Magnons, their Solitonic Avatars and the Pohlmeyer Reduction
aa r X i v : . [ h e p - t h ] F e b Preprint typeset in JHEP style - HYPER VERSION
Magnons, their Solitonic Avatars and thePohlmeyer Reduction
Timothy J. Hollowood
Department of Physics,University of Wales Swansea,Swansea, SA2 8PP, UK.E-mail: [email protected] and J. Luis Miramontes
Departamento de F´ısica de Part´ıculas and IGFAE,Universidad de Santiago de Compostela15782 Santiago de Compostela, SpainE-mail: [email protected]
Abstract:
We study the solitons of the symmetric space sine-Gordon theories thatarise once the Pohlmeyer reduction has been imposed on a sigma model with thesymmetric space as target. Under this map the solitons arise as giant magnons thatare relevant to string theory in the context of the AdS/CFT correspondence. Inparticular, we consider the cases S n , C P n and SU ( n ) in some detail. We clarifythe construction of the charges carried by the solitons and also address the possibleLagrangian formulations of the symmetric space sine-Gordon theories. We showthat the dressing, or B¨acklund, transformation naturally produces solitons directlyin both the sigma model and the symmetric space sine-Gordon equations withoutthe need to explicitly map from one to the other. In particular, we obtain a newmagnon solution in C P . We show that the dressing method does not produce themore general “dyonic” solutions which involve non-trivial motion of the collectivecoordinates carried by the solitons. . Introduction The AdS/CFT correspondence [1] is remarkable in so many ways. For example,there is an underlying integrable structure that allows one to interpolate from weakto strong coupling, and which enables many quantitave checks of the conjecturedduality by exploring both sides of the correspondence (see [2] and the referencestherein). An example is provided by the “giant magnons” and the “dyonic giantmagnons”, introduced by Hofman and Maldacena [3] and Dorey [4], respectively.They describe string configurations on curved space-times of the form R t × M , with M = F/G a symmetric space; for example, S n = SO ( n + 1) /SO ( n ). Then, theclassical motion of the string is described by a sigma model with target space M ,and the Virasoro constraints, in a particular gauge, lead to the Pohlmeyer reductionof that sigma model [5, 6]. In turn, this gives rise to an associated integrable systemthat is a generalization of the sine-Gordon theory. These are the symmetric spacesine-Gordon theories (SSSG), and giant magnons can be mapped into the solitonsolutions to their equations-of-motion. Moreover, when the symmetric space is ofindefinite signature, like AdS n = SO (2 , n − /SO (1 , n − M × S ,or even M . In the context of the AdS/CFT correspondence, giant magnons havebeen extensively used to study many aspects of superstrings in certain subspaces of AdS × S [3, 7, 8] and AdS × C P [9–11].This work is a companion to [12], which provided a systematic study of thegroup theoretical interpretation of the Pohlmeyer reduction and the associated SSSGtheories for symmetric spaces of definite, or indefinite, signature. The present workextends this to a discussion and construction of a class of soliton solutions usingthe dressing transformation method [13]. An important result that we establish isthat the dressing method produces both the giant magnon and its soliton avatar inthe SSSG theory at the same time, without the need to map one to other via thePohlmeyer constraints. This is particularly useful because, in general, it is not easyto perform the map.For cases including S n the giant magnon solutions produced by the dressingmethod have been studied in [14] and they correspond to embeddings of the Hofman–Maldacena giant magnon [3] associated to S ⊂ S . One major shortcoming of thedressing method is that, in the context of S n = SO ( n +1) /SO ( n ), it does not producethe Dorey’s dyonic giant magnon [4]. Nevertheless, this more general solution canbe constructed by the dressing procedure by using the alternative formulation of S as the symmetric space SU (2) × SU (2) /SU (2), which is isomorphic to the Liegroup SU (2). Embedding this solution back in the original formulation in terms– 1 –f the symmetric space S n = SO ( n + 1) /SO ( n ) shows that the dyonic solutioninvolves non-trivial geodesic motion in the space of collective coordinates carried bythe magnon/soliton. In this sense, the solutions have much in common with thedyonic generalization of the monopole in four-dimensional gauge theories coupled toan adjoint Higgs field (for example see [16].Other examples that we consider in this work are the complex projective spaces C P n , which are realized as the symmetric spaces SU ( n + 1) /U ( n ). The case C P is relevant to the AdS/CFT correspondence involving a spacetime AdS × C P [17].The known giant magnon solutions for this case [9–11, 18, 19] have all been obtainedfrom the Hofman-Maldacena solution and the dyonic generalization of Dorey viaembeddings of S and S in C P n , respectively. Our results provide a class of newmagnon/soliton solutions which cannot be obtained from embeddings of those for S n .In addition, we show that there should exist an equivalent class of dyonic solutionsin addition to the embeddings of Dorey’s dyon.Finally, we consider the SU ( n ) principal chiral models, which can be formulatedas a symmetric space SU ( n ) × SU ( n ) /SU ( n ). For n >
2, these models admit sev-eral non-equivalent Pohlmeyer reductions and, therefore, they give rise to differentSSSG theories. In this work we only consider the simplest cases, which correspondto the (parity invariant) homogeneous sine-Gordon theories [20]. The solitons ofthese theories have been studied in [21] using a different formulation of the dressingtransformation method based on representations of affine Lie algebras. Our resultsprovide new expressions for them involving collective coordinates that clarify theircomposite nature in terms of basic SU (2) solitons. More general reductions of theprincipal chiral models will be discussed elsewhere.Notice that all the examples that we consider involve symmetric spaces of definitesignature. In future work we will describe the generalization to symmetric spaces ofindefinite signature relevant to discussing AdS n , for example.The plan of the paper is a follows. In Section 2, we will formulate the sigmamodel with target space a symmetric space F/G in terms of a constrained F -valuedfield without introducing gauge fields. The relationship between this formulation andthe approach used in [12] is summarized in Appendix A. Using that formulation, inSection 3 we will describe the Pohlmeyer reduction of the sigma model, and recoverthe formulation of the SSSG equations as zero-curvature conditions on a left-rightasymmetric coset of the form G/H ( − ) L × H (+) R proposed in [12]. We will also addressthe possible Lagrangian formulations of these equations and clarify their symmetriesand conserved quantities, which play an important rˆole in the description of thesoliton solutions. In Section 4, we will review the already known giant magnons in– 2 –he context of S n and C P n , and we will discuss the relation between them and theirrelativistic SSSG solitonic avatars. In Section 5, we will use the dressing transforma-tion method to construct magnons and solitons following the approach of [22]. Animportant result of this section is that the dressing transformation is compatible withthe Pohlmeyer reduction, and that this method provides directly both the magnonand its SSSG soliton without the need to map one into the other. In Section 6, wewill apply the method to the SU ( n ) principal chiral model, and for n = 2 we willrecover Dorey’s dyonic giant magnon. In Sections 7 and 8, we will apply the methodto C P n and S n , respectively. In Section 9, we discuss the possibility of finding so-lutions similar to Dorey’s dyonic giant magnon by making the collective coordinatestime dependent. Finally, Section 10 contains our conclusions, and there are fourappendices.
2. Symmetric Space Sigma Model
Our story begins with a sigma model in 1 + 1 dimensions whose target space isa symmetric space, that is a quotient of two Lie groups
F/G equipped with aninvolution σ − of F that fixes G ⊂ F : σ − ( g ) = g , ∀ g ∈ G . (2.1)Acting on f , the Lie algebra of F , the automorphism σ − gives rise to the canonicalorthogonal decomposition f = g ⊕ p , with [ g , g ] ⊂ g , [ g , p ] ⊂ p , [ p , p ] ⊂ g , (2.2)where g and p are the +1 and − σ − , respectively, and g is the Liealgebra of G . Then, the sigma model with target space a symmetric space can bedescribed as a sigma model with a field f ∈ F where the G action f → f g − , g ∈ G , is gauged. For instance, this is the approach described in the prequel [12](see also [23]). However, for present purposes, we find it more convenient to workdirectly in the coset F/G by defining the F -valued field F = σ − ( f ) f − (2.3)and working directly with F instead of f , which we can think of as an F -valued fieldsubject to the constraint σ − ( F ) = F − . (2.4)In this formalism there is no need to introduce gauge fields and this simplificationturns out to be useful. Of course, our approach in terms of involutions can easily betranslated into the gauged sigma model language if need be (see Appendix A).– 3 –e will also consider the principal chiral model which can either be consideredas a symmetric space G × G/G as above, with σ − being the involution that exchangesthe two G factors, or we can simply take the target space to be F = G itself, in whichcase the involution σ − is not required. We will take the latter point of view in whatfollows.The Lagrangian of the sigma model is simply L = − κ Tr J µ J µ , (2.5)where J µ = ∂ µ F F − . (2.6)Note that F → σ − ( F ) = F − is a symmetry of the action and equations-of-motionand therefore it is consistent to impose it by hand on the field F from the start and,as mentioned above, in this formalism there are no gauge fields. The equations-of-motion for the group field are ∂ µ J µ = 0 . (2.7)In other words, J µ provides the conserved currents corresponding to the global F L × F R symmetry of the sigma model with target space F (the principal chiral model)under which F → U F V for any U, V ∈ F . The left and right currents are J Lµ = ∂ µ F F − = J µ and J Rµ = F − ∂ µ F = F − J µ F , (2.8)and we can define the corresponding conserved charges Q L = Z ∞−∞ dx ∂ F F − and Q R = Z ∞−∞ dx F − ∂ F . (2.9)In the principal chiral model these charges are independent. However, in the F/G models the F L × F R symmetry is reduced by the constraint (2.4) so that they areinvariant only under F → σ − ( U ) F U − with U ∈ F . Taking (A.4) into account, thesetransformations correspond to f → U f , which specifies the global symmetries of thesymmetric space sigma model in the gauged sigma model language. Consequently,in the
F/G models the two charges are related by σ − ( Q L ) = −Q R .Since J ± = ∂ ± F F − , these currents trivially satisfy the Cartan-Maurer condi-tions ∂ + J − − ∂ − J + − [ J + , J − ] = 0 . (2.10)Then, the equations-of-motion (2.7), along with the identity (2.10), can be writtenin the form of a zero curvature condition: h ∂ + − J + λ , ∂ − − J − − λ i = 0 , (2.11) In out notation x + = t + x and x − = t − x – 4 –here λ is a spectral parameter. The residues at λ = ± ∓ ∂ ± J ∓ + [ J + , J − ] = 0 , (2.12)respectively, which are equivalent to (2.7) and (2.10).
