Fermionic T-duality in massive type IIA supergravity on AdS_{10-k} x M_k

Fermionic T-duality transformation is studied for the N=1 supersymmetric solutions of massive type IIA supergravity with the metric AdS_{10-k} x M_k for k=3 and 5. We derive the Killing spinors of these backgrounds and use them as an input for the fermionic T-duality transformation. The resulting dual solutions form a large family of supersymmetric deformations of the original solutions by complex valued RR fluxes. We observe that the Romans mass parameter does not change under fermionic T-duaity, and prove its invariance in the k=3 case.


Introduction
Families of new solutions in type II supergravity of the form AdS 10−k × M k were found in [1,2,3] for k = 3, 4, 5. For k = 3 and 5 the solutions belong to the massive type IIA supergravity [4], while for k = 4 they solve type IIB field equations. In the AdS 7 × M 3 solutions the internal manifold M 3 is topologically a sphere. The requirement of unbroken supersymmetry was demonstrated to fix M 3 to be a fibration of S 2 over the interval. The background fields of these solutions were given by the system of first order differential equations, which the authors of [1] were solving numerically. An infinite family of solutions was obtained, which have embedded D6/D8 brane systems. The holographic interpretation of these theories was investigated in [5,6].
Analytic solutions to these equations were found later, together with a map that relates them to the AdS 5 and AdS 4 solutions in massive type IIA [3,7]. The AdS 5 solutions that we will be concerned with in the present paper are geometrically where Σ 2 is a Riemann surface of genus g ≥ 2, and M ′ 3 is a three-manifold related in a certain way to M 3 . A recent review of these and related developments can be found in [8].
The aim of this paper is to study the effect of fermionic T-duality on the AdS 10−k × M k solutions for k = 3 and k = 5. It is a well known fact that the transformation rules of the background fields under ordinary T-duality (known also as the Buscher rules [9,10,11]) can be represented in a way which manifests the role of the Killing vector as the T-duality transformation parameter [12]. Fermionic T-duality is a more recent development [13,14], which generalises T-duality to the case of fermionic isometries in superspace. The role of a Killing vector is played by a Killing spinor, which parameterises an unbroken supersymmetry of the initial background. The fermionic T-dual background can then be constructed according to the fermionic Buscher rules, which depend explicitly on the Killing spinor. The key difference from the ordinary T-duality rules is that the metric and the NSNS 2-form field b do not change, whereas the RR fluxes are transformed in a certain way that depends on the Killing spinors of the original background. Fermionic Tduality plays a key role in the self-duality of various solutions of maximal d = 10 supergravity that are important from the AdS/CFT correspondence point of view.
In order to construct fermionic T-duals of the solutions of [1,3] we study the unbroken supersymmetries of these backgrounds and solve the Killing spinor equations in full generality. Note that concise expressions for the AdS 7 × M 3 Killing spinors have appeared in [7,19], while the Killing spinor structure of the AdS 6 × M 4 solutions has been studied in detail in [20]. Fermionic T-duality preserves supersymmetry and the metric of the solution [13], hence using the Killing spinors we are able to generate new supersymmetric solutions with the same metric AdS 10−k × M k , k = 3, 5. The new solutions presented here are essentially deformations of the original solutions by complex valued RR fluxes, akin to the deformations of the D-brane solutions found earlier in [21].
The behaviour of massive type IIA supergravity solutions under fermionic T-duality is an open question since the early work [22], where a fermionic duality symmetry of type II supergravity action has been found that includes fermionic Tduality as a special case. While formally applicable to both ordinary and massive type IIA supergravity, the analysis of [22] assumed vanishing mass parameter m before the duality transformation, and resulted in keeping m zero after the duality as well. It was later reported [23], that the extension of the transformation of [22] to a nonzero Romans mass, when applied to characteristic solutions of massive type IIA, such as D8-branes and the warped product AdS 6 × S 4 [24], yields no change in the mass parameter, and that the entire transformation is trivial in that case.
In the current study we will be using the original fermionic T-duality formalism developed in [13]. We will give a proof that the Romans mass of the AdS 7 × M 3 solutions of [1] never changes under fermionic T-duality. In the less tractable case of the AdS 5 × M 5 solutions of [3] evidence will be given that the Romans mass does not change, although this will not be proved rigorously.
The rest of this article is organised as follows. We begin in the section 2 by studying the simpler case of the AdS 7 × M 3 solutions. Then in the section 3 the AdS 5 × M 5 solutions are considered. In both cases we briefly review the solutions, then formulate the Killing spinor equations and solve them. Section 4 presents the fermionic T-duals after a concise review of the fermionic Buscher rules. We briefly discuss the results in the final section 5. Our notation and conventions are summarised in the appendix B.
2 AdS 7 × M 3 solution The AdS 7 ×M 3 background of [1] is an N = 1 supersymmetric solution in massive type IIA supergravity. The metric is given by (2.1) The ten spacetime coordinates x µ are split into the AdS 7 coordinates (x 0 , . . . , x 5 , ρ) and the three coordinates (r, β, θ) on the internal manifold. M 3 is an S 2 fibration over an interval that is parameterised by the coordinate r. The S 2 fibre shrinks at the ends of the interval, so that M 3 is topologically a 3-sphere. The warping function A(r), as well as the dilaton φ(r), and the parameter x(r) of the internal metric, depend on r only. The function x(r) is related to the volume of the S 2 fibre.
These three functions are defined by the following system of differential equations: where m is a constant mass parameter of Romans supergravity. The authors of [1] study numerical solutions to this system. Later an analytic solution of these equations has been found in [3]; it describes backgrounds with D6 or D8 branes. For our purposes the equations (2.2) will be enough, and we will not spell out the details of the explicit solutions.
The metric is diagonal, and we can choose the vielbein in the simple form: We underline the world indices and assume that dx µ = (dx 0 , . . . , dx 5 , dρ, dr, dβ, dθ).
Then the vielbein e a = e a µ dx µ corresponds to the metric (2.1), ds 2 = η ab e a e b . With this choice of the vielbein the nonvanishing components of internal spin connection are: The other nonvanishing fields of the supergravity background are the RR 2-form and the NSNS H flux: H = H 789 e 7 ∧ e 8 ∧ e 9 = − 6e −A + xme φ e 7 ∧ e 8 ∧ e 9 . (2.5)

