Effective action of type II superstring theories at order $\alpha'^3$: NS-NS couplings

Recently, it has been shown that the minimum number of gauge invariant couplings for $B$-field, metric and dilaton at order $\alpha'^3$ is 872. These couplings, in a particular scheme, appear in 55 different structures. In this paper, up to an overall factor, we fix all parameters in type II supertirng theories by requiring the reduction of the couplings on a circle to be invariant under T-duality transformations. We find that there are 445 non-zero couplings which appear in 15 different structures. The couplings are fully consistent with the partial couplings that have been found in the literature by the four-point S-matrix element and by the non-linear Sigma model methods.


Introduction
String theory is a candidate quantum theory for gravity which includes 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 the massless fields [1,2]. The effective action has a double expansions. The genus-expansion which includes the classical tree-level and a tower of quantum loop-level corrections, and the string-expansion which is an expansion in terms of higher derivative couplings at each loop level. The effective action may be found by applying various techniques in string theory. One of them is T-duality which shows up when one compactifies the D-dimensional theory on d-dimensional tours T d [3,4].
The massless fields in the 26-dimensional bosonic sting theory and in the 10-dimensional heterotic string theory after truncating the Yang-Mills gauge fields, are metric, dilaton and B-field. The corresponding effective actions at the tree-level and at the two-derivative order, reveal O(d, d, R) symmetry after one dimensionally reduces them on T d [5,6,41]. The number and form of couplings in the effective action at the higher orders of derivatives are not unique. They are changed under field redefinitions, i.e., they are scheme dependent [8]. It is shown in [9,10] that the O(d, d, R) symmetry also appears explicitly at the four-derivative order when one uses a specific scheme for the D-dimensional effective action. Using closed string field theory, it is proved in [11] that this symmetry in fact exists in the full tree-level effective actions including all higher derivative couplings. This global symmetry may then be used to constrain the possible higher derivative couplings in the classical effective action of the string theory.
At the most simple case when one dimensionally reduces the effective action on a circle, the O(1, 1) symmetry or Z 2 symmetry has been used in [12,13,50,15,16,17] to find many already known and unknown couplings in the classical effective actions. In particular, it has been shown in [15,16] that this Z 2 symmetry as well as various gauge symmetries corresponding to the massless fields fix all four-and six-derivative couplings in the bosonic string theory up to an overall factor. The Z 2 -transformations are the Buscher rules [18,19] plus their higher derivative corrections. The form of these corrections, however, depend on the scheme that one uses for the D-dimensional couplings [15]. The O(d, d, R) symmetry for d > 1 has been also used in [20,21,22,23] to construct the higher derivative couplings in the D-dimensional effective action. It has been shown in [22,23] that for d > 1 the Green-Schwarz type mechanism is required to have O(d, d, R) symmetry at four-derivative order. Another T-duality based framework for constructing the higher derivative couplings in string theory is the Double Field Theory [24,25,26,27,28,29,30,31,32,33] in which the dimension of spacetime is doubled and the O(D, D, R) symmetry is imposed on the 2D-dimensional theory before using dimensional reduction.
The massless fields in the effective action of type II superstring theory appear in four sectors: NS-NS sector which has bosonic fields metric, dilaton and B-field, R-R sector which has bosonic n-form with n = 0, 1, 2, 3, 4, and fermionic sectors NS-R and R-NS in which we are not interested. This effective action at the classical level should also have the Z 2 symmetry after dimensionally reducing it on a circle. At the two-derivative order, the NS-NS couplings are the same as the couplings in the bosonic string theory, hence they have the expected symmetry. It has been shown in [34] that the R-R couplings at two-derivative order can be written in the Z 2 -invariant form after dimensionally reducing them on a circle. The first higher derivative corrections in this case are at eight-derivative order. In terms of order of R-R field strength, these couplings appear in 5 parts, i.e., the couplings with 0, 2, 4, 6 and 8 R-R field strengths. They may be found by imposing the Z 2 symmetry. Under the Z 2 -transformations, the order of R-R fields are not changed [35], hence, the couplings in each part must be separately invariant under the Z 2 -transformations. In this paper, we are going to find the couplings in the first part, i.e., the NS-NS couplings, by imposing the Z 2 symmetry as well as various gauge symmetries, and leave the construction of the couplings involving the R-R fields to the future works.
The outline of the paper is as follows: In section 2, we write the gauge invariant couplings at order α ′3 in a minimal scheme that have been found in [38]. In section 3, we impose the Tduality symmetry on the gauge invariant couplings to find their corresponding parameters. 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. We have found that there are 445 non-zero couplings in the effective action which appear in 15 different structures. A few of them which have at lest four NS-NS fields are fully consistent with the S-matrix element of four NS-NS vertex operators and with non-linear sigma model. In section 4, we briefly discuss our results.

