aa r X i v : . [ h e p - t h ] A ug MIT-CTP-4288
August 2011
Massive Type II in Double Field Theory
Olaf Hohm and Seung Ki Kwak
Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridge, MA 02139, USA [email protected], sk [email protected]
Abstract
We provide an extension of the recently constructed double field theory formulation of thelow-energy limits of type II strings, in which the RR fields can depend simultaneously onthe 10-dimensional space-time coordinates and linearly on the dual winding coordinates.For the special case that only the RR one-form of type IIA carries such a dependence,we obtain the massive deformation of type IIA supergravity due to Romans. For T-dualconfigurations we obtain a ‘massive’ but non-covariant formulation of type IIB, in whichthe 10-dimensional diffeomorphism symmetry is deformed by the mass parameter.
Introduction
Double field theory is an approach to make T-duality manifest at the level of effective space-time theories by doubling the coordinates. It introduces the usual ‘momentum’ (space-time)coordinates x i together with new ‘winding’ coordinates ˜ x i and a covariant constraint that locallyeliminates half of the coordinates. Originally accomplished for the bosonic string [1–3], morerecently it has been extended to heterotic [4] and to type II superstrings [5, 6]. (For earlierand related work see [7–16].) In particular, the type II double field theory provides a unifieddescription of the massless type IIA and type IIB theories. In this paper we will show thata minimal extension of this theory exists that also contains massive deformations [17]. (Forearlier results on massive type IIA within ‘duality-covariant’ frameworks see [18–20].)The massive extension of type IIA supergravity due to Romans [17] can be motivated asfollows. If one introduces for each RR p -form the dual 8 − p form, type IIA contains all oddforms with p = 1 , . . . ,
7. We can also introduce a 9-form potential, but imposing the standardfield equations sets its field strength F (10) = dC (9) to a constant and so a 9-form carries nopropagating degrees of freedom. We can think of massive type IIA as obtained by choosing thisintegration constant to be non-zero and equal to the mass parameter m . In the resulting theory, m enters as a cosmological constant and deforms the gauge transformations corresponding tothe NS-NS b -field such that the RR 1-form transforms with a St¨uckelberg shift symmetry. Itdoes not admit a maximally symmetric vacuum, but its most symmetric solution is the D8brane solution that features 9-dimensional Poincar´e invariance [21].The type II double field theory of [5, 6] is formulated in terms of a Majorana-Weyl spinor ofthe ‘T-duality group’ O (10 , ∗ b F (10) = − b F (0) ≡ b F (0) due to the absenceof ‘( − − C ( − and then set m = F (0) = dC ( − , ashas been done in [22], but so far it has been unclear how to find a mathematically satisfactoryinterpretation of such objects. In this note we will show that a non-trivial 0-form field strength(and thus a mass parameter) is naturally included in the type II double field theory by assumingthat the RR 1-form depends linearly on the winding coordinates, C (1) ( x, ˜ x ) = C i ( x ) dx i + m ˜ x dx , (1.2)where C i and all other fields depend only on the 10-dimensional coordinates. We will see thatthe second term in (1.2) effectively acts as a ( − ∂ M ∂ M A = η MN ∂ M ∂ N A = 0 , ∂ M A ∂ M B = 0 , η MN = ! , (1.3)1or all fields and gauge parameters A, B . Here,
M, N, . . . are fundamental O (10 ,
10) indices,with invariant metric η MN , and the derivative ∂ M = ( ˜ ∂ i , ∂ i ) combines the partial momentumand winding derivatives. This constraint is necessary for gauge invariance of the action andclosure of the gauge algebra. In its weak form, which requires ∂ M ∂ M = 2 ˜ ∂ i ∂ i to annihilate allfields and parameters, it is a direct consequence of the level matching condition of closed stringtheory, and it allows for field configurations such as (1.2) that depend locally both on x and ˜ x .The double field theory constructions completed so far, however, impose the stronger form thatrequires also all products of fields and parameters to be annihilated by ∂ M ∂ M , correspondingto the second equation in (1.3). In this form the constraint implies that locally all fields dependonly on half of the coordinates, and so (1.2) violates the strong constraint. Remarkably, as wewill show here, the gauge transformations can be reformulated on the RR fields so that thestrong constraint can be relaxed. It cannot be relaxed to the weak constraint as formulatedabove, but it is sufficient for the ansatz (1.2) to be consistent. In particular, this formulationguarantees that in the action and gauge transformations the linear ˜ x dependence drops out,such that the resulting theory has a conventional 10-dimensional interpretation.This paper is organized as follows. In sec. 2 we briefly review the type II double field theory of[5,6], and we give a rewriting of the gauge transformations that requires only a weaker constraintin a sense to be made precise. In sec. 3 we evaluate the double field theory for (1.2) and showthat it reduces to the democratic formulation of massive type IIA. We then discuss T-dualityfor massive type II by evaluating the double field theory for a coordinate dependence in whichone of the space-time coordinates is interchanged with a winding coordinate, corresponding toa single T-duality inversion. In the massless case this maps type IIA to type IIB, in agreementwith their equivalence upon compactification on a circle. In the massive case, however, it mapstype IIA into a non-covariant ‘massive’ version of type IIB. In this formulation, type IIB hasonly manifest 9-dimensional covariance but is fully diffeomorphism invariant, with the 10thdiffeomorphism being deformed by the mass parameter. We start by reviewing the double field theory formulation of massless type II theories. TheNS-NS fields are the space-time metric g ij , the Kalb-Ramond 2-form b ij and the dilaton φ . Themetric and b -field are encoded in the generalized metric H MN = g ij − g ik b kj b ik g kj g ij − b ik g kl b lj ! , (2.1)which is an element of SO (10 , H T = H and transforming covari-antly under O (10 ,
