T-duality diagram for a weakly curved background

In one of our previous papers we generalized the Buscher T-dualization procedure. Here we will investigate the application of this procedure to the theory of a bosonic string moving in the weakly curved background. We obtain the complete T-dualization diagram, connecting the theories which are the result of the T-dualizations over all possible choices of the coordinates. We distinguish three forms of the T-dual theories: the initial theory, the theory obtained T-dualizing some of the coordinates of the initial theory and the theory obtained T-dualizing all of the initial coordinates. While the initial theory is geometric, all the other theories are non geometric and additionally nonlocal. We find the T-dual coordinate transformation laws connecting these theories and show that the set of all T-dualizations forms an Abelian group.


Introduction
T-duality is a property of string theory that was not encountered in any point particle theory [1,2,3,4]. Its discovery was surprising, because it implies that there exist theories, defined for essentially different geometries of the compactified dimensions, which are physically equivalent. The origin of T-duality is seen in a possibility that unlike point particle the string can wrap around compactified dimensions. But, no matter if one dimension is compactified on a circle of radius R or rather on a circle of radius l 2 s /R, where l s is the fundamental string length scale, the theory will describe the string with the same physical properties. The investigation of T-duality does not cease to provide interesting new physical implications. *  The prescription for obtaining the equivalent T-dual theories is given by the Buscher T-dualization procedure [5,6]. The procedure is applicable along isometry directions, what allows the investigation of a backgrounds which do not depend on some coordinates. It is obtained that T-duality transforms geometric backgrounds to the non-geometric backgrounds with Q flux which are locally well defined, and these to a different types of nongeometric backgrounds, backgrounds with R flux which are not well defined even locally [7,8]. The similar prescription can be used to obtain fermionic T-duality [9]. It is argued that the better understanding of T-duality should be sought for by doubling the coordinates, investigating the theories in which the background fields depend on both the usual space-time coordinates and their doubles [10,11,12,13], which would make the Tduality a manifest symmetry.
T-duality enables the investigation of the closed string non-commutativity. The coordinates of the closed string are commutative when the string moves in a constant background. In a three dimensional space with the Kalb-Ramond field depending on one of the coordinates, successive T-dualizations along isometry directions lead to a theory with Q flux and the non-commutative coordinates [14,15,16]. The novelty in the research is the generalized T-dualization procedure, realized in [17], addressing the bosonic string moving in the weakly curved background -constant gravitational field and coordinate dependent Kalb-Ramond field with infinitesimal field strength. The non-commutativity characteristics of a closed string moving in the weakly curved background was considered in [18].
The generalized procedure is applicable to all the space-time coordinates on which the string backgrounds depend. In Ref. [17], it was first applied to all initial coordinates, which produces a T-dual theory; it was then applied to all the T-dual coordinates and the initial theory was obtained. In this paper, we will investigate the application of the generalized T-dualization procedure to an arbitrary set of coordinates. Let us mark the T-dualization along direction x µ by T µ and the Tdualization along dual direction y µ by T µ . Choosing d arbitrary directions, we mark where µ n ∈ (0, 1, . . . , D − 1), and • marks the composition of T-dualizations. We will apply T-dualizations (1) to the the initial theory, and T-dualizations (2) to its completely T-dual theory (obtained in [17]). We will prove the following composition laws where 1 marks the identical transformation (T-dualization not performed). So, elements 1, T a and T a , with d = 1, . . . , D, form an Abelian group. We will find the explicit form of the resulting theories and the corresponding T-dual coordinate transformation laws. These results complete the T-dualization diagram connecting all the theories T-dual to the initial theory. Because the Kalb-Ramond field depends on all the coordinates, all T-dual theories except the initial one are non geometric and nonlocal unlike the non geometric theories with Q flux, which have local geometric description. To all of these theories there corresponds the flux which is of the same type as the R flux. The obtained relations are the generalization of the Tdualization chain presented in Refs. [14,15,16]. Putting D = 3, d = 1, 2 with B µν depending on x 3 = Z we reproduce the T-duality chain of the Refs. [14,15,16]. Because the Kalb-Ramond field depends only on the third coordinate, a T-dualization along one of the first two coordinates leads to a geometric theory with f flux, while T-dualizing both isometry directions one obtains a non-geometric theory with Q flux. Both theories have local geometric description. Once T-dualization along all three coordinates is performed, the nonlocal and non geometric theory with R flux is obtained.
The generalized T-dualization procedure originates from the Buscher T-dualization procedure. The first rule in the prescription is to replace the derivatives with the covariant derivatives. The new point in the prescription is the replacement of the coordinates in the background fields argument with the invariant coordinates. The invariant coordinates are defined as the line integrals of the covariant derivatives of the original coordinates. Both covariant derivatives and invariant coordinates are defined using the gauge fields. These fields should be nonphysical, so one requires that their field strength should be zero. This is realized by adding the corresponding Lagrange multipliers terms. As a consequence of the translational symmetry one can fix the coordinates along which the T-dualization is performed and obtain a gauge fixed action. An important crossway in the T-dualization procedure is determined by the equations of motion of the gauge fixed action. Two equations of motion obtained varying this action are used to direct the procedure either back to the initial action or forward to the T-dual action. For the equation of motion obtained varying the action over the Lagrange multipliers, the gauge fixed action reduces to the initial action. For the equation of motion obtained varying the action over the gauge fields one obtains the T-dual theory. Comparing the solutions for the gauge fields in these two directions, one obtains the T-dual coordinate transformation laws.

