# Effective action of bosonic string theory at order \(\alpha '^2\)

## Abstract

Recently, it has been shown that the gauge invariance requires the minimum number of independent couplings for *B*-field, metric and dilaton at order \(\alpha '^2\) to be 60. In this paper we fix the corresponding 60 parameters in string theory by requiring the couplings to be invariant under the global T-duality transformations. The Riemann cubed terms are exactly the same as the couplings that have been found by the S-matrix calculations.

## 1 Introduction

String theory is a quantum theory of gravity with a finite number of massless fields and a tower of infinite number of massive fields reflecting the stringy nature of the gravity. An efficient way to study different phenomena in this theory is to use an effective action which includes only massless fields. The effects of the massive fields appear in the action as the higher derivatives of the massless fields. This effective action may be found by imposing various symmetries/dualities in the string theory. There are various gauge symmetries in the effective actions which are corresponding to the various massless fields, e.g., the diffeomorphism symmetry corresponds to the metric and the gauge symmetry corresponds to the Kalb-Ramond field or *B*-field. In the bosonic string theory which has only metric, dilaton and *B*-field, they are the only local symmetries of the effective action. Imposing only these symmetries, one finds the effective action has three couplings at order \(\alpha '^0\) (two-derivative order), has 8 couplings at order \(\alpha '\) (four-derivative order) up to field redefinitions [1], has 60 couplings at order \(\alpha '^2\) (six-derivative order) [2] and so on. The gauge symmetries, however, can not determine the coefficients of the couplings. These parameters may be found by S-matrix calculations [3, 4], by sigma-model calculations [5, 6, 7] or by imposing global symmetries of the string theory in which we are interested.

One of the global symmetries of the string theory is T-duality [8, 9]. This duality like the above gauge symmetries may be imposed at the action level to fix the parameters of the effective action at any order of \(\alpha '\). One approach for imposing this symmetry is the double field theory (DFT) [10, 11, 12, 13, 14] in which the *D*-dimensional effective action is extended to 2*D*-space. In this theory, the gauge transformations are deformed to receive \(\alpha '\)-corrections whereas the T-duality symmetry is imposed without deformation simply by writing the couplings as *O*(*D*, *D*) scalars [14, 15, 16, 17, 18]. Another approach is to reduce the *D*-dimensional gauge invariant theory on a circle and impose the T-duality symmetry by constraining the couplings in the \((D-1)\)-dimensional spacetime to be \(Z_2\) scalars [19] where \(Z_2\)-group is the Buscher rules [20, 21] plus their \(\alpha '\)-deformations [22, 23, 24]. Using this approach for the case that *B*-field is zero, the known gravity and dilaton couplings in the effective actions at orders \(\alpha ',\alpha '^2,\alpha '^3\) have been found in [25, 26], up to some overall factors. Moreover, when *B*-field is non-zero, the known couplings at order \(\alpha '\) and their corresponding corrections to the Buscher rules have been found in [27]. In this paper, we are going to use this approach to find the couplings at order \(\alpha '^2\) for the case that *B*-field is non-zero. These couplings, except its Riemann cubed couplings, have not been found by any other methods in string theory.

It is known that the effective action at order \(\alpha '^2\) depends on the scheme that one uses for the effective action at order \(\alpha '\) [28]. In the T-duality approach, this is reflected to the T-duality transformations at order \(\alpha '\). It has been observed in [27] that the T-duality transformations at order \(\alpha '\) depends on the scheme that one uses for the effective action at order \(\alpha '\). The T-duality transformation corresponding to the effective action at order \(\alpha '\) which has only first time derivative [29], is given in [24]. The T-duality transformations at order \(\alpha '\) corresponding to the effective action at order \(\alpha '\) in an arbitrary scheme have been found in [27]. In this paper we are going to find the effective action at order \(\alpha '^2\) that correspond to the effective action at order \(\alpha '\) which has minimum number of couplings [1].

The outline of the paper is as follows: In Sect. 2, we write the known minimum number of couplings at orders \(\alpha '\) and \(\alpha '^2\) that the gauge symmetry can fix up to field redefinitions. In Sect. 3, we impose the T-duality symmetry on the gauge invariant couplings to find their corresponding parameter. The calculations at order \(\alpha '\) have been already done in [27]. That calculations produce the known couplings in the literature and the corresponding T-duality transformations. The calculations at order \(\alpha '^2\) are new. We have found both the effective action and the corresponding T-duality transformations. However, since the expressions for the T-duality transformations are very lengthy we will write only the effective action (see (40)). We have found that there are only 27 non-zero couplings in the effective action at order \(\alpha '^2\). Two of them have already been found by the S-matrix calculations [30]. All other terms are new couplings that the T-duality constraint produces. In Sect. 4, we briefly discuss our results.