3. The Pohlmeyer Reduction
The Pohlmeyer reduction, at an algebraic level, involves imposing the conditions (seeAppendix A and [12]) ∂ ± F F − = f ± Λ ± f − ± , (3.1)where Λ ± are constant elements in a maximal abelian subspace a of p in (2.2) and f ± ∈ F . The natural degree-of-freedom left after the reduction is γ = f − − f + whichis valued in G ⊂ F . In order to see this, we act on (3.1) with σ − . The left-handsides become σ − (cid:0) ∂ ± F F − (cid:1) = ∂ ± F − F = −F − ∂ ± F (3.2)while the right-hand sides transform into σ − (cid:0) f ± Λ ± f − ± (cid:1) = − σ − ( f ± )Λ ± σ − ( f − ± ) . (3.3)The two can be made consistent by requiring σ − ( f ± ) = F − f ± , (3.4)and so σ − ( γ ) = f − − F F − f + = γ , (3.5)which shows that γ ∈ G . Actually, it is clear that f ± are ambiguous since we couldalways right-multiply by the group of elements that commute with Λ ± , respectively.We shall soon see that this freedom leads to a gauge symmetry in the reduced model.Notice that once the reduction has been imposed the “left” F charges can be written Q L = Z ∞−∞ dx (cid:0) f + Λ + f − + f − Λ − f − − (cid:1) . (3.6)Using (2.12) with J ± = f ± Λ ± f − ± , we have (cid:2) f − ∂ − f + − γ − Λ − γ, Λ + (cid:3) = 0 . (3.7) In [12] the right-hand side had scales multipliers µ ± . In the present work, we will not indicatethese factors. We can either re-introduce them by scaling x ± , or one can think of them as havingbeen absorbed into Λ ± (see Appendix A). – 5 –his implies that f − ∂ − f + − γ − Λ − γ = A ( R ) − (3.8)where A ( R ) − is an unknown element that satisfies [ A ( R ) − , Λ + ] = 0 and, using (3.4), σ − (cid:0) A ( R ) − (cid:1) = A ( R ) − . Therefore, A ( R ) − takes values in h + , which is the Lie algebra of thesubgroup H (+) ⊂ G of elements that commute with Λ + . Similarly, we have (cid:2) − f − ∂ + f + + γ − ∂ + γ + 12 Λ + , γ − Λ − γ (cid:3) = 0 , (3.9)which implies that f − ∂ + f + − γ − ∂ + γ −
12 Λ + = γ − A ( L )+ γ . (3.10)Here, [ A ( L )+ , Λ − ] = 0 and, using (3.4) once more, σ − (cid:0) A ( L )+ (cid:1) = A ( L )+ . This shows that A ( L )+ ∈ h − , which is the Lie algebra of the subgroup H ( − ) ⊂ G of elements thatcommute with Λ − .On the other hand, the integrability condition for (3.1) implies (cid:2) ∂ + − f + Λ + f − , ∂ − − f − Λ − f − − (cid:3) = 0 , (3.11)from which we deduce (cid:2) ∂ + + f − ∂ + f + − Λ + , ∂ − + f − ∂ − f + − γ − Λ − γ (cid:3) = 0 . (3.12)Using (3.8) and (3.10), it gives (cid:2) ∂ + + γ − ∂ + γ + γ − A ( L )+ γ −
12 Λ + , ∂ − + A ( R ) − − γ − Λ − γ (cid:3) = 0 , (3.13)which are the zero-curvature form of the Symmetric Space sine-Gordon (SSSG)equations-of-motion. Notice that, as a consequence of (3.1), this set of equationshas a natural H ( − ) L × H (+) R gauge symmetry under which f ± −→ f ± h − ± , (3.14)where h ± are local group elements in the subgroups H ( ± ) ⊂ G . Under this symmetry γ −→ h − γh − (3.15)and A ( R ) − −→ h + (cid:0) A ( R ) − + ∂ − (cid:1) h − , A ( L )+ −→ h − (cid:0) A ( L )+ + ∂ + (cid:1) h − − . (3.16)This is exactly the result of [12]. – 6 –he SSSG equations (3.13) are integrable and lead to an infinite set of con-served quantities which, as discussed in Appendix B, include charges correspondingto the global part of the gauge group, and the energy and momentum. Since theconserved charges play an important rˆole, we will describe their construction in somedetail. First of all, by projecting (3.13) onto h + and γ · · · γ − onto h − yields the zerocurvature conditions [ ∂ + + A ( R/L )+ , ∂ − + A ( R/L ) − ] = 0 , (3.17)where we have defined the “missing” components of the gauge connections, A ( R )+ = P h + (cid:0) γ − ∂ + γ + γ − A ( L )+ γ (cid:1) ,A ( L ) − = P h − (cid:0) − ∂ − γγ − + γA ( R ) − γ − (cid:1) . (3.18)Eq. (3.17) gives rise to the conserved quantities associated to the global versionof the H ( − ) L × H (+) R gauge transformations. Moreover, it enables the gauge fixingconditions that relate the SSSG equations to the non-abelian affine Toda equations( A ( R/L ) ± = A ( R/L ) ∓ = 0), and the gauge fixing conditions required for their Lagrangianformulation (see (3.34)). Then, it is important to notice that (3.17) holds providedthat Λ ± give rise to the orthogonal decompositions f = Ker (cid:0) Ad Λ ± (cid:1) ⊕ Im (cid:0) Ad Λ ± (cid:1) (3.19)and, consequently, that (cid:2) Ker (cid:0) Ad Λ ± (cid:1) , Ker (cid:0) Ad Λ ± (cid:1)(cid:3) ⊂ Ker (cid:0) Ad Λ ± (cid:1) , (cid:2) Ker (cid:0) Ad Λ ± (cid:1) , Im (cid:0) Ad Λ ± (cid:1)(cid:3) ⊂ Im (cid:0) Ad Λ ± (cid:1) . (3.20)This is always true if the symmetric space F/G is of definite signature ( G compact),which is the only case considered in this work. An example were the decomposi-tion (3.19) is not satisfied is provided by the “ lightlike ” Pohlmeyer reduction of thesigma model with target space AdS n discussed in [12].Under the gauge transformations (3.15)–(3.16), A ( R ) ± −→ h + (cid:0) A ( R ) ± + ∂ ± (cid:1) h − , A ( L ) ± −→ h − (cid:0) A ( L ) ± + ∂ ± (cid:1) h − − . (3.21)Then, in order to construct gauge invariant conserved quantities, we will trans-form (3.17) into gauge invariant equations. First of all, we choose a gauge slice γ such that any field γ can be written as γ = φ L γ φ − R , (3.22)with φ L ∈ H ( − ) and φ R ∈ H (+) . Under gauge transformations, γ remains invariantwhile φ L → h − φ L and φ R → h + φ R . Then, it can be easily checked that˜ A ( R ) ± = φ − R (cid:0) A ( R ) ± + ∂ ± (cid:1) φ R and ˜ A ( L ) ± = φ − L (cid:0) A ( L ) ± + ∂ ± (cid:1) φ L (3.23)– 7 –re gauge invariant and, moreover, that[ ∂ + + ˜ A ( R/L )+ , ∂ − + ˜ A ( R/L ) − ] = 0 . (3.24)In general, H ( ± ) will be of the form U (1) p ± × H ( ± )ss , where p ± are positive integersand H ( ± )ss are semi-simple factors. This allows one to write φ R/L = e α R/L ϕ R/L , (3.25)where e α R/L ∈ U (1) p ± and ϕ R/L ∈ H ( ± )ss . Then, the projection of (3.24) on the Liealgebras of H ( ± )ss and U (1) p ± provide two different types of gauge invariant conservedquantities. Namely, the projection of (3.24) on the Lie algebra of U (1) p ± shows thatthe currents J µR/L = ǫ µν P u (1) p ± (cid:0) ˜ A ( R/L ) ν (cid:1) = ǫ µν (cid:16) P u (1) p ± (cid:0) A ( R/L ) ν (cid:1) + ∂ ν α R/L (cid:17) (3.26)are conserved. They lead to the “local” gauge invariant conserved quantities Q R/L = Z + ∞−∞ dx J R/L = α R/L (+ ∞ ) − α R/L ( −∞ ) + Z + ∞−∞ dx P u (1) p ± (cid:0) A ( R/L )1 (cid:1) (3.27)which take values in the (abelian) Lie algebra of U (1) p ± . On the other hand, theprojection of (3.24) on the Lie algebra of H ( ± )ss provide the “non-local” conservedquantities given by the path ordered exponentialsΩ R/L = P exp (cid:18) − Z ∞−∞ dx P h ( ± )ss (cid:16) ˜ A ( R/L )1 (cid:17)(cid:19) = ϕ − R/L (+ ∞ ) P exp (cid:18) − Z ∞−∞ dx P h ( ± )ss (cid:16) A ( R/L )1 (cid:17)(cid:19) ϕ R/L ( −∞ ) , (3.28)which take values in H ( ± )ss . Notice that the conserved charges (3.27) and (3.28) are not the same as the conserved charges of the original sigma model Q R and Q L . Inparticular, the former are Lorentz invariant (see Appendix B) while the latter are not.In certain circumstances, and in particular for the soliton solutions, it can transpirethat for particular configurations P h ( ± )ss (cid:0) A ( R/L ) ± (cid:1) take values in an abelian subalgebraof h ( ± )ss , and ϕ R/L in the corresponding abelian subgroup of H ( ± )ss (for all x ). In thiscase, the path ordering in (3.28) is unnecessary and we can write Ω R/L = exp Q (ss) R/L for abelian charges Q (ss) R/L taking values in the relevant abelian subalgebras of h ( ± )ss .It is worth remarking that φ L and φ R are subject to an ambiguity whenever aparticular field configuration is invariant under a certain subgroup of H ( − ) L × H (+) R .As an example, consider the vacuum configuration itself, γ = 1 with A ( R/L ) ± = 0.– 8 –t is invariant under the global vector subgroup of H ( − ) L × H (+) R , which means that φ L and φ R are uniquely defined only modulo φ L → φ L U and φ R → φ R U , with U ∈ H ( − ) L ∩ H (+) R . Consequently, the local charges carried by the vacuum solutionare unambiguously defined only up to Q L ∼ Q L + ρ and Q R ∼ Q R + ρ , for ρ ∈ u (1) p − ∩ u (1) p + . So, in a sense, only the combination Q L − Q R is an unambiguouslywell-defined charge. The significance of this and its relation to spontaneous symmetrybreaking will become clearer when we discuss the Lagrangian formulation of the SSSGequations later in this section.The energy-momentum tensor is constructed in Appendix (B), and leads to thefollowing expression for the energy of a configuration E = 12 Z dx Tr h − (cid:0) ∂ + γγ − + A ( L )+ (cid:1) + A ( R )+ 2 − (cid:0) γ − ∂ − γ − A ( R ) − (cid:1) + A ( L ) − + Λ + γ − Λ − γ − Λ + Λ − i , (3.29)relative to E = 0 for γ = 1. We will find that the dressing procedure always producessoliton solutions of (3.13) which have vanishing gauge fields A ( R ) − = A ( L )+ = 0 andwhich satisfy the conditions P h + (cid:0) γ − ∂ + γ (cid:1) = 0 , P h − (cid:0) ∂ − γγ − (cid:1) = 0 , (3.30)and hence A ( R ) µ = A ( L ) µ = 0. Then, the conserved charges only get contributions fromthe boundary terms φ R/L ( ±∞ ) and, within the examples discussed in the followingsections, these turn out to be non-trivial only in the principal chiral models (seeSection 6). As a consequence only the principal chiral model solitons are chargedunder the SSSG H ( − ) L × H (+) R symmetry. In contrast, the solitons do always carrysigma model charge Q L,R . In Section 9, we will see how to produce solitons in thereduced symmetric space sigma models which carry non-trivial H ( − ) L × H (+) R charges;however, one needs to go beyond the dressing transformation to produce them. Lagrangian formulations
It is only natural to search for a relativistically invariant Lagrangian formulationof the SSSG equations (3.13). However, as we shall see and as has been pointed outelsewhere [12, 24, 25] there are problems that arise in pursuing this idea, and it maybe that the SSSG equations themselves should be used as a basis for a canonicalquantization without recourse to a Lagrangian.Lagrangian formulations are only known when H ( − ) and H (+) are isomorphicand of the form [12, 24] H (+) R = ǫ R ( H ) , H ( − ) L = ǫ L ( H ) , (3.31)– 9 –here H is a Lie group and ǫ L,R : H → G are two “anomaly-free” group homo-morphisms that descend to embeddings of the corresponding Lie algebras h and g . Then, each non-equivalent choice of ǫ L and ǫ R gives rise to a different Lagrangianformulation. This is obtained by writing A ( L )+ = ǫ L ( A + ) , A ( R ) − = ǫ R ( A − ) , (3.32)where A ± take values in h , and imposing the constraints P h + (cid:16) γ − ∂ + γ + γ − ǫ L ( A + ) γ (cid:17) = ǫ R ( A + ) , P h − (cid:16) − ∂ − γγ − + γǫ R ( A − ) γ − (cid:17) = ǫ L ( A − ) , (3.33)which can be viewed as a set of partial gauge fixing conditions [12, 25]. They canbe written as A ( L ) − = ǫ L ( A − ) , A ( R )+ = ǫ R ( A + ) , (3.34)where A ( L ) − and A ( R )+ are the “missing” components defined in (3.18). These condi-tions reduce the H ( − ) L × H (+) R gauge symmetry (3.15) to γ −→ ǫ L ( h ) γǫ R ( h − ) , h ∈ H , (3.35)under which A µ transforms as a gauge connection: A µ −→ h (cid:0) A µ + ∂ µ (cid:1) h − . (3.36)In addition, the gauge conditions (3.33) leave a residual symmetry under the global(abelian) transformations γ −→ e ǫ L ( ρ ) γe + ǫ R ( ρ ) , A µ −→ A µ , (3.37)where e ρ is in the centre of H .The gauge-fixed equations-of-motion are then (cid:2) ∂ + + γ − ∂ + γ + γ − ǫ L ( A + ) γ, ∂ − + ǫ R ( A − ) (cid:3) = 14 [Λ + , γ − Λ − γ ] (3.38)and these follow as the equations-of-motion of the Lagrangian density L = L W ZW ( γ ) + 12 π Tr (cid:16) − ǫ L ( A + ) ∂ − γγ − + ǫ R ( A − ) γ − ∂ + γ + γ − ǫ L ( A + ) γǫ R ( A − ) − ǫ L ( A + ) ǫ L ( A − ) −
14 Λ + γ − Λ − γ (cid:17) , (3.39) Here, anomaly free simply means that Tr (cid:0) ǫ L ( a ) ǫ L ( b ) (cid:1) = Tr (cid:0) ǫ R ( a ) ǫ R ( b ) (cid:1) for all a, b ∈ h . In [12], it was shown that this interpretation is consistent provided that the orthogonal decom-positions (3.19) hold, which is always true if the symmetric space is of definite signature. – 10 –here L W ZW ( γ ) is the usual WZW Lagrangian density for γ . In fact this theory isthe asymmetrically gauged WZW model for G/H specified by ǫ R/L with a potential.Notice that the partial gauge-fixing constraints (3.33) now appear as the equations-of-motion of the gauge connection. If we take the Lagrangian (3.39) as the basis fora QFT then many questions arise. For instance are the resulting QFTs independentof the choice of the form of the gauge group; i.e. , independent of ǫ L and ǫ R ? In manycases, it can be shown that different theories are actually related by a target spaceT-duality symmetry [26], hinting that they are equivalent at the quantum level.Now we turn to the symmetries of the Lagrangian theory and the relation withthe conserved charges Q L and Q R of the SSSG equations. Since our primary interestis in the soliton solutions, it is a fact that the transformations φ R/L that bring γ tothe gauge slice (3.22) lie in an abelian subgroup of H ( − ) L and H (+) R . As a consequencethere are associated local conserved currents and charges. Then, for our purposes,it will be enough to restrict the following discussion to the case of abelian H . Then,the Lagrangian (3.39) is symmetric under the (abelian) global transformations γ −→ e ǫ L ( u ) γe − ǫ R ( v ) , A µ −→ A µ , (3.40)where u, v take values in h . For u = v this is just a global gauge transformationof the form (3.35) while for u = − v it is a global symmetry transformation of theform (3.37). Following standard means, we can derive the corresponding Noethercurrents as follows (for instance, see [27]). Consider the variation of the Lagrangianaction S = R d x L under an infinitesimal transformation of the form γ − δγ = γ − ǫ L ( u ) γ − ǫ R ( v ) , δ A µ = 0 , (3.41)with u = u ( t, x ) and v = v ( t, x ). It reads δS = Z d x Tr (cid:16)(cid:2) ∂ + + γ − ∂ + γ + γ − ǫ L ( A + ) γ −
