Stringy Unification of Type IIA and IIB Supergravities under N=2 D=10 Supersymmetric Double Field Theory

To the full order in fermions, we construct D=10 type II supersymmetric double field theory. We spell the precise N=2 supersymmetry transformation rules as for 32 supercharges. The constructed action unifies type IIA and IIB supergravities in a manifestly covariant manner with respect to O(10,10) T-duality and a pair of local Lorentz groups, or Spin(1,9) \times Spin(9,1), besides the usual general covariance of supergravities or the generalized diffeomorphism. While the theory is unique, the solutions are twofold. Type IIA and IIB supergravities are identified as two different types of solutions rather than two different theories.


Introduction
Strings perceive spacetime in a different way than particles do through Riemannian geometry. While the fundamental object in Riemannian geometry is the metric, string theory puts the Kalb-Ramond B-field and a scalar dilaton on an equal footing along with the metric, since they form a multiplet of T-duality [1][2][3], a genuine stringy property which is not present in ordinary particle theory.
Although type IIA and IIB supergravities provide low energy effective descriptions of closed superstrings, once formulated within the Riemannian setup, they appear unable to capture the full stringy structure like T-duality or to explain the appearance of enhanced symmetries after dimensional reductions [4,5]. String theory seems to urge us to look for a novel mathematical framework, such as Generalized Geometry [6][7][8] or Double Field Theory (DFT) [9][10][11][12] (see also [13,14] for relevant pioneering works).
While generalized geometry combines tangent and cotangent spaces giving a geometric meaning to the Bfield [15,16], DFT doubles the spacetime dimension, from D to D + D in order to manifest the O(D, D) T-duality group structure [13,14,17,18]. With an additional requirement of so called strong constraint or section condition, DFT reduces to a known string theory effective action in D-dimension. The section condition means that all the DFT-fields live on a D-dimensional null hyperplane such that, the O(D, D) invariant d'Alembertian operator is trivial acting on arbitrary fields as well as their products, DFT unifies the B-field gauge symmetry and the diffeomorphism, as both are generated by generalized Lie derivative [6,19] (see also [20] for finite transformations), Further, recent study of the Scherk-Schwarz reduction in DFT has shown that, by relaxing the section condition (1) -and hence in a truly non-Riemannian set up-one may derive all the known gauged supergravities in lower than ten dimensions [21][22][23][24][25]. This seems to indicate the potential power of DFT and motivates further explorations.
In this work, we construct N = 2 D = 10 supersymmetric double field theory (SDFT). We carry out the construction employing genuine SDFT field-variables which are subject to the section condition (1) and differ a priori from Riemannian, or supergravity variables. For example, ordinary zehnbeins and various form-fields will never enter in our construction. We tend to believe that the usage of the genuine SDFT field-variables is quite crucial and it essentially ensures the following properties of the final results.
• The supersymmetric completion is fulfilled to the full order in fermions.
• Further, N = 2 D = 10 SDFT unifies type IIA and IIB supergravities: while the theory is unique, the solutions are twofold, type IIA and type IIB.

Field Content
We postulate the fundamental fields of type II SDFT to be strictly, from [32][33][34][35][36][37], We wish to stress that, for the sake of the full covariance and the (relatively) compact way of full order supersymmetric completion, it is crucial to set the fundamental fields to be precisely those above. Although some of them may be parametrized in terms of Riemannian zehnbeins and form-fields, the parametrization is not unique, may render "non-geometric" interpretations, and will certainly becloud the whole symmetry structure listed in Table 1.
Firstly for the NS-NS sector, the DFT-dilaton, d, gives rise to a scalar density with weight one, e −2d [10]. The DFT-vielbeins, V Ap ,V Ap , satisfy the following four defining properties [34,35]: In particular, they generate a pair of orthogonal and complete projections, satisfying The DFT-vielbeins, V Ap ,V Ap , are O(D, D) vectors as the index structure indicates. They are the only field variables in (3) which are O(D, D) non-singlet. As a solution to (4), they can be parametrized in terms of ordinary zehnbeins and B-field, in various ways up to O(D, D) rotations and field redefinitions [37]. Yet, in order to maintain the clear manifestation of the O(D, D) covariance, it is necessary to work with the parametrization-independent and O(D, D) covariant DFT-vielbeins, i.e. V Ap andV Ap , rather than the Riemannian variables, i.e. ordinary zehnbeins and B-field.

