Deformed solitons: The case of two coupled scalar fields
Alvaro de Souza Dutra
备注：
Work presented as a talk in XI International Conference on Symmetry Methods in Physics (SYMPHYS-11)held in Prague, the Czech Republic, from 21 to 24 June 2004. This work is also available from the link: http://www.jinr.dubna.su/publish/Proceedings/Burdik-2004/index.html
In this work, we present a general procedure, which is able to generate new exact solitonic models in 1+1 dimensions, from a known one, consisting of two coupled scalar fields. An interesting consequence of the method, is that of the appearing of nontrivial extensions, where the deformed systems presents other BPS solitons than that appearing in the original model. Finally we take a particular example, in order to check the above mentioned features.
We consider recently developed Cohomological Field Theory soliton counting diagram technique for Khovanov and Khovanov-Rozansky invariants [1,2]. Although the expectation to obtain a new way for computing the invariants has not yet come true, we demonstrate that soliton counting technique can be totally formalized at an intermediate stage, at least in particular cases. We present the corresponding algorithm, based on the approach involving deformed $\mathcal{R}$-matrix and minimal positive division, developed previously in [3]. We start from a detailed review of the minimal positive division a…
We describe solitons that live on the world--volumes of D5 branes wrapped on deformed $A_2$ singularities fibered over $C(x)$. We show that monopoles are D3 branes wrapped on a node of the deformed singularity and stretched along $C(x)$. F and D--term strings are D3 branes wrapped on a node of a singularity that is deformed and resolved respectively. Domain walls require deformed $A_3$ singularities and correspond to D5 branes wrapped on a node and stretched along $C(x)$.
We study dual geometries to a deformed ABJM model with spatially dependent source functions at finite temperature. These source functions are proportional to the mass function $m(x)= m_0 \sin k x$ and its derivative $m'(x)$. As dual geometries, we find hairy black branes and AdS solitons corresponding to deconfinement phase and confining phase of the dual field theory, respectively. It turns out that the hairy AdS solitons have lower free energy than the black branes when the Hawking temperature is smaller than the confining scale. Therefore the dual system undergoes the first order phase tran…
We study scattering properties of topological solitons in two classes of models, which are generalizations of the Sine-Gordon model and which have recently been proposed by Bazeia et al. These two classes of models depend on an integer parameter n which, when n=2(for the first class) and n=1 (for the second class), reduce to the Sine-Gordon model. We take the soliton solutions of these models (generalizations of the 'kink' solution of the Sine-Gordon model) and consider their scattering on potential holes and barriers. We present our results for n=1,...6. We find that, like in the Sine Gordon …
We have examined the deformation of a generic non-Abelian gauge theory obtained by replacing its Lie group by the corresponding quantum group. This deformed gauge theory has more degrees of freedom than the theory from which it is derived. By going over from point particles in the standard theory to solitonic particles in the deformed theory, it is proposed to interpret the new degrees of fredom as descriptive of a non-locality of the deformed theory. It also turns out that the original Lie algebra gets replaced by two dual algebras, one of which lies close to and approaches the original Lie a…
Moyal-deformed hierarchies of soliton equations can be extended to larger hierarchies by including additional evolution equations with respect to the deformation parameters. A general framework is presented in which the extension is universally determined and which applies to several deformed hierarchies, including the noncommutative KP, modified KP, and Toda lattice hierarchy. We prove a Birkhoff factorization relation for the extended ncKP and ncmKP hierarchies. Also reductions of the latter hierarchies are briefly discussed. Furthermore, some results concerning the extended ncKP hierarchy a…
We demonstrate the construction of solitons for a time-space Moyal-deformed integrable U(n) sigma model (the Ward model) in 2+1 dimensions. These solitons cannot travel parallel to the noncommutative spatial direction. For the U(1) case, the rank-one single-soliton configuration is constructed explicitly and is singular in the commutative limit. The projection to 1+1 dimensions reduces it to a noncommutative instanton-like configuration. The latter is governed by a new integrable equation, which describes a Moyal-deformed sigma model with a particular Euclidean metric and a magnetic field.