10) according to its index structure. Moreover, we introduce an O (10 , e − d = √ ge − φ , where g = | det g ij | . We can think of H as thefundamental field and view (2.1) as a particular parametrization, but in [5] we argued that weshould rather view the fundamental field as an hermitian element of the two-fold covering group2pin(10 , S , we require S = S † , S ∈ Spin(10 , , (2.2)and define the generalized metric via the group homomorphism ρ : Spin(10 , → SO (10 , H = ρ ( S ). With (2.2) we infer H T = ρ ( S † ) = ρ ( S ) = H , and thus, as required, H is asymmetric group element which generally can be parametrized as in (2.1).In the democratic formulation to be employed here, the RR fields consist of differentialforms of all odd (even) degrees for type IIA (IIB) and can be encoded in a Majorana-Weylspinor χ of O (10 , , (cid:8) Γ M , Γ N (cid:9) = 2 η MN . (2.3)Due to the off-diagonal form of the O (10 ,
10) metric in (1.3), this algebra implies that ψ i ≡ √ i , ψ i ≡ √ i , (2.4)with ( ψ i ) † = ψ i , satisfy the canonical anti-commutation relations of fermionic raising andlowering operators ψ i and ψ i , respectively, { ψ i , ψ j } = δ ij , { ψ i , ψ j } = 0 , { ψ i , ψ j } = 0 . (2.5)Thus, the spinor representation can be constructed by introducing a Clifford vacuum | i , sat-isfying ψ i | i = 0 for all i , and acting with the raising operators ψ i . A spinor is then given by ageneral state χ = X p =0 p ! C i ...i p ψ i . . . ψ i p | i , (2.6)with coefficients that are fully antisymmetric tensors and that will be identified with the RR p -forms C ( p ) . A spinor satisfying a chirality condition consists either of only odd or only evenforms, and in the following we assume that χ has a fixed chirality. The Dirac operator /∂ ≡ √ M ∂ M = ψ i ∂ i + ψ i ˜ ∂ i , (2.7)acts naturally on spinors and can be viewed as the O (10 ,
10) invariant extension of the exteriorderivative. First, for ˜ ∂ = 0 it only acts via differentiating with respect to x i and increasing theform degree by one. Second, it squares to zero, /∂ = 12 Γ M Γ N ∂ M ∂ N = 12 η MN ∂ M ∂ N = 0 , (2.8)using (2.3) and the constraint (1.3).The type II double field theory, whose independent fields are S , χ and d , is defined by theaction S = Z d x d ˜ x (cid:16) e − d R ( H , d ) + 14 ( /∂χ ) † S /∂χ (cid:17) , (2.9)supplemented by the self-duality constraint /∂χ = −K /∂χ , K ≡ C − S , (2.10)3here C denotess the charge conjugation matrix of Spin(10 , O (10 ,
10) invariant scalar R ( H , d ) see eq. (4.24) in [3].Let us next review the symmetries of this theory. It has a global T-duality invariance underSpin + (10 , δ λ χ = /∂λ , (2.11)where λ is a spinor of Spin(10 ,
10) with a fixed chirality opposite to the one of χ , such that odd(even) p -forms transform with even (odd) ( p − /∂ = 0. The double field theory is also invariant undera ‘generalized diffeomorphism’ symmetry spanned by a parameter ξ M = ( ˜ ξ i , ξ i ) that combinesthe usual diffeomorphism parameter ξ i and the b -field gauge parameter ˜ ξ i into a fundamental O (10 ,
10) vector. This gauge symmetry acts on the fundamental fields as δ ξ (cid:0) e − d (cid:1) = ∂ M (cid:0) ξ M e − d (cid:1) ,δ ξ K = ξ M ∂ M K + 12 (cid:2) Γ MN , K (cid:3) ∂ M ξ N ,δ ξ χ = ξ M ∂ M χ + 12 ∂ M ξ N Γ M Γ N χ , (2.12)where Γ MN = [Γ M , Γ N ], and we have written the gauge variation of S in terms of K defined in(2.10). The gauge invariance under ξ M transformations is non-trivial and requires the strongconstraint (1.3). We stress that, in contrast, the identity (2.8) and thus the invariance under λ transformations requires only the weak constraint, i.e., that ∂ M ∂ M annihilates χ and its gaugeparameter, but not necessarily their products, which will be used below.Next we discuss the double field theory evaluated for fields depending only on x i , i.e.,setting the winding derivatives to zero, ˜ ∂ i = 0. In order to compare the resulting theory withthe conventional formulation, we need to choose a parametrization of S in terms of g and b , asfor H in (2.1). We do so by decomposing (2.1) H = b ! g − g ! − b ! ≡ h Tb h g − h b , (2.13)and then introducing spin representatives for each of the SO (10 ,
10) elements on the right-handside. We then set S = S H ≡ S † b S − g S b = e b ij ψ i ψ j S − g e − b ij ψ i ψ j , (2.14)where we introduced the spin representative S b of h b in terms of fermionic oscillators [5]. Theexplicit form of the spin representative S g of h g can be found in [5]. By construction, S = S H projects under the group homomorphism ρ to H in SO (10 , ∂ i = 0 the Dirac operator /∂ acts like the exteriorderivative and so /∂χ reduces to the conventional field strengths of the RR potentials in (2.6), /∂χ (cid:12)(cid:12)(cid:12) ˜ ∂ =0 = X p p ! ∂ j C i ...i p ψ j ψ i · · · ψ i p | i = X p p + 1)! F i ...i p +1 ψ i · · · ψ i p +1 | i , (2.15)4here we introduced in the last equation the components of the ( p + 1)–form field strength, F ( p +1) = dC ( p ) . (2.16)In the action (2.9) the exponentials of b in (2.14) lead to the modified field strength [5, 6] b F = e − b (2) ∧ F = e − b (2) ∧ dC , (2.17)where we use a notation that combines all p -forms into a formal sum. Without repeating thederivation, we recall that the S g factor in (2.14) effectively arranges the proper contractionswith the space-time metric g in the action, so that the RR part of (2.9) reads [5, 6] S RR = − Z d x √ g X p p ! g i j · · · g i p j p b F i ...i p b F j ...j p = 14 X p Z b F ( p ) ∧ ∗ b F ( p ) . (2.18)Similarly, the self-duality constraint (2.10) reduces to the conventional duality relations b F ( p ) = − ( − p ( p +1) ∗ b F (10 − p ) . (2.19)Thus, for the RR sector the double field theory action reduces to the sum of kinetic terms forall p -forms, which is supplemented by duality relations. This is the ‘democratic formulation’originally introduced in [23], and which is on-shell equivalent to the conventional formulationof type IIA for odd forms or type IIB for even forms.Let us finally review the gauge transformations for ˜ ∂ i = 0. The gauge variations (2.11),with parameter λ = X p p ! λ i ...i p ψ i · · · ψ i p | i , (2.20)reduce to δ λ χ = ψ j ∂ j λ = X p p − ∂ [ i λ i ...i p ] ψ i · · · ψ i p | i , (2.21)which amounts to the standard abelian gauge symmetry of the RR p -forms, δ λ C = dλ . (2.22)The gauge symmetry (2.12) parameterized by ξ M = ( ˜ ξ i , ξ i ) gives for ξ i δ ξ χ = (cid:16) ξ j ∂ j + ∂ j ξ k ψ j ψ k (cid:17) X p p ! C i ...i p ψ i · · · ψ i p | i , (2.23)which by use of the oscillator algebra (2.5) implies δ ξ C i ...i p = ξ j ∂ j C i ...i p + p ∂ [ i ξ j C | j | i ...i p ] ≡ L ξ C i ...i p . (2.24)This is the conventional diffeomorphism symmetry, acting infinitesimally via the Lie deriva-tive. From (2.12) it also follows that the C ( p ) transform non-trivially under the b -field gaugeparameter ˜ ξ i , δ ˜ ξ χ = ∂ k ˜ ξ l ψ k ψ l χ = X p p ! ∂ [ i ˜ ξ i C i ...i p +2 ] ψ i · · · ψ i p +2 | i , (2.25)which implies δ ˜ ξ C = d ˜ ξ ∧ C . (2.26)This means that the C ( p ) are redefinitions by the b -field of the more conventional RR fields,which are invariant under ˜ ξ i . These are exactly the expected gauge symmetries for masslesstype II theories. 