Killing spinors
Let us solve the Killing spinor equations for this background. This requires finding a spinor ǫ such that the supersymmetry variations of the type IIA fermions vanish, δ ǫ λ = 0 = δ ǫ ψ µ . Our supersymmetry and spinor conventions are summarised in the appendix B. After decomposing the dilatino supersymmetry variation (B.3) with respect to the Weyl components of the Killing spinor (B.7), we obtain: where ε 1,2 are 16-component Weyl spinors, defined in (B.7). Note that ∂ 7 φ = ∂ r φ (2.2), since e 7 r = 1, and the values of H 789 , F 89 can be read off from (2.5). Using the decomposition of ε 1,2 in terms of the AdS 7 spinors ζ and M 3 spinors χ 1,2 (B.7), δλ may be brought to the form: where (2.8) Requiring that both δ 1 λ and δ 2 λ vanish imposes four linear equations on the four components of the internal Killing spinors χ 1 = a ′ b ′ , χ 2 = a b . The resulting system has rank two, and can be solved in a straightforward manner. For example, we can express a ′ , b ′ in terms of a, b: where we have denoted the coefficients in (2.8) as: (2.10) Using the explicit values of A, B, C, and D from (2.2) and (2.5), we observe that the expressions simplify considerably: where x = x(r) was defined in (2.2). To summarize, the dilatino supersymmetry variation vanishes for the following values of the internal spinors: (2.13) Under the 7 + 3 split of (B.7) the AdS 7 spinor ζ factors out, where The covariant derivative acting on a spinor is D µ = ∂ µ + 1 2 / ω µ . Note that since all the mixed components ω i+6,mn of the spin connection vanish (2.4), the derivative of Using the values of H 789 , F 89 from (2.5) and the internal spinors χ 1,2 found in (2.12), after some algebra one can represent the equations (2.15) as (2.17) Despite the presence of arbitrary functions A(r), x(r), this system can be solved exactly: where ϕ is a new internal variable related to r by sin 2ϕ = x(r). The solution is parameterised by the two constants c 1,2 . Together with (2.12), this completely determines the internal part of the Killing spinor, χ 1,2 .
Simplifying the AdS 7 part of the gravitino Killing spinor equation δψ m = 0, we get the standard expression for the AdS 7 Killing spinor, as briefly reviewed in the appendix C: Here ζ 0 is an arbitrary 8 component complex spinor parameter, and ζ 0 = ζ 0 + + ζ 0 − is its decomposition into eigenvectors of α 6 , which is the gamma matrix corresponding to the AdS 7 radial direction ρ. The complete Killing spinors are then given by (B.7): (2.20) It is easy to see that there is a total of 16 Killing spinors ǫ a , a = 1, . . . , 16, which means that the solution of [1] preserves half of the maximal supersymmetry. To check this, note that we can represent the eigenvectors of α 6 (B.9) as

