The Affine Hidden Symmetry and Integrability of Type IIB Superstring in AdS 5 S 5

Transcription

1 The Affine Hidden Symmetry and Integrability of Type IIB Superstring in AdS 5 S 5 arxiv:hep-th/ v1 25 Jun 2004 Bo-Yu Hou a, Dan-Tao Peng ab, Chuan-Hua Xiong a, Rui-Hong Yue a a Institute of Modern Physics, Northwest University, Xi an, , China b Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui, , P. R. China Abstract In this paper, we motivate how the Hodge dual related with S-duality gives the hidden symmetry in the moduli space of IIB string. Utilizing the static κ- symmetric killing gauge, if we take the Hodge dual of the vierbeins keeping the connection invariant, the duality of Maure-Cartan equations and the equations of motion becomes manifest. Then by twist transforming the EOM, we express the BPR Lax connection by a unique spectral parameter. Then we construct the generators of the infinitesimal dressing symmetry, the related symmetric algebra is affine gl(2,2 4) (1), which can be used to find the classical r matrix. 1 Introduction Maldacena [1] proposed the AdS/CFT correspondence between the classical bulk theory of AdS 5 S 5 SUGRA and the quantum conformal SYSY theory on the boundary. byhou@nwu.edu.cn dtpeng@nwu.edu.cn or dtpeng@ustc.edu.cn chxiong@phy.nwu.edu.cn rhyue@nwu.edu.cn 1

2 In the last years, the existence of integrability structures on both side of the correspondence have been pointed out [2, 3, 4, 5]. Bena, Polchinski and Roiban found the Lax-connection of the Green-Scharz [6] string in the background AdS 5 S 5 [4]. With the help of this Lax connection, one can find transfer matrix U. The expansion of U with respecting to the spectral parameter will gives an infinite conserved laws. Dolan, Nappi and Witten discuss the corresponding infinite-dimensional non-local current [5]. They argue that such infinite dimensional symmetry is the Yangian symmetry. For this infinite dimensional symmetry, the quantization procedure will preserve it. But, to quantize such system, one has to handle infinite number of constrains given by the Dirac quantization method. The ghost will appear, BRST method is needed and the symmetry e.g. κ symmetry will not be manifest. We try to disclose the complete symmetries of this classical system and to construct the Lax-pair. Then the fundamental poisson bracket will give out the classical r matrix. Further, the quantization gives the quantum doubled R matrix. The amplitudes will be given in terms of Bethe Ansatz method. In this paper, we investigate the duality between the Maurer-Cartan equations (MCE) and the equations of motion (EOM) and obtain a new Lax-matrix by using the twist dual transformation which represents a dreesing symmetry in AdS 5 S 5 GS string. Physically, the invariance of re-parametrization admits such twist transformation. But, the action is not invariant since the conformal gauge is used. If we take the improved stress-energy tensor such symmetry will recovered. Hence, the full symmetric algebra becomes gl(2, 2 4) (1). 2 The Metsaev-Tseytlin action of Green-Shwarz superstring in AdS 5 S 5 [7] The AdS 5 S 5 is a coset space SO(4,2) SO(6) SO(4,1) SO(5). It also preserves the full supersymmetry of the SUGRA and corresponds to the maximally supersymmetric background vacuum of IIB SUGRA. Combining the bosonic SO(4, 2) SO(6) isometry symmetry with the full supersymmetry, the symmetry turns to be the PSU(2, 2 4) acting on the super 2

