Quantum Bound States in Yang-Mills-Higgs Theory

We give rigorous proofs for the existence of infinitely many (non-BPS) bound states for two linear operators associated with the Yang-Mills-Higgs equations at vanishing Higgs self-coupling and for gauge group SU(2): the operator obtained by linearising the Yang-Mills-Higgs equations around a charge one monopole and the Laplace operator on the Atiyah-Hitchin moduli space of centred charge two monopoles. For the linearised system we use the Riesz-Galerkin approximation to compute upper bounds on the lowest 20 eigenvalues. We discuss the similarities in the spectrum of the linearised system and the Laplace operator, and interpret them in the light of electric-magnetic duality conjectures.Comment: minor corrections implemented; to appear in Communications in Mathematical Physic

Some Exact Solutions for Maximally Symmetric Topological Defects in Anti de Sitter Space

We obtain exact analytical solutions for a class of SO($l$) Higgs field theories in a non-dynamic background $n$-dimensional anti de Sitter space. These finite transverse energy solutions are maximally symmetric $p$-dimensional topological defects where $n=(p+1)+l$. The radius of curvature of anti de Sitter space provides an extra length scale that allows us to study the equations of motion in a limit where the masses of the Higgs field and the massive vector bosons are both vanishing. We call this the double BPS limit. In anti de Sitter space, the equations of motion depend on both $p$ and $l$. The exact analytical solutions are expressed in terms of standard special functions. The known exact analytical solutions are for kink-like defects ($p=0,1,2,\dotsc;\, l=1$), vortex-like defects ($p=1,2,3;\, l=2$), and the 'tHooft-Polyakov monopole ($p=0;\, l=3$). A bonus is that the double BPS limit automatically gives a maximally symmetric classical glueball type solution. In certain cases where we did not find an analytic solution, we present numerical solutions to the equations of motion. The asymptotically exponentially increasing volume with distance of anti de Sitter space imposes different constraints than those found in the study of defects in Minkowski space.Comment: 45 pages, 19 figures. In version 2: added two paragraphs about how our double BPS limit automatically gives a solution to the Yang-Mills equation, and related it to Yang-Mills solutions in AdS_4 that appeared on the same day in eprint 1708.0636

On the dynamics of topological solitons

This thesis investigates the dynamics of lump-like objects in non-integrable field theories, whose stability is due to topological considerations. The work concerns three different low dimensional ((1 + 1)- and (2 + l)-dimensional) systems and addresses the questions of how the topology and metric structure of physical space, the quantum mechanics of the basic field quanta and intersoliton interactions affect soliton dynamics. In chapter 2 a sine-Gordon system in discrete space, but with continuous time, is presented. This has some novel features, namely a topological lower bound on the energy of a kink and an explicit static kink which saturates this bound. Kink dynamics in this model is studied using a geodesic approximation which, on comparison with numerical simulations, is found to work well for moderately low kink speeds. At higher speeds the dynamics becomes significantly dissipative, and the approximation fails. Some of the dissipative phenomena observed are explained by means of a dispersion relation for phonons on the spatial lattice. Chapter 3 goes on to quantize the kink sector of this model. A quantum induced potential called the kink Casimir energy is computed numerically in the weak coupling approximation by quantizing the lattice phonons. The effect of this potential on classical kink dynamics is discussed. Chapter 4 presents a study of the low-energy dynamics of a CP(^1) lump on the two-sphere in the geodesic approximation. By considering the isometry group inherited from globalsymmetries of the model, the structure of the induced metric on the unit-charge moduli space is so restricted that the metric can be calculated explicitly. Some totally geodesic submanifolds are found, and the qualitative features of motion on these described. The moduli space is found to be geodesically incomplete. Finally, chapter 5 contains an analysis of long range intervortex forces in the abelian Higgs model, a massive field theory, extending a point source. approximation previously only used in massless theories. The static intervortex potential is rederived from a new viewpoint and used to model type II vortex scattering. Velocity dependent forces are then calculated, providing a model of critical vortex scattering, and leading to a conjecture for the analytic asymptotic form of the metric on the two-vortex moduli space

