Reconstruction of Type II Supergravities via $O(d) \times O(d)$ Duality Invariants

We reconstruct type II supergravities by using building blocks of $O(d) \times O(d)$ invariants.These invariants are obtained by explicitly analyzing $O(d) \times O(d)$ transformations of 10 dimensional massless fields. Similar constructions are done by employing double field theory or generalized geometry, but we completed the reconstruction within the framework of the supergravities.


Introduction
Dualities among superstring theories play important roles to reveal both perturbative and non-perturbative aspects of superstring theories.Especially, type IIA superstring theory is related to type IIB superstring theory by T-duality, which interchanges Kaluza-Klein modes (KK modes) and winding modes of a compactified circle direction [1,2].In the low energy limit, massive modes in the type II superstring theories are decoupled, and the effective actions are well described by corresponding type II supergravity theories [3,4].The T-duality transformations of background massless fields are well-known as Buscher rule [5,6].
When the superstring theories are toroidally compactified on T d , the duality transformation can be generalized to O(d, d) duality [7,8].Actually it is argued in ref. [9] that, by assuming all fields depend only on a time coordinate, NS-NS sector in the low energy effective action, which consist of a graviton, a dilaton and Kalb-Ramond B field (B field), can be rewritten in manifestly O(d, d) invariant expression.And O(d, d) invariance of the NS-NS sector in general background was confirmed in refs.[10,11].Furthermore, it is also proven that O(d, d) invariance can be extended to all orders in α ′ corrections to the low energy effective action [12].
O(d, d) transformation of R-R sector has been investigated in refs.[13]- [20].One approach is to note that R-R potentials fill up a spinor representation of SO(d, d) duality transformation [13,14].The spinor representation of R-R potentials combined with B field was explicitly constructed when the compactified space was T 3 [15], and general case of T d compactification was completed in ref. [16].Another approach was done by Hassan in refs.[17]- [20], where the consistency of the duality transformation with local supersymmetry transformation is imposed.In this approach, the O(d, d) transformations of dilatinos and gravitinos are explicitly written in terms of 10 dimensional forms, and those of R-R potentials are derived in a bispinor form.In the type II superstring theories, the formulation of superspace which is compatible with T-duality was discussed in ref. [21], and inclusion of R-R fields and an application to AdS background were investigated in refs.[22,23].Generalization of the ref.[20] to non-abelian T-duality was done in refs.[24,25].
Although the type II supergravities possesses O(d, d) duality invariance, forms of the action are not manifestly invariant in terms of 10 dimensional fields.There are two formalism to improve this point.First one is a double field theory, which treats internal coordinates of winding modes and KK modes simultaneously [26,27,28].O(d, d) transformation is realized as a rotation among these 2d coordinates and fields are generalized to behave as tensors under this coordinate transformation.O(d, d) invariant forms of the type II supergravities are discussed in the framework of the double field theory in refs.[29,30,31].Second one is a generalized geometry, which treats tangent and cotangent bundles of compactified manifold on equal footing [32,33,34].Lie bracket of two vector fields are also modified to Courant bracket to incorporate B field transformation with general coordinate one.O(d, d) invariant forms of the type II supergravities are discussed in the framework of the generalized geometry in ref. [35].
The double field theory or the generalized geometry played important roles to reveal the O(d, d) invariant structure, however, it is not so clear to derive such structure within the framework of the type II supergravities.In this paper, we revisit O(d)×O(d) subgroup of the duality transformation discussed in the ref.[20] to construct O(d) × O(d) invariants within the framework of the type II supergravities.We review that O(d) × O(d) transformations of NS-NS fields and fermionic fields are completely written in terms of 10 dimensional fields, and construct O(d)×O(d) invariants by evaluating these.The actions of the type II supergravities are completely written by combinations of these building blocks, which are consistent with ones obtained in refs.[31,35].
This paper is organized as follows.In section 2, we review the O(d) × O(d) duality transformations of fields shown in the ref.[20].Especially we show that these transformations can be written by using 10 dimensional fields 1 .In section 3, we construct O(d) × O(d) duality invariants for NS-NS fields and fermionic ones.We also check that these duality invariants in the background of fundamental strings and wave solutions, or NS5-branes and KK monopoles.In section 4, we construct NS-NS bosonic terms in the type II supergravities by using the duality invariants.In section 5, we construct fermionic bilinear terms in the type II supergravities by duality invariants.Section 6 is devoted to conclusions and discussions.In Appendix A, we review the actions of the type II supergravities for NS-NS sector and fermionic bilinear terms.
and the O(d, d) transformation O for massless NS-NS fields is defined by [9] Dimensionally reduced dilaton field Φ − 1 4 log detg is invariant under the The case of S = R corresponds to a part of general linear coordinate transformation.From the eq.( 2), it is possible to extract duality transformations of dimensionally reduced fields.And these are gathered into duality transformations of original 10 dimensional fields.Below we summarize O(d) × O(d) transformations of fields in 10 dimensions [18].By introducing 10 × 10 matrices as the 10 dimensional inverse metric transforms as Since the duality invariant which includes the dilaton field is written by Φ − 1 4 log detG, the duality transformation of the dilaton field is given by The ± sign originates from actions to the world-sheet left and right moving modes, respectively.From the eq.( 5), it is possible to define O(d) × O(d) transformation of the vielbein as Notice that E ′M (±)A are related by local Lorenz transformation of Thus local Lorentz frame of the left moving sector is obtained by twisting that of the right moving sector by Λ A B .So invariants under local Lorentz transformation, which are constructed out of E ′M (+)A , can always be written in terms of E ′M (−)A .Since two kinds of vielbein can be used after the duality transformation, the 3-form field strength Here W ± M A B are connections defined by using torsionless spin connection Ω M A B as and the duality transformations are calculated as Notice that W ′± A (±)M B are constructed out of E ′M (±)A , respectively.Similarly Γ ±K M N are connections defined by using affine connection Γ K M N as and the duality transformations are derived as Since the vielbein is not used in the eq.( 12), there are no (±) subscription in the above.Next let us summarize duality transformations of gravitinos Ψ ±M and dilatinos λ ± .In ref. [20], these transformations are derived so as to be consistent with the local supersymmetry (60).It is easy to check this for the gravitino Ψ −M , and the result is To derive the above, we used Q N ±M ∂ N = ∂ M .This holds because the derivatives of fields with respect to the compact directions are zero.For the gravitino Ψ +M , the duality transformation Ψ ′ +M is defined by using E ′M (−)A and the susy transformation becomes where Γ A is a gamma matrix in 10 dimensions.In the above we ignored R-R fields, and used local Lorentz transformation to change The eq. ( 15) is compatible with the duality transformation if we define Finally we consider O(d) × O(d) duality transformations of dilatinos.As in the case of the gravitinos, the duality transformations are derived so as to be consistent with the local supersymmetry (60).
In the second equality, we used the eq.( 9) and employed the fifth line of the eq.( 27).Thus the duality transformation of the dilatino λ − is compatible with the local supersymmetry if we define As in the case of the gravitino, the duality transformation λ ′ + is defined by using E ′M (−)A .By taking into account the local Lorentz transformation, we obtain

