Supersymmetric gauged Double Field Theory: Systematic derivation by virtue of \textit{Twist}

In a completely systematic and geometric way, we derive maximal and half-maximal supersymmetric gauged double field theories in lower than ten dimensions. To this end, we apply a simple twisting ansatz to the $D=10$ ungauged maximal and half-maximal supersymmetric double field theories constructed previously within the so-called semi-covariant formalism. The twisting ansatz may not satisfy the section condition. Nonetheless, all the features of the semi-covariant formalism, including its complete covariantizability, are still valid after the twist under alternative consistency conditions. The twist allows gaugings as supersymmetry preserving deformations of the $D=10$ untwisted theories after Scherk-Schwarz-type dimensional reductions. The maximal supersymmetric twist requires an extra condition to ensure both the Ramond-Ramond gauge symmetry and the $32$ supersymmetries unbroken.

The Spin(1, D−1) L × Spin(D−1, 1) R local Lorentz symmetry is only forced at the whole action level.
On the other hand, in the semi-covariant formulation of DFT [34,35], once proper 'projections' are imposed, the semi-covariant derivatives and the semi-covariant curvatures all become completely covariant with respect to both diffeomorphisms and the local Lorentz symmetry, besides the O(D, D) T-duality.
Within this setup, the maximal as well as half-maximal D = 10 supersymmetric double field theories (SDFT) have been constructed to the full order in fermions [36,37] where each term in the Lagrangians is completely covariant, see also the earlier formulation within Generalized Geometry [25,26]. Berman and Lee then modified the semi-covariant formalism to be apt for the twisted generalized Lie derivative by introducing torsionful semi-covariant derivative connections [29]. However, it is fair to say that while these proposals all opened up novel aspects of the section condition and hence DFT itself, many ingredients were introduced ad hoc by hand. Deeper systematic understanding has been desirable.
It is the purpose of the present paper to propose such a geometric scheme to twist the maximal and the half-maximal supersymmetric double field theories of Refs. [36,37] and systematically derive the gauged supersymmetric double field theories. Essentially, as our main results, we show that the semi-covariant formalism itself can be twisted by the Scherk-Schwarz ansatz, without any arbitrariness. This enables us to address readily the supersymmetric completions. The twisted and hence gauged maximal as well as half-maximal supersymmetric double field theories are then completely fixed by requiring the supersymmetry to be unbroken. Each term in the constructed Lagrangian is completely covariant with respect to the twisted diffeomorphisms, the Spin(1, 9) × Spin(9, 1) local Lorentz symmetries, and a subgroup of O(10, 10) which preserves the structure constant. This complete covariance also ensures the internal coordinate independence.
The organization of the paper is as follows.
• In section 2, we revisit with care the semi-covariant formulation of the ungauged or untwisted double field theory [34,35] and its supersymmetric extensions [36][37][38]. While reviewing them in a selfcontained manner, we spell, for later use of twist, all the relevant exact formulas which hold without assuming any section condition. Such formulas have not been fully spelled elsewhere before.
• In section 3, we twist the double field theory with a simple Scherk-Schwarz ansatz. Following closely Grana and Marques [27], we analyze a set of consistency conditions for the closure of the twisted generalized Lie derivatives, which we call twistability conditions. We show that all the nice properties of the semi-covariant formalism, including its complete covariantizability, are still valid after the twist under the twistability conditions. In particular, we verify that the consistent definition of the twisted Ramond-Ramond cohomology requires one additional condition which is, after the diagonal gauge fixing of the twofold local Lorentz symmetries, consistent with the previous work by Geissbühler et al. [28].
• Section 4 contains our main results. Readers may want to have a glance of our final results therein, before reading the preparatory sections, 2 and 3. We present the maximal and the half-maximal supersymmetric gauged double field theories as the twists of the N = 2 and the N = 1, D = 10 supersymmetric double field theories [36,37]. In particular, we show the twisted maximal supersymmetric invariance calls for the same extra condition which the twisted R-R gauge symmetry demands as well.
• In section 5 we conclude with comments.
Although our supersymmetry analyses are explicit only up to the leading order, we argue in section 3 that the full order supersymmetric completions are guaranteed to work, as the higher order fermionic terms are immune to the "relaxation" of the section condition.
Conventions. Equations which hold due to the original section condition (1.1) and the alternative twistability conditions are denoted differently with the two distinct symbols, ' ∼ ' and ' ≡ ' respectively, besides the strict equality, ' = '. For the sake of simplicity we shall often adopt a matrix notation to suppress contracted indices, e.g. (P ∂ A P ) B C = P B E ∂ A P E C . Our index conventions follow [37] and are summarized in Table 2. In Table 1, we also list various derivatives which are explained and used throughout the paper.

