O-plane couplings at order $\alpha'^2$: one R-R field strength

It is known that the anomalous Chern-Simons (CS) coupling of O$_p$-plane is not consistent with the T-duality transformations. Compatibility of this coupling with the T-duality requires the inclusion of couplings involving one R-R field strength. In this paper we find such couplings at order $\alpha'^2$. By requiring the R-R and NS-NS gauge invariances, we first find all independent couplings at order $\alpha'^2$. There are $1,\, 8,\,30,\,20,\, 19,\, 2$ couplings corresponding to the R-R field strengths $F^{(p-4)}$, $\,F^{(p-2)}$, $\,F^{(p)}$, $\,F^{(p+2)}$, $\,F^{(p+4)}$ and $F^{(p+6)}$, respectively. We then impose the T-duality constraint on these couplings and on the CS coupling $C^{(p-3)}\wedge R\wedge R$ at order $\alpha'^2$ to fix their corresponding coefficients. The T-duality constraint fixes almost all coefficients in terms of the CS coefficient. They are fully consistent with the partial couplings that have been already found in the literature by the S-matrix method.


Introduction
The best candidate for quantum gravity is the superstring theory in which the graviton appears as a specific mode of a relativistic superstring at weak coupling [1,2]. Superstring has massless and infinite tower of massive states which appear in the low energy effective action, as higher derivative corrections to the supergravity. Study of these higher derivative corrections are important because they signal the stringy nature of the quantum gravity.
One of the most exciting discoveries in perturbative string theory is the T-duality which has been observed first in the spectrum of string when one compactifies theory on a circle [3,4]. This symmetry may be used to construct the effective action of string theory including its higher derivative corrections, in the Double Field Theory formalism in which the T-duality transformations are standard whereas the gauge transformations are non-standard [5,6,7]. It has been also speculated that the invariance of the effective actions of string theory and its non-perturbative objects, i.e., D-branes and O-planes, under standard gauge transformations and non-standard T-duality transformations may be used as a constraint to construct the effective actions [8]. In this approach, one first constructs the most general gauge invariant and independent couplings at a given order of α ′ with arbitrary parameters. Then the parameters may be found for the string theory by imposing the T-duality symmetry on the couplings. That is, one reduces the couplings on a circle and requires them to be consistent with the T-duality transformations which are the standard Buscher rules [9,10] plus their α ′ -corrections [11,12,13,14]. Using this approach, the effective action of the bosonic string theory at order α ′ and α ′2 have been found in [14,15]. It has been shown in [16,17] that the leading order effective action of type II superstring theories, including the Gibbons-Hawking-York boundary term [18,19], can also be rederived by the T-duality constraint. The couplings involving metric and dilaton in both heterotic string and in superstring theories at order α ′3 have been also redeived by the T-duality constraint in [20]. There are many other approaches for constructing the effective actions including the S-matrix approach [21,22], the sigma-model approach [23,24,25], and the supersymmetry approach [26,27,28,29] The T-duality approach for constructing the effective action of D p -brane is such that one first writes all gauge invariant and independent D p -brane world-volume couplings at a specific order of α ′ with some unknown p-independent coefficients. Then one reduces the world-volume theory on the circle. There are two possibilities for the killing coordinate. Either it is along or orthogonal to the brane. The reduction of the world-volume theory when the killing coordinate is along the brane (the world-volume reduction), is different from the reduction of the worldvolume theory when the killing coordinate is orthogonal to the brane (the transverse reduction). However, the T-duality transformation of the world-volume reduction of D p -brane should be the same as the transverse reduction of the D p−1 theory, up to some total derivative terms which can be ignored for closed spacetime manifold [8].
The orientifold projection on D p -brane effective action produces O p -plane effective action [1]. This projection removes the open string couplings, the couplings that have odd number of transverse indices on metric and dilaton and their corresponding derivatives, and the couplings that have even number of transverse indices on B-field and its corresponding derivatives. Hence, the T-duality constraint may also be used to construct the O p -plane effective action which is much simpler than the D p -brane effective action. The T-duality constraint has been used in [30,31] to find the effective action of O p -planes of type II superstring at order α ′2 for NS-NS fields. In this paper, we are interested in applying the T-duality constraint on the effective action of O p -plane when there is one R-R field strength.
The O p -plane CS action at the leading order of α ′ is given by the orientifold projection of the CS action of D p -brane [1], i.e., where C = 8 n=0 C (n) is the R-R potential. It is invariant under the R-R gauge transformation where Λ = 7 n=0 Λ (n) and H = dB. Note that the last term in δC is projected out by the orientifold projection when all indices of the R-R potential are world-volume, as in (1). The curvature corrections to this action has been found by requiring that the chiral anomaly on the world-volume of intersecting D-branes cancels with the anomalous variation of the CS action [32,33,34]. This correction for O-plane is where A(R T,N ) is the A-roof genus of the tangent and normal bundle curvatures respectively, where R T,N are the curvature 2-forms of the tangent and normal bundles respectively. This action at order α ′2 in component form is 3

S
(2) The above couplings have been confirmed by the S-matrix element calculations in [35,36,37]. Using the cyclic symmetry of the Riemann curvature, one can verify that the above diffeomorphism invariant action is also invariant under R-R gauge transformation. As it has been argued in [38,39], the above couplings, however, are not consistent with the T-duality transformations.
