Flat Currents of Green Schwarz Superstring in AdS 2 S 2 Background

Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 663 668 c International Academic Publishers Vol. 45, No. 4, April 15, 2006 Flat Currents of Green Schwarz Superstring in AdS 2 S 2 Background WANG Xiao-Hui, WANG Zhan-Yun, CAI Xiao-Lin, SONG Pei, HOU Bo-Yu, and SHI Kang-Jie Institute of Modern Physics, Northwest University, Xi an 710069, China (Received August 25, 2005) Abstract From the κ symmetric action of IIB string in AdS 2 S 2 background given by Zhou, we derive the equations of motion. By using the twisted dual transformation which was introduced by Hou, we construct the flat currents, conserving non-local charge with one free parameter, for the superstring in AdS 2 S 2. PACS numbers: 11.25.Hf Key words: Green Schwarz superstring, integrability, AdS 2 S 2 1 Introduction Recently, motivated by the AdS/CFT correspondence, [1 3] there has been much interest in the role of integrability in the world-sheet theory of type IIB strings in AdS 5 S 5. In Ref. [4], Bena, Polchinski, and Roiban constructed a hierarchy of infinite nonlocal conserved charges for the Green Schwarz superstring in AdS 5 S 5 spacetime, implying that the world-sheet sigma model is completely integrable. Subsequently Vallilo showed [5] that such charges also exist in the pure-spinor formalism for the superstring. These charges are the analogues of the nonlocal charges which have long been known to exist in the sigma models on symmetric spaces, [6 8] and their discovery allows integrable field theory to be applied to the world-sheet theory of superstrings in AdS 5 S 5. [9,10] In the pure-spinor formalism it has been argued that the charges survive quantum-mechanically. [11,12] It is important to study the AdS/CFT correspondence in the cases with lower-dimensional space, since as a toy model, it is easier to handle for better understanding the correspondence. In an important work [13] Metsaev and Tseytlin (MT) have provided a method to construct the Green Schwarz (GS) superstring action in AdS 5 S 5 background, then Bena, Polchinski, and Roiban [4] gave a method to construct the flat currents of this model in the coset base. Soon after Hou et al. [14,15] used another way to construct the flat currents in the original group base. In fact those two methods are equivalent. Recently Bin Chen et al. have studied the AdS 3 S 3 string in Ref. [16]. In this paper we focus on the AdS 2 S 2 string. Our main aim is to construct explicitly a one-parameter family of flat currents which would lead to classical nonlocal charges. The type IIB GS superstring action is constructed in AdS 2 S 2 background in terms of supercoset formalism, which was introduced by Zhou. [17] This action possesses global PSU(1, 1 2) super-invariance, has κ-symmetry and 2D reparametrization invariance as its local symmetries, and reduces to the conventional type IIB GS superstring action in the flat background limit. For comparison with the original work of MT, we change the basis of the algebra psu(1, 1 2), and give the Maurer Cartan equations, then rewrite the GS superstring action via supercoset method, and derive the equations of motion. The action and the equations of motion have the same form as