O(d) × O(d) Duality Invariants
In this section we construct O(d) × O(d) duality invariants.In order to find these, let us prepare useful relations for Q M ± N .Q ± are defined in the 10 × 10 matrix notation as (4), and by noting S T S = R T R = 1, we obtain It is often useful to express the above as follows.
On the other hand, from the eq.( 4), Q + is written by By multiplying Q −1 ∓ from the left, we find Combining the eqs.( 21) and ( 23), we obtain a useful relation This relation is often used to construct O(d) × O(d) duality invariants.

Duality Invariants S ± ABC and T ± A for NS-NS Bosonic Fields
Now we construct duality invariants for NS-NS bosonic fields.O(d) × O(d) transformation of H ABC in 10 dimensions is evaluated as follows. where In the second line, we used the eq.( 24).Thus we find O(d) × O(d) duality invariant of the form Note that these do not behave like tensors under general coordinate transformation and E ′M (+) A is used for the + mode of the dual theory.This means that S ′± (±)ABC = S ± ABC .O(d) × O(d) transformation of the dilaton is given by the eq.( 6), and the derivative of that equation is evaluated as The eq. ( 24) is used in the 6th line, and where Notice that in the dual theory E ′M (+) A is used for the + mode, so we obtained Invariants which are similar to S ± ABC and T ± A are also constructed in the flux formulation of the double field theory [36].