A linear system, which generates a Moyal-deformed two-dimensional soliton equation as integrability condition, can be extended to a three-dimensional linear system, treating the deformation parameter as an additional coordinate. The supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter, the flow of which commutes with the flow of the deformed soliton equation. In this way, a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations. We prove the extended hierarch…
The isotropy of Newton potential in noncommutative space disappears as it deforms to a momentum dependent one. We generalize the earlier derivation of such a deformed potential to the relativistic regime by calculating the 2 scalar-1 graviton scattering amplitude by taking into account the noncommutativity of space.
We study a model describing some aspects of the dynamics of biopolymers. The models involve either one or two finite chains with a number N of sites that represent the "units" of a biophysical system. The mechanical degrees of freedom of these chains are coupled to the internal degrees of freedom through position dependent excitation transfer functions. We reconsider the case of the one chain model discussed by Mingaleev et al. and present new results concerning the soliton sector of this model. We also give new (preliminary) results in the two chain model in which case we have introduced an i…
We study dynamics of scalar fields on a large class of geometries described by integrable sigma models. Although equations of motion are not separable due to absence of isometries and Killing tensors, we completely determine the spectra using algebraic and group-theoretic methods.
We study the propagation of a scalar, the trace of $h_{ij}$ in the deformed Hořava-Lifshitz gravity with coupling constant $λ$. It turns out that this scalar is not a propagating mode in the Minkowski spacetime background. In this work, we do not choose a gauge-fixing to identify the physical degrees of freedom and instead, make it possible by substituting the constraints into the quadratic Lagrangian.
In this letter the nontopological scalar solitons with a self-interaction potential are investigated in a static de Sitter spacetime and their field equations are derived. In particular, with series expansion we prove analytically the nonexistence of the solutions under the boundary conditions, which show that some nontopological solitons can not live in the background.
We study scalar solitons on the fuzzy sphere at arbitrary radius and noncommutativity. We prove that no solitons exist if the radius is below a certain value. Solitons do exist for radii above a critical value which depends on the noncommutativity parameter. We construct a family of soliton solutions which are stable and which converge to solitons on the Moyal plane in an appropriate limit. These solutions are rotationally symmetric about an axis and have no allowed deformations. Solitons that describe multiple lumps on the fuzzy sphere can also be constructed but they are not stable.
We observe that the equation of motion for a free scalar field in a closed universe with radiation and a positive cosmological constant is given by Lamé's equation. Computing the exact power spectrum of scalar field perturbations, the presence of both curvature and radiation produces a red tilt weakly dependent on the amount of radiation.
In this letter the nontopological scalar solitons are investigated in an anti de Sitter spacetime. We find analytically that the solitons obeying the necessary conditions $m=\frac{2}{\sqrt{3}}|Λ|^{1/2}$ or $m=\sqrt{\frac{2(n+3)(2n+3)}{3}}|Λ|^{1/2}$ can exist in the background by means of the series expansion.
We canonically quantize multi-component scalar field theories in the presence of solitons. This extends results of Tomboulis to general soliton moduli spaces. We derive the quantum Hamiltonian, discuss reparameterization invariance and explicitly show how, in the semiclassical approximation, the dynamics of the full theory reduce to quantum mechanics on the soliton moduli space. We emphasize the difference between the semiclassical approximation and a truncation of the dynamical variables to moduli. Both procedures produce quantum mechanics on moduli space, but the two Hamiltonians are generic…
We obtain the primitively divergent diagrams in $κ$-deformed scalar field in four-dimensional spacetime with quartic self-interaction in order to investigate the effect of the fundamental length $q=1/(2κ)$ on such diagrams. Thanks to $κ$-deformation, we find that the dimensionally regularized forms of the diagrams lead to finite results in the limit of space-time dimension four. The effect of the deformation appears as a displacement of the poles in the complex plane.
In this paper we use the deformation procedure introduced in former work on deformed defects to investigate several new models for real scalar field. We introduce an interesting deformation function, from which we obtain two distinct families of models, labeled by the parameters that identify the deformation function. We investigate these models, which identify a broad class of polynomial interactions. We find exact solutions describing global defects, and we study the corresponding stability very carefully.