3 coset space PSU(2,2 4). In what follows, we adapt the conventions introduced by [7]: SO(4,1) SO(5) a, b, c = 0, 1,, 4 so(4, 1) vector indices (AdS 5 tangent space) a, b, c = 5,, 9 so(5) vector indices (S 5 tangent space) â,ˆb, ĉ = 0, 1,, 9 combination of (a, a ), (b, b ), (c, c ) (D = 10 vector indices) α, β, γ, δ = 1,, 4 so(4, 1) spinor indices (AdS 5 ) α, β, γ, δ = 1,, 4 so(5) spinor indices (S 5 ) ˆα, ˆβ, ˆγ = 1,, 32 D = 10 Majorana-Weyl spinor indices I, J, K, L = 1, 2 SO(2) labels of the N = 2 two sets of spinors The generators of the so(4, 1) and so(5) Clifford algebras are 4 4 matrices γ a and γ a γ (a γ b) = η ab = ( ), γ (a γ b ) = η a b = ( ). ˆγ a γ a, ˆγ a iγ a (1) satisfying (γ a ) = γ 0 γ a γ 0, (γ a ) = γ a respect to the two supercharges and the Majorana condition is diagonal with Q αα I (Q ββ I ) (γ 0 ) β α δβ α = Q ββ I C βα C β α. (2) Here C = (C αβ ) and C = (C α β ) are the charge conjugation matrices of the so(4, 1) and so(5) Clifford algebras Q αα I Q ββ I C βα C β α. The bosonic generators are antihermitean: P a = P a, P a = P a, J ab = J ab, J a b = J a b. The SO(2) 2 2 matrices are ǫ IJ = ǫ JI, ǫ 12 = 1, and s IJ diag(1, 1). The 10-dimensional Dirac matrices Γâ of SO(9, 1) (Γ (â Γˆb) = ηâˆb) and the corresponding charge conjugation matrix C can be represented as Γ a = γ a I σ 1, Γ a = I γ a σ 2, C = C C iσ 2, (3) where I is the 4 4 unit matrix and σ i are the Pauli matrices. The chirality is the eigenvalue of σ 3 in the last factor. The generators of superalgebra g su(2, 2 4) are divided into the even generators B two pairs of translations and rotations (P a, J ab ) (h, k) for AdS 5 and (P a, J a,b ) (h, k ) for S 5 respectively; the odd generators F are the two D = 10 Majorano-Weyl spinors Q αα I F I. 3

4 The commutation relations for the generators T A = (P a, P a ; J ab, J a b Q αα I) (h, h ; k, k F I ) (ĥ;ˆk F) are [J ab, J cd ] = η bc J ad + 3 terms, [h, h] h, (4) [J a b, J c d ] = η b c J a d + 3 terms, [h, h ] h, (5) [P a, P b ] = J ab, [k, k] h (6) [P a, P b ] = J a b, [k, k ] h (7) [P a, J bc ] = η ab P c η ac P b, [h, k] k, (8) [P a, J b c ] = η a b P c η a c P b, [h, k ] k. (9) [Q I, P a ] = i 2 ǫ IJQ J γ a, [F I, k] ǫ IJ F J, (10) [Q I, P a ] = 1 2 ǫ IJQ J γ a, [F I, k ] ǫ IJ F J, (11) [Q I, J ab ] = 1 2 Q Iγ ab, [F I, h] F I, (12) [Q I, J a b ] = 1 2 Q Iγ a b, [FI, h ] F I. (13) {Q αα I, Q ββ J} = δ IJ [ 2iC alpha β (Cγa ) αβ P a + 2C αβ (C γ a ) α β P a ] The left-invariant Cartan 1-forms + ǫ IJ [ C α β (Cγab ) αβ J ab C αβ (C γ a b ) α β J a b ] [F, F] B., (14) L A = dx M L A M, X M = (x, θ) (15) 4

5 are given by G 1 dg = L A T A L a P a + L a P a Lab J ab La b J a b + Lαα I Q αα I, (16) where G = G(x, θ) is a coset representative in PSU(2, 2 4), e.g. by using the S-gauge given by [8] or the KRP gauge used by Roiban and Siegel [9]. We coin the name Ĥ(H, H ), ˆK(K, K ), K F forms respectively for the Cartan connections L ab and L a b, the super-beins Lâˆα including 5-beins L a and L a and the 2-spinor 16-beins L αα I. They satisfy the Maurer-Cartan (MC) equation, i.e. the structure equation of basic one forms on the superspace SU(2,2 4) SO(4,1) SO(5) d(g 1 dg) + (G 1 dg) (G 1 dg) = 0. (17) Then the super Gauss equations of the induced curvatures F ab and F a b F = dh + H H are defined by F ab dl ab + L ac L cb = L a L b + ǫ IJ LI γ ab L J, (18) F a b dl a b + L a c L c b = L a L b ǫ IJ LI γ a b L J. (19) The super Coddazi equation for the vector 5-beins are dl a + L b L ba = il I γ a L I, dl a + L b L b a = L I γ a L I, (20) and the super Coddazi equation for the spinor 16-beins are dl I 1 4 γab L I L ab 1 4 γa b L I L a b = 1 2 γa ǫ IJ L J L a ǫij γ a L J L a. (21) In the super Gauss equations and the Coddazi equations, the terms on the left hand side are the usual gauge covariant exterior derivative d+h, while the right hand side are the curvature and torsion contributions by the fermions. To embed the IIB superstring into the super coset space M, we should pull back the Cartan form down to the world sheet Σ(σ, τ) as L A = L A M dxm = L A M ix M dσ i = L A i dσi L A 1 dτ + LA 2 dσ. (22) Then the MC 1-form becomes G 1 i G = L A i P A = L a i P a + L a i P a (Lab 5 i J ab + L a b i J a b ) + Lαα I i Q αα I, (23)