On the other hand, there are many other gauge invariant couplings at this order which can not be found by the anomaly analysis. The R-R gauge symmetry requires all such couplings to be in terms of the nonlinear R-R field strength, i.e., which is invariant under the R-R gauge transformation (2). Some of these couplings involving one R-R field strength F (p−2) and two NS-NS fields have been found in [40,41,42,43] by linear T-duality and by the S-matrix method. The complete couplings involving one R-R field strength F (p) , F (p+2) or F (p+4) and one NS-NS field have been found in [44] by the S-matrix method and have been shown that they are invariant under the linear T-duality. However, the above couplings are not invariant under the full nonlinear T-duality either. Hence, the T-duality of the CS coupling (5) may require adding couplings involving one R-R field strength and an arbitrary number of NS-NS fields at order α ′2 in which we are interested in this paper. An outline of the paper is as follows: In section 2, we write all gauge invariant couplings involving one R-R field strength. We use the Bianchi identities and total derivative terms to write the minimum number of gauge invariant couplings. There are 1, 10, 46, 53, 47, 2 such couplings corresponding to the R-R field strengths F (p−4) , F (p−2) , F (p) , F (p+2) , F (p+4) , F (p+6) , respectively. However, since the couplings involve the volume form ǫ a 0 ···ap , there are still some relations between the couplings when one imposes the constraint that the world-volume indices must be only 0, 1, · · · , p. By writing the world-volume indices explicitly as 0, 1, · · · , p, we find all independent couplings which are 1, 8, 30, 20, 19, 2 couplings corresponding to the R-R field strengths F (p−4) , F (p−2) , F (p) , F (p+2) , F (p+4) , F (p+6) , respectively. We then reduce them in section 3 to impose the T-duality constraint on them.
An appropriate method for reducing a gauge invariant coupling to 9-dimensional base space has been presented in [15]. In this method one keeps the U(1) × U(1) gauge invariant part in the reduction of the Riemann curvature and other components of a given coupling and removes all other terms. In section 3, we extend this method for the reduction of the couplings involving R-R fields as well, i.e., we find the U(1) × U(1) gauge invariant part of the reduction of R-R field strength and its first derivatives. In section 4, we impose the T-duality constraint on the independent couplings to fix their parameters. In this section, we show that the T-duality can fix almost all parameters of the gauge invariant couplings in terms of an overall factor, and they are consistent with the partial couplings that have been already found in the literature by the S-matrix method. Apart from the overall factor, there is only one unfixed parameter which is coefficient of two couplings that their reduction on the circle is zero. In section 5, we present the final form of the gauge invariant couplings and briefly discuss our results.

Gauge invariant couplings
In this section we would like to find all independent couplings on the world-volume of O p -plane involving one R-R field strength and an arbitrary number of NS-NS fields at order α ′2 , i.e., where L n is the Lagrangian which includes all independent couplings involving one R-R field strength F (n) . As it has been argued in [30], since we are interested in O p -plane as a probe, it does not have back reaction on the spacetime. As a result, the massless closed string fields must satisfy the bulk equations of motion at order α ′0 . Using the equations of motion, one can rewrite the terms in the world-volume theory which have contraction of two transverse indices, e.g., ∇ i ∇ i Φ, or R iA i B in terms of contraction of two world-volume indices, e.g., ∇ a ∇ a Φ, or R aA a B . This indicates that the former couplings are not independent. The couplings involving the Riemann curvature and its derivative and the couplings involving derivatives of H and derivatives of R-R field strength satisfy the following Bianchi identities Moreover, the couplings involving the commutator of two covariant derivatives of a tensor are not independent of the couplings involving the contraction of this tensor with the Riemann curvature, i.e., This indicates that if one consider all couplings involving the Riemann curvature, then only the symmetric combination of two covariant derivatives of a tensor can be consider as independent coupling. Using the symmetries of the R-R field strength F (n) , H and the Riemann curvature, one can easily verify that it is impossible to have non-zero contractions of one F (n) and some R, H, ∇Φ at order α ′2 for n < p − 4 and n > p + 8. Moreover, the parity of the coupling (5) indicates that the couplings of the R-R field strength F (p−2) are non-zero when there are even number of B-field. The consistency with linear T-duality then indicates that the couplings of the R-R field strength F (p−4) , F (p) and F (p+4) are non-zero when there are odd number of B-field, and the couplings of the R-R field strength F (p+2) , and F (p+6) are non-zero when there are even number of B-field. There are similar parity selection rule for the corresponding S-matrix elements [47]. For n = p − 4 there is only one non-zero independent coupling 4 , i.e., where the transverse indices are raised by the tensor ⊥ ij = G ij (see next section for the definition of tensor ⊥), and coefficient a is an arbitrary parameter at this point. This parameter may be fixed by studying the disk-level S-matrix element of one R-R and three NS-NS vertex operators which is a very lengthy calculation. We expect this parameter to be related to the coupling (5) by the T-duality constraint.
There is no derivative on the R-R field strength and on the B-field strength in the above coupling. Hence, there is no Bianchi identity involved here. Moreover, there is no total derivative term here. This is not the case for n > p − 4 cases. Let us discuss each of the cases n = p − 2, n = p, n = p + 2, n = p + 4 and n = p + 6 separately.

n = p − 2 case
To find all gauge invariant and independent couplings corresponding to one R-R field strength F (p−2) , we first consider all contractions of one ǫ a 0 ···ap , one F , ∇F or ∇∇F , even number of H and ∇H, and any number of ∇Φ, ∇∇Φ, ∇∇∇Φ, R, ∇R at four-derivative order. We then remove the terms which are projected out by the orientifold projection and by the equations of motion. We call the remaining terms, with coefficients b ′ 1 , b ′ 2 , · · ·, the Lagrangian L p−2 . Not all terms in this Lagrangian are independent. Some of them are related by total derivative terms. To remove such redundancy, we write all total derivative terms at order α ′2 which involve the R-R field strength F (p−2) . To this end we first write all contractions of one ǫ a 0 ···ap , one F , ∇F , even number of H and ∇H, and any number of ∇Φ, ∇∇Φ, R at three-derivative order. Then we remove the terms which are projected out by the orientifold projection and by the equations of motion. We call the remaining terms, with arbitrary coefficients, the vector I p−2 a . The total derivative terms are then where g ab = G ab is inverse of the pull-back metric (see next section for the definition of the pull-back metric). Adding the total derivative terms to L p−2 , one finds the same Lagrangian but with different parameters b 1 , b 2 , · · ·. We call the new Lagrangian L p−2 . Hence where ∆ p−2 = L p−2 − L p−2 is the same as L p−2 but with coefficients δb 1 , δb 2 , · · · where δb i = b i − b ′ i . Solving the above equation, one finds some linear relations between only δb 1 , δb 2 , · · · which indicate how the couplings are related among themselves by the total derivative terms. The above equation also gives some relation between the coefficients of the total derivative terms and δb 1 , δb 2 , · · · in which we are not interested.