AdS 5 × M 5 solutions
The AdS 5 × M 5 solution of [3] has the metric given by (3.1) Σ g is a Riemann surface of Gaussian curvature k = −3 and genus g ≥ 2. Its metric is written in terms of the coordinate z = x 1 + ix 2 of the complex upper half-plane.
Coordinates on the Riemann surface x 1 , x 2 are not to be confused with the AdS 5 coordinates x 0 , . . . , x 3 , ρ (we will never raise or lower indices of coordinates).
The M 3 subspace of the internal manifold is fibered over the Riemann surface Σ g , which is reflected by the long derivative Dψ = dψ + ρ appearing in the metric which for the above metric takes the form The warping function A(r), as well as the parameter x(r) of the internal metric, and the dilaton φ(r), only depend on the coordinate r, which runs over an interval.
All of these are defined by the following system of ODEs: where m is a constant mass parameter of Romans supergravity. Note there are slight differences in the coefficients as compared to the AdS 7 × M 3 case (2.2). As was already mentioned there, we will not need the explicit form of the solutions to this system of equations.
We can choose the following simple vielbein: We underline the world indices and assume that dx µ = (dx 0 , . . . , dx 3 , dρ, dx 1 , dx 2 , dr, dθ, dψ). Then the vielbein e a = e a µ dx µ corresponds to the metric (3.1), ds 2 = η ab e a e b . With this choice of the vielbein the nonvanishing components of internal spin connection are: The nonvanishing fluxes of the solution are (3.7)

Killing spinors
Let us construct the Killing spinors for the above background. As in the section 2, we start with the variation of the dilatino (B.3). Plugging in the values of the fluxes (3.7) and using the 5 + 5 decomposition of the 16-component Weyl spinors , we find that the dilatino variation can be written: where 9) and the coefficients are given by: (3.10) Their explicit values can be read off from (3.7), (3.4). Note that we are using a different (5 + 5) gamma-matrix decomposition in the present case, see appendix B.
Requiring that δ 1 λ, δ 2 λ in (3.9) vanish results in a system of eight linear homogeneous equations for the eight unknown components of χ 1 = (f 1 , . . . , f 4 ) and . . , f 8 ). Determinant of this system vanishes due to a rather non-trivial relationship between the coefficients: The rank of the corresponding matrix is six, which means that the system may be solved, for instance, for (f 1 , . . . , f 6 ) in terms of (f 7 , f 8 ): (3.12) At this point f 7 , f 8 are two arbitrary functions of the coordinates (x 1 , x 2 , r, θ, ψ) on M 5 . Their values are fixed by the Killing spinor equations that follow from δψ i+4 = 0 for i = 1, . . . , 5 (ψ i+4 are the components of the gravitino with the vector index along the internal manifold). Going through the same steps as in the AdS 7 × M 3 case, we obtain the following system of PDEs for the functions f 7 , f 8 : Despite explicit dependence on many arbitrary functions, this system can be solved exactly. The solution is parameterised by the two numbers c 1 , c 2 : where ϕ is a new internal variable related to r by x(r) = sin ϕ. We have also defined Note the following identity, which can be obtained by integrating the constraint (3.3) with respect to x 1 and x 2 : This relationship can be employed in order to represent χ in different ways.
From the AdS 5 part of the gravitino Killing spinor equation δψ m = 0 we get the standard expression for the AdS 5 Killing spinor, as briefly reviewed in the appendix C: Here ζ 0 is an arbitrary 4 component complex spinor parameter, and ζ 0 = ζ 0 + +ζ 0 − is its decomposition into the eigenspinors of α 4 , which is the gamma matrix corresponding to the AdS 5 radial direction ρ. The complete Killing spinors are then given by (B.7): It is easy to see that there is a total of 16 Killing spinors ǫ a , a = 1, . . . , 16, which means that the solution of [3] preserves half of the maximal supersymmetry. To check this, note that we can represent the eigenvectors of α 4 (B.13) in the form

