Generalization of the Lunin-Maldacena transformation on the AdS(5) x S-5 background

全文

(1)PHYSICAL REVIEW D 72, 106008 (2005). Generalization of the Lunin-Maldacena transformation on the AdS5 S5 background R. C. Rashkov* Department of Physics, Sofia University, 1164 Sofia, Bulgaria. K. S. Viswanathan† Department of Physics, Simon Fraser University, Burnaby BC, Canada. Yi Yang‡ Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan (Received 15 September 2005; published 23 November 2005) In this paper we consider a simple generalization of the method of Lunin and Maldacena for generating new string backgrounds based on TsT transformations. We study multishift Ts sT transformations applied to backgrounds with at least two U(1) isometries. We prove that the string currents in any two backgrounds related by Ts sT transformations are equal. Applying this procedure to the AdS5 S5 , we find a new background and study some properties of the semiclassical strings. DOI: 10.1103/PhysRevD.72.106008. PACS numbers: 11.25.Mj, 04.65.+e. I. INTRODUCTION In this paper we present a simple generalization of the method for obtaining deformed string backgrounds proposed by Lunin and Maldacena [1] and developed in detail by Frolov [2]. The method in the above papers is based on T-duality on one of the U(1) variables, shift of another U(1) variable, and T-duality back on the first U(1) variable (called TsT transformation).1 Our method consists in multishifts at the second step which allows one to obtain new string backgrounds (we call this Ts1 sn T transformation). We prove also that the U(1) string currents in any two backgrounds related by Ts1 sn T transformations are equal. We present also an application of our method to string theory in AdS5 S5 background. In the past few years, the main efforts in string theory were directed towards establishing string/gauge theory correspondence. The vast majority of papers were on qualitative and quantitative descriptions of N 4 supersymmetric Yang-Mills (SYM) theory with a SUN gauge group by making use of the string sigma model on AdS5 S5 [3–5]. The AdS/CFT correspondence implies that the energy of closed string states is equal to the anomalous dimensions of certain local SYM operators [6,7]. At supergravity level this correspondence has been checked in a number of cases (for review, see for instance [8]) but the match between the string energy and the anomalous dimensions beyond that approximation still remains a challenge. The first important step in establishing AdS/CFT correspondence is to obtain the spectrum of the anomalous dimensions of the primary operators made up of local *Electronic address: rash@phys.uni-sofia.bg † Electronic address: kviswana@sfu.ca ‡ Electronic address: yiyang@mail.nctu.edu.tw 1 See the discussion in Sec. II.. 1550-7998= 2005=72(10)=106008(13)$23.00. gauge fields. On the string theory side, it requires one not only to solve the theory at the classical level but also include its quantization. The main challenge in quantizing string theory is that it is highly nonlinear and thus difficult to manage. The only option available so far is to look at the semiclassical region of large quantum numbers where the results are reliable. On the gauge theory side, the derivation of the anomalous dimensions is also a difficult task. A breakthrough in this direction has been the observation of Minahan and Zarembo that a one loop dilatation operator restricted to the bosonic sector of N 4 SYM theory can be interpreted as the Hamiltonian of the integrable spin chain [9]. This observation raised the question about the dilatation operator in N 4 SYM theory and integrability (for a recent review, see for instance [10] and references therein). On the other hand, the question of reduction of the string sigma model to particular integrable systems and the question of integrability of string theory at the classical and the quantum level was considered in a number of papers [11– 14]. The intensive study of ‘‘nearly’’ Bogomol’nyi-PrasadSommerfield saturated (BPS), or Berenstein-MaldacenaNastase (BMN) type, quantum strings and non-BPS ones gives a remarkable match with the results from the gauge theory side at least at the first few loops [9,10,12–20]. This match however is not a coincidence. In the above papers it was suggested that certain spin chains should describe particular string sectors and thus should allow the comparison to the gauge theory computations. Subsequently, it has been found that the match between string theory and SYM theory in the examples discussed above lies in the Yangian symmetries responsible to a large extent for the integrability on both sides [21,22]. Since in this paper we will consider the string theory side, we refer the reader to the above papers for details on this connection.. 106008-1. © 2005 The American Physical Society.

(2) R. C. RASHKOV, K. S. VISWANATHAN, AND YI YANG. PHYSICAL REVIEW D 72, 106008 (2005). From the picture emerging from the above studies, one can conclude that the integrable structures play an important role in establishing the AdS/CFT correspondence at the classical and hopefully at the quantum level as well. Although we already have some understanding of string/ gauge theory correspondence in the case of AdS5 S5 background and N 4 SYM, much less is known in the case of theories with less than maximal supersymmetry. There have been some studies of AdS/CFT correspondence for less supersymmetric string backgrounds [23– 44]. However, it is not quite clear how exactly to implement the correspondence. The main obstacles lie in knowing if and how Kaluza-Klein modes naturally present in such backgrounds contribute to the string energy, which corner in the space of gauge operators is described by these strings, and if these subsectors are closed under the renormalization group flow. An important step towards a deeper understanding of AdS/CFT correspondence in its less supersymmetric sector was recently given by Lunin and Maldacena [1]. From the gauge theory point of view, the possible deformations of N 4 SYM gauge theory that break the supersymmetry were studied by Leigh and Strassler [45]. It should be mentioned that the deformations of N 4 SYM theory and integrable spin chains have been considered in some detail in [46]. In [1] Lunin and Maldacena found the gravity dual to the -deformations of N 4 SYM theory studied in [45]. They demonstrated that a certain deformation of the AdS5 S5 background corresponds to a gauge theory with less supersymmetry classified in [45]. This deformation of the string background can be obtained applying two T-dualities accompanied by certain shift parametrized by (TsT transformation). For real values of , Frolov obtained the Lax operator for the deformed background which proves the integrability at classical level [2]. String theory in this background was studied in [47,48] and its pp-wave limit was investigated in [49,50]. The -deformations of more complicated (non)supersymmetric backgrounds was considered also in [51,52]. The aim of this paper is to consider a simple extension of the transformations considered in [1,2] and to prove that, under TsT transformations applied to any background possessing U(1) symmetries, the corresponding currents before and after the transformation are equal. The paper is organized as follows. In the next section we give a brief review of the -deformations of the N 4 gauge theory and its gravity dual. In Sec. III we consider a general background with at least two U(1) isometries. We show that the U(1) currents are equal after Ts1 s2 sn T transformations where s1 sn means multishifts along the remaining U(1) variables. In the next section, as an example for multishift procedure, we consider AdS5 S5 and find a new background parametrized by two real parameters. We show that the new background reduces to those found in [1,2] when one of the parameters vanishes. We also consider the limit of pointlike string which corre-. sponds actually to the BMN limit. In the concluding section, we comment on the results found in the paper. II. LUNIN-MALDACENA BACKGROUND In this section we give a very brief review of the procedure of Lunin and Maldacena for obtaining the gravity dual of the -deformed SYM theory considered in [45]. Let us consider the N 4 SYM gauge theory in terms of N 1 supersymmetry. The theory contains a vector multiplet V and three chiral multiplets i . The superpotential is given by the expression W g0 Tr1 ; 2 3 : The action then can be written as Z i egV i S Tr d4 xd4 egV 0 1 Z 4 2 g 2 d xd W W c:c: 3! 2g Z d4 d2 "ijk i j ; k c:c: :. (2.1). (2.2). We note that the N 4 theory is conformal at any value of the complex coupling . 4i. 2 g2YM. (2.3). and the deformations that change this value are exactly marginal. In [45] Leigh and Strassler considered deformations of the superpotential of the form W h Trei 1 2 3 ei 1 3 2 h0 Tr31 32 33 :. (2.4). Let us focus on the h0 0 case. The symmetries are: one U(1) R-symmetry group and two global U1 U1 groups acting as follows: U 11 :. 1 ; 2 3 ! 1 ; ei’1 2 ; ei’1 3 . U12 :. 1 ; 2 3 ! ei’2 1 ; ei’2 2 ; 3 :. (2.5). Since the theory is periodic in , one can think of as living on a torus with complex structure s and the SL2; Z duality group acts on it and as follows: s !. as b ; cs d. !. cs d. (2.6). 1 s : As a result of all of this, we end up with a N 1 supersymmetric conformal field theory. The gravity dual for real can be obtained in three steps [2]. Consider the S5 part of the AdS5 S5 background. In the first step, we perform a T-duality with respect to one of. 106008-2.