A system of coupled scalar fields is introduced which possesses a spectrum of massive single-soliton solutions. Some of these solutions are unstable and decay into lower mass stable solitons. Some properties of the solutions are obtained using general principles including conservations of energy and topological charges. Rest energies are calculated via a variational scheme, and the dynamics of the coupled fields are obtained by solving the field equations numerically.
In this article, we study kink soliton configurations in interacting scalar field theories containing two fields without $SO(2)$ invariance. We study a class of such theories, the well-known Montonen-Sarker-Trullinger-Bishop model is one of them. These models are interesting since the $U(1)$ current is not conserved in them due to the presence of explicit symmetry breaking terms in the action. The existence of kink soliton configurations is shown in terms of a system of first-order ordinary differential equations. Although $U(1)$ current in these models are non-conserved, our approach is gener…
We investigate the behavior of the noncommutative scalar soliton solutions at finite noncommutative scale $θ$. A detailed analysis of the equation of the motion indicates that fewer and fewer soliton solutions exist as $θ$ is decreased and thus the solitonic sector of the theory exhibits an overall hierarchy structure. If the potential is bounded below, there is a finite $θ_c$ below which all the solitons cease to exist even though the noncommutativity is still present. If the potential is not bounded below, for any nonzero $θ$ there is always a soliton solution, which becomes singular only at…
We consider solitonic solutions of coupled scalar systems, whose Lagrangian has a potential term (quasi-supersymmetric potential) consisting of the square of derivative of a superpotential. The most important feature of such a theory is that among soliton masses there holds a Ritz-like combination rule (e.g. $M_{12}+M_{23}=M_{13}$), instead of the inequality ($M_{12}+M_{23}<M_{13}$) which is a stability relation generally seen in N=2 supersymmetric theory. The promotion from N=1 to N=2 theory is considered.
We present a method to obtain soliton solutions to relativistic system of coupled scalar fields. This is done by examining the energy associated to static field configurations. In this case we derive a set of first-order differential equations that solve the equations of motion when the energy saturates its lower bound. To illustrate the general results, we investigate some systems described by polynomial interactions in the coupled fields.
We make an analysis about several aspects of localization of a scalar field and a Kalb-Ramond gauge field in a specific four dimensional AdS membrane embedded in a five dimensional space-time. The membrane is generated from a deformation of the $λϕ^4$ potential and belongs to a new class of defect solutions. The study of deformed defects is important because they contain internal structures and these may have implications to the way the background space-time is constructed and the way its curvature behaves. The main objective is to observe the contributions of the deformation procedure to the …
The light-cone gauge approach to $T\overline{T}$ deformed models is used to derive the $T\overline{T}$ deformed matrix nonlinear Schrödinger equation, the Landau--Lifshitz equation, and the Gardner equation. Properties of one-soliton solutions of the $T\overline{T}$ deformed nonlinear Schrödinger and Korteweg--de Vries equations are discussed in detail. The NLS soliton exhibits the recently discussed phenomenon of widening/narrowing width of particles under the $T\overline{T}$ deformation. However, whether the soliton's size is increasing or decreasing depends not only on the sign of the defor…
Interactions of noncommutative waves and solitons in 2+1 dimensions can be analyzed exactly for a supersymmetric and integrable U(n) chiral model extending the Ward model. Using the Moyal-deformed dressing method in an antichiral superspace, we construct explicit time-dependent solutions of its noncommutative field equations by iteratively solving linear equations. The approach is illustrated by presenting scattering configurations for two noncommutative U(2) plane waves and for two noncommutative U(2) solitons as well as by producing a noncommutative U(1) two-soliton bound state.
