Democratic actions for type II supergravities

We present a universal democratic Lagrangian for the bosonic sector of ten-dimensional type II supergravities, treating"electric"and"magnetic"potentials of all RR fields on equal footing. For type IIB, this includes the five-form whose self-duality equation is derived from the Lagrangian. We also present an alternative form of the action for type IIB, with manifest $SL(2,\mb R)$ symmetry.


INTRODUCTION
A large part of our knowledge on string theory comes from its low-energy limit -10d supergravities. The rich symmetry structures of the latter theories are governed by generalized geometry [1][2][3][4][5] (see also [6,7]; for a review, see, e.g., [8]) operating not only with metric tensor but also form fields. It is therefore very useful to have descriptions of supergravities that manifest large number of underlying symmetries of string theory. We start investigation in this direction by developing a new Lagrangian formulation for type II supergravities adapted to generalised geometry. That is, they treat the electric and magnetic degrees of freedom in equal footing for the RR sector. This formulation we put forward in this work comes with a large number of new symmetries that will be explored in more detail elsewhere.
We present here a new O(10, 10)-adapted approach that treats all RR fields and their duals in a democratic manner [36,39] and is universal for both type II supergravities. We also derive an action for type IIB case with manifest SL(2, R) symmetry.

TYPE II SUPERGRAVITY ACTION IN THE
O(10, 10)-ADAPTED FORM We follow the exposition from [5] which uses the interpretation of the Ramond-Ramond fields in terms of O(10, 10)-spinors [40].
We consider 10-dimensional spacetime with a metric of Lorentzian signature. In addition, we assume the presence of a closed 3-form flux H. In this setup, we can then define the following operations on the space of (inhomogeneous) differential forms.
First, there is a natural pairing (Mukai pairing), valued in the top-degree differential forms, given by where ⌊ · ⌋ denotes the integer part of a number, deg α is the degree of the differential form α and ( · ) top stands for the top form part. Then, we define the "reflection" operator ⋆ and the "differential" D by where * is the Hodge star for the Lorentzian metric. These operations satisfy the following easy-to-check properties: • ⋆ 2 = 1 (⋆ serves as a natural replacement of * in the context of Ramond-Ramond fields), • the pairing (α, ⋆β) is symmetric in α and β (this serves as a replacement for α ∧ * β), • D 2 = 0 (D replaces the de Rham differential), • D(f α) = f Dα + df ∧ α, for any function f .
If B is a 2-form, we can define the operation e B by Conjugating the operator D = d + H by e B we then get Since H is locally exact, we can always locally relate the operator D to the ordinary d in this way.
In particular, suppose that α satisfies Dα = 0. Locally, writing H = dB, we get 0 = Dα = e −B d(e B α), which implies that e B α is closed and hence we can write e B α = dβ + c, where c is a constant [57]. Thus In particular, if α has no degree zero component and is D-closed, then it is also locally D-exact.

Ramond-Ramond fields
In type II string theory, the (field strengths of the) Ramond-Ramond fields correspond to either purely even (in type IIA) or purely odd (IIB) inhomogeneous differential forms F satisfying The self-duality condition (7) ensures that only half of the degrees of freedom are present. This makes the contact with the original description of the RR sector, where in type IIA one has a 0-form (called the Romans mass), 2-form, and a 4-form, while in the IIB case one has a 1-form, 3-form, and an (anti-)self-dual 5-form. For simplicity, we will make the usual restriction and set the Romans mass to zero. (Otherwise we would run into some cohomological complications, stemming from the extra term in (5).) We thus have Let us now describe the potentials for the Ramond-Ramond fields. First, choose a cover of M by contractible open sets U i . Then the potentials are given by a collection of purely odd (in type IIA case) or purely even (IIB) differential forms A i on U i , which satisfy DA i = DA j on overlaps U i ∩ U j . The DA i 's then glue together into a globally defined field strength F . When varying the potentials, we assume that on overlaps we have δA i = δA j , and so the variation corresponds to a globally defined form δA -correspondingly, we have δF = DδA. Similarly, the gauge transformations are given by δA i = Dc, with c a globally defined differential form.
In particular, since F = DA, the equation (6) is automatically satisfied. Our aim is now to derive the second equation (7) from an action principle.