The semi-covariant formulation of ungauged DFT
In this preparatory section, we revisit the semi-covariant formulation of ungauged or untwisted double field theory [34,35] and its supersymmetric extensions [36][37][38]. Our goal is threefold: to review them in a self-contained manner, to locate the exact places where the original section condition is assumed, and to collect, for later use of twist, precise formulas which hold without assuming any section condition. Such formulas have not been fully spelled in the literature before. Every formula which holds up to the original section condition will be denoted by the symbol, ' ∼ ', rather than by the strict equality, ' = '.
In particular, we pay attention to two strictly-different yet section-condition-equivalent semi-covariant four-index curvatures, namely G ABCD and S ABCD , and analyze their differences exactly without assuming the section condition.

Coordinate gauge symmetry, section condition and diffeomorphism
• Doubled-yet-gauged spacetime. The spacetime is formally doubled, being (D+D)-dimensional.
However, the doubled spacetime coordinates are gauged: the coordinate space is equipped with an equivalence relation, which is called 'coordinate gauge symmetry' [8,10]. In ( A, B, · · · Untwisted O(10, 10) vector Table 2: Index for each symmetry representation and the corresponding "metric" which raises or lowers its position. Only the capital O(10, 10) indices are to be twisted. The ' + ' subscripts of the charge conjugation matrices indicate that they are chosen to be symmetric. The doubling of the local Lorentz symmetries, Spin(1, 9) → Spin(1, 9) × Spin(9, 1), is crucial to achieve the unification of IIA and IIB supergravities within the unique N = 2, D = 10 untwisted SDFT [37].
denoted here collectively by Φ, are invariant under the coordinate gauge symmetry shift [8,10], This invariance is equivalent, i.e. sufficient [8] and necessary [10] to the section condition, Acting on arbitrary functions, Φ, Φ ′ , and their products, the section condition leads to the weak constraint, ∂ A ∂ A Φ ∼ 0 as well as the strong constraint, ∂ A Φ∂ A Φ ′ ∼ 0.
• Diffeomorphism. Diffeomorphism symmetry in DFT is generated by a generalized Lie derivative [1,39,40], where ω denotes the weight of the field, T A 1 ···An . In particular, the generalized Lie derivative of the The commutator of the generalized Lie derivatives is closed by C-bracket [1,41] up to the section condition, since the following strict equality holds without resorting to the section condition [34], of which the right hand side clearly vanishes upon the section condition. We shall come back to this expression when we perform the twist. The vielbeins satisfy the following four defining properties [35,42] (see also [1,40]):

Dilaton, vielbeins and projectors
That is to say, they are normalized, orthogonal and complete. The vielbeins are O(D, D) vectors as their indices indicate. In fact, they are the only O(D, D) non-singlet field variables even in the supersymmetric extensions of DFT [36,37]. As a solution to (2.9), they can be parametrized in terms of ordinary zehnbeins and B-field, in various ways up to O(D, D) rotations and field redefinitions, e.g. [38,43,44].
Due to the defining properties of (2.9), arbitrary variations of the vielbeins meet ] . (2.10) • Projectors. The vielbeins generate a pair of symmetric, orthogonal and complete two-index projectors, 1 (2.12) Further, the two-index projectors generate a pair of six-index projectors, 1 The difference of the two projectors, , PAB −PAB = HAB, corresponds to the "generalized metric" in [4], which can be also independently defined as a symmetric O(D, D) element, i.e. HAB = HBA, HA B HB C = δ C A . However, in the 'full order' supersymmetric extensions of DFT [36,37] where e.g. the 1.5 formalism works, it appears that the projectors are more fundamental than the "generalized metric". which satisfy the 'projection' property, symmetric and traceless properties, as well as further properties like (2. 16) In addition to the six-index projection operators (2.13), we also set for later use, (2.17)