12 Λ + ,∂ − + ǫ R ( A − ) − γ − Λ − γ (cid:3) γ − δγ (cid:17) = Z d x Tr (cid:18)(cid:0) ∂ + A − − ∂ − A + (cid:1) ( u − v )++ ∂ − (cid:16) P h + (cid:0) γ − ∂ + γ + γ − ǫ L ( A + ) γ (cid:1) − ǫ R ( A + ) (cid:17) ǫ R ( v )++ ∂ + (cid:16) P h − (cid:0) − ∂ − γγ − + γǫ R ( A − ) γ − (cid:1) − ǫ L ( A − ) (cid:17) ǫ L ( u ) (cid:19) (3.42) For a more general configuration, we would have to separate out the abelian factors in H = U (1) p × H ss in an obvious way, as we did in the last section, and describe the semi-simple partin terms of non-local conserved charges. However, for the soliton solutions this technology isunnecessary. – 11 –hen, the condition that δS vanishes for any u, v provides the conservation equationswe are looking for. Using the constraints (3.33), they read ∂ + A − − ∂ − A + = 0 , (3.43)which are the conservation equations of the current J µ = ǫ µν A µ . (3.44)Notice that, since δS = 0 for u = v , J µ is the Noether current associated to theabelian global transformations (3.37), and there is no conserved current associatedto global gauge transformations. J µ is clearly not invariant under the gauge transformations (3.35)–(3.36), whichin this (abelian) case are of the form γ −→ e ǫ L ( u ) γe − ǫ R ( u ) , A µ −→ A µ − ∂ µ u . (3.45)In order to construct gauge invariant conserved quantities, we write the SSSG gaugeslice (3.22) as γ = φ L γ φ − R = e ǫ L ( α + β ) γ e − ǫ R ( α − β ) (3.46)such that, under (3.45), α → α + u while β and γ remain fixed. Then, the gaugeinvariant Noether current associated to the abelian global transformations (3.37) is˜ J µ = ǫ µν (cid:16) A ν + ∂ ν α (cid:17) , (3.47)which provides the Noether charge Q N = α (+ ∞ ) − α ( −∞ ) + Z + ∞−∞ dx A . (3.48)Similarly to the case of Q R/L discussed in the previous section, the definition of Q N is subject to an ambiguity whose form can be found by looking at the vacuumconfiguration γ vac = 1. Namely, since it is invariant under γ → e ρ γe − ρ , the field α in (3.46) is only defined up to α → α + η for any field η ∈ h such that ǫ L ( η ) = ǫ R ( η ).Consequently, the Noether charge is defined only modulo Q N −→ Q N + q for each q ∈ h such that ( ǫ L − ǫ R )( q ) = 0 . (3.49)In the Lagrangian formulation, this ambiguity has a physical interpretation. Noticethat each constant ρ ∈ h such that ( ǫ L − ǫ R )( ρ ) = 0 generates a symmetry trans-formation of the form (3.37) that changes γ vac = 1; namely, 1 → e ǫ L ( ρ ) . Then, theambiguity reflects the impossibility of defining a Noether charge for global symmetrytransformations that do not leave the vacuum configuration invariant.– 12 –he relationship between the SSSG conserved quantities Q R/L (or Q (ss) R/L ) and Q N can be easily derived by taking into account (3.27), (3.28), (3.32) and, accordingto (3.46), α R/L = ǫ R/L ( α ∓ β ). It reads Q L = ǫ L (cid:0) Q N + Q T (cid:1) , Q R = ǫ R (cid:0) Q N − Q T (cid:1) , (3.50)where Q T = β (+ ∞ ) − β ( −∞ ) (3.51)is a kind of (gauge invariant) topological, or kink, charge. The definition of Q T only makes sense if the global symmetry (3.37), which corresponds to β → β + ρ in (3.46), changes the vacuum configuration and gives rise to non-trivial boundaryconditions for γ . Consequently, the definition of Q T is also subject to an ambiguitywhose form can be found by looking again at γ vac = 1. Since it is invariant under γ → e ρ γe − ρ , the field β in (3.46) is defined only up to β → β + η for any η ∈ h suchthat ǫ L ( η ) = − ǫ R ( η ), which means that the topological charge is defined modulo Q T −→ Q T + q for each q ∈ h such that ( ǫ L + ǫ R )( q ) = 0 . (3.52)To summarize, in the Lagrangian formulation the soliton configurations are ex-pected to carry both Noether Q N and topological Q T charges. It is worth noticingthat the combination of the SSSG charges that is free of ambiguities reads Q L − Q R = (cid:0) ǫ L − ǫ R (cid:1) ( Q N ) + (cid:0) ǫ L + ǫ R (cid:1) ( Q T ) (3.53)which, not surprisingly, is also free of the ambiguities (3.49) and (3.52). Looking atthis equation, it is worthwhile to recall that the different Lagrangian formulations ofa set of SSSG equations are related by H ( − ) L × H (+) R gauge transformations, and that Q L − Q R is gauge invariant and, hence, independent of the choice of ǫ R/L . Moreover,since T-duality transformations interchange Noether and topological charges, (3.53)is consistent with the expectation that the different Lagrangian theories are indeedrelated by T-duality symmetries.The physical meaning of the charges and their ambiguities becomes clearer oncewe consider examples of particular gaugings. The most obvious kind of gauging thatcan always be chosen is ǫ L ( α ) = ǫ R ( α ) = α , (3.54)which corresponds to gauging the vector subgroup of H ( − ) L × H (+) R . Then, the value of Q N is meaningless, and the solitons are kinks characterized by the topological charge Q T . In this case, the Noether current corresponds to the axial transformations γ → e ρ γe ρ , which do not leave the vacuum invariant. This means that at the– 13 –lassical level the symmetry is spontaneously broken. Of course at the quantumlevel this would have to be re-evaluated.Since we are assuming that H is abelian, one can also gauge the axial vectorsubgroup by taking ǫ L ( α ) = − ǫ R ( α ) = α , (3.55)In this case, Q N is free of ambiguities. It is the Noether charge corresponding tovector transformations γ → e β γe − β that leave the vacuum invariant and, therefore,do not break the symmetry. In contrast, since the vacuum configuration is unique upto (axial) gauge transformations, the topological charge Q T is arbitrary. Therefore,solitons are similar to Q-balls. Other choices of ǫ R/L give rise to different interpre-tations of solitons as some sort of dyons that carry both Noether and topologicalcharge.As we have mentioned, the dressing procedure always produces soliton solutionsof (3.13) which have vanishing gauge fields A ( R ) − = A ( L )+ = 0. This means that thesoliton solutions are valid solutions of the gauged WZW model (3.38) and (3.33) forany choice of gauging with A µ = 0. Consequently, as is clear from (3.48) and (3.51),the charges can be calculated in terms of boundary values of α and β or, equivalently, α R/L . The mass of the solitons can be calculated from the energy-momentum tensorof the gauged WZW theory which leads (up to an overall factor) to (3.29) with theidentifications (3.32). Notice that, as a consequence of the anomaly free condition,Tr (cid:0) A ( R/L ) ± (cid:1) = Tr (cid:0) A ( L/R ) ± (cid:1) = Tr (cid:0) A ± (cid:1) . C P example In order to have an explicit example of the SSSG equations and their Lagrangianformulation consider M = C P . Since the symmetric space C P = SU (3) /U (2)has rank one, there is a unique Pohlmeyer reduction for which we can take (up toconjugation) Λ + = Λ − ≡ Λ = − . (3.56)In this case H ( ± ) = U (1) and we can use both vector or axial gauging to achieve aLagrangian formulation. We can parameterize the group element and gauge field as(see (3.22)) γ = e a L h θe iϕ sin θ − sin θ cos θe − iϕ e − a R h , (3.57)– 14 –here h = i diag(1 , , −
2) is the generator of h , and α R/L = a R/L h .If we choose vector gauging, then solving the conditions (3.33) for A µ and writing α L + α R = 2 α like in (3.46), yields the gauge invariant Noether current for axialtransformations:˜ J ( V ) µ = ǫ µν (cid:16) A ν + ∂ ν α (cid:17) = 13 (cid:16)(cid:0)
12 + 2 cot θ ) ∂ µ ( a L − a R ) + cot θ∂ µ ϕ (cid:17) h . (3.58)The two remaining equations, also depend only on α L − α R , as one expects sincethe combination α L + α R has been gauged away, and so it is convenient to define a L − a R = ψ/ ψh corresponds to 4 β in (3.46)): ∂ µ ∂ µ ψ = − θ sin ϕ ,∂ µ ∂ µ θ + cos θ sin θ ∂ µ ( ϕ + ψ ) ∂ µ ( ϕ + ψ ) = − sin θ cos ϕ . (3.59)These equations along with the continuity of the Noether current follow from theLagrangian L = ∂ µ θ∂ µ θ + 14 ∂ µ ψ∂ µ ψ + cot θ∂ µ ( ψ + ϕ ) ∂ µ ( ψ + ϕ ) + 2 cos θ cos ϕ . (3.60)The Lagrangian manifests the axial symmetry ψ → ψ + a . In this case, the vacuumconfiguration is degenerate, γ vac = e hψ/ , i.e. θ = ϕ = 0 with 0 ≤ ψ < π , andthe definition of the Noether charge does not make sense. Then, the solitons arecharacterized by the charge Q T , which is simply the kink charge Q T = 14 (cid:2) ψ ( ∞ ) − ψ ( −∞ ) (cid:3) h . (3.61)Classically the axial symmetry would be spontaneously broken. In the quantumtheory, this would have to re-evaluated since the theory is defined in 1+1-dimensionalspacetime, Goldstone’s Theorem does not apply and the would-be Goldstone modesshould be strongly coupled giving rise to a mass gap. A related issue is the factthat the Lagrangian does not have a good expansion in terms of fields around theirvacuum values due to the cot θ term in the Lagrangian.On the other hand, if we choose axial gauging, then α L − α R = 2 α , and thegauge invariant conserved Noether current for vector transformations is˜ J ( A ) µ = ǫ µν (cid:16) A ν + ∂ ν α (cid:17) = 11 + 4 cot θ (cid:16) ∂ µ (cid:0) a L + a R (cid:1) − cot θǫ µν ∂ ν ϕ (cid:17) h . (3.62)Then, we can define a L + a R = ˜ ψ/ ψh = 4 β in (3.46)) and the correspondingLagrangian is L = ∂ µ θ∂ µ θ + 11 + 4 cot θ (cid:16) ∂ µ ˜ ψ∂ µ ˜ ψ + cot θ∂ µ ϕ∂ µ ϕ − θǫ µν ∂ µ ˜ ψ∂ ν ϕ (cid:17) + 2 cos θ cos ϕ , (3.63)– 15 –hich manifests the vector symmetry ˜ ψ → ˜ ψ + a . In this case the vacuum is non-degenerate, γ vac = 1, because ˜ ψ is not a good coordinate around θ = ϕ = 0, andthe vacuum is invariant under the (vector) symmetry. Consequently, the definitionof Q T does not make sense. In contrast, Q N = Z + ∞−∞ dx ˜ J ( A )0 (3.64)is unambiguously defined. Moreover, in this case the Lagrangian does have a goodfield expansion around the vacuum.
4. Giant Magnons and their Solitonic Avatars “Giant magnon” is the name given to a soliton of the reduced
F/G model in thecontext of string theory. In particular, the examples of S and C P are directlyrelevant to the AdS/CFT correspondence for AdS × S [1] and AdS × C P [17],respectively. In this section, we review the known giant magnons in the context of S n and C P n and use them to illustrate some of the more important ideas discussed inthe previous section in a more concrete way. In particular, we will discuss the relationbetween the giant magnons and their relativistic solitonic avatars in the associatedSSSG equations.The sphere S n corresponds to the symmetric space SO ( n + 1) /SO ( n ) while thecomplex symmetric space C P n corresponds to SU ( n + 1) /U ( n ). In both cases theassociated involution is σ − ( F ) = θ F θ − , (4.1)where θ = diag (cid:0) − , , . . . , (cid:1) . (4.2)In Appendix C we explain how to map the spaces S n and C P n , expressed in termsof their usual coordinates, into the group field F . For the spheres, parameterized bya real unit n + 1-vector X with components X a , | X | = 1, we have F = θ (cid:0) − XX T (cid:1) , (4.3)while for the complex projective spaces C P n we have the complex n + 1 vector Z whose components are the complex projective coordinates Z a , a = 1 , . . . , n + 1, so In the same way that the polar angle it not a good coordinate around r = 0. – 16 –hat C P n is identified by modding out by complex re-scalings Z a ∼ λZ a , λ ∈ C . Inthis case F = θ (cid:18) − ZZ † | Z | (cid:19) . (4.4)The giant magnons can be though of as excitations around a “vacuum” whichis the simplest solution to the equations-of-motion and Pohlmeyer constraints, (2.7)and (3.1). The vacuum solution has f ± = 1 and F = exp (cid:2) x + Λ + + x − Λ − (cid:3) , (4.5)where, up to overall conjugation and generalizing (3.56),Λ + = Λ − ≡ Λ = − . (4.6)This solution corresponds to the following solution for the sphere and projectivecoordinates, X = Z = e cos t − e sin t . (4.7)Here, e , . . . , e n +1 are a set of orthonormal vectors in R n +1 . The physical interpre-tation is clear: the string is collapsed to a point which traverses a great circle on S n defined by the plane spanned by e and e at the speed of light. The vacuumsolution actually carries infinite sigma model charge and so the physically meaningfulcharge is actually the charge relative to the vacuum,∆ Q L = Z ∞−∞ dx (cid:0) ∂ F F − − ∂ F F − (cid:1) . (4.8)In particular, the component of this charge along the Lie algebra element Λ, up toscaling, is identified with ∆ − J , the difference between the scaling dimension and R charge of the associated operator in the boundary CFT:∆ − J = √ λ π Tr (cid:0) Λ∆ Q L (cid:1) , (4.9)where λ is the ’t Hooft coupling.For the sphere case, we can express the conserved charge (4.8) directly in termsof X : Q L,ab = Z ∞−∞ dx (cid:0) ∂ X a X b − X a ∂ X b (cid:1) . (4.10) Notice that the plane is determined by the choice of representative Λ. – 17 –n particular, ∆ − J = √ λ π ∆ Q L, . There is an analogous equation for the complexprojective spaces in terms of the projective coordinates. S n giant magnons The original giant magnon was described by Hofman and Maldacena [3]. It isa solution which takes values in the subspace S ⊂ S n picked out by 3 mutuallyorthonormal vectors { e , e , Ω } . The vectors e and e are already fixed by thechoice of vacuum solution, however the direction of Ω , which describes an S n − ⊂ S n ,plays the rˆole of an internal collective coordinate of the magnon. If ( θ, φ ) are polarcoordinates on S , then the solution written down by Hofman and Maldacena iscos θ = sin p cosh x ′ , tan( φ − t ) = tan p x ′ . (4.11)Here, and in the following, we define the Lorentz boosted coordinates t ′ and x ′ : x ′ = x cosh ϑ − t sinh ϑ , t ′ = t cosh ϑ − x sinh ϑ , (4.12)where ϑ is the rapidity ( v = tanh ϑ ). For the Hofman-Maldacena magnon,tanh ϑ = cos p . (4.13)Notice that the magnon is not relativistic in the sense that the moving solution is not the Lorentz boost of the stationary solution. The reason is that the Pohlmeyer con-straints (3.1) are not Lorentz covariant (this is discussed in more detail in AppendixB). In terms of the unit vector X , we can write this solution as X = (cid:2) sin t sin p tanh x ′ − cos t cos p (cid:3) e + (cid:2) cos t sin p tanh x ′ + sin t cos p (cid:3) e + sin p sech x ′ Ω . (4.14)It has sigma model charge ∆ Q L = − (cid:12)(cid:12) sin p (cid:12)(cid:12) Λ , (4.15)relative to the vacuum.The solitonic avatar of the Hofman-Maldacena giant magnon in the reducedSSSG model has vanishing gauge fields A ( L )+ = A ( R ) − = 0, while the non-vanishingelements of γ are γ = − T − cos θ ( x ) sin θ ( x ) Ω T sin θ ( x ) Ω 1 + (cos θ ( x ) − ΩΩ T , (4.16)– 18 –here we have highlighted the 2 × e and e . In the above, θ ( x ) (not to be confused with the polar angle above or the rapidity) is the solitonsolution to the sine-Gordon equation ∂ µ ∂ µ θ = − sin θ , (4.17)which can be written θ = 4 tan − ( e x ) . (4.18)Since the sine-Gordon equation is relativistic, in the sense that the moving solutionis the Lorentz boost of the static solution, it is sufficient to write the solution abovein the soliton rest frame. This soliton has vanishing charges Q L = Q R = 0. The second kind of solution is Dorey’s dyonic giant magnon [4]. The relationof Dorey’s magnon to the Hofman-Maldacena magnon is analogous to the relationbetween the dyon and monopole solutions in gauge theories in 3+1 dimensions. In thelatter case, the dyon is obtained by allowing the charge angle, a collective coordinatetalking values in S , to move around the circle with constant velocity. The non-trivialaspect of this is that the angular motion has a back-reaction on the original monopole.One way to think of what is happening is in terms of Manton’s picture of geodesicmotion [28]. The charge angle is an internal collective coordinate of the monopoleand the idea is that one can make a time-dependent solution by allowing the internalcollective coordinates to be time dependent. For low velocities the motion is simplygeodesic motion on the moduli space defined by a metric which is constructed fromthe inner-product of the zero modes associated to the collective coordinates. In thepresent setting, it is not clear whether Manton’s analysis applies directly because theHofman-Maldacena giant magnon is a time-dependent solution rather than a time-independent one like the monopole. We have seen that the Hofman-Maldacena giantmagnon has an internal collective coordinate Ω which parameterizes an S n − . Dorey’ssolution corresponds to allowing Ω to move around a great circle (the geodesic) in S n − . We can describe this motion by picking out two orthonormal vectors Ω ( i ) ,orthogonal to e and e , and then take (in the magnon’s rest frame) Ω ( t ) = cos( t sin α ) Ω (1) + sin( t sin α ) Ω (2) . (4.19)The parameter α sets the angular velocity. This motion has a back-reaction on theoriginal solution and we can write the complete moving solution as X = (cid:0) − cos t cos p + sin t sin p tanh( x ′ cos α ) (cid:1) e + (cid:0) sin t cos p + cos t sin p tanh( x ′ cos α ) (cid:1) e + sin p sech( x ′ cos α ) Ω ( t ′ ) , (4.20) Since H = SO ( n −