Gauge invariance constraint
The classical effective action of type II superstring theories has the following string-expansion or α ′ -expansion in the string frame: The effective action must be invariant under the coordinate transformations, under the B-field and R-R gauge transformations. L 0 is the Lagrangian of type II supergravity. Its bosonic part in democratic form is given by [36] where H is field strength of the B-field and F (n) is the R-R field strength. The first part has NS-NS couplings and the second part has metric and two R-R field strengths. The R-R gauge symmetry dictates that the α ′3 -corrections has the following five parts: where L 0 3 has only NS-NS couplings, L 2 3 has NS-NS and two R-R fields, L 4 3 has NS-NS and four R-R fields, and so on. Even though the one-loop effective action has both odd and even parity couplings [37], the couplings in the tree-level effective action have all even parity. In this paper we are interested only in the tree-level couplings in the NS-NS sector.
The NS-NS gauge symmetries dictate that there are 23996 couplings at order α ′3 in 202 different structures, i.e., H 8 -structure, R 4 -structure, (∇Φ) 8 -structure, and so on. However, many of couplings are related to each other by Bianchi identities, total derivative terms and field redefinitions. It has been shown in [38] that the minimum number of independent gauge invariant couplings is 872. These couplings in one particular scheme which appear in 55 different structures, have been found in [38]. They are where c 1 , · · · , c 872 are parameters that the gauge symmetry can not fix them. Except the coupling with coefficient c 520 , all other couplings have no term with three derivatives. We refer the interested reader to [38] for the explicit form of all couplings. A few of these couplings have been found in [38] by comparing the above couplings with the α ′ -expansion of the S-matrix element of four NS-NS vertex operators [39,40]. In the next section, we show that imposition of the Z 2 -symmetry on the above couplings can fix all 872 parameters in the type II superstring theory in terms of an overall factor.

T-duality invariance constraint
The T-duality constraint on the D-dimensional effective action S eff , in the most simple form, is first to dimensionally reduce theory on a circle with U(1) isometry to produce its corresponding (D −1)-dimensional effective action S eff (ψ) where ψ 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 eff (ψ ′ ) where ψ ′ represents the T-duality transformations of the massless fields in the base space. The T-duality invariance constraint is then whereḡ ab ,φ are the metric and dilaton in the base space, and J a is an arbitrary covariant vector made of the (D − 1)-dimensional fields and their derivatives. It has the following α ′ -expansion: where J a n is an arbitrary covariant vector at order α ′n made of the (D − 1)-dimensional fields. The T-duality transformation ψ ′ also has α ′ -expansion, i.e., where ψ ′ 0 is the Buscher rules, ψ ′ 1 contains corrections to the Buscher rules at order α ′ and so on. The T-duality transformation forms a Z 2 -group, i.e., This Z 2 -symmetry alone can not fix completely the corrections to the Buscher rules. However, this symmetry as well as the T-duality constraint (5) may fix the corrections completely. This indicates that the form of T-duality transformations depended on the scheme that one uses for the effective action [15].
where S n (ψ) is reduction of S n in (1). However, S eff (ψ ′ ) in (5) has two α ′ -expansions. One is the same as the above expansion which is inherited from the expansion (1), and the other one is corresponding to the α ′ -expansion of the T-duality transformations (7). Using the following α ′ -expansion: where δ (0) S n = S n , one can write the T-duality constraint (5) as J a n (ψ)) (11) Using the above T-duality constraint at orders α ′0 , α ′ , α ′2 , · · ·, one may find the parameters in the D-dimensional effective actions S 0 , S 1 , S 2 , · · ·, as well as the corresponding corrections to the Buscher rules and the boundary terms. We are, however, interested only on fixing the parameters in the gauge invariant couplings in the D-dimensional effective action. This constraint in the bosonic string theory fixes the effective actions at orders α ′ and α ′2 up to an overall pre-factor [15,16]. The calculations in [15,16] reveal that the constraint relations between the parameters of the D-dimensional effective action are independent of the geometry of the base space, i.e., they are the same whether or not the base space is curved. Hence to simplify the calculations it is convenient to assume the base space is flat, i.e.,ḡ ab = η ab .
To have a background with U(1) isometry, it is convenient to use the following background for the metric, B-field and dilaton [41]: whereb ab is the B-field in the base space, and g a , b b are two vectors in this space. Inverse of the above D-dimensional metric is where η ab is inverse of the base metric which raises the index of the vectors.
