The Poisson algebra of the Lax matrix associated with the Pohlmeyer reduction
of the AdS_5 x S^5 superstring is computed from first principles. The resulting
non-ultralocality is mild, which enables to write down a corresponding lattice
Poisson algebra.Comment: 5 page

B. Vicedo

Bakas

Beisert

Bena

Bowcock

Delduc

F. Delduc

Faddeev

Freidel

Grigoriev

Gubser

Kulish

M. Magro

Magro

Maillet

Maldacena

Metsaev

Mikhailov

Pohlmeyer

Semenov-Tian-Shansky

Vicedo

Witten

Zamolodchikov

Physics Letters B

English

arXiv

5 pagesInternational audienceThe Poisson algebra of the Lax matrix associated with the Pohlmeyer reduction of the AdS_5 x S^5 superstring is computed from first principles. The resulting non-ultralocality is mild, which enables to write down a corresponding lattice Poisson algebra

Delduc, Francois

Magro, Marc

Vicedo, Benoit

HAL: Hyper Article en Ligne

A lattice Poisson algebra for the Pohlmeyer reduction of the AdS_5 x S^5 superstring

HAL-ENS-LYON

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A lattice Poisson algebra for the Pohlmeyer reduction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="italic">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>S</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:math> superstring

The Poisson algebra of the Lax matrix associated with the Pohlmeyer reduction of the AdS5×S5 superstring is computed from first principles. The resulting non-ultralocality is mild, which enables to write down a corresponding lattice quadratic Poisson algebra, a key step towards the integrable quantization of this theory from first principle

University of Hertfordshire Research Archive

arXiv:1204.2531v1  [hep-th]  11 Apr 2012A lattice Poisson algebra for the Pohlmeyerreduction of the AdS5 × S5 superstringF. Delduc 1, M. Magro 1, B. Vicedo 21) Laboratoire de Physique, ENS Lyon et CNRS UMR 5672, Université de Lyon,46, allée d’Italie, 69364 LYON Cedex 07, France2) Department of Mathematics, University of York,Heslington, York, YO10 5DD, United KingdomFrancois.Delduc@ens-lyon.fr, Marc.Magro@ens-lyon.fr, Benoit.Vicedo@gmail.comAbstract. The Poisson algebra of the Lax matrix associated with the Pohlmeyer reduction ofthe AdS5 × S5 superstring is computed from first principles. The resulting non-ultralocality ismild, which enables to write down a corresponding lattice Poisson algebra.1 IntroductionWe recently showed in [1] that the Poisson algebra of the Lax matrix associated with symmetricspace sine-Gordon models, defined through a gauged Wess-Zumino-Witten action with an inte-grable potential [2], admits an integrable lattice discretization. In the present letter we computethe r/s-matrix structure [3] associated with the Pohlmeyer reduction of AdS5 × S5 superstringtheory [4, 5] directly from its representation in terms of a fermionic extension of a gauged WZWaction with an integrable potential. We similarly find that it is precisely of the type which, afterregularization as in [6], admits an integrable lattice discretization of the general form identified in[7, 8].2 Canonical analysis and HamiltonianTo begin with we briefly recall some usual notations. We refer the reader to [4] for more detailsconcerning this setup. The superalgebra f = psu(2, 2|4) admits a Z4-grading, f = f(0)⊕f(1)⊕f(2)⊕f(3)where g = f(0) = so(4, 1) ⊕ so(5). Let G denote the corresponding Lie group. The supertrace iscompatible with the Z4-grading, in the sense that Str(A(m)B(n)) = 0 for m + n 6= 0 mod 4. The1reduced theory relies on the element T = i2diag(1, 1,−1,−1, 1, 1,−1,−1) ∈ f(2). It defines a Z2-grading of f with f[0] = Ker(AdT ) and f[1] = Im(AdT ). Elements of f[0] commute with T while thoseof f[1] anti-commute with T and we have Str(A[0]B[1]) = 0. Finally, projectors on f[0] and f[1] aregiven respectively by P [0] = −[T, [T, · ]+]+ and P [1] = −[T, [T, · ]]. Let h = g[0] be the subalgebrain g of elements commuting with T . The corresponding Lie group H is [SU(2)]4.Our starting point is the field theory introduced in [4]. It corresponds to a fermionic extensionof a G/H gauged WZW with a potential term. The action we start with is, taking ǫτσξ = 1,S = 12∫dτdσ Str(g−1∂+gg−1∂−g) +13∫dτdσdξǫαβγ Str(g−1∂αgg−1∂βgg−1∂γg)−∫dτdσ Str(A+∂−gg−1 − A−g−1∂+g + g−1A+gA− − A+A−)+ 12∫dτdσ Str(ψL[T,D+ψL] + ψR[T,D−ψR])+∫dτdσ(µ2 Str(g−1TgT ) + µ Str(g−1ψLgψR)).The fields g, ψR, ψL and the gauge fields A± respectively take values in G, f(1)[1], f(3)[1] and in h.The covariant derivatives are D± = ∂± − [A±, ] with ∂± = ∂τ ± ∂σ.Generalizing the analysis of [9] to the case considered here, one finds that the phase space isspanned by the fields (g,JL, A±, P±, ψL, ψR). The field JL corresponds to the left-invariant WZWcurrent. Alternatively, one can use instead the right-invariant current JR, related to JL byJR = −2∂σgg−1 + gJLg−1.The fields P± are the canonical momenta of A±. The non-vanishing Poisson brackets are{JL1(σ),JL2(σ′)} = [C(00)12 ,JL2]δσσ′ + 2C(00)12∂σδσσ′ ,{JR1(σ),JR2(σ′)} = −[C(00)12 ,JR2]δσσ′ − 2C(00)12∂σδσσ′ ,{JL1(σ), g2(σ′)} = −g2C(00)12 δσσ′{JR1(σ), g2(σ′)} = −C(00)12 g2δσσ′{A±1(σ), P±2(σ′)} = C(00)[00]12 δσσ′ ,{ψR1(σ), ψR2(σ′)} =[T2, C(13)12]δσσ′ ,{ψL1(σ), ψL2(σ′)} =[T2, C(31)12]δσσ′ .In these expressions C(ij)12∈ f(i) ⊗ f(j) are the components of the tensor Casimir (see [10] for itsproperties) in the decomposition C12 = C(00)12+C(13)12+C(22)12+C(31)12with respect to the Z4-grading.The component C(00)[00]12is defined in a similar way relative to the Z2-grading.2The standard analysis shows that there is a total of four constraints,χ1 = P+, χ2 = P−, (2.2a)χ3 = J [0]R + A+ − A− − 12 [ψL, [T, ψL]], χ4 = J[0]L + A+ − A− + 12 [ψR, [T, ψR]]. (2.2b)The extended Hamiltonian, which has weakly vanishing Poisson brackets with the constraints (2.2),isH =∫dσ(14 Str(JL2 + JR2)+ Str(J [0]R A+ − J[0]L A−)+ 12 Str[(A+ − A−)2]− 12 Str(ψL[T, ∂σψL − [A+, ψL]])− 12 Str(ψR[T,−∂σψR − [A−, ψR]])(2.3)− µ2 Str(g−1TgT )− µ Str(g−1ψLgψR) + v+P+ + v−P− + λ(χ3 − χ4))with v+ − v− = ∂σ(A+ +A−)− [A+, A−]. The combination χ3 − χ4 of the constraints generates