas em um parâmetro pertrubativo adequado. / In this thesis, through the use of suitable perturbative methods, rigorous results are obtained for two specific dynamical systems. First, we present a mathematical investigation of the phenomenon of dynamical localization in a class of quasi-periodically and periodically time-dependent two-level systems. Our results are based on an interative procedure of elimination of polynomial terms from the Lindstedt series, which is proposed as a solution of a certain associated Riccati equation. Such a procedure is developed here in a systematic way in order to adapt it to the effect of localization in any perturbative order. In the quasi-periodic case, this procedure leads only to a formal well defined Lindstedt series. In the periodic case, a convergent perturbative solution is obtained and, in particular, a convergent perturbative expansion for the secular frequency is presented. The particular case of a monochromatic field is discussed in detail, where numerical computations of the solutions are presented and results are exhibited in terms of certain transition probabilities between the two eigenstates of the system. Second, we consider a perturbed Hill\'s equation of the form + (p0(t) + p1(t)) = 0, where p0 is real analytic and periodic, p1 is real analytic and quasi-periodic and R is small. Assuming Diophantine conditions on the frequencies of the decoupled system i.e., thr frequencies of the external potentials p0nd p1 and the proper frequency of the unperturbed ( = 0) Hills equation and making only one specific non-degeneracy assumption on the perturbating potential p1, we prove that quasi-periodic solutions of the unperturbed equation are stable if lies in a Cantor set of relatively large measure in [-0,0] C R where 0 is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a genrelized Riccati equation associated to Hills problem. Finally, we stress that the two systems above are mathematically related. Indeed, both pass through the solutions of certain strongly related Riccati euqations. Such solutions are scarched in terms of Lindstedt series expandend in a suitable pertrubative parameter.