Derivatives
Another essential ingredient is so called master semi-covariant derivative from [35], which contains generically three kinds of connections: Γ A for the DFT-diffeomorphism or the generalized Lie derivative (2), Φ A for Spin (1,9) andΦ A for Spin(9, 1) local Lorentz symmetries. Contracted with the projections (6) or the DFT-vielbeins properly, it can produce various fully covariant derivatives, and hence the name, 'semi-covariant' [34,35,37]. By definition, the master derivative (9) is required to be compatible with all the constants in Table 1 ("metrics" and gamma matrices), and further to annihilate the whole NS-NS sector, The connections are then related to each other through where we put Especially, as the DFT analogy of the Riemannian Christoffel connection, the torsionless connection, Γ 0 A , can be uniquely singled out [34,37] (c.f. [39]): such that a generic torsionful DFT-diffeomorphism connection assumes the following general form: where ∆ C[pq] and∆ C[pq] correspond to torsions. Explicitly we shall employ four different kinds of torsions: (21) for the curvature, (22) for the fermionic kinetic terms, (23) for the supersymmetry, and (29) for the equations of motion.
The R-R field strength, F αᾱ , is defined from [37], where D 0 + corresponds to one of the two fully covariant and nilpotent differential operators, D 0 ± , which are set by the torsionless connection (12), and may act on an arbitrary Pin(1, 9) × Pin(9, 1) bi-fundamental field, T αβ : where we put 1

Curvature
The final ingredient we shall employ is the semi-covariant DFT-curvature, S ABCD , from [34], which is defined through the standard (yet never-covariant) field strength of the DFT-diffeomorphism connection (13), Again, with the help of the projections, it can produce fully covariant curvatures, such as Ricci (28) and scalar, 1 Strictly speaking, due to the presence of γ (11) in (15), the R-R field strength, F = D 0 + C, is covariant -up to the flipping of the chirality-with respect to, not Pin(1, 9) × Pin(9, 1) but Spin(1, 9) × Pin(9, 1). For the opposite equivalent choice, see eq.(2.25) in [37].

The Lagrangian and Supersymmetry
The Lagrangian of N = 2 D = 10 SDFT we construct in this work is the following, As they are contracted with the DFT-vielbeins properly, each term in the Lagrangian is fully covariant with respect to O(10, 10) T-duality, Spin(1, 9) × Spin(9, 1) local Lorentz symmetry and the DFTdiffeomorphism. With the charge conjugation of the R-R field strength,F =C −1 (19) is over the Spin(1, 9) spinorial indices.
The N = 2 supersymmetry transformation rules are

Torsions
Presenting our main results above, (19) and (20), we have organized all the higher order fermionic terms into various torsions. Firstly, with (16), the DFT-curvature, S ABCD , in the Lagrangian is given by the connection, Secondly, the master derivatives in the fermionic kinetic terms are twofold: D ⋆ A for the unprimed fermions and D ′⋆ A for the primed fermions. They are set by the following twin connections, Similarly, for the supersymmetry transformations (20), we takê The connection, Γ ABC given in (21) and also appearing in (22), (23), has been fixed by requiring the 1.5 formalism to work, see (25). The additional parts of the connections in (22) and (23) are then uniquely determined from the full order supersymmetric completion.

Self-duality and Equations of Motion
The type II SDFT Lagrangian (19) is pseudo: An additional self-duality relation needs to be imposed by hand on the R-R field strength combined with fermions, Under arbitrary infinitesimal variations of all the fields, the Lagrangian transforms, up to total derivatives, Each line then corresponds to the equation of motion of N = 2 D = 10 SDFT. In particular, the on-shell Lagrangian vanishes, L Type II = 0, and the DFT-generalization of the Einstein equation follows The self-duality (24) implies the equation of motion for the R-R potential, D 0 −F− = 0. Further, as in the N = 1 SDFT [36], the 1.5 formalism, 'δΓ ABC × 0', nicely works here with the connection spelled in (21).
Writing (25), we set some shorthand notations: For the arbitrary variations of the fields, and for the Ricci curvature, We also set the derivatives,D A ,D ′ A appearing in (25), bỹ which are designed to serve as common connections for all the equations of motion, see Appendix A.
Under the N = 2 supersymmetry (20), disregarding total derivatives, the Lagrangian transforms concisely, This verifies, to the full order in fermions, the supersymmetric invariance of the type II SDFT action modulo the self-duality (24), see Appendix B for details. For a nontrivial consistency check, the supersymmetric variation of the self-duality relation (24) is, to the full order precisely, closed by the equations of motion for fermions, especially the gravitinos (25),

