# T-dualization of type II superstring theory in double space

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## Abstract

In this article we offer a new interpretation of the T-dualization procedure of type II superstring theory in the double space framework. We use the ghost free action of type II superstring in pure spinor formulation in approximation of constant background fields up to the quadratic terms. T-dualization along any subset of the initial coordinates, \(x^a\), is equivalent to the permutation of this subset with subset of the corresponding T-dual coordinates, \(y_a\), in double space coordinate \(Z^M=(x^\mu ,y_\mu )\). Requiring that the T-dual transformation law after the exchange \(x^a\leftrightarrow y_a\) has the same form as the initial one, we obtain the T-dual NS–NS and NS–R background fields. The T-dual R–R field strength is determined up to one arbitrary constant under some assumptions. The compatibility between supersymmetry and T-duality produces a change of bar spinors and R–R field strength. If we dualize an odd number of dimensions \(x^a\), such a change flips type IIA/B to type II B/A. If we T-dualize the time-like direction, one imaginary unit *i* maps type II superstring theories to type \(\hbox {II}^\star \) ones.

## 1 Introduction

T-duality is a fundamental feature of string theory [1, 2, 3, 4, 5, 6, 7, 8]. As a consequence of T-duality there is no physical difference between string theory compactified on a circle of radius *R* and circle of radius 1 / *R*. This conclusion can be generalized to tori of various dimensions.

The mathematical realization of T-duality is given by Buscher T-dualization procedure [4, 5]. If the background fields have global isometries along some directions then we can localize that symmetry introducing gauge fields. The next step is to add the new term in the action with Lagrange multipliers which forces these gauge fields to be unphysical. Finally, we can use gauge freedom to fix initial coordinates. Varying this gauge fixed action with respect to the Lagrange multipliers one gets the initial action and varying with respect to the gauge fields one gets the T-dual action.

Buscher T-dualization can be applied along directions on which background fields do not depend [4, 5, 6, 7, 8, 9, 10]. Such a procedure was used in Refs. [11, 12, 13, 14, 15, 16, 17, 18] in the context of closed string noncommutativity. There is a generalized Buscher procedure which deals with background fields depending on all coordinates. The generalized procedure was applied to the case of bosonic string moving in the weakly curved background [19, 20]. It leads directly to closed string noncommutativity [21].

The Buscher procedure can be considered as the definition of T-dualization. But there are also other frameworks in which we can represent T-dualization which must be in accordance with the Buscher procedure. Here we talk about the double space formalism which was the subject of the articles about 20 years ago [22, 23, 24, 25, 26]. Double space is spanned by coordinates \(Z^M=(x^\mu ,y_\mu )\) \((\mu =0,1,2,\ldots ,D-1)\), where \(x^\mu \) and \(y_\mu \) are the coordinates of the *D*-dimensional initial and T-dual space-time, respectively. Interest for this subject emerged again with Refs. [27, 28, 29, 30, 31, 32, 33, 34], where T-duality is related with *O*(*d*, *d*) transformations. The approach of Ref. [22] has been recently improved when the T-dualization along some subset of the initial and corresponding subset of the T-dual coordinates has been interpreted as permutation of these subsets in the double space coordinates [35, 36].

Let us motivate our interest in this subject. It is well known that T-duality is important feature in understanding M-theory. In fact, five consistent superstring theories are connected by a web of T and S dualities. In the beginning we are going to pay attention to the T-duality. To obtain formulation of M-theory it is not enough to find all corresponding T-dual theories. We must construct one theory which contains the initial theory and all corresponding T-dual ones.

We have succeeded to realize such program in the bosonic case, for both constant and weakly curved background. In Refs. [35, 36] we doubled all bosonic coordinates and obtain the theory which contains the initial and all corresponding T-dual theories. In such theory T-dualization along an arbitrary set of coordinates \(x^a\) is equivalent to replacement of these coordinates with the corresponding T-dual ones, \(y_a\). Therefore, T-duality in double space becomes symmetry transformation with respect to permutation group.