Semi-covariant derivatives, curvatures and their complete covariantizations
• Semi-covariant derivative and the torsionless connection. The semi-covariant derivative is defined by [34,35] It satisfies the Leibniz rule and is compatible with the O(D, D) invariant constant metric, We choose the connection to be the torsionless one from Ref. [35]: 2 which is a unique solution to the following five constraints [35]: The last formulae (2.25) are projection conditions which ensure the uniqueness.
While the torsionless connection satisfies all the five constraints, (2.21 -2.25) and thus uniquely determined, a generic torsionful connection meets only the first three conditions, (2.21), (2.22), (2.23), and decomposes into the torsionless connection and torsions [25,36], In order to maintain (2.22), the torsions must satisfy In the full order supersymmetric extensions of DFT [36,37], they are given by quadratic fermions.
It is worth while to note P I AP J • Spin connections and semi-covariant master derivative. The master semi-covariant derivative [42], By definition, it is compatible with the vielbeins, and, from (2.22), also with the dilaton, The connections are then related to each other by and Consequently, their generic infinitesimal variations satisfy (2.36) The master semi-covariant derivative is also compatible with all the constant metrics and the gamma matrices in Table 2, (2.37) The well known relation between the spinorial and the vectorial representations of the spin connections follows • Semi-covariant four-index curvatures. The usual "field strengths" of the three connections, This implies Following [46], replacing the ordinary or the naked derivatives in (2.39) by the semi-covariant derivatives we define 42) which are, with the torsion-free condition (2.24), related to (2.39) by 43) and appear in the commutators of the master semi-covariant derivatives, (2.44) Further, they can be rewritten in terms of the master semi-covariant derivatives, then to carry some opposite signs in comparison to (2.42), (2.45) Hence, contracted with the vielbeins -which are compatible with D A but not with ∇ A -we may write (2.46) Now we are ready to define two kinds of semi-covariant four-index curvatures: -Semi-covariant four-index curvature of the spin connections, c.f. [28], -Semi-covariant Riemann curvature of the diffeomorphic connection [34,35], These two four-index curvatures are closely related to each other, such that upon the section condition we have As a bonus, this implies that, up to the section condition G ABCD is local Lorentz invariant as S ABCD is so. Note that while F ABpq andF ABpq are local Lorentz covariant, F ABpq andF ABpq are not.
A notable difference between G ABCD and S ABCD is that while the latter can be expressed in terms of the dilaton and the projectors, the former cannot be defined thoroughly by them: it requires the vielbeins. In the following section, we shall see that it is G ABCD rather than S ABCD that survives to serve as the semi-covariant curvature after the twist.
It is worth while to note that, in the expressions of Φ Apq ,Φ Apq (2.34), F ABpq ,F ABpq (2.42) and G ABCD (2.47), the ordinary naked derivative and the Γ-connection are completely 'confined' into the semi-covariant derivative. On the other hand, it is not the case with R ABCD , F ABpq ,F ABpq and S ABCD .
A crucial defining property of the semi-covariant Riemann curvature is that, under arbitrary transformation of the connection, it transforms as Surely for the torsion-free connection (2.20), the last two terms are absent and only the first two total derivative terms remain, Yet, in the full order supersymmetric extensions of DFT [36,37], the connection includes bifermionic torsions and the above general relation (2.51) enables the '1.5 formalism' to work.
Without necessity of the section condition, S ABCD satisfies [34], and, especially for the torsionless connection, a Bianchi identity, 3 Further, for the torsionless connection (2.20), one can show by a brute-force method, 4 of which the right hand sides all vanish upon the section condition, It follows, from (2.50), that identical relations hold for G ABCD , either by the strict equality or up to the section condition, for example, • Complete covariantizations. The ordinary derivative of a covariant tensor is no longer covariant under diffeomorphisms. The difference between its actual diffeomorphic transformation and the generalized Lie derivative reads precisely, (2.57) 3 See Eq.(2.46) of [34] for a simple proof of the Bianchi identity. 4 To obtain (2.55), we have used the computer algebra, Cadabra [47,48].
Especially for the connection we have (2.58) It follows that we may obtain an exact expression of the diffeomorphic anomaly of the semi-covariant derivative, into which (2.58) can be readily substituted.
Lastly for the semi-covariant Riemannian curvature of the torsionless connection, from we get an exact formula, and for the four-index curvatures, The anomalous terms can be easily projected out through appropriate contractions with the two-index projectors. In this manner, the completely covariant derivatives are given by (2.67) These can be also freely pull-backed by the vielbeins to take the form: Similarly we obtain completely covariant two-index as well as zero-index curvatures from the semicovariant four-index curvatures.
-Completely covariant "Ricci" curvatures, 5 In fact, the two "Ricci" curvatures agree to each other upon the section condition: From the identity (2.55), their difference reads exactly, and hence, Similarly the two scalar curvatures are related to each other: From (2.55), their sum reads exactly, and hence, upon the section condition, The other formulas in (2.55) also imply a pair of 'trivial' four-index covariant quantities, (2.76)
All the fermions, i.e. dilatinos, gravitinos and supersymmetry parameters, are not twenty, but tendimensional Majorana-Weyl spinors. The chirality of the theory reads with two arbitrary sign factors, (2.78) A priori, there are four different sign choices. But, they are all equivalent up to the field redefinitions through Pin(1, 9) × Pin(9, 1) rotations. That is to say, N = 2 D = 10 SDFT is chiral with respect to both Spin (1,9) and Spin(9, 1), and the theory is unique. Without loss of generality, henceforth we set Although the theory is unique, the Riemannian solutions are twofold and can be identified as type IIA or IIB supergravity backgrounds [37]. The theory admits also non-Riemannian backgrounds [10] (c.f. math literature [49]).
The R-R field strength, F αᾱ , and its charge conjugation are defined by [38] F : Here D + corresponds to one of the two completely covariant differential operators, D ± , which are defined by the torsionless connection (2.20) to act on an arbitrary Spin(1, 9) × Spin(9, 1) bifundamental field, T αβ : The crucial property of the differential operators, D ± , is that upon the section condition they are nilpotent [38]. Straightforward computation can show (2.82) Thus, up to the section condition, with (2.49), (2.56), (2.55), (2.74), each term on the right hand side above vanishes and the nilpotency of the differential operators follows This defines the R-R cohomology consistently coupled to the NS-NS sector in an O(D, D) covariant manner [38]. In particular, the R-R gauge transformations are given by the same nilpotent differential operator, In a similar fashion to (2.65), upon the section condition, the spin connections transform anomalously under diffeomorphisms, (2.85) Thus, like (2.67), these anomalous terms can be easily projected out, such that the following modules of the spin connections are completely covariant under diffeomorphisms, The completely covariant Dirac operators are then, with respect to both diffeomorphisms and local Lorentz symmetries, as follows [36,42].
One can also show that D ± T (2.81) are completely covariant too.