Unification
As stressed before, one of the characteristic features in our construction of N = 2 D = 10 SDFT is the usage of the covariant fundamental fields, identified in (3). However, the relation to an ordinary supergravity can be established only after we solve the defining algebraic relations of the DFT-vielbeins (4) and parametrize the solution in terms of zehnbeins and B-field: Up to O(10, 10) rotations and field redefinitions, the generic solution reads [34,37] where e µ p andē νp are two copies of zehnbeins which must constitute a common spacetime metric, e µ p e ν q η pq = −ē µpēνqηpq = g µν .
Namely, the chirality remains the same if det(e −1ē ) = +1, while it changes the sign if det(e −1ē ) = −1. Therefore, it depends on each specific background or each individual solution of the theory whether the chirality changes or not. That is to say, formulated in terms of the covariant fields, i.e. V Ap ,V Ap , C αᾱ , etc. the N = 2 D = 10 SDFT is simply a chiral theory with respect to the pair of local Lorentz groups. All the possible chirality choices are equivalent and hence the theory is unique. We may safely put c ≡ c ′ ≡ +1 without loss of generality. However, the theory contains two 'types' of solutions. All the solutions are classified into two groups, cc ′ det(e −1ē ) = +1 : type IIA , Conversely, making full use of the above Pin(9, 1) rotation, any solution in type IIA and type IIB supergravities can be mapped to a solution of N = 2 D = 10 SDFT of fixed chirality e.g. c ≡ c ′ ≡ +1. The single unique N = 2 D = 10 SDFT unifies type IIA and IIB supergravities.
The diagonal "gauge" fixing (36) inevitably modifies the O(10, 10) T-duality transformation rule to call for a compensating Pin(9, 1) local Lorentz rotation [37], such that the fermions and the R-R sector are no longer O(10, 10) singlets. In particular, the R-R sector can be mapped to the O(10, 10) spinor in [28][29][30][31]. Moreover, the modified O(10, 10) T-duality transformation, or more precisely the compensating Pin(9, 1) local Lorentz rotation, may flip the chirality of the theory, resulting in the usual exchange of IIA and IIB.
However, a priori T-duality is not a Noether symmetry. It becomes so only if it acts on an isometry direction. Hence, as is well known, within the supergravity setup the equivalence between IIA and IIB can be established only when the background admits an isometry. This is compared to the 'background independent' unification of the two supergravities by N = 2 D = 10 SDFT, discussed in this work.
The Appendices contain some details of the computations for (25) and (30).

A Variation of the Lagrangian under arbitrary transformations of fields
For arbitrary variations of fields, the identities below hold either strictly (' = ') or up to total derivatives and the section condition (' ≃ ').
For the double-vielbein, generic (torsionful) connection and curvature, Hence, for the R-R sector of the Lagrangian, we obtain For the fermionic kinetic terms, from (A.1), we have (A.6) Here we let the connections assume the following generic forms: It is easy to check that, a ′ 1 and a ′ 3 decouple from the fermionic kinetic terms (A.6), and only the linear combination, a ′ 2 − 1 2 a ′ 4 alone is relevant among the four primed coefficients, {a ′ 1 , a ′ 2 , a ′ 3 , a ′ 4 }. Without loss of generality, henceforth we put We proceed to compute the variations of Γ ⋆ ABC and Γ ′⋆ ABC (A.7), for which we first note (A.9) Yet, with (A.8) taken, we just need ′qγp ψ ′rψq γ pqr ψq + δρ − 1 4 δV Bpρ γ Bp 1 4 γ rst ρ(−a 1 + 1 16 a 2 )ψpγ rst ψp and similarly (A.11) The variation of the fermionic kinetic terms (A.6) now assumes the desired expression: 12) and the Lagrangian transforms up to total derivatives as δL Type II ≃ −2δd × L Type II +δΓ ABC × 0 Here we set genericallỹ and of which the coefficients must satisfy the following nine constraints, A particularly simple solution is given by Specifically, for Γ ⋆ ABC and Γ ′⋆ ABC given in (22) as we achieve (29), (A.21) Alternatively, in a similar fashion to Ref. [36], we might set (A.22) That is to say, there are various ways of absorbing the higher order fermionic terms into the torsions as long as the constraint (A.17) is satisfied. In this paper, we choose (A.21) and hence (29) such that, the entire equations of motion (25) can be written in terms of only two kind of torsions: one for the unprimed fermions and the other for the primed fermions.

B N = 2 supersymmetric invariance of the action
Here, we sketch our verification of the N = 2 supersymmetric invariance of the action as in (30), order by order in fermions. We substitute the N = 2 supersymmetry transformation rules (20) into (25) and organize the supersymmetric variation of the Lagrangian as δ ε L Type II = δ ε L [1] Type II + δ ε L [3] Type II + δ ε L [5] Type II , where δ ε L [1] Type II , δ ε L [3] Type II and δ ε L [5] Type II denote respectively the linear, cubic and quintic order terms in fermions which are DFT-dilatinos and gravitinos.
First of all, we focus on the linear order terms which decompose into four parts: where, disregarding the total derivative terms, we have 3) We show, up to the level matching section constraint (1), each of them vanishes except the last one, We first note Then, due to the identities [37], we obtain (c.f. [26,43]) These simplify ∆ ρ as and finally from the identity [34,37], Then, from the identity [34,37], we verify ∆ ψ ≃ 0.

(B.12)
Hence, from the chirality of the fermions and the nilpotent property [37], we note ∆ F ≃ 0.
(B. 16) In fact, the cubic order terms decompose into two parts: one involving the R-R field strength, F, and the other with the torsionless master derivative, D 0 A . The former reduces to (B.15) and the latter turns out to be a total derivative which we neglect. The computation of the quintic order terms is genuinely algebraic.
At last, adding up (B.14), (B.15) and (B.16), we obtain the final expression (30). This completes our verification of the N = 2 supersymmetric invariance of the action, modulo the self-duality (24), to the full order in fermions.