Fermionic T-duality
The Killing spinors that we have found can be used to study fermionic T-duals of the supergravity solutions of the sections 2, 3. We will proceed to this after briefly reviewing the rules that link fermionic T-dual backgrounds (fermionic Buscher rules).
For more detail on the basics of fermionic T-duality see the reviews [23,25] or the original derivation [13].
Fermionic T-duality only transforms the RR fluxes and the dilaton φ; there is no change in the metric nor in the antisymmetric b field. In type IIA supergravity it is convenient to unify the RR field strengths F µν and F µνρσ together with the Romans mass parameter m into a bispinor F α β (for the spinor and gamma matrix conventions see appendix B): Fermionic T-duality transformation rules are: where C IJ is the matrix defined by The transformation parameters ǫ I are the Killing spinors of the original background.
Indices I, J run over the subset of the Killing spinors that we have chosen to Tdualise. In particular, one may choose to do fermionic T-duality with respect to just one Killing spinor, in which case the I, J indices become redundant and C IJ is no longer a matrix but just some scalar function C. For consistency of the above transformation, the Killing spinors must obey the so called abelian constraint which comes from the requirement that the corresponding supersymmetries anticommute [13]. Alternatively, the abelian constraint may be interpreted as integrability condition for (4.3) [22]. Killing spinors ǫ a . For a generic complex linear combination ǫ = ǫ a + iǫ b (assuming that a = b) the abelian constraint takes the form: is a valid parameter for a fermionic T-duality transformation of (4.   This is a characteristic form of a fermionic T-dual solution whenever the complexified Killing spinor is constructed as explained above. Apart from ǫ = ǫ 1 + iǫ 2 one may consider, e.g. ǫ 3 + iǫ 4 , ǫ 5 + iǫ 6 , and ǫ 7 + iǫ 8 . Each of these gives a fermionic T-dual RR flux same as above, but the values of the indices abc are slightly different every time. The resulting fermionic T-dual may be simplified considerably if these four Killing spinors are dualised at the same time. To achieve this, take the Killing spinor ǫ I that appears in (4.2), (4.3) to assume the values just listed,  The argument after the equation (4.6) allows for multiple other complexification patterns. We note here one more specific case that leads to a simple fermionic T-dual, and later we will draw certain conclusions valid for any choice of complexification. Consider the following Killing spinors:  Again we are free to choose C IJ to be a unit matrix, hence the background is the same as before, up to 12 new RR 4-form components:  Killing spinor as well. These are the Killing spinors whose AdS part has dependence on the flat AdS coordinates, which result from keeping the ζ 0 + part of the complete AdS Killing spinor (2.19), (3.17). The resulting fermionic T-dual RR fluxes are rather intricate and we will not give their explicit form here. These expressions are akin to what was classed as the 'complicated' fermionic T-dual case in [21], or as a T-dual with respect to the supernumeracy Killing spinors in [15]. They are similar to the fluxes of (4.14), additionally multiplied by a polynomial of degree up to 4 in the AdS coordinates. No matter what kind of a Killing spinor we use, the fermionic T-duality parameter C appears always to be a constant.

