Complex sine-Gordon Research Articles

Towards a quadratic Poisson algebra for the subtracted classical monodromy of symmetric space sine-Gordon theories

Symmetric space sine-Gordon theories are two-dimensional massive integrable field theories, generalising the sine-Gordon and complex sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution.

On bosonic Thirring model in Minkowski signature

We present the way to continue the bosonic Thirring model or βγ-system with quartic interaction to Minkowski signature, based on the symmetries of this model. It is shown that the considered Minkowski version of the model is one-loop renormalizable. Based on this, we find the amplitudes of scattering of the excitations corresponding to the γ and γ¯ fields up to the one-loop order. In particular, it was computed that the 2→2 amplitudes of these excitations possess property of reflectionless scattering and thus the corresponding S-matrix of such excitations satisfies the Yang-Baxter equation. The obtained S-matrix elements for γ and γ¯ are shown to coincide with the corresponding S-matrix elements of the solitons in the complex sine-Gordon model proposed by Dorey and Hollowood.

Classical solutions of the degenerate fifth Painlevé equation

In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to be expressible in terms of solutions of the third Painlevé equation. The classification and description of the classical solutions of the degenerate fifth Painlevé equation is done using the Hamiltonian associated with third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.

The two-component complex coupled integrable dispersionless equations: Darboux transformation and soliton solutions

The complex coupled integrable dispersionless (CID)system is an important system. It has been shown that complex CID equation is gauge equivalent to the Pohlmeyer–Lund–Regge model (a complex sine-Gordon equation) and can also be transformed to the complex short pulse equation. CID system can describe a current-fed string interacting with an external magnetic field. In this paper, we investigate the two-component complex coupled integrable dispersionless system. By using the Darboux transformation, we obtain a new type of solution (the breather-like soliton solution) with zero boundary condition and a large number of solutions (the breather-II type solution, bright–dark soliton solution, the resonance solutions and the semi-rational solutions) with non-zero boundary condition. In particular, these solutions are not available for the scalar complex integrable dispersionless system.

Non-Abelian Toda field theories from a 4D Chern-Simons theory

We derive non-abelian Toda field theories (NATFTs) from a 4d Chern-Simons (CS) theory with two order defects by employing a certain asymptotic boundary condition. The 4d CS theory is characterized by a meromorphic 1-form ω. We adopt ω with two simple poles and no zeros, and each of the order defects is located at each pole. As a result, an anisotropy parameter β2 can be identified with the distance between the two defects. As examples, we can derive the (complex) sine-Gordon model and the Liouville theory.

Algebraic structure of classical integrability for complex sine-Gordon

The algebraic structure underlying the classical rr-matrix formulation of the complex sine-Gordon model is fully elucidated. It is characterized by two matrices aa and ss, components of the rr matrix as r=a-sr=a−s. They obey a modified classical reflection/Yang–Baxter set of equations, further deformed by non-abelian dynamical shift terms along the dual Lie algebra su(2)^*su(2)*. The sign shift pattern of this deformation has the signature of the twisted boundary dynamical algebra. Issues related to the quantization of this algebraic structure and the formulation of quantum complex sine-Gordon on those lines are introduced and discussed.

Complex short-pulse solutions by gauge transformation

In this article we apply the gauge transformation method for solving the Heisenberg ferromagnet equation by using the NLS Jost solutions to derive solutions of the complex short-pulse equation from the Jost functions of the complex sine-Gordon equation. The matrix triplet method is used to compute the most general reflectionless complex short-pulse solutions by solving the Marchenko integral equation for the complex sine-Gordon equation.

Complex solitons with real energies

Using Hirota’s direct method and Bäcklund transformations we construct explicit complex one and two-soliton solutions to the complex Korteweg–de Vries (KdV) equation, the complex modified KdV (mKdV) equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be -symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex -symmetric solutions to the KdV equation may also be constructed alternatively from real solutions to the mKdV by means of Miura transformations. The multi-soliton solutions obtained from Hirota’s method break the -symmetric, whereas those obtained from Bäcklund transformations are -invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are -symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space–time and the fact that any of our N-soliton solutions may be decomposed into N separate -symmetrizable one-soliton solutions.