1) is semi-simple for n ≥
4, these charges provide examples of the conservedquantities Q (ss) R/L defined in the paragraph after (3.28). – 19 –he parameter α and the rapidity ϑ are determined by two parameters p and r viacot α = 2 r − r sin p , tanh ϑ = 2 r r cos p . (4.21)The Hofman-Maldacena magnon corresponds to the limit r → α → x ′ → x ′ cos α . The dyonic giant magnon carries charge Q L = − r ) r (cid:12)(cid:12) sin p (cid:12)(cid:12) Λ − − r ) r (cid:12)(cid:12) sin p (cid:12)(cid:12) h , (4.22)relative to the vacuum, where h is the generator of SO ( n + 1) corresponding torotations in the plane picked out by Ω ( i ) : h = Ω (1) Ω (2) T − Ω (2) Ω (1) T . (4.23)In the reduced SSSG model, the dyonic magnon gives a soliton for which thegauge fields do not vanish: A ( L )+ = − A ( R ) − = cos α sin α cos(2 α ) − cosh(2 x cos α ) h . (4.24)In addition, the “missing” components defined in (3.18) are A ( R )+ = − A ( R ) − , A ( L ) − = − A ( L )+ . (4.25)which means that only the temporal components of the currents J µR/L are non-vanishing. In addition, we have A ( L ) µ = − A ( R ) µ which, in the case when H is abelian,corresponds to the condition for axial gauging (3.32) in the Lagrangian formulation.The group field (in the rest frame) generalizes (4.16) in an obvious way: γ = − T − cos θ ( x ) sin θ ( x ) Ω ( t ) T sin θ ( x ) Ω ( t ) + (cos θ ( x ) − Ω ( t ) Ω ( t ) T . (4.26)where cos θ ( x ) = 1 − α sech ( x cos α ) , (4.27)which includes the effects of the back reaction of the geodesic motion. The solutioncarries charges (3.27) Q L = − Q R = Z ∞−∞ dx cos α sin α cos(2 α ) − cosh( x cos α ) h = (cid:16) α − π (cid:17) h . (4.28)– 20 –n particular, notice that the non-vanishing combination Q L − Q R is the unambigu-ously defined charge according to the discussion in Section 3. Notice that for thesesolutions there is no contribution from the boundary terms in (3.27). In addition,when M = S n , n >
3, the subgroup H = SO ( n −
1) is non-abelian. However, we canstill define local conserved currents and associated charges because φ R/L and gaugefields A ( R/L ) µ lie in an abelian subgroup SO (2) ⊂ H . The dyonic soliton has a mass M = 4 cos α . (4.29)The Lagrangian interpretation of these dyons depends of the choice of ǫ R/L , whichfixes the form of the group of gauge transformations. In the gauged WZW La-grangian formulation with vector gauging, which can be achieved for any n , thesedyons would carry non-vanishing topological charge Q T . However, for the particularcase of M = S , when H = SO (2) is abelian, it is also possible to define an axiallygauged WZW theory, in which case the dyons carry non-vanishing Noether charge Q N corresponding to vector SO (2) transformations. In both cases, Q T and Q N arerelated to Q R/L by means of (3.53). C P n giant magnons Motivated by its application to the investigation of the AdS/CFT correspon-dence for
AdS × C P [17], the C P case has been discussed in some detail in theliterature [9–11, 18, 19]. The giant magnon solutions described so far are all obtainedby embeddings of the Hofman-Maldacena giant magnon and Dorey’s dyonic magnon.The Hofman-Maldacena giant magnon can be embedded in C P n in two distinct ways.Firstly, by taking S ≃ C P ⊂ C P n [9]. If θ ( x, t ) and φ ( x, t ) is the solution in termsof polar coordinates in (4.11), then the projective coordinates are Z = e iφ (2 x, t ) / sin (cid:0) θ (2 x, t ) / (cid:1) e + e − iφ (2 x, t ) / cos (cid:0) θ (2 x, t ) / (cid:1) e . (4.30)The scaling of the spacetime coordinates here is necessary in order to be consistentwith the scaling of the Pohlmeyer constraints in (3.1). In addition, in order that thesolution is oriented with respect to the choice of vacuum in (4.5), we have to rotateit with an element of SU (2) ⊂ SU ( n ), Z → U Z , U = 1 √ e πi/ e iπ/ e iπ/ e − iπ/
00 0 1 . (4.31)Notice that this solution has no internal collective coordinates. The charge carriedby the magnon is ∆ Q L = − (cid:12)(cid:12) sin p (cid:12)(cid:12) Λ , (4.32)– 21 –elative to the vacuum.This magnon corresponds to a soliton solution of the SSSG equations of the form γ = e iψ e − iψ
00 0 1 , (4.33)with A ( L )+ = A ( R ) − = 0. The field ψ then satisfies the sine-Gordon equation ∂ µ ∂ µ ψ = 2 sin(2 ψ ) . (4.34)The giant magnon solution (4.30) corresponds to the sine-Gordon kink ψ = 2 tan − ( e x ) + π . (4.35)The solution carries H ( − ) L × H (+) R charge Q L = − Q R = 12 (cid:0) ψ ( ∞ ) − ψ ( −∞ ) (cid:1) = π . (4.36)The inequivalent embedding of the Hoffman-Maldacena giant magnon is via thesubspace R P ⊂ C P n [10]. The solution for Z is exactly equal to X in (4.14),however the vector Ω can now be taken to be a complex vector with | Ω | = 1. Consequently the soliton has internal collective coordinates associated to an S n − .The fact that the solution is valued in R P is because the solution (4.14) is itselfvalued in S and this fixes the complex scaling freedom, Z → λ Z , up to the discreteelement Z → − Z and a further quotient by this gives R P . The dyonic magnon canbe embedded in an analogous way when n ≥ R P ⊂ C P n [11].
5. Magnons and Solitons by Dressing the Vacuum
One way to construct the magnon/soliton solutions is to use an approach knownas the dressing transformation [13] which is closely related to the B¨acklund trans-formation. For magnons in string theory this approach has been described in detailin [14,15]. For the SSSG theories, and in particular their gauged WZW formulations,such an approach was described in [24, 29]. Schematically the transformation takes aknown solution—for example a trivial kind of solution that we call the “vacuum”—and adds in a soliton. It is an important fact that the dressing transformation In addition, we should replace Ω T by Ω † in (4.16). In Section 6, we will show that in the principal chiral model cases multiple soliton solutions aresometimes produced where the solitons are all mutually at rest. – 22 –s consistent with the Pohlmeyer reduction in the sense that if the original solutionsatisfies the constraint (3.1) then so will the dressed solution. In fact we shall seethat the dressing transformation constructs both the magnon and its SSSG solitonavatar at the same time without the need to map one into the other.The dressing procedure has been described for general symmetric space sigmamodels in [22] and we shall draw heavily on results derived there. The procedurebegins by identifying a “vacuum” solution. In the present context, our vacuumsolution will be the simplest solution that satisfies the Pohlmeyer constraint (3.1).This identifies it as (4.5) which naturally satisfies the Pohlmeyer constraints (3.1)with f ± = 1. In Appendix D, we show that the dressing dressing transformationdirectly produces a soliton of the SSSG equations (3.13) with vanishing A ( L )+ = A ( R ) − = 0: ∂ − (cid:0) γ − ∂ + γ (cid:1) = 14 [Λ + , γ − Λ − γ ] . (5.1)In addition, we find that the conditions γ − ∂ + γ (cid:12)(cid:12)(cid:12) h + = ∂ − γγ − (cid:12)(cid:12)(cid:12) h − = 0 (5.2)are satisfied. Notice that this means that the solitons automatically satisfy theequations of the gauged WZW model (3.39) for any choice of gauging.The strategy of [22] begins by defining the symmetric space sigma model
F/G interms of initially the subgroup F ⊂ SL ( n, C ) via one, or possibly more, involutionsthat we denote collectively as σ + . In order to pick out the coset F/G ⊂ F , the secondpart of the construction involves the extra involution σ − whose explicit form for S n and C P n have been given in (4.1). The dressing transformation is then constructedin SL ( n, C ) and the involutions give constraints that ensure that the transformationis restricted to the quotient F/G . In this work, we shall focus on three examples inorder to be concrete:(i)
F/G = SU ( n ) × SU ( n ) /SU ( n ). These are the principal chiral models and inthis case it is more convenient to formulate the sigma model directly in terms of an SU ( n )-valued field F ( x ). In this case, there is only a single involution σ + ( F ) = F †− (5.3)required, and the involution σ − is absent.(ii) The complex projective spaces C P n = SU ( n + 1) /U ( n ). In this case, thereare two involutions σ + ( F ) = F †− and σ − ( F ) = θ F θ , the latter defined in (4.1). This latter result makes use of the orthogonal decompositions (3.19), which hold in general forsymmetric spaces of definite signature. – 23 –iii) S n = SO ( n + 1) /SO ( n ). In this case, there are three involutions σ (1)+ ( F ) = F †− , σ (2)+ ( F ) = F ∗ and σ − ( F ) = θ F θ , the latter defined in (4.1).Starting in SL ( n, C ), the equations-of-motion for the sigma model have the zerocurvature form (2.11) which are the integrability conditions for the associated linearsystem ∂ + Ψ( x ; λ ) = ∂ + F F − λ Ψ( x ; λ ) ,∂ − Ψ( x ; λ ) = ∂ − F F − − λ Ψ( x ; λ ) . (5.4)Notice that the group field is simply F ( x ) = Ψ( x ; 0) . (5.5)The dressing transformation involves constructing a new solution Ψ of the linearsystem of the form Ψ( x ; λ ) = χ ( x ; λ )Ψ ( x ; λ ) (5.6)in terms of an old one Ψ , which in our case corresponds to the vacuum solution in(4.5): Ψ ( x ; λ ) = exp h x + λ Λ + + x − − λ Λ − i , (5.7)By picking out the residues of ∂ ± Ψ( λ )Ψ( λ ) − at λ = ∓ ( x ; λ )) it follows that ∂ ± F F − = χ ( ∓ ± χ ( ∓ − . (5.8)This is a key result because it means that the dressing transformation preserves thePohlmeyer reduction and, in addition, we have f ± = χ ( ∓ , (5.9)where Φ is a, as yet, unknown element that commutes with Λ ± :[Λ ± , Φ] = 0 , (5.10)which will be chosen so that γ = f − − f + = Φ − χ (+1) − χ ( − G ⊂ F . This will then guarantee that the dressing transformation willgive a soliton solution in the associated SSSG model. In addition, we will see that f ± satisfy (3.4). We will find below thatΦ = F / = exp h x + Λ + x − Λ − i . (5.12)– 24 –e now briefly review the construction of the dressing factor χ ( λ ) following [22].The general form is χ ( λ ) = 1 + X i Q i λ − λ i , χ ( λ ) − = 1 + X i R i λ − µ i (5.13)where the residues Q i and R i are matrices of the form Q i = X i F † i , R i = H i K † i , (5.14)for vectors X i , F i , H i and K i . Taking the residues of χ ( λ ) χ ( λ ) − = 1 at λ = λ i and µ i , gives, respectively, Q i + Q i R j λ i − µ j = 0 , R i + Q j R i µ i − λ j = 0 , (5.15)which can be used to solve for X i and K i : X i Γ ij = H j , K i (Γ † ) ij = − F j , (5.16)where the matrix Γ ij = F † i H j λ i − µ j . (5.17)It follows from the linear system (5.4) that ∂ ± F F − = (1 ± λ ) ∂ ± χχ − + χ Λ ± χ − . (5.18)Since the left-hand side is independent of λ , the residues of the right-hand side at λ = λ i and µ i must vanish, giving(1 ± λ i )( ∂ ± Q i ) (cid:16) R j λ i − µ j (cid:17) + Q i Λ ± (cid:16) R j λ i − µ j (cid:17) = 0 , − (1 ± µ i ) (cid:16) Q j µ i − λ j (cid:17) ∂ ± R i + (cid:16) Q j µ i − λ j (cid:17) Λ ± R i = 0 , (5.19)which are solved by(1 ± λ i ) ∂ ± F † i = − F † i Λ ± , (1 ± µ i ) ∂ ± H i = Λ ± H i . (5.20)The solutions of these equations are F i = (cid:0) Ψ ( λ i ) † (cid:1) − ̟ i , H i = Ψ ( µ i ) π i , (5.21) Notice that i is not the vector index but rather labels a set of vectors associated to the polesof χ ± ( λ ). In the following we will assume that λ i = µ j for any pair i, j . If the contrary is true thenadditional conditions must be imposed as we shall see later with an example. – 25 –or constant complex n -vectors ̟ i and π i .It can then be shown by tedious computation (re-produced in Appendix D) thatthe generic solution that arises from the dressing procedure gives γ , as in (5.11) withΦ as in (5.12), i.e. γ = F − / χ (+1) − χ ( − F / (5.22)satisfies the equation-of-motion (5.1). Using the explicit formulae for χ ( λ ) and itsinverse, we find that the G -valued field of the reduction is γ = 1 − − µ i )(1 + λ j ) F − / H i (cid:0) Γ − (cid:1) ij F † j F / . (5.23)So we see that the data of the dressing transformation constructs both the sigmamodel magnon and the soliton in the SSSG. There are simple formulae for the charges Q L,R of the dressed solution defined in (2.9), and for Q L the formula follows directlyfrom (5.18): since the right-hand side is independent of λ we can evaluate it at λ = ∞ , which gives ∂ ± F F − = ± ∂ ± X i Q i + Λ ± . (5.24)The final term here is precisely ∂ ± F F − and so the meaningful quantity to calculateis the charge of the dressed solution relative to the vacuum solution, and it followsdirectly that ∆ Q L = X i Q i (cid:12)(cid:12)(cid:12) x = ∞ − X i Q i (cid:12)(cid:12)(cid:12) x = −∞ . (5.25)In order to calculate the charge Q R , we use the fact that F − = F (cid:12)(cid:12)(cid:12) λ i → λ − i ,µ i → µ − i , Λ ± →− Λ ± , (5.26)which can be proved directly. Hence, it follows that the charge relative to the vacuumsolution