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T-duality as coordinates permutation in double space ∗ B. Sazdović † 6 Institute of Physics, 1 University of Belgrade, 0 2 11001 Belgrade, P.O.Box 57, Serbia c e December 6, 2016 D 3 ] h Abstract t - p Weintroducethe2DdimensionaldoublespacewiththecoordinatesZM =(xµ,y ) µ e which components are the coordinates of initial space xµ and its T-dual y . We shall h µ [ showthatinthisextendedspacetheT-dualitytransformationscanberealizedsimply by exchanging places of some coordinates xa, along which we want to perform T- 3 v dualityandthecorrespondingdualcoordinatesy . Insuchapproachitisevidentthat a 4 T-dualityleadsto the physicallyequivalenttheoryandthatcomplete setofT-duality 2 0 transformationsformsubgroupofthe 2D permutationgroup. So,in the double space 1 we are able to represent the backgrounds of all T-dual theories in unified manner. 0 . 1 0 5 1 Introduction 1 : v T-duality of the closed string has been investigated for a long time [1, 2, 3, 4]. It trans- i X forms the theory of a string moving in a toroidal background into the theory of a string r moving in different toroidal background. Generally, one suppose that background has a some continuous isometries which leaves the action invariant. In suitable adopted coordinates, where the isometry acts as translation, it means that background does not depend on some coordinates. In the paper [5] the new procedure for T-duality of the closed string, moving in D dimensional weakly curved space, has been considered. The generalized approach allows one to perform T-duality along coordinates on which the Kalb-Ramond field depends. In thatarticleT-duality transformationshasbeenperformedsimultaneouslyalongallcoordi- nates. It corresponds to Tfull = T0 T1 ... TD−1 -duality relation with transformation ◦ ◦ ◦ ∗WorksupportedinpartbytheSerbianMinistryofEducationandScience,undercontractNo. 171031. †e-mail: [email protected] 1 of the coordinates y = y (xµ) connecting the beginning and the end of the T-duality µ µ chain Π , xµ ⇀↽T1 Π , xµ ⇀↽T2 Π , xµ ⇀↽T3 ... T⇀↽D Π = ⋆Π , xµ = y . (1.1) ±µν 1±µν 1 2±µν 2 D±µν ±µν D µ T1 T2 T3 TD µ Here Π and x , (i = 1,2, ,D) are background fields and the coordinates of the i±µν i ··· correspondingconfigurations. Applyingthe proposedprocedureT = T T ... T full 0 1 D−1 ◦ ◦ ◦ totheT-dualtheoryonecanobtaintheinitialtheoryandtheinversedualityrelation xµ = xµ(y ), connecting the end and the beginning of the T-duality chain. For simplicity, in µ the article [5] T-duality has been performed along all directions. The nontrivial extension of this approach, compared with the flat space case, is a source of closed string non- commutativity [6, 7, 8]. In D-dimensional space it is possible to perform T-duality along any subset of coordinates xa : Ta = T0 T1 ... Td−1, and along corresponding T-dual ones y : T = a a ◦ ◦ ◦ T T ... T ,(a = 0,1, ,d 1). Inthepaper[9]thiswas doneforthestringmoving 0 1 d−1 ◦ ◦ ◦ ··· − in the weakly curved background. For each case the T-dual actions, T-dual background fields and T-duality transformations has been obtained. Let us stress that T-dualization a = Ta T of the present paper in the 2D dimensional double space contains two T- a T ◦ dualizations in terminology of Ref.[9]. In fact D dimensional T-dualizations Ta and T of a the present paper are denoted a and in Ref.[9]. a T T The introduction of the extended space of double dimensions with coordinates ZM = (xµ,y ) wil help us to reproduce all results of Ref.