Symmetries and soliton solutions of the Galilean complex Sine-Gordon equation

We discuss a new equation, the Galilean version of the complex Sine-Gordon equation in 1+1 dimensions, Ψxx(1−Ψ⁎Ψ)+2imΨt+Ψ⁎Ψx2−Ψ(1−Ψ⁎Ψ)2=0, derived from its relativistic counterpart via Galilean covariance. We determine its Lie point symmetries, discuss some group-invariant solutions, and examine some soliton solutions. The reduction under Galilean symmetry leads to an equation similar to the stationary Gross–Pitaevskii equation. This work is motivated in part by recent applications of the relativistic complex Sine-Gordon equation to the dynamics of Q-balls.

On deformations of AdS n × S n supercosets

We study the deformed AdS 5 × S 5 supercoset model of arXiv:1309.5850 which depends on one parameter κ and has classical quantum group symmetry. We confirm the conjecture that in the “maximal” deformation limit, κ → ∞, this model is T-dual to “flipped” double Wick rotation of the target space AdS 5 × S 5, i.e. dS 5 × H 5 space supported by an imaginary 5-form flux. In the imaginary deformation limit, κ → i, the corresponding target space metric is of a pp-wave type and thus the resulting light-cone gauge S-matrix becomes relativistically invariant. Omitting non-unitary contributions of imaginary WZ terms, we find that this tree-level S-matrix is equivalent to that of the generalized sine-Gordon model representing the Pohlmeyer reduction of the undeformed AdS 5 × S 5 superstring model. We also study in some detail similar deformations of the AdS 3 × S 3 and AdS 2 × S 2 supercosets. The bosonic part of the deformed AdS 3 × S 3 model happens to be equivalent to the symmetric case of the sum of the Fateev integrable deformation of the SL(2) and SU(2) principal chiral models, while in the AdS 2 × S 2 case the role of the Fateev model is played by the 2d “sausage” model. The κ = i limits are again directly related to the Pohlmeyer reductions of the corresponding AdS n × S n supercosets: (2,2) super sine-Gordon model and its complex sine-Gordon analog. We also discuss possible deformations of AdS 3 × S 3 with more than one parameter.

Bi-dimensional soliton-like solutions of the nonlinear complex sine-Gordon system

Bi-dimensional soliton-like solutions of the nonlinear complex sine-Gordon system

Integrability vs supersymmetry: Poisson structures of the pohlmeyer reduction

We construct recursively an infinite number of Poisson structures for the su- persymmetric integrable hierarchy governing the Pohlmeyer reduction of superstring sigma models on the target spaces AdS n × S n , n = 2,3,5. These Poisson structures are all non-local and not relativistic except one, which is the canonical Poisson structure of the semi-symmetric space sine-Gordon model (SSSSG). We verify that the superposition of the first three Poisson structures corresponds to the canonical Poisson structure of the reduced sigma model. Using the recursion relations we construct commuting charges on the reduced sigma model out of those of the SSSSG model and in the process we explain the integrable origin of the Zukhovsky map and the twisted inner product used in the sigma model side. Then, we compute the complete Poisson superalgebra for the conserved Drinfeld-Sokolov supercharges associated to an exotic kind of extended non-local rigid 2d supersymmetry recently introduced in the SSSSG context. The superalgebra has a kink central charge which turns out to be a generalization to the SSSSG models of the well-known central extensions of the N = 1 sine-Gordon and N = 2 complex sine-Gordon model Poisson superalgebras computed from 2d superspace. The computation is done in two different ways concluding the proof of the existence of 2d supersymmetry in the reduced sigma model phase space under the boost invariant SSSSG Poisson structure.

On the perturbative S-matrix of generalized sine-Gordon models

Motivated by its relation to the Pohlmeyer reduction of AdS_5 x S^5 superstring theory we continue the investigation of the generalized sine-Gordon model defined by SO(N+1)/SO(N) gauged WZW theory with an integrable potential. Extending our previous work (arXiv:0912.2958) we compute the one-loop two-particle S-matrix for the elementary massive excitations. In the N = 2 case corresponding to the complex sine-Gordon theory it agrees with the charge-one sector of the quantum soliton S-matrix proposed in hep-th/9410140. In the case of N > 2 when the gauge group SO(N) is non-abelian we find a curious anomaly in the Yang-Baxter equation which we interpret as a gauge artifact related to the fact that the scattered particles are not singlets under the residual global subgroup of the gauge group.