We study solitons in scalar theories with polynomial interactions on the fuzzy sphere. Such solitons are described by projection operators of rank k, and hence the moduli space for the solitons is the Grassmannian Gr(k,2j+1). The gradient term of the action provides a non-trivial potential on Gr(k,2j+1), thus reducing the moduli space. We construct configurations corresponding to well-separated solitons, and show that although the solitons attract each other, the attraction vanishes in the limit of large j. In this limit, it is argued that the moduli space is CP^k. For the k-soliton bound stat…
In a series of recent works Ishihara and Ogawa have investigated non-topological solitons (Q-balls) in a spontaneously broken Abelian gauge theory coupled to two complex scalar fields. The present paper extends their investigations to the most general U(1)$\times$U(1) symmetric quartic potential. Also a new class of charged Q-ball solutions with vanishing self-interaction terms is investigated and some of their remarkable properties is exhibited.
We consider the Abelian Higgs model as well as the SU(2) Georgi-Glashow model in which the gauge field action is replaced by a non linear Born-Infeld action. We study soliton solutions arising in these models, namely the vortex and monopole solutions, respectively. We construct formulas which provide good approximations for the mass of the Born-Infeld deformed solitons using only the data of the undeformed solutions. The results obtained indicate that in the self-dual limit, the Born-Infeld interaction leads to bound vortices, while for monopoles it gives rise to repulsion.
In this note we study soliton, breather and shockwave solutions in certain two dimensional field theories. These include: (i) Heisenberg's model suggested originally to describe the scattering of high energy nucleons (ii) $T\bar T$ deformations of certain canonical scalar field theories with a potential. We find explicit soliton solutions of these models with sine-Gordon and Higgs-type potentials. We prove that the $T\bar T$ deformation of a theory of a given potential does not correct the mass of the soliton of the undeformed one. We further conjecture the form of breather solutions of these …
The Poisson bracket algebra corresponding to the second Hamiltonian structure of a large class of generalized KdV and mKdV integrable hierarchies is carefully analysed. These algebras are known to have conformal properties, and their relation to $\cal W$-algebras has been previously investigated in some particular cases. The class of equations that is considered includes practically all the generalizations of the Drinfel'd-Sokolov hierarchies constructed in the literature. In particular, it has been recently shown that it includes matrix generalizations of the Gelfand-Dickey and the constraine…
In this paper we study the quantisation of scalar field theory in $κ$-deformed space-time. Using a quantisation scheme that use only field equations, we derive the quantisation rules for deformed scalar theory, starting from the $κ$-deformed equations of motion. This scheme allows two choices; (i)a deformed commutation relation between the field and its conjugate which leads to usual oscillator algebra, (ii) an undeformed commutation relation between field and its conjugate leading to a deformed oscillator algebra. This deformed oscillator algebra is used to derive modification to Unruh effect…
We determine the stability conditions for a radially symmetric noncommutative scalar soliton at finite noncommutivity parameter $θ$. We find an intriguing relationship between the stability and existence conditions for all level-1 solutions, in that they all have nearly-vanishing stability eigenvalues at critical $θm^2$. The stability or non-stability of the system may then be determined entirely by the $ϕ^3$ coefficient in the potential. For higher-level solutions we find an ambiguity in extrapolating solutions to finite $θ$ which prevents us from making any general statements. For these stab…
We investigate BPS soliton solutions of U(N) Chern-Simons gauge theory coupled to a scalar field in noncommutative plane. With a scalar field in the fundamental representation, we show that the BPS equation becomes that of abelian Chern-Simons theory in the unitary gauge. We also find a class of particular solutions for the BPS equation with scalar field in the adjoint representation.
We study massive graviton propagations of scalar, vector, and tensor modes in the deformed Hořava-Lifshitz gravity by introducing Lorentz-violating mass term. It turns out that vector and tensor modes are massively propagating on the Minkowski spacetime background. However, adding the mass term does not cure a ghost instability in the Hořava scalar.
We study one- and two-soliton solutions of noncommutative Chern-Simons theory coupled to a nonrelativistic or a relativistic scalar field. In the nonrelativistic case, we find a tower of new stationary time-dependent solutions, all with the same charge density, but with increasing energies. The dynamics of these solitons cannot be studied using traditional moduli space techniques, but we do find a nontrivial symplectic form on the phase space indicating that the moduli space is not flat. In the relativistic case we find the metric on the two soliton moduli space.