• Completely covariant curvatures from completely covariant derivatives
From (2.30), (2.44) and the relation, In a similar fashion to (2.44), we may obtain the expressions for the commutators of the master semi-covariant differential operators which act on spinors, ε α and ε ′ᾱ , in Spin(1, D−1) L and (2.90) These immediately imply and further give (2.92) Then, combined with the following relations, we can derive the following identities, (2.94) Therefore, upon the section condition the completely covariant "'Ricci"' and scalar curvatures (2.69), (2.71) are related to the completely covariant Dirac operators (2.87), c.f. Generalized Geometry [25], and

DFT action and supersymmetric extensions
• Pure DFT action. The bosonic action of the untwisted DFT for the NS-NS sector, or the pure DFT, is given by the fully covariant scalar curvature, where the integral is taken over a section, Σ D . The dilaton and the projector equations of motion correspond to the vanishing of the scalar curvature i.e. the Lagrangian itself and the "Ricci" curvature, S pq , respectively.
It is precisely this expression of (2.97) that ensures the '1.5 formalism' in the full order supersymmetric extensions of DFT with torsionful connections [36,37]. In fact, without imposing the section condition, the scalar curvature in the Lagrangian (2.97) precisely agrees with the original DFT Lagrangian ( [4]) written in terms of the generalized metric, H = P −P , These agree with (2.97) upon the section condition, yet strictly differ by section-condition-vanishing purely bosonic terms: and The second equality of (2.100) follows from (2.9), (2.11) and an identity, • The full order supersymmetric extensions. Based on the semi-covariant formalism revisited above, the N = 2 (maximal) D = 10 supersymmetric double field theory has been constructed to the full order in fermions [37], By truncating the R-R potential and the primed fermions, the N = 1 (half-maximal) D = 10 supersymmetric double field theory [36] is also readily obtainable, Generically, the supersymmetric double field theory Lagrangians decompose into three parts, . In particular, the supersymmetry of the N = 1, D = 10 SDFT [36] amounts to the following algebraic identities, such that the Lagrangian is invariant up to total derivatives and the section condition, On the other hand, the supersymmetry of the N = 2 D = 10 SDFT [37] means the invariance of the Lagrangian up to total derivatives, the section condition and the self-duality of the R-R field strength, whereF − is the self-dual part of the R-R field strength (2.80) defined, to the full order in fermions, andF − denotes, like (2.80), its charge conjugation, A crucial fact about the section-condition-vanishing terms, [2,1] , which is common in (2.108) and (2.109), is that they are strictly linear in the fermions (dilatinos and gravitinos), and hence they are fully obtainable just from the leading order supersymmetry transformation rules.
It is also crucial to note that the above form of the algebraic identities, (2.108) and (2.109), still holds, i.e. the supersymmetry is unbroken upon the section condition, even if we deform the Lagrangian by adding arbitrary section-condition-vanishing terms, which, counting the mass dimensions, should be purely bosonic. Examples include the replacement of S ABCD by G ABCD and adding G pq pq + Gpqpq (2.74) to the N = 1 (but not N = 2) D = 10 SDFT Lagrangian, which we shall take below, for the supersymmetry preserving twist.