is ∆ Q R = − ∆ Q L (cid:12)(cid:12)(cid:12) λ i → λ − i ,µ i → µ − i , Λ ± →− Λ ± . (5.27)Up till now we have described the B¨acklund transformation for SL ( n, C ). How-ever, as mentioned above we have to impose involution conditions in order to describea particular symmetric space. As described in [22] for each choice of symmetric spacethere are a set of involutions that must be imposed. First of all, there is an involution(or possibly more than one) σ + that picks out F ⊂ SL ( n, C ): F = (cid:8) F ∈ SL ( n, C ) (cid:12)(cid:12) σ + ( F ) = F (cid:9) . (5.28)Then there is a further involution σ − , that is detailed above for the explicit ex-amples we have in mind, that picks out F/G parameterized by F (as explained inAppendix C): F/G ≃ (cid:8) F ∈ F (cid:12)(cid:12) σ − ( F ) = F − (cid:9) . (5.29)– 26 –otice also that the quotient group is identified as G = (cid:8) γ ∈ F (cid:12)(cid:12) σ − ( γ ) = γ (cid:9) . (5.30)This allows us to prove that (5.22) is, as claimed, valued in G ⊂ F . Using σ − ( F ± / ) = F ∓ / and σ − ( χ ( ± F − χ ( ± F gives σ − ( γ ) = F / · F − χ (+1) − · F − χ ( − F · F − / = γ . (5.31)The involutions are each of of the following four types: σ ( F ) = θ F θ − , σ ( F ) = θ F ∗ θ − ,σ ( F ) = θ ( F T ) − θ − , σ ( F ) = θ F †− θ − , (5.32)where θ is either a symmetric, antisymmetric, hermitian or anti-hermitian matrix.The involutions ( σ , σ ) are holomorphic while ( σ , σ ) are anti-holomorphic.The correct way to impose these conditions on Ψ( x ; λ ) areΨ( λ ) = σ + (cid:0) Ψ(˜ λ ) (cid:1) , Ψ(1 /λ ) = F σ − (cid:0) Ψ(˜ λ ) (cid:1) , (5.33)where ˜ λ = λ, λ ∗ , if σ ± is holomorphic or anti-holomorphic, respectively. Notice thatif we take λ = 0 and use F = Ψ(0) and Ψ( ∞ ) = 1, yields the correct conditions σ + ( F ) = F , σ − ( F ) = F − . (5.34)Furthermore it is easy to see that the vacuum solution (5.7) satisfies these conditionssince Λ ± ∈ a ⊂ p .Written in terms in terms of χ ( λ ) the conditions (5.33) become χ ( λ ) = σ + (cid:0) χ (˜ λ ) (cid:1) ,χ (1 /λ ) = F σ − (cid:0) χ (˜ λ ) (cid:1) F − , (5.35)which means that the two sets of poles { λ i } and { µ i } must be separately invariantunder λ → ˜ λ (for σ + ), or λ → / ˜ λ (for σ − ), for σ and σ , and mapping into eachother for σ and σ .Rather than describe all the different cases, we specialize in this work to thethree examples (i)-(iii). Notice that in all our examples the group F is compactand the elements Λ †± = − Λ ± . Since we are principally interested in the applicationto string theory, there are some additional conditions on Λ ± . The vacuum solution– 27 – should be a t -dependent, but x -independent, solution. This immediately requiresthat Λ + = Λ − . Notice that in cases (ii) and (iii), the symmetric space has rank 1 andso either Λ + = Λ − ≡ Λ or Λ + = − Λ − ≡ Λ, for a fixed element Λ (up to conjugation).For case (i), the principal chiral model, there are more general models with Λ + = Λ − that will be discussed elsewhere.The issue of relativistic invariance is quite subtle. With the choice Λ + = Λ − ,the vacuum solution is t -dependent. Clearly, if we boost this solution then it willno longer satisfy the Pohlmeyer constraints (3.1). This fact is then inherited by thedressed solution. As we shall argue, although the solution is localized in the sense,for example, that the density of its charges J L and J R , relative to the vacuum, islocalized at a certain position in space moving with a certain velocity, the solutionswith different velocities are not related by boosts. On the contrary, the solution inthe reduced SSSG theory does respect Lorentz transformations in the sense that thesolutions with different velocities are related by boosts (see Appendix B).Let us identify the velocity of the dressed solution. The dependence on x isvia Ψ ( ξ ), where ξ is one of the λ i or µ i . The localized nature of the soliton arisesbecause when ξ has a imaginary part, Ψ ( ξ ) has an exponential dependence on x .Assuming that Λ is anti-hermitian, the relevant dependence isexp h i Im (cid:16) x + ξ + x − − ξ (cid:17) Λ i = exp h i Im (cid:16) t − ξx − ξ (cid:17) Λ i (5.36)and this leads to exponential fall-off of the energy/charge density away from thecentre which is located at the solution ofIm (cid:16) t − ξx − ξ (cid:17) = 0 . (5.37)The velocity of the soliton is therefore v = Im (1 − ξ ) − Im ξ (1 − ξ ) − = 2 r r cos p , (5.38)where ξ = re ip/ . Roughly speaking, the dressed solution describes N solitons (for i, j = 1 , . . . , N ) and λ i is a parameter that determines the velocity of the i th solitonvia (5.38) with ξ = λ i . However, for the cases S n and C P n , the additional constraintsmean that the solution actually represents less than N independent solitons.On the other hand, in the reduced SSSG model, the complete dependence on t and x is through the combination F − / Ψ ( ξ ) = exp h(cid:16) (1 + ξ ) t − ξ − ξx − ξ (cid:17) Λ i . (5.39)– 28 –n this case, the model does have relativistic invariance and the expression above canbe written exp h(cid:16) − t ′ sin α − ix ′ cos α (cid:17) Λ i , (5.40)where ( t ′ , x ′ ) are the boosted coordinates defined in (4.12) and the parameter α andthe rapidity ϑ are determined by r and p as in (4.21). The angle α sets both thesize and the internal angular velocity of the soliton. In the rest frame of the soliton, p = π or, equivalently, ξ = ri .
6. The Principal Chiral Models
As described in Section 2, we can either think of these theories as symmetric spacesigma models on M = G × G/G (so as a theory on G × G with an involution σ − that exchanges the two G factors) or more directly as a sigma model defined on theLie group G (and thus not needing a σ − involution). We shall follow the secondoption and, hence, we formulate the theory in terms of a G -valued field F definedas a subgroup of SL ( n, C ) by the involution(s) σ + , and in this approach there is noinvolution σ − .In this work we will only consider the choice G = SU ( n ), where there is a singleinvolution σ + ( F ) = F †− (6.1)that, in the classification of [22], is of type σ with θ = I . Invariance under σ + requires Ψ( λ ) = Ψ( λ ∗ ) †− , (6.2)which is satisfied by imposing µ i = λ ∗ i that in turn implies that H i = F i , K i = X i (6.3)in (5.14). This means that π i = ̟ i for each i and, moreover, that X i = K i = F j (cid:0) Γ − (cid:1) ji , Γ ij = F † i F j λ i − λ ∗ j , (6.4)and χ ( λ ) = 1 + F i (cid:0) Γ − (cid:1) ij F † j λ − λ j , χ ( λ ) − = 1 − F i (cid:0) Γ − (cid:1) ij F † j λ − λ ∗ i . (6.5)Then, using (5.5) and (5.6), the SU ( n ) principal chiral model magnon is F = χ (0) F = F − F i (cid:0) Γ − (cid:1) ij F † j F λ j , (6.6)– 29 –hile, according to (5.23), its solitonic avatar in the associated SSSG theory reads γ = 1 − − λ ∗ i )(1 + λ j ) F − / F i (cid:0) Γ − (cid:1) ij F † j F / , (6.7)where F i = Ψ ( λ ∗ i ) ̟ i . (6.8)In general, we will have to multiply (6.6) and (6.7) by constant phase factors in orderto enforce det F = 1 and det γ = 1, respectively.The rank of the symmetric space M = G × G/G coincides with the rank of theLie group G . Therefore, unless G = SU (2), it gives rise to different Pohlmeyer reduc-tions whose interpretation in the context of string theory is still to be understood.They are specified by two elements Λ ± of the Cartan subalgebra of g which, in thedefining representation, are anti-hermitian diagonal matrices. In this work we willonly consider the reductions corresponding toΛ + = Λ − = i diag (cid:0) ζ a (cid:1) = Λ (6.9)so that F only depends on t . Then, the vacuum solution of the associated linearsystem is Ψ ( λ ) = diag (cid:0) e Θ a ( λ ) (cid:1) , (6.10)where Θ a ( λ ) = iζ a x + λ + iζ a x − − λ = iζ a (cid:16) t − λ − λx − λ (cid:17) . (6.11)Furthermore, we will restrict ourselves to the cases with ζ a = ζ b for a = b so that H (+) = H ( − ) = U (1) n − , which correspond to the so-called (parity symmetric) ho-mogeneous sine-Gordon models [20]. Then, without loss of generality, we can orderthe ζ a according to ζ > ζ > · · · ζ n . (6.12)More general reductions with Λ + = Λ − will be discussed elsewhere.One-soliton solutions are obtained by considering a single pole in χ ( λ ), and so χ ( λ ) = 1 + ξ − ξ ∗ λ − ξ F F † F † F , (6.13)where F = Ψ ( ξ ∗ ) ̟ (6.14) For those who are familiar with the HSG theories, this is equivalent to taking Λ + = Λ − insidethe principal Weyl chamber with respect to the standard choice of the basis of simple roots. – 30 –or a complex n -vector ̟ . In terms of components F a = e Θ a ( ξ ∗ ) ̟ a . (6.15)The complex n -vector ̟ represents a set of collective coordinates for the solitons.Since χ ( λ ) and, hence, the soliton solutions are explicitly invariant under complexre-scalings ̟ → λ ̟ , with λ ∈ C , these collective coordinates span a C P n − . Noticethat constant shifts of the solitons in space and time act on the collective coordinatesvia ̟ −→ exp h δx + ξ ∗ Λ + δx − − ξ ∗ Λ i ̟ . (6.16)So some of the collective coordinates fix the position of the soliton in space and deter-mine the temporal origin. The interpretation of the remaining “internal” collectivecoordinates will emerge when we analyze the solutions in more detail.First of all, we think of the SU ( n ) principal chiral model magnons. Using (6.6),the group-valued field F is given by F ab = e ip/n (cid:18) δ ab e Θ a (0) − ξ − ξ ∗ ξ e Θ a ( ξ ∗ ) ̟ a ̟ ∗ b e − Θ b ( ξ )+Θ b (0) P c | ̟ c | e Θ c ( ξ ∗ ) − Θ c ( ξ ) (cid:19) , (6.17)where we have multiplied F by the phase e ip/n in order to enforce det F = 1. Thismagnon carries SU ( n ) L × SU ( n ) R charges Q R/L whose value relative to the vac-uum solution can be calculated using (5.25) and (5.27). The result is ∆ Q R/L =diag (cid:0) ∆ Q aR/L (cid:1) with ∆ Q aL = − i (cid:12)(cid:12) r sin p (cid:12)(cid:12) (cid:0) δ a, min − δ a, max (cid:1) ∆ Q aR = − i (cid:12)(cid:12) r − sin p (cid:12)(cid:12) (cid:0) δ a, min − δ a, max (cid:1) , (6.18)where ξ = re ip/ . Next, we look at the solitonic avatar of the magnon (6.17) in theSSSG model which is provided by (6.7). It reads γ ab = e iC (cid:18) δ ab − ξ − ξ ∗ )(1 − ξ ∗ )(1 + ξ ) · e Θ a ( ξ ∗ ) − Θ a (0) / ̟ a ̟ ∗ b e − Θ b ( ξ )+Θ b (0) / P c | ̟ c | e Θ c ( ξ ∗ ) − Θ c ( ξ ) (cid:19) , (6.19)where we have multiplied γ by the constant phase e iC = (cid:16) − r + 2 ri sin p − r − ri sin p (cid:17) /n (6.20)to enforce det γ = 1. This field configuration satisfies the equations-of-motion (3.13)with A ( L ) − = A ( R )+ = 0. Then, the value of the unambiguously well-defined Lorentzinvariant SSSG charge Q L − Q R carried by this soliton is provided by γ (+ ∞ ) γ − ( −∞ ) = e Q L − Q R , (6.21)– 31 –hich follows from (3.22), (3.27), and γ vac = 1. The result is Q L − Q R = diag (cid:0) Q aL − Q aR (cid:1) with Q aL − Q aR = 2 i arctan (cid:16) | r | r − (cid:17) (cid:0) δ a, min − δ a, max (cid:1) . (6.22)Finally, the mass of this SSSG soliton can be calculated using (3.29), which leads to M = 4 | r | ( r + 1) (cid:0) ζ min − ζ max (cid:1) . (6.23)Eqs. (6.22) and (6.23) show that all the non-trivial SSSG solutions are obtainedwith r >
0, and that charge conjugation corresponds to r → /r . Moreover,eqs. (6.18), (6.22), and (6.23) unravel the rˆole of the collective coordinates ̟ . Thissolution is actually a superposition of “max”–“min” basic solitons, all mutually atrest, which exhibits that, with the special choice Λ + = Λ − , there are no forces be-tween them. The basic solitons are associated to the pairs ( a, a + 1), with only ̟ a and ̟ a +1 non-vanishing. Those with a knowledge of root systems will appreciatethat these basic solitons are naturally associated to the simple roots of SU ( n ) anda particular SU (2) ⊂ SU ( n ). Then, the rˆole of the collective coordinates ̟ a is tofix the relative space-time positions of the basic solitons.Let us analyze the SU (2) case in more detail, since SU (2) ≃ S and, in any case,this describes the basic solitons of the SU ( n ) theory. We take ζ = − ζ = 12 ⇒ Λ = i (cid:0) , − (cid:1) . (6.24)Shifting x ± as in (6.16), and using the overall scaling symmetry, allows us to fixwithout-loss-of-generality ̟ = (1 , ξ = re ip/ determines the velocity of the soliton as well as the angular velocity of the internalmotion, the former as in (5.38). The solution has the explicit form F = e it (cid:0) cos p + i sin p tanh( x ′ cos α ) (cid:1) − i sin p e − it ′ sin α sech( x ′ cos α ) − i sin p e + it ′ sin α sech( x ′ cos α ) e − it (cid:0) cos p − i sin p tanh( x ′ cos α ) (cid:1)! , (6.25)where x ′ = x cosh ϑ − t sinh ϑ and t ′ = t cosh ϑ − x sinh ϑ are the boosted coordinates.The parameter α and the rapidity ϑ are determined by the two parameters p and r via (4.21). Notice, that the moving solution is not the boost of the solution at restbecause of the e ± it factors. Using (6.18), the SU (2) L × SU (2) R charges relative tothe vacuum carried by this magnon can be written as∆ Q L = − | r sin σ | Λ , ∆ Q R = − | r − sin σ | Λ . (6.26) The composite nature of these solutions was already noticed in [21] in the context of thehomogeneous sine-Gordon theories. – 32 –t is not difficult to check that (6.25) corresponds to Dorey’s dyonic magnon (4.20). Next we turn to the soliton avatar of (6.25). In the rest frame ( ξ = ri , or p = π ),it is γ = 1 r + 1 r − − ri tanh( x cos α ) ire − it sin α sech( x cos α )2 ire + it sin α sech( x cos α ) r − ri tanh( x cos α ) ! , (6.28)along with A ( L )+ = A ( R ) − = 0. Using (4.21) with p = π , in this equationcos α = 2 r r , sin α = 1 − r r . (6.29)Then, the charge and mass carried by this SSSG soliton can be written as Q L − Q R = 4 arctan (cid:16) | r | r − (cid:17) Λ , M = 8 | r | ( r + 1) = 4 (cid:12)(cid:12) sin (cid:0)
12 Tr[Λ( Q L − Q R )] (cid:1)(cid:12)(cid:12) . (6.30)Eq. (6.28) provides the well known one soliton solutions of the complex sine-Gordonequation [30]. Notice that the r = 1 soliton is static. It is the embedding of theusual sine-Gordon soliton in the reduced SU (2) principal chiral model. For thisconfiguration, the charge Q L − Q R is uniquely defined only modulo 4 π Λ, a featurethat played an important rˆole in the construction of the CSG scattering matrixproposed in [31].In [26], it was shown that this SSSG soliton saturates a Bogomol’nyi-type bound,which explains the explicit relationship between mass and charge shown in (6.30).If we choose axial gauging, then Q L − Q R corresponds to a U (1) Noether chargeand these solutions provide two-dimensional examples of Q -balls, which has been re-cently exploited to investigate some aspects of the dynamics of that type of extendedsolutions in quantum field theories [32].