(3) GENERALIZATION OF THE LUNIN-MALDACENA . . .. PHYSICAL REVIEW D 72, 106008 (2005). 2. the U(1) isometries parametrized by the angle ’1 . The second step consists in performing a shift ’2 ! ’2 ’1 where ’2 parametrizes another U(1) isometry of the background and is a real parameter. In the last step we T-dualize back on ’1 . The resulting geometry is described by " X 2 2 dsstr R ds2AdS5 dr2i Gr2i d2i . ~2 r21 r22 r23 . X. !2 # ;. (2.7). G1 1 2 r21 r22 r22 r23 r21 r23 ;. ~ R2 : (2.8). di. where. We start with the general action p d Z S @ X @ X G. d 2 2

(4) @ X @ X B :. We will assume that G and B do not depend on X1 and X2 allowing to perform TsT transformation. In what follows we use the notations 1; . . . ; d, i 2; . . . ; d, a 3; . . . ; d. We will prove the statement in several steps. Step 1: T-duality on X1 . For completeness we write again the T-duality rules and relations4 ~ 11 1 ; G G11 ~ 1i B1i ; G G11. The other fields are correspondingly3 e2 e20 G;. (2.9). ~2 R2 Gr21 r22 d1 d2 r22 r23 d2 d3 BNS r23 r21 d3 d1 ; C2 316Nw1 d ; C4 16Nw4 Gw1 d1 d2d3 ;. (3.1). (2.10). ~ ij Gij G1i G1j B1i B1j ; G G11 G1i B1j B1i G1j (3.2) B~ij Bij ; G11 G B~1i 1i ; G11.

(5) @ X~1 @ XM G1M

(6) @ XM B1M ;. (3.3). @ X~1

(7) @ X G1 @ X B1 ;. (3.4). ~ 1 @ X~ B~1 ; @ X1

(8) @ X~ G. (3.5). (2.11) (2.12) X~ i Xi :. F5 16NwAdS5 GwS5 :. (2.13). Using the fact that the currents J before the TsT transformations are equal to the currents J~ after the transformations, Frolov obtained the Lax operator for the deformed geometry, thus proving the integrability at the classical level. The properties of string theory in this background were further studied in [47,48]. The Penrose limit of the Lunin-Maldacena background was investigated in [49,50]. III. U(1) CURRENTS AND TST TRANSFORMATION As mentioned in the previous section, based on the observation that the string U(1) currents before and after TsT transformation are equal, Frolov was able to obtain the Lax operator of the theory in the deformed background. He also conjectured that the equality of the currents holds for any two backgrounds related by TsT transformation. Below we prove the following. Proposition.—The U(1) currents of strings in any two backgrounds related by TsT transformation are equal.. (3.6). The T-dual action has the same form but with transformed background fields: p d ~ ~ ~ Z S~ @ X @ X G. d 2 2

(9) @ X~ @ X~ B~ : (3.7) Step 2 consists in shift of X~2 ^ x1 ; X~ 2 x~2 ~. X~1 x~1 ;. X~a x~a :. (3.8). Note that the background remains independent of X~1 and X~2 . The shift described above produces the following transformations of the metric ~ 22 ; ~ 11 2^ G ~ 12 ^ 2 G g~11 G ~ 1i ^ G ~ 2i ; ~ ij ; g~ij G g~1i G. (3.9). and for the B~ we get b~ ij B~ij ;. b~1i ! B~1i ^ B~2i :. (3.10). The relations (3.3), (3.4), and (3.5) are also changed; for 2 3. See the appendix for general U(1) T-duality. See [1,2] for details.. 4. 106008-3. See also the appendix..