U-twisted double field theory
Here we twist the double field theory formulated within the semi-covariant formalism. Our twist is a DFT generalization of the Scherk-Schwarz twist, based on [21,22,27,28], and will be from time to time referred to as U-twist. In section 3.1, we introduce our ansatz of the twist. It involves a scalar and a local O(D, D) group element which may not obey the section condition. In section 3.3, following closely Grana and Marques [27], from the closure of the U-twisted generalized Lie derivative we derive a set of consistency conditions which we call twistability conditions. They generalize the original section condition and slightly differ from [27]. In section 3.4, we perform the U-twist on the semi-covariant formalism and verify that, with the replacement of S ABCD by G ABCD , essentially all the nice properties of the semi-covariant formalism, including the complete covariantizability, survive after the twist, subject to the twistability conditions. We also verify that both the N = 2 supersymmetric invariance and the nilpotency of the differential operators which define the twisted Ramond-Ramond cohomology commonly require an extra condition. Consequently, the maximal supersymmetric twist of the N = 2 D = 10 SDFT requires one more twistability condition compared to the half-maximal supersymmetric twist of the N = 1 D = 10 SDFT.

Ansatz for U-twist
it satisfies While the dotted (twisted) metric,JṀṄ , may coincide numerically with the undotted (untwisted) metric, J M N (c.f. Table 2), hereafter we deliberately distinguish them. In particular, the two different kinds of indices will never be contracted.
U-twist prescribes substituting the following expression for each untwisted (undotted) field, T A 1 ···An , into the D = 10 ungauged DFT Lagrangians, c.f. [27], Equivalently, twisted fields are defined to carry dotted O(D, D) indices with a relevant weight factor, The derivatives of the untwisted fields then assume a generic form, the U-derivative,ḊĊ, is defined to act on a twisted field bẏ In particular, the twist of the N = 1 or the N = 2, D = 10 SDFT amounts to inserting the following expressions for the dilaton and the vielbeins into the untwisted Lagrangian, Twisted SDFT (JȦḂ,ḊȦ,ḋ,VȦ p ,VȦp, C, ρ, ψp, ρ ′ , ψ ′ p ) . Before doing so, in the next subsection we pause to collect some useful properties of the U-derivative,ḊĊ .

Properties of the U-derivative and its connection
Since U AḂ is an O(D, D) element, we have Hence, the U-derivative "connection" is skew-symmetric for the last two indices, It is worth while to note and 6∂Ȧ ΩḂḂȦ = − 1 2 ΩȦȦḂΩĊĊḂ − 1 2 ΩĊḂȦΩḂĊȦ . Pulling back the dotted derivative index to a undotted index, it is useful to consider where naturally we put This corresponds to "pure gauge" and thus its "field strength" vanishes identically, (3.18) By construction, the U-derivative (3.8) can be rewritten aṡ and is compatible with the U matrix itself, Furthermore, from the U-derivatives are all commutative, This is a crucial result. It means that there is no ordering ambiguity of the U-derivatives, as one might worry while performing the twist, (3.10). Namely, the 'field strength' and the 'torsion' of the U-derivative are all trivial. It is also worth while to compare with the Weitzenböck connection, e.g. [30]. Although it appears formally similar to our Ω, there is a crucial difference: we intentionally distinguish the dotted indices from the undotted indices, while the Weitzenböck connection and hence the corresponding Weitzenböck derivative do not. Consequently, the Weitzenböck derivatives do not commute, unlike (3.24), and the Weizenböck connection is torsionful.
The dilaton, d, corresponds to the logarithm of a weightful scalar density. Its U-derivative is then Further, its second order derivatives are 7 27) and thus, in general, Now, following [27], we define two key quantities out of the twisting data, In particular, this implies We shall make use of these identities shortly below.
Finally, from (3.5), the divergence of a vector density with weight one becomes after the twist, In order to ensure the supersymmetry to be unbroken after the twist, we need to show that these terms vanish up to the twistability conditions. Fortunately, as discussed in section 2.5 and demonstrated in section 4 later, these anomalous terms can be all sufficiently obtained just from the leading order supersymmetry.

