T-duality as coordinates permutation in double space for weakly curved background

In the paper [1] we showed that in double space, where all initial coordinates $x^\mu$ are doubled $x^\mu \to y_\mu$, the T-duality transformations can be performed by exchanging places of some coordinates $x^a$ and corresponding dual coordinates $y_a$. Here we generalize this result to the case of weakly curved background where in addition to the extended coordinate we will also transform extended argument of background fields with the same operator $\hat {\cal T}^a$. So, in the weakly curved background T-duality leads to the physically equivalent theory and complete set of T-duality transformations form the same group as in the flat background. Therefore, the double space represent all T-dual theories in unified manner.


Introduction
The T-duality is one of the stringy properties, because it has no analogy in particle physics. Its distinguishing features are unification of equations of motion with Bianchi identity. The standard way to construct T-dual theory is Buscher's proscription [2,3,4,5]. In order to apply such approach it is necessary that background has some continuous isometries which leaves the action invariant. Then, in some adopted coordinates, the background does not depend on these coordinates. For the backgrounds which depend on the coordinates such approach is not applicable.
The simplest coordinate depending background is the weakly curved background. There the metric G µν is constant and the Kalb-Ramond field B µν is linear in coordinates with infinitesimal coefficient. In the paper [6] the new procedure for T-duality, adopted for the case of the weakly curved background, has been introduced. This approach generalize Buscher's one and makes it possible to carry out T-duality along coordinates on which the Kalb-Ramond field depends. In that article T-duality transformations has been performed simultaneously along all coordinates x µ : T f ull = T 0 • T 1 • . . . • T D−1 , (µ = 0, 1, · · · , D), while in the article [7] it has been performed along any subset of coordinates x a : T a = T 0 • T 1 • . . . • T d−1 , (a = 0, 1, · · · , d − 1). The first case connects the beginning and the end of the T-duality chain, while the second one connects the beginning with the arbitrary node. Here Π i±µν and x µ i , (i = 1, 2, · · · , D) are background fields and the coordinates of the corresponding configurations and we will also use notation Π D±µν = ⋆ Π ±µν and x µ D = y µ . The nontrivial extension of T-duality transformations in this approach, compared with the flat space case, is a source of closed string non-commutativity [8,9,10].
The T-duality in the extended space has been investigated in Refs. [11]- [17]. In Ref. [11] all coordinates are doubled and T-duality relation between the beginning and the end of the chain has been established for the flat space. In Ref. [12] only coordinates along which T-duality is performed are doubled and background fields do not depend on them. The relation with our approach has been discussed in Ref. [1].
In paper [1] the extended space with coordinates Z M = (x µ , y µ ), which contains all the coordinates of the initial and T-dual spaces, has been introduced. It was shown that in such double space T-duality has a simple interpretation. Arbitrary T-duality T a = T a •T a , along some initial coordinates x a : T a = T 0 • T 1 • . . . • T d−1 , and along corresponding T-dual ones y a : T a = T 0 • T 1 • . . . • T d−1 , can be realized by exchanging their places, x a ↔ y a . It has been proven for constant background fields, the metric G µν and the Kalb-Ramond field B µν , when Buscher's approach can be applied. This interpretation shows that T-duality leads to the equivalent theory, because replacement of coordinates does not change the physics.
In the present article, following idea of Ref. [1], we are going to offer similar interpretation of T-duality in the weakly curved background. The main difference, comparing with the flat space case, is that in the weakly curved background the background fields depend on the coordinates. So, together with changing the coordinates, we should change the arguments of the background fields, also.
Let us stress that we doubled all the coordinates. We rewrite T-duality transformations connected beginning and end of the chain (1.1) in the double space. We obtain the fundamental expression, where the generalized metric depend on both initial and T-dual coordinates. We will show that this expression is enough to find background fields from all nodes of the chain (1.1) and T-duality transformations between arbitrary nodes. In such a way, as well as in the flat background, we unify all T-dual theories of the chain (1.1).