However, to find the correct independent terms in L p−2 , one has to consider the terms in ∆ p−2 and J p−2 which are not related to each other by the Bianchi identities (8). To impose the first two Bianchi identities, one may rewrite the terms in (12) in the local frame in which the first derivative of metric is zero. To impose the third Bianchi identity, one can rewrite the terms in (12) which have derivatives of H in terms of B-field, i.e., H = dB. In this way, the Bianchi identities satisfy automatically [46]. The last Bianchi identity in (8) relates the couplings involving derivative of F (p−2) to themselves and to the couplings involving F (p−4) . However, the independent couplings involving F (p−4) has been already fixed in (10). Hence, the last Bianchi identity in (8) should relate only the couplings involving F (p−2) , i.e., one should use the identity dF (p−2) = 0. To impose this identity on the couplings in (12) as well, one has to rewrite the terms involving the derivatives of the R-R field strength F (p−2) in terms of the R-R potential, i.e., F (p−2) = dC (p−3) .
Using the above steps, one can rewrite the different terms on the left-hand side of (12) in terms of independent but non-gauge invariant couplings. The solution to the equation (12) then has two parts. One part is a relations between only δb i 's, and the other part is a relation between the coefficients of the total derivative terms and δb i 's in which we are not interested. The number of relations in the first part gives the number of independent couplings in L p−2 . One can set some of the coefficients in L p−2 to zero, however, after replacing the non-zero terms in (12), the number of relations between only δb i 's should not change. In the present case this number is 10. We set the coefficients of the terms that have world-volume derivative on the R-R field strength, to be zero. After setting this coefficients to zero, there are still 10 relations between δb i 's. This means we are allowed to remove these terms. We choose some other coefficients to zero such that the remaining coefficients satisfy the 10 relations δb i = 0. In this way one can find the minimum number of gauge invariant couplings. One particular choice for the 10 couplings is the following: where the world-volume indices are raised by the first fundamental form G ab = G ab (see next section for the definition of the first fundamental form), and the b 1 , · · · , b 10 are arbitrary coefficients. These coefficients do not depend on p. In fact the p-dependence of the couplings has been written explicitly by 1/n! where n is the number of indices of the R-R field strength that are contracted with ǫ a 0 ···ap . These couplings are consistent with the linear T-duality for the special case that the world-volume killing index of ǫ a 0 ···ap contracts with the R-R field strength. That is, where the dots before the index a m in the R-R field strength are the world-volume or transverse indices that contract with other parts of the coupling, i.e., contract with (· · ·). In the first line we assume one of the world-volume indices is the killing index y, and in the second line we have used the linear T-duality transformation for the linearised R-R field strength, i.e., F (n) ···y = F (n−1) ··· , and the identity ǫ a 0 ···a p−1 y = ǫ a 0 ···a p−1 . The couplings (13) for arbitrary coefficients b 1 , · · · , b 10 , however, are not consistent with the linear T-duality when the killing index is not carried by the R-R field strength. We are interested in constricting these coefficients and the coefficients of other R-R field strengths that we will find in the subsequent subsections, by requiring the couplings to be consistent with nonlinear T-duality.
In finding the couplings in (13), we have not assumed that the number of world-volume indices in each coupling must be the same as the world-volume indices of ǫ a 0 ···ap . It has been observed in [44] that imposing this constraint, one may find some relations between couplings involving ǫ a 0 ···ap . Some of these relations for the simple case of two-field couplings, have been found in [44]. To impose this constraint on the couplings (13), we write them explicitly in terms of the values that each world-volume index can take, e.g., a 0 = 0, 1, 2, · · · , p. In terms of these explicit values, one finds that the couplings b 3 , b 6 can be written in terms of couplings b 2 , b 4 , b 5 , so they are not independent. Hence, there are the following 8 independent terms: The 8 coefficients may be found by the T-duality constraint. Note that there is no term in (14) which involves only one NS-NS field. This is consistent with the S-matrix calculation of one R-R and one NS-NS vertex operator which has no such term [44]. The disk-level S-matrix element of one R-R potential C (p−3) and two B-field vertex operators has been calculated in [42,43] from which the couplings of one F (p−2) and two H has been found for D p -brane. The orintifold projection of the couplings found in [43] are the same as the above couplings with the following coefficients: where we have also used the Bianchi identity dH = 0 to relate the couplings found in [43] to the couplings in (14). The above independent couplings, however, are not the most general gauge invariant couplings because they do not include the Riemann curvature. The gauge invariant couplings involving the Riemann curvature are the couplings in the CS action (5) which are found by the anomaly cancellation mechanism. The T-duality constraint should reproduce these couplings as well. Hence, we include in this subsection the following gauge invariant couplings with arbitrary coefficients: The two parameters α 1 , α 2 which are known from the anomaly cancellation mechanism and also from the S-matrix calculation, should be fixed by the T-duality constraint as well.