the ones in the AdS 5 S 5 string, [13] but the charge conjugation matrix C and C are both symmetric. Instead of the method given in Ref. [4], we use the dual twisted transformation which was introduced by Hou and construct a family of flat currents which would naturally lead to a hierarchy of classical conserved nonlocal charges. With the help of the world-sheet Hodge dual of the equations of motion, we show that these new currents are really flat. This paper is organized as follows. We first review the AdS 2 S 2 string. In Sec. 2 we give the Lie algebra of the superalgebra psu(1, 1 2) in a new base, then we define the Maurer Cartan 1-forms with respect to these generators and write down the Maurer Cartan equations. In Sec. 3 we rewrite the action of AdS 2 S 2 string using the Maurer Cartan 1-forms of the new base and then derive the equations of motion. Then in Sec. 4 we explicitly construct a one-parameter family of the flat currents in AdS 2 S 2 superstring. At last, we summarize our results with discussion in Sec. 5. 2 psu(1, 1 2) Superalgebra We begin with the psu(1, 1 2) superalgebra. The target space of string theory in AdS 2 S 2 with 8 supersymmetry generators is the supercoset manifold PSU(1, 1 2) SO(1, 1) SO(2), The project supported by National Natural Science Foundation of China under Grant No. 90403019 E-mail: xhwang@nwu.edu.cn

664 WANG Xiao-Hui, WANG Zhan-Yun, CAI Xiao-Lin, SONG Pei, HOU Bo-Yu, and SHI Kang-Jie Vol. 45 whose bosonic part is SO(1, 2) SO(3) SO(1, 1) SO(2) = AdS 2 S 2. The bosonic generators of this supergroup are the monenta and Lorentz transformation on AdS 2 and S 2, P a, J ab and P a, J a b, and the fermionic generators are the two D = 4 Majorana Weyl spinors Q αα I (α, α = 1, 2; I = 1, 2). In what follows we use the same notations as that in Ref. [13]. The ordinary Latin letters a, b = 0, 1 are the SO(1,1) vector indices (AdS 2 tangent space). The primed Latin letters a, b = 2, 3 are the SO(2) vector indices. The ordinary Greek letters α, β = 1, 2 are the SO(1,1) spinor indices. The primed Greek letters α, β = 1, 2 are the SO(2) spinor indices. We also use the 2 2 matrices ɛ IJ = ɛ JI, ɛ 12 = 1, and S IJ diag (1, 1) to contract the indices I = 1, 2. The 4-dimensional Dirac matrices Γâ and the corresponding charge conjugation matrix C can be represented as Γ a = γ a I, Γ a = I γ a, C = C C, (1) where I is the 2 2 unit matrix. The generators of the SO(1,1) and SO(2) Clifford algebras are 2 2 matrices γ a and γ a, ( ) ( ) i γ 0 =, γ 1 i =, (2) i i ( ) ( ) γ 2 1 i =, γ 3 =, (3) 1 i and the charge conjugation matrices are ( ) C = C 1 =. (4) 1 Then the conjugate supercharge Q αα I is defined by Q I = (Q I ) t CC. (5) Thus psu(1, 1 2) superalgebra is given by [P a, P b ] = J ab, (6) [P a, P b ] = J a b, (7) [J ab, J cd ] = [J a b, J c d ] = 0, (8) [P a, J bc ] = η ab P c η ac P b, (9) [P a, J b c ] = η a b P c η a c P b, (10) [Q I, P a ] = i 2 ɛ IJQ J γ a, (11) [Q I, P a ] = 1 2 ɛ IJQ J γ a, (12) [Q I, J ab ] = 1 2 Q Iγ ab, (13) [Q I, J a b ] = 1 2 Q Iγ a b, (14) {Q αα I, Q ββ J} = δ IJ [ 2i C α β (Cγa ) αβ P a + 2C αβ (C γ a ) α β P a ] + ɛ IJ [C α β (Cγab ) αβ J ab The left-invariant Cartan 1-forms are given by C αβ (C γ a b ) α β J a b ]. (15) L A = dx M L A M, X M = (x, θ) (16) G 1 dg = L A T A L a P a + L a P a + 1 2 Lab J ab + 1 2 La b J a b + Lαα I Q αα I, (17) where G = G(x, θ) is a coset representative in PSU(1, 1 2). The Maurer Cartan 1-form satisfies the zero-curvature equation d(g 1 dg) = G 1 dg G 1 dg. Then decompose it according to the generators of the Lie algebra, we get the following Maurer Cartan equations: dl a = L b L ba i L I γ a L I, (18) dl a = L b L b a + L I γ a L I, (19) dl ab = L a L b + ɛ IJ LI γ ab L J, (20) dl a b = L a L b ɛ IJ LI γ a b L J, (21) dl I = i 2 γ aɛ IJ L J L a + 1 2 γ a ɛ IJL J L a + 1 4 γ abl I L ab + 1 4 γ a b LI L a b. (22) In this paper we set L αα I Q ββ J = Q ββ JL αα I, which is the same as in Ref. [13] but different from the convention in Ref. [17]. In Eqs. (18) (21) we set ( L I ) αα = L µµ I C µα C µ α = (LI CC ) αα according to Eq. (5). 3 κ Symmetric Action and Equations of Motion The AdS 2 S 2 GS superstring action is given as superspace sigma model, [13] PSU(1, 1 2) SO(1, 1) SO(2) S = S 0 + S 1, (23) S 0 = 1 d 2 σ gg ij (L a i L a j + L a i L a j ), (24) 2 M 3 S 1 = i L W Z (25) M 3 = i S IJ (L a L I γ a L J + il a L I γ a L J ). (26) M 3 Here g = det g ij, g ij g jk = δ ik, i, j = 0, 1 and g ij is the metric of the world-sheet. This action is invariant with respect to the local κ-transformations in terms of δx a δx M L a M, δx a δx M L a M, δθ I δx M L I M,

No. 4 Flat Currents of Green Schwarz Superstring in AdS 2 S 2 Background 665 i.e. G 1 δg δx a P a + δx a P a + 1 2 δxab J ab + 1 2 δxa b J a b + δθαα I Q αα I, δ κ x a = 0, δ κ x a = 0, (27) δ κ θ I = 2(L a i γ a il a i γ a )κ ii, (28) δ κ ( gg ij ) = 16i g(p jk L 1 kκ i1 + P jk + L 2 kκ i2 ). (29) Here P ij ± (1/2)(g ij ±(1/ g)ɛ ij ) are the projection operators, and 4-component spinor κ ii satisfy the (anti-) selfduality constraints, P ij κ 1 j = κ i1, P ij + κ 2 j = κ i2, (30) which can be written as (1/ g)ɛ ij κ 1 j = κ i1, (1/ g) ɛ ij κ 2 j = κi2, i.e. (1/ g)ɛ ij κ I j = SIJ κ ij. To derive the EOM and also to check the κ invariance, we decompose the following equation, δ(g 1 dg) = [G 1 dg, G 1 δg ] + d(g 1 δg) (31) according to the generators of the Lie algebra and obtain the variations of the Cartan 1-forms: δl a = dδx a + L ab δx b + L b δx ba + 2 i L I γ a δθ I, (32) δl a = dδx a + L a b δx b + L b δx b a 2 L I γ a δθ I, (33) δl I = dδθ I + i 2 ɛ IJ(δx a γ a + iδx a γ a )L J i 2 ɛ IJ(L a γ a + il a γ a )δθ J 1 4 (δxab γ ab + δx a b γ a b )L I + 1 4 (Lab γ ab + L a b γ a b )δθ I. (34) The WZ term L WZ in Eq. (25) is also a closed 3-form and its variation is given by δl WZ = dλ, Λ = S IJ ( L I γ a L J δx a + i L I γ a L J δx a + 2L a L I γ a δθ J + 2iL a L I γ a δθ J ). (36) From the variation of action (23), the equations of motion are obtained, The variation of the metric g ij gives the Virasoro constraint, i ( gg ij L a j ) + gg ij L ab i L b j + iɛ ij S IJ LI i γ a L J j = 0, (37) i ( gg ij L a j ) + gg ij L a b i L b j ɛ ij S IJ LI i γ a L J j = 0, (38) (γ a L a i + iγ a L a i )( gg ij δ IJ ɛ ij S IJ )L J j = 0. (39) (35) L a i L a j + L a i L a j = 1 2 g ijg kl (L a kl a l + L a k L a l ). (40) From the variation of action one can check that the κ symmetry is manifest. 