Scattering and duality in the 2 dimensional OSp(2|2) gross neveu and sigma models

We write the thermodynamic Bethe ansatz for the massive OSp(2|2) Gross Neveu and sigma models. We find evidence that the GN S matrix proposed by Bassi and Leclair [12] is the correct one. We determine features of the sigma model S matrix, which seem highly unconventional; we conjecture in particular a relation between this sigma model and the complex sine-Gordon model at a particular value of the coupling. We uncover an intriguing duality between the OSp(2|2) GN (resp. sigma) model on the one hand, and the SO(4) sigma (resp. GN model) on the other, somewhat generalizing to the massive case recent results on OSp(4|2). Finally, we write the TBA for the (SUSY version of the) flow into the random bond Ising model proposed by Cabra et al. [43], and conclude that their S matrix cannot be correct.

A new and elementary ℂPn} Dyonic magnon

We show that the dressing transformation method produces a new type of dyonic Pn magnon in terms of which all the other known solutions are either composites or arise as special limits. In particular, this includes the embedding of Dorey's dyonic magnon via P3⊂Pn. We also show how to generate Dorey's dyonic magnon directly in the Sn sigma model via the dressing method without resorting to the isomorphism with the SU(2) principle chiral model when n = 3. The new dyon is shown to be either a charged dyon or topological kink of the related symmetric-space sine-Gordon theories associated to Pn and in this sense is a direct generalization of the soliton of the complex sine-Gordon theory.

Defects and dressed boundaries in complex sine-Gordon theory

In this paper we constructed integrable defects in complex sine-Gordon theory. Soliton and particle interactions with the defect are analysed. Defects are used to dress Dirichlet boundaries to create a wider class of integrable boundary conditions to ones previously studied. Properties of these new boundaries including soliton and particle reflections with the boundaries are presented.

Quantum complex sine-Gordon dressed boundaries

In this paper we investigate the quantum reflection factor for the CSG dressed boundary, previously constructed by dressing the Dirichlet boundary with the integrable CSG defect [1]. We analyse classical bound states and use semi-classical methods to investigate the quantum boundary spectrum. We conjecture a fully quantum reflection matrix for a particle reflecting from an unexcited boundary. By using the reflection and boundary bootstrap equations, the reflection matrix for a charge Q = +n soliton reflecting from the mth excited boundary is constructed. Evidence supporting our conjecture is given by checking that the bootstrap closes and that the reflection matrices agrees with known results in the classical limit. A partial analysis of the poles in the reflection matrices which arise from Coleman-Thun diagrams is given.

Finite-size effects of membranes onAdS4×S7

We consider semi-classical solution of membranes on the AdS_4 x S^7. This is supposed to be dual to the N=6 super Chern-Simons theory with k=1 in a planar limit recently proposed by Aharony, Bergmann, Jafferis, and Maldacena (ABJM). We have identified giant magnon and single spike states on the membrane by reducing them to the Neumamm - Rosochatius integrable system. We also connect these to the complex sine-Gordon integrable model. Based on this approach, we find finite-size membrane solutions and obtain their images in the complex sine-Gordon system along with the leading finite-size corrections to the energy-charge relations.

Finite-size effects for single spike

We use the reduction of the string dynamics on R_t x S^3 to the Neumann-Rosochatius integrable system to map all string solutions described by this dynamical system onto solutions of the complex sine-Gordon integrable model. This mapping relates the parameters in the solutions on both sides of the correspondence. In the framework of this approach, we find finite-size string solutions, their images in the (complex) sine-Gordon system, and the leading finite-size effects of the single spike E-\Delta\phi relation for both R_t x S^2 and R_t x S^3 cases.

REDUCED MODEL FOR SUPERSTRINGS ON AdSn × Sn

We review the Pohlmeyer reduction of the superstring sigma model on AdSn × Sn leading to a gauged WZW model with an integrable potential. In particular, we consider the case of GS superstrings on AdS3 × S3 supported by RR flux. The bosonic part of the reduced Lagrangian is given by the sum of the complex sine-Gordon Lagrangian and its sinh-Gordon counterpart. We determine the corresponding fermionic part and discuss possible 2d supersymmetry of the reduced action.