Supersymmetry for Gauged Double Field Theory and Generalised Scherk-Schwarz Reductions

Previous constructions of supersymmetry for double field theory have relied on the so called strong constraint. In this paper, the strong constraint is relaxed and the theory is shown to possess supersymmetry once the generalised Scherk-Schwarz reduction is imposed. The equivalence between the generalised Scherk-Schwarz reduced theory and the gauged double field theory is then examined in detail for the supersymmetric theory. As a byproduct we write the generalised Killing spinor equations for the supersymmetric double field theory.


Introduction
Double field theory has been through a recent rebirth. After its orginal inception [1,2] and development [3][4][5] there has been a huge number of works by a variety of groups extending the formalism in numerous directions and exploring its consequences . See the following and references therein for a review of the subject [63,64].
In double field theory one doubles the dimension of the space to make the O(D, D) symmetry manifest on on a 2D dimensional space and then imposes a seperate so called section condition that restricts to a D dimensional submanifold. Different choice of solutions to this section condition produce different Tduality frames. If one may pick a global choice for the solution to the section condition ie. there is a global choice for the T-duality frame then one is ultimately left with a normal supergravity theory and although this reformulation may be interesting we are only rewriting the theory.
This section condition is intimately tied to the consistency of the theory, the algebra of generalised Lie derivatives depends on the section condition for its closure; the supersymmetric formulations of the theory rely on the section conditions for supersymmetry to work; and various geometric aspects such as tensoral properties appeared to depend directly on the obeying of the section condition.
One of the most exciting aspects of double field theory is to examine to what extent one may relax the section condition and remain a consistent theory. Remarkabley, it is known that the Scherk-Schwarz ansatz allows one to do exactly this [16][17][18][19][20]. That is we relax the section condition and allow dependence on both the usual coordinaates and their duals simultaneously. However, the geometry is not unconstriained; the generalised metric must obey the so called Scherk-Schwarz factorisation (we will describe this subsequently). It has been shown how all the consistency checks such as closure of the local algebra and the obeying of the Jacobi identity are satisfied even though there is explicit dependence on all the extended coordinates [16][17][18][19][20]. The generalised Scherk-Schwarz reduced theory then produces a gauged supergravity theory. The embedding tensor which determines the gauging then becomes related to the twist matrics of the Scherk-Schwarz anstaz. This result filled a lacuna in M-theory; now all known supergravities theories (with appropriate amounts of supersymmetry) have lifts to a single theory-although that theory neccessarily has novel extended dimensions. So far there have been different approaches to studying the geometry of these Scherk-Schwarz reduced theories [31][32][33]. In this paper we wish to examine the Scherk-Schwarz reduced theories in the context of the supersymmetric formulation of double field theory developed by [21][22][23][24][25][26][27] where one has a semicovariant formulation (the choices of formalism and their relevant various properties is discussed in [31]).
Using this semicovariant formulation we develop how supersymmetry works in double field theory once we remove the section condition. As a by product we will produce the BPS equations (ie. Killing spinor equations) for double field theory in the absence of section condition. Solving these might have substantial applications for future directions in exploring new and novel solutions to double field theory outside that of usual supergravity.
We begin by describing the geometry for gauged double field theory and then its supersymmetric extension. The generalised Scherk-Schwarz ansatz is described and related to the gauged double field theory in the supersymmetric formalism. Finally we write down the Scherk-Schwarz reduced Killing Spinor equations for double field theory.

Gauged double field theory
As explained in the introduction, this paper is motivated by seeing how one can remain consistent and yet relax the the physical section condition. In previous work the section condition was crucial for different aspects of the theory to work; this includes the local algebra of generalised diffeomorphisms and importantly supersymmetry. In what follows we will review how the section condition can be replaced by the Scherk-Schwarz case which is equivalent to gauging the theory.
We start by recalling gauged double field theory [15,16] and defining conventions. Let V M N is an arbitrary rank-2 tensor for gauged DFT. The gauge symmetry for gauged DFT is given by a twisted generalised Lie derivative which is defined by L 0 X is the ordinary generalised Lie derivative defined in ungauged DFT by, where f M N P are the structure constants for Yang-Mills gauge group. The adjoint representation for the gauge parameter X M N by may introduced as follows Then the previous generalised Lie derivatives may be written in the following suggestive form, For consistency of the algebra (ie. closure), arbitrary fields and gauge parameters are required to obey the section condition as in the ordinary DFT. The section condition also known as the strong constraint is given by: The structure constants f M N P should then satisfy Jacobi identity, It is also convenient to impose an orthogonality condition on the structure constants f M N P This means the gauge symmetry will be orthogonal to the ordinary generalised Lie derivative.
Remarkabley one may write the action for gauged double field theory in a very compact form as follows: S 0 M N H M N is the generalised Ricci scalar for ungauged DFT, and V is the potential for gauged DFT as given in [17,18],