Weakly curved background
Requirement for the quantum conformal invariance of the world-sheet results in the space-time equations of motion for the background fields. In the lowest order in the slope parameter α ′ these equations are Here B µνρ = ∂ µ B νρ + ∂ ν B ρµ + ∂ ρ B µν is the field strength of the field B µν , and R µν and D µ are the Ricci tensor and the covariant derivative with respect to the space-time metric. We will consider one of the simplest coordinate dependent solutions of (5), the weakly curved background. This background was considered in Refs. [19,20,21], where the influence of the boundary conditions on the non-commutativity of the open bosonic string has been investigated. The same approximation was considered in [15,18] in context of the closed string non-commutativity. The weakly curved background is defined by with b µν , B µνρ = const. This background is the solution of the space-time equations of motion if the constant B µνρ is taken to be infinitesimal and all the calculations are done in the first order in B µνρ , so that the curvature R µν can be neglected as the infinitesimal of the second order. The assumption that B µνρ is infinitesimal, means that we consider the D-dimensional torus so large, as to for any choice of indices holds [15] where R µ are the radii of the torus. In this paper we will investigate the T-dualization properties of the action (4) describing the closed string moving in the weakly curved background. Taking the conformal gauge g αβ = e 2F η αβ , the action (4) becomes with the background field composition equal to and the light-cone coordinates given by