n = p case
To find all gauge invariant and independent couplings involving one R-R field strength F (p) , we first consider all contractions of one ǫ a 0 ···ap , one F , ∇F or ∇∇F , odd number of H, ∇H and ∇∇H, and any number of ∇Φ, ∇∇Φ, ∇∇∇Φ, R, ∇R at four-derivative order. We then remove the terms which are projected out by the orientifold projection, the equations of motion, the total derivative terms and by the Bianchi identities with the same strategy that is discussed in the previous subsection. In this manner one finds 46 couplings with coefficients c 1 , c 2 , · · · , c 46 (see Appendix for the explicit form of these couplings). We call the corresponding Lagrangian L p . There are still some relations between these couplings when one imposes the constraint that the world-volume indices must be only 0, 1, · · · , p. In this case one finds there are only 30 independent couplings. One particular form for them is the following: where c 2 , · · · , c 46 are the arbitrary coefficients that do not depend on p. They may be found by the T-duality constraint. The coefficients c 2 , c 3 can be also fixed by the tree-level S-matrix element of one R-R and one NS-NS vertex operators [44]. One finds In finding this result we write the two-field terms in (17) and the couplings found in [44] in terms of independent structures, and then force them to be the same.

n = p + 2 case
To find all gauge invariant and independent couplings involving one R-R field strength F (p+2) , we consider all contractions of one ǫ a 0 ···ap , one F , ∇F or ∇∇F , even number of H and ∇H, and any number of ∇Φ, ∇∇Φ, ∇∇∇Φ, R, ∇R at four-derivative order. We then remove the terms which are projected out by the orientifold projection, the equations of motion, the total derivative terms and by the Bianchi identities with the same strategy that is discussed in the subsection 2.1. In this case one finds 53 couplings with coefficients d 1 , d 2 , · · · , d 53 (see Appendix for the explicit form of these couplings). These 53 terms, however, are not independent. When one writes the world-volume indices of the 53 couplings explicitly in terms of 0, 1, · · · , p, only 20 independent couplings will be found. One particular form for the independent couplings is the following: where the p-independent coefficients d 2 , · · · , d 48 may be found by the T-duality constraint. The coefficients d 11 , d 12 , d 15 can be also fixed by the tree-level S-matrix element of one R-R and one NS-NS vertex operators [44]. They are In finding the above result, we have imposed the first Bianchi identity in (8) on the two-field couplings found in [44]. Note that as observed in [44] the above results indicates that the curvature R iaj a and ∇ i ∇ j Φ appear in the O-plane action as ij-component of the following combination: where A, B are 10-dimensional bulk indices. Note that the transverse contraction of the Riemann curvature, i.e., R AiB i has been removed at the onset by imposing the equations of motion. This dilaton-Riemann curvature appears also in NS-NS couplings of O-plane action at order α ′2 [31]. We speculate that the second derivative of dilaton appears in all O-plane and D-brane couplings in above combination.

n = p + 4 case
Performing the same steps as in subsection 2.1, one finds there are 47 couplings on the worldvolume of O p -plane that are not related to each other by the Bianchi identities and the total derivative terms (see Appendix for the explicit form of these couplings). Not all of them, however, are independent. When we write the world-volume indices of the 47 couplings explicitly in terms of 0, 1, · · · , p, we will find that there are only 19 independent couplings. One particular form for the independent couplings is the following: where the p-independent coefficients e 1 , · · · , e 47 may be found by the T-duality constraint. The coefficient e 1 has been fixed by the tree-level S-matrix element of one R-R and one NS-NS vertex operators [44], i.e., The proposal that the combination (21) should appear in the world-volume couplings, dictates that the T-duality should fix the coefficient e 8 to be the same as e 37 .

n = p + 6 case
Similar calculation for the couplings involving one R-R field strength F (p+6) gives the following two independent coupling: where f 1 , f 2 are two arbitrary coefficients that may be found by the T-duality constraint. There are no couplings involving one NS-NS field which is consistent with the tree-level S-matrix element of one R-R and one NS-NS vertex operators [44]. The above two coefficients have been fixed in [42] by the corresponding S-matrix element of one R-R and two NS-NS vertex operators to be zero. Therefore, there are 82 independent couplings at order α ′2 which have one R-R field. These gauge invariant couplings are the appropriate couplings on the world-volume of O p -plane for some specific values for the 82 parameters. They may be found by the S-matrix or other methods in string theory. We are going to find these parameters in the next section by the T-duality constraint. We will find that all 82 parameters are fixed up to an overall factor.

T-duality transformations
When compactifying the superstring theory on a circle with radius ρ and with the coordinate y, the full nonlinear T-duality transformations at the leading order of α ′ for the NS-NS and R-R fields are given in [9,10,48], i.e., where µ, ν denote any direction other than y. Our notation for making antisymmetry is such that e.g., C In above transformations the metric is in the string frame. If one assumes fields are transformed covariantly under the coordinate transformations, then the above transformations receive corrections at order α ′3 in the superstring theory [20] in which we are not interested because the couplings in this paper are at order α ′2 .
To impose the T-duality constraint on the effective action, one should first write all independent gauge invariant couplings of O p -plane, as we have done in the previous section, and then reduce them on the circle when O p -plane is along the circle. The T-duality transformation of the reduced action should be the same as the reduction of O p−1 -plane when it is orthogonal to the circle, up to some total derivative terms. To impose the T-duality constraint on the effective action, however, it is convenient to use the following background for the metric, B-field, dilaton and the R-R potentials [49,17]: whereḡ µν ,b µν ,φ andc (n) are the metric, B-field, dilaton and the R-R potentials, respectively, in the 9-dimensional base space, and g µ , b µ are two vectors in this space. In this parametrization, inverse of metric becomes whereḡ µν is the inverse of the base metric which raises the indices of the vectors. The nonlinear T-duality transformations (25) in the parametrizations (26) then become remarkably the following linear transformations: and all other 9-dimensional fields remain invariant under the T-duality transformation. Note that the T-duality transformation of the base space R-R potentialc (n) is trivial in the parametrization (26), however, the R-R gauge transformation of this potential in which we are not interested in this paper, seems to be non-trivial.