## 2 Gauge invariance constraint

*B*-field gauge transformations. So the metric \(G_{\mu \nu }\), the antisymmetric

*B*-field and dilaton \(\Phi \) must appear in the Lagrangian \( \mathcal {L}_n\) trough their field strengths and their covariant derivatives. This requires the effective action at order \(\alpha '^0\) to have the following couplings:

Up to this point, the above couplings are valid for any higher derivative theory which includes metric, *B*-field and dilaton. In the string theory, however, the parameters in (2), (3) and (6) may be fixed by imposing some other specific constraints which are valid only in the string theory. For example, one may construct the appropriate S-matrix elements with the above couplings and then compare them with the \(\alpha '\)-expansion of the corresponding sphere-level S-matrix elements in the string theory to fix the parameters. This method has been used in [1] to find the parameters in (2), (3). The parameters \(c_1,c_2\) in (6) have been also found by the S-matrix method in [30]. The S-matrix method for fixing all parameters in (6), however, requires one to calculate six-point function in string theory in full details which has not been done yet.

Instead of comparing the S-matrix elements of above couplings with the corresponding S-matrix elements in the string theory, one may impose some other symmetries of the string theory to fix the parameters in (2), (3) and (6). The bosonic string theory has the global T-duality symmetry as well as the gauge symmetries that have been used to find the couplings in (2), (3) and (6). It has been shown in [27] that the T-duality symmetry can fix correctly the couplings in (2), (3) up to overall factors at each order of \(\alpha '\). In the next section, we show that imposing the T-duality on the couplings in (6) can also fix all 60 parameters in terms of the overall factor at order \(\alpha '\).

## 3 T-duality invariance constraint

*D*-dimensional effective action \(\mathbf{S }_\mathrm{eff}\), in the most simple form, is to reduce the theory on a circle with

*U*(1) isometry to find the \((D-1)\)-dimensional effective action \(S_\mathrm{eff}(\psi )\) where \(\psi \) represents all massless fields in the \((D-1)\)-dimensional base space. Then one has to transform it under the T-duality transformations to produce \(S_\mathrm{eff}(\psi ')\) where \(\psi '\) represents the T-duality transformations of the massless fields in the base space. The T-duality invariance constraint is then

*U*(1) isometry, it is convenient to use the following background for the metric,

*B*-field and dilaton:

*D*-dimensional metric is

*x*is any field in the base space. At higher orders of \(\alpha '\), the above transformations receive higher derivative corrections, i.e.,

### 3.1 T-duality constraint at orders \(\alpha '^0,\, \alpha '\)

*U*(1) gauge field \(g_{a}\), i.e., \(V_{ab}=\nabla _{a}g_{b}-\nabla _{b}g_{a}\), and \(W_{\mu \nu }\) is field strength of the

*U*(1) gauge field \(b_{a}\), i.e., \(W_{ab}=\nabla _{a}b_{\nu }-\nabla _{b}b_{a}\). The three-form \({\bar{H}}\) is defined as \({\bar{H}}_{abc}=\tilde{H}_{abc}-g_{a}W_{bc}-g_{c}W_{ab}-g_{b}W_{ca}\) where the three-form \(\tilde{H}\) is field strength of the two-form \({\bar{b}}_{ab}+\frac{1}{2}b_{a}g_{b}-\frac{1}{2}b_b g_a \) in (11). The three-form \({\bar{H}}\) is invariant under the Buscher rules and is not the field strength of a two-form. It satisfies the following Bianchi identity [24]:

*D*-dimensional action [27], i.e.,

### 3.2 T-duality constraint at order \( \alpha '^2\)

*H*and 1-forms

*g*,

*b*is \(dH=-(3/2)dg\wedge db\). The T-dual fields should satisfy this identity as well, i.e.,

Therefore, the second order corrections \(\Delta \varphi ^{(2)}, \Delta g_a^{(2)},\, \Delta b_a^{(2)}, \,\Delta {\bar{g}}_{ab}^{(2)}\) and \(\Delta \bar{\phi }^{(2)}\) should be all contractions of \(\nabla \varphi ,\nabla \bar{\phi }, e^{\varphi /2}V, e^{-\varphi /2}W,{\bar{H}},{\bar{R}}\) and their covariant derivatives at order \(\alpha '^2\) with unknown coefficients. The correction \(\Delta {\bar{H}}_{abc}^{(2)}\) is then can be calculated from (34). All corrections should satisfy the \(Z_2\)-relations (29). They produce some algebraic equations between the parameters of the corrections at order \(\alpha '^2\) and the parameter \(b_1\) in the corrections at order \(\alpha '\), i.e., (24). These parameters and the 60 parameters in the action (6) should satisfy the constraint (28) as well.