Duality Invariants Θ ± for Fermionic Fields
where U − = 1 and U + is a spinor representation of local Lorentz transformation whose corresponding vector representation is given by Λ We used the eq.( 24) in the second line, and U + Γ A U −1 + = Λ −1A B Γ B in the 4th line.Thus we find duality invariants up to local Lorentz transformation U ± .
Notice that the dual theory is written by E ′M (−) A for Θ ± .This means Θ ′ (−)± = U ± Θ ± , which is different from S ± ABC and T ± A .Similar expressions to the above are also obtained in the framework of the double field theory [31] or generalized geometry [35].First we consider classical solutions of fundamental strings and waves.The solution of the fundamental strings is given by

Check of Duality Invariants for Classical Solutions
where r 2 = 8 i=1 (X i ) 2 .And nontrivial components of O(d)×O(d) invariants for this solution are evaluated as On the other hand, the dual solution of the wave along X 9 direction is given by And nontrivial components of O(d) × O(d) invariants for this solution are calculated as Here we used hats for local Lorentz indices.If we set h 1 = h w , we obtain Second we consider classical solutions of smeared NS5-branes and KK monopoles.The solution of the NS5-branes smeared along X 9 direction is given by where r 2 = i=6,7,8 (X i ) 2 .And nontrivial components of O(d) × O(d) invariants for this solution are evaluated as On the other hand, the dual solution of the KK monopoles is given by

And nontrivial components of O(d) × O(d) invariants for this solution become
Here we used hats for local Lorentz indices.If we set h 5 = h m , we obtain

Construction of NS-NS Bosonic Terms in Type II Supergravity via Duality Invariants
Let us construct NS-NS bosonic terms in the type II supergravities by using duality invariants.Building blocks are S ± ABC , T A and W ± M AB .The action consists of two derivative terms, so candidates are Next we calculate Ee −2Φ T A T A . 4Ee where In the 4th equality, we used Now we require invariance under general coordinate transformation.This means that HW ± and W ±2 terms should be removed by combining Ee −2Φ S ± ABC S ±ABC , Ee −2Φ T A T A and Ee −2Φ W ±ABC W ± ABC .This uniquely constrains the form of the combination up to overall factor and the result becomes

Construction of Fermionic Bilinear Terms in Type II Supergravities via Duality Invariants
Let us construct fermionic bilinear terms in the type II supergravities by using duality invariants.First we consider bilinear terms of the dilatinos.Since the duality invariant forms of the dilatinos are given by Θ ± , we would like to construct duality invariants which partially contain These are not duality invariants nor scalars under local Lorentz transformation.In order to recover the latter covariance, we add the connection S ± ABC as follows.
Here we used Θ ± Γ A Θ ± = 0 for Majorana fermions, and D M is a covariant derivative with respect to the connection of Ω M AB .In this case, Thus the terms in the eq.( 45) are scalars under local Lorentz transformation.Furthermore, these are O(d) × O(d) duality invariant as we show below.The dual theory is written by E ′M (−) A for the vielbein, and the dual of the above is written as In the 3rd equality, we used local Lorentz covariance for the + mode, such as . Thus the terms of the eq.( 45) are O(d, d) invariant.
Next we consider two derivative terms which consists of Ψ ±M and Θ ± .The duality invariants should partially contain These are not duality invariants nor scalars under local Lorentz transformation.In order to make scalars under local Lorentz transformation, we need to add the connection term to the above.
Furthermore, these are duality invariants as we show below.
Thus the terms of the eq.( 48) are O(d, d) invariant.Finally let us investigate two derivative terms which are bilinear of Majorana gravitinos.These should partially contain following terms.
These are not duality invariants nor scalars under local Lorentz transformation.In order to recover the latter covariance, we add connection terms S ± ABC and Γ ∓K M N as follows.
Note that These are scalars under local Lorentz transformation.The first two terms are similar to the eq.( 46), so the transformations under O(d) × O(d) are also similar.One difference is on the derivatives of Q −1 ∓ , which is written as On the other hand, the O(d) × O(d) transformations of the connections Γ ∓K M N are given by the eq.( 13), and the third term in the eq.( 51) transforms as In the second equality, we used Thus we see that the last term in the eq.( 52) is cancelled by the last term in the eq.( 53).The combinations of the eq.( 51) are O(d, d) invariant.
So far we constructed O(d, d) invariants of ( 45), (48) and (51).Then up to overall factor the Lagrangian is expressed as In the last equality, if we choose c 1 = −2 and c 2 = −1, it is possible to express the derivative of the Majorana gravitinos as field strengths of D [M Ψ ±N ] up to partial integral.Since this prescription is important to realize local supersymmetry, we employ these values.Then the O(d, d) invariant action of the fermionic bilinear is uniquely determined as Of course, a linear combination of these terms is consistent with the type II supergravities.Thus we showed that fermionic bilinears without R-R fields can be written in terms of the duality invariants within the framework of the type II supergravities.Invariant forms of fermionic bilinears with R-R fluxes are obtained in the framework of the double field theory [31] or generalized geometry [35].