One can easily verify that the CS action at order α ′0 is invariant under the T-duality. If the killing coordinate y is a world volume, then the T-duality transformation of the reduction of O p -plane action in the parametrization (26) becomes where we have used the relation 2πρT p = T p−1 and ǫ a 0 ···a p−1 y = ǫ a 0 ···a p−1 . On the other hand, the reduction of the O p−1 -plane action in the parametrization (26) when the y-coordinate is transverse to the O p−1 -plane is Using the fact that g a p−1 is the component of the 10-dimensional metric which has one y-index and y is a transverse index in this case, the last term above is projected out by the orientifold projection. The rest is the same as the action (29).
There is no such symmetry for the CS action at higher orders of α ′ because the Riemann curvature is not invariant under the T-duality transformations. As a result, one has to add some other terms to this action to make it T-duality invariant as in the leading order term. Since the new couplings involve R-R and NS-NS field strengths and their covariant derivatives, it is convenient to first find the reduction of these field strengths and then apply them to find the reduction of each gauge invariant coupling.
Using the reductions (26), it is straightforward to calculation reduction of the Riemann curvature, H, ∇H, ∇Φ or ∇∇Φ. As it has been argued in [15], after writing the reductions in terms ofH which is defined asH where W = db and V = dg, they have two parts. One part includes terms which are invariant under U(1) × U(1) gauge transformations corresponding to the gauge fields g µ , b µ . They have been found in [15] and removes all other terms in the reduction. In this way one can find the reduction of any gauge invariant bulk coupling. However, the metric G AB is not used in constructing the O pplane couplings in the previous section. The world-volume couplings in fact are constructed by contracting the tensors with the first fundamental form G AB = ∂ a X A ∂ b X B g ab which projects the spacetime tensors to the world-volume directions, and with ⊥ AB = G AB − G AB which projects the tensor to the transverse directions. In the first fundamental form, g ab is inverse of the pull-back metric For the O p -plane at X i = 0, one has G ij = G ai = G ia = 0, and G ab = g ab , g ab = G ab . When O p -plane is orthogonal to the the killing coordinate, the first fundamental form and world-volume components of the inverse of the spacetime metric have no component along the y-direction, because y is a transverse direction. Hence, in this case ⊥ ab = 0. Moreover ⊥ ai = G ai = 0 by the orientifold projection. The reduction of the non-zero terms in this case are The gauge field gã does not appear in the reduction of G ab , however, it appears in the reduction of ⊥ ij . As in (32), we have ignored it because we have ignored the non-gauge invariant terms in the reduction of the Riemann curvature, H, ∇H, ∇Φ and ∇∇Φ.
On the other hand, when O p -plane is along the killing coordinate, both the first fundamental form and world-volume components of the inverse of the spacetime metric have component along the y-direction, however, because G ab = G ab one again has ⊥ ab = 0. In this case the reduction of the non-zero terms are The gauge field gã does not appear in the reduction of ⊥ ij , however, it appears in the reduction of G ab that we have again removed it. Using the reduction of the R-R potential in (26), one can find the reduction of R-R field strength and its first derivative which appear in the couplings in the previous section. They have again two parts. One part is not invariant under the U(1) × U(1) gauge transformations which is cancelled in the gauge invariant couplings, hence we ignore it. The U(1) × U(1) gauge invariant part of the reduction is where the covariant derivatives on the right-hand side are 9-dimensional andF = dc. One can check that the reduction of ∇H found in [15] can be found from the above reduction when one uses H W (2) = W and H V (3) =H. Obviously, the U(1) × U(1) gauge invariant part of the reduction of the R-R potential C is Using the above U(1) × U(1) gauge invariant part of the reduction of the field strengths, one can calculate the reduction of any 10-dimensional gauge invariant coupling. The result would be the same as writing the coupling in terms of ordinary derivatives of metric, B-field, dilaton and R-R potential and then using the reductions (26). For example, using the above reduction for the R-R field strength, one finds the following reduction for the gauge invariant coupling F 2 : which is the correct reduction that has been found in [17] by writing the R-R field strength in terms of R-R potential and using the reductions (26). It is obvious that the left-hand side is invariant under the 10-dimensional R-R gauge transformations, hence, the right-hand side should be also invariant under the 9-dimensional R-R gauge transformations. This might be used to define the gauge transformation of the base space R-R potentialc (n) in which we are not interested in this paper. As another example, the O p -plane world-volume reduction of the CS terms in (5) are In finding the above result we have separated the world-volume indices to y and the world indices which do not include the y-index, then we have used the reduction for each tensors. We have assumed the 9-dimensional base space is flat, and removed the terms that are projected out by the orientifold projection, e.g., we have removed V ai because g i is related to G iy and y is world-volume index, hence, it is projected out. Note that the world-volume indices on the right-hand side do not include the y-index.
The O p−1 -plane transverse reduction of the CS terms are In finding the above result we have separated the transverse indices to y and the transverse indices which do not include the y-index, then we have used the reduction for each tensors.
Here, we have also removed the terms that are projected out by the orientifold projection, e.g., we have removed V ab because g a is related to G ay and y is transverse index, hence, it is projected out. Note that the transverse indices on the right-hand side do not include the y-index. Similar calculations as above can be done for all couplings in the previous section. Writing the reduced couplings in terms of the base fieldsc, V, · · ·, one can easily transform them under the T-duality transformations (28).