7. Complex Projective Space
In this case the target space of the sigma model is the symmetric space SU ( n +1) /U ( n ). As we have described in Section 5, this is picked out from the universalconstruction in SL ( n, C ) by two involutions; σ + ( F ) = F †− , along with σ − in (4.1). The explicit relationship reads X = − Re( F ) e + Im( F ) e − Im( F ) Ω (1) − Re( F ) Ω (2) . (6.27) – 33 –otice that σ − is of type σ in the list (5.32) and is consequently holomorphic. Thevacuum solution is defined in (4.5).Turning to the dressing transformation, invariance under σ − requires thatΨ(1 /λ ) = F θ Ψ( λ ) θ − (7.1)and this means that the poles { λ i } must come in pairs ( λ i , λ i +1 = 1 /λ i ) and we canthink of a single soliton as being a pair of the basic solitons of the SU ( n ) principalchiral model. In addition, the fact that the poles come in pairs, requires associatedconditions for each pair i = 1 , , . . . : ̟ i +1 = θ ̟ i , (7.2)which in turn means that F i +1 = Ψ (1 /λ ∗ i ) θ ̟ i = F θ Ψ ( λ ∗ i ) ̟ i = F θ F i . (7.3)Let us consider in more detail the one soliton solution obtained from a singlepair of poles { ξ, /ξ } . The dressing factor is χ ( λ ) = 1 + Q λ − ξ + Q λ − /ξ , (7.4)and the matrix Γ ij has componentsΓ = βξ − ξ ∗ , Γ = ξ ∗ γ | ξ | − , Γ = − ξγ | ξ | − , Γ = − | ξ | βξ − ξ ∗ , (7.5)where we have defined the two real numbers β = F † F , γ = F † F θ F , (7.6)where F ≡ F . Therefore Q = 1∆ h − | ξ | βξ − ξ ∗ F F † + ξγ | ξ | − F θ F F † i ,Q = 1∆ h βξ − ξ ∗ F θ F F † θ F † − ξ ∗ γ | ξ | − F F † θ F † i . (7.7)In the above, we have defined∆ = det Γ = | ξ | γ ( | ξ | − − | ξ | β ( ξ − ξ ∗ ) . (7.8)– 34 –he solution depends on the complex vector ̟ ≡ ̟ and the complex number ξ .In addition, χ ( λ ) − = 1 + R λ − ξ ∗ + R λ − /ξ ∗ , (7.9)where R = 1∆ h | ξ | βξ − ξ ∗ F F † + ξ ∗ γ | ξ | − F F † θ F i ,R = 1∆ h − βξ − ξ ∗ F θF F † θ F † − ξγ | ξ | − F θF F † i . (7.10)The magnon solution is obtained from F = χ (0) F . It corresponds to the pro-jective coordinates Z = (cid:0) ˜ α + θ F F † θ (cid:1) Z , (7.11)where ˜ α = − ξβξ − ξ ∗ − γ | ξ | − . (7.12)The complex n + 1-vector ̟ represents a set of collective coordinates for themagnon. In fact, it is easy to see that only this vector up to complex re-scalings ̟ → λ ̟ lead to inequivalent solutions. By making shifts in x ± , as in (6.16), wecan set always set, say, ̟ = 0 and then use the scale symmetry to set ̟ = i , sothat ̟ = i e + Ω , Ω · e = Ω · e = 0 , (7.13)where the constant vector Ω is the internal collective coordinates of the magnon.The explicit solution is rather cumbersome to write down, Z = Z e + Z e + Z Ω , (7.14)where Z = ˜ α cos t + cos (cid:18) − e ip/ rx + 2 e ip te ip − r (cid:19) cos (cid:18) − e ip/ rx + ( r + 1) e ip t − e ip r (cid:19) Z = − ˜ α sin t − sin (cid:18) − e ip/ rx + 2 e ip te ip − r (cid:19) cos (cid:18) − e ip/ rx + ( r + 1) e ip t − e ip r (cid:19) Z = − cos (cid:18) − e ip/ rx + ( r + 1) e ip t − e ip r (cid:19) , (7.15) This is similar to the Euclidean space formulae in [33]. The fact that we choose i here will make it simpler to relate the solution to the case M = S n . – 35 –nd ˜ α = e ip − e ip (cid:20) | Ω | + cos (cid:18) ir sin p ( − ( r + 1) x + 2 rt cos p )2 r cos p − − r (cid:19)(cid:21) + 11 − r (cid:20) | Ω | − cos (cid:18) r − rx cos p − (1 + r ) t )2 r cos p − − r (cid:19)(cid:21) (7.16)The magnon carries SU ( n ) charge which can be extracted from (5.25). The compu-tation is simplified by noticing that the off-diagonal elements in Q i = F i (Γ − ) ij F † j (those with j = i ) vanish as x → ±∞ and so do not contribute to the charge. Thisis because as x → ±∞ , β , as defined in (7.6), diverges exponentially, while γ , alsodefined in (7.6), remains bounded. The remaining two contribution to the charge arethen easily evaluated to give∆ Q L = − r r | sin p | Λ . (7.17)The magnon solution that we have constructed above is apparently singularwhen | ξ | = 1, i.e. r = 1 or α = 0. However, a regular solution in this limit can beconstructed by imposing the additional condition that γ = F † F θF = ̟ † θ ̟ = 0 , (7.18)which can be written as a condition on the internal collective coordinates, | Ω | = 1 . (7.19)In this case, the matrix Γ is diagonal and the dressing transformation has the simplerform: χ ( λ ) = 1 + ξ − ξ ∗ λ − ξ F F † β − ξ − ξ ∗ λ − ξ ∗ F θ F F † θ F † β , (7.20)The solution can also be obtained from (7.14) by setting | Ω | = 1 and taking the limit r →
1. It is not difficult to see that up to a re-scaling by − cosh x ′ sin p , (7.21)the solution is precisely an embedding of the Hofman-Maldacena magnon in (4.11).With reference to the discussion in Section 4, it is the one associated to R P ⊂ C P n . In addition, it is necessary that ̟ = 0. – 36 –he solitonic avatar of the magnon (7.14) in the SSSG theory is γ = 1 + 2∆ h | ξ | β ( ξ − ξ ∗ )(1 − ξ ∗ )(1 + ξ ) F − / F F † F / + ξγ ( | ξ | − − /ξ ∗ )(1 + ξ ) F / θ F F † F / + ξ ∗ γ ( | ξ | − − ξ ∗ )(1 + 1 /ξ ) F − / F F † θ F − / − β ( ξ − ξ ∗ )(1 − /ξ ∗ )(1 + 1 /ξ ) F / θ F F † θ F − / i . (7.22)In the rest frame, p = π , this solution has the explicit form γ = γ T γ γ Ω † γ Ω 1 + ( γ − ΩΩ † , (7.23)where γ = e iη ( r − i ) | Ω | + 2 ir cos 2 T + ( r −
1) cosh 2 X ( r + i ) | Ω | − ir cos 2 T + ( r −
1) cosh 2
X ,γ = e iη ( r − i ) e − iη | Ω | − ir cos 2 T + ( r −
1) cosh 2 X ( r + i ) | Ω | + 2 ir cos 2 T + ( r −
1) cosh 2
X ,γ = e − iη/ ( r + i ) e iη | Ω | − ir cos 2 T + ( r −
1) cosh 2 X ( r − i ) | Ω | + 2 ir cos 2 T + ( r −
1) cosh 2
X ,γ = − e − iη/ r sin( T + iX )( r − i ) | Ω | + 2 ir cos 2 T + ( r −
1) cosh 2
X ,γ = − e − iη/ r sin( T − iX )( r − i ) | Ω | + 2 ir cos 2 T + ( r −
1) cosh 2
X , (7.24)where e iη = ( r + i ) / ( r − i ), and where T = r − r + 1 t , X = 2 rr + 1 x . (7.25)These solutions have vanishing SSSG charges Q L = Q R = 0. The mass of the solutioncan be computed using the expression for the energy in (3.29) and one finds M = 8 r r = 4 cos α . (7.26)Notice that it is more meaningful to write the result in terms of the parameter α defined in (4.21). The energy of the general moving solution (3.29) is E = 8 r r (cid:12)(cid:12) sin p (cid:12)(cid:12) . (7.27) In general this solution has det γ = e iC , for a constant C and so in order that γ ∈ G we shouldre-scale it by an appropriate compensating factor, as is done below in the explicit expressions. – 37 –he solution with | ξ | = 1, is obtained by first taking the limit | Ω | → r → γ = − T − ( x ) 2 tanh( x ) sech ( x ) Ω † x ) sech ( x ) Ω 1 − ( x ) ΩΩ † , (7.28)which is a static solution.