5 .2 Reformulation of gauge symmetries The gauge invariance of the type II double field theory requires for ξ M transformations thestrong form of the constraint (1.3), but for λ transformations only the weak constraint. Here,we will perform a change of basis for the gauge parameters such that, for the RR sector, the ξ M transformations are consistent with a weaker form of the constraint.We start by rewriting the ξ M gauge transformation of χ as follows δ ξ χ = ξ M ∂ M χ + 12 ∂ M ξ N Γ M Γ N χ = ξ M ∂ M χ −
12 Γ M Γ N ξ N ∂ M χ + 12 Γ M ∂ M (cid:0) ξ N Γ N χ (cid:1) . (2.27)The last term is of the form of a field-dependent λ gauge transformation /∂λ and can thereforebe ignored. We then use the Clifford algebra in the second term, δ ξ χ = ξ M ∂ M χ − (cid:0) η MN − Γ N Γ M (cid:1) ξ N ∂ M χ = 12 Γ N Γ M ξ N ∂ M χ . (2.28)Using the ‘slash’ notation (2.7), /∂ = 1 √ M ∂ M , /ξ = 1 √ M ξ M , (2.29)we finally get δ ξ χ = /ξ /∂χ , (2.30)which is the form of the ξ M gauge transformations we will use from now on.We will show that, starting from (2.30), gauge invariance of the RR action and closure ofthe gauge algebra uses only the constraint η MN ∂ M ∂ N A = ∂ M ∂ M A = 0 , A = (cid:8) χ, λ, ξ M (cid:9) . (2.31)In particular, for this computation we do not need to use ∂ M A∂ M B = 0. This observation doesnot imply, however, that the RR sector is ‘weakly constrained’ in the sense that fields but nottheir products need to satisfy the constraint. In fact, (2.30) is not a consistent transformationrule assuming that χ and ξ are weakly constrained. Before discussing this in more detail, weinvestigate some consequences of the form (2.30) of the gauge transformations.The original gauge transformations have the property that a gauge parameter of the form ξ M = ∂ M Θ is ‘trivial’ in that it generates no gauge transformation. After the above redefinition,this statement is modified. We compute δ ∂ Θ χ = 12 Γ N Γ M ∂ N Θ ∂ M χ = 12 Γ N ∂ N (cid:0) Θ Γ M ∂ M χ (cid:1) , (2.32)assuming only the weaker form (2.31) of the constraint. Thus, the gauge variation (2.32) takesthe form of a field-dependent λ gauge transformation, δ ∂ Θ χ = /∂λ , λ = Θ /∂χ . (2.33)Therefore, the statement that ξ M = ∂ M Θ leads to a trivial gauge transformation leaving thefields invariant has to be relaxed to the statement that it leaves the fields invariant up to a λ auge transformation , but it has the advantage that in this weaker form only the constraint(2.31) is required.We compute next the gauge variation of /∂χ under (2.30), which is needed in order to verifygauge invariance, δ ξ (cid:0) /∂χ (cid:1) = /∂ (cid:0) /ξ /∂χ (cid:1) = /∂/ξ /∂χ + 12 √ M Γ N Γ P ξ N ∂ M ∂ P χ = /∂/ξ /∂χ + 12 √ (cid:0) η MN − Γ N Γ M (cid:1) Γ P ξ N ∂ M ∂ P χ . (2.34)The last term contains Γ ( M Γ P ) = η MP and therefore vanishes by (2.31), while the second termreduces to ξ M ∂ M /∂χ . In total we have δ ξ (cid:0) /∂χ (cid:1) = ξ M ∂ M /∂χ + /∂/ξ /∂χ . (2.35)This result agrees with the variation under the original form of the gauge transformationsdetermined in [5] (as it should be, because the modification is a λ gauge transformation thatleaves /∂χ invariant), but in the original derivation the strong constraint was used. As the proofof gauge invariance of the action and the self-duality constraint given in [5] requires only thetransformation rule (2.35), we conclude that this proof uses only the weaker constraint (2.31).Let us verify that also closure of the gauge transformations on χ requires only this weakerconstraint. First, for the modified form of the gauge transformations there is no non-vanishingcommutator between λ and ξ gauge transformations because /∂χ is λ -invariant. Thus, it remainsto verify closure of the ξ M transformations, for which we find (cid:2) δ ξ , δ ξ (cid:3) χ = δ ξ χ + δ λ χ . (2.36)Here, ξ M = (cid:2) ξ , ξ (cid:3) M C = ξ N ∂ N ξ M − ξ N ∂ M ξ N − (1 ↔ , (2.37)which is given by the ‘C-bracket’ that characterizes the closure of ξ M transformations on theNS-NS fields [1, 2], and λ = − (cid:0) /ξ /ξ − /ξ /ξ (cid:1) /∂χ . (2.38)The verification of (2.36) is a straightforward though somewhat tedious exercise in gammamatrix algebra, which we defer to the appendix. The computation makes repeated use ofthe constraint (1.3), but only in its relaxed form (2.31). Thus, on the RR field χ all gaugesymmetries close using only this weaker constraint.We close this section by computing the form of these redefined ξ M gauge transformations(2.30) for ˜ ∂ i = 0. For the diffeomorphism parameter ξ i we find δ ξ χ = ξ i ψ i ψ j ∂ j χ = X p p ! ξ i ∂ j C i ··· i p ψ i ψ j ψ i · · · ψ i p | i . (2.39)Using the oscillator algebra (2.5) to simplify this, we obtain δ ξ C i ··· i p = ( p + 1) ξ j ∂ [ j C i ··· i p ] = ξ j F ji ··· i p . (2.40)7or the b -field gauge parameter ˜ ξ i one obtains δ ˜ ξ χ = ˜ ξ i ψ i ψ j ∂ j χ = X p p ! ˜ ξ i ∂ j C i ...i p ψ i ψ j ψ i · · · ψ i p | i , (2.41)from which we read off δ ˜ ξ C = ˜ ξ ∧ F . (2.42)The diffeomorphism symmetry in the form (2.40) is sometimes referred to as ‘improveddiffeomorphisms’. They can be introduced for any p -form gauge field by adding to the familiardiffeomorphism symmetry (2.24) a field-dependent gauge transformation with ( p − λ i ...i p − = − ξ j C ji ...i p − . (2.43)Similarly, (2.42) is obtained from the original ˜ ξ transformation (2.26) by adding an abelian gaugetransformation with parameter λ = − ˜ ξ ∧ C . Thus, the redefinition of the gauge transformationsleading to (2.30) is precisely the double field theory analogue of the improved diffeomorphismsin conventional gauge theories. In this form the gauge field appears only under a derivative,which will be instrumental