Massive nonlinear sigma models and BPS domain walls in harmonic superspace

Four-dimensional massive nonlinear sigma models and BPS wall solutions are studied in the off-shell harmonic superspace approach in which supersymmetry is manifest. The general nonlinear sigma model can be described by an analytic harmonic potential which is the hyper-Kähler analog of the Kähler potential in theory. We examine the massive nonlinear sigma model with multi-center four-dimensional target hyper-Kähler metrics and derive the corresponding BPS equation. We study in some detail two particular cases with the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two separate limits, both Taub-NUT and Eguchi–Hanson metrics. We find that domain wall solutions exist only in the double Taub-NUT case including its Eguchi–Hanson limit.     

Local theory of solutions for the complex grassmannian andCP N−1 sigma models

We study the classical solutions of the complex Grassmannian nonlinear sigma models and of theCP ^N?1 model in two euclidean dimensions. Exact solutions of various types, which seem to be complete, are constructed explicitly in an elementary way, namely in terms of holomorphic functions and the Gramm-Schmidt orthonormalization procedure. A new type of discrete symmetry transformations which map one solution into another is presented.     

Solutions of theU(N)σ models with the Wess-Zumino term

We study finite action classical solutions of the Euclidean two-dimensionalU(N) sigma models with the Wess-Zumino term. We show that these solutions are related to (and so can be derived from) the solutions of the Lax-pair problem for the correspondingU(N) sigma model (without the Wess-Zumino term). We discuss the value of the action for these solutions and prove that all these solutions are unstable.     

Solutions of the supersymmetricU(N) σ models with the Wess-Zumino-Witten term

We present finite action classical solutions of the supersymmetric (Susy)U(N) sigma models with and without the Wess-Zumino-Witten (WZW) term. We derive solutions of the model with the WZW-term from the solutions of the Lax-pair problem for the corresponding model without this term. We study some properties of these solutions such as the value of their action and show that this value is related to that of the purely bosonicU(N) σ-model. Moreover, we prove that all derived solutions are unstable.     

On stability of solutions of theU(N) chiral model in two dimensions

We discuss the stability properties of classical solutions of theU(N) sigma models in two Euclidean dimensions. We show that all nontrivial solutions are unstable. For a general case we exhibit one mode of instability; in some special cases (corresponding to a grassmannian solution and an instantonic grassannian embedding) we exhibit two such independent modes.     

A theorem on the existence of dyon solutions

We prove the existence of finite energy dyon solutions to Yang-Mills-Higgs equations satisfying the Julia-Zee ansatz, and the generalization to SU(N) gauge groups. This rigorously establishes the existence of a model for the particles having electric and magnetic charge conjectured by Schwinger. We also prove that the solutions are real analytic on (0, ∞) and C^∞ at r = 0. To establish our result we prove a new abstract theorem that allows one to study singular constrained minimization problems without the introduction of Lagrange multipliers.     

A gauged nonlinear sigma-model realization of the symplectic blowing up

In this paper a two dimensional non-linear sigma model with a general symplectic manifold with isometry as target space is used to study symplectic blowing up of a point singularity on the zero level set of the moment map associated with a quasi-free Hamiltonian action. We discuss in general the relation between symplectic reduction and gauging of the symplectic isometries of the sigma model action. In the case of singular reduction, gauging has the same effect as blowing up the singular point by a small amount. Using the exponential mapping of the underlying metric, we are able to construct symplectic diffeomorphisms needed to glue the blow-up to the global reduced space which is regular, thus providing a transition from one symplectic sigma model to another one free of singularities.Alexander von Humboldt fellow, on leave from Zhejiang University. Institute of Modern Physics, Hangzhou, China. Address after 1October, 1994; Department of Mathematical Sciences University of Durham, South Road, Durham, England.     

New instanton and meron solutions of a generally covariant four-dimensional sigma model

We construct new solutions of the fourdimensional sigma model coupled to the metric tensor field and having an internalO invariance. Our solutions interpolate continuously between the known instanton and meron solutions depending upon a parameterf. We find that the typical domain for the instantons is 2<f≦3 while for the merons is 0≦f≦2.     

Solutions of theU(N)σ models with the Wess-Zumino term

We study finite action classical solutions of the Euclidean two-dimensionalU(N) sigma models with the Wess-Zumino term. We show that these solutions are related to (and so can be derived from) the solutions of the Lax-pair problem for the correspondingU(N) sigma model (without the Wess-Zumino term). We discuss the value of the action for these solutions and prove that all these solutions are unstable.     

Nonlinear dynamics of the classical isotropic Heisenberg antiferromagnetic chain: The sigma model sector and the kink sector

We identify two distinct low-energy sectors in the classical isotropic antiferromagnetic Heisenberg spin-S chain, in the continuum limit. We show that two types of rotation generators arise for the field in each sector. Using these, the Lagrangian for sector I is shown to be that of the nonlinear sigma model. Sector II has a null Lagrangian. Its Hamiltonian density is just the Pontryagin term. Exact solutions are found in the form of magnons and precessing pulses in I and moving kinks in II. The kink has ‘spin’ S. Sector I has a higher minimum energy than II.     