Complete T-dualization
The T-dualization of the closed string theory in the weakly curved background was presented in [17]. The procedure is related to a global symmetry of the theory The symmetry still exists in the presence of the nontrivial Kalb-Ramond field (6), but only in the case of the trivial mapping of the world-sheet into the space-time, because in that case the variation of the action (8) equals zero.
The T-dual picture of the theory, obtained applying the T-dualization procedure to all the coordinates, is given by where with being the effective metric and the non-commutativity parameter in Seiberg-Witten terminology of the open bosonic string theory [22]. The T-dual background fields are equal to and their argument is given by Here Θ µν 0± is the zeroth order value of the field composition Θ µν ± defined in (14) and g µν = G µν − 4b 2 µν and θ µν 0 = − 2 κ (g −1 bG −1 ) µν are the zeroth order values of the effective fields (15). The variable ∆ỹ µ is the double of the dual variable ∆y µ = y µ (ξ) − y µ (ξ 0 ), defined as the following line integral taken along the path P , from the point The initial theory (8) and its completely T-dual theory (13) are connected by the T-dual coordinate transformation laws (eq.(42) of Ref. [17]) and its inverse (eq.(66) of Ref. [17]) where and therefore In this section, we will learn what theory is obtained if one chooses to apply the T-dualization procedure to the action (8), along arbitrary d coordinates x a , The closed string action in the weakly curved background (6) has a global symmetry (11). One localizes the symmetry for the coordinates x a , by introducing the gauge fields v a α and substituting the ordinary derivatives with the covariant derivatives The covariant derivatives are invariant under standard gauge transformations In the case of the weakly curved background, in order to obtain the gauge invariant action one should additionally substitute the coordinates x a in the argument of the background fields with their invariant extension, defined by where To preserve the physical equivalence between the gauged and the original theory, one introduces the Lagrange multipliers y a and adds term 1 2 y a F a +− to the Lagrangian, which will force the field strength F a 01 to vanish. In this way, the gauge invariant action is obtained, where the last term is equal to 1 2 y a F a +− up to the total divergence. Now, we can fix the gauge taking x a (ξ) = x a (ξ 0 ) and obtain the gauge fixed action This action reduces to the initial one for the equations of motion obtained varying over the Lagrange multipliers. The T-dual action is obtained for the equations of motion for the gauge fields.

Regaining the initial action
Varying the gauge fixed action (28) over the Lagrange multipliers y a one obtains the equations of motion On this solution the background fields argument ∆V a defined in (26) is path independent and reduces to The gauge fixed action (28) reduces to the initial action (8), but the background fields argument is ∆V a instead of x i . However, the action (8) is invariant under the constant shift of coordinates, so shifting coordinates by x a (ξ 0 ) one obtains the exact form of the initial action.

The T-dual action
Using the equations of motion for the gauge fields, we eliminate them and obtain the T-dual action. The equations of motion obtained varying the gauge fixed action (28) over the gauge fields v a ± are where is the contribution from the background fields argument ∆V a , defined in a same way as in Ref. [17], by Multiplying the equations (32) by 2κΘ ab ∓ , defined in (126), the inverse of the background fields composition Π ±ab , one obtains Substituting (34) into the action (28), we obtain the Tdual action where In order to find the explicit value of the background fields argument ∆V a (x i , y a ), it is enough to consider the zeroth order of the equations of motion for the gauge HereΘ ab 0± and Π 0∓bi stand for the zeroth order values of Θ ab ± and Π ∓bi , and they are defined in (130).
Substituting (37) into (26) we obtain Here are the variables T-dual to the coordinates y a and x i in the zeroth order in B µνρ , for b µν = 0, which we call the double variables. So, we obtain the explicit form of the T-dual action and conclude that it is given in terms of the original coordinates x i and the dual coordinates y a originating from the Lagrange multipliers. However, the background fields argument depend not only on these variables but on their doubles as well. Because of this the theory is nonlocal as the double variablesx i andỹ a are defined as line integrals.
The action (35) can be obtained from the initial action (8) under the following substitutions of the coordinate derivatives and the background fields where the dual background fields are with Π +ij , Π +µν andΘ ab − defined in (36), (9) and (126). The argument of all T-dual background fields is [x i , V a (x i , y a )]. According to (26) and (38), it is nonlocal and consequently non geometric. Calculating the symmetric and antisymmetric part of the T-dual field compositions (42), we obtain that the T-dual metric and Kalb-Ramond field are equal to whereG Eab andθ ab are defined in (125) and (129)

The inverse T-dualization
In this section we will show that T-dualization of the action S[x i , y a ], given by (35), along already treated directions y a leads to the original action, T a : So, let us localize the global symmetry of the coordinates y a δy a = λ a , of the action (35). Note that this is the symmetry, despite the coordinate dependence of the metric (43), due to the invariance of the background fields argument [17]. Following the T-dualization procedure, we substitute the ordinary derivatives with the covariant ones where u ±a are gauge fields which transform as δu ±a = −∂ ± λ a . We also substitute coordinates y a in the background fields argument with the invariant coordinates where In this way, adding the Lagrange multiplier term which makes the introduced gauge fields nonphysical, we obtain the gauge invariant action which after fixing the gauge by y a (ξ) = y a (ξ 0 ) becomes where ∆V a is defined in (38) and ∆U a in (48).