Performing T-duality in supersymmetric case generates new problems. In the present paper we are going to extend such an approach to the type II theories. In fact, doubling all bosonic coordinates we have unified types IIA, IIB as well as type \(\hbox {II}^\star \) [37] (obtained by T-dualization along time-like direction) theories. We expect that such a program could be a step toward better understanding M-theory.

In the present article we apply the approach of Refs. [35, 36] in the cases of complete (along all bosonic coordinates) and partial (subset of the bosonic coordinates) T-dualization of the type II superstring theory [1, 2, 3]. We use ghost free type II superstring theory in pure spinor formulation [33, 38, 39, 40, 41, 42, 43, 44] in the approximation of constant background fields and up to the quadratic terms. This action is obtained from the general type II superstring action [45] which is given in the form of an expansion in powers of fermionic coordinates \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \). In the first step of our consideration we will limit our analysis to the basic term of the action neglecting \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \) dependent terms. Later, in the discussion of proper fermionic variables, using an iterative procedure [45], we take into consideration higher power terms and restore the supersymmetric invariants \(\Pi _\pm ^\mu \), \(d_\alpha \) and \({\bar{d}}_\alpha \) as variables in the theory.

Rewriting the T-dual transformation laws in terms of the double space coordinates \(Z^M\) we introduce the generalized metric \(\mathcal{H}_{MN}\) and the generalized current \(J_{\pm M}\). The permutation matrix \((\mathcal{T}^a)^M{}_N\) exchanges the places of \(x^a\) and \(y_a\), where the index *a* marks the directions along which we make T-dualization. The basic request is that T-dual double space coordinates, \({}_a Z^M=(\mathcal{T}^a)^M{}_N Z^N\), satisfy the transformation law of the same form as initial coordinates, \(Z^M\). It produces the expressions for the T-dual generalized metric, \({}_a \mathcal{H}_{MN}=(\mathcal{T}^a\mathcal{H}\mathcal{T}^a)_{MN}\), and the T-dual current, \({}_a J_{\pm M}=(\mathcal{T}^a J_{\pm })_M\). This is equivalent to the requirement that transformations of the coordinates and background fields, \(Z^M\rightarrow {}_a Z^M\), \(\mathcal{H}_{MN}\rightarrow {}_a\mathcal{H}_{MN}\) and \(J_{\pm M}\rightarrow {}_a J_{\pm M}\), are symmetry transformations of the double space action. From transformation of the generalized metric we obtain T-dual NS–NS background fields and from transformation of the current we obtain T-dual NS–R fields.

The supersymmetry case includes the new features in both the Buscher and the double space T-duality approaches. In the bosonic case the left and right world-sheet chiralities have different T-duality transformations. It implies that in T-dual theory two fermionic coordinates, \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \), and corresponding canonically conjugated momenta, \(\pi _\alpha \) and \(\bar{\pi }_\alpha \) (with different world-sheet chiralities), have different supersymmetry transformations. As shown in [46, 47] it is possible to make a supersymmetry transformation in T-dual theory unique if we change one world-sheet chirality sector. Therefore, compatibility between supersymmetry and T-duality can be achieved by action on the bar variables with the operator \({}_a\Omega \), \({}^\bullet \bar{\pi }_\alpha ={}_a\Omega _\alpha {}^\beta \;{}_a\bar{\pi }_\beta \). As a consequence of the relation \(\Gamma ^{11}\;{}_a\Omega =(-1)^d \;{}_a\Omega \Gamma ^{11}\) it follows that such transformations for odd *d* change space-time chiralities of the bar spinors. In such a way the operator \({}_a\Omega \) for odd *d* maps type IIA/B to type IIB/A theory. Here *d* denotes the number of T-dualized directions.