T-duality in the weakly curved background
The propagation of the closed bosonic string in D-dimensional space-time is described by the action [18] S Here x µ (ξ), µ = 0, 1, ..., D − 1 are the coordinates of the string moving in the background, defined by the space-time metric G µν and the Kalb-Ramond field B µν . The intrinsic worldsheet metric we denote by g αβ . The integration goes over two-dimensional world-sheet Σ with coordinates ξ α (ξ 0 = τ, ξ 1 = σ). The space-time equations of motion, in the lowest order in slope parameter α ′ , for the constant dilaton field Φ = const have the form µν is the field strength of the field B µν , and R µν and D µ are Ricci tensor and covariant derivative with respect to space-time metric. The equations of motion are consequence of the world-sheet conformal invariance on the quantum level. We will consider the simplest coordinate dependent solutions of (2.2), the so-called weakly curved background, defined as This background satisfies the space-time equations of motion, if the constant B µνρ is taken to be infinitesimally small (for more details see [9]). Then all the calculations can be performed in the first order in B µνρ , when the Ricchi tensor can be neglected as the infinitesimal of the second order.

Sigma-model approach to T-duality in the weakly curved background
T-dualization along all the coordinates in the weakly curved background, has been obtained in Ref. [6]. The T-dual action is where Its symmetric and antisymmetric parts are the inverse of the effective metric G E µν and the non-commutativity parameter θ µν They depend on the expression where ∆y µ = y µ (ξ) − y µ (ξ 0 ) and ∆ỹ µ = (dτ y ′ µ + dσẏ µ ). (2.10) We also introduced flat space effective metric and non-commutativity parameter g µν = (G − 4bG −1 b) µν and θ µν 0 = − 2 κ (g −1 bG −1 ) µν as well as their combinations θ µν Consequently, both T-dual background fields in the case of weakly curved background depend on the coordinates ∆V (y) and have a form Note that the dual effective metric is inverse of the initial metric and hence it is coordinate independent while the following combination depend on the coordinates (2.13)

T-duality transformations in the weakly curved background
As well as in Ref. [1] we will start with the T-duality transformations between all initial coordinates x µ and all T-dual coordinates y µ . For the closed string propagating in the weakly curved background they have been derived in ref. [6] ∂ (2.14) Here V µ is defined in (2.9) and the functions β ± µ have a form If B µν (x) does not depend on some coordinate x µ 1 , then the corresponding β µ 1 functions are equal to zero, β ± µ 1 = β 0 µ 1 = β 1 µ 1 = 0. Because in that case the standard Buscher approach can be applied, from now on we will suppose that B µν (x) depend on all coordinates.
The transformations (2.14) are inverse to one another. Using the fact that we can reexpress these T-duality transformations as (2.17) Here with diamond we denoted redefined background fields, where infinitesimally small parts are rescaled and Let us explain the origin of the coefficient 3 2 . The T-duality relations (2.14) have been obtained varying the gauge fixed action of ref. [6] with respect to v µ ± . From the quadratic terms in v µ ± we obtain the coefficient 2 and from that of third degree in v µ ± we obtain the coefficient 3. Note that equality of some redefined background fields is equivalent to the equality of corresponding initial background fields because both finite and infinitesimal parts are equal. Similarly, all relations between background fields as (2.12) and (2.13) also valid with diamond.

Finally, we can rewrite above T-duality transformations in a form
where the terms with world-sheet antisymmetric tensor ε α β (ε ± ± = ±1) are on the left hand side. In the double space, which contains all initial and T-dual coordinates these T-duality relations obtain the simple form Here we introduced and the coordinate dependent generalized metric In Double field theory [19]- [22] it is usual to call Ω M N the O(D, D) invariant metric and denote with η M N . The argument of the generalized metric was not written manifestly in double form. We can rewrite generalized metric as where the zeroth order generalized metric and infinitesimal coefficient are constant. We also introduced the double space vector according the rule that all background fields in the upper D rows of (2.24) depend on V µ while all background fields in the lower D rows of (2.24) depend on x µ . With the help of (2.9) we can conclude that V µ and consequently Z M arg depend on both y µ and its doublẽ y µ . It is important that we can not express Z M arg in terms of Z M becauseỹ µ is not linear function on y µ . We can relate theirs derivatives as but the arguments of background fields does not appear with derivatives. This is significant differences in relation to a series of papers [19]- [22], where the arguments of background fields depend on Z M . The finite part (the zeroth order) of the T-dual transformations (2.14) have a form The solution of these relation are Note that solution for x µ coincides with V µ in (2.9), so that we also can write The generalized metric satisfies the condition because the generalized metric is symmetric, H T ∼ = H. As noticed in Refs. [11,12], the last relation shows that there exists manifest O(D, D) symmetry. The inverse of the generalized metric has a form In the zero order it takes the form

