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C. Adam

Publications and source records attributed to C. Adam.

At least 109 records · Page 6Linked to original sources
arXiv2008

Conservation laws in Skyrme-type models

The zero curvature representation of Zakharov and Shabat has been generalized recently to higher dimensions and has been used to construct non-linear field theories which either are integrable or contain integrable submodels. The Skyrme model, for instance, contains an integrable subsector with infinitely many conserved currents, and the simplest Skyrmion with baryon number one belongs to this subsector. Here we use a related method, based on the geometry of target space, to construct a whole class of theories which are either integrable or contain integrable subsectors (where integrability means the existence of infinitely many conservation laws). These models have three-dimensional target space, like the Skyrme model, and their infinitely many conserved currents turn out to be Noether currents of the volume-preserving diffeomorphisms on target space. Specifically for the Skyrme model, we find both a weak and a strong integrability condition, where the conserved currents form a subset of the algebra of volume-preserving diffeomorphisms in both cases, but this subset is a subalgebra only for the weak integrable submodel.

hep-th↗
arXiv2008

Integrable subsystem of Yang--Mills dilaton theory

With the help of the Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2) Yang-Mills field, we find an integrable subsystem of SU(2) Yang-Mills theory coupled to the dilaton. Here integrability means the existence of infinitely many symmetries and infinitely many conserved currents. Further, we construct infinitely many static solutions of this integrable subsystem. These solutions can be identified with certain limiting solutions of the full system, which have been found previously in the context of numerical investigations of the Yang-Mills dilaton theory. In addition, we derive a Bogomolny bound for the integrable subsystem and show that our static solutions are, in fact, Bogomolny solutions. This explains the linear growth of their energies with the topological charge, which has been observed previously. Finally, we discuss some generalisations.

hep-th↗
arXiv2007

A first integration of some knot soliton models

Recently it has been shown that there exists a sector within the Faddeev-Niemi model for which the equations of motion may be reduced to first order equations. However, no solutions to that sector have been given. It is not even known whether this sector contains topologically nontrivial solutions, at all. Here, we show that two models with analytically known Hopf solitons, namely the Nicole and the Aratyn-Ferreira-Zimerman models, possess sectors which can be integrated to first order partial differential equations. The main result is that these sectors are topologically nontrivial. In fact, all analytically known hopfions belong to them.

hep-th↗
arXiv2007

Comment on ``Reduction of static field equation of Faddeev model to first order PDE'', arXiv:0707.2207

The authors of the article Phys. Lett. B 652 (2007) 384, (arXiv:0707.2207), propose an interesting method to solve the Faddeev model by reducing it to a set of first order PDEs. They first construct a vectorial quantity $\bm α$, depending on the original field and its first derivatives, in terms of which the field equations reduce to a linear first order equation. Then they find vectors $\bm α_1$ and $\bm α_2$ which identically obey this linear first order equation. The last step consists in the identification of the $\bm α_i$ with the original $\bm α$ as a function of the original field. Unfortunately, the derivation of this last step in the paper cited above contains an error which invalidates most of its results.

hep-th↗
arXiv2007

Integrability in Theories with Local U(1) Gauge Symmetry

Using a recently developed method, based on a generalization of the zero curvature representation of Zakharov and Shabat, we study the integrability structure in the Abelian Higgs model. It is shown that the model contains integrable sectors, where integrability is understood as the existence of infinitely many conserved currents. In particular, a gauge invariant description of the weak and strong integrable sectors is provided. The pertinent integrability conditions are given by a U(1) generalization of the standard strong and weak constraints for models with two dimensional target space. The Bogomolny sector is discussed, as well, and we find that each Bogomolny configuration supports infinitely many conserved currents. Finally, other models with U(1) gauge symmetry are investigated.

hep-th↗
arXiv2006

New Integrable Sectors in Skyrme and 4-dimensional CP^n Model

The application of a weak integrability concept to the Skyrme and $CP^n$ models in 4 dimensions is investigated. A new integrable subsystem of the Skyrme model, allowing also for non-holomorphic solutions, is derived. This procedure can be applied to the massive Skyrme model, as well. Moreover, an example of a family of chiral Lagrangians providing exact, finite energy Skyrme-like solitons with arbitrary value of the topological charge, is given. In the case of $CP^n$ models a tower of integrable subsystems is obtained. In particular, in (2+1) dimensions a one-to-one correspondence between the standard integrable submodel and the BPS sector is proved. Additionally, it is shown that weak integrable submodels allow also for non-BPS solutions. Geometric as well as algebraic interpretations of the integrability conditions are also given.

hep-th↗
arXiv2006

Soliton stability in some knot soliton models

We study the issue of stability of static soliton-like solutions in some non-linear field theories which allow for knotted field configurations. Concretely, we investigate the AFZ model, based on a Lagrangian quartic in first derivatives with infinitely many conserved currents, for which infinitely many soliton solutions are known analytically. For this model we find that sectors with different (integer) topological charge (Hopf index) are not separated by an infinite energy barrier. Further, if variations which change the topological charge are allowed, then the static solutions are not even critical points of the energy functional. We also explain why soliton solutions can exist at all, in spite of these facts. In addition, we briefly discuss the Nicole model, which is based on a sigma-model type Lagrangian. For the Nicole model we find that different topological sectors are separated by an infinite energy barrier.