Remark
Let us briefly comment on the more conceptual viewpoint of the above constructions. First, differential forms can be regarded as spinors (or more precisely spinor halfdensities) of Spin(10, 10), while purely even and purely odd forms correspond to chiral spinors. The pairing can be interpreted as the invariant spinor pairing. The operator D corresponds to the generating Dirac operator of [41][42][43]. The choice of Lorentzian metric breaks the group Spin(10, 10) down to Spin(9, 1) × Spin (1,9). Selfdual and anti-self-dual differential forms (in the sense of (7)) live in (16,16) and (16,16) of this subgroup. The reflection operator ⋆ coincides with the chirality operator for Spin(9, 1) ⊂ Spin(10, 10) [44].

The RR-sector action
We start from the following action for RR fields: where a is a scalar field, and R is of the same type as A: a collection of even/odd forms for type IIB/IIA supergravity. R does not have to be globally defined, similarly to A.
The equations of motion are given by the vanishing of The action is invariant under the usual gauge transformations δA = DΛ and δR = DΩ. We also have the symmetry (analogous to PST formulation, see, e.g., [45,46]) Finally, any shift of a which does not violate the condition (da) 2 = 0 (see below) can be compensated by a change in A and R such that the action remains invariant [47,48].
Plugging this back in E A = 0 we get This in turn implies that locally where Ω and Ψ are some (inhomogeneous) differential forms. Thus, R is locally pure gauge and can be gauged away. Note, that the field a also does not contain any physical degrees of freedom, as can be seen from its offshell shift symmetry.
[59] Hence, if the spacetime is topologically trivial, all the degrees of freedom are encoded in F = DA, which satisfies the self-duality condition (7). If the topology is non-trivial, one still obtains the self-dual combination F + aQ, but there might be finitely many residual degrees of freedom associated to the topology. Note also, that the Lagrangian is invariant with respect to the symmetry (15) only up to boundary terms [60], and a global transformation to aQ = 0 everywhere is not always allowed. Given that F + aQ is invariant with respect to the transformation (15) and closed due to (18), it is more natural to think of the degrees of freedom being encoded in F + aQ rather than F . A consequence of this will be discussed in the Conclusions.

The full action
The full pseudoaction for the bosonic sector of both type II supergravities can be given as (following the conventions of [5]): Here H is a field strength of the Kalb-Ramond 2-form field B, ϕ is the dilaton, and R is the Ricci scalar for the metric g. In the last expressions we have upper/lower sign for type IIA/IIB case. In order to pass from the pseudoaction to the action we simply replaceŜ R (22) with S R (10). The full action is then given as: Here again, the upper/lower sign corresponds to type IIA/IIB supergravity. In this formulation the SL(2, R) symmetry of type IIB supergravity is not explicit. We will proceed now to derive another form of the action, adapted to the SL(2, R) symmetry.