Equations of motions as consistency condition of T-duality relations
As was discussed in [11,23,6,9] 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. So, we will show that the consistency conditions of the relations (2.22) are T-dual to the equations of motion for both initial and T-dual theories.
We are going to multiply the last equation from the left with H −1 . So, we will need expression .
where in the first row we chose x µ dependence and in the second row V µ dependence. Multiplying (2.37) from the left with H −1 and separating first and second rows we obtain (for simplicity here we omit the indices) where all variables in the first equation depend on x µ and in the second equation on V µ . Using he zeroth order T-dual transformations (2.30) these equations turn to where in analogy with (2.18) we introduce with ∆θ ± (V ) = θ ± (V ) − θ 0± . So, they are just equations of motion for initial and T-dual theories, respectively. Note that the second one can be rewritten as which is the form of the equations of motion of T-dual theory from the Ref. [6]. The expression (2.37) originated from conservation of the topological currents i αM = ε αβ ∂ β Z M , which is often called Bianchi identity. So, we proved that T-duality transformations in the double space (2.22), for weakly curved background, unites equations of motion and Bianchi identities.

T-duality as coordinates permutations in flat double space
The present article is generalization of the paper [1] for the case of weakly curved background. So, in this section we will repeat some notation and the results we are going to use.
Let us split coordinate index µ into a and i ( a = 0, · · · , d − 1, i = d, · · · , D − 1), and denote T-dualization along direction x a and y a The main result of the paper [1] is the proof that exchange the places of some coordinates x a with its T-dual y a , in the flat double space produce the T-dual background fields where all notation are introduced in App. A.1. The symmetric and antisymmetric parts of these expressions are T-dual metric and T-dual Kalb-Ramond field. This is in complete agreement with the Ref. [24,7]. The similar way to perform T-duality in the flat space-time for D = 3 has been described in App. B of Ref. [9]. Consequently, exchange the places of coordinates is equivalent to T-dualization along these coordinates. As was shown in [1] eliminating y i from zero order T-duality transformations (2.22) gives Similarly, eliminating y a from the same relation produces The equation (3.4) is zero order of the T-duality transformations for x a (eq. (44) of ref. [7]) and (3.5) is its analogue for x i .

The complete T-duality chain in the weakly curved background
Following the line of paper [1] we will show that, in the case of the weakly curved background, the complete T-duality chain can be obtained by by exchanging the places of coordinates in the double space. Due to the fact that background fields depend on the coordinates this conjecture will be proven iteratively.
We start with the T-duality relations in the form (2.22) with generalized metric (2.24) In the case of weakly curved background, comparing with the flat case one, the argument dependence is a new feature and will be discussed in Subsecs.4.2 and 4.3.

T-duality transformations as coordinates permutations
In the flat space, permutation of the coordinates x a with the corresponding T-dual y a , we realized by multiplying double space coordinate (2.21) by the matrix We are going to apply the procedure of paper [1] to the case of weakly curved background. Then, beside double space coordinate (4.3) we should also transform extended coordinates of the arguments of background fields (2.28) (4.7) We will require that the T-duality transformations (4.1) are invariant under transformations of the double space coordinates Z N and Z N arg with the same matrix T a . The new coordinates should satisfy the same form of the equation which produced the expression for the dual generalized metric in terms of the initial one a H(T a Z arg ) = T a H(Z arg ) T a . (4.10)

The T-dualities in the double space along all coordinates
To learn what is going on with the arguments of background fields we will suppose that the first relation (4.8) valid and we will derive the second one in the simplest case of complete T-dualization. In that case we have P a → ⋆ P = 1, so that (4.4) turns to T a → ⋆ T = 0 1 1 0 and the first relation (4. y µ x µ . The generalized metric with the help of (4.10) and (4.2) obtains the form .
(4.11) Using the analogy with known expressions for dual background fields (2.11), (2.12) and (2.13), but this time according to (2.19) with diamond, we have That is exactly what we expected, because ⋆ H has the same form as the initial one (4.2) but with T-dual background fields. We additionally learned that we also should exchange x µ with V µ , because all background fields in upper D rows depend on x µ and all background fields in lower D rows depend on V µ . So, ⋆ T does not work only on Z M , as in the case of flat background, but it should be applied to the arguments of background fields, Z M arg as well. It exchanges x µ with V µ as ⋆ Z M arg = ⋆ T M N Z N arg which is just the second eq.(4.8) for P a → ⋆ P , or explicitly So, we should transform both Z M and Z M arg with same matrix ⋆ T M N .