hep-th↗
arXiv2006

Hopf solitons and Hopf Q-balls on S^3

Field theories with a $S^2$-valued unit vector field living on $S^3 \times \RR$ space-time are investigated. The corresponding eikonal equation, which is known to provide an integrable sector for various sigma models in different spaces, is solved giving static as well as time-dependent multiply knotted configurations on $S^3$ with arbitrary values of the Hopf index. Using these results, we then find a set of hopfions with topological charge $Q_H=m^2$, $m \in \mathbf{Z}$, in the integrable subsector of the pure $CP^1$ model. In addition, we show that the $CP^1$ model with a potential term provides time-dependent solitons. In the case of the so-called "new baby Skyrme" potential we find, e.g., exact stationary hopfions, i.e., topological $Q$-balls. Our results further enable us to construct exact static and stationary Hopf solitons in the Faddeev--Niemi model with or without the new baby Skyrme potential. Generalizations for a large class of models are also discussed.

hep-th↗
arXiv2006

Investigation of the Nicole model

We study soliton solutions of the Nicole model - a non-linear four-dimensional field theory consisting of the CP^1 Lagrangian density to the non-integer power 3/2 - using an ansatz within toroidal coordinates, which is indicated by the conformal symmetry of the static equations of motion. We calculate the soliton energies numerically and find that they grow linearly with the topological charge (Hopf index). Further we prove this behaviour to hold exactly for the ansatz. On the other hand, for the full three-dimensional system without symmetry reduction we prove a sub-linear upper bound, analogously to the case of the Faddeev-Niemi model. It follows that symmetric solitons cannot be true minimizers of the energy for sufficiently large Hopf index, again in analogy to the Faddeev-Niemi model.

hep-th↗
arXiv2005

Integrability from an abelian subgroup of the diffeomorphism group

It has been known for some time that for a large class of non-linear field theories in Minkowski space with two-dimensional target space the complex eikonal equation defines integrable submodels with infinitely many conservation laws. These conservation laws are related to the area-preserving diffeomorphisms on target space. Here we demonstrate that for all these theories there exists, in fact, a weaker integrability condition which again defines submodels with infinitely many conservation laws. These conservation laws will be related to an abelian subgroup of the group of area-preserving diffeomorphisms. As this weaker integrability condition is much easier to fulfil, it should be useful in the study of those non-linear field theories.

hep-th↗
arXiv2005

Knot soliton models, submodels, and their symmetries

For some non-linear field theories which allow for soliton solutions, submodels with infinitely many conservation laws can be defined. Here we investigate the symmetries of the submodels, where in some cases we find a symmetry enhancement for the submodels, whereas in others we do not.

hep-th↗
arXiv2005

Generalized integrability conditions and target space geometry

In some higher dimensional nonlinear field theories integrable subsectors with infinitely many conservation laws have been identified by imposing additional integrability conditions. Originally, the complex eikonal equation was chosen as integrability condition, but recently further generalizations have been proposed. Here we show how these new integrability conditions may be derived from the geometry of the target space and, more precisely, from the Noether currents related to a certain class of target space transformations.

hep-th↗
arXiv2005

The symmetries of the Dirac--Pauli equation in two and three dimensions

We calculate all symmetries of the Dirac-Pauli equation in two-dimensional and three-dimensional Euclidean space. Further, we use our results for an investigation of the issue of zero mode degeneracy. We construct explicitly a class of multiple zero modes with their gauge potentials.

hep-th↗
arXiv2004

Symmetries of generalized soliton models and submodels on target space $S^2$

Some physically relevant non-linear models with solitons, which have target space $S^2$, are known to have submodels with infinitly many conservation laws defined by the eikonal equation. Here we calculate all the symmetries of these models and their submodels by the prolongation method. We find that for some models, like the Baby Skyrme model, the submodels have additional symmetries, whereas for others, like the Faddeev--Niemi model, they do not.

hep-th↗
arXiv2004

Hopf maps as static solutions of the complex eikonal equation

We demonstrate that a class of torus-shaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation. Further, we explore the geometric structure behind these solutions, explaining thereby the reason for their existence. As this equation shows up as an integrability condition in certain non-linear field theories, the existence of such solutions is of some interest.

math-ph↗
arXiv2003

Comment on "Vacuum Photon Splitting in Lorentz-Violating Quantum Electrodynamics"

It has been argued in hep-ph/0212382 that there may be observable effects from photon triple splitting in an extended version of quantum electrodynamics with Lorentz violation in the fermion sector. The argument is based on an analogy with photon splitting via collinear parametric down-conversion in optically active crystals. In this Comment, we point out that the analogy is misleading and that, at the order considered, the probability of on-shell photon triple splitting is strictly zero.

hep-ph↗
arXiv2003

Chern-Simons action for zero-mode supporting gauge fields in three dimensions

Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree the conjecture that the Chern-Simons action admits only certain quantized values for gauge fields that lead to zero modes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing an explicit counter-example.

hep-th↗