Regaining the T-dual action
The equations of motion obtained varying the gauge fixed action (50) over the Lagrange multipliers z a have the solution On this solution the variable ∆U a defined by (48) is path independent and reduces to ∆U a (ξ) = y a (ξ) − y a (ξ 0 ), and the gauge fixed action (50) reduces to the action (35).

Regaining the initial action
The equations of motion obtained varying the gauge fixed action (50) over the gauge fields u ±a are where termsΘ ab 0∓ β ± b are the contribution from the variation over the background field argument Here β ± a is of the same form as (33) andΘ ab 0∓ is defined in (130).
Let us show that for the equations of motion (54), the gauge fixed action (50) will reduce to the initial action (8). Using the fact thatΘ ab ∓ is inverse to 2κΠ ±ab , these equations of motion can be rewritten as Substituting (56) into (50), using the definition (36) and the first relation in (141) one obtains The explicit form of the argument of the background fields is obtained substituting the zeroth order of the equations (56) into (48) Consequently, the argument of the background fields ∆V a , defined in (38), is just So, the action (57) is equal to the initial action (8) with Comparing the solutions for the gauge fields (52) and (56), we obtain the T-dual transformation law Substituting ∂ ∓ y a to (44) with the help of (59) one finds ∂ ± x a = ∂ ± z a . So, (60) is the transformation inverse to (44), which confirms the relation T a • T a = 1.

T-dualization along all undualized coordinates
In this section we will T-dualize the action (35), applying the T-dualization procedure to the undualized coordinates x i . Substituting the ordinary derivatives ∂ ± x i with the covariant derivatives where the gauge fields w i ± transform as δw i ± = −∂ ± λ i , substituting the coordinates x i in the background field arguments by and adding the Lagrange multiplier term, we obtain the gauge invariant action Substituting the gauge fixing condition x i (ξ) = x i (ξ 0 ) one obtains where ∆W µ = ∆W i , ∆V a (∆W i , y a ) with ∆W i defined by and ∆V a = ∆V a (∆W i , y a ) is defined in (38), where argument x i is replaced by ∆W i .

Regaining the T-dual action
The equations of motion for the Lagrange multipliers y i are and they have the solution On this solution the background field argument ∆W i defined in (65) reduces to so that the argument ∆V a becomes and therefore the gauge fixed action (64) reduces to the action (35).

From the gauge fixed action to the completely T-dual action
The equations of motion obtained varying the gauge fixed action (64) over w i ± are where Terms Π ±ij Θ jµ ∓ β ± µ (W ) are the contribution from the background fields argument defined by calculated using (134), (135) and (38). Using the fact that the background field composition Π ±ij is invese to 2κΘ ij ∓ defined by (141), we can rewrite the equation of motion (70) expressing the gauge fields as Using the second relation in (142), we obtain Substituting (74) into the gauge fixed action (64), we obtain Using (141), (146) and (148) one can rewrite this action as In order to find the background fields argument ∆W i , we consider the zeroth order of the equations (74), and conclude that Using (147) and (142), we obtain that ∆V a (∆W i , y a ) defined in (38) equals ∆V a (∆W i , y a ) = −κθ aµ 0 ∆y µ + (g −1 ) aµ ∆ỹ µ .
Therefore, we conclude that the background fields argument is equal to (17), so that the action (76) is the completely T-dual action (13), which is in agreement with Ref. [17]. Comparing the solutions for the gauge fields (67) and (74), we obtain the T-dual transformation law One can verify that two successive T-duality transformations (44) and (79) correspond to the total T-duality transformation (19). Indeed, the relation (79) is just the i-th component of this transformation. Substituting ∂ ± x i from (79) into (44), using (144) and (148), we obtain which is just the a-th component of the complete Tduality transformation. So, we confirm that T a •T i = T .
6 Inverse T-dualization along arbitrary subset of the dual coordi- Finally, in this section we will show that the T-dualization of the completely T-dual action (13), along arbitrary subset of the dual coordinates y i leads to T-dual action (35). So, let us start with the T-dual action which is globally invariant to the constant shift of coordinates y µ δy µ = λ µ .
We localize this symmetry for the coordinates y i and obtain the locally invariant action where D ± y i = ∂ ± y i + u ±i are the covariant derivatives. The gauge fields u ±i transform as δu ±i = −∂ ± λ i and the invariant coordinates are defined by y inv i = P (dξ + D + y i + dξ − D − y i ). After fixing the gauge by y i (ξ) = y i (ξ 0 ), the action becomes where ∆U i = P (dξ + u +i + dξ − u −i ).