4 Flat Currents of AdS 2 S 2 String 4.1 Hodge Dual of Cartan Forms L a and L a To construct a one-parameter family of nonlocal currents from the κ-symmetric actions of Green Schwarz superstring in AdS 2 S 2 background, we firstly introduce the world-sheet Hodge dual of the Maurer Cartan 1-forms L a and L a. Let that is gg ij L a i ɛ jk L a k, (41) gg i1 L a i ɛ 12 L a 2 = L a 2, (42) gg i2 L a i ɛ 21 L a 1 = L a 1. (43) Thus the EOM (37) becomes ɛ jk j L a k + L ab j ɛ jk L b k + is IJ ɛ jk LI j γ a L J k = 0. (44) Then multiplying by dσ j dσ k on both sides of the above equation, we get d L a + L ab L b + is IJ LI γ a L J = 0. (45) In a similar way the EOM (38) and (39) become d L a + L a b L b S IJ LI γ a L J = 0, (46) δ IJ ( L a γ a + i L a γ a ) L J + S IJ (L a γ a + il a γ a ) L J = 0. (47) The above three equations, which are very useful to constructing the flat currents, are the EOM in terms of the Hodge dual of the Maurer Cartan 1-forms L a and L a. [14]

666 WANG Xiao-Hui, WANG Zhan-Yun, CAI Xiao-Lin, SONG Pei, HOU Bo-Yu, and SHI Kang-Jie Vol. 45 4.2 Twisted Dual Transformations Let us now introduce the twisted dual transformations [14,15,18] on psu(1, 1 2) superalgebra, L a (λ) = 1 2 (λ2 + λ 2 )L a + 1 2 (λ2 λ 2 ) L a, (48) L a (λ) = 1 2 (λ2 + λ 2 )L a + 1 2 (λ2 λ 2 ) L a, (49) L ab (λ) = L ab, L a b (λ) = L a b, (50) L 1 (λ) = λl 1, L 2 (λ) = λ 1 L 2. (51) 4.3 Construction of Flat Currents Using the above dual twisted transformation, we have obtained the new Cartan 1-forms L(λ), which we use to construct the flat currents. The most important thing left to do is to check whether these Cartan 1-forms L(λ) satisfy the Maurer Cartan equations (18) (22). Now we check the new Maurer Cartan equations one by one. With the help of MC equation (18), the dual twisted transformation equations (48) (51) and Eq. (45), we get Next, for the MC equation (20) dl a (λ) = 1 2 (λ2 + λ 2 )dl a + 1 2 (λ2 λ 2 )d L a ( 1 = 2 (λ2 + λ 2 )L b + 1 2 (λ2 λ 2 ) L b) L ba iλ 2 L1 γ a L 1 iλ 2 L2 γ a L 2 = L b (λ) L ba (λ) i L I (λ)γ a L I (λ). dl ab (λ) = dl ab = L a L b + ɛ IJ LI γ ab L J (52) = L a (λ) L b (λ) + ɛ IJ LI (λ)γ ab L J (λ). (53) Here the second term (ɛ IJ LI γ ab L J = ɛ IJ LI γ ab L J ) is obvious whereas the first term is not straightforward. To identify L a L b = L a (λ) L b (λ), (54) we have used the following properties of Cartan 1-forms: In a similar way we can check the MC equations (19) and (21) A B = A B, (55) A B = A B. (56) dl a (λ) = L b (λ) L b a (λ) + L I (λ)γ a L I (λ), (57) dl a b (λ) = L a (λ) L b (λ) ɛ IJ LI (λ)γ a b L J (λ). (58) Then there remains the MC equation (22) to check. Since the fermionic Cartan 1-forms L 1 and L 2 have different transformation rules, the best way is to write down them separately. The Hodge dual of the third equation of motion is also useful, so we write Eq. (47) more explicitly, To check the MC equation (22) we first consider the term (L a γ a + il a γ a + L a γ a + i L a γ a ) L 1 = 0, (J = 1), (59) (L a γ a + il a γ a L a γ a i L a γ a ) L 2 = 0, (J = 2). (60) i 2 γ aɛ IJ L J (λ) L a (λ) + 1 2 γ a ɛ IJL J (λ) L a (λ) [( λ 2 = i 2 ɛ IJ 2 ( λ 2 + 2 ) (L a γ a + il a γ a + L a γ a + i L a γ a ) L J (λ) ) (L a γ a + il a γ a L a γ a i L a γ a ) L J (λ) ]. (61) Substituting Eqs. (59) and (60) into the above equation, we obtain { i 2 γ aɛ IJ L J (λ) L a (λ) + 1 i 2 γ a ɛ IJL J (λ) L a 2 (λ) = λ(la γ a + il a γ a ) L 2, (I = 1), i 2 λ 1 (L a γ a + il a γ a ) L 1, (I = 2). Therefore we get dl 1 (λ) = λdl 1 (62)