for the generalization we discuss next. In the previous section we have seen that the proof of gauge invariance and closure of the gaugealgebra uses only the weaker constraint (2.31) for the RR sector. Naively, this would allow forfield configurations like χ ( x, ˜ x ) = χ ( x ) + χ (˜ x ) , (3.1)where χ , are arbitrary functions of their arguments, and similarly for the gauge parameters.However, as mentioned above, there is a subtlety, because the gauge variations (2.30) are notconsistent assuming only the weak constraint. In fact, δ ξ χ on the left-hand side should satisfythe constraint, but with χ and ξ being weakly constrained their product on the right-handside in general does not satisfy the constraint. Rather, one should introduce a projector thatrestricts to the part satisfying the weak constraint [1], while our computation above did notkeep track of these projectors. After the insertion of projectors, the gauge invariance of theaction and closure of the gauge algebra does not follow from our computation (and is mostlikely not true). Moreover, the RR fields interact with the NS-NS sector that is still stronglyconstrained, and so it is presumably inconsistent to have a weakly constrained RR sector. Thus,a complete relaxation of the strong constraint must await a resolution of this problem for theNS-NS sector. However, if we only assume the function χ in (3.1) to depend linearly on ˜ x , theresulting gauge variations and field equations are independent of ˜ x , and therefore the constraintis satisfied without insertion of projectors. (In particular, the energy-momentum tensor of theRR fields depends only on /∂χ [5] and is thereby independent of ˜ x .) An ansatz with linear ˜ x dependence is therefore consistent, and we will investigate its consequences in what follows.8e will show that the type II double field theory defined by (2.9) and (2.10) leads tomassive type IIA if we assume that the RR spinor χ depends on the 10-dimensional space-timecoordinates and, in its 1-form part, also linearly on a winding coordinate. We thus write χ ( x, ˜ x ) = (cid:16) X p p ! C i ...i p ( x ) ψ i . . . ψ i p + m ˜ x ψ (cid:17) | i , (3.2)where we assume that χ is of negative chirality such that the sum extends only over odd p .Here we have singled out a particular (winding) coordinate direction, but we stress that thischoice is immaterial for the final result: we could have chosen any linear combination of the ˜ x i ,which would merely amount to a rescaling of the mass parameter m . Let us also note that itwould be consistent to allow for a linear ˜ x dependence in other p -form parts, both in χ and inits gauge parameter λ . We will comment on this more general case below.Let us next evaluate the field strength /∂χ for (3.2). In contrast to (2.15), the term ψ i ˜ ∂ i in /∂ acts now non-trivially, /∂χ = X p p ! ∂ j C i ...i p ψ j ψ i · · · ψ i p | i + ψ j ˜ ∂ j ( m ˜ x ) ψ | i = X p p + 1)! ( p + 1) ∂ [ i C i ...i p +1 ] ψ i · · · ψ i p +1 | i + m | i≡ X p p + 1)! ( F m ) i ...i p +1 ψ i · · · ψ i p +1 | i , (3.3)where we used the oscillator algebra (2.5). We observe that the non-trivial action of ψ i ˜ ∂ i leadsto a reduction of the form degree such that the ‘1-form potential’ precisely leads to a non-vanishing 0-form field strength or, in other words, that the ˜ x dependent part acts effectivelylike a ‘( − m -deformed field strengths defined in the last line of (3.3) then read F (0) m = m , F ( p +1) m = F ( p +1) = dC ( p ) for p ≥ . (3.4)In the action the modified field strengths (2.17) enter, which are now deformed according to(3.4), b F m = e − b (2) ∧ ( dC + m ) . (3.5)This reads explicitly b F (0) m = m b F (2) m = F (2) − mb (2) b F (4) m = F (4) − b (2) ∧ F (2) + 12 mb (2) ∧ b (2) , etc . (3.6)These are precisely the m -deformed field strengths appearing in massive type IIA, see, e.g., [21].We turn now to the gauge symmetries acting on (3.2), starting with the λ -transformations(2.11). In analogy to (3.2) it is natural to allow here also for a linear ˜ x dependence in the0-form part of λ , but such a contribution will be annihilated by /∂ due to ψ i | i = 0. We note,however, that a linear ˜ x dependence in the higher-form components of λ can lead to a rigid λ gauge transformations are unchanged compared to themassless case (2.22). The ξ M transformation (2.30) evaluated for the diffeomorphism parameter ξ i yields no new contribution since δ ξ i χ (cid:12)(cid:12)(cid:12) ∂ i =0 = ξ i ψ i ψ j ˜ ∂ j χ = 0 , (3.7)due to the action of two annihilation operators ψ i on (3.2). Thus, the diffeomorphism symmetryis given by (2.40), as for m = 0. Finally, the gauge transformation of the b -field gauge parameter˜ ξ i receives a non-trivial modification, δ ˜ ξ i χ (cid:12)(cid:12)(cid:12) ∂ i =0 = ˜ ξ i ψ i ψ j ˜ ∂ j χ = m ˜ ξ i ψ i ψ ψ | i = m ˜ ξ i ψ i | i . (3.8)Together with the gauge transformation (2.42) for m = 0 we thus obtain δ ˜ ξ C = ˜ ξ ∧ dC + m ˜ ξ . (3.9)Therefore, for m = 0 the RR 1-form C (1) transforms with a St¨uckelberg shift symmetry underthe b -field gauge transformations, which is precisely the expected result for massive type IIA [21].The modified field strengths b F m are manifestly invariant under the λ gauge transformations.The invariance under ˜ ξ transformations can be easily verified with δ ˜ ξ b (2) = d ˜ ξ , δ ˜ ξ b F m = δ ˜ ξ (cid:0) e − b (2) ∧ ( dC + m ) (cid:1) = − d ˜ ξ ∧ b F m + e − b (2) ∧ d (cid:0) ˜ ξ ∧ dC + m ˜ ξ (cid:1) = − d ˜ ξ ∧ b F m + e − b (2) ∧ d ˜ ξ ∧ ( dC + m ) = − d ˜ ξ ∧ b F m + d ˜ ξ ∧ b F m = 0 . (3.10)Let us now consider the double field theory action and duality relations (2.9) and (2.10),evaluated for (3.2), and compare with the dynamics of massive type IIA. As in (2.18), the actionreduces to the sum of kinetic terms, but here for the modified field strengths (3.5), L RR = 14 X p =0 b F ( p ) m ∧ ∗ b F ( p ) m = 14 X p ≥ b F ( p ) m ∧ ∗ b F ( p ) m + 14 m ∗ . (3.11)The action contains now also the 0-form field strength, which contributes a cosmological termproportional to m , as made explicit in the second equation. Moreover, we can use theSt¨uckelberg gauge symmetry (3.9) with parameter ˜ ξ to set C (1) = 0. From the second equationin (3.6) we then infer that the kinetic term for C (1) reduces to a mass term for the b -field. Thus,the b -field becomes massive by ‘eating’ the RR 1-form.The self-duality constraint (2.10) reduces to the same duality relations as in (2.19), againwith all field strengths being m -deformed, b F ( p ) m = − ( − p ( p +1) ∗ b F (10 − p ) m . (3.12)This democratic formulation is equivalent to the conventional formulation of massive type IIA.In the following we compare the two formulations in a little more detail.10he RR action of massive type IIA in the standard formulation is given by [17, 21] S RR = 12 Z (cid:16) b F (2) m ∧ ∗ b F (2) m + b F (4) m ∧ ∗ b F (4) m + m ∗ (cid:17) + 12 Z (cid:16) b (2) ( dC (3) ) − ( b (2) ) dC (1) dC (3) + 13 ( b (2) ) ( dC (1) ) + 13 m ( b (2) ) dC (3) − m ( b (2) ) dC (1) + 120 m ( b (2) ) (cid:17) , (3.13)where for simplicity we have omitted all wedge products between forms in the topological Chern-Simons terms S CS in the second and third line. We note in passing that this Chern-Simonsaction simplifies significantly if we formally introduce a ( − C ( − and then define b A = e − b (2) ∧ (cid:0) C + C ( − (cid:1) , (3.14)where C still represents the formal sum of all (odd) p -forms with p ≥
1. The Chern-Simonsaction can then simply be written as S CS = 12 Z b (2) ∧ d b A (3) ∧ d b A (3) . (3.15)More precisely, expanding (3.15) according to (3.14), the resulting action can be written, upto total derivatives, such that C ( − enters only under an exterior derivative, and then setting m = dC ( − reproduces precisely the Chern-Simons terms in (3.13). Formally, this drasticsimplification can be understood as a consequence of the b -field gauge transformations (3.9),which we rewrite here as δ ˜ ξ C = ˜ ξ ∧ d (cid:0) C + C ( − (cid:1) = d ˜ ξ ∧ (cid:0) C + C ( − (cid:1) − d (cid:0) ˜ ξ ∧ (cid:0) C + C ( − (cid:1)(cid:1) . (3.16)The last term takes the form of a field-dependent λ gauge transformation and can thus beignored. The b A defined in (3.14) is then ˜ ξ gauge invariant, δ ˜ ξ b A = − d ˜ ξ ∧ e − b (2) ∧ (cid:0) C + C ( − (cid:1) + e − b (2) ∧ (cid:0) d ˜ ξ ∧ (cid:0) C + C ( − (cid:1)(cid:1) = 0 , (3.17)where we have taken C ( − to be gauge invariant. From this we infer that (3.15) is the onlyterm invariant under ˜ ξ gauge transformations (up to a boundary term). Note that we couldhave included the ( − p -forms, in which case the gaugetransformations would be formally as in the massless case.We have verified the exact equivalence between the equations of motion following from (3.13)and those derived by varying (3.11) and then supplementing them by the duality relations(3.12). For the Einstein equations this is easy to see because the Chern-Simons terms that arepresent in the conventional formulation do not contribute to the variation of the metric. Theenergy-momentum tensor then agrees for both formulations owing to the relative factor of between the kinetic terms in (3.11) and (3.13), which compensates for the doubling of fields in This action differs from eq. (2.8) of [21] in certain numerical factors, which is due to different conventionsregarding differential forms. Moreover, there is a mismatch of a relative factor of between kinetic and Chern-Simons terms, but (3.13) is consistent with [17]. p -forms the on-shell equivalence is aconsequence of the Bianchi identities d b F ( p ) m = − H (3) ∧ b F ( p − m , (3.18)following from (3.5). More precisely, the duality relations yield the second-order field equationsas integrability conditions of d = 0, including the required source terms originating fromthe Chern-Simons terms in the conventional formulation. Thus, the double field theory leadsprecisely to massive type IIA. We discuss now the double field theory evaluated for fields depending on coordinates that resultfrom the 10-dimensional space-time coordinates x i by a T-duality inversion. The O (10 ,
10) in-variance of the constraint (1.3) implies that fields resulting by an O (10 ,
10) transformation fromfields depending only on the x i (thereby satisfying the constraint) also satisfy the constraint.For instance, we may perform a single T-duality inversion in one direction, which exchanges a‘momentum coordinate’ x i with the corresponding ‘winding coordinate’ ˜ x i . The double fieldtheory evaluated for this field configuration then reduces to the T-dual theory. If it reduces totype IIA in one ‘T-duality frame’, it reduces to type IIB in the other frame, when expressedin the right T-dual field variables [5]. The mapping of (massless) type IIA into type IIB underT-duality can therefore be discussed without reference to dimensional reduction, while in theusual approach this relation is inferred from the equivalence of type IIA and type IIB uponreduction on a circle [24].Our task is now to see how this generalizes in the massive case. The usual point of viewis as follows [21]. Massive type IIA reduced on a circle leads to a massive N = 2 theory innine dimensions, but there is no corresponding massive deformation of type IIB that could leadto the same nine-dimensional theory upon standard reduction. Rather, to identify the properT-duality rules one has to perform a Scherk-Schwarz reduction [25] of massless type IIB, whichintroduces a mass parameter and leads to the same massive N = 2 theory in nine dimensions.In contrast, in double field theory the T-dual theory is identified without any dimensionalreduction, as we discussed above, and so the puzzle arises what the T-dual to massive type IIAis if there is no massive type IIB in ten dimensions.In order to address this issue let us analyze the double field theory evaluated for fields inwhich one space-time coordinate, say x , is replaced by the corresponding winding coordinate.We split the coordinates as x i = ( x µ , x ), µ = 1 , . . . ,
9, and replace (3.2) by the ansatz χ ( x, ˜ x ) = (cid:16) X p p ! C i ...i p ( x µ , ˜ x ) ψ i · · · ψ i p + m ˜ x ψ (cid:17) | i , (3.19)where again the sum extends over all odd p . In the massless case the double field theory reducesto type IIB, which can be made manifest by performing a field redefinition that takes the formof a T-duality inversion in the 10th direction [5]. This T-duality transformation acts on the Here we assume that x is a space-like direction, g , >
0. For T-dualities along time-like directions thedual theories are the so-called type II ∗ theories [5, 6], which have a reversed sign for the RR kinetic terms [26].Similarly, the double field theory discussed here contains also a massive type IIA ∗ .