T-duality constraint on the couplings
It has been observed in [14,15] that the T-duality constraints on the couplings in the bosonic string theory at order α ′ and α ′2 are the same whether or not the base space is flat. In fact, the constraints that one finds between the coefficients of effective action when base space is flat are exactly the same constraints as one finds for the curved base space. So it is convenient to consider the reduction of the couplings in section 2 on the flat base space, and then impose the T-duality constraint on them to find the unknown coefficients of the couplings. It is evident that the transformation of couplings in (37) under the T-duality transformation (28) are not the same as the couplings in (38). This indicates that the CS couplings (5) at order α ′2 are not consistent with the T-duality. However, the combination of the CS action and the couplings in section 2 may be invariant under the T-duality transformations. This constraint way fix the coefficients of all couplings in section 2.
It can easily be observed that the T-duality constraint fixes the coefficient of the coupling F (p−4) to be zero. To see this, we first note that the reduction of F (n) , involves the base space fieldsc (n−1) ,c (n−2) ,c (n−3) andc (n−4) . So the world-volume reduction of O p -plane and the transverse reduction of O p−1 -plane produces the following 9-dimensional R-R potentials: The T-duality of the world-volume reduction of O p -plane couplings should be the same as the transverse reduction of O p−1 -plane couplings up to some total derivative terms. We look at the term in the reduction which producesc (p−9) . This term is produced only by the reduction of the coupling (10) when one of the transverse indices of the R-R field strength carries the y-index. The reduction of this term, however, is zero after imposing the orientifold projection. So this can not constraint the coefficient of the coupling (10). We consider instead the reductions which producec (p−8) . When the O p -plane is along the circle, it produces the following reduction: where dots represent some other terms which do not includec (p−8) . On the other hand, when O p−1 -plane is orthogonal to the circle, the reduction of the coupling (10) produces the following terms: where dots represent some other terms which do not includec (p−8) . The difference between this term and the T-duality transformation of (40) produces the following term which involves c (p−8) : This term can not be cancelled by total derivative terms, so the T-duality constraint predicts the coefficient of the coupling (10) to be zero, i.e., Hence, the T-duality constraint force the coupling (10) to be zero. It is a nontrivial result which would be very difficult to confirm with the S-matrix element of one R-R and three NS-NS vertex operators. It can be also easily observed that the T-duality constraint fixes the coefficients of the F (p+6) -couplings to be zero. In this case we look at the term in the reduction which produces c (p+5) . This term is produced only by the world-volume reduction of the couplings in (24). The T-duality transformation of this term produces the following term forc (p+5) : which can not be cancelled by a gauge invariant total derivative term. Hence, the T-duality constraint forces the above term to be zero, i.e., 3f 1 −f 2 = 0. To fix these coefficients completely, we look also at the terms in the reduction which producec (p+4) . The difference between the O p−1 -plane and the T-duality of O p -plane produces many terms involvingc (p+4) . Here we focus on the terms involvingc (p+4) and ∇ϕ. One can easily find that only the reduction of the second term in (24) produces such term. The T-duality of the reduction of O p -plane produces F p+5 ∇ϕHH, whereas, the reduction of O p−1 -plane producesF p+5 ∇ϕW W . They can not cancel each other unless the coefficient of the second term in (24) to be zero, i.e., f 2 = 0. Combining with the previous constraint, one finds This is the result that the S-matrix calculation produces [42].
Since the coefficient of the F (p−4) -coupling is zero, the next simple case to look at is the terms involvingc (p−7) . One findsc (p−7) is produced only by the transverse reduction of the couplings F (p−2) in (14) which have R-R field strength with transverse indices. Since only the couplings with coefficients b 1 , b 7 in (14) involves the R-R field strength with the transverse indices, and the transverse reduction of these terms produces non-zero results which are not total derivative terms, one finds that the T-duality constraint fixes these coefficients to be zero, i.e., If one looks at the terms involvingc (p−6) , one would find that only the reductions of the terms with coefficients b 1 , b 7 survived the orientifold projection. The T-duality constraint then again forces these coefficients to be zero. This result is consistent with the S-matrix calculation (15). The surviving terms in (14) have R-R field strength with only world-volume indices. One finds that the reduction of these terms produce terms involvingc (p−5) , however, they removed by the orientifold projections. Having noc (p−5) -term from the reduction of F (p−2) -couplings, one concludes that the transverse reduction of F (p) -couplings on the O p−1 -plane which also producesc (p−5) , must be zero. So one has to consider the R-R field strengths F (p) , ∇F (p) in (17) which have transverse indices because only those terms producec (p−5) . In fact all terms in (17) have such structure. However, the transverse reduction of those terms that have only one transverse index, produceH ∧c (p−5) with only world-volume indices which is removed by the orentifold projection. Therefore, they produce no non-zero term after reduction. The terms in (17) which have more than one transverse indices, i.e., c 21 , c 23 , c 34 , c 40 , however, produce non-zero result after orientifold projection. The T-duality constraint requires these terms to be zero, i.e., Since the reduced couplings involve onlyc (p−5) there is no total derivative terms connecting the reduced couplings. Moreover, since they involve no derivative of field strengthH, there is no Bianchi identity relation between the reduced couplings. Hence, the coefficients of all terms must be zero, as we have set in above equation.