Twistability conditions: closure of the diffeomorphisms
Acting on the dotted twisted fields, twisted diffeomorphism is generated by the U-twisted generalized Lie derivative, In an identical manner to the twisting ansatz (3.4), this expression is related to the untwisted generalized Lie derivative (2.4) byLẊṪȦ The commutator of the U-twisted generalized Lie derivatives, without employing any section condition, reads readily from (2.7), c.f. Grana and Marques [27], where [Ẋ,Ẏ ]Ċ denotes the U-twisted C-bracket, Clearly, if the condition of (3.11) were imposed, the right hand side of (3.39) would vanish. Yet, we are after other way of ensuring the closure. To this end, we dismantle the U-derivative and display its connection explicitly: the U-twisted generalized Lie derivative (3.37) and the U-twisted C-bracket (3.39) can be rewritten, in terms of fȦ (3.29) and fȦḂĊ (3.30), aṡ and Similarly, the right hand side of the equality in (3.39) reads  are as follows, c.f. [27,29]. 8 1. The section condition for all the dotted twisted fields, 2. The orthogonality between the connection and the derivatives of the dotted twisted fields, We stress that these five constraints, (3.46) -(3.50), are the natural requirement for the closure (3.45) directly read off from (3.44). 9 In principle, we should solve these constraints. While we are currently lacking the most general form of the solutions, a class of solutions are well known which involve dimensional reductions. If we assume the U matrix to be in a block diagonal form,  Further, from the integrability of the last condition (3.50), we get The U-twisted generalized Lie derivative (3.41) reduces, upon the twistability conditions, tô which clearly has no 'internal' coordinate dependency 10 and decomposes into the external diffeomorphism and internal gauge symmetry [27,29] (see also [50]). 10 With the internal/external splitting (3.51), the∂Ȧ derivatives of the dotted fields are independent of the internal coordinates.

