An integrable deformation of the type IIB AdS5xS5 superstring action is presented. The deformed field equations, Lax connection, and k-symmetry transformations are given. The original psu (2,2\4) symmetry is expected to become q deformed.Peer reviewedSubmitted Versio

B. Vicedo

E. Witten

F. Delduc

J. M. Maldacena

M. Magro

Physical Review Letters

English

arXiv

Crossref

Integrable Deformation of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>AdS</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">S</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>Superstring Action

An integrable deformation of the type IIB AdS5xS5 superstring action is presented. The deformed field equations, Lax connection, and k-symmetry transformations are given. The original psu (2,2\4) symmetry is expected to become q deformed

Magro, Marc

Delduc, Francois

Vicedo, Benoit

University of Hertfordshire Research Archive

arXiv:1309.5850v1  [hep-th]  23 Sep 2013An integrable deformation of the AdS5 × S5 superstring actionF. Delduc,1 M. Magro,1 and B. Vicedo21Laboratoire de Physique, ENS Lyon et CNRS UMR 5672, Université de Lyon,46, allée d’Italie, 69364 LYON Cedex 07, France2School of Physics, Astronomy and Mathematics, University of Hertfordshire,College Lane, Hatfield AL10 9AB, United KingdomAn integrable deformation of the type IIB AdS5×S5 superstring action is presented. The deformedfield equations, Lax connection, and κ-symmetry transformations are given. The original psu(2, 2|4)symmetry is expected to become q-deformed.INTRODUCTIONIntegrability plays a central role in the study of theAdS/CFT correspondence [1] between type IIB super-string theory on the AdS5 × S5 background [2] and themaximally supersymmetric Yang-Mills gauge theory infour dimensions (see [3] for a review). On the Anti-deSitter side of this correspondence, integrability enteredthe scene with the discovery that the lagrangian fieldequations of the AdS5 × S5 theory can be recast in thezero curvature form [4]. This implies the existence of aninfinite number of conserved quantities.It is quite natural to seek deformations of the AdS5×S5superstring which preserve this integrable structure. Animportant example of such an integrable deformation isthe so called β-deformation associated with strings on theLunin-Maldacena background [5]. The integrability ofthis model was shown in [6, 7] (see also the review [8] andreferences therein). Here we shall take a more systematicapproach to the construction of integrable deformationsby demanding the deformed theory to be integrable fromthe very outset. This requires approaching the problemfrom the hamiltonian perspective.Let us recall that in order to prove integrability in thehamiltonian formalism, one must show the existence ofan infinite number of conserved quantities in involution.More precisely, this follows at once if the Poisson bracketof the hamiltonian Lax matrix can be shown to take thespecific form in [9, 10]. This was achieved in the case ofthe AdS5 × S5 superstring in [11].The algebraic structure underpinning this property ofthe AdS5×S5 superstring was identified in [12]. By util-ising this structure, an alternative Poisson bracket withthe same property was subsequently constructed in [13].Moreover, this second Poisson bracket is compatible withthe original one, giving rise to a one-parameter family ofPoisson brackets sharing the same property [9, 10] whichensures integrability.These features of the superstring theory are in factshared with bosonic integrable σ-models [14]. In this lat-ter context, the two compatible Poisson brackets wereused very recently in [15] as a building block for con-structing integrable q-deformations of the principal chi-ral model associated