[9] and offer simple explanation for T- µ duality. In the present article we will demonstrate this for the flat background, while for the weakly curved background it will be presented elsewhere [10]. For example, T-duality Tµ1 (along fixed coordinate xµ1) and T-duality T (along corresponding dual coordinate µ1 y ) can be performed simply by exchanging the places of the coordinates xµ1 and y in µ1 µ1 the double space. It can be realized just multiplying ZM with constant 2D 2D matrix. × Similarly, arbitrary T-duality a = Ta T can berealized by exchanging the places of the a T ◦ coordinatesxµ1,xµ2,···,xµd−1 withthecorrespondingdualcoordinatesyµ1,yµ2,···,yµd−1. From this explanation it is clear that T-duality leads to the equivalent theory, because permutation of the coordinates in double space can not change the physics. Similarapproach toT-duality, asatransformationindoublespace, appearedlongtime ago [11]-[15]. Interest in this topic emerged again with the articles [16, 17]. In the paper [11] the beginning and the nd of the chain (1.1) has been established. The relation of our approach and Ref.[16] will be discussed in Sec.4. The basic tools in our approach are T-duality transformations connected beginning and end of the chain. Rewriting these transformations in the double space we obtain the fundamental expression, where the generalized metric relate derivatives of the extended coordinates. We will show that this expression is enough to find background fields from 2 every nodes of the chain and T-duality transformations between arbitrary nodes. In such a way we unify the beginning and all corresponding T-dual theories of the chain (1.1). 2 T-duality in the double space Let us consider the closed bosonic string which propagates in D-dimensional space-time described by the action [18] 1 ǫαβ S[x]= κ d2ξ√ g gαβG [x]+ B [x] ∂ xµ∂ xν, (ε01 = 1). (2.1) µν µν α β − 2 √ g − ZΣ h − i The string, with coordinates xµ(ξ), µ = 0,1,...,D 1 is moving in the non-trivial − background, defined by the space-time metric G and the Kalb-Ramond field B . Here µν µν g is intrinsic world-sheet metric and the integration goes over two-dimensional world- αβ sheet Σ with coordinates ξα (ξ0 = τ, ξ1 = σ). The requirement of the world-sheet conformal invariance on the quantum level leads to the space-time equations of motion, which in the lowest order in slope parameter α′, for the constant dilaton field Φ = const are 1 R B B ρσ = 0, D Bρ = 0. (2.2) µν − 4 µρσ ν ρ µν Here B = ∂ B +∂ B +∂ B is the field strength of the field B , and R and µνρ µ νρ ν ρµ ρ µν µν µν D are Ricci tensor and covariant derivative with respect to space-time metric. µ We will consider the simplest solutions of (2.2) G = const, B = const, (2.3) µν µν which satisfies the space-time equations of motion. Choosing the conformal gauge g = e2Fη , and introducing light-cone coordinates αβ αβ ξ± = 1(τ σ), ∂ = ∂ ∂ , the action (2.1) can be rewritten in the form 2 ± ± τ ± σ S = κ d2ξ ∂ xµΠ ∂ xν, (2.4) + +µν − ZΣ where 1 Π = B G . (2.5) ±µν µν µν ± 2 2.1 Standard sigma-model T-duality Applying the T-dualization procedure on all the coordinates, we obtain the T-dual action [5] κ2 S[y] = κ d2ξ ∂ y ⋆Πµν∂ y = d2ξ ∂ y θµν∂ y , (2.6) + µ + − ν 2 + µ − − ν Z Z 3 where 2 1 θµν (G−1Π G−1)µν = θµν (G−1)µν. (2.7) ± ≡ −κ E ± ∓ κ E Here we consider flat background and omit argument dependence of. Ref. [5]. The symmetric and antisymmetric parts of θµν are the inverse of the effective metric GE and ± µν the non-commutativity parameter θµν 2 GE G 4(BG−1B) , θµν (G−1BG−1)µν. (2.8) µν ≡ µν − µν ≡ −κ E Consequently, the T-dual background fields are κ ⋆Gµν =(G−1)µν, ⋆Bµν = θµν. (2.9) E 2 Note that the dual effective metric is just inverse of the initial one ⋆Gµν ⋆Gµν 4(⋆B⋆G−1⋆B)µν =(G−1)µν, (2.10) E ≡ − and the following relations valid (⋆B⋆G−1)µ = (G−1B)µ , (⋆G−1⋆B) ν = (BG−1) ν. (2.11) ν ν µ µ − − 2.2 T-duality transformations The T-duality transformations between all initial coordinates xµ and all dual coordinates y of the closed string theory have been derived in ref.