There is one difference compared with the bosonic string case [35, 36] where all results from the Buscher procedure were reproduced. In the T-dual transformation laws of type II superstring theory the R–R field strength \(F^{\alpha \beta }\) does not appear. The reason is that R–R field strength couples only with the fermionic degrees of freedom, which are not dualized. This is in analogy with the term \(\partial _+ x^i \Pi _{+ ij}\partial _- x^j\) in the bosonic case, where background field \(\Pi _{+ij}\) couples only with coordinates \(x^i\), which are undualized [27, 28, 29]. To reproduce the Buscher form of the T-dual R–R field strength we should make some additional assumptions.

There is an appendix, which contains the block-wise expressions for the tensors used in this article and useful relations.

## 2 Buscher T-dualization of type II superstring theory

In this section we will consider type II superstring action in pure spinor formulation [38, 43, 44] in the approximation of constant background fields and up to the quadratic terms. Then we will give the overview of the results obtained by Buscher T-dualization procedure [9, 10, 46, 47].

### 2.1 Type II superstring in pure spinor formulation

The action from which we start (2.7) could be considered as an expansion in powers of \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \). In the iterative procedure presented in [45] it has been shown that each component in the expansion can be obtained from the previous one. Therefore, for practical reasons (computational simplicity), in the first step we limit our considerations to the basic component i.e. we neglect all terms in the action containing \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \). As a consequence the \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \) terms disappear from \(\Pi _\pm ^\mu \), \(d_\alpha \) and \({\bar{d}}_\alpha \) and in the solutions for the physical superfields just *x*-dependent supergravity fields survive. Therefore we lose explicit supersymmetry in such approximation. Later, when we discuss proper fermionic variables, we would go further in the expansion and take higher power terms, which means that supersymmetric invariants, \(\Pi _\pm ^\mu \), \(d_\alpha \) and \({\bar{d}}_\alpha \), would play the roles of \(\partial _\pm x^\mu \), \(\pi _\alpha \) and \(\bar{\pi }_\alpha \), respectively.

We are going to perform T-dualization along some subset of bosonic coordinates \(x^a\). Therefore, we will assume that these directions are Killing vectors. Since \(\partial _\pm x^a\) appears in \(\Pi _\pm ^\mu \), \(d_\alpha \) and \({\bar{d}}_\alpha \), it essentially means that corresponding superfields (\(A_{ab}\), \(\bar{E}_a{}^\alpha \), \(E^\alpha {}_a\), \(\mathrm{P}^{\alpha \beta }\)) should not depend on \(x^a\). This assumption regarding Killing spinors could be extended on all space-time directions \(x^\mu \), which effectively means, in the first step, that physical superfields are constant. All auxiliary superfields can be expressed in terms of space-time derivatives of physical supergravity fields [45]. Then, in the first step, the auxiliary superfields are zero, because all physical superfields are constant. On the other hand, having constant physical superfields means that their field strengths, \(\Omega _{\mu ,\nu \rho }(\Omega _{\mu \nu ,\rho })\), \(C^\alpha {}_{\mu \nu }(\bar{C}_{\mu \nu }{}^\alpha )\) and \(S_{\mu \nu ,\rho \sigma }\), are zero. In this way, in the first step, we eliminated from the action terms containing variables \(N_+^{\mu \nu }\) and \({\bar{N}}_-^{\mu \nu }\) (2.5).

*S*is

### 2.2 T-dualization along arbitrary number of coordinates

### 2.3 Relation between left and right chirality in T-dual theory

One can see from (2.21) and (2.22) that the left and right chiralities transform differently in T-dual theory. As a consequence, in T-dual theory we will have two types of vielbeins, two types of \(\Gamma \)-matrices, two types of spin connections and two types of supersymmetry transformations. We want to have a single geometry in T-dual theory. Therefore, we will show that all these different representations of the same variables can be connected by Lorentz transformations [46, 47].

#### 2.3.1 Two sets of vielbeins in T-dual theory

*a*denotes the T-dualization along \(x^a\) directions. For coordinates which contain both \(x^i\) and \(y_a\) we will use “hat” indices \(\hat{\mu }, \hat{\nu }\). The matrices

*d*is the number of dimensions along which we perform T-duality.