with a compact Lie group and of theσ-model on a symmetric space F/G with F compact. Inthe case of the principal chiral model, the deformationcoincides with the Yang-Baxter σ-model introduced byKlimč́ık in [16]. A key characteristic of this procedureis that the integrability of the deformed theories is au-tomatic since it is used as an input in the construction.Moreover, an interesting output is that the symmetryassociated with left multiplication in the original mod-els is deformed into a classical q-deformed Poisson-Hopfalgebra.It is possible to generalize the method developed in [15]to deform the AdS5 × S5 superstring theory. The wholeconstruction is carried out at the hamiltonian level andwill be presented in detail elsewhere. The purpose of thisletter is to present the deformed action and indicate itsproperties.SETTINGWe begin by recalling the necessary ingredients fordefining the AdS5 × S5 superstring action (see [17] formore details). Define the projectors Pαβ± =12(γαβ ± ǫαβ)where γαβ is the worldsheet metric with det γ = −1 andǫ01 = 1. Worldsheet indices are lowered and raised withthe two-dimensional metric. Let f denote the Grassmannenvelope of the superalgebra su(2, 2|4), namely the Liealgebraf = Gr[0] ⊗ su(2, 2|4)[0] ⊕ Gr[1] ⊗ su(2, 2|4)[1],where Gr is a real Grassmann algebra. Introduce the two-dimensional field g(σ, τ) taking value in the Lie group Fwith Lie algebra f. The corresponding vector currentAα = g−1∂αg belongs to f. The integrability of theAdS5 × S5 superstring action relies heavily on the ex-istence of an order 4 automorphism which induces a Z4-grading of the superalgebra su(2, 2|4), and thus of f. Wedenote by f(i) the subspace of f with grade i = 0, · · · , 3.The projector on f(i) shall be denoted by Pi and we alsowrite M (i) = PiM for the projection of M ∈ f on f(i).The invariant part f(0) is the Lie algebra so(4, 1)⊕ so(5),and the corresponding Lie group isG = SO(4, 1)×SO(5).The supertrace is compatible with the Z4-grading, whichmeans that Str(M (m)N (n)) = 0 for m+ n 6= 0 mod 4.2The extra ingredient needed to specify the deformationis a skew-symmetric (non-split) solution of the modifiedclassical Yang-Baxter equation on f. Specifically, this isan R-linear operator R such that, for M,N ∈ f,[RM,RN ]−R([RM,N ] + [M,RN ])= [M,N ] (1)and Str(MRN)= − Str(RMN). We shall take R to bethe restriction to su(2, 2|4) of the operator acting on thecomplexified algebra by −i on generators associated withpositive roots, +i on generators associated with negativeroots, and 0 on Cartan generators. We will make useof the operator Rg = Ad−1g ◦ R ◦ Adg which is also askew-symmetric solution of (1). Finally, we define thefollowing linear combinations of the projectors,d = P1 +21− η2P2 − P3, d̃ = −P1 +21− η2P2 + P3.The operator d̃ is the transpose operator of d and thussatisfies Str(M d(N))= Str(d̃(M)N). The real variableη ∈ [0, 1[ will play the role of the deformation parameter.DEFORMED ACTIONAs pointed out in the introduction, we will restrictourselves in this letter to presenting the deformed actionand summarising its most important properties. In thissection we shall write down this action and indicate theproperties it shares with the undeformed action. Prop-erties which depend on the deformation parameter η arepresented in the next section.ActionThe action, which can be obtained by generalising themethod developed in [15] to the case at hand, readsS[g] =∫dσdτL withL = −(1 + η2)22(1− η2)Pαβ− Str(Aα d ◦11− ηRg ◦ d(Aβ)). (2)The operator 1 − ηRg ◦ d is invertible on f for all val-ues of the deformation parameter η ∈ [0, 1[. As inthe undeformed case, there is an abelian gauge invari-ance g(σ, τ) → g(σ, τ)eiθ(σ,τ) under which the vectorfield Aα transforms as Aα → Aα + ∂αθ . 1. Indeed, thisleaves the action associated with (2) invariant becauseStr(1.M) = 0 for any M in su(2, 2|4). This invariancemeans that physical degrees of freedom do not belong tothe whole group F but rather to the projective group PF .From now on the commutators that will appear shouldbe considered as commutators of the projective algebrapf, and the adjoint action of g, Adg, is that of the projec-tive group PF . This peculiarity already appears in theundeformed case and the reader is referred for instanceto the review [17] for more details.Original Metsaev-Tseytlin actionThe undeformed action corresponds to η = 0. Indeed,when η vanishes, the Lagrangian (2) simply becomesL|η=0 = −12Pαβ− Str(Aαd|η=0(Aβ)),= − 12Str(γαβA(2)α A(2)β + ǫαβA(1)α A(3)β).One therefore recovers at η = 0 the type IIB superstringaction on the AdS5 × S5 background. This celebratedMetsaev-Tseytlin action [2] is that of a σ-model on thesemi-symmetric space PSU(2, 2|4)/G with Wess-Zuminoterm (see for instance the reviews [17, 18]) [19].SO(4, 1)× SO(5) gauge invarianceThe action corresponding to (2) has a gauge invarianceg(σ, τ) → g(σ, τ)h(σ, τ) where the function h(σ, τ) takesvalues in the subgroup G. This can be easily shown usingthe corresponding transformationsAα → h−1∂αh+Ad−1h (Aα),d(Aα) → Ad−1h ◦ d(Aα),Rg → Ad−1h ◦Rg ◦Adh.This gauge transformation does not depend on the de-formation parameter η.PROPERTIES OF THE DEFORMED ACTIONTo present the properties of the action (2), we willfollow the approach presented in the review [17] for theundeformed case.Equations of motionThe equations of motion are most conveniently writtenin terms of the vectorsJα =11− ηRg ◦ d(Aα),J̃α =11 + ηRg ◦ d̃(Aα)and their projections, Jα− = Pαβ− Jβ and J̃α+ = Pαβ+ J̃β. Inthe following, we shall often use the fact that the com-ponents J0− and J1− are proportional to each other. Onehas in particular [Jα−, Jβ−] = 0 (and similarly for J̃α+). Theequations of motion arising from the Lagrangian (2) aregiven by E = 0 whereE := d(∂αJα−)+d̃(∂αJ̃α+)+ [J̃+α, d(Jα−)]+[J−α, d̃(J̃α+)].3It is easy to check that the projection E(0) of E onto f(0)vanishes, in accordance with the gauge invariance of theaction described above.Rewriting the Maurer-Cartan equationWe now wish to address the question of integrability ofthe theory defined by (2). Recall that in the undeformedcase, in deriving the Lax connection one makes use of theMaurer-Cartan equation Z = 0 satisfied by Aα, whereZ := 12ǫαβ(∂αAβ − ∂βAα + [Aα, Aβ ]).To find a Lax connection we therefore start by rewritingZ in terms of Jα− and J̃α+. The resulting expression isa quadratic polynomial in η. Using equation (1) for theoperator Rg, one can rewrite the coefficient of η2 of thispolynomial to obtainZ = ∂αJ̃α+−∂αJα−+[J−α, J̃α+]+η2[d(J−α), d̃(J̃α+)]+ηRg(E).Anticipating the result, let us note here that choosing Rto be a non-split solution of the modified classical Yang-Baxter equation is essential in order to preserve integra-bility as we deform the theory. Before constructing theLax connection, let us remark that the field equations inthe odd sector P1,3(E) = 0 may be greatly simplified byconsidering the combinationsP1 ◦ (1− ηRg)(E) + P1(Z) = −4[J̃(2)+α, Jα(3)− ], (3a)P3 ◦ (1 + ηRg)(E) − P3(Z) = −4[J(2)−α, J̃α(1)+ ]. (3b)As a consequence, one can take as field equations in theodd sector[J̃(2)+α, Jα(3)− ] = 0, [J(2)−α, J̃α(1)+ ] = 0,which have the same form as those of the undeformedmodel written in terms of ordinary currents.Lax connectionWe define the two vectorsLα+ = J̃α(0)+ + λ√1 + η2J̃α(1)+ + λ−2 1 + η21− η2J̃α(2)++ λ−1√1 + η2J̃α(3)+ ,Mα− = Jα(0)− + λ√1 + η2Jα(1)− + λ2 1 + η21− η2Jα(2)−+ λ−1√1 + η2Jα(3)− ,where λ is the spectral parameter. Then, the whole set ofequations of motion E = 0 and zero curvature equationsZ = 0 are equivalent to∂αLα+ − ∂αMα− + [M−α, Lα+] = 0. (4)One may define an unconstrained vectorLα = L+α +M−α,in terms of which the equation (4) becomes an ordinaryzero curvature equation∂αLβ − ∂βLα + [Lα,Lβ ] = 0.The existence of this Lax connection shows that the dy-namics of the deformed action admits an infinite numberof conserved quantities.Virasoro constraintsIt is clear that each term in the Lagrangian (2) is pro-portional either to the metric γαβ or to ǫαβ . The part ofthe action proportional to the metric takes the formSγ = −12(1 + η21− η2)2 ∫dσdτγαβ Str(J (2)α J(2)β), (5a)= − 12(1 + η21− η2)2 ∫dσdτγαβ Str(J̃ (2)α J̃(2)β). (5b)To obtain this result, the skew-symmetry of Rg has beenused. The Virasoro constraints are then found to beStr(J̃α(2)+ J̃β(2)+)≈ 0, Str(Jα(2)− Jβ(2)−)≈ 0.Kappa symmetryThe invariance under κ-symmetry is a characteristic ofthe Green-Schwarz formulation. We now want to showthat the kappa invariance is essentially unchanged afterdeformation. To do this, consider an infinitesimal righttranslation of the field, δg = gǫ, where the parameter ǫtakes the formǫ = (1 − ηRg)ρ(1) + (1 + ηRg)ρ(3).The fields ρ(1) and ρ(3), whose expressions will be deter-mined shortly, respectively take values in f(1) and f(3).Then the variation of the action with respect to g readsδgS =(1+η2)22(1−η2)∫dσdτ Str(ρ(1)P3 ◦ (1 + ηRg)(E)+ ρ(3)P1 ◦ (1− ηRg)(E)).We may then use equations (3) to write this variation asδgS = −2(1+η2)2(1−η2)∫dσdτ Str(ρ(1)[J(2)−α, J̃α(1)+ ]++ ρ(3)[J̃(2)+α, Jα(3)− ]).4In full analogy with the undeformed case (see [17]), wetake the following ansatz for ρ(1) and ρ(3):ρ(1) = iκ(1)+αJα(2)− + Jα(2)− iκ(1)+α,ρ(3) = iκ(3)−αJ̃α(2)+ + J̃α(2)+ iκ(3)−α,where κ(1)+ and κ(3)− are constrained vectors of respectivegradings 1 and 3. Note that we are using the standardconvention for the real form su(2, 2|4) (see for instanceappendix C of [20]). Then, a short calculation leads toStr(ρ(1)[J(2)−α, J̃α(1)+ ])= Str(Jα(2)− Jβ(2)− [J̃(1)+α, iκ(1)+β]),Str(ρ(3)[J̃(2)+α, Jα(3)− ])= Str(J̃α(2)+ J̃β(2)+ [J(3)−α, iκ(3)−β]).At this point, we use the standard property (see [17])that the square of an element of grade 2 only containsa term proportional to W = diag(14,−14) and a termproportional to the identity which does not play a role inthe case at hand. We finally obtainδgS = −(1+η2)24(1−η2)∫dσdτ(Str(Jα(2)− Jβ(2)− )××Str(W [J̃(1)+α, iκ(1)+β])+Str(J̃α(2)+ J̃β(2)+ ) Str(W [J(3)−α, iκ(3)−β])).This expression comes from the variation of the field gin the action. It may be compensated by another termcoming from the variation of the metric γ. To determinethis variation we use the result (5). We are then led tochooseδγαβ =1− η22Str(W [iκα(1)+ , J̃β(1)+ ] +W [iκα(3)− , Jβ(3)− ])for the transformation of the metric in order to ensureκ-symmetry.CONCLUSIONThe Lagrangian (2) is a semi-symmetric space gener-alisation of the one obtained in [15] by deforming thesymmetric space σ-model on F/G. In the latter case, itwas shown that the original FL symmetry is deformedto a Poisson-Hopf algebra analogue of Uq(f). The samefate is confidently expected for the psu(2, 2|4) symmetryof the AdS5 × S5 superstring. Hence, the q-deformationproposed here generalizes the situation which holds forthe squashed sphere σ-model [21, 22].As mentioned in the introduction, the construction ofthe deformed theory relies on the existence of a secondcompatible Poisson bracket. The latter is known to berelated [13] to the Pohlmeyer reduction of the AdS5×S5superstring [20, 23]. In fact, one motivation for deform-ing the superstring action comes from the q-deformed S-matrix appearing in this context [24–26], built from theq-deformed R-matrix of [27]. It would therefore be veryinteresting to make contact between these two