TYPE IIB SUPERGRAVITY ACTION IN THE SL(2, R)-ADAPTED FORM
If in type IIB case we pass to the Einstein frame and rename (and rescale) the variables, we obtain the follow-ing pseudoaction [16].
where the field content is now given by a metric g, two scalars φ and ℓ, two 2-forms B and B ′ with curvatures H and H ′ , respectively, and a 4-form C with curvature F = dC. Finally, we set M : The group SL(2, R) acts on the complex scalar τ := ℓ + ie −φ by fractional linear transformations; similarly H ′ and H form an SL(2, R)-doublet.
We can now introduce the new type IIB action replacing only the sector of the pseudoaction containing the self-dual gauge field C supplemented with interactions involving both 2-forms B and B ′ with (Q = dR): where X = 1 2 (B ∧ H ′ − B ′ ∧ H) is the SL(2, R) symmetric Chern-Simons interaction contribution [49]. The equations of motion are given by the vanishing of In particular, we get Assuming (da) 2 = 0 (almost) everywhere, we obtain * (F + aQ + X) = F + aQ + X.
Plugging this back in E A = 0 we get da ∧ dR = 0 (here R is only a four-form), which locally implies that for some 4-forms ω and ψ, and so we can gauge it away. Let us check that the equations of motion for B and B ′ given by the new action coincide with the original ones. Varying (26) w.r.t. B, we get a term Using the self-duality (31), the vanishing of Q, and the identity 2H ′ ∧ X = B ′ ∧ dX, this reduces to Similarly, the contribution to the equation of motion for This matches with the original variation, provided we choose the normalisation as in (26). Note that the action (26) is also gauge invariant under the familiar gauge transformations: An important observation is that the self-duality equation can be amended by interactions with other fields via the mechanism put forward in [48], that is, adding to the Lagrangian a term (1 + * )(F + aQ) ∧ Y , to modify the self-duality equation (31) as: similarly to the addition of interactions with B and B ′ encoded in X. This suggests an immediate way to incorporate the quadratic fermion interactions (the expression for the fermionic bilinear correction Y to the self-duality equation is given, e.g., in [17]).

CONCLUSIONS
We constructed a novel Lagrangian (23) for type II supergravities which is universal for both type IIA and type IIB. The fundamental fields of the RR sector are collections of odd/even forms for type IIA/IIB, described by a democratic Lagrangian using the formulation of [47]. We also constructed a Lagrangian where only the selfdual field of the type IIB theory is treated as in [32], manifesting the SL(2, R) symmetry also present in the non-democratic pseudoaction. One interesting question would be to look for a formulation adapted to both O(10, 10) and SL(2, R) symmetries. This might require to reformulate the scalar and the two-form of the NS-NS sector in a democratic manner, introducing dual (eightand six-form) potentials for them.
As discussed, e.g., in [36,50], the democratic formulation is useful also for coupling branes to the supergravity.
One can perform dimensional reduction of the actions presented here to arrive at novel democratic Lagrangians of maximal supergravities in different lower dimensions d = 11 − n (n > 1), with E n(n) duality symmetries (see, e.g., [51,52], [18] and references therein).
Another application of the actions presented here would be to use the extra symmetries involved in them compared to the pseudoactions (together with supersymmetry) to constrain higher-order α ′ -corrections to the string effective action (see [53,54] and references therein). The unique quartic invariant of the self-dual four-form field in ten dimensions was already identified in [48] (see also [45] where the quartic vertex in the perturbative expansion was found in PST formulation), allowing to define a large class of full non-linear theories of selfinteracting chiral four-form (see also [55]). These findings can have a direct application to the type IIB case. Higher-order interactions are less constrained for a single chiral form, but can be much more constrained given the O(10, 10)−adapted structure works at higher orders.
An interesting aspect of type IIB supergravity was discussed in [56]. A resolution of the puzzle related to the on-shell value of the action on product manifolds was suggested, which is related to a boundary term, specific to the background. Another potential resolution can be derived from the Lagrangian (26). There, a solution [61] of the equations of motion with a background spacetime M 5 × X 5 can be given by F being proportional to the volume form of M 5 and aQ being proportional to the volume form of X 5 , with the same coefficient, so that F +aQ is self-dual, solving the equation (17). Then, the on-shell value of the action is given by the second term F ∧ aQ in the Lagrangian (26) and is proportional to the volume of the space-time. Note, that aQ is closed on-shell and therefore F ∧ aQ becomes a boundary term on-shell, satisfying the same conditions as the topological term added in [56], except that now this term is already encoded in the 10d covariant action and is not background-specific.