6 and e.g. the super Coddazi equations for the vector 5-beins (20) become ǫ ij ( i L a j + L ab i L b j) + iǫ ij LI i γ a L I j = 0, (24) ǫ ij ( i L a j + L a b i L b j ) ǫij LI i γ a L I j = 0. (25) Turn to the string dynamics. the AdS 5 S 5 GS superstring action is given as SU(2,2 4) SO(4,1) SO(5) I = 1 2 superspace sigma model [7, 8]. d 2 σ gg ij (L a i La j + La i La j ) + i s IJ (L a L I γ a L J + il a M 3 M 3 L I γ a L J ). This action is invariant with respect to the local κ-transformations in terms of δx a δx M L a M, δxa δx M L a M, δθi δx M L I M (26) δ κ x a = 0, δ κ x a = 0, δ κ θ I = 2(L a i γ a il a i γ a )κ ii (27) δ κ ( gg ij ) = 16i g(p jk L 1 k κi1 + P jk + L 2 k κi2 ). (28) Here P ij ± 1 2 (gij ± 1 g ǫ ij ) are the projection operators, and 16-component spinor κ ii (the corresponding 32-component spinor has opposite chirality to that of θ) satisfy the (anti) self duality constraints P ij κ 1 j = κi1, P ij + κ 2 j = κi2, (29) which can be written as 1 g ǫ ij κ 1 j = κi1, 1 g ǫ ij κ 2 j = κi2, i.e. 1 g ǫ ij κ I j = SIJ κ ij. From the variation of action (26), the equations of motion (EOM) are obtained [7] gg ij ( i L a j + Lab i Lb j ) + iǫij s IJ LI i γ a L J j = 0, (30) gg ij ( i L a j + L a b i L b j ) ǫij s IJ LI i γ a L J j = 0, (31) (γ a L a i + iγ a L a i )( gg ij δ IJ ǫ ij s IJ )L J j = 0, (32) where i is the g ij -covariant derivative on the worldsheet Σ(σ, τ). The MC equations for the vierbeins describes the geometric behavior for the embedding of the type IIB string worldsheet into the target space AdS 5 S 5. While the equations of motion govern the dynamic of the string. 6

7 3 The Hodge dual and duality between MC equations and EOM The equations of motion (30) and (31) can be rewritten as g ij ( i ( gl a j ) + Lab i Lb j ) + iǫij S IJ LI i γ a L J j = 0, (33) g ij ( i ( gl a j ) + L a b i L b j ) ǫ ij S IJ LI i γ a L J i = 0, (34) where we have used i Lâi = 1 g i ( glâi ). (35) In order to disclose the duality between the MC equations and the equations of motion, we first describes the Hodge dual of bosonic and fermionic forms. As usual, the Hodge dual of the coordinates of world-sheet is given by (dz i ) = 1 g ǫ ij dz j, (dz 1 ) = dτ, (dz 2 ) = dσ, ǫ 12 = ǫ 21 = ǫ 21 = ǫ 12 = 1. (36) L iâ L iâ ǫij g Lâj, Lâi Lâi ǫ ij gl jâ. (37) In the static κ symmetric killing gauge [10], we have P ij L 1 j = L 1i, P ij + L 2 j = L 2i, P κ 1 = κ 1, P + κ 2 = κ 2, θ I S IJ θ J. (38) Then the MC equations (24) and (25) can be rewritten as i ( g L ai ) + L ab i g L bi iǫ ij LI i γ a L I j = 0, (39) i ( g L a i ) + L a b i g L b i + ǫ ij LI i γ a L I j = 0. (40) Applying the above transformations, it is easy to find that the equations of motion (30) and (31) are the dual of the 5-bein super Codazzi equations (39) and (40) respectively. It is clear that the GS string action is invariant under the above dual transformation. There exists no dual between MC eq.21) and EOM eq.(32), because the L I just appears 7