(10) R. C. RASHKOV, K. S. VISWANATHAN, AND YI YANG. PHYSICAL REVIEW D 72, 106008 (2005). j1 J1 p @ X G1

(11) @ X B1 p : (3.18). instance, (3.5) becomes ~ 1 @ x~ B~1. @ X1

(12) @ x~ G ~ 12 @ ^

(13) @ x~1 G ^ x~1 B~12 : . (3.11) (b) We turn now to the case of Ji and ji i 2; . . . ; d. In this case there are more transformations to be performed but all of them are based on (3.2), (3.3), (3.4), and (3.5). Note that it is crucial that the background is independent of X1 and X2 , otherwise we cannot perform a T-duality back on x~1 . In the new variables the action is given by p Z S~ d d @ x~ @ x~ g~ . 2 2

(14) @ x~ @ x~ b~ :. ji p @ x gi

(15) @ x bi. @ x1 gi1 @ x~j gij

(16) @ x1 b1i

(17) @ x~j bij. (3.12). @ x~1 g~i1 @ x~j g~ij

(18) @ x~1 b~1i

(19) @ x~j b~ij : (3.19). x~1 .. In step 3 we T-dualize back on The action again has the standard form. Now we go to the X~ variables by making the inverse shift. p d Z ~~ @ x @ x g. S d 2 2

(20). . @ x @ x b ;. ji ~ i

(21) @ X~ B~i : (3.20) p @ X~ G Since X~i Xi , we separate X~1 and X~i dependent parts and find. (3.13). where g and b are obtained from g~ and b~ by making use of the standard rule equations (3.2), (3.3), (3.4), and (3.5). and j obtained Now we will prove that the currents J. from (3.1) and (3.13) respectively are equal, i.e., J. . j ;. ji ~ i1

(22) @ X~1 B~i1 p @ X~1 G ~ ij

(23) @ X~j B~ij @ X~j G. (3.14). @ X1 Gi1

(24) @ X1 Bi1 @ Xj Gij. where p p j @ x g

(25) @ x b ; p p @ x G . J. .

(26) @ x B :.

(27) @ Xj Bij :. ji Ji p @ X Gi

(28) @ X Bi p ; (3.23). (3.16). We will prove the statement directly, but in two steps. (a) First we will prove the equality (3.14) for J1 and j1 and then for Ji and ji.

(29) @ x~ g~1 @ x~ b~1 g~11 g~ b~ @ x~i 1i

(30) @ x~i 1i g~11 g~11 g~1. g~

(31) @ x~.

(32) @ x~i 1i g~11 g~11 .

(33) @ x~1 :. (3.22). Therefore. (3.15). j1 p @ x1 g11 @ xi g1i

(34) @ xi b1i @ x1 g11 @ x~i g1i

(35) @ x~i b1i. (3.21). which proves the statement (3.14). IV. Ts1 sd T TRANSFORMATIONS In this section we make a simple generalization of the TsT transformation. We proceed as follows. First we make a T-duality on X1 after which the original action p d Z @ X @ X G. d S 2 2

(36) @ X @ X B . (4.1). becomes (3.17). Now we use (3.3) and find. 106008-4. p d ~ ~ ~ Z @ X @ X G. d S 2 2

(37) @ X~ @ X~ B~ ;. (4.2).

(38) GENERALIZATION OF THE LUNIN-MALDACENA . . .. PHYSICAL REVIEW D 72, 106008 (2005). where the tilde variables are defined in (3.2), with the relations (3.4) and (3.6) satisfied. The second step consists in applying multishifts along the U(1) isometries unaffected by the T-duality in the previous step. This slightly generalizes the MaldacenaLunin procedure described in the previous section, X~ i. . x~i. i x~1 ;. ~1. X . x~1 ;. (4.3). ~ A~ or X x where 0 ~1 1 X .. C B ~ @ X . A; X~N. 0. 1. B B 2 B AB . B @ .. N. 0. . 1 ... 0. . . 0. 1 0 .. C .C C C C: (4.4) 0A 1. and therefore j1 J1 :. Let us show that this equality is satisfied for the other currents. One can easily show that ji j~i :. g~ 11. ~ ij ; g~ij G. b~1i B~1i j B~ij ;. ji p @ x~ g~i

(39) @ x~ b~i. ~ 1i j G ~ ij ~ ij @ x~j j @ x~1 G @ x~1 G (4.11)

(40) @ x~1 B~1i j B~ij

(41) @ x~j j @ x~1 B~ij (4.12). ~1i j G ~ ij ; g~1i G ~ 1

(42) @ x~ B~1 @ x~ G. b~ij B~ij : (4.5) ~1. The last step consists in T-dualization back on x . The resulting action is p d Z @ x @ x g. d S 2 2

(43) @ x @ x b :. (4.6). As in the case of TsT transformation, for the generalization described above we prove below. Proposition.—The U(1) currents of strings in any two backgrounds related by Ts1 Sn T transformation are equal. Proof.—One can first consider j1 and using the relations between the variables write them in terms of the original coordinates j1 p @ x g1

(44) @ xi b1i @ x1 g11 @ x~i g1i

(45) @ x~i b1i

(46) @ x~ g~1 @ x~ b~1 g~11 g~ b~ @ x~i 1i

(47) @ x~i 1i g~11 g~11 ~ g g~1i 1.

(48) @ x~.

(49) @ x~i

(50) @ x~1 : g~11 g~11. (4.10). Let us see how j~i is related to J~i. Under these multishifts the background fields take the form ~ 11 2i G ~ 1i i j G ~ ij ; G. (4.9). J~i p : . Simple calculations now lead to Ji J~i . This proves that ji Ji :. (4.14). Although the proof is straightforward, it may have important consequences. For instance, if the theory in the initial background is integrable, one can study integrability of the second theory by making use of the above relation. We will comment on this issue in the next section. The equality between the currents in the AdS5 S5 background and its deformation relate the boundary conditions imposed on the fields in the initial and the transformed backgrounds. It remains to examine how the boundary conditions for x and X in our case are related. 0 First we notice that the time component of J. , i.e. J. , is. just the momentum conjugated to X . The equality of j0. 0 and J. means that the two momenta are equal and constant (due to the isometry). Therefore this property, observed first in [2], continues to hold in the general case of TsT and multishift transformations. To examine the boundary conditions, we will use the relation @ x1

(51) @ x~ g~1 @ x~ b~1 :. . (4.7). (4.13). (4.15). To simplify the calculation, we choose the conformal gauge for the 2D metric diag1; 1 and

(52) 01 1. Let us compute the boundary conditions for x1 . To do this we need expressions for the metric components g~ in terms of the original metric G . Using the transformation properties, we find g~ 11 . But J1 p @ X G1

(53) @ X B1

(54) @ x~1 ; (4.8). 106008-5. g~ 1i . G ; G11. B1i j Gij G11 G1i G1j B1i B1j ; G11. (4.16). (4.17).