#### 2.3.2 Two sets of \(\Gamma \)-matrices in T-dual theory

#### 2.3.3 Two sets of spin connections in T-dual theory

#### 2.3.4 Single form of supersymmetry invariants in T-dual theory and new spinor coordinates

So far we used the action from Ref. [45] which is an expansion in powers of \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \). We performed the procedure of bosonic T-dualization using the first term in the expansion i.e. \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \) independent part of the action. Consequently, the supersymmetric invariants, \(\Pi _\pm ^\mu \), \(d_\alpha \) and \({\bar{d}}_\alpha \), in that approximation became \(\partial _\pm x^\mu \), \(\pi _\alpha \) and \(\bar{\pi }_\alpha \). But if we would take higher power terms into consideration, then these invariants would appear again in the theory. Consequently, we can use these invariants to find proper spinor variables.

From the compatibility between supersymmetry and T-duality we will find appropriate spinor variables changing the bar ones. We are not going to apply such a procedure to background fields which transformation we will find from T-dualization. In Sect. 2.5 we will check that both T-dual gravitinos satisfy a single supersymmetry transformation rule.

#### 2.3.5 Spinorial representation of the Lorentz transformation

*ab*component of \(G_{\mu \nu }\). Then from (2.33) it follows that

*a*dependent projector on the \(\underline{a} \underline{b}\) subspace \({}_a P^{\underline{a}}{}_{ \underline{c}} \,\, {}_a P^{\underline{c}}{}_{ \underline{b}} = {}_a P^{\underline{a}}{}_{ \underline{b}}\). If we introduce the \(\Gamma \)-matrices in curved space

*d*, along which we perform T-dualizations. Therefore we have

When the number of coordinates along which we perform T-duality is even \((d=2k)\), we have \({}_a \Omega \,\, = (-1)^{\frac{d}{2}}{\sqrt{\prod _{i=1}^{d} G_{a_i a_i} }} \,\, {}_a \Gamma \). As a consequence of the relation \(\Gamma ^{11} \, {}_a \Omega = (-1)^d \, \, {}_a \Omega \, \, \Gamma ^{11}\) we can conclude that in that case bar spinors preserve chirality. When the number of coordinates along which we perform T-duality is odd \((d=2k+1)\), we have \({}_a \Omega \,\, = (-1)^{\frac{d-1}{2}}{\sqrt{\prod _{i=1}^{d} G_{a_i a_i} } } \, i\, \, {}_a \Gamma \, \Gamma ^{11}\). As a consequence of the above relation such a transformation changes the chirality of the bar spinors.

### 2.4 Choice of the proper fermionic coordinates and T-dual background fields

The dilaton transformation in the term \(\Phi R^{(2)}\) originates from quantum theory and will be discussed in Sect. 2.6.

### 2.5 Supersymmetry transformations of T-dual gravitinos

Note that in the expressions for the T-dual fields \({}_a\bar{\Psi }^{\alpha a}\), \({}_a\bar{\Psi }^\alpha _i\) and \({}_a F^{\alpha \beta }\) the matrix \({}_a\Omega \) appears as a consequence of the T-dualization procedure and adoption of the bullet spinor coordinates. In Refs. [46, 47] it appears as a consequence of the compatibility between supersymmetry and T-duality.

### 2.6 Transformation of pure spinors

In this subsection we will find transformation laws for pure spinors, \(\lambda ^\alpha \) and \({\bar{\lambda }}^\alpha \), which are the main ingredient of the pure spinor formalism.

### 2.7 T-dual transformation of antisymmetric fields: from IIB to IIA theory

*D*-directions. Then it is necessary to perform T-dualization along the time-like direction. Here the above square root has important consequences. For our signature \((+,-,-,\ldots ,-)\), the square of the field strength \(({}_a F^{\alpha \beta })^2\) and, consequently, the square of all antisymmetric fields will change the sign when we perform T-dualization along the time-like direction. This is just what we need to obtain type \(II^\star \) theories in accordance with Ref. [37].