12R spinor via the spin representative S = ψ + ψ , i.e., we define χ ′ = S χ = (cid:16) X p p ! C ′ i ...i p ψ i · · · ψ i p + m ˜ x ( ψ + ψ ) ψ (cid:17) | i , (3.20)where in the first term we introduced redefined variables denoted by C ′ . As S is linear inthe fermionic oscillators the sum extends now over all even p . Specifically, one finds (compareeq. (6.41) in [6]) C ′ i ...i p = ( C µ ...µ p if i = 10, i = µ , . . . , i p = µ p C µ ...µ p if i = µ , . . . , i p = µ p . (3.21)Thus, the dual field variables are obtained by adding or deleting the special index, therebymapping odd forms into even forms, as required for the transition from type IIA to type IIB.By performing this field redefinition (and renaming the coordinates) one infers that evaluatingthe theory for fields depending on x µ and ˜ x is equivalent to evaluating the theory for fieldsdepending on x i , but with the opposite chirality for the spinor, i.e., replacing odd forms byeven forms. (See sec. 6.2 in [6] for more details.) Now, in the massive case we have to take intoaccount the second term in (3.20), which reduces to m ˜ x ψ ψ . Thus, our task is to evaluatethe double field theory for χ ( x, ˜ x ) = (cid:16) X p p ! C i ...i p ( x ) ψ i · · · ψ i p + m ˜ x ψ ψ (cid:17) | i , (3.22)dropping the primes from now on. In other words, we have to evaluate the double field theoryfor a field configuration in which the 2-form part depends now linearly on ˜ x , (cid:0) χ ( x, ˜ x ) (cid:12)(cid:12) − form (cid:1) ij = C ij ( x ) + 2 m ˜ x δ [ i δ j ]1 , (3.23)with all other fields still depending only on the 10-dimensional space-time coordinates.We start by computing the field strength F = /∂χ = F m =0 − ψ ˜ ∂ ( m ˜ x ) ψ ψ | i = F m =0 − mψ | i . (3.24)Therefore, the field strength of the RR 0-form C (0) gets modified in the 10th component, F (1) = dC (0) − mdx ⇔ F i = ∂ i C (0) − mδ i , (3.25)while all other field strengths F ( p ) , p = 1, remain unchanged. The ‘hatted’ field strength (2.17)then receives corresponding modifications, b F = e − b (2) ∧ (cid:0) dC − mdx (cid:1) , (3.26)and thus in components b F (3) = F (3) − b (2) ∧ dC (0) + mb (2) ∧ dx , etc . (3.27)The dynamics is described by the same action (2.18) and duality relations (2.19) as before, butwith all field strengths replaced by their m -deformed version (3.26).13his theory breaks manifest 10-dimensional covariance in that the 10th coordinate is treatedon a different footing in (3.25). We observe, however, that this theory can be obtained fromstandard (covariant) type IIB by performing the redefinition C (0) → C (0) − mx , (3.28)as is apparent from (3.25). Thus, the ‘deformation’ induced by the m -dependent 2-form contri-bution in (3.22) can be absorbed into a redefinition of the lower RR form C (0) , and thereforethe obtained theory is nothing but standard type IIB after a somewhat peculiar (non-covariant)redefinition. For this reason we do not introduce a new symbol for the ‘deformed’ field strengths.In order to understand the consequences of the non-covariance let us inspect the gaugesymmetries. As above, the λ gauge transformations are unchanged compared to the masslesscase. The gauge transformations (2.30) parametrized by ξ M applied to (3.22) give δ ξ χ = (cid:0) ψ i ˜ ξ i + ψ i ξ i (cid:1) /∂χ = δ ξ χ (cid:12)(cid:12)(cid:12) m =0 − m ( ˜ ξ i ψ i ψ + ξ i ψ i ψ ) | i = δ ξ χ (cid:12)(cid:12)(cid:12) m =0 − m ( ˜ ξ µ ψ µ ψ + ξ ) | i . (3.29)We read off the m -deformed gauge transformations which are modified on C µ , δ ˜ ξ C µ = 2 ˜ ξ [ µ F ,m =0 − m ˜ ξ µ = 2 ˜ ξ [ µ F , (3.30)and on C (0) δ ξ C (0) = ξ j ∂ j C (0) − mξ = ξ j F j , (3.31)where we used (3.25) for both equations in the last step. Thus, the nine-component parameter˜ ξ µ acts as a St¨uckelberg symmetry on the off-diagonal RR 2-form components, while the 10thdiffeomorphism parameter ξ acts as a St¨uckelberg symmetry on the RR 0-form. The fieldstrength of C µ read off from (3.27), b F µν = 2 ∂ [ µ C ν ]10 + mb µν + ∂ C µν − b µν ∂ C (0) − b µ ∂ ν ] C (0) , (3.32)is invariant under the ˜ ξ µ shift symmetry. Moreover, (3.25) is invariant under ξ , i.e., the theoryis diffeomorphism invariant under x → x − ξ ( x ) and (3.31), δ ξ F (1) = − mdξ + mdξ = 0 . (3.33)Thus, despite the non-covariant formulation that treats the 10th direction on a different footing,the theory is still fully diffeomorphism invariant, as it should be in view of the fact that it resultsfrom standard type IIB by the redefinition (3.28). Since this invariance under non-covariantdiffeomorphisms is somewhat unconventional, let us also verify this for the component formgiven in (3.25), δ ξ F i = ∂ i (cid:0) ξ j ∂ j C (0) − mξ (cid:1) = ξ j ∂ j (cid:0) ∂ i C (0) (cid:1) + ∂ i ξ j ∂ j C (0) − m∂ i ξ = ξ j ∂ j F i + ∂ i ξ j (cid:0) ∂ j C (0) − mδ j (cid:1) = ξ j ∂ j F i + ∂ i ξ j F j . (3.34)Thus, the m -deformed field strength transforms under the m -deformed diffeomorphisms (3.31)with the usual Lie derivative of a 1-form field strength. Therefore, the action and dualityrelations build with this field strength are diffeomorphism invariant.14o summarize, we have identified the 10-dimensional theory that is the T-dual to massivetype IIA and that can be seen as a ‘massive’ formulation of type IIB. It is unconventional inthat the 10-dimensional diffeomorphism symmetry is not realized in the usual way, but non-linearly in the 10th direction. This is, however, analogous to the deformation of the gaugetransformation of C (1) under the b -field gauge parameter in massive type IIA, and since thediffeomorphisms and b -field gauge symmetries are on the same footing in double field theorythis result is not surprising.Let us now discuss the physical content. We can choose a gauge for the ˜ ξ µ St¨uckelbergsymmetries by setting C µ = 0. From (3.32) we then infer that their kinetic terms give massterms for the 9-dimensional components of the b -field, rendering these components massive.This is analogous to massive type IIA, but in the latter case the full 10-dimensional b -fieldbecomes massive, carrying 36 massive degrees of freedom, while here only the 9-dimensionalcomponents become massive, carrying 28 massive degrees of freedom. It turns out that the8 missing degrees of freedom are carried instead by the Kaluza-Klein vector field. In orderto see this, let us perform a Kaluza-Klein decomposition of the kinetic term involving C (0) (but we stress that we are not performing a reduction in that the fields still depend on all 10coordinates). The standard Kaluza-Klein decomposition of the (inverse) metric reads g