[5] µ ∂ xµ = κθµν∂ y , ∂ y = 2Π ∂ xν. (2.12) ± ∼ − ± ± ν ± µ ∼ − ∓µν ± Theyareinversetooneanother. Weomitargumentdependenceandβ± functionsbecause µ they appear only in the weakly curved background. We can put above T-duality transformations in a useful form, where on the left hand side we put the terms with world-sheet antisymmetric tensor ε β (note that ε ± = 1) α ± ± ∂ y = GE ∂ xν 2[BG−1] ν∂ y , ± ± µ ∼ µν ± − µ ± ν ∂ xµ = 2[G−1B]µ ∂ xν +(G−1)µν∂ y . (2.13) ± ∼ ν ± ± ν ± Let us introduce the 2D dimensions double target space, which will play important role in the present article. It contains both initial and T-dual coordinates xµ ZM = . (2.14) yµ ! Here, as well as in Double field theory (for recent reviews see [19]-[22]), all coordinates are doubled. It differs from approach of Ref.[16] where only coordinates on the torus along 4 which we perform T-dualization are doubled. The relation of our and that of Ref.[16] will be discussed in Sec.4. In terms of double space coordinate we can rewrite the T-duality relations (2.13) in the simple form ∂ ZM = ΩMN ∂ ZK, (2.15) ± ∼ NK ± ± H where 0 1 ΩMN = , (2.16) 1 0 ! is a constant symmetric matrix and we introduced so called generalized metric as GE 2B (G−1)ρν = µν − µρ . (2.17) HMN 2(G−1)µρBρν (G−1)µν ! It is easy to check that TΩ = Ω. (2.18) H H As noticed in Ref.[11], the relation (2.18) shows that there exists manifest O(D,D) sym- metry. In Double field theory it is usual to call ΩMN the O(D,D) invariant metric and denote with ηMN. 2.3 Equations of motions as consistency condition of T-duality relations It is well known that the equation of motion and the Bianchi identity of the original theory are equal to the Bianchi identity and the equation of motion of the T-dual theory [11, 23, 5, 7]. The consistency conditions of the relations (2.15) ∂ [ ∂ ZN]+∂ [ ∂ ZN]= 0, (2.19) + HMN − − HMN + ∼ in components take a form ∂ ∂ xµ = 0, ∂ ∂ y = 0. (2.20) + − ∼ + − ν ∼ They are the equations of motion for both initial and T-dual theories. The expression (2.19) originated from conservation of the topological currents iαM = εαβ∂ ZM. It is often called Bianchi identity. In this sense T-duality in the double space β unites equations of motion and Bianchi identities in a single relation (2.19) as is shown in [11]. We can write the action κ S = d2ξ ∂ ZM ∂ ZN , (2.21) + MN − 4 H Z which variation produce the eq.(2.19). 5 3 T-duality as coordinates permutations in double space Let us mark the T-dualization along some direction xµ1 by Tµ1, and its inverse along correspondingdirectiony byT . Uptonowwecollected theresultsfromT-dualizations µ1 µ1 along all directions xµ(µ = 0,1, ,D 1), Tfull = T0 T1 TD−1 and from its ··· − ◦ ◦··· ◦ inverse along all directions y T = T T T . So, the relation (2.15) in fact µ full 0 1 D−1 ◦ ◦···◦ contains T-dualizations along all directions xµ and y = Tfull T . µ full T ◦ In this section we will show that relation (2.15) contains information about any in- dividual T-dualizations along some direction xµ1 and corresponding one y for fixed µ µ1 1 ( µ1 = Tµ1 T ). Applying the same procedure to the arbitrary subset of directions we T ◦ µ1 will be able to obtain all possible T-dualizations. It means that we are able to connect any two backgrounds in the chain (1.1) and treat all theories connected by T-dualities in a unified manner. Let us split coordinate index µ into a and i ( a = 0, ,d 1, i = d, ,D 1), and ··· − ··· − perform T-dualization along direction xa and y a a = Ta T , Ta T0 T1 Td−1, T T T T . (3.1) a a 0 1 d−1 T ◦ ≡ ◦ ◦···◦ ≡ ◦ ◦···◦ We will show that such T-dualization can be obtained just by exchanging places of coordinates xa and y . Note that the double space contains coordinates of two spaces which a are totally dual relative to one another. In the beginning these two theories are the initial one S(xµ) and its T-dual along all coordinates S(y ). Arbitrary T-dualization in the µ double space along d coordinate with index a, a, transforms at the same time S(xµ) to T S[y ,xi] and S(y ) to S[xa,y ]. The obtained theories are also totally T-dual relative to a µ i one another. 