8 in the Wess-Zumino-Witten term and has no dynamical contribution to the action. Under the dual transformation, it changes into (γ a L a i + iγa L a i )( gg ij δ IJ ǫ ij s IJ )L J j = 0. (41) Namely, only the first factor takes the dual form. For the Lâˆb, it does not change under duality since it is not dynamical and does not appear in the GS string action. Remark that this duality is the generalization of the usual bosonic Hodge dual. The symplectic structure (Käller 1-form) of bosonic pseudo-sphere and the matric 1-form are generalized as following. For the pseudo-sphere with negative constant curvature dω 12 = kω 1 ω 2, (42) where ω 12 is connection 1-form and the constant curvature k = 1/ g. The metric 1-form is ds 2 = dω dω 2 2. Under the same moving frame, one has With the help of eq.(43), the eq.(42) changes into ω i = ǫ ij dω 3j. (43) dω 12 = ω 13 ω 23, (44) i.e. the Maure-Cardan equation [11, 12]. The geodesic motion on pseudo-sphere gives Sine-Gordon equation. The image of the normal line of pseudo-sphere gives the nonlinear σ model on the sphere. For AdS 5, the conformal matric is the poincaré matric 3 i=1 dx2 i +dr2. The eq.(39) and (40) can be considered as the generalization of eq.(43) in r 2 AdS 5 S 5 with the κ gauge. 4 The twisted dual and integrablity Now we introduce the twisted dual transformation (continuous dual transformation). The duality discussed in previous section will be considered as a special case of it. On the world sheet Σ(σ, τ), the re-scale transformations along the two directions of the positive and negative light-cone τ ± σ with the scale factors λ and λ 1 correspond to Lorentz rotating, boost the even vierbein forms Lâ by ±2φ oppositely, and rotating the odd vierbein forms L I by ±ϕ together with κ I. The linear combination of these transformations can give out the twisted dual transformation. 8

9 For the even vierbein form part, we have Lâi = exp 2ϕ P ij + Lâj + exp 2ϕ P ij Lâj = 1 2 (λ + λ 1 )Lâi (λ λ 1 ) Lâi, (45) here λ = exp 2ϕ. The transformations of odd vierbein form part are L ii = exp ϕ P ij + L I j + exp ϕ P ij L I j, (46) Then, the eq.(46) and eq.(47) become as L ii = exp ϕ P ij + L I j + exp ϕ P ij L I j. (47) L 1i L 2i = e ϕ P ij L 1 j = λ 1 2 L 1 j, (48) = e ϕ P ij + L 2 j = λ 1 2 L 2 j, (49) Notice that the re-parametrization invariance of action implies the loop group symmetry, while the WZW term is a 2-cocycle and gives the center extension. Furthermore, the GL(1) GL(1) supplies an axial symmetry A of the extended N = 2SUSY to GS string in ref.[7], and affine the U(1) S-duality of SL(2,R) super-gravity. The U(1) dual twist breaks the invariance of the action in conformal gauge ( Ta a 0) superficially. SU(1,1) U(1) But, one can recover it by using the improved stress-energy tensor since the center and grade derivative operators of affine algebra are still commutative with the algebraic generators. This is not the apparent symmetry of action, but the hidden symmetry in the modula space. The continuous spectral parameter describes such symmetry. For the Euclidean 2-dimension σ model, ( on the complexified worldsheet) we may assume that exp 2ϕ = i, (50) we have Lâi = i Lâi (51) L i1 = ( i) 1 2 L i1, L i2 = i 1 2 L i2 (52) L i1 = ( i) 1 2 Li1, Li2 = i 1 2 Li2. (53) 9