(55) R. C. RASHKOV, K. S. VISWANATHAN, AND YI YANG. PHYSICAL REVIEW D 72, 106008 (2005). and G1i j Bij G11 G1i B1j B1i G1j ; b~ 1i G11. (4.18). p Z d ~~ i @ ’ ~~ j S d @ ri @ ri gij @ ’ 2 2 r2i 1; (5.1) where the metric gij and the antisymmetric 2-form field bij are. where G 1 2i B1i i j Gij G11 G1i G1j B1i B1j . g11 r22 r23 ;. (4.19) (all others are not changed by the shifts and are given in the appendix). Substituting the above expressions for g~ and b~ in (4.15) and using the inverse transformations relating x~. withX , we find @1 x1 @1 X1 i Ji0 ;. i 2; . . . ; N:. (4.20). The boundary conditions for the other coordinates are easily obtained from @ xi @ x~i i @ x~1 :. i. i. j. i. @1 x @1 X @0 x G1 @1 x B1j @1 x . i J10 : (4.22). Therefore, the boundary conditions for the fields in the deformed background are twisted as follows: @1 x1 @1 X1 i Ji0 ;. (4.23). @1 xi @1 Xi i J10 :. (4.24). g13 r22 r23 ;. i Ji ;. (4.25). xi 2 xi 0 2ni i J1 ;. (4.26). X 2 X 0 2n ;. (4.27). . x1 0. 2n1 . where. and the current. g23 r22 r21 ; (5.2). and is a Lagrangian multiplier which ensures the constraint X r2i 1: (5.3) This action is related to the one used in [1] by the following change of the variables: ~~ 1 1’^ 1 ’^ 2 2’^ 3 ; ’ 3 ~~ 2 12’^ 1 ’^ 2 ’^ 3 ; ’ 3. ~~ 3 1’^ 1 ’^ 2 ’^ 3 ; ’ 3 (5.4). which leads to the following relations between the old and new angular momenta:. Integrating over we find x1 2. g33 1;. bij 0:. (4.21). Using the relation (3.11) and (4.3), we get i. g12 r22 ;. g22 r21 r22 ;. J~~ 1 J^2 J^3 ;. (5.5). J~~ 2 J^2 J^1 ;. (5.6). J~~ 3 J^1 J^2 J^3 :. (5.7). We next make the T-duality transformation on the circle parametrized by ’1 ; the action becomes p d Z @ ri @ ri g~ij @ ’ d S ~ i @ ’ ~j 2 2

(56) b~ij @ ’ ~ i @ ’ ~ j r2i 1; (5.8) where. J . Z d J0 : 2. (4.28). g~11 . In the next section we will apply these results to the AdS5 S5 background and analyze the implications of these transformations to string theory.. g~33 g~23 . V. ^ 2 ; ^ 3 -DEFORMATION. 1 ; 2 r2 r23. g~22 . r22 r23 r22 r23 2 ; r22 r23. 2r22 r23 r21 r22 r21 r23 ; r22 r23 2. We start with the S5 part of string action as in [2] with i 1; 2; 3:. 2. r r3 ; b~13 22 r2 r23. A. Supergravity solution. r21 r22 r21 r23 r22 r23 ; r22 r23 g~12 g~13 0; b~12 . r22 ; r22 r23. b~23 0:. ~~ i as follows: The T-dual variables ’ ~ i are related to ’. 106008-6. (5.9).

(57) GENERALIZATION OF THE LUNIN-MALDACENA . . .. ~~ 1

(58) @ ’ @ ’ ~ 1 g~11 @ ’ ~ i b~1i ; ~~ 2 ’ ~~ 3 ’ ’ ~2; ’ ~3:. PHYSICAL REVIEW D 72, 106008 (2005). G1 1 ^ 22 r21 r22 r21 r23 r22 r23 (5.10). Next, we make the following shift of the angle variables ’ ~2 and ’ ~ 3 simultaneously: ’ ~2 !’ ~ 2 ^ 2 ’ ~ 1;. ’ ~3 ! ’ ~ 3 ^ 3 ’ ~1;. (5.11). ^ 23 r22 r23 r22 r23 2 2^ 2 ^ 3 2r22 r23 r21 r22 r21 r23 ;. and we have used the constraint (5.3). The variables ’ ~ i are related to the T-dual variables ’i as follows: @ ’ ~ 1

(59) @ ’ ~ i G1i @ ’ ~ i B1i ;. where ^ 2 and ^ 3 are two arbitrary constants. The metric transforms in the following way under the above shift:. ’ ~ 2 ’2 ;. (5.17). ’ ~ 3 ’3 :. Equations (5.10), (5.13), and (5.17) allow us to determine ~~ i and the following relations between the angle variables ’ the TsT-transformed variables ’i :. g~11 ! g~11 ^ 22 g~22 ^ 23 g~33 2^ 2 g~12 2^ 3 g~13 2^ 2 ^ 3 g~23 ; g~12 ! g~12 ^ 2 g~22 ^ 3 g~23 ; g~13 ! g~13 ^ 2 g~23 ^ 3 g~33 ; b~12 ! b~12 ^ 3 b~23 ; b~13 ! b~13 ^ 2 b~23 ;. (5.12). ~~ 1 ~ @ ’ g11 G1i ^ 2 b~12 ^ 3 b~13 B1i b~1i @ ’i ^ 2 b~12 ^ 3 b~13 G1i g~11 B1i

(60) @ ’i ; (5.18) ~~ 2 @ ’2 ^ 2 B1i @ ’i ^ 2 G1i

(61) @ ’i ; @ ’ (5.19). and the variables ’ ~ i transform into. ~~ 3 @ ’3 ^ 3 B1i @ ’i ^ 3 G1i

(62) @ ’i ; @ ’. ~~ 1

(63) @ ’ @ ’ ~ 1 g~11 @ ’ ~ i b~1i ^ 2 @ ’ ~ 1 b~12 ^ 3 @ ’ ~ 1 b~13 ; ~~ 2 ’ ’ ~2;. ~~ 3 ’ ’ ~3:. (5.16). (5.20) (5.13). Finally, we make the T-duality transformation on the circle parametrized by ’ ~ 1 again. After the TsT transformation, the ^ 2 ; ^ 3 -deformed background becomes p d Z @ ri @ ri Gij @ ’i @ ’i d S 2 2