ij = γ µν − A µ − A ν ℓ − + A ρ A ρ ! , (3.35)where γ µν denotes the 9-dimensional metric, A µ is the Kaluza-Klein vector and ℓ the Kaluza-Klein scalar. If we choose a gauge for the ξ St¨uckelberg symmetry by setting C (0) = 0, weinfer with (3.25) that the relevant term in the Lagrangian reads L = − √ g g ij F i F j = − √ gg , F F = − m √ γ √ ℓ (cid:0) ℓ − + A µ A µ (cid:1) . (3.36)Therefore, the Kaluza-Klein vector receives a mass term and so becomes massive by ‘eating’the RR scalar C (0) , thus carrying 8 massive degrees of freedom.We have to point out that the above analysis of the physical content was somewhat naive.In fact, one may wonder why this theory, if obtained from massless type IIB by the mereredefinition (3.28), exhibits a spectrum that is rather different from the usual physical contentof type IIB, e.g., with (parts of) the b -field becoming massive and a cosmological term in (3.36).The point is that such a classification of the masses of various fields is only meaningful withrespect to a particular background. Type IIB admits a 10-dimensional Minkowski solution,with all field strengths zero in the background, and it is with respect to this background thatthe b -field is massless. Now, after the redefinition (3.28) the theory of course still admits thesame Minkowski vacuum, but now we have to switch on a ‘background flux’ in order to realizethis solution, h g ij i = η ij , h dC (0) i = mdx , (3.37)because only then we have h b F i = 0 in the Einstein equations, as follows with (3.25). Aroundthis background, the b -field is still massless.Thus, there is no conflict of our above analysis of ‘massive’ type IIB with the usual way typeIIB is presented. The presence of massive fields just means that the background space-time we15onsider is not flat space, but rather a background that is appropriate for the comparison tothe T-dual massive type IIA. In fact, massive type IIA does not admit a Minkowski (or AdS)vacuum, but instead the D8-brane solution that is invariant under the 9-dimensional Poincar´egroup corresponding to its world-volume [21]. The T-dual configuration is the D7-brane solutionof type IIB, which is only invariant under the 8-dimensional Poincar´e group [21], and the aboveanalysis has to be understood with respect to such a background.Let us close this section by comparing our result with the usual story that relates massivetype IIA to the Scherk-Schwarz reduction of massless type IIB [21, 22]. In Scherk-Schwarzreduction one allows some fields to depend non-trivially on the internal coordinates in such away that this dependence drops out in the effective lower-dimensional theory. For the Scherk-Schwarz reduction of type IIB to nine dimensions relevant for T-duality, the Kaluza-Klein ansatzallows for a linear x dependence for the RR scalar C (0) , C (0) ( x µ , x ) = c (0) ( x µ ) − mx , (3.38)where c (0) denotes the nine-dimensional field. For all other fields the ansatz is as for circlereductions, i.e., the fields are simply assumed to be independent of x . In the resulting actionthe dependence on x drops out, leaving a massive deformation of the usual circle reductionof type IIB.Instead of this Scherk-Schwarz reduction one may first perform the redefinition (3.28) andthen employ a standard reduction, as is apparent by comparing (3.38) with (3.28). We concludethat the Scherk-Schwarz reduction of massless type IIB gives the same 9-dimensional theoryas the conventional reduction of the ‘massive’ formulation of type IIB. Thus, our results areconsistent with [21, 22], and the formulation of type IIB that appears naturally in double fieldtheory is already adapted to the Scherk-Schwarz reduction. In this paper we have shown that the type II double field theory defined by (2.9) and (2.10)can be extended by slightly relaxing the constraint (1.3) such that the RR fields may dependsimultaneously on all 10-dimensional space-time coordinates and linearly on the winding coor-dinates. In case that only the RR 1-form carries such a dependence, the double field theoryreduces precisely to the massive type IIA theory. We have shown that the T-dual configura-tion corresponds to the case that the RR 2-form (3.23) of type IIB carries such a dependence.This gives rise to a ‘massive’ version of type IIB, whose circle reduction to nine dimensionsyields the same theory as the Scherk-Schwarz reduction of conventional type IIB. This massiveformulation of type IIB is still invariant under 10-dimensional diffeomorphisms, with the 10thdiffeomorphism being deformed by the mass parameter.Here we have only considered a non-trivial ˜ x dependence for the RR 1-form of type IIA and,in the T-dual situation, for the RR 2-form of type IIB. It is also consistent with the relaxedconstraint to have a linear ˜ x dependence for all higher RR forms C ( p ) . Such an ansatz wouldlead to a multiple parameter family of ‘massive’ and non-covariant type II theories, but as for We thank Eric Bergshoeff for discussions on this point. C ( p − , as in (3.28). The only exception is the RR 1-form which isdistinguished because it leads to a covariant massive deformation in that it merely deforms the0-form field strength by the scalar mass parameter. This deformation cannot be absorbed into aredefinition, precisely because there is no ‘( − − − N = 1 supersymmetry [7]. Moreover, the recent work [12]presents a rewriting of the N = 2 fermionic terms and supersymmetry variations in the contextof generalized geometry. While this does not yet prove the existence of an N = 2 supersym-metric extension of double field theory, since the coordinates are not doubled in generalizedgeometry, it provides strong evidence. These matters are currently under investigation. Acknowledgments
We would like to thank E. Bergshoeff, C. Hull, and P. Townsend for helpful discussions andcorrespondence. We are especially indebted to B. Zwiebach for numerous comments and forcarefully reading the manuscript.This work is supported by the U.S. Department of Energy (DoE) under the cooperativeresearch agreement DE-FG02-05ER41360. The work of OH is supported by the DFG – TheGerman Science Foundation, and the work of SK is supported in part by a Samsung Scholarship.
A Proof of the gauge algebra