The arguments of the dual background fields
Because, all arguments are multiplied by infinitesimal coefficient B µνρ we can calculate them using the zero order of the T-duality transformations (2.22). So, the solution for x a we proclaim as V a and from (3.4) we obtain which coincide with eq.(37) of Ref. [7]. Similarly, we proclaim solution for x i as V i and from (3.5) we obtain This is in fact the same relation (4.14) with altered indices i, j ↔ a, b.

The T-dual background fields in the weakly curved background
The dual generalized metric (4.10) has explicit form For simplicity, we omit the argument dependence which was discussed in previous subsections in details. As well as in the case of the flat background we will require that the dual generalized metric has the same form as the initial one (2.24) but with T-dual background fields where V i = V i (x a , y i ) and V a = V a (x i , y a ) are defined in (4.15) and (4.14) respectively. The argument dependence has been obtained using (2.25) and the second relation (4.8).
Again, it is useful to consider background fields 18) and express them in terms of initial fields. From lower D rows of expressions (4.16) and (4.17) we have and (4.20) where we used (A.3), (A.10), (A.12) and (A.13) to obtain the second equalities. To calculate the inverse of the last expression we will use the general expression for block wise inversion matrices So, we find that where we introduced It can be shown that where the last expression is definition (A.14). Similarly as in the flat space case, ⋄GE ab , is just ab component of ⋄ G E µν , while the ⋄ĜE ab has the same form as effective metric G E µν but with all components (G, ⋄B ) defined in d dimensional subspace with indices a, b.
With the help of (4.24) we can put the first equation (4.23) in the form ( ⋄ĜE ) ab = ( ⋄GE ) ab − 4( ⋄ β 1γ −1⋄ β T 1 ) ab . Multiplying it on the left with ( ⋄G E −1 ) ab and on the right which help us to verify that .
(4.27) After lengthy computation using (A.5), (A.12) and (A.13) we find where ⋄ Π ±ab and ⋄θab ± are defined in (A.15). Comparison of the upper D rows of expressions (4.16) and (4.17) yields the same result. In component notation we can write The equality of two redefined background fields with diamond means that both finite and infinitesimal parts are equal. According to (2.19) it follows that the same relations are valid for background fields without diamond The arguments of all background fields are [V a (x i , y a ), x i ], where V a is defined in (4.14). Consequently, we obtained the T-dual background fields in the weakly curved background after dualization along directions x a (a = 0, 1, · · · , d − 1). They are in complete agreement with eq.(42) of Ref. [7]. So, the generalized metric H M N (Z arg ) contains sufficient information about background fields in each node of the chain (1.1).
The double space contains coordinates of two spaces totally dual relative to one another. The starting theories are: the initial one described by the action S(x µ ) and its T-dual along all coordinates with the action S(y µ ). Arbitrary T-dualization T a , in the double space along d coordinate with index a, transforms at the same time S(x µ ) to S[V a (x i , y a ), x i ] and S(y µ ) to S[x a , V i (x a , y i )]. The obtained theories are also totally dual relative to one another.

T-duality group for the weakly curved background
Although, in the weakly curved background not only the double coordinate Z M but also the argument of background field Z M arg should be transformed, they are transformed with the same matrix T a . So, the corresponding T-duality group is the same. Successively T-dualization can be represent by matrix multiplications (T a 1 T a 2 ) M N = (T a ) M N . The set of matrices T a , form a commutative group with respect to matrix multiplication. Consequently, the set of all T-duality transformations form the same group with respect to the operation •.
This is a subgroup of the 2D permutational group and T-duality along 2d coordinates x a and y a can be represent as (4.33) or in the cyclic notation (1, D + 1)(2, D + 2) · · · (d, D + d) . (4.34) So, T-duality group is the same for the flat and weakly curved background. It is a global symmetry group of equations of motion (2.37).

