Classical Supersymmetric Mechanics

We analyse a supersymmetric mechanical model derived from (1+1)-dimensional field theory with Yukawa interaction, assuming that all physical variables take their values in a Grassmann algebra B. Utilizing the symmetries of the model we demonstrate how for a certain class of potentials the equations of motion can be solved completely for any B. In a second approach we suppose that the Grassmann algebra is finitely generated, decompose the dynamical variables into real components and devise a layer-by-layer strategy to solve the equations of motion for arbitrary potential. We examine the possible types of motion for both bosonic and fermionic quantities and show how symmetries relate the former to the latter in a geometrical way. In particular, we investigate oscillatory motion, applying results of Floquet theory, in order to elucidate the role that energy variations of the lower order quantities play in determining the quantities of higher order in B.Comment: 29 pages, 2 figures, submitted to Annals of Physic

Platonic hyperbolic monopoles

We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in R 4 that are invariant under a circle action, and in most cases the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Unlike for Euclidean monopoles, the formulae for the squared Higgs field magnitude in the examples we construct are rational functions of the coordinates. Using these formulae, we compute and illustrate the energy density of the monopoles. We also prove, for particular monopoles, that the number of zeros of the Higgs field is greater than the monopole charge, confirming numerical results established earlier for Euclidean monopoles. We also present some one-parameter families of monopoles analogous to known scattering events for Euclidean monopoles within the geodesic approximation

Spinning particles in Taub-NUT space

The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analysed. The constants of motion are expressed in terms of Killing-Yano tensors. Some previous results from the literature are corrected.Comment: LaTeX, 8 page

Massless monopoles and the moduli space approximation

We investigate the applicability of the moduli space approximation in theories with unbroken non-Abelian gauge symmetries. Such theories have massless magnetic monopoles that are manifested at the classical level as clouds of non-Abelian field surrounding one or more massive monopoles. Using an SO(5) example with one massive and one massless monopole, we compare the predictions of the moduli space approximation with the results of a numerical solution of the full field equations. We find that the two diverge when the cloud velocity becomes of order unity. After this time the cloud profile approximates a spherical wavefront moving at the speed of light. In the region well behind this wavefront the moduli space approximation continues to give a good approximation to the fields. We therefore expect it to provide a good description of the motion of the massive monopoles and of the transfer of energy between the massive and massless monopoles.Comment: 18 pages, 5 figure

More on scattering of Chern-Simons vortices

I derive a general formalism for finding kinetic terms of the effective Lagrangian for slowly moving Chern-Simons vortices. Deformations of fields linear in velocities are taken into account. From the equations they must satisfy I extract the kinetic term in the limit of coincident vortices. For vortices passing one over the other there is locally the right-angle scattering. The method is based on analysis of field equations instead of action functional so it may be useful also for nonvariational equations in nonrelativistic models of Condensed Matter Physics.Comment: discussion around Eq.(45) is generalised, one more condition for the local right-angle scattering is adde

First Order Vortex Dynamics

A non-dissipative model for vortex motion in thin superconductors is considered. The Lagrangian is a Galilean invariant version of the Ginzburg--Landau model for time-dependent fields, with kinetic terms linear in the first time derivatives of the fields. It is shown how, for certain values of the coupling constants, the field dynamics can be reduced to first order differential equations for the vortex positions. Two vortices circle around one another at constant speed and separation in this model.Comment: 22pages, no figures, tex fil

Hyperkahler Metrics from Periodic Monopoles

Relative moduli spaces of periodic monopoles provide novel examples of Asymptotically Locally Flat hyperkahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four-dimensional, this construction yields interesting examples of metrics with self-dual curvature (gravitational instantons). We discuss their topology and complex geometry. An alternative construction of these gravitational instantons using moduli spaces of Hitchin equations is also described.Comment: 23 pages, latex. v2: an erroneous formula is corrected, and its derivation is given. v3 (published version): references adde

Moduli of vortices and Grassmann manifolds

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are smooth at least in the local case r=n. For abelian local vortices we prove that, if a certain "quantization" condition is satisfied, the embedding can be chosen in such a way that the induced Fubini-Study structure realizes the Kaehler class of the usual L^2 metric of gauged vortices.Comment: 22 pages, LaTeX. Final version: last section removed, typos corrected, two references added; to appear in Commun. Math. Phy

Looking for defects in the 2PI correlator

Truncations of the 2PI effective action are seen as a promising way of studying non-equilibrium dynamics in quantum field theories. We probe their applicability in the non-perturbative setting of topological defect formation in a symmetry-breaking phase transition, by comparing full classical lattice field simulations and the 2PI formulation for classical fields in an O($N$) symmetric scalar field theory. At next-to-leading order in 1/N, the 2PI formalism fails to reproduce any signals of defects in the two-point function. This suggests that one should be careful when applying the 2PI formalism for symmetry breaking phase transitions.Comment: 22 pages, 6 figure

Bogomol'nyi Bounds for Gravitational Cosmic Strings

We present a new method for finding lower bounds on the energy of topological cosmic string solutions in gravitational field theories. This new method produces bounds that are valid over the entire space of solutions, unlike the traditional approach, where the bounds obtained are only valid for cylindrically symmetric solutions. This method is shown to be a generalisation of the well-known Bogomol'nyi procedure for non-gravitational theories and as such, it can be used to find gravitational Bogomol'nyi bounds for models wherever the traditional Bogomol'nyi procedure can be applied in the non-gravitational limit. Furthermore, this method yields Bogomol'nyi equations that do not rule out the existence of asymmetric bound-saturating solutions.Comment: 17 pages - final version (accepted for publication in JHEP