The T-duality transformations at the leading order of α ′ on the (D − 1)-dimensional fields are given by the Buscher rules [18,19]. In the above parametrisation, they become the following Z 2 -transformations: The reduction of field strength of B-field in the parametrizations (12) becomes where W is field strength of the U(1) gauge field b a , i.e., W = db, and the three-formH which is torsion in the base space, is defined as whereĤ is field strength of the two-formb ab and V is field strength of the U(1) gauge field g µ , i.e.,Ĥ = db, V = dg. The three-formH is invariant under the T-duality and under various gauge transformations. SinceH is not exterior derivative of a two-form, it satisfies anomalous Bianchi identity, whereas the W, V satisfy ordinary Bianchi identity, i.e., Our notation for making antisymmetry is such that e.g., At the higher orders of α ′ , the Z 2 -transformations (14) receive higher derivative corrections. Since the higher derivative corrections to the leading order supergravity start at order α ′3 in type II supersting theory, the higher derivative corrections to the above Buscher rules also start at order α ′3 in this theory, i.e., The deformations ∆g (3) a (ψ) and ∆H (3) abc (ψ) are odd under the parity and all other deformations are even under the parity. They must satisfy the Z 2 -transformation. This produces the following constraint between the corrections at order α ′3 : The deformations ∆b (3) a , ∆g (3) a and ∆H (3) abc must also satisfy the Bianchi identity (17), i.e., This relation at order α ′0 gives the Bianchi identity (17). At order α ′3 it gives the following relation between the deformations at order α ′3 : whereH (3) is a gauge invariant closed 3-form, i.e., dH (3) = 0, at order α ′3 which is odd under parity. The 3-formH (3) and the deformation ∆g (3) a contain all odd-parity contractions and the defor- ab contain all even-parity contractions of ∂ϕ, ∂φ, e ϕ/2 V, e −ϕ/2 W , H and their derivatives at order α ′3 with unknown coefficients. Using the package "xAct" [42], one findsH (3) has 171551 terms, ∆ϕ (3) has 3371 terms, ∆φ (3) has 3371 terms, ∆b (3) a has 9054 terms, ∆g (3) a has 9054 terms, and ∆ḡ (3) ab has 17581 terms. The terms inH (3) must satisfy the ab must satisfy the Z 2 -constraints (19). To impose these constraint to find some relations between the parameters of the deformations, one has to also use the Bianchi identities (17). These constraints produce many relations between the parameters, however as expected, they can not fix them all, because the corrections to the Buscher rules depend on the scheme that one uses for the effective action [15]. Imposing these relations into the deformations, one finds the deformations that are consistent with the Z 2 -symmetry and satisfy the Bianchi identity (20). They should then be used in the T-duality constraint (11) at order α ′3 .

T-duality at order α ′0
The T-duality constraint (11) at order α ′0 when the base space is flat, is The left-hand side is odd under the Buscher rules, hence, the vector J a 0 on the right-hand side must be also odd under the Buscher rules.
Reduction of the NS-NS couplings at order α ′0 are the following: Then the reduction of S 0 becomes It is even under the parity. It satisfies the T-duality constraint (22) for the total derivative term with vector J a 0 = −2∂ a ϕ which is odd under the Buscher rules and even under the parity, as expected. If spacetime has no boundary the total derivative term becomes zero using the Stokes's theorem. On the other hand, if the spaetime has boundary this total derivative term dictates the Hawking-Gibbons boundary term [43]. In this paper we assume the spacetime is closed, hence, the total derivative terms can be ignored.

T-duality at order α ′3
The T-duality constraint (11) at order α ′3 when the base space is flat, is where S 3 (ψ) is reduction of the gauge invariant couplings (4) on a circle, S 3 (ψ ′ 0 ) is its transformation under the Buscher rules (14) and δ (3) S 0 (ψ ′ 0 ) is transformation of the perturbation of (24) under the Buscher rules (14), i.e., where we have used the relations (19) and removed some total derivative terms in which we are not interested in this paper. In finding the above result for ∆ḡ ab we assumed the metric of the base space is not flat. After perturbing the metric, we set itḡ ab = η ab . Note that the extra factors of e ϕ/2 and e −ϕ/2 in (18) make it possible to have a factor of e ϕ/2 in front of each V and a factor of e −ϕ/2 in front of each W . Note that δ (3) S 0 (ψ ′ 0 ) is odd under the Buscher rules and it is even under parity. Hence, the left-hand side of (25) must be odd under the Buscher rules and even under the parity, i.e., the vector J a 3 (ψ) must be even under parity and satisfies the following relation: The vector J a 3 (ψ) contains all even-parity contractions of ∂ϕ, ∂φ, e ϕ/2 V, e −ϕ/2 W ,H and their derivatives at order α ′3 with unknown coefficients. Using the package "xAct", one finds it has 71678 terms. They should satisfy the above Z 2 -constraint.