(64) Bij @ ’i @ ’i r2i 1; (5.14). which gives the boundary conditions ~~ 01 ’01 ^ 2 J20 ^ 3 J30 ; ’. ~~ 02 ’02 ^ 2 J10 ; ’. ~~ 03 ’03 ^ 3 J10 ; ’. (5.21). which are consistent with the boundary conditions (4.23) and (4.24). It is easy to see that, when ^ 3 0, the above background reduces to the Lunin-Maldacena background [1,2]. We can check that the Virasoro constraint, ~~_ i ’ ~~_ j ’ ~~ 0j Gij ’_ i ’_ j ’0i ’0j ; ~~ 0i ’ gij ’. where. (5.22). is satisfied as expected. G1i Gg1i ; B. The dual field theory. G22 Gg22 9^ 23 r21 r22 r23 ; G33 Gg33 9^ 22 r21 r22 r23 ; G23 Gg23 9^ 2 ^ 3 r21 r22 r23 ; B12 G^ 2 r21 r22 r21 r23 r22 r23 ^ 3 2r22 r23 r21 r22 r21 r23 ; B13 G^ 2 2r22 r23 r21 r22 r21 r23 ^ 3 r22 r23 r22 r23 2 ; B23 G^ 2 2r21 r22 r21 r23 r22 r23 ^ 3 g13 g23 g12 ; (5.15) where. According to the AdS/CFT duality, string theory in the background (5.15) is dual to a field theory on the boundary of the AdS space. This field theory is a deformed theory from N 4 SYM theory by the deformation ^ 2 ; ^ 3 , so we will call it ^ 2 ; ^ 3 -deformed N 4 SYM theory. Now the question is: what is this dual field theory? To answer this question, let us look at the symmetries of the deformed background (5.15). We try first to find how many supersymmetries are preserved in the dual field theory. To derive the background (5.15), we wrote the S5 part of AdS S5 as (5.1). The metric has manifestly a U1 U1 U1 isometry, of which a U1 U1 preserves the Killing spinors. In the case of the Lunin-Maldacena background, a very special. 106008-7.

(65) R. C. RASHKOV, K. S. VISWANATHAN, AND YI YANG. PHYSICAL REVIEW D 72, 106008 (2005). torus was chosen to compactify the 10D string theory. The TsT transformation only breaks the supersymmetry corresponding to the Killing spinor associated to U1 U1 so that the deformed background preserves 1=4 supersymmetries. The left U(1) remains an R-symmetry in the dual N 1 SYM theory. In our case, TssT transformation breaks all U1 U1 U1 isometry so that no Killing spinor is preserved. Therefore the dual field theory has no supersymmetry. Next we try to learn more about the dual field theory from the gravity side. Let us recall the relation between the TsT transformation of the supergravity background and the star product of the dual field theory in the case of the Lunin-Maldacena background [1]. SL2; R acts on the parameter p B12 i g; (5.23). background is equivalent to a shift of the noncommutative parameters by 12 12 and 13 13 in the dual field theory. Since the modification only affects the directions 1 ; 2 ; 3 , the action of the dual field theory will be the same as the one of the N 4 SYM theory except the superpotential term, which can be obtained from the undeformed one by replacing the usual product i j by the associative star product i

(66) j . Obviously, we will not be able to write down the action by using the N 1 superfields since all supersymmetries are broken in the process. C. Semiclassical analysis A classical solution of the sigma model associated with the background (5.15) is obtained as t ;. as ! 0 . 1 . or. 1 1 1 ! 0 : . Schematically, 1= can be written as [53] ij 1 1 ij. Gij open ; g B Gij open. (5.24). (5.25). ’3 3 ;. 1 23;. where is the open string metric and is the noncommutative parameter. Then the result of the SL2; R transformation (5.24) is just to introduce a noncommutativity parameter 12 . This analogy can be seen more precisely if we define a 2 2 matrix as 1 1 0 g B0 g B . Gopen . It is easy to get the matrix 0 . 0. : 0. :. ’1 1 ; ’2 2 ; s ^ 2 2^ 3 arccos ; ; 4 4^ 3 ^ 2 (5.29). where. ij. G0open. 0;. 2 43;. 3 13:. (5.30). The angular momenta and the energy corresponding to this state are J1 0; J2 . (5.26) J3 (5.27). 3^ 3 C; ^ 2 4^ 3. 3^ 2 C; ^ 2 4^ 3. (5.31). (5.32). (5.33). and. Thus the TsT transformation of the supergravity background is equivalent to a shift of the noncommutative parameter by 12 in the dual field theory. Now let us look at the ^ 2 ; ^ 3 -deformed background which we found in the previous section. We can similarly define a 3 3 matrix as in (5.26). Straightforward calculation leads to the following5 : 0 1 0 12 13 C @ 12 (5.28) B 0 0 A: 13 0 0 Thus in our case the TsT transformation of the supergravity 5 Here we define new symbols 12 ; 13 which are related to the symbols we used in the previous section as 12 ^ 2 =R2 and 13 ^ 3 =R2 , where R is the radius of S5 .. E 1 J1 2 J2 3 J3 3C;. (5.34). where C/ N is a constant. From the relations of angular momenta (5.5), (5.6), and (5.7), we can see that this solution is associated to the state with ^ 2^ 3 ^ 2 ^ 3 ^ ^ 3 J^1 ; J^2 ; J^3 2 C; C; 2 C : ^ 2 4^ 3 ^ 2 4^ 3 ^ 2 4^ 3 (5.35) It is easy to see that the above state reduces to the J; J; J state when ^ 3 0 and to the J; 0; 0 state when ^ 2 ^ 3 with J E=3. Next, let us consider the fluctuations around the above classical solution (5.29) with large ’t Hooft coupling gYM N R4 =02 as. 106008-8.