8. The Spheres
In this case the target space of the sigma model is the symmetric space S n ≃ SO ( n +1) /SO ( n ), and the symmetric space is picked out by the three involutions σ (1)+ ( F ) = F †− , σ (2)+ ( F ) = F ∗ , σ − ( F ) = θ F θ − , (8.1)where θ is given in (4.2). Notice that σ − is of type σ in the list (5.32) and isconsequently holomorphic. The Pohlmeyer reduction is defined by taking Λ ± as in(4.6). If we compare with the discussion of C P n the only difference is the realitycondition F ∗ = F .The simplest magnon solution is obtained by considering the dressing transfor-mation with a pair of poles ξ and 1 /ξ , where ξ is a phase. The constraints on thecollective coordinates are (with ̟ = ̟ ) ̟ = θ ̟ , ̟ ∗ = θ ̟ , ̟ † θ ̟ = 0 . (8.2)These are precisely the same conditions on the magnon of the C P n case with r = 1,with an additional reality condition. So just as in (7.13) we have ̟ = i e + Ω , (8.3)where now Ω is a real unit vector orthogonal to e and e . Hence, the magnon hasan internal collective coordinate taking values in S n − . This magnon is precisely theHofman-Maldacena magnon (4.11). The soliton in the associated SSSG theory isprecisely the r = 1 solution in the C P n case (7.28) with the additional restrictionthat Ω is real. – 38 – . Dyonic Magnons/Solitons One characteristic feature of the magnon/soliton solutions that we have generatedusing the dressing transformation acting on the vacuum solution is that they carrya non-trivial moduli space of internal collective coordinates [3, 14].Usually when solitons have internal collective coordinates one expects there aremore general solutions for which the collective coordinates become time dependent.For a static soliton, the resulting motion is simply geodesic motion on the modulispace corresponding to a metric which is constructed from the inner products of theassociated zero modes. When the moduli space arises from the action of a globalsymmetry then the metric will be invariant under the symmetry. The situation isfamiliar for BPS monopoles in gauge theories. In this case the monopoles carry aninternal S moduli space which can be thought of as the U (1) charge orientation of themonopole. A more general solution, the dyon, exists where the angle parameterizingthe S rotates with constant angular velocity. An important lesson for our presentsituation is that the dyon solution now carries electric charge as a consequence of themotion. Finding the dyon is not easy because the motion of the collective coordinatehas a non-trivial back-reaction on the original solution.In the present context, it is important to understand the action of the symmetrieson the collective coordinates. First of all, recall that the sigma model with targetspace a symmetric space has a global F symmetry under which F → U F σ − ( U − ), U ∈ F . Once the Pohlmeyer reduction is performed, this symmetry correspondsto f ± → U f ± , which leaves the SSSG field γ = f − − f + invariant. Notice that thevacuum solution is invariant under the subgroup H ⊂ G ⊂ F . Hence, the transfor-mations U ∈ H on a magnon have a well defined action on the collective coordinates ̟ → U ̟ , i.e. Ω → U Ω in the C P n and S n cases. On the other hand, the SSSGtheory exhibits a global H L × H R symmetry that acts as f ± → f ± h − ± or, equiva-lently, γ → h − γh − , where h ± ∈ H . In particular, the vector subgroup γ → U γU − of transformations leaves the vacuum invariant and acts as a transformation on thesoliton’s collective coordinates in the same way as above: ̟ → U ̟ . So the sym-metry group H action on the collective coordinates can be interpreted in terms ofa transformation of both the magnon’s and soliton’s collective coordinates where H ⊂ F and H ⊂ G , respectively. This symmetry will play an important rˆole infixing the geometry on the moduli space of collective coordinates.For example, for the cases M = C P n , the general magnon/soliton, (7.14) and We are assuming here that H ( ± ) , the subgroups of G that commute with Λ ± , are equal to H since in this paper we have Λ + = Λ − . – 39 –7.23), has an internal collective coordinate Ω which is a complex n − n + 1-vector orthogonal to e and e ). For the particular solutionwith r = 1, we have the additional constraint | Ω | = 1, so that the moduli space ofcollective coordinates is S n − . In both cases there is a natural action of H = U ( n − M = S n the soliton has a moduli space of collective coordinatesequal to S n − parameterized by the real unit length n − Ω (again presentedas an n + 1-vector orthogonal to e and e ) on which there is a natural action of H = SO ( n − M = S = SO (4) /SO (3). In this case Ω is a unit2-vector in the subspace spanned by e and e . Allowing it to rotate with constantangular velocity, Ω ( t ) = cos( t sin α ) e + sin( t sin α ) e , (9.1)leads to Dorey’s dyonic magnon. However, in order to compute the complete back-reacted solution, it is more convenient to notice that there is another realization ofthe S = SO (4) /SO (3) model as the principal chiral model for G = SU (2). Theexplicit map is F = (cid:18) X + iX iX + X iX − X X − iX (cid:19) ∈ SU (2) . (9.2)In the SU (2) formulation, the dyonic magnon is just the ordinary magnon solutionwhich we described in Section 6. In particular, we wrote (9.1) in such a way thatthe parameter α is the same as the one that appears as a parameter of the SU (2)magnon.The dyonic magnon gives a dyonic generalization of the SSSG soliton as wedescribed in Section 4 for the more general case with M = S n , n >
3. In particular,the solution has non-trivial gauge fields A (+) L and A ( − ) R , and, as we also explained inSection 4, the dyonic solution can also be embedded in C P n , for n ≥ S → R P → C P n . However, because the symmetry H = U ( n −
1) is notlarge enough to fix the metric on S n − there should exist another inequivalent classof dyonic solutions. In more detail, invariance under U ( n −
1) fixes the metric to bea linear combination ds = d Ω † · d Ω + ξ ( d Ω † · Ω − Ω † · d Ω ) , (9.3)up to overall scaling. When ξ = 0 we have the usual spherically symmetric metricand in this case there are no new dyon solutions. However, when ξ = 0, the new– 40 –lass of dyon solutions are associated to geodesics of the form Ω ( t ) = e ht p , (9.4)where we can choose the overall orientation so that p = (1 , , . . . , h can be found by solving the geodesic equations for the metric (9.3).There are two classes of solution, firstly h = i (cid:18) w † w (cid:19) , (9.5)where w is a complex n − h = i v w T w (8 ξ − v w w T / | w | ! , (9.6)where v is a real number and w is a real n − Ω ( t ) = (cid:0) e ivt , , . . . , (cid:1) . (9.7)Such dyons will carry charge lying in the abelian subalgebra defined by h . Forexample in the case M = C P considered at the end of Section 3, Ω = Ω is just acomplex 1-vector (or number) and only the new class of dyons with Ω( t ) = e ivt willexist. In terms of the Lagrangian formulation via axial gauging in (3.63), the dyonwill correspond to a solution for which ˜ ψ = vt . Finding the back-reaction on thefields ϕ ( x ) and θ ( x ) is a difficult challenge that we will not solve here.
10. Conclusions and outlook
In this work we have considered the interplay between the magnons in the sigmamodel describing string motion of certain symmetric spaces and the solitons of therelated SSSG equations. A notable result is that the dressing procedure produces themagnon and soliton at the same time without the need to implement the complicatedmap between the two systems. We have also described how the dressing procedure inits current understanding cannot produce the more general dyonic magnon/solitonsolutions which involve the non-trivial motion of the internal collective coordinates.It would be interesting to try to find a generalization of the dressing method whichproduces such dyonic solutions directly from the vacuum. In this work we haverestricted ourselves to the simplest compact symmetric spaces and also to the simplestsingle magnon/soliton solutions: generalizations will be presented elsewhere.– 41 – cknowledgments
JLM thanks the Galileo Galilei Institute for Theoretical Physics for the hospital-ity and the INFN for partial support while this work was in progress. His work waspartially supported by MICINN (Spain) and FEDER (FPA2008-01838 and FPA2008-01177), by Xunta de Galicia (Consejer´ıa de Educaci´on and PGIDIT06PXIB296182PR),and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042).TJH would like to acknowledge the support of STFC grant ST/G000506/1 and thehospitality of the Department of Particle Physics and IGFAE at the University ofSantiago de Compostela.
Appendix A: Relation to the Gauged Sigma ModelApproach
In this appendix we summarize the relationship between the approach describedin [12] (see also [23]) and the formulation of the
F/G symmetric space sigma modelused in Section 2 in terms of the principal chiral model for F . In [12], the F/G symmetric space sigma model is formulated with two fields f ∈ F and B µ ∈ g subject to the gauge symmetry f → f g − , B µ → g ( B µ + ∂ µ ) g − , g ∈ G . (A.1)If the Lie group F is simple, the nonlinear sigma model is defined by the Lagrangian L = − κ Tr (cid:0) J µ J µ (cid:1) , (A.2)where the current J µ = f − ∂ µ f − B µ → gJ µ g − is covariant under gauge transfor-mations.The relationship between the two formulations relies on the fact that the solutionspace of the F/G sigma model can be realized as a subspace of the solution space ofthe F principal chiral model, which is a consequence of the following result due toCartan [34]: The smooth mappingΦ : F/G → F , with f G Φ( f G ) = σ − ( f ) f − , (A.3)is a local diffeomorphism of F/G onto the closed totally geodesic submanifold M = { f ∈ F : σ − ( f ) = f − } , where σ − is the involution of F that fixes G ⊂ F andgives rise to the canonical decomposition (2.2). Examples of this map can be found– 42 –n Appendix C. Taking (A.3) into account, the explicit connection between the twomodels was worked out in [34] making use of the gauge-invariant field F = σ − ( f ) f − (A.4)that trivially satisfies the constraint (2.4); namely, σ − ( F ) = F − . Notice thatin (A.2) the gauge fields B µ are just Lagrangian multipliers whose equations-of-motion are J µ (cid:12)(cid:12) g = 0, which is equivalent to B µ = f − ∂ µ f (cid:12)(cid:12) g and J µ = f − ∂ µ f (cid:12)(cid:12) p . (A.5)Then, it is easy to check that J µ = ∂ µ F F − = − σ − ( f ) J µ σ − ( f − ) , (A.6)and the Lagrangian (A.2) becomes L = − κ Tr (cid:0) J µ J µ (cid:1) = − κ Tr (cid:0) J µ J µ (cid:1) , (A.7)which is the Lagrangian of the F principal chiral model subject to the constraint (2.4).Moreover, using the identity D µ J ν = ∂ µ J ν + [ B µ , J ν ] = − σ − ( f − ) (cid:18) ∂ µ J ν −
12 [ J µ , J ν ] (cid:19) σ − ( f ) , (A.8)the equations-of-motion of the F/G symmetric space sigma model become D ± J ∓ = 0 ⇒ ∂ ± J ∓ − (cid:2) J ± , J ∓ (cid:3) = 0 , (A.9)which are just (2.12).Now, taking (A.6) into account, the constraints that specify the Pohlmeyer re-duction of the model in terms of constrained principal chiral model field F can beimported directly from the Eqs. (3.11) and (3.17) of [12]: ∂ ± F F − = − σ − ( f ) J ± σ − ( f − ) = − σ − ( f ) (cid:16) g ± (cid:0) µ ± Λ ± (cid:1) g − ± (cid:17) σ − ( f − ) , (A.10)where g ± ∈ G , Λ ± ∈ a , and a is a maximal abelian subspace of p in (2.2). Theycorrespond to (3.1) with f ± = σ − ( f ) g ± ∈ F where, for simplicity and without loss ofgenerality, we have fixed µ ± = − . One can think of these overall scales multipliersas having been absorbed into Λ ± . Moreover, since g ± ∈ G , it is straightforward tocheck that σ − ( f ± ) = F − f ± , and that γ = g − − g + = f − − f + takes values in G , inagreement with (3.4) and (3.5), respectively.– 43 – ppendix B: Integrability, Conserved Currents,Energy-Momentum Tensor and Lorentz Transformations In order to uncover the integrability of the SSSG equations (3.13), it is useful toformulate them as the zero curvature condition[ L + , L − ] = 0 , (B.1)with the components of the Lax operator L µ given by L + = ∂ + + γ − ∂ + γ + γ − A ( L )+ γ − z Λ + ≡ L + ( x ± , γ, A ( L )+ ; z ) , L − = ∂ − + A ( R ) − − z − γ − Λ − γ ≡ L − ( x ± , γ, A ( R ) − ; z ) . (B.2)In the above z , the spectral parameter , is an arbitrary auxiliary parameter whoseintroduction plays a key rˆole in establishing the integrability of the theory. The zerocurvature condition gives rise to an infinite number of conserved densities labeled bytheir spin. The ones corresponding to spin 1 and 2 provide the usual Noether currentsand the components of the stress-energy tensor, respectively. It is important to recallthat the zero curvature condition is subject to the gauge symmetry transformations γ → h − γ h − , A ( R ) − → h + (cid:0) A ( R ) − + ∂ − (cid:1) h − , A ( L )+ → h − (cid:0) A ( L )+ + ∂ + (cid:1) h − − . (B.3)We can deduce the form of those conserved densities using the “Drinfeld-Sokolovprocedure” [35]. In order to do that, we notice that, with the introduction of thespectral parameter, the Lax operator can be written in terms of the affine algebra f (1) = X k ∈ Z (cid:16) z k ⊗ g + z k +1 ⊗ p (cid:17) = M k ∈ Z f (1) k (B.4)by means of z Λ + ≡ z ⊗ Λ + ∈ f (1)1 , z − Λ − ≡ z − ⊗ Λ − ∈ f (1) − , A ( L/R ) ± ≡ ⊗ A ( L/R ) ± ∈ f (1)0 . (B.5)Moreover, γ takes values in G that is the group associated to the Lie algebra f (1)0 .Next, we introduce Φ (+) ∈ exp( f (1) < ), and solveΦ (+) (cid:16) ∂ + + γ − ∂ + γ + γ − A ( L )+ γ − z Λ + (cid:17) Φ − = ∂ + − z Λ + + h (+) , (B.6)with h (+) = X k ≤ z − k h (+) − k ∈ Ker (cid:0)