The dimensional reduction of each gauge invariant term in (4) is a very lengthy calculation. However, the final result for the reduction of each term when it is written in terms of the physical torsionH abc , must be an invariant expression under the U(1) × U(1) gauge transformations where the first U(1) corresponds to g a gauge transformation and the second one corresponds to the b a gauge transformation. This fact has been used in [16] to simplify greatly the calculations of the reduction of the couplings at order α ′2 . We use the same simplification for calculating reduction of the couplings at order α ′3 .
The couplings in (4) have only Riemann curvature, H, ∇H, ∇∇H, ∇Φ and ∇∇Φ. So we need to reduce these terms and then contract them with the metric (13). 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 one drops those terms in the reduction of R µναβ , H µνα , ∇ µ H ναβ , ∇ ρ ∇ µ H ναβ , ∇ µ Φ, ∇ µ ∇ ν Φ and G µν which have the gauge field g a . Using this simplification, the reduction of ∇ ρ ∇ µ H ναβ becomes The indices are raised by η ab . The reduction of R µναβ , H µνα , ∇ µ H ναβ , ∇ µ Φ, and ∇ µ ∇ ν Φ are calculated in [16]. The reduction of inverse of the D-dimensional metric becomes Using above reductions, one can calculate the reduction of different gauge invariant terms in (4) to find S 3 (ψ) and its corresponding S 3 (ψ ′ 0 ). Then using the constraint (25), one finds some equations involving the 872 parameters in (4), the arbitrary parameters in J a 3 and in ∆ϕ (3) ab , ∆φ (3) ,H (3) . To solve this constraint, one has to also impose the Bianchi identities (17). To impose the last two Bianchi identities in (17), we write W and V in the derivative terms which appear in (25), i.e., in ∂W , ∂∂W and so on, in terms of potential, i.e., To impose the first identity in (17), we make all even-parity contraction of ∂ [aHbcd] + V [ab W cd] and its derivatives with ∂ϕ, ∂φ, e ϕ/2 V, e −ϕ/2 W ,H and their derivatives to produce terms at order α ′3 . We then add them with arbitrary coefficients to the constraint (25).
One then finds 143146 linear algebraic equations involving all the parameters. Solving the resulting equations, one finds many relations between the parameters. Since there are too many terms in the vector J a 3 and in the T-duality corrections ∆ϕ (3) ab , ∆φ (3) andH (3) , we are not interested in their corresponding parameters. Interestingly, when one solves the linear algebraic equations, one would find there are 871 relations between only the parameters in the D-dimensional Lagrangian (4). It means the T-duality constraint can fix the parameters in (4) up to an overall factor. In the next subsection we write these relations, and compare some of the non-zero couplings with the partial couplings that have been found by other methods.

Results
Our calculation indicates that 427 parameters in (4) are zero. All other parameters can be written in terms of one of them. The 445 non-zero couplings appear in only 15 structures. We find that when there is no B-field, there are only two non-zero couplings. Writing the effective action as where c is the overall parameter, the couplings involving only metric and dilaton are the following: It is exactly the couplings that have been found by the S-matrix and sigma-model calculations [40,46,44,45,47,48,49] provided that one chooses the overall parameters to be c = −ζ(3)/2 6 . The above couplings have been also found in [50] by the T-duality constraint when metric is diagonal and B-field is zero. All other 443 non-zero couplings involve B-field. We have found 8 couplings with structure (∇H) 4 . They are the following: They are exactly the couplings that are produced by the S-matrix element of four NS-NS vertex operators [40] using the scheme (4) for the field theory couplings [38].
We have found 22 couplings with structure R 2 (∇H) 2 , i.e., They are consistent with the couplings that are produced by the S-matrix element of four NS-NS vertex operators using the scheme (4) for the field theory couplings [38]. However, the four-point S-matrix calculation can not fix all 22 parameters in (4) which involve R 2 (∇H) 2 couplings [38]. They can be fixed by studying five-point S-matrix elements. The T-duality constraint, however, fixes all 22 parameters in these couplings. It would be interesting to study the S-matrix element of five NS-NS vertex operators and check that its low energy limit reproduces the above couplings. This five-point S-matrix element has been calculated in [51] from which the couplings with structure H 2 R 3 have been extracted. Our calculation produce no couplings involving two dilatons and two B-fields, and no couplings involving one dilaton, one graviton and two B-fields in the string frame. They are consistent with four-point S-matrix element which produces no such couplings in the string frame [38].