Regaining the T-dual action
The equations of motion obtained varying the gauge fixed action (83) over the Lagrange multipliers have the solution On this solution the variable ∆U i reduces to and therefore ∆V µ (∆U i , y a ) = ∆V µ (y).

Obtaining the T-dual action
The equations of motion obtained varying the action (83) over u ±i are where β ± µ are given by (71). The terms with beta function come from the variation over the argument U i and are calculated using (134) and (17). Using the fact that 2κΠ ∓ij is the inverse of Θ ij ± , the equation (88) can be rewritten as Substituting (90) into the gauge fixed action (83), using (144) we obtain which with the help of (148) becomes In order to find the argument of the background fields ∆V (∆U i , y a ), one considers the zeroth order of the equations (90) and obtains where the double variables are defined in analogy with (39). Substituting (93) into (17), we obtain and ∆V a (∆U i , y a ) = −κ Θ ab which is exactly (38) with z i = x i . So, we can conclude that the action (92) is equal to the T-dual action (35). Comparing the solutions for the gauge fields (85) and (90), we obtain the T-dual transformation law These transformations are inverse to (79), so that T i • T i = 1. Successively applying (96) and (60), using (148) and (144), we obtain the i-th component of the inverse law of the total T-dualization (20). Its a-th component is (60), so we confirm that T a • T i =T .

Group of the T-dual transformation laws
In this section we will recapitulate the coordinate transformation laws between the theories considered. In section 3, we performed T-dualization procedure along coordinates x a and obtained the following coordinate transformation law (44) where V a and β ± a are given by (38) and (33). In the zeroth oder this law implies In section 4, starting from the action S[x i , y a ] we performed T-dualization procedure along coordinates y a and obtained the transformation law (60) which is the law inverse to (98) and in the zeroth order it implies Multiplying the transformation law (98) from the left side by Π ±ca (x) ∼ = Π ±ca x i , ∆V a (x i , y a ) , using (99), we obtain the transformation law (101). So, we confirm that T a • T a = 1.
In the section 5, starting once again from the action S[x i , y a ], we performed T-dualization procedure along the undualized coordinates x i and obtained the coordinate transformation law (79) where V µ and β ± µ are given by (17) and (71). In the zeroth order it gives Two successive T-duality transformations (98) and (104) give the complete transformation (19), so that T a •T i = T . In section 6, starting from the completely T-dual action S[y], we performed T-dualization procedure along coordinates y i and obtained (96) with V a , U i and β ± µ given by (78), (93) and (71). In the zeroth order this law implies Multiplying (107) from the left by using (105), we obtain the transformation law (104), so that T i • T i = 1. Successively applying (107) and (101), using (148) and (144), we obtain the i-th component of the inverse law of the complete T-dualization (20). Its a-th component is (101), so we confirm that T a • T i =T . We can conclude that the elements 1, T a and T a , with d = 1, . . . , D, form an Abelian group. The element T a is the inverse of the element T a .