Conclusion and Discussion
In As we have checked the duality invariants in the background of strings and wave solutions, or NS5-branes and KK monopoles, it is easy to apply to other nongeometric backgrounds [37]- [40].It is interesting to see corrections to the nongeometric background which was studied from the viewpoint of world-sheet instantons [41].It is also interesting to investigate βtwisted solutions of the double field theory [42] by evaluating O(d) × O(d) invariants in this paper.
Since we have constructed O(d)×O(d) duality invariants within the framework of the type II supergravities, it is natural to generalize these formulation to higher derivative corrections in the type II superstring theories.However, this is not a simple task and it is shown that higher derivative corrections in bosonic or heterotic string theory cannot be written in terms of generalized metric [43,44].We should take into account total derivative terms and field redefinitions which consist of dimensionally reduced fields.Constraint on R 2 terms via cosmological ansatz was investigated in ref. [45], was executed via T-duality in refs.[46], and was done via O(d, d) duality in ref. [47].In our formalism, the difficulty can be seen by duality transformation of the Riemann tensor (41), which is calculated as where If we consider R ± ABCD S ±ABE S ±CD E , which exists as a part of higher derivative terms in bosonic string theory, the duality transformation of this term contains ±2R ± ABN [C X N ∓D] S ±ABE S ±CD E .However, this cannot be cancelled by other terms even if we consider total derivatives and field redefinitions of 10 dimensional fields.
Although we should decompose the eq.( 56) in terms of dimensionally reduced fields, S ABC , T A and W ± µAB are invariant under O(d, d) transformations.Thus we should only take care of W ± αAB .It is also useful to consult a frame formalism of the double field theory [48].If we find nice structure on total derivatives and field redefinitions in terms of these fields, it will be possible to apply our O(d) × O(d) construction to higher derivative terms such as R 4 terms[49]- [55].supersymmetry are given by Again we ignored contributions of R-R fields.ǫ is a Majorana fermion and satisfy In the case of the type IIB supergravity, ǫ ± should satisfy

2
Brief Review of O(d) × O(d) Transformations In this section, we briefly review O(d) × O(d) transformations of massless fields in the type II supergravities.We denote the 10 dimensional spacetime indices as K, L, M, N, • • • .Non compact spacetime directions are labeled by µ, ν, • • • and compact d dimensions are done by α, β, • • • .On the other hand, local Lorentz indices are denoted as A, B, C, D, • • • .Non compact local Lorentz indices are labeled by i, j, • • • and those for compact d dimensions are done by a, b, • • • .NS-NS fields of the type II supergravities are the graviton G M N , the Kalb-Ramon field B M N and the dilaton Φ. Corresponding fields with compact spatial indices g αβ and B αβ are gathered into

O
(d) × O(d) transformations of the dilatinos in 10 dimensions are given by Since we have constructed O(d) × O(d) invariants, let us evaluate these values for classical solutions which exchange under T-duality.

)
Thus we construct NS-NS bosonic terms of the type II supergravities via O(d)× O(d) duality invariants.The Lagrangian is O(d, d) invariant since it behaves as a scalar under general coordinate transformation and invariant under a constant shift of B field.Notice that the dual theory for + mode is written in terms of E ′M (+)A , but it is possible to use local Lorentz transformation E ′M (+)A = E ′M (−)B Λ B A to write the + mode of the dual theory in terms of E ′M (−)A .
this paper, within the framework of the type II supergravities, we have constructed O(d) × O(d) duality invariants of the eqs.(26) (28) and (30) by examining O(d) × O(d) transformations of 3-form H field, dilaton and dilatino.These invariants are checked in the background of fundamental strings and wave solutions, or NS5-branes and KK monopoles.By using these duality invariants, we reconstructed the actions of type II supergravities in a manifestly O(d) × O(d) invariant form in section 4 and 5. Since these actions are also invariant under linear GL(d) transformation and shift of the B field, these are exactly O(d, d) invariant.As for the kinetic terms on R-R fields, SO(d, d) invariant construction was already discussed within the framework of the type II supergravities in the ref.[20].