All other couplings that the T-duality constraint produces involve more than four fields which can not be compared with the four-point S-matrix elements. We find there are no couplings involving more than one dilaton. The couplings involving ∇∇Φ appears in three structures. We have found 10 couplings with structure H 6 ∇∇Φ, i.e., They can be reproduced by studying six-point S-matrix element. We have found 15 couplings with structure H 2 R 2 ∇∇Φ, i.e., They can be reproduced by studying the five-point S-matrix element. The couplings involving ∇Φ appear in three structures. We have found 30 couplings with structure H 5 ∇H∇Φ, i.e.,  [51] using a scheme for the couplings which is different from the scheme (4) that we use in this paper for finding the above couplings.

Discussion
In this paper, we have shown that imposition the gauge symmetries and the T-duality symmetry on the effective action of type II superstring theories for NS-NS fields at order α ′3 , can fix the effective action, i.e., (30), up to an overall factor. In fact, the gauge symmetries require to have 872 couplings at order α ′3 with unfixed coefficients [38], and the T-duality symmetry fixes these 872 parameters in terms of only one parameter.
Most of the couplings in (30) are new couplings which have not been found in the literature by other methods in string theory. When B-field is zero, the couplings (30) reduce to two Riemann quartet 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 [40,44,45] by the S-matrix and the sigma-model methods. These couplings are invariant under the field redefinitions. However, the couplings which have B-field are not invariant under the field redefinitions, except the parameters c 1 , c 2 , c 5 , c 7 , c 8 in the couplings with structure H 8 [38]. The couplings (30) that we have found are in one particular scheme. To compare these couplings with the couplings in the literature that are found by the S-matrix elements, one has to first reproduce the field theory S-matrix elements using the couplings in the scheme (4) and compare them with the corresponding string theory S-matrix elements to fix the parameters in (4). Then the fixed couplings should be compared with the couplings (30) that the T-duality produces. Using the scheme (4), some of the parameters in (4) have been found in [38] by S-matrix element of four NS-NS vertex operators. Those couplings are exactly the same as the couplings that the T-duality constraint produces. It would be interesting to fix some of other parameters in (4) by comparing them with the string theory S-matrix element of five NS-NS vertex operators calculated in [51] and compare the resulting couplings with the couplings in (30).
The number of gauge invariant couplings in the minimal scheme (4) is 872. However, it is not guaranteed that the number of couplings in the string theory, i.e., the couplings(30), is minimum. One can use field redefinitions, total derivative terms and Bianchi identities to rewrite the couplings (30) in other schemes that may have less couplings than in (30). One way to find such couplings is to use a minimal scheme in which there are maximum number of couplings involving gravity and dilaton. In the scheme (4) there are 36 such couplings. On the other hand, it has been argued in [50] that the T-duality constraint makes the coefficients of these terms in the minimal schemes to be zero. In fact the couplings (30) and the couplings in the bosonic string theory at order α ′ , α ′2 [15,16] have no such couplings. In that scheme the number of T-duality invariant couplings may be less than 445 that we have found in the scheme (4). It would interesting to write the couplings (30) in a scheme which has minimum number of couplings.
The gravity couplings (31) can be written in another scheme as where now the constant factor is a = −ζ(3)/(3 × 2 13 ) and the tensors ǫ 8 ǫ 8 and t 8 are defined as where M 1 , · · · , M 4 are four arbitrary antisymmetric matrices. It has been speculated in [52,51] that the B-field couplings may be included in above action by extending the curvature tensor to include the torsion, i.e., where the generalized curvature R is This proposal has been supported by studying couplings with structure H 2 R 3 in [51]. Having all couplings in scheme (4), i.e., (30), one may use appropriate field redefinitions, total derivative terms and Bianchi identities to rewrite the couplings (30) in the scheme (47). It has been observed in [38] that the couplings with coefficients c 1 , c 2 , c 5 , c 7 , c 8 in (4) do not change under the above field redefinitions. The T-duality constraint produces c 2 = c 5 = c 7 = 0 and c 1 = 9/128, c 8 = −1/48. In fact the second and the last terms in (39) do not change under field redefinitions. Hence one can check the proposal (47) with these coefficients. If one replaces R µν αβ in (47) with 1 2 H [µ αγ H ν]γ β , one would find c 2 = 0 and c 8 = 0 which are different from what the T-duality constraint produces. Hence, our calculations indicates that the effective action (30) can not be written in any other scheme as (47).