(67) GENERALIZATION OF THE LUNIN-MALDACENA . . .. t. PHYSICAL REVIEW D 72, 106008 (2005). The difference between energy and angular momenta is. 1 ~t; 1=4. 1 1=4 ; ~ . E 1 J1 2 J2 3 J3 . 1 ’ ~ 1; 1=4 1 ’2 2 1=4 ’ ~ 2; 1 ’3 3 1=4 ’ ~ 3; s ^ 2 2^ 3 1 ~ arccos 1=4 ; 4^ 3 ^ 2 s ^ 2 4^ 3 ~ 1 : 1=4 4 2^ 2 ^ 3 . E Pt . (5.36). Ji P’i . S ; ’_ i. S ; t_. i 1; 2; 3;. (5.38). (5.39). and H is the corresponding Hamiltonian. By using the Virasoro constraints Taa Gmn @a Xm @a Xn 0;. where we have defined r2 sin cos;. (5.37). where the energy and angular momenta are defined as. ’1 1 . r1 cos;. 1 Z 2 d H 2 0 2. r3 sin sin:. (5.40). and keeping the terms up to quadratic order, the transverse Hamiltonian can be obtained as. ~ a ~ H @a~t@a~t 2 @a @a 4G^ 2 2^ 3 ^ 2 ^ 3 2 ~ 2 @a @ ~ a ~ 4G^ 2 2^ 3 ^ 2 ^ 3 2 ~2 @a @ ~ 1 @a ’ ~1 2G^ 3 ^ 2 @a ’. 3G^ 3 3^ 32 ^ 3 ^ 22 9^ 2 ^ 33 6^ 43 8^ 2 ^ 3 16^ 23 @a ’ ~ 2 @a ’ ~2 4^ 3 ^ 2 2. G 9^ 52 64^ 33 18^ 22 ^ 33 12^ 22 ^ 3 ^ 32 27^ 32 ^ 23 48^ 2 ^ 23 @a ’ ~ 3 @a ’ ~ 3 2G^ 3 ^ 2 @a ’ ~ 1 @a ’ ~2 4^ 3 ^ 2 2 2G. 15^ 22 ^ 3 18^ 2 ^ 43 16^ 33 27^ 22 ^ 33 2^ 32 9^ 42 ^ 3 24^ 2 ^ 23 @a ’ ~ 2 @a ’ ~3 4^ 3 ^ 2 2 s q 2^ 3 ^ 2 ~ 0 0 0 2G^ 3 ^ 2 2^ 3 ^ 2 ^ 2 2^ 3 ~’ ~ 2 2’ ~ 1 2G^ 3 ^ 2 ^ 2 2^ 3 ~ 03 : (5.41) ’ ~ 2 4’ 4^ 3 ^ 2. where we have made a change of coordinates ; ~ 3 ! , 1; 2; 3; 4, and G1 ^ 32 ^ 2 3^ 2 ^ 23 2^ 23 4^ 3 :. (5.42). We diagonalize the Hamiltonian by making the following coordinate transformations: 1 ’ ~ 1 1 2 ; 2. ’ ~ 2 2 ;. 3 . 15^ 22 ^ 3 18^ 2 ^ 43 16^ 33 27^ 22 ^ 33 2^ 32 9^ 42 ^ 3 24^ 2 ^ 23 2 : 9^ 52 64^ 33 18^ 22 ^ 33 12^ 22 ^ 3 ^ 32 27^ 32 ^ 23 48^ 2 ^ 23. Then ~ a ~ ~ 2 @a @ ~ a ~ 4G^ 2 2^ 3 ^ 2 ^ 3 2 ~2 @a @ H @a~t@a~t 2 @a @a 4G^ 2 2^ 3 ^ 2 ^ 3 2 2G^ 3 ^ 2 @a 1 @a 1 . 9G2 ^ 3 ^ 2 4^ 3 ^ 2 @a 2 @a 2 9^ 52 64^ 33 18^ 22 ^ 33 12^ 22 ^ 3 ^ 32 27^ 32 ^ 23 48^ 2 ^ 23. 2G 9^ 52 64^ 33 18^ 22 ^ 33 12^ 22 ^ 3 ^ 32 27^ 32 ^ 23 48^ 2 ^ 23 @a 3 @a 3 4^ 3 ^ 2 q q 4G^ 2 ^ 3 ^ 2 4^ 3 2^ 2 ^ 3 ^ 2 2^ 3 ~ 01 2G^ 2 ^ 3 ^ 2 2^ 3 2^ 2 ^ 3 ^ 2 4^ 3 9^ 2 4^ 3 ^ 2 0 40 : ~ (5.43) 3 9^ 52 64^ 33 18^ 22 ^ 33 12^ 22 ^ 3 ^ 32 27^ 32 ^ 23 48^ 2 ^ 23 2. 106008-9.

(68) R. C. RASHKOV, K. S. VISWANATHAN, AND YI YANG. PHYSICAL REVIEW D 72, 106008 (2005). Since the coefficients are constants, the Hamiltonian can be quantized to get the string spectrum as discussed in [52,54]. VI. CONCLUSIONS In this paper, we consider a deformation of the AdS5 S5 background of string theory. We propose a simple generalization of the Lunin-Maldacena procedure for obtaining a so-called beta deformed theory which, from the gauge theory side, corresponds to a deformation of YangMills theory studied by Leigh and Strassler. For real deformation parameter , the Lunin-Maldacena background can be thought of as a T-duality on one of the angles 1 corresponding to one of the three U(1) isometries of the AdS5 S5 background , a shift on another isometry variable, followed by T-duality again of 1 . It was proven in the original paper by Lunin and Maldacena that this procedure does not produce additional singularities except for only those in the original background. Our generalization consists in additional shifts on the other U(1) variables in the intermediate step. In this way, one can obtain a new deformed background which depends on more parameters 1 n . Since this procedure consists only in additional shifts, the resulting background again contains only the singularities descended from the original one. In Sec. II, we reviewed the Lunin-Maldacena background and the TsT transformation procedure. In the next section we have proved that the currents for any two backgrounds related by TsT transformations are equal (which was conjectured in [2]). In the next section, we consider Ts sT transformations. We find that due to these transformations the boundary conditions for the U(1) variables are twisted. We prove also that the U(1) currents in any two backgrounds related by Ts sT transformations are equal. This property is important since, as it is discussed in [2], it means that the theory preserves the nice property of integrability. The integrability can be proved along the lines of the paper by Frolov [2]. In Sec. V, we apply the TssT transformation to AdS5 S5 background. The obtained background is new and the string theory on it is integrable. We argue that the supersymmetry is broken and the background is less supersymmetric than that of Lunin and Maldacena. After short comments on the gauge theory side, we perform a semiclassical analysis of string theory in 2 3 deformed AdS5 S5 background. We study the theory in the BMN limit and obtain the corresponding conserved quantities important for AdS/CFT correspondence. It is important to note that for 3 0 the background and therefore string theory reduce to that studied by Lunin and Maldacena. In the appendix we give for. completeness a detailed derivation of the T-duality transformations. There are several ways to develop the results obtained in this paper. First of all, one can study multispin solutions in our background along the lines of [47]. To clarify the AdS/ CFT correspondence, one must consider the gauge theory side in more detail. It would be interesting to see what kind of spin chain should describe the string and gauge theory in this case. One can use then the powerful Bethe ansatz technique to study the correspondence on both sides. We leave these questions for further study.. ACKNOWLEDGMENTS We would like to thank J. Maldacena and C. Nu´n˜ez for reading our draft and giving many useful suggestions. We also thank S. Frolov for pointing out a mistake in our first version. R. R. and Y. Y. thank Simon Fraser University for kind hospitality. This work is supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.. APPENDIX: T-DUALITY TRANSFORMATIONS In this appendix we give a detailed derivation of the T-duality transformation. We start with the general string theory action: p d Z S @ XM @ XN GMN Xi d 2 2