Comparison with the existing facts
In this section we will compare our results with the Tdualization chain of the Ref. [15]. The coordinates of the D = 3 dimensional torus will be denoted by x 1 , x 2 , x 3 . Because of the different notation, the background fields considered in this paper and those considered in [15], which will be marked by G and B, are related by Nontrivial components of the background considered in Ref. [15] are which in our notation corresponds to the background fields Let us first compare the results in the case d = 1, corresponding to the transition Therefore and so our result is in agreement with that of Ref. [15]. Now, let us make the comparison in the case d = 2 which corresponds to the transition T 1 • T 2 : torus with H flux → Q flux non − geometry.
Instead to perform T 2 dualization, from twisted torus to Q-flux non-geometry as in [15], we will start from the initial background with H-flux and perform T-dualizations along x 1 and x 2 , T 1 • T 2 : S[x] → S[y 1 , y 2 , x 3 ]. The indices take the following values a, b ∈ {1, 2} and i, j ∈ {3}. Because the only nontrivial contribution to the Kalb-Ramond field B ab is B 12 = − 1 2 Hx 3 , the effective background fields areG E ab = δ ab ,Ḡ E ij = δ ij and the only nonzero component ofθ ab isθ 12 = 1 κ Hx 3 . The T-dual background fields linear in H are therefore and Consequently so the results of this paper and [15] in this case coincide.

Conclusion
In this paper, we considered the closed string propagating in the weakly curved background (6), composed of a constant metric G µν and linearly coordinate dependent Kalb-Ramond field B µν , with infinitesimal field strength. We investigated the application of the generalized T-dualization procedure on the arbitrary set of coordinates and obtained the following T-duality diagram: Let us stress that generalized T-dualization procedure enables the T-dualization along arbitrary direction, even if the background fields depend on these directions. The consequence of this procedure is that the arguments of the background fields, such as ∆V a , are nonlocal. They are nonlocal by definition, as they are the line integrals of the gauge fields. Once the explicit form is obtained the non locality is seen in a fact that they depend on double coordinatesx andỹ, which are the line integrals of the τ and σ derivatives of the original coordinates. To all the theories considered, except the initial theory, there corresponds the non geometric, nonlocal flux.
The generalized T-dualization procedure was first applied along arbitrary d (d = 1, . . . , D − 1) coordinates x a = {x µ 1 , . . . , x µ d }. We obtained the T-dual action S[x i , y a ], given by eq. (35) with the dual background fields equal to The argument of all background fields, [x i , V a (x i , y a )], depends nonlinearly on coordinates x i , y a through their doublesx i ,ỹ a (see (38) and (39)). All actions S[x i , y a ] are physically equivalent, but are described with coordinates x i = {x µ d+1 , . . . , x µ D }, for the untreated directions and dual coordinates y a = {y µ 1 , . . . , y µ d }, for the dualized directions. The case d = D corresponds to the completely T-dual action with the T-dual fields κ 2 Θ µν − V (y) and the case d = 0 to the initial action with the background Π +µν (x).
Applying the procedure to the T-dual action along dual directions y a = {y µ 1 , . . . , y µ d } we obtained the initial theory, and applying it to the untreated directions x i = {x µ d+1 , . . . , x µ D } we obtained the completely T-dual theory. All these derivations confirmed that the set of all T-dualizations forms an Abelian group. The neutral element of the group is the unexecuted Tdualization, while the T-dualizations along some subset of original directions T a is inverse to the T-dualizations along the set of the corresponding dual directions T a . and Π ∓ib = −2κΠ ∓ij Θ ja ± Π ∓ab , Π ∓aj = −2κ Π ∓ab Θ bi ± Π ∓ij .
Let us derive some useful relations between these quantities. The relation (124), for µ = a, ν = i and µ = i and ν = a becomes while taking µ = a, ν = b and µ = i and ν = j we obtain Multiplying the relation (146) from the left withΘ ca ∓ and from the right withΠ ∓ik we get the relation while multiplying the relation (147) from the right with Θ ki ∓ and from the left withΠ ±ac , we obtain