Symmetric Space sine-Gordon Theories 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.

The semi-classical spectrum of solitons and giant magnons

In this note, we summarize recent progress in constructing and then semi-classically quantizing solitons, or non-abelian Q-balls, in the symmetric space sine-Gordon theories. We then consider the images of these solitons in the related constrained sigma model, which are the dyonic giant magnons on the string theory world-sheet. Focussing on the case of the symmetric space S 5, we perform a semi-classical quantization of the solitons and magnons and show that both lead to Chern-Simons quantum mechanics on the internal moduli space which is a real Grassmannian SO(4)/SO(2) × SO(2) but — importantly — with a different coupling constant. Quantizing this system shows that both the Q-balls and magnons come in a tower of states transforming in symmetric representations of the SO(4) symmetry group; however, the former come in a finite tower whereas the latter come in the well-known infinite tower of dyonic giant magnons.

The AdS5 × S5 semi-symmetric space sine-Gordon theory

The generalized symmetric space sine-Gordon theories are a series of 1+1-integrable field theories that are classically equivalent to superstrings on symmetric space spacetimes F/G. They are formulated in terms of a semi-symmetric space as a gauged WZW model with fermions and a potential term to deform it away from the conformal fixed point. We consider in particular the case of PSU(2,2|4)/Sp(2,2)xSp(4) which corresponds to AdS(5)xS(5). We argue that the infinite tower of conserved charges of these theories includes an exotic N=(8,8) supersymmetry that is realized in a mildy non-local way at the Lagrangian level. The supersymmetry is associated to a double central extension of the superalgebra psu(2|2)+psu(2|2) and includes a non-trivial R symmetry algebra corresponding to global gauge transformations, as well as 2-dimensional spacetime translations. We then explicitly construct soliton solutions and show that they carry an internal moduli superspace CP(2|1)xCP(2|1) with both bosonic and Grassmann collective coordinates. We show how to semi-classical quantize the solitons by writing an effective quantum mechanical system on the moduli space which takes the form of a co-adjoint orbit of SU(2|2)xSU(2|2). The spectrum consists of a tower of massive states in the short, or atypical, symmetric representations, just as the giant magnon states of the string world sheet theory, although here the tower is truncated.

Classical and quantum solitons in the symmetric space sine-Gordon theories

We construct the soliton solutions in the symmetric space sine-Gordon theories. The latter are a series of integrable field theories in 1+1-dimensions which are associated to a symmetric space F/G, and are related via the Pohlmeyer reduction to theories of strings moving on symmetric spaces. We show that the solitons are kinks that carry an internal moduli space that can be identified with a particular co-adjoint orbit of the unbroken subgroup H of G. Classically the solitons come in a continuous spectrum which encompasses the perturbative fluctuations of the theory as the kink charge becomes small. We show that the solitons can be quantized by allowing the collective coordinates to be time-dependent to yield a form of quantum mechanics on the co-adjoint orbit. The quantum states correspond to symmetric tensor representations of the symmetry group H and have the interpretation of a fuzzy geometric version of the co-adjoint orbit. The quantized finite tower of soliton states includes the perturbative modes at the base.

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.

Magnons, their solitonic avatars and the Pohlmeyer reduction

We study the solitons of the symmetric space sine-Gordon theories that arise once the Pohlmeyer reduction has been imposed on a sigma model with the symmetric space as target. Under this map the solitons arise as giant magnons that are relevant to string theory in the context of the AdS/CFT correspondence. In particular, we consider the cases Sn, Pn and SU(n) in some detail. We clarify the construction of the charges carried by the solitons and also address the possible Lagrangian formulations of the symmetric space sine-Gordon theories. We show that the dressing, or Backlund, transformation naturally produces solitons directly in both the sigma model and the symmetric space sine-Gordon equations without the need to explicitly map from one to the other. In particular, we obtain a new magnon solution in P3. We show that the dressing method does not produce the more general ``dyonic'' solutions which involve non-trivial motion of the collective coordinates carried by the solitons.

Massive symmetric space sine-Gordon soliton theories and perturbed conformal field theory

The perturbed conformal field theories corresponding to the massive Symmetric Space sine-Gordon soliton theories are identified by calculating the central charge of the unperturbed conformal field theory and the conformal dimension of the perturbation. They are described by an action with a positive-definite kinetic term and a real potential term bounded from below, their equations of motion are non-abelian affine Toda equations and, moreover, they exhibit a mass gap. The unperturbed CFT corresponding to the compact symmetric space G/G_0 is either the WZNW action for G_0 or the gauged WZNW action for a coset of the form G_0/U(1)^p. The quantum integrability of the theories that describe perturbations of a WZNW action, named Split models, is established by showing that they have quantum conserved quantities of spin +3 and -3. Together with the already known results for the other massive theories associated with the non-abelian affine Toda equations, the Homogeneous sine-Gordon theories, this supports the conjecture that all the massive Symmetric Space sine-Gordon theories will be quantum integrable and, hence, will admit a factorizable S-matrix. The general features of the soliton spectrum are discussed, and some explicit soliton solutions for the Split models are constructed. In general, the solitons will carry both topological charges and abelian Noether charges. Moreover, the spectrum is expected to include stable and unstable particles.