3.1 The coordinates permutations in double space Permutation of the initial coordinates xa with its T-dual y we can realize by multiplying a double space coordinate (2.14), now written as xa ZM =  xi  , (3.2) y  a     yi      by the constant symmetric matrix ( a)T = a T T 0 0 1 0 a aM = 1−Pa Pa =  0 1i 0 0  . (3.3) N T Pa 1−Pa !  1a 0 0 0     0 0 0 1i      6 Here P is D D projector with d units on the main diagonal a × 1 0 a P = , (3.4) a 0 0 ! where 1 and 1 are d and D d dimensional identity matrices. In Ref.[3] this transfor- a i − mation is called factorized duality. Note also that ( a a)M = δM , (Ω aΩ)M =( a)M , aΩ a = Ω. (3.5) N N N N T T T T T T The last relation means that a SO(D,D). More precisely, we will see that a is in T ∈ T fact element of permutation group, which is a subgroup of SO(D,D). We will require that the dual extended space coordinate, y a ZM = aM ZN =  xi  , (3.6) a T N xa      yi      satisfy the same form of the T-duality transformations (2.15) as the initial one ∂ ZM = ΩMN ∂ ZK. (3.7) ± a ∼ ± aHNK ± a Consequently, with the help of second equation (3.5) we find the dual generalized metric = a a, (3.8) a H T HT or explicitly (G−1)ab 2(G−1b)a 2(G−1b)a (G−1)aj j b  2(bG−1)ib gij gib 2(bG−1)ij  = − − . (3.9) aHMN 2(bG−1) b g g 2(bG−1) j  − a aj ab − a   (G−1)ib 2(G−1b)ij 2(G−1b)ib (G−1)ij      3.2 Explicit form of T-duality transformations Rewriting eq. (3.7) in components we get ∂ y = 2(bG−1) b∂ y +g ∂ xj +g ∂ xb 2(bG−1) j∂ y ± ± a ∼ − a ± b aj ± ab ± − a ± j ∂ xi = (G−1)ib∂ y +2(G−1b)i ∂ xj +2(G−1b)i ∂ xb+(G−1)ij∂ y ± ± ∼ ± b j ± b ± ± j ∂ xa = (G−1)ab∂ y +2(G−1b)a ∂ xj +2(G−1b)a ∂ xb+(G−1)aj∂ y ± ± ∼ ± b j ± b ± ± j ∂ y = 2(bG−1) b∂ y +g ∂ xj +g ∂ xb 2(bG−1) j∂ y . (3.10) ± ± i ∼ − i ± b ij ± ib ± − i ± j 7 Eliminating y from the second and third equations we find i 1 Π ∂ xb+Π ∂ xi + ∂ y = 0. (3.11) ∓ab ± ∓ai ± 2 ± a ∼ Multiplication with 2κθâb, which according to (A.10) is the inverse of Π , gives ± ∓ab ∂ xa = 2κθâbΠ ∂ xi κθâb∂ y . (3.12) ± ∼ ± ∓bi ± ± ± b − − Similarly, eliminating y from the second and third equations we get a 1 Π ∂ xj +Π ∂ xa+ ∂ y = 0, (3.13) ∓ij ± ∓ia ± 2 ± i ∼ which after multiplication with 2κθîj, the inverse of Π , produces ± ∓ij ∂ xi = 2κθîjΠ ∂ xa κθîj∂ y . (3.14) ± ∼ ± ∓ja ± ± ± j − − The equation (3.12) is the T-duality transformations for xa (eq. (44) of ref. [9]) and (3.14) is its analogue for xi. 3.3 T-dual background fields Requiring that the dual generalized metric (3.9) has the form (2.17) but with T-dual background fields, (denoted by lower index a on the left of background fields) gµν 2( b G−1)µ a a a ν = − , (3.15) aHMN 2(aG−1ab)µν (aG−1)µν ! we can find expressions for the T-dual background fields in terms of the initial ones. It is useful to consider the combination of the dual background fields in the form 1 1 Πµν ( b G)µν = Gµρ[( G−1 b) ν δν] . (3.16) a ± ≡ a ± 2a a a a ρ ± 2 ρ Comparing lower D rows of expressions (3.9) and (3.15) we find (bG−1) b 1g β˜ 1gT ( G−1 b) ν = − a 2 aj − 2 , (3.17) a a µ 21(G−1)ib (G−1b)ij ! ≡ 21γ −β¯T ! and g 2(bG−1) j g˜ 2β ( G−1) = ab − a − 1 . (3.18) a µν 2(G−1b)ib (G−1)ij ! ≡ −2β1T γ¯ ! The notation in the second equalities which has been obtained using (A.3), (A.4), (A.6) and (A.7) will simplify calculations. 