Constant fermionic T-duality parameter
All the fermionic T-dual backgrounds described above have a property that the duality parameter C is a constant, ∂ µ C = 0. Vanishing of the corresponding Killing Killing spinors we arrive at the following equations for µ = 7, 8, 9:

Constant Romans mass parameter
The behaviour of IIA mass parameter m under fermionic T-duality is a long standing question [23] which was one of the motivations for this work. Recall that we have incorporated m into the bispinor of RR fields (4.1), which under fermionic T-duality has the transformation law: Taking the trace of this relation we remove the 2-form and the 4-form terms in the gamma-matrix expansion of F and F ′ . This leaves us with the transformation law of the mass parameter under fermionic T-duality: Thus, m is shifted by the trace part of the Killing spinor matrix A. In particular, this shift might in principle generate mass in some type IIA background that was originally massless. The rescaling by a factor of (det C) −1/2 in the present case is trivial as C IJ = const, but in general C IJ can be a coordinate dependent function.
Note that the Romans mass parameter is intrinsically a constant quantity, hence a nontrivial coordinate dependent C IJ would require a very special form of tr A, so as to keep both m and m ′ constants.  (4.16). Assume that we choose some nonzero constant value for the parameter C IJ . Explicit computation then shows that the trace of the matrix A α β = C −1 IJ ǫ α I ǫ Jβ is expressed in terms of the same polynomials that appear in the constraints (4.16).
Hence, the anticommutation constraint for the supersymmetries implies that the only possible transformation of the mass parameter in the present case is rescaling by a constant (det C) −1/2 . where ρ is the AdS radial coordinate. In the self-dual cases this contribution can be canceled by fermionic T-duality, which adds to the dilaton an extra term

Discussion
However, in the section 4.3 we have shown that the parameter C for the solutions of [1,3] can only assume constant values, hence the standard self-T-duality scheme does not work here. This agrees with the classifica-tion of self-T-dual backgrounds constructed in [26,27].
We have seen in the section 4 that the RR 2-form flux rarely appears in the fermionic T-dual. In fact, for the AdS 7 × M 3 solutions none of the Killing spinors produce any contributions to the RR 2-form. On the contrary, in AdS 5 × M 5 the 2form flux routinely appears after fermionic T-duality and we have seen the example of it in (4.14).
The new components of the RR 4-form that appear after fermionic T-duality in the AdS 7 × M 3 background could not be found in the original study of [1] because the Ansatz of the AdS 7 × M 3 solution that was used there intentionally did not include any 4-form flux. This restriction was put in order to protect the AdS 7 symmetry, because a nonvanishing 4-form necessarily would a have at least one leg off the internal M 3 manifold. The fermionic T-dual solutions with the 4-form flux found here nevertheless keep the same AdS 7 geometry. This is possible essentially because the fermionic T-dual fluxes that we have found do not backreact: they have vanishing energy-momentum tensor and therefore do not contribute to the gravity field equations. Vanishing stress-energy of some fermionic T-dual fluxes is a feature already observed in [21] for the fermionic T-duals of D-branes. Note also that the D-brane dimension is not modified by fermionic T-duality [13], thus the fact that When a spinor ǫ appears without an index, it means the full 32 component spinor as above. Using the explicit gamma-matrix realizations below, it is easy to verify that we are dealing with Weyl representation as the d = 10 chirality operator is in its conventional form Γ 11 = Γ 0 . . . Γ 9 = 1 16 ⊗ σ 3 .
In order to find the Killing spinors of a supergravity background we require that the supersymmetry variations of all fermionic fields in the theory vanish. Variations of the massive type IIA fermions in the conventions of [28] are given by Note that since in both gamma-matrix representations below Γ 11 = 1 ⊗ σ 3 , the chirality constraints immediately imply that σ 3 v ± = ±v ± , hence v + = 1 0 , v − = 0 1 . Charge conjugation for the component spinors ζ c , χ c is defined according to the decomposition of the charge conjugation matrix B.
Occasionally it proves more convenient to work with 16-component spinors ε 1,2 defined as (B.7) B.1 Gamma-matrices for d = 7 + 3 spacetime In choosing the representation we mostly follow the conventions outlined in [1].