deforma-tions.Let us end on a more conjectural note by commentingon the limit η → 1 of the deformed model. The analogouslimit in the case of the deformed SU(2)/U(1) σ-modelcorresponds to a SU(1, 1)/U(1) σ-model [15]. If sucha property were to generalise to the case at hand, weexpect that the cosets AdS5 ≃ SO(4, 2)/SO(4, 1) andS5 ≃ SO(6)/SO(5) would respectively be replaced in thislimit by SO(5, 1)/SO(4, 1) ≃ dS5 and SO(5, 1)/SO(5) ≃H5. Such cosets have already been considered in [28].This point certainly requires closer investigation and wewill come back to it from the hamiltonian point of viewelsewhere.[1] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998),arXiv:hep-th/9711200 [hep-th]; S. Gub-ser, I. R. Klebanov, and A. M.Polyakov, Phys. Lett. B428, 105 (1998),arXiv:hep-th/9802109 [hep-th]; E. Witten,Adv. Theor. Math. Phys. 2, 253 (1998),arXiv:hep-th/9802150 [hep-th].[2] R. R. Metsaev and A. A. Tseytlin,Nucl. Phys. B533, 109 (1998), arXiv:hep-th/9805028.[3] N. Beisert and al., Lett. Math. Phys. 99, 3 (2012),arXiv:1012.3982 [hep-th].[4] I. Bena, J. Polchinski, and R. Roiban,Phys. Rev. D69, 046002 (2004), arXiv:hep-th/0305116.[5] O. Lunin and J. M. Maldacena, JHEP 0505, 033 (2005),arXiv:hep-th/0502086.[6] S. A. Frolov, R. Roiban, and A. A. Tseytlin, JHEP 07,045 (2005), arXiv:hep-th/0503192.[7] S. Frolov, JHEP 0505, 069 (2005),arXiv:hep-th/0503201.[8] K. Zoubos, Lett. Math. Phys. 99, 375 (2012),arXiv:1012.3998 [hep-th].[9] J. M. Maillet, Phys. Lett. B162, 137 (1985).[10] J. M. Maillet, Nucl. Phys. B269, 54 (1986).[11] M. Magro, JHEP 0901, 021 (2009),arXiv:0810.4136 [hep-th].[12] B. Vicedo, Lett. Math. Phys. 95, 249 (2011),arXiv:1003.1192 [hep-th].[13] F. Delduc, M. Magro, and B. Vicedo,JHEP 1210, 061 (2012), arXiv:1206.6050 [hep-th].[14] F. Delduc, M. Magro, and B. Vicedo,JHEP 1208, 019 (2012), arXiv:1204.0766 [hep-th].[15] F. Delduc, M. Magro, and B. Vicedo, (2013), 1308.3581.[16] C. Klimcik, JHEP 0212, 051 (2002),arXiv:hep-th/0210095 [hep-th].[17] G. Arutyunov and S. Frolov,J. Phys. A42, 254003 (2009), arXiv:0901.4937 [hep-th].[18] M. Magro, Lett. Math. Phys. 99, 149 (2012),arXiv:1012.3988 [hep-th].[19] To be precise, note that one must take the universal coverof PSU(2, 2|4) (see e.g. [29]).[20] M. Grigoriev and A. A. Tseytlin,Nucl. Phys. B800, 450 (2008),arXiv:0711.0155 [hep-th].5[21] I. Kawaguchi, T. Matsumoto, and K. Yoshida, JHEP1204, 115 (2012), arXiv:1201.3058 [hep-th].[22] I. Kawaguchi, T. Matsumoto, and K. Yoshida, JHEP1206, 082 (2012), arXiv:1203.3400 [hep-th].[23] A. Mikhailov and S. Schäfer-Nameki,JHEP 05, 075 (2008), arXiv:0711.0195 [hep-th].[24] B. Hoare and A. Tseytlin,Nucl. Phys. B851, 161 (2011),arXiv:1104.2423 [hep-th].[25] B. Hoare, T. J. Hollowood, and J. L. Miramontes,JHEP 1203, 015 (2012), arXiv:1112.4485 [hep-th].[26] B. Hoare, T. J. Hollowood, and J. L. Miramontes,JHEP 1210, 076 (2012), arXiv:1206.0010 [hep-th].[27] N. Beisert and P. Koroteev,J.Phys.A A41, 255204 (2008), arXiv:0802.0777 [hep-th].[28] C. Hull, JHEP 9807, 021 (1998),arXiv:hep-th/9806146 [hep-th].[29] N. Beisert, Lett. Math. Phys. 99, 529 (2012),arXiv:1012.4004 [hep-th].

An integrable deformation of the AdS5 x S5 superstring action

5 pagesInternational audienceAn integrable deformation of the type IIB AdS_5 x S^5 superstring action is presented. The deformed field equations, Lax connection, and kappa-symmetry transformations are given. The original psu(2,2|4) symmetry is expected to become q-deformed

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An integrable deformation of the AdS_5 x S^5 superstring action

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An integrable deformation of the AdS5 x S5 superstring action

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