Conclusion
In the paper [1] we offered simple formulation for T-duality transformations. We introduced the extended 2D dimensional space with the coordinates Z M = (x µ , y µ ), which beside initial D dimensional space-time coordinates x µ contains the corresponding T-dual coordinates y µ . We showed that in that double space the T-duality along some subset of coordinates x a (a = 0, 1, · · · , d − 1) and corresponding dual coordinates y a is equivalent to replacing their places.
Here we generalize this result to the case of weakly curved background where in addition to the extended coordinate we should also transform extended argument of the background fields. We define particular permutation of the coordinates, realized by matrix T a , which in the weakly curved background has two roles. First, it exchanges the places of some subset of the coordinates x a and the corresponding dual coordinates y a along which we perform T-dualization. Second, in arguments of background fields Z M arg = [V µ (y), x µ ] it exchanges x a with its T-dual image V a (y).
We require that the obtained double space coordinates satisfy the same form of Tduality transformations as the initial one, or in other words that this permutation is a global symmetry of the T-dual transformation. We show that this permutation produce exactly the same T-dual background fields as in the generalized Buscher approach of Ref. [7].
In the flat space-time, this statement has been proved by direct calculations in Ref. [1]. In the weakly curved background, thanks to the arguments of background fields, we made nontrivial generalization. But in that case we should solve the problem iteratively. We start with T-duality transformations (4.1), with 2D dimensional variables Z M and Z M arg . The zero order does not depend on Z M arg , because it appears with infinitesimal coefficient. Elimination of y i from T-duality equations produce solution x a (x i , y a ), eq.(3.4), while elimination of y a produce solution x i (x a , y i ), eq.(3.5). The solution for x a we proclaim the V a (x i , y a ) and solution for x i we proclaim the V i (x a , y i ). Thanks to the plus minus sign in front of T-duality relations (4.1) these solutions depend on both y a and its doubleỹ a . The zero order background field arguments Z (0)M arg consists of V µ and x µ , see (4.7). It is the arguments of the first order transformations, because it appears with infinitesimal coefficient. So, we obtain the first order transformations, which according to (2.37), produce equations of motion.
In the standard Buscher formulation T-duality transforms the initial theory to the equivalent one, T-dual theory. The double space formulation contains both initial and Tdual theory and T-duality becomes the global symmetry transformation. With the help of (4.10) it is, easy to see that equations of motion ( where d = 0 formally corresponds to the neutral element (no permutations) and d = D corresponds to the case when T-dualization is performed along all coordinates. It is well known that all theories in the chain (1.1), except the initial one, are nongeometric. In the case of the weakly curved background, the source of the nongeometry is nonlocality of the arguments, which itself is a line integral. It depend on two variables, the Lagrangian multiplier y µ and its doubleỹ µ . So, because our approach unify all nodes of the chain (1.1) it unify geometric with non-geometric theories. It is also clearly explains that T-duality is unphysical, because it is equivalent to the permutation of some coordinates.
Our approach is essentially different from the standard one of the double field theory, where all background fields depend on the same variable Z M . Here, it formally depends on two kind of double space coordinates Z M and Z M arg . The argument Z M arg appears with infinitesimal coefficient and it is solution for Z M in one order smaller approximation. So, it essentially depend only on variable Z M , but we should solve the problem iteratively.
A Block-wise expressions for background fields In order to simplify notation and to write expressions without indices (as matrix multiplication) we will introduce notations for component fields. The similar relations valid in ij subspace.
With the help of (4.25) one can prove the relation where D ij is defined in (4.23).

A.2 Block-wise expressions for weakly curved background fields
For the effective metric tensor and the Kalb-Ramond background fields (2.18) we define and It is also useful to introduce new notation for expressions (A. 12) and We will use the effective metric and non commutativity parameter but with all contributions from ab subspace (A.14) Note that ⋄ĜE ab = ⋄GE ab because ⋄GE ab is projection of ⋄ G E µν on subspace ab. It is also useful to introduce background field combinations in weakly curved background 15) which are inverse to each other ⋄θac ± ⋄ Π ∓cb = 1 2κ δ a b .
(A. 16) The similar relations valid in ij subspace.