Since the coefficients of the couplings involving F (p+6) are zero, i.e., (45), the next simple case to consider is to look at the terms involvingc (p+3) . One findsc (p+3) is produced only by the world-volume reduction of the couplings in (22) which have R-R field strength with no y index. So all terms in (22), except the terms in which the R-R field strength carries the world-volume indices a 0 , · · · , a p , producec (p+3) . The T-duality constraint makes the coefficients of all these terms to be zero, i.e., e 13 = 0, e 26 = 0, e 32 = 0, e 44 = 0 (48) In finding the above result, we have added all possible total derivative terms and write the world-volume indices explicitly as 0, 1, · · · , p − 1. We find that there is no total derivative term involved here. There are still further T-duality constraint on the non-zero couplings involving F (p+4) . The world-volume reduction of O p -plane and the transverse reduction of O p−1 -plane producec (p+2) . Imposing the T-duality of the former to be the same as the latter, one finds the following relations for the other coefficients: In finding the above results we have imposed the Bianchi identities dH = − 3 2 V ∧ W and dV = dW = 0 by writing the field strengthsH, W, V in terms of potentials g, b,b. Here also we have added all possible total derivative terms and write the world-volume indices explicitly as 0, 1, · · · , p − 1. In this case we find that there is some total derivative term involved in which we are not interested in this paper. Up to an overall coefficient e 1 , then all terms in (22) are fixed by the T-duality constraint that we have considered so far. It is interesting that the coefficients e 8 , e 37 are identical which is in accord with the proposal that the second derivative of dilaton appears in the world-volume action as the dilaton-Riemann curvature (21). Moreover, the first derivative of dilaton appears only in the term with coefficient e 3 . Using an integration by part on the first term in (22), and the relation e 3 = e 1 , one finds that the first derivative of dilaton appears in the following extension of ∇ a ∇ a H ABC : We will see that this structure appears in all couplings that the T-duality produces. Note that the transverse contraction of two derivatives, i.e., ∇ i ∇ i has been removed at the onset by imposing the equations of motion. Imposing the constraints that we have found so far, i.e., (43), (45), (47), (48), and (49), the remaining reductions in (39) are The next case that we are going to consider in the reductions (51), isc (p+1) . Since one part of the reduction involve the F (p+2) -couplings, the T-duality constraint should relate the remaining constant e 1 in F (p+4) -couplings to the d-parameters in (19). The T-duality constraint in this case remarkably fixes e 1 and all d's in terms of one overall parameter, i.e., In finding the above results we have imposed the Bianchi identities and we have added all possible total derivative terms and write the world-volume indices explicitly as 0, 1, · · · , p − 1.
In this case also, the T-duality constraint requires some total derivative terms in which we are not interested. The coefficients d 12 , d 15 in (52) are consistent with the S-matrix result (20). Moreover, the relation between e 1 and d 11 is also consistent with the S-matrix results (20) and (23). As pointed out before, since d 11 = d 12 the second derivative of dilaton appears as the dilaton-Riemann curvature (21). The first derivative of dilaton also appears as dilatonderivative extension of world-volume derivative contraction with Riemann curvature and with H, i.e., Note that the transverse derivative contraction with the Riemann curvature and with H have been removed by the equations of motion. We will see that this extension appears in other couplings that the T-duality produces.
Since all e-parameters and d-parameters are fixed up to the overall factor d 11 , one does not need to considerc (p) because this term is produced only by F (p+4) -and F (p+2) -couplings. In fact, we have checked that the T-duality constraint onc (p) reproduces only the relations in (49) and (52). Hence, for the next case we considerc (p−1) in the reductions (51). The T-duality constraint on this term should give some relations between F (p+4) -, F (p+2) -and F (p) -couplings. Since the parameters in the first two set of couplings are fixed, this constraint should fix the c-parameters in (17). The T-duality constraint in this case fixes d 11 and almost all c's in terms of one overall parameter c 12 , i.e., Here again we have imposed the Bianchi identities and we have added all possible total derivative terms and write the world-volume indices explicitly as 0, 1, · · · , p − 1 to find the above results. In this case also there are some total derivative terms in which we are not interested in this paper because we assumed the spacetime manifold has no boundary. The reason that c 29 is not fixed is that the couplings with coefficients c 29 and c 30 produce zero world-volume reduction when c 30 = −c 29 /4. The coefficients c 2 , c 3 in (54) are consistent with the S-matrix result (18). Moreover, the relation between d 11 and c 2 is also consistent with the S-matrix results (18) and (20). The coefficients c 12 , c 46 are not identical, so one may conclude that the corresponding couplings in (17) are not in accord with the proposal that the second derivative of dilaton appears in the world-volume action as the dilaton-Riemann curvature (21). However, using the R-R Bianchi identity (8), one can write where we have used the orientifold projection on H and the fact that there is an overall tensor ǫ a 0 ···ap . Then up to a total derivative term, one can write the term in (17) with coefficient c 5 as 1 The first term on the right hand side then has the same structure as the term with coefficient c 12 . Since c 12 + c 5 = c 46 , one can write the corresponding couplings in (17) as the dilaton-Riemann curvature (21). The second term on the right hand side can be combined with the first term in (17) to write them as dilaton-derivative combination (50). The last term should be added to the b 9 -coupling in (14). The coefficients c 3 , c 8 are identical, hence, the corresponding couplings can be combined as the dilaton-derivative (50). It seems, however, that the second derivative of dilaton in the coupling with coefficient c 13 in (17) can not be combined with any coupling with structure F HR to be written as the dilaton-Riemann curvature. This steams from the fact that when we have written the independent couplings in (17), we had not paid attention on the proposal (21). Now that we have found the couplings we may use appropriate identity to write the couplings as the dilaton-Riemann curvature. In fact, writing the world-volume indices explicitly as 0, 1, · · · , p, one can find the following identity: Using this identity, one finds that the couplings in (17) with coefficients c 13 , c 38 , c 39 can be written as the dilaton-Riemann curvature (21).
The T-duality constraint onc (p−2) should reproduce only the relations in (54). We have checked it explicitly.