(69) @ XM @ XN BMN Xi ;. (A1). where (a) M; N 1; . . . ; d 1, i 2; . . . ; d 1, and (b) the background fields GMN and BMN do not depend on X1 . The equation of motion for X1 tells us that there exists conserved current J : p @L : @ J 0 , J 2 @@ X1 . . (A2). Let us define p as. p @ XN G1N

(70) @ XN B1N :. (A3). The action (5.3) can be rewritten in terms of p as follows:. 106008-10.

(71) GENERALIZATION OF THE LUNIN-MALDACENA . . .. PHYSICAL REVIEW D 72, 106008 (2005). p Z d 1 @ X1 @ X1 G11 @ X1 @ Xi G1i

(72) @ X1 @ XN B1N S d 2 2 1 @ Xi @ Xj Gij

(73) @ Xi @ Xj Bij 2 p Z d 1 @ X1 @ XN G1N

(74) @ XN B1N @ X1 @ X1 G11 d 2 2 1 @ Xi @ Xj Gij

(75) @ Xi @ Xj Bij 2 p Z d 1 1 p @ X1 @ X1 @ X1 G11 @ Xi @ Xj Gij

(76) @ Xi @ Xj Bij : d 2 2 2. (A4). Let us consider the second term in the above expression: 1 @ X1 @ X1 G11 @ X1 G11 @ X1 G11 : 2G11 2. (A5). In order to perform T-duality, we have to eliminate X1 which enters the action only through @ X1 G11 . From the definition of p , p @ X1 G11 @ Xi G1i

(77) @ Xi B1i ;. (A6). @ X1 G11 p @ Xi G1i

(78) @ Xi B1i :. (A7). we find. Substituting for @ X1 G11 in (A5) we find @ X1 G11. @ X1 G11 @ X1 G11 @ X1 G11 2G11 2G11 p @ Xi G1i

(79) @ Xi B1i 2G11 p p G1i G1j G1i B1i 1 i i p @ X

(80) @ X @ Xi @ Xj 2 2G11 G11 G11 G11 B1i B1j

(81) B1i G1j 1 @ Xj @ Xi

(82)

(83) @ Xi @ Xj. 2 G11 2 G11 G B

(84) 1i 1j @ Xi @ Xj. 2 G11 p p G1i G1j B1i B1j G B 1 p @ Xi 1i

(85) @ Xi 1i @ Xi @ Xj 2 2G11 G11 G11 G11 G B G B 1i 1j 1j 1i

(86) @ Xi @ Xj : G11 p @ Xi G1i

(87) @ Xi B1i . (A8). Substitution of (A8) into (A4) gives p Z p p 1 G1i G1j B1i B1j d N G1N N B1N i j p @ X

(88) @ X @ X @ X Gij S d 2 2 G11 G11 2G11 G11 1 G B G1N B1M

(89) @ XM @ XN BMN 1M 1N : (A9) 2 G11 We will use now the conservation of p : @ p 0; to write down the general solution to (A10) as. p

(90) @ X~1 ; (A10). (A11). where X~1 is a scalar field which is the T-dual of X1 . If we substitute for p from (A11) into its definition (A3), we find the relation. 106008-11.

(91) R. C. RASHKOV, K. S. VISWANATHAN, AND YI YANG.

(92) @ X~1 @ XM G1M

(93) @ XM B1M :. PHYSICAL REVIEW D 72, 106008 (2005). (c) The third term in (A9) can be written as. (A12). . Now we can derive the T-dual action by substituting for p the expression (A11). Let us consider the different terms separately. Obviously ij components remain the same since X~i i X. (a) The first term in (A9) becomes. Summing up all the terms we derived above, we find. G p @ XN G1N p @ X1 p @ X~i 1i G11 G11 p @ X1

(94) @ X~1 @ X~i.

(95) @ X~1 @ X1 G11 @ X~i G1i (A14). and therefore 1 B @ X

(96) @ X~1

(97) @ X~i 1i G11 G11 G @ X~i 1i : (A15) G11 1. . Substituting (A15) into (A13) we get 1 G11 B @ X~1 @ X~i1 1i G11 G

(98) @ X~1 @ X~i 1i : G11.

(99) ~1 ~i G1i 1 ~1 ~1 1 @ X @ X @ X @ X G11 2 G11 2 1 ~1 ~i B1i @ X @ X : 2 G11. G1i ; G11 (A13). where we substitute p in the second term with

(100) @ X~1 . We need also expression for @ X1 in terms of X~M . From (A12) we have

(101) @ X~i B1i ;. 1 p p 1

(102)

(103) @ X~1 @ X~ 2 2 G11 G11 1 1 @ X~1 @ X~i : 2 G11 (A18). All the other terms in the action remain unchanged. The final action has the same form as (A1) but with new background fields p d ~M ~N ~ Z S @ X @ X GMN d 2 2

(104) @ X~M @ X~N B~MN ; (A20) with the following transformation laws for the background fields: ~ 11 1 ; G G11 ~ 1i B1i ; G G11. p @ X1 @ X~1 @ X~1. ~ ij Gij G1i G1j B1i B1j ; G G11 G1i B1j B1i G1j B~ij Bij ; (A21) G11 G B~1i 1i ; G11. and the following relations between the variables: X~i Xi ;

(105) @ X~1 @ XM G1M

(106) @ XM B1M ;. (A16). (A22). or, equivalently,. (b) The second term in (A9) becomes p

(107) @ X~i. (A19). B1i B

(108)

(109) @ X~i 1i G11 G11 1 @ X~. 1 B @ X1

(110) @ X~1

(111) @ X~i 1i G11 G11 G @ X~i 1i G11 ~1M @ X~M B~1M :

(112) @ X~M G. B @ X~1 @ X~i 1i : G11 (A17). This completes transformations.. 106008-12. the. derivation. of. the. (A23) T-duality.