We start by discussing the classical noncommutative (NC) Q-ball solutions near the commutative limit, then generalize the virial relation. Next we quantize the NC Q-ball canonically. At very small theta quantum correction to the energy of the Q-balls is calculated through summation of the phase shift. UV/IR mixing terms are found in the quantum corrections which cannot be renormalized away. The same method is generalized to the NC GMS soliton for the smooth enough solution. UV/IR mixing is also found in the energy correction and UV divergence is shown to be absent. In this paper only (2+1) dim…
Solitonic objects play a central role in gauge and string theory (as, e.g., monopoles, black holes, D-branes, etc.). Certain string backgrounds produce a noncommutative deformation of the low-energy effective field theory, which allows for new types of solitonic solutions. I present the construction, moduli spaces and dynamics of Moyal-deformed solitons, exemplified in the 2+1 dimensional Yang-Mills-Higgs theory and its Bogomolny system, which is gauge-fixed to an integrable chiral sigma model (the Ward model). Noncommutative solitons for various 1+1 dimensional integrable systems (such as sin…
We investigate the Casimir effect at finite temperature for a charged scalar field in the presence of an external uniform and constant magnetic field, perpendicular to the Casimir plates. We have used a boundary condition characterized by a deformation parameter $θ$; for $θ=0$ we have a periodic condition and for $θ=π$, an antiperiodic one, for intermediate values, we have a deformation. The temperature was introduced using the imaginary time formalism and both the lagrangian and free energy were obtained from Schwinger proper time method for computing the effective action. We also computed th…
We refute the claim that previous works on the one-loop quantum mass of solitons had incorrectly dropped a surface term from a partial integration. Rather, the paper quoted in the title contains a fallacious derivation with two compensating errors. We also remark that the $ϕ^2\cos^2\ln(ϕ^2)$ model considered in that paper does not have solitons at the quantum level because at two-loop order the degeneracy of the vacua is lifted. This may be remedied, however, by a supersymmetric extension.
The (1+1)-dimensional gauge model of two complex self-interacting scalar fields that interact with each other through an Abelian gauge field and a quartic scalar interaction is considered. It is shown that the model has nontopological soliton solutions describing soliton systems consisting of two Q-ball components possessing opposite electric charges. The two Q-ball components interact with each other through the Abelian gauge field and the quartic scalar interaction. The interplay between the attractive electromagnetic interaction and the repulsive quartic interaction leads to the existence o…
I agree with the authors of hep-th/0211149 that the claim made in Phys.Lett. B542, 282 (2002) is incorrect and that the derivation of its main formula, although correct, contains two compensating errors. In this reply the main formula of Phys.Lett. B542, 282 (2002) is rederived. This new derivation shows that not only the energy momentum cut off regularization method still works in the calculation of the soliton quantum mass corrections, but also that the so called mode number regularization emerges naturally from it.
Deformed sine-Gordon (DSG) models $\partial_ξ\partial_η\, w + \frac{d}{dw}V(w) = 0$, with $V(w)$ being the deformed potential, are considered in the context of the Riccati-type pseudopotential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for DSG model, and provide an infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have rece…
In this paper we consider a class of systems of two coupled real scalar fields in bidimensional spacetime, with the main motivation of studying classical or linear stability of soliton solutions. Firstly, we present the class of systems and comment on the topological profile of soliton solutions one can find from the first-order equations that solve the equations of motion. After doing that, we follow the standard approach to classical stability to introduce the main steps one needs to obtain the spectra of Schrödinger operators that appear in this class of systems. We consider a specific syst…
In a previous paper \cite{lp}, supersymmetric $p$-brane solutions involving one dilatonic scalar field in maximal supergravity theories were classified. Although these solutions involve a number of participating field strengths, they are all equal and thus they carry equal electric or magnetic charges. In this paper, we generalise all these solutions to multi-scalar solutions in which the charges become independent free parameters. The mass per unit $p$-volume is equal to the sum of these Page charges. We find that for generic values of the Page charges, they preserve the same fraction of the …