*H*, \(\nabla H\), \(\nabla \Phi \) and \(\nabla \nabla \Phi \). So we need to reduce these terms and then contract them with the metric (12). In the reduction of these terms, there are many terms which contains gauge field \(g_a\) without its field strength. These terms must be cancelled at the end of the day for the scalar couplings. Hence, to simplify the calculation we drop those terms in the reduction of \(R_{\mu \nu \alpha \beta }\), \(H_{\mu \nu \alpha }\), \(\nabla _{\mu } H_{\nu \alpha \beta }\), \(\nabla _{\mu }\Phi \), \(\nabla _{\mu }\nabla _{\nu }\Phi \) and \(G^{\mu \nu }\) which have the gauge field \(g_a\). Using this simplification, the reduction of Riemann curvature becomes

^{1}

*H*become

*D*-dimensional metric in this case also becomes

*V*and

*W*, we rewrite them in terms of their gauge fields, i.e., \(V_{ab}=\partial _a g_{b}-\partial _b g_{a} \) and \(W_{ab}=\partial _a b_{b}-\partial _b b_{a} \).

The algebraic equations also fix some of the parameters in the T-duality transformations and the parameters of total derivative terms at order \(\alpha '^2\) in terms of \(b_1\), and leave many of them to be arbitrary. Some of the arbitrary parameters in the T-duality transformations may be removed by the Bianchi identities and some of them are related to the coordinate transformations at order \(\alpha '^2\). Even when all the arbitrary parameters are set to zero, there are still too may terms in the T-duality transformations at order \(\alpha '^2\), so we do not write them explicitly. On the other hand, those corrections are only needed if one would like to extend the above couplings to the order \(\alpha '^3\) in the bosonic theory in which we are not interested in this paper. The important part of the calculations is that there are 60 relations between the 60 parameters in (6) and the parameter \(b_1\), i.e., the T-duality constraint fixes all 60 parameters at order \(\alpha '^2\) in terms of the overall factor of the couplings at order \(\alpha '\)! This ends our illustration of the fact that the T-duality constraint on the effective action can fix uniquely the effective action of bosonic string theory at order \(\alpha '^2\).

## 4 Discussion

In this paper, we have shown that imposing the gauge symmetries and the T-duality symmetry on the effective action of string theory for metric, *B*-field and dilaton at order \(\alpha '^2\), can fix the effective action, i.e., (40), up to an overall factor which is the overall factor of the effective action at order \(\alpha '\). This is extension of the similar calculation at order \(\alpha '\) done in [27] which fixes the effective action at order \(\alpha '\) up to the overall factor \(b_1\), i.e., (23). In fact, the gauge symmetries require to have 60 couplings at order \(\alpha '^2\) with unfixed coefficients [2], and the T-duality symmetry which is imposed on the reduction of the effective action on a circle, fixes these 60 parameters.

In the base space, we have done the calculations in the local frame in which the first derivatives of the base metric is zero. After solving the constraints, we have imposed the solution for the parameters in the constraints (28), (29), (34) and found that they are satisfied even when the first derivative of metric is non-zero. It is as expected, because the constraints are some covariant identities. If they satisfy in one particular frame like the local inertial frame, they would satisfy in all other frames as well.

Most of the couplings in (40) are new couplings which have not been found in the literature by other methods in string theory. When *B*-field is zero, the couplings (40) reduce to two Riemann cubed terms that their coefficients, after using the cyclic symmetry of the Riemann curvature, become exactly the same as the coefficients that have been found in [30] by the S-matrix method. These couplings are invariant under the field redefinitions. However, the couplings which have *B*-field are not invariant under the field redefinitions. When *B*-field is non-zero, one may check the couplings involving four fields with the corresponding four-point S-matrix elements in bosonic string theory. To check this comparison, one has to use a field redefinition that change the Riemann squared terms in (23) to the Gauss-Bonnet combination in which the propagators do not receive \(\alpha '\)-correction. That field redefinitions would then change the form of the couplings in (40). The resulting couplings then may be checked with the corresponding S-matrix elements. We leave the details of this calculation for the future works.

