42 research outputs found
Quantum lump dynamics on the two-sphere
No full textIt is well known that the low-energy classical dynamics of solitons of Bogomol'nyi type is well approximated by geodesic motion in M_n, the moduli space of static n-solitons. There is an obvious quantization of this dynamics wherein the wavefunction evolves according to the Hamiltonian H_0 equal to (half) the Laplacian on M_n. Born-Oppenheimer reduction of analogous mechanical systems suggests, however, that this simple Hamiltonian should receive corrections including k, the scalar curvature of M_n, and C, the n-soliton Casimir energy, which are usually difficult to compute, and whose effect on the energy spectrum is unknown. This paper analyzes the spectra of H_0 and two corrections to it suggested by work of Moss and Shiiki, namely H_1=H_0+k/4 and H_2=H_1+C, in the simple but nontrivial case of a single CP^1 lump moving on the two-sphere. Here M_1=TSO(3), a noncompact kaehler 6-manifold invariant under an SO(3)xSO(3) action, whose geometry is well understood. The symmetry gives rise to two conserved angular momenta, spin and isospin. A hidden isometry of M_1 is found which implies that all three energy spectra are symmetric under spin-isospin interchange. The Casimir energy is found exactly on the zero section of TSO(3), and approximated numerically on the rest of M_1. The lowest 19 eigenvalues of H_i are found for i=0,1,2, and their spin-isospin and parity compared. The curvature corrections in H_1 lead to a qualitatively unchanged low-level spectrum while the Casimir energy in H_2 leads to significant changes. The scaling behaviour of the spectra under changes in the radii of the domain and target spheres is analyzed, and it is found that the disparity between the spectra of H_1 and H_2 is reduced when the target sphere is made smaller
Moduli Spaces of Topological Solitons
No full textThis thesis presents a detailed study of phenomena related to topological solitons (in -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 lumps is a - dimensional manifold of cohomogeneity one. In this thesis, we discuss the charge 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 lumps. We also discuss the implications for lump decay. We discuss interesting families of moduli spaces of charge lumps using the symmetry property and Riemann-Hurwitz formula. We discuss the K\"ahler potential for lumps and find an explicit formula on the -dimensional charge 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 - and some submanifolds of -vortices. We construct collinear hyperbolic - and -vortices and derive explicit formula of their corresponding metrics. We find interesting subspaces in both - and -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 of the moduli space of hyperbolic -vortices. In this thesis, we discuss the K\"{a}hler potential on 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 , in particular when and . We evaluate the vortex scattering angle-impact parameter relation and discuss the vortex scattering of the space . Moreover, we study the vortex scattering of the space . We also compute the scalar curvature of .
Finally, we discuss vortices with impurities and calculate explicit metrics in the presence of impurities
Quantization of Skyrmions
No full textThe Skyrme model is a nonlinear classical field theory which models the strong interaction between atomic nuclei. In order to compare the predictions of the Skyrme model with nuclear physics, it has to be quantized. We show, summarizing earlier work, how the rational map ansatz can be employed to calculate the Finkelstein-Rubinstein constraints which arise during quantization. Then we give an overview of current results on the quantum ground states in the Skyrme model. We end with an outlook on future work
Fermions coupled to skyrmions on S**3
No full textThis paper discusses Skyrmions on the 3-sphere coupled to fermions. The resulting Dirac equation commutes with a generalized angular momentum G. For G = 0 the Dirac equation can be solved explicitly for a constant Skyrme configuration and also for a SO(4) symmetric hedgehog configuration. We discuss how the spectrum changes due to the presence of a non-trivial winding number, and also consider more general Skyrme configurations numerically
Finkelstein-Rubinstein constraints for the Skyrme model with pion masses
No full textThe Skyrme model is a classical field theory modelling the strong interaction between atomic nuclei. It has to be quantized in order to compare it to nuclear physics. When the Skyrme model is semi-classically quantized it is important to take the Finkelstein-Rubinstein constraints into account. Recently, a simple formula has been derived to calculate the these constraints for Skyrmions which are well-approximated by rational maps. However, if a pion mass term is included in the model, Skyrmions of sufficiently large baryon number are no longer well-approximated by the rational map ansatz. This paper addresses the question how to calculate Finkelstein-Rubinstein constraints for Skyrme configurations which are only known numerically
Fermions, Skyrmions and the 3-sphere
No full textThis paper investigates a background charge one Skyrme field chirally coupled to light fermions on the 3-sphere. The Dirac equation for the system commutes with a generalized angular momentum or grand spin. It can be solved explicitly for a Skyrme configuration given by the hedgehog form. The energy spectrum and degeneracies are derived for all values of the grand spin. Solutions for non-zero grand spin are each characterized by a set of four polynomials. The paper also discusses the energy of the Dirac sea using zeta-function regularization
Moduli Spaces of Lumps on Real Projective Space
No full textHarmonic maps that minimize the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on 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 three lumps is a D2-symmetric 7-dimensional manifold of cohomogeneity one. In this paper, we discuss the charge three moduli spaces of lumps from two perspectives: discrete symmetries of lumps and the Riemann-Hurwitz formula. We then calculate the metric and find explicit formula for various geometric quantities. We also discuss the implications for lump decay
Negative Baryon density and the Folding structure of the B=3 Skyrmion
No full textThe Skyrme model is a non-linear field theory whose solitonic solutions, once quantised, describe atomic nuclei. The classical static soliton solutions, so-called Skyrmions, have interesting symmetries and can only be calculated numerically. Mathematically, these Skyrmions can be viewed as maps between two three-manifolds and, as such, their stable singularities can only be folds, cusps and swallowtails. Physically, the occurrence of singularities is related to negative baryon density. In this paper, we calculate the charge three Skyrmion to a high resolution in order to examine its singularity structure in detail. Thereby, we explore regions of negative baryon density. We also discuss how the negative baryon density depends on the pion mass
Homotopy of rational maps and the quantization of Skyrmions
No full textThe Skyrme model is a classical field theory which models the strong interaction between atomic nuclei. It has to be quantized in order to compare it to nuclear physics. When the Skyrme model is semi-classically quantized it is important to take the Finkelstein-Rubinstein constraints into account. The aim of this paper is to show how to calculate these FR constraints directly from the rational map ansatz using basic homotopy theory. We then apply this construction in order to quantize the Skyrme model in the simplest approximation, the zero mode quantization. This is carried out for up to 22 nucleons, and the results are compared to experiment
Recent Developments in the Skyrme Model
No full textIn this talk, we describe recent developments in the Skyrme model. Our main focus is on discussing various effects which need to be taken into account, when calculating the properties of light atomic nuclei in the Skyrme model. We argue that an important step is to understand "spinning Skyrmions" and discuss the theory of relative equilibria in this context