10 It is obvious that (51) (53) are the dual transformations of the MC and the EOM. We can construct the Lax connection A i with the spectral parameter λ as A i (λ) = H + K + F = 1 2 Lâˆb i Jâˆb + Lâi Pâ + L αα I Q αα I = 1 2 (Lab i J ab + L a b J a b ) (λ + λ 1 )(L a i P a + L a P a ) + 1 [ ] 2 (λ λ 1 ) (L a i )P a + (L a i )P a +λ 1 2 L αα 1 Q αα 1 + λ 1 2 L αα 2 Q αα 2. (54) Based on the Lax connection A i, we can give out the transfer matrices U i U = A i U. (55) It is easy to verify that the connections A i satisfy the zero curvature (flat connection) condition: Then we prove the integrability of the system. i A j j A i + [A i, A j ] = 0. (56) Given the Lax connection A i, the time-component (τ component) A of A i satisfy the poisson bracket A A 0 (λ) = A 0 (ϕ), (57) {A(ϕ), A(ψ)} = [r(ϕ ψ), A(ϕ) A(ψ)], (58) where r(ϕ ψ) is the classical r-matrix with spectral parameter. In fact, the r-matrix may be derived directly from the symplectic form given by the action with WZW term, i.e. by the poisson Lie algebra of the symmetry. It is just the classical double algebra, given by our dressing transform structure due to the GS action can be obtained by some reduction condition of the WZW model. Since the r matrix does not depend on the detail of the constrain, the r matrix should be given by the standard one of the algebra su(2, 2 4) double. After quantizing the fundamental field, the classical r matrix structure should be replaced by the Yang-Baxter relation R 12 (φ ψ)l 1 (φ)l 2 (ψ) = L 2 (ψ)l 1 (φ)r 12 (φ ψ). (59) Based on this, one can get the quantization version of the commutative relations involving vertex operators. 10

11 5 Conclusion In this paper, we have introduced a twist dual transformation which depends on a free parameter. Replacing the variation δg [T, G] of the coset element G by δg [U(λ) 1 TU(λ), G], we get a conserved Noether cuurent depending on a parameter. Later, e.g. Dolan expands it. The expansion to the spectral λ gives out an infinite number of conserved laws. This is quite different from ones given by Dolan, Nappi and Witten [5]. In fact, the quantity U(λ) 1 TU(λ) produces the infinitesimal dressing symmetry. On the other hand, the dressing transformation can be given by the gauged WZW theory. It is well-known that the gauged WZW is equivalent to an affine Toda theory[13]. Thus, the the dressing transformation gives all the soliton solution, i.e. the moduli space of the string background states. After the quantization, the system will become the quantum Affine Toda. The fundamental poisson bracket with classical r matrix will be replaced by the quantum Yang-Baxter relation with quantum R and L matrices. The scattering amplitude can also be obtained. The corresponding vertex operators relation is governed by the double scaling limit of the quantum Virasoso and W algebra, which is equivalent to the q-deformation of Yangian double at the critical level. Thus the dressing symmetry revealed in this paper discloses that the integrable structure of GS string is given by the Affine Toda system. e.g. why the Seiberg-Witten curve is given by the spectral determinant of affine Toda. We will give the detail in further paper. Acknowledgments We are grateful to Bo-Yuan Hou, Xing-Chang Song and Kang-Jie Shi for helpful discussions. References [1] J. M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, Adv.Theor.Math.Phys. 2 (1998) ; Int.J.Theor.Phys. 38 (1999) [arxiv:hep-th/ ] 11

12 [2] G. Mandal, N. V. Suryanarayana and S. R. Wadia, Aspects of semiclassical strings in AdS(5), Phys. Lett. B 543 (2002) 81 [arxiv:hep-th/ ]. [3] L. F. Alday, JHEP 0312 (2003) 033 [4] I. Bena, J. Polchinski, R. Roiban, Phys.Rev. D69 (2004) [arXiv:hep-th/ ] [5] L. Dolan, C.R. Nappi, E. Witten, A Relation Between Approaches to Integrability in Superconformal Yang-Mills Theory, JHEP 0310 (2003) 017, [arxiv:hep-th/ ]; Yangian Symmetry in D=4 Superconformal Yang-Mills Theory, arxiv:hep-th/ [6] M. B. Green, J. H. Shwarz, Phys. Lett. B 136 (1984) , Nucl. Phys. B 243 (1984) 285. [7] R. R. Metsaev, A. A. Tseytlin, Nucl. Phys. B 533 (1998) , [arxiv:hep-th/ ]. [8] R. R. Metsaev JHEP 0011 (2000) 014. [9] R. Roiban and W. Siegel JHEP 0011(2000) 024. [10] R. Kallosh Superconformal Actions in Killing Gauge, arxiv:hep-th/ [11] B.Yu. Hou, B.Yuan. Hou and P. Wang, J. Phys. A: Math. Gen 18 (1985) [12] B.Yu. Hou, B.Yuan. Hou, Differential Geometry for Physicists, World Scientific Publishing Company(April 1, 1997). [13] O. Babelon and D. Bernard, Int. J. Mod. Phys. A 8 (1993)