8 To obtain background field (3.16) we need the inverse of last expression. We will use the general expression for block wise inversion matrices −1 A B (A BD−1C)−1 A−1B(D CA−1B)−1 = − − − . (3.19) C D ! D−1C(A BD−1C)−1 (D CA−1B)−1 ! − − − It produces (A−1)ab 2(g˜−1β D−1)a ( G)µν = 1 j , (3.20) a 2(γ¯−1β1TA−1)ib (D−1)ij ! where A = (g˜ 4β γ¯−1βT) , Dij = (γ¯ 4βTg˜−1β )ij. (3.21) ab − 1 1 ab − 1 1 After some direct calculations it can be shown that A = (G˜ 4˜bG˜−1˜b) gˆ , (3.22) ab ab ab − ≡ where gˆ has been defined in (A.8). Note that unlike g˜ , which is just ab component ab ab of g , the gˆ has the same form as effective metric g but with all components (G˜,˜b) µν ab µν defined in d dimensional subspace with indices a,b. Using result (3.22) we can rewrite the first equation (3.21) in the form gˆ = g˜ ab ab − 4(β γ¯−1βT) . Multiplying it on the left with (g˜−1)ab and on the right with (gˆ−1)ab we get 1 1 ab (g˜−1)ab = (gˆ−1)ab 4(g˜−1β γ¯−1βTgˆ−1)ab. (3.23) − 1 1 With the help of this relation we can verify that (D−1) = (γ¯−1+4γ¯−1βTgˆ−1β γ¯−1) , (3.24) ij 1 1 ij is inverse of the second equation (3.21). Now, we are able to calculate background field (3.16) g˜−1β D−1γ A−1(β˜ 1) 1A−1gT 2g˜−1β D−1(β¯T 1) Πµν = 1 − ∓ 2 2 − 1 ∓ 2 . (3.25) a ± 12D−1γ −2γ¯−1β1TA−1(β˜∓ 21) γ¯−1β1TA−1gT −D−1(β¯T ∓ 21) ! After tedious calculations using (A.5)-(A.7) and (A.11) we can obtain κθâb κθâbΠ Πµν = 2 ∓ ∓ ±bi , (3.26) a ± κΠ±ibθˆ∓ba Π±ij 2κΠ±iaθˆ∓abΠ±bj ! − − where θâb has been defined in (A.9). ± Itstillremainstocheck thatupperD rowsof(3.9)and(3.15)producethesameexpres- sions for T-dual background fields. The field ( b G−1)µ is just transpose of ( G−1 b) ν. a a ν a a µ It is useful to express gµν in the form a gµν =( G)µρ[δν 4( G−1 b) σ( G−1 b) ν]. (3.27) a a ρ a a ρ a a σ − 9 Then using (3.20), (3.17), (3.22), and (A.11) we can show that (G−1)ab 2(G−1b)a gµν = j , (3.28) a 2(bG−1)ib gij ! − which is in agreement with (3.9). Consequently, we obtained the T-dual background fields in the flat background after dualization along directions xa, (a = 0,1, ,d 1) ··· − Πab = κθâb, Πa = κθâbΠ , a ± 2 ∓ a ±i ∓ ±bi Π a = κΠ θˆba, Π = Π 2κΠ θâbΠ . (3.29) a ±i ±ib ∓ a ±ij ±ij ±ia ∓ ±bj − − The symmetric and antisymmetric parts of these expressions produce T-dual metric and T-dual Kalb-Ramond field. This is in complete agreement with the Refs.[9, 24]. The similar way to perform T-duality in the flat space-time for D = 3 has been described in App. B of ref. [7]. This proves that exchange the places of some coordinates xa with its T-dual y in the a flat double space represents T-dualities along these coordinates. In Sec. 4.1. of Ref.[3] the Buscher’s T-dualities has been derived in eq.(4.9) in the case when there is only one isometry direction. For such a case it was concluded that ”the dual background is related to the original one by the action of factorized duality”. There is essential difference between such eq.(4.9) and relation (3.29) of the present article, where the general case of T-dualityes along arbitrary sets of coordinates has been derived and proof its equivalence with the action of factorized duality. For proof of expression (3.29) with mathematical induction eq.(4.9) is just first step for n = 1. The next step from n to n+1 is nontrivial because then we have three kind of variables (beside isometry one θ there are a set of original variables and a set of variables along which we already performed duality transformations). This leads to the formulae different from eq.(4.9). For example, when we performed T-dualization along more then one coordinate (lets say along xa,a = 1,2) in expression for T-dual background fields it is not carried out division with G as in eq.(4.9) but with G +2B which was recorded aa ab ab in expression θâb of (3.29). − 3.4 T-duality group Successively T-dualization along disjunct sets of directions a1 and a2 will produce T- T T dualization along all directions a = a a 1 2 ∪ a1 a2 = a. (3.30) T ◦T T This can berepresent by matrix multiplications ( a1 a2)M = ( a)M , which is easy to N N T T T check because the projectors satisfy the relations P2 = P , P2 = P , P P = 0 and a1 a1 a2 a2 a1 a2 P +P = P . a1 a2 a 10