Connection
To construct a geometry we must make a will not be a problem. At the moment there is no formalism that has all the properties we would like.
We follow closely the construction given in [21,22]. First we introduce a semi-covariant derivative which acts on a generic quantity carrying O(D, D) vector indices where ω is a wight factor of each tensor T N 1 ···Nn . To determine the connection we assume the following constraints analogous with ungauged DFT: First, we assume that the semi-covariant derivative preserves the O(D, D) metric J M N , then it follows that the connection is anti-symmetric for last two indices Second, we impose the compatibility condition for all NS-NS sector fields, where P M N andP M N are projections defined as, satisfying (2.15) Further, we require a generalised torsion free condition: Since the righthandside is a gauge transformation of T M , in ordinary DFT language, it means the geometry is torsion free up to gauge transformation. The difference betweenL ∇ and from (2.16) and (2.17), the modified torsion-free condition implies One can construct the connection in terms of P ,P , d and the structure constants f M N P , which satisfy the compatibility conditions and modified torsion free condition, where Γ 0 P M N is the connection for ordinary DFT [22], 20) and P andP are rank-six projection operators which are symmetric and traceless, Here the superscript 0 indicates a quantity defined in the higher dimensional parent DFT.
The connection transforms under the (2.4) as (2.23) As in ungauged DFT, the derivative (2.10) combined with the projections can be used to form generate various covariant quantities such as: This is the whole point of the so called semi-covariant formalism. Some of the quantities are not fully covariant but we can build actions by using the fully covariant projected quantities as building blocks.

Spin Connections
Let us consider a local frame for gauged DFT. Following the same structure as ungauged DFT we introduce the double local Lorentz group, Spin(1, D − 1) × Spin(D − 1, 1) and corresponding double-vielbeins, V M m andV Mm . These satisfy the following defining properties [22], (2.25) Here unbared indices, m, n, p, q · · · , represent Spin(1, D − 1) vectors and bared indices,m,n,p,q · · · , represent Spin(D − 1, 1) vectors. Hence the double-vielbeins form a pair of rank-two projections [21], and further meet and Spin(D − 1, 1) representations as follows We then impose the generalised vielbein compatibility condition for these double-vielbeins V M m and V Mm , and for the metric of Spin(1, D − 1) and Spin(D − 1, 1), η mn andηmn respectively, From the compatibility of η mn andηmn, we can deduce that the spin-connections are antisymmetric, In addition, because of the compatibility of the double-vielbeins (2.29), the spin-connections are antisymmetric for last two indices, and using (2.23), these spin-connections are semi-covariant as well, (2.33) Crucially, we can then form fully covariant quantities by contracting the semi-covariant quantities with projection operators or double-vielbeins as shown below: This willl be a reoccuring trick that the formalism uses. One produces fully covariant objects by contracting semicovariant objects with projection operators.

Curvature
Again following [21,22], we may construct a rank-4 quantity R P QM N which is generated by the commutator of the semi-covariant derivatives as below, The curvature, R P QM N is given by Note, that unlike ordinary DFT, an additional term is introduced in R P QM N . However it does satisfy the same properties as ungauged DFT, namely that, We can then define the semi-covariant curvature, S M N P Q , by Just as for an ordinary Riemann curvature tensor, the semi-covariant curvature satisfies the following symmetry properties on its indices, The Jacobi identity for the structure constants implies the Bianchi identities as well, The variation of S M N P Q is given by  It is worth noting that our curvature scalars are equivalent to the half-maximal and maximal gauged supergravity potential (2.9a) and (2.9b). For the half-maximal supersymmetric case, the NS-NS sector Lagrangian is given by This is exactly same potential as (2.9a). For maximal supersymmetric case, the NS-NS sector Lagrangian is given by and it can be rewritten as This is the potential (2.9b).
The generalised Ricci tensor can be defined from the equations of motion for projection operators. Under an arbitrary infinitesimal variation of the S M N P Q , we have and from this result, we may define the generalized Ricci tensor, which is a rank-two tensor from (2.41).

Supersymmetric Gauged Double Field Theory
We are now ready to consider supersymmetric gauged double field theory having half-maximal supercharges from 10D minimal SDFT [24,27]. The bosonic sector of the supersymmetric gauged DFT consists of DFTdilaton, d, and double-vielbeins, V M m ,V Mm .
• Bosons -Gravitino: ψ ᾱ p . The Dirac operators for Spin (1,9) spinors are denoted by [23] γ m D m ρ , Dmρ , γ m D m ψn . (3.5) The explicit form for these is then given by, From these fields we construct a supersymmetric action with half-maximal supersymmetry.
This action is invariant under the following SUSY transformations up to leading order in fermions, where SUSY parameter ε is a Spin (1,9) spinor with positive chirality,