## 3 Double space formulation

In this section we will introduce double space, doubling all bosonic coordinates \(x^\mu \) by corresponding T-dual ones \(y_\mu \). We will rewrite the transformation laws in double space and show that both the equations of motion and the Bianchi identities can be written by that single equation.

### 3.1 T-dualization along all bosonic directions

### 3.2 Transformation laws in double space

*D*dimensions. Let us stress that the matrix \({}_a \Omega \) and \(\Omega ^{MN}\) are different quantities.

*SO*(

*D*,

*D*) invariant metric and denoted by \(\eta ^{MN}\).

### 3.3 Equations of motion and double space action

## 4 T-dualization of type II superstring theory as a permutation of coordinates in double space

In this section we will derive the transformations of the generalized metric and current, which are a consequence of the permutation of some subset of the bosonic coordinates with the corresponding T-dual ones. First we will present the method in the case of the complete T-dualization (along all bosonic coordinates) and find the expressions for T-dual background fields. Then we will apply the results to the case of partial T-dualization.

### 4.1 The case of complete T-dualization

Consequently, using double space we can easily reproduce the results of T-dualization, Eqs. (3.7) and (3.2). The problem with T-dualization of the R–R field strength \(F^{\alpha \beta }\) will be discussed in Sect. 5.3.

### 4.2 The case of partial T-dualization

*a*and

*i*(\(a=0,\ldots ,d-1\), \(i=d,\ldots ,D-1\)) and denote T-dualization along direction \(x^a\) and \(y_a\) by

*a*means dualization along the \(x^a\) directions.

## 5 T-dual background fields

In this section we will show that permutation of some bosonic coordinates leads to the same T-dual background fields as standard Buscher procedure [9]. The transformation of the generalized metric (4.11) produces expressions for NS–NS T-dual background fields (\(G_{\mu \nu }\) and \(B_{\mu \nu }\)). They are the same as in bosonic string case obtained in Ref. [35]. Therefore, we will just shortly repeat these results. From the transformation of the current \(J_{\pm M}\) (4.12) we will find T-dual background fields of the NS–R sector (\(\Psi ^\alpha _\mu \) and \(\bar{\Psi }^\alpha _\mu \)). Because R–R field strength \(F^{\alpha \beta }\) does not appear in T-dual transformations, we will find its T-dual under some assumptions.

### 5.1 T-dual NS–NS background fields \(G_{\mu \nu }\), \(B_{\mu \nu }\)

*g*and \(\tilde{g}\) in (A.5), while \(\beta _1\), \(\tilde{\beta }\) and \(\bar{\beta }\) are defined in (A.7). The quantities

*A*and

*D*are given in (A.11) and (A.13), respectively. In more compact form we have

### 5.2 T-dual NS–R background fields \(\Psi ^\alpha _\mu \), \(\bar{\Psi }^\alpha _\mu \)

*a*and

*i*components because in the T-dual picture the index

*a*has a different position, it is now up. T-dual currents are written between the brackets to make a distinction between a left subscript

*a*denoting partial T-dualization and summation indices in the subspace spanned by \(x^a\).

*D*components of the above equation. In order to find the solution of these equations it is more practical to rewrite them using the block-wise form of matrices given in the appendix and Ref. [35],

*ab*block component of Eq. (5.2). Therefore, with the help of (5.3) it is easy to see that

*a*components of the T-dual NS–R fields are of the form

The upper *D* components of Eq. (5.6) produce the same result for T-dual background fields.