Ad(Λ + ) (cid:1) ∪ f (1) ≤ . (B.7)– 44 –orrespondingly,Φ (+) (cid:16) ∂ − + A ( R ) − − z − γ − Λ − γ (cid:17) Φ − = ∂ − + I (+) , I (+) ∈ f (1) ≤ . (B.8)Then, the zero curvature condition implies (cid:2) ∂ + − z Λ + + h (+) , ∂ − + I (+) (cid:3) = 0 , (B.9)The components of h (+) and I (+) on Cent (cid:0) Ker (cid:0)
Ad(Λ + ) (cid:1) provide an infinite set oflocal conserved densities, while the other components provide non-local conservedones. A second set of conserved quantities can be constructed starting from γ (cid:0) ∂ − + A ( R ) − − z − γ − Λ − γ (cid:1) γ − = ∂ − − ∂ − γγ − + γA ( R ) − γ − − z − Λ − (B.10)instead of L + .The explicit expression of the densities of spin 1 and 2 can be found by writingΦ (+) = exp (cid:16)X k ≥ z − k y − k (cid:17) , z − k y − k ∈ f (1) − k , (B.11)and looking at the first components of (B.6), which read h (+)0 −
12 [Λ + , y − ] = γ − ∂ + γ + γ − A ( L )+ γ ≡ q (B.12a) h (+) − −
12 [Λ + , y − ] = − ∂ + y − + [ y − , q ] −
14 [ y − , [ y − , Λ + ]] (B.12b) · · · · · · · · · · · · · · · · · · Using (3.19), eq. (B.12a) provides y − ∈ Im (cid:0) Ad(Λ + ) (cid:1) , h (+)0 = P h + (cid:0) γ − ∂ + γ + γ − A ( L )+ γ (cid:1) = A ( R )+ , (B.13)where we have also used (3.18). In turn, (B.8) gives I (+)0 = A ( R ) − . (B.14)Therefore, the 0-grade component of (B.9) on Ker (cid:0) Ad(Λ + ) (cid:1) reads (cid:2) ∂ + + A ( R )+ , ∂ − + A ( R ) − (cid:3) = 0 , (B.15)which is one of the two equations in (3.17). The other is obtained is a similar waystarting from (B.10) instead of L + . The local and non-local conserved quantitiesprovided by these equations and their interpretation are extensively discussed inSection 3. – 45 –he components of the stress-energy tensor are found by looking at the compo-nents of h (+) − and I (+) − along Λ + . Using (B.12b),Tr (cid:0) Λ + h (+) − (cid:1) = Tr (cid:16) Λ + (cid:0) [ y − , q ] −
14 [ y − , [ y − , Λ + ]] (cid:1)(cid:17) = − Tr (cid:16) ( q − h (+)0 ) (cid:17) ≡ T ++ . (B.16)Correspondingly, (B.8) providesTr (cid:0) Λ + I (+) − (cid:1) = −
12 Tr (cid:16) Λ + γ − Λ − γ (cid:17) ≡ − T − + , (B.17)and (B.9) leads to ∂ + T − + + ∂ − T ++ = 0 . (B.18)The component T −− is obtained is a similar fashion starting from (B.10) instead of L + . Then, the complete set of components of the energy-momentum tensor can bewritten as T ++ = −
12 Tr (cid:16) ( q − h (+)0 ) (cid:17) = −
12 Tr h(cid:0) ∂ + γγ − + A ( L )+ (cid:1) − A ( R )+ 2 i (B.19a) T −− = −
12 Tr h(cid:0) γ − ∂ − γ − A ( R ) − (cid:1) − A ( L ) − i (B.19b) T − + = T − + = + 14 Tr h Λ + γ − Λ − γ i , (B.19c)and it can be easily checked that these expressions are gauge invariant.The formulation in terms of the Lax operator L ± is also useful to discuss thebehaviour of the reduced equations under Lorentz transformations. The SSSG equa-tions (3.13) are Lorentz invariant, which means that given a solution γ = γ ( x + , x − ) , A ( L )+ = A ( L )+ ( x + , x − ) , A ( R ) − = A ( R ) − ( x + , x − ) (B.20)we can generate a boosted one by simply γ → γ λ = γ ( λ − x + , λx − ) ,A ( L )+ → A ( L )+ λ = λ − A ( L )+ ( λ − x + , λx − ) ,A ( R ) − → A ( R ) − λ = λ +1 A ( R ) − ( λ − x + , λx − ) . (B.21)This is equivalent to saying that the zero-curvature condition is invariant under thetransformations x ± → λ ± x ± , γ → γ, A ( L )+ → λ − A ( L )+ , A ( R ) − → λ +1 A ( R )+ . (B.22)Correspondingly, the Lax operators (B.2) transform as L + ( x ± , γ, A ( L )+ ; z ) → λ − L + ( x ± , γ, A ( L )+ ; λz ) , L − ( x ± , γ, A ( R ) − ; z ) → λ +1 L − ( x ± , γ, A ( R ) − ; λz ) . (B.23)– 46 –n other words, the Lorentz transformation (B.22) is equivalent to the re-scaling ofthe spectral parameter z → λz , and the zero-curvature condition is invariant becauseit does not depend on z . Then, in (B.6) the Lorentz transformation (B.23) inducesthe following transformation on the conserved densities: h (+) − j → λ − − j h (+) − j (B.24)which, in particular, shows that h (+)0 is of spin 1 (currents) and, therefore, that thecorresponding conserved charges are Lorentz invariant.In contrast to the SSSG equations, the Pohlmeyer reduced sigma model is notLorentz invariant, as a consequence of the constraints (3.1). However, we can usethe formulation of the former in term of the Lax operators L ± to deduce a formalexpression for the action of Lorentz transformations on the solutions to the reducedsigma model equations-of-motion. Consider the solutions to the z -dependent auxil-iary linear problem L + ( x ± , γ, A ( L )+ ; z )Υ − ( z ) = L − ( x ± , γ, A ( R ) − ; z − )Υ − ( z ) = 0 (B.25)where Υ( z ) ≡ Υ (cid:0) x ± , γ, A ( L )+ , A ( R ) − ; z (cid:1) , whose integrability conditions are provided bythe zero-curvature equation (B.1). As explained in [12], in the gauged sigma modelapproach the reduced sigma model configuration corresponding to a given SSSGsolution (cid:8) γ, A ( L )+ , A ( R ) − (cid:9) is specified by the solution to (B.25) for z = 1; namely, f = Υ(1). Then, (B.23) shows that under a Lorentz transformation Υ( z ) → Υ( λz ),which induces the following transformation of the reduced sigma model configuration: f = Υ(1) −→ f λ = Υ( λ ) . (B.26) Appendix C: The Spheres and Complex Projective Spaces
In this appendix, we explain how to map the spaces S n and C P n , expressed interms of their usual coordinates, into the group valued field F given by (A.4).A generic f ∈ SO ( n + 1) satisfies f f T = 1 which is equivalent to f ac f bc = δ ab .Then, F = σ − ( f ) f − = θf θf T (C.1)which, in terms of components, reads F ab = θ ac (cid:16) δ cb − f c f b (cid:17) . (C.2)Now, for a symmetric space M = F/G , we have to use that F = I ( M ) is the identitycomponent of the group of isometries of M , and that it acts transitively on M = F/G .– 47 –his means that M = F · p for an arbitrary point p ∈ M and, moreover, that G is the isotropy group (or little group) of p . In our case, for S n = SO ( n + 1) /SO ( n )we can take p = (1 , , . . . , f is X = f · p ⇒ X a = f a (C.3)Then, (C.2) becomes F = θ (cid:16) − XX T (cid:17) , (C.4)which is the parameterization we are looking for in terms of the unit vector X , | X | = 1. Notice that the map X → F , which provides a particular example of (A.3),is surjective but not injective.A similar argument can be followed for the case of the complex projective spaces,in which case (C.4) is replaced by F = θ (cid:16) − ZZ † | Z | (cid:17) , (C.5)where Z is a vector whose components are the usual n + 1 projective coordinates of C P n . In this case, the map Z → F is one-to-one. Appendix D: The SSSG Equations-of-Motion
In this appendix we prove that the dressing procedure produces solutions of theSSSG equations-of-motion (3.13) with vanishing gauge fields. To start with, using(5.4) along with (5.6) and (5.8) one quickly deduces ∂ ± χ ( λ ) χ ( λ ) − = χ ( ∓ ± χ ( ∓ − − χ ( λ )Λ ± χ ( λ ) − ± λ (D.1)from which it follows that ∂ ± χ ( ± χ ( ± − = (cid:0) χ ( ∓ ± χ ( ∓ − − χ ( ± ± χ ( ± − (cid:1) . (D.2)By writing γ = F − / χ (+1) − χ ( − F / , and using ∂ ± F = Λ ± F , we have γ − ∂ + γ = − F − / χ ( − − χ (+1)Λ + χ (+1) − χ ( − F / + Λ + − F − / χ ( − − ∂ + χ (+1) χ (+1) − χ ( − F / + F − / χ ( − − ∂ + χ ( − F / . (D.3)Using the upper-sign identity (D.2), one sees that the third term cancels the firsttwo, to leave γ − ∂ + γ = F − / χ ( − − ∂ + χ ( − F / . (D.4)– 48 –hen ∂ − (cid:0) γ − ∂ + γ (cid:1) = − Λ − F − / χ ( − − ∂ + χ ( − F / + F − / χ ( − − ∂ + χ ( − F / Λ − + F − / χ ( − − ∂ + (cid:0) ∂ − χ ( − χ ( − − (cid:1) χ ( − F / (D.5)Next, we use the lower-sign identity (D.2) to re-write the third term as ∂ + (cid:0) ∂ − χ ( − χ ( − − (cid:1) = − ∂ + χ ( − − χ ( − − + χ ( − − χ ( − − ∂ + χ ( − χ ( − − + ∂ + χ (+1)Λ − χ (+1) − − χ (+1)Λ − χ (+1) − ∂ + χ (+1) χ (+1) − . (D.6)The first two terms cancel the first two terms in (D.5) to leave ∂ − (cid:0) γ − ∂ + γ (cid:1) = F − / χ ( − − ∂ + χ (+1)Λ − χ (+1) − χ ( − F / − F − / χ ( − χ (+1)Λ − χ (+1) − ∂ + χ (+1) χ (+1) − χ ( − F / . (D.7)Finally, we use the upper-sign identity in (D.2) again and the fact that [Λ + , Λ − ] = 0,to end up with ∂ − (cid:0) γ − ∂ + γ (cid:1) = 14 [Λ + , γ − Λ − γ ] . (D.8)This is (3.13) with A ( L )+ = A ( R ) − = 0.The next thing to prove is that the constraints (3.30) are satisfied. Taking theresidue of the upper sign in (D.1) at λ = −
1, gives ∂ + χ ( − χ ( − − = − ∂ λ χ ( − + χ ( − − + χ ( − + χ ( − − ∂ λ χ ( − χ ( − − . (D.9)Substituting this in (D.4), gives γ − ∂ + γ = − [ F − / χ ( − − ∂ λ χ ( − F / , Λ + (cid:3) . (D.10)with a similar expression for ∂ − γγ − . Hence, γ − ∂ + γ and ∂ − γγ − are in the image ofthe adjoint action of Λ ± . Then, provided that the orthogonal decompositions (3.19)hold, which is always true if the symmetric space has definite signature, the con-straints (3.30) are satisfied. References [1] J. M. Maldacena, Adv. Theor. Math. Phys. (1998) 231 [Int. J. Theor. Phys. (1999) 1113] [arXiv:hep-th/9711200]. – 49 –
2] K. Zarembo, Comptes Rendus Physique (2004) 1081 [Fortsch. Phys. (2005)647] [arXiv:hep-th/0411191];M. Staudacher, JHEP (2005) 054 [arXiv:hep-th/0412188];J. A. Minahan, J. Phys. A (2006) 12657;N. Gromov, V. Kazakov and P. Vieira, PoS SOLVAY (2006) 005[arXiv:hep-th/0703137];N. Beisert, PoS
SOLVAY (2006) 002 [arXiv:0704.0400 [nlin.SI]];G. Arutyunov and S. Frolov, arXiv:0901.4937 [hep-th].[3] D. M. Hofman and J. M. Maldacena, J. Phys. A (2006) 13095[arXiv:hep-th/0604135].[4] N. Dorey, J. Phys. A , 13119 (2006) [arXiv:hep-th/0604175];H. Y. Chen, N. Dorey and K. Okamura, JHEP , 024 (2006)[arXiv:hep-th/0605155].[5] K. Pohlmeyer, Commun. Math. Phys. (1976) 207.[6] A. A. Tseytlin, arXiv:hep-th/0311139.[7] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B (1998) 105[arXiv:hep-th/9802109].[8] S. Frolov and A. A. Tseytlin, JHEP (2002) 007 [arXiv:hep-th/0204226].[9] D. Gaiotto, S. Giombi and X. Yin, arXiv:0806.4589 [hep-th].[10] G. Grignani, T. Harmark and M. Orselli, arXiv:0806.4959 [hep-th].[11] M. C. Abbott and I. Aniceto, arXiv:0811.2423 [hep-th].[12] J. L. Miramontes, JHEP (2008) 087 [arXiv:0808.3365 [hep-th]].[13] V. E. Zakharov and A. V. Mikhailov, Commun. Math. Phys. , 21 (1980);V. E. Zakharov and A. V. Mikhailov, Sov. Phys. JETP , 1017 (1978) [Zh. Eksp.Teor. Fiz. , 1953 (1978)].[14] M. Spradlin and A. Volovich, JHEP (2006) 012 [arXiv:hep-th/0607009];A. Jevicki, C. Kalousios, M. Spradlin and A. Volovich, JHEP (2007) 047[arXiv:0708.0818 [hep-th]];C. Kalousios, M. Spradlin and A. Volovich, JHEP (2007) 020[arXiv:hep-th/0611033].[15] A. Jevicki, C. Kalousios, M. Spradlin and A. Volovich, JHEP (2007) 047[arXiv:0708.0818 [hep-th]].[16] G. W. Gibbons and N. S. Manton, Nucl. Phys. B (1986) 183. – 50 –
17] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, JHEP (2008) 091[arXiv:0806.1218 [hep-th]];M. Benna, I. Klebanov, T. Klose and M. Smedback, JHEP (2008) 072[arXiv:0806.1519 [hep-th]].[18] D. Astolfi, V. G. M. Puletti, G. Grignani, T. Harmark and M. Orselli,arXiv:0807.1527 [hep-th].[19] G. Grignani, T. Harmark, M. Orselli and G. W. Semenoff, arXiv:0807.0205 [hep-th].[20] C. R. Fernandez-Pousa, M. V. Gallas, T. J. Hollowood and J. L. Miramontes, Nucl.Phys. B (1997) 609 [arXiv:hep-th/9606032];J. L. Miramontes and C. R. Fernandez-Pousa, Phys. Lett. B (2000) 392[arXiv:hep-th/9910218];P. Dorey and J. L. Miramontes, Nucl. Phys. B (2004) 405[arXiv:hep-th/0405275].[21] C. R. Fernandez-Pousa and J. L. Miramontes, Nucl. Phys. B (1998) 745[arXiv:hep-th/9706203].[22] J. P. Harnad, Y. Saint Aubin and S. Shnider, Commun. Math. Phys. (1984) 329.[23] H. Eichenherr and M. Forger, Commun. Math. Phys. (1981) 227.[24] I. Bakas, Q. H. Park and H. J. Shin, Phys. Lett. B (1996) 45[arXiv:hep-th/9512030].[25] M. Grigoriev and A. A. Tseytlin, Nucl. Phys. B (2008) 450 [arXiv:0711.0155[hep-th]].[26] J. L. Miramontes, Nucl. Phys. B (2004) 419 [arXiv:hep-th/0408119].[27] J. Zinn-Justin, “Quantum field theory and critical phenomena”, Oxford UniversityPress: International Series of Monographs in Physics (2002).[28] N. S. Manton, Phys. Lett. B (1982) 54.[29] Q. H. Park, Phys. Lett. B (1994) 329 [arXiv:hep-th/9402038].[30] F. Lund and T. Regge, Phys. Rev. D (1976) 1524 ;F. Lund, Phys. Rev. Lett. (1977) 1175;B. S. Getmanov, Pisma Zh. Eksp. Teor. Fiz. , 132 (1977);Q. H. Park and H. J. Shin, Phys. Lett. B (1995) 125 [arXiv:hep-th/9506087].[31] N. Dorey and T. J. Hollowood, Nucl. Phys. B (1995) 215[arXiv:hep-th/9410140].[32] P. Bowcock, D. Foster and P. Sutcliffe, arXiv:0809.3895 [hep-th].[33] R. Sasaki, J. Math. Phys. (1985) 1786. – 51 –
34] H. Eichenherr and M. Forger, Nucl. Phys. B (1980) 528 [Erratum-ibid. B (1987) 745].[35] M. F. De Groot, T. J. Hollowood and J. L. Miramontes, Commun. Math. Phys. (1992) 57 ;J. L. Miramontes, Nucl. Phys. B , 623 (1999) [arXiv:hep-th/9809052];, 623 (1999) [arXiv:hep-th/9809052];