Here we prove that the gauge transformations (2.30) close according to (2.36), using only theweaker constraint (2.31). We compute (cid:2) δ ξ , δ ξ (cid:3) χ = δ ξ (cid:0) Γ N Γ M ξ N ∂ M χ (cid:1) − (1 ↔ N Γ M Γ P Γ Q ξ N ∂ M (cid:0) ξ P ∂ Q χ (cid:1) − (1 ↔ . (A.1)Let us work out structures with ∂χ and ∂ χ separately. Consider18 Γ N Γ M Γ P Γ Q ξ N ξ P ∂ M ∂ Q χ = 18 (cid:0) Γ P Γ Q Γ N Γ M + [Γ N Γ M , Γ P Γ Q ] (cid:1) ξ N ξ P ∂ M ∂ Q χ = 18 (cid:0) Γ P Γ Q Γ N Γ M − (cid:0) η MP Γ QN + η NQ Γ P M (cid:1)(cid:1) ξ N ξ P ∂ M ∂ Q χ , (A.2)17here we used the constraint (2.31) for χ and the symmetry in M, Q . If we antisymmetrize in1 ↔ M, Q , we infer that the term on the left-hand side in the firstline is minus the first term in the second line. Similarly, upon antisymmetrization 1 ↔
2, thefinal two terms in the second line are equal. Rearranging terms, we thus get14 Γ N Γ M Γ P Γ Q ξ N ξ P ∂ M ∂ Q χ − (1 ↔
2) = 12 Γ M Γ P ξ N ξ P ∂ M ∂ N χ − (1 ↔ M ∂ M (cid:0) ξ N Γ P ξ P ∂ N χ (cid:1) −
12 Γ M Γ P ∂ M (cid:0) ξ N ξ P (cid:1) ∂ N χ − (1 ↔ , (A.3)where we used in the first equality that a term with η MP is zero by the antisymmetry 1 ↔ λ gauge transformation.We turn now to the terms proportional to ∂χ in (A.1),14 Γ N Γ M Γ P Γ Q ξ N ∂ M ξ P ∂ Q χ = 12 Γ P Γ Q ξ M ∂ M ξ P ∂ Q χ −
14 Γ M Γ N Γ P Γ Q ξ N ∂ M ξ P ∂ Q χ , (A.4)where we used the Clifford algebra for Γ N Γ M . The first term on the right-hand side takes theform of a ξ M gauge transformation (2.30). The second term can be re-written as −
14 Γ M Γ N Γ P Γ Q ξ N ∂ M ξ P ∂ Q χ = −
14 Γ M (cid:0) Γ Q Γ N Γ P + [Γ N Γ P , Γ Q ] (cid:1) ξ N ∂ M ξ P ∂ Q χ = −
14 Γ M Γ Q Γ N Γ P ξ N ∂ M ξ P ∂ Q χ + 12 Γ M (cid:0) η QN Γ P − η QP Γ N (cid:1) ξ N ∂ M ξ P ∂ Q χ . (A.5)Antisymmetrizing 1 ↔
2, the last term in the second line gives12 Γ M Γ P ξ N ∂ M ξ P ∂ N χ −
12 Γ M Γ N ξ N ∂ M ξ P ∂ P χ − (1 ↔ M Γ P ∂ M (cid:0) ξ N ξ P (cid:1) ∂ N χ − (1 ↔ , (A.6)which cancels against the same structure in (A.3). The first term in the second line of (A.5),antisymmetrized in 1 ↔
2, can be simplified as follows −
14 Γ M Γ Q (cid:0) η NP + Γ NP (cid:1)(cid:0) ξ N ∂ M ξ P − ξ N ∂ M ξ P (cid:1) ∂ Q χ = −
14 Γ M Γ Q Γ NP ∂ M (cid:0) ξ N ξ P (cid:1) ∂ Q χ −
14 Γ M Γ Q (cid:0) ξ N ∂ M ξ N − ξ N ∂ M ξ N (cid:1) ∂ Q χ . (A.7)The second term is of the form of a ξ M gauge transformation. Commuting gamma matricesand using the weak constraint, the first term can be rewritten as a λ gauge transformation, −
14 Γ M ∂ M h(cid:0) Γ NP Γ Q + 2 (cid:0) η QN Γ P − η QP Γ N (cid:1)(cid:1) ξ N ξ P ∂ Q χ i = − /∂ h(cid:0) /ξ /ξ − /ξ /ξ (cid:1) /∂χ i −
12 Γ M ∂ M h(cid:0) ξ N Γ P ξ P − ξ N Γ P ξ P (cid:1) ∂ N χ i . (A.8)The second term in here cancels against the the first term in (A.3).Summarizing, the surviving structures are the ξ M gauge transformations in (A.4) and (A.7)and the λ transformation in (A.8), combining into (cid:2) δ ξ , δ ξ (cid:3) χ = 12 Γ P Γ Q (cid:16) ξ M ∂ M ξ P − ξ M ∂ P ξ M − (1 ↔ (cid:17) ∂ Q χ + /∂λ , (A.9)with the λ parameter (2.38). Thus, the gauge algebra closes as stated in (2.36).18 eferences [1] C. Hull, B. Zwiebach, “Double Field Theory,” JHEP , 099 (2009). [arXiv:0904.4664[hep-th]],“The Gauge algebra of double field theory and Courant brackets,” JHEP , 090 (2009).[arXiv:0908.1792 [hep-th]].[2] O. Hohm, C. Hull and B. Zwiebach, “Background independent action for double fieldtheory,” JHEP (2010) 016 [arXiv:1003.5027 [hep-th]].[3] O. Hohm, C. Hull and B. Zwiebach, “Generalized metric formulation of double field the-ory,” JHEP (2010) 008 [arXiv:1006.4823 [hep-th]].[4] O. Hohm, S. K. Kwak, “Double Field Theory Formulation of Heterotic Strings,” JHEP , 096 (2011). [arXiv:1103.2136 [hep-th]].[5] O. Hohm, S. K. Kwak, B. Zwiebach, “Unification of Type II Strings and T-duality,”[arXiv:1106.5452 [hep-th]].[6] O. Hohm, S. K. Kwak, B. Zwiebach, “Double Field Theory of Type II Strings,”[arXiv:1107.0008 [hep-th]].[7] W. Siegel, “Superspace duality in low-energy superstrings,” Phys. Rev. D , 2826 (1993)[arXiv:hep-th/9305073],“Two vierbein formalism for string inspired axionic gravity,” Phys. Rev. D , 5453 (1993)[arXiv:hep-th/9302036].[8] A. A. Tseytlin, “Duality Symmetric Formulation Of String World Sheet Dynamics,” Phys.Lett. B , 163 (1990); “Duality Symmetric Closed String Theory And Interacting ChiralScalars,” Nucl. Phys. B , 395 (1991).[9] S. K. Kwak, “Invariances and Equations of Motion in Double Field Theory,” JHEP (2010) 047 [arXiv:1008.2746 [hep-th]],O. Hohm, S. K. Kwak, “Frame-like Geometry of Double Field Theory,” J. Phys. A A44 ,085404 (2011). [arXiv:1011.4101 [hep-th]],O. Hohm, “T-duality versus Gauge Symmetry,” arXiv:1101.3484 [hep-th],O. Hohm, “On factorizations in perturbative quantum gravity,” JHEP , 103 (2011).[arXiv:1103.0032 [hep-th]].[10] P. West, “ E , generalised space-time and IIA string theory,” Phys. Lett. B , 403 (2011)[arXiv:1009.2624 [hep-th]].A. Rocen and P. West, “E11, generalised space-time and IIA string theory: the R-R sector,”arXiv:1012.2744 [hep-th].[11] D. C. Thompson, “Duality Invariance: From M-theory to Double Field Theory,”[arXiv:1106.4036 [hep-th]].[12] A. Coimbra, C. Strickland-Constable, D. Waldram, “Supergravity as Generalised Geome-try I: Type II Theories,” [arXiv:1107.1733 [hep-th]].1913] D. S. Berman and M. J. Perry, “Generalized Geometry and M theory,” arXiv:1008.1763[hep-th].D. S. Berman, H. Godazgar and M. J. Perry, “SO(5,5) duality in M-theory and generalizedgeometry,” Phys. Lett. B , 65 (2011) [arXiv:1103.5733 [hep-th]].[14] I. Jeon, K. Lee and J. H. Park, “Differential geometry with a projection: Application todouble field theory,” JHEP (2011) 014, arXiv:1011.1324 [hep-th],“Double field formulation of Yang-Mills theory,” arXiv:1102.0419 [hep-th],“Stringy differential geometry, beyond Riemann,” arXiv:1105.6294 [hep-th].[15] N. B. Copland, “Connecting T-duality invariant theories,” [arXiv:1106.1888 [hep-th]].[16] D. Andriot, M. Larfors, D. Lust, P. Patalong, “A ten-dimensional action for non-geometricfluxes,” [arXiv:1106.4015 [hep-th]].[17] L. J. Romans, “Massive N=2a Supergravity in Ten-Dimensions,” Phys. Lett. B169 , 374(1986).[18] I. Schnakenburg and P. C. West, “Massive IIA supergravity as a nonlinear realization,”Phys. Lett. B , 137 (2002) [arXiv:hep-th/0204207].[19] A. Kleinschmidt and H. Nicolai, “E(10) and SO(9,9) invariant supergravity,” JHEP ,041 (2004) [arXiv:hep-th/0407101].[20] M. Henneaux, E. Jamsin, A. Kleinschmidt and D. Persson, “On the E10/Massive Type IIASupergravity Correspondence,” Phys. Rev. D (2009) 045008 [arXiv:0811.4358 [hep-th]].[21] E. Bergshoeff, M. de Roo, M. B. Green, G. Papadopoulos, P. K. Townsend, “Duality oftype II 7 branes and 8 branes,” Nucl. Phys. B470 , 113-135 (1996). [hep-th/9601150].[22] I. V. Lavrinenko, H. Lu, C. N. Pope, K. S. Stelle, “Superdualities, brane tensions andmassive IIA / IIB duality,” Nucl. Phys.
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