Unitary dressing transformations and exponential decay below threshold for quantum spin systems. Parts I and II

We consider a class of quantum spin systems defined on connected graphs of which the following HeisenbergXY-model with a variable magnetic field gives an example: $$H_\lambda = \sum\limits_{x \in \mathbb{Z}^d } {h_x \sigma _x^{(3)} + \lambda } \sum\limits_{< x,y > \subset \mathbb{Z}^d } {(\sigma _x^{(1)} \sigma _y^{(1)} + \sigma _x^{(2)} \sigma _y^{(2)} )} .$$ We treat first the case in whichh _x =±1 for all sitesx and we introduce a unitary dressing transformation to control the spectrum for λ small. Then, we consider a situation in which |h _x | can be less than one forx in a finite setL and prove exponential decay away fromL of dressed eigenfunctions with energy below the one-quasiparticle threshold. If the ground state is separated by a finite gap from the rest of the spectrum, this result can be strengthened and one can compute a second unitary transformation that makes the ground state of compact support. Finally, a case in which the singular setL is of finite density, is considered. The main technical tools we use are decay estimates on dressed Green's functions and variational inequalities.     

On Spin(7) holonomy metric based on SU(3)/U(1): II

We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first-order system of Spin(7) instanton equation. One is the flag manifold SU(3)/T² also known as the twistor space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distinguished by the choice of S¹ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of the possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyper-Kähler metric, the other is a new family of metrics on a line bundle over the twistor space of CP(2).     

Pseudo-Hyperkähler Geometry and Generalized Kähler Geometry

We discuss the conditions for additional supersymmetry and twisted super-symmetry in N = (2, 2) supersymmetric nonlinear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We find that additional linear supersymmetry has no interesting solution, whereas additional linear twisted supersymmetry has solutions with interesting geometrical properties. We solve the conditions for invariance of the action and show that these solutions correspond to a bi-hermitian metric of signature (2, 2) and a pseudo-hyperkähler geometry of the target space.     

Generalized nonlinear Doebner-Goldin Schrödinger equation and the relaxation of quantum systems

We propose a new class of nonlinear homogeneous extension of the Doebner-Goldin Schrödinger equation, valid for arbitrary representations and operators, chosen in accordance with the investigated physical problem. We verify that the nonlinearity simulates an environment, thence, the new model leads to simple exact solutions as, for instance, the time-dependent squeezed coherent states and a special class of stationary states that we call pseudothermal, reached after relaxation. We illustrate the use of the new equation with applications to problems such as, the relaxation of a two-level or spin-1/2 system, and of the harmonic oscillator (HO) or equivalently, the emission-absorption process of photons in an electromagnetic cavity. Furthermore, in order to compare solutions for the HO example we introduce two different representations in the new equation, one continuous (positional representation) and the other discrete (Fock states).     

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

A global identity for nonlinear σ-models

An identity is presented for nonlinear sigma models (harmonic maps) which is a generalization of Green's identity. This identity provides us with a useful tool for analysing questions of the bifurcations and uniqueness of solutions of nonlinear sigma models on noncompact symmetric spaces. As an example of an application, a short proof of the no-hair theorem for charged black holes is given. The uniqueness problem of axisymmetric monopoles is discussed also.     

On the complete integrability of the static axially symmetric self-dual gauge field equations for an arbitrary group

We present an analysis of static axially symmetric gauge fields for an arbitrary gauge group G. Two ansätze are considered. The full ansatz involves a total of 4d(d = dim G), the reduced ansatz only 2d functions of (?, z). Imposing self-duality is shown to reduce the problem to a sigma model in the curved two-dimensional (?, z) space over the coset spaces G?/G for the full, and G¹/K for the reduced ansatz. G? is the complexification of G. ¹ is a particular non-compact form of G, and K the local form-preserving symmetry group of the reduced ansatz. We give explicitly the Lax-pair type representations (linear scattering problem) of the sigma model, indicating that the standard methods available for certain non-linear two-dimensional problems can be used to generate solutions. Our procedure has the advantage that only real fields over a real manifold enter the analysis.     

Singularities in Einstein–conformally coupled Higgs cosmological models

The dynamics of Einstein–conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman–Robertson–Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (‘Small Bang’), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (‘Big Bang’). Thus, in the present EccH model there is a new kind of physical spacetime singularity (‘Small Bang’). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout–Englert–Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.     

Topological features of the nonlinear sigma model with Hopf term using chiral lagrangians

We rewrite the O(3) nonlinear sigma model in 2 + 1 dimensions in terms of SU(2) matrices, thereby solving the constraint. The lagrangian has the symmetry SU(2)_Global×U(1)_Local. Static soliton solutions to this lagrangian have energy 4πN as usual. We then show that the Hopf instantons, in the formalism of principle chiral fields, are just the skymions of QCD in 3 + 1 dimensions.     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.