Single View Modeling and View Synthesis

This thesis develops new algorithms to produce 3D content from a single camera. Today, amateurs can use hand-held camcorders to capture and display the 3D world in 2D, using mature technologies. However, there is always a strong desire to record and re-explore the 3D world in 3D. To achieve this goal, current approaches usually make use of a camera array, which suffers from tedious setup and calibration processes, as well as lack of portability, limiting its application to lab experiments. In this thesis, I try to produce the 3D contents using a single camera, making it as simple as shooting pictures. It requires a new front end capturing device rather than a regular camcorder, as well as more sophisticated algorithms. First, in order to capture the highly detailed object surfaces, I designed and developed a depth camera based on a novel technique called light fall-off stereo (LFS). The LFS depth camera outputs color+depth image sequences and achieves 30 fps, which is necessary for capturing dynamic scenes. Based on the output color+depth images, I developed a new approach that builds 3D models of dynamic and deformable objects. While the camera can only capture part of a whole object at any instance, partial surfaces are assembled together to form a complete 3D model by a novel warping algorithm. Inspired by the success of single view 3D modeling, I extended my exploration into 2D-3D video conversion that does not utilize a depth camera. I developed a semi-automatic system that converts monocular videos into stereoscopic videos, via view synthesis. It combines motion analysis with user interaction, aiming to transfer as much depth inferring work from the user to the computer. I developed two new methods that analyze the optical flow in order to provide additional qualitative depth constraints. The automatically extracted depth information is presented in the user interface to assist with user labeling work. In this thesis, I developed new algorithms to produce 3D contents from a single camera. Depending on the input data, my algorithm can build high fidelity 3D models for dynamic and deformable objects if depth maps are provided. Otherwise, it can turn the video clips into stereoscopic video

Existence and stability of viscoelastic shock profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure

The scattering of baby Skyrmions off potential obstructions, in a Landau-Lifshitz model

The dynamics of baby skyrmions of the (2+1) new baby Skyrme model, in a Landau - Lifshitz model, was studied in the presence of various potential obstructions of varying geometries. The potential obstructions were created by introducing a localised inhomogeneity in the new baby Skyrme model's potential coefficient. The size and shape of the potential obstruction was varied and two systems were investigated, namely the symmetric and asymmetric systems. In the symmetric system the trajectory of the baby skyrmions, as they traverse the barrier, was deformed from the normal circular trajectory, during which time the skyrmions sped up. For critical values of the barrier height, the baby skyrmions no longer formed a bound state and were free to separate. In the case of a potential hole, the baby skyrmions no longer formed a bound state and moved asymptotically along the edge of the hole. In the asymmetric barrier system the baby skyrmions behaved the same as the skyrmions of the symmetric obstructions. Away from the obstruction the baby skyrmions orbited the boundary of the system. In the potential hole system the bound skyrmions moved along the edge of the hole. For critical values of the hole depth, the bound state between the skyrmions was broken, resulting in one of the skyrmions remaining stationary and the other traversing the edge of the hole. During our investigations into this system it was found that the definition of the angular momentum must be modified to ensure overall conservation. It was shown how these modifications arise and how they are calculated

Doctor of Philosophy

dissertationThe statistical study of anatomy is one of the primary focuses of medical image analysis. It is well-established that the appropriate mathematical settings for such analyses are Riemannian manifolds and Lie group actions. Statistically defined atlases, in which a mean anatomical image is computed from a collection of static three-dimensional (3D) scans, have become commonplace. Within the past few decades, these efforts, which constitute the field of computational anatomy, have seen great success in enabling quantitative analysis. However, most of the analysis within computational anatomy has focused on collections of static images in population studies. The recent emergence of large-scale longitudinal imaging studies and four-dimensional (4D) imaging technology presents new opportunities for studying dynamic anatomical processes such as motion, growth, and degeneration. In order to make use of this new data, it is imperative that computational anatomy be extended with methods for the statistical analysis of longitudinal and dynamic medical imaging. In this dissertation, the deformable template framework is used for the development of 4D statistical shape analysis, with applications in motion analysis for individualized medicine and the study of growth and disease progression. A new method for estimating organ motion directly from raw imaging data is introduced and tested extensively. Polynomial regression, the staple of curve regression in Euclidean spaces, is extended to the setting of Riemannian manifolds. This polynomial regression framework enables rigorous statistical analysis of longitudinal imaging data. Finally, a new diffeomorphic model of irrotational shape change is presented. This new model presents striking practical advantages over standard diffeomorphic methods, while the study of this new space promises to illuminate aspects of the structure of the diffeomorphism group