Twisted semi-covariant formalism
The twisting of the semi-covariant formalism is straightforward. The U-twisted master semi-covariant   Now, from (2.49), it is useful to notė and thus, upon the twistability conditions, In the above, for sure, we seṫ (3.64) Thus, in contrast to the untwisted case (2.50), GȦḂĊḊ differs fromṠȦḂĊḊ after the twist. In the twisted SDFT to be constructed below, we shall disregard the latter and employ the former only. The former will be shown to be semi-covariant, while the latter is not.
Starting from the strict equality of (3.62) and using (2.52), one can easily show nevertheless that the infinitesimal transformation ofĠȦḂĊḊ induced by the variations of its constituting all the twisted fields coincides with that ofṠȦḂĊḊ, up to the twistability conditions, This should be a naturally expected result, if we focus on the variation of the equivalence relation (3.63) rather than the strict equality (3.62). Since ΩȦḂĊ is not a field variable but rather a fixed data for a given internal manifold, it is not taken to transform but must be inert under any 'symmetry', 11 δU AḂ = 0 , δΩĊȦḂ = 0 , δ(ḊĊ ) = 0 . • Complete covariantizations after the twist.
Here we focus on the twisted diffeomorphism. We twist the relation (2.57) in order to obtain the difference between the actual transformation of the U-derivative of a twisted field and its twisted generalized Lie derivative, (3.67) Writing the first equality above, we have implicitly assumed (3.66). It follows for the twisted connection (3.57), To simplify this expression up to the twistability conditions, we use (3.31) and an identity, Hence, upon all the twistability conditions, finally we obtain   Ḋ p Tq 1 ···qn ,ḊpT q 1 ···qn ,Ḋ p T pq 1 ···qn ,ḊpTp q 1 ···qn ,Ḋ pḊ p Tq 1 ···qn ,ḊpḊpT q 1 ···qn , and γ pḊ p ρ , γ pḊ p ψp ,Ḋpρ ,Ḋpψp ,ψȦγ p (ḊȦψq − 1 2Ḋq ψȦ) ,  We conclude thatṠȦḂĊḊ is of no use in the twisted double field theory. We discard it and keep GȦḂĊḊ only.
• Identities which still hold after the twist.
Straightforward yet useful implications of the twistability conditions includė That is to say, replacingṠȦḂĊḊ byĠȦḂĊḊ, almost all the properties of the four-index curvature (2.55) still hold after the twist, up to the twistability conditions. The only exception is (3.88) and this is also crucial. 13 Putting the three relations (3.65), (3.66) and (3.82) together, we conjecture an equivalence relation, of which a direct proof is desirable.
The relations between the completely covariant curvatures and the completely covariant derivatives (2.89), (2.95), (2.96) still hold after the twist, (3.89) But, in contrast to (2.83), we get after the twist, This indicates that in addition to the twistability conditions, in the presence of the R-R sector, in order to ensure the nilpotency of the differential operators,Ḋ ± , which should define the twisted R-R gauge symmetry or the 'twisted R-R cohomology' consistently, we should separately impose For a relevant previous work we refer the readers to [28] where the R-R sector was treated as an O(10, 10) spinor which can be related to our treatment after the diagonal gauge fixing of the twofold local Lorentz symmetries [38]. We shall see shortly that this extra condition is also required for the supersymmetric invariance of the twisted maximal SDFT.
• "Effective connection" and internal coordinate independence. We may view the last two lines of (3.93) as "effective torsions". They satisfy the desired properties (2.28). In this "effective" point of view, the "torsion" should be defined from the difference, L X (∇) −L X (∂), instead of the trivial one (3.59),L X (∇) −L X (Ḋ)= 0. For further related discussion, we refer readers to section 3.4.2 of [25].
We also note that only the last line, i.e. the six-index projection of ΩḊĖḞ , may depend on the internal coordinates, which could be problematic. However, as seen in ( The above expression of the effective connection (3.93) is also comparable to the torsionful connection proposed by Berman and Lee in [29]. With the intention of handling the twisted generalized Lie derivative [27,29], i.e. (3.54), they introduced torsions by clever guess work. Their torsionful connection differs from our effective connection (3.93). Yet, it nevertheless satisfies (3.94) and the difference amounts to certain six-index projection terms. Accordingly, their proposal is practically consistent with our result. A novel contribution of the present work is to derive the effective connection (3.93) straightforwardly by applying the U-twisting ansatz (3.3) to the semi-covariant formalism, without any ambiguity.

Twisted supersymmetric double field theory
Here we present explicitly half-maximal (i.e. sixteen) and maximal (i.e. thirty two) supersymmetric gauged

Half-maximal supersymmetric gauged double field theory
After replacing S ABCD by G ABCD and adding section-condition-vanishing purely bosonic terms of (2.75), we twist the N = 1 D = 10 SDFT which was constructed in [36] to the full order in fermions. The twist leads to a half-maximal supersymmetric gauged double field theory of which the Lagrangian iṡ Each term in the Lagrangian is completely covariant with respect to the twisted diffeomorphisms, (3.37) or (3.54), the Spin(1, 9) × Spin(9, 1) local Lorentz symmetries, and a subgroup of O(10, 10) which preserves the structure constant, fȦḂĊ . Being completely covariant, each term is also independent of the internal coordinates.
The leading order half-maximal (i.e. sixteen) twisted supersymmetry transformation rules are, for the twisted bosons, and for the 'untwisted' fermions, The supersymmetry works, as the induced leading order variation of the Lagrangian vanishes, up to total derivatives and the twistability conditions, thanks to (3.89), As discussed at the end of section 3.2, the leading order supersymmetric invariance is sufficient to guarantee the full order completion. Outsourcing from the full order untwisted N = 1 D = 10 SDFT [36], we only need to add the quartic fermions therein to the twisted Lagrangian (4.1) and the cubic fermions to the twisted supersymmetry transformation rules for the fermions (4.3).
As in the untwisted SDFT [36], the conventional Rarita-Schwinger term is forbidden, and this is due to the hybrid nature of the gravitino indices, ψ ᾱ p : one Spin(9, 1) vectorial and the other Spin (1,9) spinorial. Simply they cannot be mixed. Nonetheless, the N = 1 D = 10 SDFT reduces consistently to the minimal supergravity in ten-dimensions after the diagonal gauge fixing, Spin(1, 9) × Spin(9, 1) → Spin(1, 9) D , see the appendix of [36] for details.
It is worth while to note from the Z 2 symmetry which exchanges the two spin groups, Spin(1, 9) ↔ Spin(9, 1), there is a parallel formulation of the half-maximal SDFT, The supersymmetry is realized by