(113) GENERALIZATION OF THE LUNIN-MALDACENA . . .. PHYSICAL REVIEW D 72, 106008 (2005). [1] O. Lunin and J. Maldacena, J. High Energy Phys. 05 (2005) 033. [2] S. Frolov, J. High Energy Phys. 05 (2005) 069. [3] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); Int. J. Theor. Phys. 38, 1113 (1999). [4] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998). [5] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428, 105 (1998). [6] D. Berenstein, J. M. Maldacena, and H. Nastase, J. High Energy Phys. 04 (2002) 013. [7] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Nucl. Phys. B636, 99 (2002). [8] Eric D’Hoker and Daniel Z. Freedman, hep-th/0201253. [9] J. A. Minahan and K. Zarembo, J. High Energy Phys. 03 (2003) 013. [10] N. Beisert, Phys. Rep. 405, 1 (2004). [11] G. Arutyunov, J. Russo, and A. A. Tseytlin, Phys. Rev. D 69, 086009 (2004). [12] G. Arutyunov and S. Frolov, J. High Energy Phys. 02 (2005) 059. [13] M. Kruczenski, Phys. Rev. Lett. 93, 161602 (2004). [14] H. Dimov and R. C. Rashkov, Int. J. Mod. Phys. A 20, 4337 (2005). [15] I. Bena, J. Polchinski, and R. Roiban, Phys. Rev. D 69, 046002 (2004). [16] N. Beisert, S. Frolov, M. Staudacher, and A. A. Tseytlin, J. High Energy Phys. 10 (2003) 037. [17] M. Kruczenski, A. V. Ryzhov, and A. A. Tseytlin, Nucl. Phys. B692, 3 (2004). [18] L. Freyhult, J. High Energy Phys. 06 (2004) 010. [19] R. Hernandez and E. Lopez, J. High Energy Phys. 04 (2004) 052. [20] R. Hernandez, E. Lopez, A. Perianez, and G. Sierra, J. High Energy Phys. 06 (2005) 011. [21] L. Dolan, C. R. Nappi, and E. Witten, hep-th/0401243. [22] A. Agarwal and S. G. Rajeev, Int. J. Mod. Phys. A 20, 5453 (2005). [23] H. Dimov, V. Filev, R. C. Rashkov, and K. S. Viswanathan, Phys. Rev. D 68, 066010 (2003). [24] N. P. Bobev, H. Dimov, and R. C. Rashkov, hep-th/ 0410262. [25] R. C. Rashkov and K. S. Viswanathan, hep-th/0211197. [26] R. C. Rashkov, K. S. Viswanathan, and Y. Yang, Phys. Rev. D 70, 086008 (2004). [27] D. Mateos, T. Mateos, and P. K. Townsend, J. High Energy Phys. 12 (2003) 017. [28] D. Mateos, T. Mateos, and P. K. Townsend, hep-th/ 0401058.. [29] S. A. Hartnoll and C. Nunez, J. High Energy Phys. 02 (2003) 049. [30] M. Alishahiha and A. E. Mosaffa, J. High Energy Phys. 10 (2002) 060. [31] M. Alishahiha, A. E. Mosaffa, and H. Yavartanoo, Nucl. Phys. B686, 53 (2004). [32] M. Alishahiha and A. E. Mosaffa, Int. J. Mod. Phys. A 19, 2755 (2004). [33] H. Ebrahim and A. E. Mosaffa, J. High Energy Phys. 01 (2005) 050. [34] K. Ideguchi, J. High Energy Phys. 09 (2004) 008. [35] F. Bigazzi, A. L. Cotrone, and L. Martucci, Nucl. Phys. B694, 3 (2004). [36] F. Bigazzi, A. L. Cotrone, L. Martucci, and L. A. Pando Zayas, Phys. Rev. D 71, 066002 (2005). [37] J. M. Pons and P. Talavera, Nucl. Phys. B665, 129 (2003). [38] D. Aleksandrova and P. Bozhilov, J. High Energy Phys. 08 (2003) 018. [39] D. Aleksandrova and P. Bozhilov, Int. J. Mod. Phys. A 19, 4475 (2004). [40] M. Schvellinger, J. High Energy Phys. 02 (2004) 066. [41] V. Filev and C. V. Johnson, Phys. Rev. D 71, 106007 (2005). [42] E. G. Gimon, L. A. Pando Zayas, J. Sonnenschein, and M. J. Strassler, J. High Energy Phys. 05 (2003) 039. [43] A. Armoni, J. L. F. Barbon, and A. C. Petkou, J. High Energy Phys. 10 (2002) 069. [44] Z. W. Chong, H. Lu, and C. N. Pope, hep-th/0402202. [45] R. G. Leigh and M. J. Strassler, Nucl. Phys. B447, 95 (1995). [46] David Berenstein and Sergey A. Cherkis, Nucl. Phys. B702, 49 (2004). [47] S. A. Frolov, R. Roiban, and A. A. Tseytlin, hep-th/ 0503192. [48] N. P. Bobev, H. Dimov, and R. C. Rashkov, hep-th/ 0506063. [49] R. de Mello Koch, J. Murugan, J. Smolic, and M. Smolic, J. High Energy Phys. 08 (2005) 072. [50] T. Mateos, J. High Energy Phys. 08 (2005) 026. [51] U. Gursoy and C. Nunez, Nucl. Phys. B725, 45 (2005). [52] S. A. Frolov, R. Roiban, and A. A. Tseytlin, J. High Energy Phys. 07 (2005) 045. [53] N. Seiberg and E. Witten, J. High Energy Phys. 09 (1999) 032. [54] M. Blau, M. O’Loughlin, G. Papadopoulos, and A. A. Tseytlin, Nucl. Phys. B673, 57 (2003).. 106008-13.

(114)