agauge invariance.3 Continuum and lattice Poisson algebrasUp to a gauge transformation, the equations of motion for the fields (JL, g, ψL, ψR) under theHamiltonian (2.3) are equivalent to the zero curvature equation {L, H} = ∂σM + [M,L] for thefollowing Lax connection [4]L(z) = −12JL − 12z√µψR − 12z2µT + 12z−1√µg−1ψLg +12z−2µg−1Tg, (3.1a)M(z) = −12JL + A− − 12z√µψR − 12z2µT − 12z−1√µg−1ψLg − 12z−2µg−1Tg. (3.1b)The field A+ entering the equations appears as an arbitrary element of h. We now have all theingredients needed to compute the Poisson bracket of the Lax matrix (3.1a). The result reads4{L1(z1),L2(z2)} = [r12(z1, z2),L1(z1) + L2(z2)]δσσ′+ [s12(z1, z2),L1(z1)− L2(z2)]δσσ′ + 2s12(z1, z2)∂σδσσ′ , (3.2)where the kernels of the r/s-matrices are given byr12(z1, z2) =z42 + z41z42 − z41C(00)12+2z1z32z42 − z41C(13)12+2z21z22z42 − z41C(22)12+2z31z2z42 − z41C(31)12, (3.3a)s12(z1, z2) = C(00)12. (3.3b)One can check explicitly that the kernels (3.3) coincide exactly with the ones that would beobtained from the generalization of the alleviation procedure proposed in [1] to semi-symmetricspace σ-models. This is simply a matter of replacing the twisted inner product on the twisted loop3algebra considered in [11] by the trigonometric one and to compute the corresponding kernels asexplained in [1].An important property of the above r/s-matrix structure is that s is simply the projection ontothe subalgebra g. In this case, the corresponding Poisson algebra (3.2) can be discretized following[6] by introducing a skew-symmetric solution α ∈ End g of the modified classical Yang-Baxterequation on g. Then the matricesa12 = (r + α)12, b12 = (−s− α)12, c12 = (−s+ α)12, d12 = (r − α)12,satisfy all the requirements of [7, 8] in order to define the following consistent lattice algebra,4{Ln1,Lm2} = a12Ln1Lm2 δmn −Ln1Lm2 d12δmn + Ln1b12Lm2 δm+1,n −Lm2 c12Ln1δm,n+1.This algebra reduces to (3.2) in the continuum limit (see [1]). The corresponding algebra for themonodromy may be found in [1].4 ConclusionWe have constructed a quadratic lattice Poisson algebra associated with the fermionic extension ofthe (SO(4, 1)× SO(5))/[SU(2)]4 gauged WZW model with an integrable potential. The fact thatone is able to write down such a lattice algebra is quite appealing and in sharp contrast with whathappens for the canonical Poisson structure of the AdS5 × S5 superstring [10]. Indeed, it bringshope of being able to construct a lattice quantum algebra related to the Pohlmeyer reduction of theAdS5×S5 superstring. The precise link of this Pohlmeyer reduction with the alleviation procedurepresented in [1] is under study.Acknowledgements We thank J.M. Maillet for useful discussions. B.V. is supported by UKEPSRC grant EP/H000054/1.References[1] F. Delduc, M. Magro, and B. Vicedo, Alleviating the non-ultralocality of coset sigma modelsthrough a generalized Faddeev-Reshetikhin procedure, arXiv:1204.0766.[2] I. Bakas, Q.-H. Park, and H.-J. Shin, Lagrangian formulation of symmetric spacesine-Gordon models, Phys. Lett. B372 (1996) 45–52, [hep-th/9512030].[3] J. M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys.B269 (1986) 54.[4] M. Grigoriev and A. A. Tseytlin, Pohlmeyer reduction of AdS5 × S5 superstring sigmamodel, Nucl. Phys. B800 (2008) 450–501, [arXiv:0711.0155].4[5] A. Mikhailov and S. Schäfer-Nameki, Sine-Gordon-like action for the Superstring inAdS5 × S5, JHEP 05 (2008) 075, [arXiv:0711.0195].[6] M. Semenov-Tian-Shansky and A. Sevostyanov, Classical and quantum nonultralocal systemson the lattice, hep-th/9509029.[7] L. Freidel and J. M. Maillet, Quadratic algebras and integrable systems, Phys. Lett. B262(1991) 278–284.[8] L. Freidel and J. M. Maillet, On classical and quantum integrable field theories associated toKac-Moody current algebras, Phys. Lett. B263 (1991) 403–410.[9] P. Bowcock, Canonical quantization of the gauged Wess-Zumino model, Nucl. Phys. B316(1989) 80.[10] M. Magro, The classical exchange algebra of Ads5 × S5 string theory, JHEP 0901 (2009)021, [arXiv:0810.4136].[11] B. Vicedo, The classical R-matrix of AdS/CFT and its Lie dialgebra structure, Lett. Math.Phys. 95 (2011) 249–274, [arXiv:1003.1192].5

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A lattice Poisson algebra for the Pohlmeyer reduction of the AdS_5 x S^5
  superstring

A lattice Poisson algebra for the Pohlmeyer reduction of the AdS_5 x S^5 superstring

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