### 5.3 T-dual R–R field strength \(F^{\alpha \beta }\)

Using the relations \({}_a\mathcal{H}=\mathcal{T}^a \mathcal{H}\mathcal{T}^a\) and \({}_a J_{\pm }=\mathcal{T}^a J_{\pm }\) we obtained the form of the NS–NS and NS–R T-dual background fields of type II superstring theory. But we know from the Buscher T-dualization procedure that the T-dual R–R field strength \({}_a F^{\alpha \beta }\) has the form given in Eq. (2.72). In this subsection we will derive this relation within the double space framework.

*ij*-term in approach of Refs. [27, 28, 29] where \(x^i\) coordinates are not doubled. Consequently, as in [27, 28, 29] we should make some assumptions. Let us suppose that the fermionic term \(L (\pi _\alpha ,\bar{\pi }_\alpha )\) is symmetric under exchange of the R–R field strength \(F^{\alpha \beta }\) with its T-dual \({}_a F^{\alpha \beta }\)

*D*different solutions,

*c*is an arbitrary constant. Consequently, when we T-dualize

*d*dimensions \(x^a\;(a=0,1,\ldots d-1)\), from (5.22) we can conclude that the T-dual R–R field strength has the form

*d*, the number of directions along which we perform T-duality, just in Refs. [27, 28, 29].

## 6 Conclusion

In this article we showed that the new interpretation of the bosonic T-dualization procedure in the double space formalism offered in [35, 36] is also valid in the case of type II superstring theory. We used the ghost free action of type II superstring theory in a pure spinor formulation in the approximation of quadratic terms and constant background fields. One can obtain this action from the action (2.7), which could be considered as an expansion in powers of fermionic coordinates. In the first part of the analysis we neglect all terms in the action containing powers of \(\theta ^\alpha \) and \({\bar{\theta }}^\alpha \). This approximation is justified by the fact that the action is a result of an iterative procedure in which every step results from the previous one. Later, when we discuss proper fermionic variables, taking higher power terms we restore supersymmetric invariants (\(\Pi _\pm ^\mu \), \(d_\alpha \), \({\bar{d}}_\alpha \)) as variables instead of \(\partial _\pm x^\mu \), \(\pi _\alpha \) and \(\bar{\pi }_\alpha \).

We introduced the double space coordinate \(Z^M=(x^\mu ,y_\mu )\) adding to all bosonic initial coordinates, \(x^\mu \), the T-dual ones, \(y_\mu \). Then we rewrote the T-dual transformation laws (3.8) in terms of double space variables (3.12) introducing the generalized metric \(\mathcal{H}_{MN}\) and the current \(J_{\pm M}\). The generalized metric depends only on the NS–NS background fields of the initial theory. The current \(J_{\pm M}\) contains fermionic momenta \(\pi _\alpha \) and \(\bar{\pi }_\alpha \), along which we do not make a T-dualization, and it depends also on NS–R background fields. The R–R background fields do not appear in T-dual transformation laws.

The coordinate index \(\mu \) is split in \(a=(0,1,\ldots d-1)\) and \(i=(d,d+1,\ldots D-1)\), where index *a* marks subsets of the initial and T-dual coordinates, \(x^a\) and \(y_a\), along which we make T-dualization. T-dualization is realized as permutation of the subsets \(x^a\) and \(y_a\) in the double space coordinate \(Z^M\). The main require is that T-dual double space coordinates \({}_a Z^M=(\mathcal{T}^a)^M{}_N Z^N\) satisfy the transformation law of the same form as the initial coordinates \(Z^M\). From this condition we found the T-dual generalized metric \({}_a \mathcal{H}_{MN}\) and the T-dual current \({}_a J_{\pm M}\). Because the initial and T-dual theory are physically equivalent, \({}_a \mathcal{H}_{MN}\) and \({}_a J_{\pm M}\) should have the same form as the initial ones, \(\mathcal{H}\) and \(J_{\pm M}\), but in terms of the T-dual background fields. It produces the form of the NS–NS and NS–R T-dual background fields in terms of the initial ones which are in full accordance with the results obtained by the Buscher T-dualization procedure [9, 10].