Finally, to relate the constant c 12 to the b-parameters in (14) and α-parameters in (16), one can consider the T-duality constraint onc (p−3) orc (p−4) . We considerc (p−3) in the reductions (51). The T-duality constraint on this term should give some relations between F (p+2) -, F (p)and F (p−2) -couplings and the couplings in (16). Since the parameters in the first two sets of couplings are fixed, this constraint should fix the b-parameters in (14), α-parameters in (16) and c 12 in terms of one overall parameter. The T-duality constraint in this case produces the following relations: In this case also there are some total derivative terms in which we are not interested in this paper. The first relation above is consistent with CS coupling (5). The coefficients b 2 , b 4 , b 5 are consistent with the S-matrix result (15). The first term in the b 5 is consistent with the proposal that the first derivative of dilaton appears in the dilaton-derivative combination. To see this we note that the last term in (57) has the same structure as b 9 -coupling. Hence, this structure has coefficient b 9 − c 5 /2 = 2α 1 which is minus of b 2 . As a result they can be combined into the dilaton-derivative combination (50). The coefficients b 8 , b 10 are not fixed. These parameters appear also in the total derivative terms. This indicates that the couplings in (14) are not really independent. In finding the couplings (14), we first found the minimum couplings (13) which are not related to each other under total derivative terms and Bianchi identities and then wrote the world-volume indices in them explicitly as 0, 1, · · · , p. If one write the world-volume indices before solving the equation (12) as 0, 1, · · · , p, then the independent coupling would not include the b 8 -coupling and b 8 -coupling. In fact these couplings are not consistent with the dilaton-derivative proposal either. Hence, we set these parameters to zero, i.e., b 8 = b 10 = 0. The parameter c 29 again remains unfixed because reduction of c 29 -coupling and c 30 -coupling is zero when c 30 = −c 29 /4.

Discussion
In this paper, imposing only the gauge symmetry and the T-duality symmetry on the effective action of O p -plane, we have found the following couplings at order α ′2 : where α 1 is an overall constant that can not be fixed by the T-duality constraint. The gauge invariant Lagrangians are the following: The second derivative of dilaton appears in the dilaton-Riemann curvature (21) and the first derivative of dilaton appears in the dilaton-derivative (50). There is also the following Lagrangian which is zero when reducing it on the circle, hence, its overall coefficient is not fixed by the T-duality constraint: Most of the couplings in (59) are new couplings which have not been found by any other method in string theory. This action is fully consistent with the partial couplings that have been already found in the literature by the S-matrix method, i.e., the couplings of one arbitrary R-R field strength and one NS-NS, and also the couplings of one R-R field strength F (p−2) and two B-fields. The disk-level S-matrix elements of one arbitrary R-R and two NS-NS vertex operators have been calculated in [47,50]. The low energy expansion of them should produce D-brane couplings at order α ′2 . The orientifold projection of those couplings then should be the same as the couplings that we have found in (59). It would be interesting to perform this comparison.
We have seen that the derivatives of dilaton appears only through the dilaton-Riemann curvature (21) and the dilaton-derivative (50). It has been shown in [44] that the dilaton-Riemann curvature is invariant under linear T-duality. The dilaton-derivative is also invariant under the linear T-duality. In fact one can write the contraction of the dilaton-derivative with an arbitrary vector at the linear order of metric perturbation as where G AB = η AB + h AB . Separating the world-volume indices to y-index and other worldvolume indices, and using the linear T-duality transformations h yy → −h yy and Φ → Φ − h yy /2, then one finds the above expression is invariant under the linear T-duality. Similar analysis has been done in [44] to show that the dilaton-Riemann curvature is invariant under the linear T-duality. The invariance of the world-volume action under linear T-duality requires the derivatives of dilaton appear in the dilaton-Riemann and dilaton-derivative combinations. However, the invariance of the effective action under nonlinear T-duality requires one R-R and an arbitrary number of NS-NS fields appear only through the combination (59). The action (59) is complete action of O p -plane at order α ′2 for α 1 = −1/4. This action however has only one R-R field. The O p -plane action for only NS-NS fields have been found in [30,31]. The O p -plane action at order α ′2 has also sets of couplings involving two, three and four R-R fields. Each set of couplings may be found by the T-duality constraint up to an overall factor. Then the S-duality may be used to relate the overall factor of three R-R couplings to the couplings (59), and the two and four R-R couplings to the couplings found in [30,31]. It would be interesting to perform this calculation to find a gauge invariant action which is also invariant under the T-duality and the S-duality.
It would be also interesting to extend the calculation in this paper to find the D p -brane couplings at order α ′2 . A difficulty in this calculation is that each coupling in the effective action at order α ′2 may have an arbitrary number of B ab . They may also have world-volume derivative of this field, i.e., ∂ a B bc which does not appear in the field strength H abc . They are consistent with the gauge symmetry because the D-brane has also open string gauge field strength f ab and the combination B ab + f ab is invariant under the gauge transformation. However, the T-duality does not relate the massless closed string fields to the massless open string fields. Hence, in the T-duality constraint for the massless closed string fields, one may have couplings that are not gauge invariant. The reduction of those couplings then would not be invariant under the U(1) × U(1) gauge transformations. That makes problem in using the trick used in section 3 to keep only the U(1) × U(1) gauge invariant part of reduction of the Riemann curvature and other field strengths.
In finding the parameters in section 4, we have ignored some total derivative terms in the base space. If O-plane are at the fixed point of closed spacetime, then there is no boundary in the base space and the total derivative terms become zero by using the Stokes's theorem. However, if the spacetime has boundary, then the base space has boundary as well. Hence, the total derivative terms in the base space can not be ignored. They produce some boundary terms in the boundary of the base space [16]. In that case, one should consider some couplings at the boundary of O-plane. The boundary terms in the boundary of the base space should be cancelled by the T-duality of the couplings on the boundary of O-plane. This constraint may fix the couplings at the boundary of the O-plane. It would be interesting to find the boundary terms in the effective action of O-plane.
where c 1 , · · · c 46 are arbitrary parameters. There are 53 couplings for R-R field strength F (p+2) , i.e., where e 1 , · · · e 47 are arbitrary parameters. These couplings, however, are not independent because the world-volume indices must be 0, 1, · · · , p which have not been imposed in finding the above couplings. Writing the world volume indices explicitly as 0, 1, · · · , p, one finds only 30 couplings for F (p) , 20 couplings for F (p+2) are 19 couplings for the R-R field strength F (p+4) .