We have found that seven dilaton couplings in (8) are non-zero. On the other hand, it is known that the couplings at order \(\alpha '^2\) depends on the effective action at order \(\alpha '\) [28]. We have used the minimal action (23) and the corresponding T-duality transformations (24). Using another scheme for the couplings at order \(\alpha '\), some of the parameters in (40) may be changed. It would be interesting to check if there is a scheme for the couplings at order \(\alpha '\) for which all the dilaton couplings in (8) become zero.

We have found the effective action (40) by imposing only the symmetries of string theory, i.e., the *B*-field gauge invariance, diffeomorphism and T-duality invariances. As a result, the effective action (40) is background independent. However, the total derivative terms are ignored in imposing the T-duality constraint. Hence the effective action (40) is valid for all backgrounds that have no boundary. It would be interesting to take into account in details the total derivative terms to find the boundary terms as well as the bulk terms for the general case that the background has boundary.

We have done the calculations in the curved base space to make sure that the constraints (28), (29), (34) are satisfied in full details. We have performed the calculations in flat base space as well and found exactly the same parameters for (6) as in (40). In the T-duality calculations at order \(\alpha '\) [27] which have correctly reproduced the effective action at order \(\alpha '\), it is also assumed that the base space is flat. Hence, for the calculations at the higher orders of \(\alpha '\) which would be very lengthy calculations, one may safely assume the base space is flat. The most simple calculations at order \(\alpha '^3\) is for superstring theory in which the T-duality transformations have no deformation at orders \(\alpha '\) and \(\alpha '^2\). It would be interesting to perform this calculations at order \(\alpha '^3\) in the superstring to find the *B*-field couplings which are not known in the literature.

If one extends the calculations in the bosonic theory to the order \(\alpha '^3\), one would find a set of couplings which are proportional to \(b_1\) and another set of couplings that their overall factor is arbitrary. The comparison with the four-point S-matrix elements dictates that this factor should be \(\zeta (3)\). At order \(\alpha '^4\), again one should find a set of couplings which are proportional to \(b_1\), a set of couplings proportial to \(\zeta (3)\) and some other sets of couplings that their overall factor may be fixed by the corresponding S-matrix elements. Continuing these logic, one would find sets of couplings at each order of \(\alpha '\) which are proportional to \(b_1\). Hence, one expects the T-duality constraint produces a set of couplings at each order of \(\alpha '\) that are proportional to \(b_1\). They form a complete set of couplings which would be invariant under the T-duality transformations at all orders of \(\alpha '\). That T-dual set of couplings may have de Sitter solution [32]. It would be interesting to find this T-dual set.

*B*-field gauge transformations which are the correct transformations in the bosonic and superstring theories. In the superstring theory \(b_1=0\), hence, the couplings (40) are zero in the superstring theory as expected. On the other hand, the 60 parameters in (40) do not dependent on the dimension of spacetime. That does not indicate the result (40) is valid also for the heterotic theory for \(b_1=1/8\). The reason is that in the heterotic theory the

*B*-field gauge transformation is deformed at order \(\alpha '\) which is resulted from the Green-Schwarz anomaly cancellation mechanism [33]. To produce the heterotic result, one has to add to the couplings (6) the fixed couplings at order \(\alpha '^2\) which are resulted from the deformed gauge transformations, i.e., \(-\frac{\alpha '^2}{12}\Omega _{\mu \nu \alpha }\Omega ^{\mu \nu \alpha }\) where \(\Omega \) is the three-form Chern–Simons which can be written in terms of spin connection,

*B*-field gauge transformations which correspond to the Chiral string theory [39]. The deformation at order \(\alpha '\) is the same as the deformation in the heterotic theory in which spin connection is replaced by the Christoffel connection [40]. The low energy effective action of this theory at the leading order, is given by the T-duality invariant action (18) and at the order \(\alpha '\), it is given by the T-duality invariant coupling \(H^{\mu \nu \alpha }\varvec{\Omega }_{\mu \nu \alpha }\) [41]

^{2}where the three-form Chern–Simons \(\varvec{\Omega }_{\mu \nu \alpha }\) is resulted from the deformed gauge transformation, i.e.,

In general, both the diffeomorphisms and the *B*-field gauge transformations may receive higher derivative deformations in a general gauge invariant higher-derivative theory. One may impose these gauge transformations and the deformed T-duality transformations to study the effective action of a higher-derivative theory which is invariant under the gauge transformations and under the T-duality transformations. The effective action at the leading order of \(\alpha '\) is given by (18). At order \(\alpha '\), the parity invariant part of the effective action would be more general than the action (23). It would be interesting to find this effective action.

## Footnotes

## Notes

### Acknowledgements

This work is supported by Ferdowsi University of Mashhad under grant 1/50251(1398/04/31).

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