Following the work in \cite{Mohammed:2012gi}, where the massive ABJM model in 2+1 dimensions was shown to have an abelian reduction to the relativistic Landau-Ginzburg, and motivated by the implications for condensed matter through AdS/CFT, we show that a FI deformation of ${\cal N}=4$ SYM in 3+1 dimensions with a mass term can also be reduced to a relativistic Landau-Ginzburg model, with the possibility of coupling it to a real scalar, whereas the simply mass deformed ${\cal N}=4$ SYM reduces only to a massive $ϕ^4$ model (scalar QED) coupled to a real scalar. We study the classical solutions…
We interpret D-strings at the bottom of the warped deformed conifold as axionic strings in the dual cascading SU(N+M) x SU(N) gauge theory. The axion is a massless pseudo-scalar glueball which we find in the supergravity fluctuation spectrum and interpret as the Goldstone boson of spontaneously broken U(1) baryon number symmetry. The existence of this massless glueball, anticipated in hep-th/0101013, supports the idea that the cascading gauge theory is on the baryonic branch, i.e. the U(1)_B global symmetry is broken by expectation values of baryonic operators. We also find a massless scalar g…
We study soliton and black hole solutions of Einstein charged scalar field theory in cavity. We examine the effect of introducing a scalar field mass on static, spherically symmetric solutions of the field equations. We focus particularly on the spaces of soliton and black hole solutions, as well as studying their stability under linear, spherically symmetric perturbations of the metric, electromagnetic field, and scalar field.
We study solitonic solutions of a deformed Wess-Zumino model in 2 dimensions, corresponding to a deformation of the usual ${\cal N}=1,D=2$ superspace to the one with non-anticommuting odd supercoordinates. The deformation turns out to add a kinetic term for the auxiliary field besides the known $F^{3}$ term coming from the deformation of the cubic superpotential. Both these modifications are proportional to the effective deformation parameter $λ\equiv \det C$, where $C$ denotes the non-anticommutativity matrix. We find a modified ``orbit'' equation which on the EOM relates the auxiliary and th…
We observe that the Faddeev-Skyrme model emerges as a low-energy limit of scalar QED with two charged scalar fields and a selfinteraction potential of a special form (inspired by supersymmetric QCD). Then we discuss possible Hopf solitons of the "twisted-toroidal" type.
We consider the quantization of a scalar kappa-deformed field up to the point of obtaining an expression for its vacuum energy. The expression is given by the half sum of the field frequencies, as in the non-deformed case, but with the frequencies obeying the kappa-deformed dispersion relation. We consider a set of kappa-deformed Maxwell equations and show that for the purpose of calculating the Casimir energy the Maxwell field, as in the non-deformed case, behaves as a pair of scalar fields. Those results provide a foundation for computing the Casimir energy starting from the the half sum of …
We show that the quantum solitons occurring in theories describing a complex scalar field in (1+1)-dimensions with a Z(N) symmetry may be identified with sine-Gordon quantum solitons in the phase of this field. Then using both the Euclidean thermal Green function of the two-dimensional free massless scalar field in coordinate space and its dual, we obtain an explicit series expression for the corresponding solitonic correlation function at finite temperature.
We compute the Hagedorn temperature of $μT \bar T + \varepsilon_+ J \bar T + \varepsilon_-T \bar J$ deformed CFT using the universal kernel formula for the thermal partition function. We find a closed analytic expression for the free energy and the Hagedorn temperature as a function of $μ$, $\varepsilon_+$, and $\varepsilon_-$ for the case of a compact scalar boson by taking the large volume limit. We also compute the Hagedorn temperature for the single trace deformed $AdS_3 \times S^1 \times T^3 \times S^3$ using holographic methods. We identify black hole configurations whose thermodynamics …
The bare one loop soliton quantum mass corrections can be expressed in two ways: as a sum over the zero-point energies of small oscillations around the classical configuration, or equivalently as the (Euclidean) effective action per unit time. In order to regularize the bare one loop quantum corrections (expressed as the sum over the zero-point energies) we subtract and add from it the tadpole graph that appear in the expansion of the effective action per unit time. The subtraction renders the one loop quantum corrections finite. Next, we use the renormalization prescription that the added tad…