No. 4 Flat Currents of Green Schwarz Superstring in AdS 2 S 2 Background 667 = λ i 2 (La γ a + il a γ a ) L 2 + λ 1 4 γ abl 1 L ab + λ 1 4 γ a b L1 L a b = i 2 γ al 2 (λ) L a (λ) + 1 2 γ a L2 (λ) L a (λ) + 1 4 γ abl 1 (λ) L ab (λ) + 1 4 γ a b L1 (λ) L a b (λ), (63) dl 2 (λ) = i 2 γ al 1 (λ) L a (λ) 1 2 γ a L1 (λ) L a (λ) + 1 4 γ abl 2 (λ) L ab (λ) + 1 4 γ a b L2 (λ) L a b (λ). (64) In general dl I (λ) = i 2 γ aɛ IJ L J (λ) L a (λ) + 1 2 γ a ɛ IJL J (λ) L a (λ) + 1 4 γ abl I (λ) L ab (λ) + 1 4 γ a b LI (λ) L a b (λ). (65) In conclusion, we have checked that all these new Cartan 1-forms L(λ) satisfy the Maurer Cartan equations. Thus the current defined as satisfies J (λ) L a (λ)p a + L a (λ)p a + 1 2 Lab (λ)j ab + 1 2 La b (λ)j a b + Lαα I (λ)q αα I (66) dj (λ) + J (λ) J (λ) = 0, (67) and J (λ) is a flat current with parameter λ. Existence of the λ-dependent flat connections easily leads to the infinite number of conserved quantities. Thus the equation T (λ) 1 dt (λ) = J (λ) (68) is integrable. This leads to the integral round a contour as shown in Fig. 1. P exp = P exp b a t2 t 1 Fig. 1 Integral round a contour. t2 J σ (λ, t 1, σ)dσ P exp J τ (λ, τ, b)dτ t 1 J τ (λ, τ, a)dτ P exp b a J σ (λ, t 2, σ)dσ. (69) For periodic case (J (λ, τ, a) = J (λ, τ, b)), we see that Tr (P exp b a J σ(λ, t, σ)dσ) is independent of t, where the trace is taken in the group PSU(1, 1 2). For infinity boundary, when J (λ, τ, σ = ± ) 0, The monodromy T (λ, t) = P exp J σ (λ, t, σ)dσ is also independent of t. In both cases, we have conserved quantities with one parameter λ, suggesting that the system is integrable. 5 Conclusion and Discussion In this paper we construct a one-parameter family of nonlocal currents from the κ-symmetric action of Green Schwarz superstring in AdS 2 S 2 background. This oneparameter family of flat currents would naturally lead to a hierarchy of classical conserved nonlocal charges by the standard method. This fact is a characteristic property that the second sigma model of AdS 2 S 2 is completely integrable at least at the classical level. The charges are constructed by first identifying a family of currents J (λ), depending smoothly on a spectral parameter λ that are valued in some Lie-algebra psu(1, 1 2) and that are flat: dj (λ) + J (λ) J (λ) = 0. One then constructs the monodromy matrix T (λ) (t) = P exp (+,t) (,t) J (λ), which is conserved by virtue of the flatness of J(λ), and the non-local charges are obtained by expanding T (λ) in powers of the spectral parameter. For the periodic boundary condition (closed string) we have ( ) Tr T (λ) (t) = Tr P exp J σ (λ, t, σ)dσ, is also a conserved quantity. In principle with these formulae one can start the heavy machinery of the inverse scattering method. But even in the bosonic case this is not straightforward because of the possible quantum anomalies. We will not proceed with it here and only notice that this hidden symmetry must manifest itself in the spectrum of the anomalous dimensions. The correspondence and the possible role

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