Explicit comparison with the untwisted case
To compare with the untwisted DFT and to identify the newly added terms after the U-twist, we dismantle the U-derivatives,ḊȦ, explicitly and obtain up to the twistability conditions, It follows that the sum,Ġ pq As expected from the consistency of the "effective connection", (4.17) and (4.18) agree with Berman and Lee [29], while (4.16) is a new result we report in this work.

Discussion
In this paper, we have successfully twisted the semi-covariant formulations of the N = 2 and the N = 1, D = 10 SDFT constructed in [36,37], and systematically derived the gauged maximal and half-maximal supersymmetric double field theories, (4.1) (4.5), (4.7), along with their supersymmetry transformation rules, (4.2), (4.3), (4.9), (4.5), (4.10). Our derivation is systematic in the sense that, we only applied the twisting ansatz (3.3) to the untwisted SDFT of [36,37], and then without any ambiguity the gauged supersymmetric double field theories were straightforwardly derived. Further, just like the untwisted SDFT yet now subject to the twistability conditions, (3.46) -(3.50) and also (3.91) for the maximal supersymmetric twist, each term in the constructed Lagrangian is completely covariant. Namely, the NS-NS curvature term, the fermionic kinetic terms and the R-R kinetic term are all completely covariant, with respect to the twisted diffeomorphisms, the Spin(1, 9) ×Spin(9, 1) local Lorentz symmetries, the R-R gauge symmetry for the maximal case, and a subgroup of O(10, 10) which preserves the structure constant. The twofold Lorentz symmetries are 'local' with respect to the dimensionally reduced external spacetime. The twisted and hence gauged SDFTs are completely fixed by requiring the supersymmetry to be unbroken, in the precisely same manner as the untwisted SDFTs.
The nilpotency of the twisted R-R cohomology differential operators (3.90), (3.91), implies the Bianchi identity for the twisted R-R flux,Ḋ +Ḟ = (Ḋ + ) 2 C ≡ 0 . As demonstrated in the section 4.3 of [38], one may take the diagonal gauge fixing of the local Lorentz symmetry, expand the R-R potential in terms of the conventional p-form fields coupled to gamma matrices in a 'democratic' manner [51], and compute the R-R field strengths explicitly. The above Bianchi identity is then naturally expected to produce the 'tensor hierarchy' [52][53][54].
It is worth while to note that, while the twist breaks the O(10, 10) T-duality to its subgroup which preserves the structure constant, fȦḂĊ , the Spin(1, 9) × Spin(9, 1) local Lorentz symmetries are still all unbroken after the twist and the dimensional reduction.
When the twisting data, U AȦ , λ, do not satisfy the original section condition, the corresponding background cannot be identified as a solution to the untwisted 'D = 10' supersymmetric double field theories.
This might well motivate one to wonder about the existence of unknown genuinely ten-dimensional "generalized double field theory" with "relaxed" section conditions. However, the twistability conditions seem to admit only lower dimensional sections, as the non-trivial solutions. In those lower dimensions, the standard section condition must be obeyed, see (3.46), and its doubled coordinates are still to be gauged.
We regard the twist not as an indication of the existence of any unknown D = 10 "generalized DFT" but as a lower dimensional deformation of the known rigid untwisted D = 10 theories, i.e. [36,37]. A well known such example is the massive supersymmetric deformations of the super Yang-Mills quantum mechanics [55,56]. The deformations do not necessarily mean that the parental super Yang-Mills field theories can be likely deformed.
In this work, the R-R sector is taken as O(10, 10) singlet and assumes the Spin(1, 9) × Spin(9, 1) local Lorentz bi-spinorial representation [25,26,37,38,[57][58][59][60]. 15 This made the twisting of the R-R sector rather trivial. Essentially, the R-R potential, C αᾱ , is not twisted, like other fermions. Only the R-R field strength,Ḟ =Ḋ + C, is influenced by the twist through the twisted nilpotent differential operator. We expect that this feature should change when the U-duality group is twisted in M-theory setup, but this goes beyond the scope of the present work.