The supersymmetry case is not a simple generalization of the bosonic one, but it requires some new interesting steps. The origin of the problem is the different T-duality transformations of the world-sheet chirality sectors. It produces two possible sets of vielbeins in the T-dual theory with the same T-dual metric. These vielbeins are related by a particular local Lorentz transformation which depends on T-duality transformation and of which the determinant is \((-1)^d\), where *d* is the number of T-dualized coordinates. Therefore, when we perform T-dualization along an odd number of coordinates then such transformation contains a parity transformation. Consistency of T-duality with supersymmetry requires changing one of two spinor sectors. We redefine the bar spinor coordinates, \({}_a{\bar{\theta }}\rightarrow {}_a^\bullet {\bar{\theta }}^\alpha ={}_a \Omega ^\alpha {}_\beta {\bar{\theta }}^\beta \), and the variable \({}_a \bar{\pi }_\alpha \), \({}_a \bar{\pi }_\alpha \rightarrow {}_a^\bullet \bar{\pi }_\alpha ={}_a\Omega _\alpha {}^\beta \bar{\pi }_\beta \). As a consequence the bar NS–R and R–R background fields include \({}_a\Omega \) in their T-duality transformations. For an odd number of coordinates *d* along which T-dualization is performed, \({}_a\Omega \) changes the chirality of the bar gravitino \(\bar{\Psi }^\alpha _\mu \) and the chirality condition for \(F^{\alpha \beta }\). We need it to relate type IIA and type IIB theories.

The transformation law (3.12) induces the consistency condition which can be considered as equation of motion of the double space action (3.21). It contains an arbitrary term depending on the undualized variables \({L}(\pi _\alpha ,\bar{\pi }_\alpha )\). This is in analogy with the term \(\partial _+ x^i \Pi _{+ij}\partial _- x^j\) in the approach presented in Refs. [27, 28, 29]. Therefore, to obtain the T-dual transformation of the R–R field strength \(F^{\alpha \beta }\) we should make some additional assumptions. Supposing that the term \({L}(\pi _\alpha ,\bar{\pi }_\alpha )\) is T-dual invariant and taking into account that two successive T-dualizations act as the identity operator, we found the form of the T-dual R–R field strength up to one arbitrary constant *c*. For \(c=4\kappa \) we get the T-dual R–R field strength \({}_a F^{\alpha \beta }\) as in the Buscher procedure [9].

A T-duality transformation of the R–R field strength \(F^{\alpha \beta }\) has two contributions in the form of square roots. The contribution of the dilaton produces the term \(\sqrt{|\prod _{i=1}^d G_{a_ia_i}|}\). On the other hand the contribution of the spinorial representation of a Lorentz transformation \({}_a\Omega \) contains the same expression without the absolute value \(i^d \sqrt{\prod _{i=1}^d G_{a_ia_i}}\). Therefore, the T-dual R–R field strength \({}_a F^{\alpha \beta }\), besides a rational expression, contains the expression \(i^d \sqrt{\hbox {sign}(\prod _{i=1}^d G_{a_i a_i})}\) (2.89). If we T-dualize along the time-like direction (\(G_{00} > 0\)), the square root does not produce an imaginary unit *i*, not canceling the one in front of the square root. Therefore, T-dualization along the time-like direction maps type II superstring theories to type \(\hbox {II}^\star \) ones [37].

The successive T-dualizations make a group called the T-duality group. In the case of type II superstring T-duality, transformations are performed by the same matrices \(\mathcal{T}^a\) as in the bosonic string case [35, 36]. Consequently, the corresponding T-duality group is the same.

If we want to find a T-dual transformation of \(F^{\alpha \beta }\) without any assumptions, we should follow the approach of [35, 36] and, besides all bosonic coordinates \(x^\mu \), double also all fermionic variables \(\pi _\alpha \) and \(\bar{\pi }_\alpha \). In other words, besides bosonic T-duality we should also consider fermionic T-duality [53, 54, 55, 56, 57, 58, 59].

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