Generalized Scherk-Schwarz reduced DFT as a gauged DFT
In this section we show how using this formalism, the gauged double field theory can be obtained from the generalised Scherk-Schwarz reduction from the higher dimensional ungauged double field theory. Let whereVM is an arbitrary O(D, D) vector and e −2d is a tensor density. The generalised Scherk-Schwarz reduction of the DFT connection is given bŷ  It is important to compare the reduced connection with the gauged DFT connection in (2.19). For consistency, we need to set f A = 0. If we calculate the difference, then we have On the right hand side, the last term is removed after contraction with a projection operator and so does not contribute to the fully covariant quantities. The first term, −(U −1 ) NQ ∂ M UQ P however does contribute. This is the difference between reduced parent DFT connectionΓ P M N in (4.3) and gauged DFT connection Γ P M N in (2.19) and it explains the additional term in the action that appeared in [16].
As discussed before, the gauged DFT action should be independent of Y I or UM M (Y) so that, However, if we carry out a generalised Scherk-Schwarz reduction on the parent DFT action, the reduced action is written in terms ofΓ P M N having explicit twist matrix dependence, UM M (Y), To get a UM M (Y) independent action, an additional term should be added which compensates the UM M (Y) dependence of the reduced action (4.7). The additional term is given by This term exactly reproduces the additional term in (2.4) of [16], It is worthwhile to compare how supersymmetry works. The difference of spin-connections is given by •μ,ν, · · · : D-dimensional vector indices, •m,n, · · · : Local Spin(1, D−1) vector indices, •m,n, · · · : Local Spin(D−1, 1) vector indices.

Reduction of ordinary supergravity
The ordinary Scherk-Schwarz reduction of two copies of the D-dimensional viellein is given bŷ The reduction of Kalb-Ramond field is then:
Let us now consider the Scherk-Schwarz reduction ansatz for the double vielbein,VMm andVMm where U I A (Y) is a generalised twist matrix. Finally, the Scherk-Schwarz ansatz of dialton is given bŷ andV where A M A is an unified gauge field, and V A a ,V Aā are n-dimensional double vielbeins parametrized as (5.14) One can find the transformation laws for various fields by substituting the reduced vielbeins defined in eq. To examine the symmetry transformations of the reduced fields, we should decompose the double gauge parameterXM as before,XM We then interpret X M as a generalised Lie derivative parameter and Y A as a gauge symmetry parameter.
The symmetry transformation of each of the reduced fields is given by Since projection operators can be written in terms of double vielbein, the Scherk-Schwarz reduction of the projection operators may be easily obtained using the reduction of the double vielbeins from (5.10) and (5.11), which yieldŝ Each component ofPMN iŝ where Similarly, each component ofPMN is given bŷ If we apply these reduction conventions, the bosonic part of the half-maximal supersymmetric gauged double field theory action (3.7) is reduced to where ω M N P is Chern-Simon 3-form, (5.28) One has thus shown that the reduced action (5.26) is exactly same as the half-maximal gauged supergravity action in [18].

Killing Spinor Equations
In the context of ordinary supergravity, the Killing spinor equations have proven very useful in finding solutions to the equations of motion that preserve some fraction of supersymmetry. Essentially they reduce the supergravity equations to be first order in derivatives and often when combined with a suitable ansatz for the metric and fields will lead to linear equations. Importantly, the Scherk-Schwarz factorisation ansatz for double field theory allows dependence on both the usual coordinates and their duals simultaneously.
Finding actual solutions that obey this ansatz is much more difficult than finding solutions that obey the strong constraint. Obviously from the connection to gauged field supergravity there is a clear interpretation of some of these solutions. However, one might wish to just try and seek solutions in double field theory with a Scherk-Schwarz ansatz and interpret this as a double field theory geometry. This then involves solving the double field theory equations of motion. Obviously this is a hard problem since the double field theory equations of motion are as hard to solve as Einstein's equations.
The natural thing is to then use the Killing spinor equations with a Scherk-Schwarz factorisation ansatz (reduction). One can then seek solutions in the full double field theory preserving different fractions of supersymmetry but that obey the Scherk-Schwarz ansatz. In what follows we write down the Killing spinor equations with the Scherk-Schwarz anstaz.
We list the reduction of the relevent spin connections that are needed for the Killing spinor equations in the appendix. Inserting these generalised Scherk-Schwarz reduced spin connections into to the Killing spinor equations, produces: for dilatino and for the gravitino.
A great deal of insight has been achieved through the analysis of the usual Killing spinor equations.
It would be very interesting to investigate these equations that come from double field theory in detail.
Hopefully one could obtain results along the lines as of the G-structure geometric spinor approach [65]. We leave this for future work. Reduction of the covariant combinations ofΦMmn yields:

A Reduction of spin-connections
andΦMmn part: