Pin(d, d) covariance of pure spinor equations for supersymmetric vacua and non-Abelian T-duality

In a supersymmetric compactification of Type II supergravity, preservation of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 supersymmetry in four dimensions requires that the structure group of the generalized tangent bundle TM ⨁ T∗M of the six dimensional internal manifold M is reduced from SO(6) to SU(3) × SU(3). This topological condition on the internal manifold implies existence of two globally defined compatible pure spinors Φ1 and Φ2 of non-vanishing norm. Furthermore, these pure spinors should satisfy certain first order differential equations. In this paper, we show that non-Abelian T-duality (NATD) is a solution generating transformation for these pure spinor equations. We first show that the pure spinor equations are covariant under Pin(d, d) transformations. Then, we use the fact NATD is generated by a coordinate dependent Pin(d, d) transformation. The key point is that the flux produced by this transformation is the same as the geometric flux associated with the isometry group, with respect to which one implements NATD. We demonstrate our method by studying NATD of certain solutions of Type IIB supergravity with SU(2) isometry and SU(3) structure.


Introduction
Non-Abelian T-duality (NATD) is an extension of Abelian T-duality, which works well as a solution generating mechanism for string backgrounds with non-Abelian isometries. Although the rules for NATD for the metric, the B-field and the dilaton field has been known for almost thirty years [1,2], those for the RR fields has been understood relatively recently [3]. Since then, NATD has been widely used to generate new supergravity solutions with interesting holographic duals, see for example [4]- [13].
Recently, NATD has been described as a coordinate dependent O(d, d) transformation, [14][15][16][17][18][19]. The transformation under NATD of the metric, the B-field and the dilaton field is determined by a coordinate dependent O(d, d) matrix, which we will be calling T NATD , and the RR fields transform under the corresponding Pin(d, d) transformation generated by S NATD . Here ρ(S NATD ) = T NATD , and ρ is the usual double covering homomorphism from Pin(d, d) to O(d, d). This approach makes it possible to view NATD as a solution generating transformation for Double Field Theory (DFT), a framework which provides an O(d, d) covariant formulation for effective string actions [20][21][22][23] by introducing dual, winding type coordinates. Since T NATD is not constant, it is not immediate that NATD should be a solution generating transformation for DFT. However, it is a special O(d, d) matrix, determined by the structure constants C k ij of the isometry algebra of the original background, and viewed as a twist matrix within the framework of Gauged Double

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Field Theory, it gives rise to geometric flux f k ij = C k ij . This is the key point in showing that NATD is a solution generating transformation in DFT, which then provides a unified framework to prove that it is a solution generating transformation for Type II (generalised) supergravity. For details, see [17]. For a similar approach, also see [18].
An important question is whether supersymmetry is preserved under NATD. This problem is addressed in various papers, notably in [11][12][13][24][25][26][27]. In [11], the transformation under NATD of the gravitino and dilatino supersymmetry variations were shown to be the same provided that the Killing spinors did not depend on the isometry directions along which NATD was applied. Equivalently, supersymmetry was shown to be preserved (at least for a large class of backgrounds with SU(2) isometry) if the Killing spinors had vanishing Kossmann-Lie derivative with respect to the Killing vector fields generating the isometry. In the papers [24][25][26][27] NATD is applied to backgrounds with N = 1 supersymmetry. For such backgrounds, conditions for supersymmetry can be described by using tools from generalised geometry [28][29][30], as was first shown in [31]. In this case, equations coming from supersymmetry variations can be shown to be equivalent to a set of differential equations to be obeyed by two globally defined pure spinors. This will be discussed in detail in section 3. It is possible to apply NATD directly on these pure spinors and check whether the transformed pure spinors still satisfy the differential equations coming from supersymmetry. This was the approach taken in [24][25][26][27]32], where various backgrounds with interesting holographic duals were examined. In each case, the geometry supports an SU(3) structure with associated pure spinors, and it was checked by direct computation that the NAT dual of these pure spinors indeed satisfied the supersymmetry equations proving that NAT dual background also preserved at least N = 1 supersymmetry. For a similar approach where one works with backgrounds supporting a G 2 structure, see [33]. It should be noted that the Kossmann derivative of the Killing spinors along the isometry directions vanish if and only if the Lie derivative of the pure spinors (constructed as bilinears of these Killing spinors) along these directions vanish. This condition was met by all the examples considered in the papers mentioned above.
In this paper, we describe the transformation of pure spinors under NATD viewed as a Pin(d, d) transformation by utilizing the tools developed in [17]. This enables us to prove that NAT dual of pure spinors of N = 1 vacua still satisfy these differential equations (and hence, the dual background will also preserve at least N = 1 supersymmetry since Bianchi identities are also satisfied as shown in [17]), provided that they have vanishing Lie derivative along the isometry directions. To this end, we will first prove that the pure spinor equations for preserved N = 1 supersymmetry are Pin(d, d) covariant, by embedding them in DFT. Among other things, using the framework of DFT makes it easier to show that the action of the exterior derivative operator and the Pin(d, d) transformation must (anti-)commute, which is the trickiest part in the proof. This is when the Pin(d, d) matrix is constant. When it is not constant, as is the case with NATD, the pure spinor equations will not be left invariant. However, NATD is generated by a very special Pin(d, d) transformation yielding geometric flux, as we discussed above and again, this becomes the key point in showing that solutions of pure spinor equations are still solutions after NATD, provided that the pure spinors have vanishing Lie derivative along the isometry JHEP12(2021)071 directions. Compared to the methods already present in the literature our method has various advantages. First of all, describing the dualisation of pure spinors as a Pin(d, d) transformation makes the computations rather direct, as it is not needed to specify an ansatz for the seed background, as long as the isometry is respected by the whole geometry, the fields and the pure spinors. In particular, our proof is valid for any isometry group, not just SU (2). We should also note that our method makes the determination of the G structure of the NAT dual background rather straightforward. In the particular examples we will study in section 4, the seed background will be assumed to support SU(3) structure, and we will see directly how the associated pure spinors are transformed to pure spinors associated with an SU(2) structure. More generally, starting with a pure spinor associated with a generic SU(3) × SU(3) structure, it is possible to work out the G-structure of the NAT dual background, as is done in [34] for Abelian T-duality. In this paper, we will focus on the invariance of the N = 1 supersymmetry equations on pure spinors under NATD and will leave the discussion of the transformation of a generic SU(3) × SU(3) structure to future study.
The plan of this paper is as follows: in section 2, we review the methods developed in [17]. In section 3, we focus on the pure spinor equations, which were shown in [31,35,36] to be equivalent to the supersymmetry equations to be satisfied by Type II vacua with at least N = 1 supersymmetry. We embed these equations in the framework of Double Field Theory, so that the covariance under a general constant Pin(d, d) transformation becomes manifest. Then, in a separate subsection we discuss the case when the Pin(d, d) transformation is coordinate dependent (as it happens for NATD), and show that whether the transformed pure spinors satisfy the differential equations or not is completely determined by the fluxes generated by the Pin(d, d) transformation. Section 4 is devoted to explicit examples. This is the section where we consider the ansatz for Type IIB supergravity studied in [25]. This ansatz is general enough to cover many examples that are important in the context of AdS/CFT duality. We transform the pure spinors associated with the SU(3) structure supported by the geometry by applying the Pin(d, d) transformation generating the NATD and show that the resulting pure spinors (as well as the resulting metric, B field and the RR fields) are in agreement with the ones found in [25]. We end with a discussion of results and future directions in section 5.

Non-Abelian T-duality as an O(d, d) transformation
The purpose of this preliminary section is to review the methods developed in [17], where it was shown that NATD of a given d dimensional Type II background with isometry G can be obtained through the action of a coordinate dependent O(d, d) matrix (also called T NATD ) obtained by embedding the following O(n, n) matrix: algebra of the isometry group G, so i, j, k = 1, . . . , n. To keep things simpler, we will always assume that the isometry group acts without isotropy and d ≥ n. 1 The presence of the Lie group O(d, d), which is the global symmetry group of DFT, makes it possible to describe the transformation under NATD of the Type II supergravity fields as a transformation in DFT. More precisely, one rewrites the supergravity fields in terms of the DFT fields H, d, χ, where H is the generalized metric that encodes the metric and the B-field, d is the generalized dilaton field and χ is the spinor field that packages the modified RR fields of Type II supergravity in the democratic formulation. These fields, being solutions of Type II supergravity also solve the DFT equations in the supergravity frame. 2 As it is assumed that the isometry is respected by all the fields in the background, it is possible to go to a non-holonomic frame so that the DFT fields, when written with respect to such a frame, are independent of the isometry coordinates. In [17] we refer to such fields as untwisted fields. Plugging the initial DFT fields in the field equations of DFT (of both the NS-NS sector and RR sector of Type II supergravity), one sees that the untwisted DFT fields satisfy the field equations of Gauged Double Field Theory (GDFT), 3 with geometric fluxes associated with isometry. It was shown in [17] When the action of the isometry group G is not free, n might be bigger than d, in fact it may even exceed ten. However, the methods that are discussed here can be generalized as described in [59], where they assume that the action is not free but that there is a local symmetry group H ⊂ G. This is the case when the target space includes a coset space G/H. They show in [59] that NATD can be applied without fixing the local H invariance and that the dual action still inherits the H invariance. This can then be used to gauge away the n − d redundant degrees of freedom after dualisation. Specifically, the background matrix E = g + B in (3.7) is taken to be n dimensional but in such a way that only the part that projects onto the coset G/H is non-trivial. For example, if the background is S 3 and the isometry group used for dualisation is SO(4) then the metric g is taken as the diagonal 6 × 6 matrix diag(1, 1, 1, 0, 0, 0), see appendix B of [59].
After transforming under (2.1), which is in O(6, 6), one ends up with a six dimensional background. As demonstrated in [59], 3 of the 6 dual coordinates ν1, · · · , ν6 of the dual background can be removed by using the local SO(3) invariance that survives dualisation so that NAT dual of S 3 is also 3 dimensional, as it has to be. Although [59] [17]. 2 DFT is consistent only when one imposes the so called strong constraint, that effectively eliminates half of the doubled coordinates. This constraint is trivially satisfied when the DFT fields and gauge parameters are independent of the winding type coordinates. In this case, the DFT fields are said to belong to the supergravity frame, since the DFT action and field equations reduce to those of Type II supergravity in the democratic formulation. 3 GDFT is a deformation of DFT, obtained from a Scherk-Schwarz reduction and the deformation is determined entirely by the fluxes associated with the Scherk-Schwarz twist matrix [38][39][40][41]. 4 Here, we call the spectator coordinates excluding the isometry directions collectively x and the doubled coordinates of the NAT dual background collectively ν.

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Here, K = C −1 d S, and C d is given in (C.7) in appendix C. S is the element in Spin − (d, d) that projects onto H under the double covering homomorphism ρ between Pin(d, d) and O(d, d), that is ρ(S) = H. Similarly, ρ(S NATD ) = T NATD and up to a sign it is given as [17] S NATD = C n S θ = S β C n . (2.6) The factors S θ and S β in S NATD are the Spin + (10, 10) elements that projects onto the SO + (10, 10) matrix that generates the B-transformations and β-shifts with θ ij = ν k C k ij and β ij = ν k C k ij , respectively. B (x, ν) that appears in (2.4) is read off from H (x, ν) in (2.2). The field σ(ν) in (2.5) and (2.4) is non-vanishing only when the isometry group is non-unimodular. For the purposes of this paper, it can be taken to be zero. The fact that the NAT dual fields can be written in terms of DFT fields as in (2.2)-(2.5) makes it straightforward to prove that NATD is a solution generating transformation for the field equations of Type II supergravity. In fact, all one has to do is to show that the fields in (2.2)-(2.5) solve the DFT equations, since the coordinates (x, ν) can be identified with the physical space-time coordinates, putting all the fields in the supergravity frame. Due to the special form of the fields, this amounts to showing that the untwisted fields H(x), d(x), F (x) appearing on the right hand side of (2.2)-(2.5) solve the field equations of GDFT, with fluxes generated by T NATD . Now, the key point is that this is exactly the same as the geometric flux associated with the isometry group, that is, f k ij = C k ij , and we already know that the untwisted fields satisfy the GDFT equations with geometric flux. As a result, one concludes that NATD is a solution generating transformation for Type II supergravity, both in the NS-NS and the RR sector, simply owing to the fact that fluxes are preserved. The idea that preservation of flux should be a guiding principle in determining whether an O(d, d) transformation is a solution generating transformation for supergravity has also been used in [42][43][44][45] and very recently in [46]. A similar approach was applied in [47] to find solution generating U-duality transformations within the framework of exceptional field theory. In the next section, we will see that the same principle also plays a key role in examining preservation of supersymmetry under NATD.

Covariance of pure spinor equations under Pin(d, d)
As was first shown in the seminal paper [35], the conditions to be obeyed by the internal space in a supersymmetric compactification of Type II supergravity can be neatly described within the framework of generalized complex geometry [28][29][30]. Demanding that the four dimensional solution preserves at least N = 1 supersymmetry implies that the structure group of the generalized tangent bundle T M ⊕ T * M of the six dimensional internal manifold M is reduced from SO(6, 6) to SU(3) × SU(3). This topological condition on the internal manifold implies the existence of two globally defined compatible pure spinors Φ 1 and Φ 2 of non-vanishing norm. These Clif f (6, 6) spinors can be constructed from the internal spinors arising from the 10 dimensonal Killing spinors generating the supersymmetry transformations in 10 dimensions. A Clif f (6, 6) spinor can be mapped to a non-homogenous differential form (a polyform) through the Clifford map. It was shown

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in [31,35,36] (also see [48]) that the Killing spinor equations coming from supersymmetry variations is equivalent to the following differential equations for the two pure spinors: For computational details on derivation of these equations, see appendix A of [36] and appendix B of [48]. For the corresponding equations for general ten dimensional supersymmetric solutions which do not necessarily involve a four dimensional Minkowski space factor, see [49] (the equations above are discussed as a special case in their section 4.1). Note that for our purposes, we have presented the equations in a form where the B field appears explicitly, rather than writing them in terms of the differential operator d H = d+H∧ as was originally done in [31,35,36]. 5 In the equations above, A is the warp factor that appears in the compactification ansatz φ is the dilaton field and * 6 is the Hodge duality on the six dimensional internal manifold. F is related to the polyform F (10) that encodes the RR fields in the democratic formulation of supergravity [50] in the following way Here, F = F 0 + F 2 + F 4 + F 6 for Type IIA and F = F 1 + F 3 + F 5 for Type IIB, and they are internal forms having components only along the six dimensional internal space. Also, for an n-form A n . As a Spin(d, d) spinor F has positive chirality for Type IIA and is of negative chirality for Type IIB. The chirality of the pure spinor Φ 1 is the same as that of the RR fluxes and the pure spinor Φ 2 has opposite chirality.
In the next two subsections, we will show that these equations are covariant under Pin(d, d) transformations.

Constant Pin(d, d) transformation
Although (2.1) is non-constant, we start by considering the transformation of the pure spinor equations under a constant O(d, d) matrix T and the corresponding Pin . 5 In fact, it is more common in the literature to express these equation in terms of the operator dH = d − H∧. This involves a field redefinition H → −H for Type IIA with respect to the conventions of [50]. In section 4 we will be looking at a IIA background with non-trivial B-field, so we prefer to agree with the conventions of [50] for Type IIA (since the conventions adopted in [17] agree with those of [50]), and this means we need the above field redefinition for Type IIB. This also means that our convention for the B-field is the opposite of that of [25]. Indeed, the B-field we find in (4.8) in section 4 has opposite sign compared to the B-field found in [25].

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The transformation of the RR fluxes F under P ∈ Pin(d, d) is [51] Here, the transformation of the B-field is read off from the antisymmetric part of the transformed background matrix E ≡ g + B: Note that this is equivalent to the aforementioned transformation of the generalized metric [52] H It is known that the pure spinors transform under Pin(d, d) in basically the same way as the RR fields transform. However, there is a slight change which makes sure that the norms of the pure spinors are kept invariant (up to a sign). The norm Φ of a pure spinor Φ is defined [31,35,36] via the Mukai pairing <, >, which is an invariant bilinear form on spinors (see appendix C): where vol is the volume form determined by the metric. As discussed in detail in appendix C, the Mukai pairing has the following transformation property under the action of certain elements P of Pin(d, d): where either P ∈ Spin(d, d) or is of the form P = C n S or P = SC n with S ∈ Spin(d, d) and C n is as in (C.7) (Recall that the NATD matrix (2.6) is of this form). On the other hand the volume form vol = d 1 transforms as This follows immediately from the transformation of the metric g which can be read off from the symmetric part of E in (3.7). Hence, the transformation of the pure spinors under Spin(d, d) must be accompanied by a scale transformation with a factor of √ G: 6

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so that the norm (3.10) remains invariant up to a sign. Note that this extra factor of √ G also ensures that since A is invariant and the dilaton field φ transforms exactly with the same √ G factor: The transformation rule (3.15) follows from the fact that the generalized dilaton field Now all we have to do is to figure out the transformation of the term involving Hodge duality on the right hand side of equation (3.2) and also to show that the action of P and the exterior derivative operator d on the Clif (d, d) spinors Φ 1,2 and F commutes.
For both purposes, we find it useful to embed these equations in Double Field Theory. Towards this we extend the exterior derivative operator d = 1 Here, the gamma matrices Γ M = (Γ i , Γ i ) are the Clifford algebra elements satisfying the following Clifford product relations: For future reference we also defined in (3.17) Also, we write λ(F ) as [21,41] Note that the equation (3.20) is valid in all even dimensions, 7 in particular for Rewriting the equations (3.1) and (3.2) we get These equations reduce to equations (3.1) and (3.2) in the supergravity frame where fields do not depend on the winding type coordinates so that∂ i = 0. The upper sign in the last term of (3.26) is for Type IIB and the lower sign is for Type IIA. This is because in six dimensions * 6 λ = λ * 6 for odd degree forms, whereas * 6 λ = −λ * 6 for even degree forms. We know that F (10) in (3.4) transforms as in (3.6). Let us discuss what this implies for the transformation of the internal forms F . We have ( 3.27) where we have used (3.24) and the fact that vol 4 , being an even form, commutes with all elements of Pin(d, d). To rewrite (3.27) in the form (3.4) we first define which is again an internal form, as all the Pin(d, d) operators on the left hand side have actions only on the internal space and then use the fact that under P ∈ Pin(d, d) the field K d transforms as Inserting a P −1 P after K in the second term of the right hand side of (3.27) and using (3.24) and (3.28), we obtain Note that F has components only along the six dimensional deformed space and the Hodge duality is taken with respect to the metric after the O(d, d) transformation. This shows us that not only the polyform F (10) that encodes the RR fields in the democratic formulation, but also the internal polyform F that appears in the pure spinor equations (3.1), (3.2) transform in the expected way as given in (3.28 for any spinor field χ. Using the relations where h is the SO(d, d) element that satisfies ρ(S −1 ) = h, we see that for constant S ∈ Spin(d, d): Then, the commutation relation (3.32) holds, if we have Note that we would have in DFT, since one also transforms X → X = hX. However in all the examples we will be looking at, the transformation generated by P will act only along the coordinates on which the pure spinors will not depend, so that we will always have X = X and hence ∂ A χ = ∂ A χ.

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Consider the Pin(d, d) elements Λ i given in (C.8). From the Clifford commutation relations (3.18) one can easily compute 8 In this paper, we will be looking at the Pin(d, d) elements that can be written as a product of Spin(d, d) elements and Λ i , simply because the NATD matrix is of this form. Our discussions here can be straightforwardly extended so as to include the Pin(d, d) elements which also involve the elements Λ + i given in (C.11), but we refrain from doing that in order to avoid equations cluttered with pluses and minuses.
Due to the relations (3.38), we see that provided that M = i or M = i , which then implies that the differential d = Γ M ∂ M and P commutes if P involves an even number of Λ i s and they anti-commute otherwise.
As discussed above, this condition is automatically satisfied for Abelian T-duality, due to the existence of d commuting isometries. This makes it possible to choose a coordinate system such that none of the fields depend on the coordinates along which the (constant) O(d, d)/Pin(d, d) transformation acts, and hence the desired commutation or anti-commutation relations hold. Therefore, we conclude that the pure spinor equations are covariant under Abelian T-duality. As for NATD, (3.35) is also satisfied with a convenient choice of coordinates (again due to existence of isometries), but we still need to discuss the situation with non-constant P , since the NATD matrix (2.1) is not constant as has been assumed above. This discussion will be carried out in the next section. Note that the covariance of the equations (3.1), (3.2) for certain cases has been discussed before, albeit in a different language. For example, in [34] the covariance of the pure spinor equations for backgrounds with U(1) isometry was shown. 9 Another example is the Lunin-Maldacena (LM) transformation (also called TsT transformation) which can be described as on O(2, 2) transformation [53][54][55]. In [56] the transformation of the pure spinors corresponding to an SU(3) structure under this O(2, 2) transformation was discussed within the framework of generalized complex geometry, as we do here. That the transformed pure spinors (now corresponding to an SU(2) structure) still satisfy the supersymmetry equations was also checked for this particular O(2, 2) transformation. 8 Note that (3.38)

Non-constant Pin(d, d) transformation
In this subsection, we extend the discussion in the previous subsection to the case where the Pin(d, d) transformation (and hence the corresponding O(d, d) transformation) depends on some of the internal coordinates. This is important, as the NATD transformation and the Yang-Baxter transformation are known to be generated by such coordinate dependent Pin(d, d) transformations. The transformation properties summarized in (3.14), (3.31) are obviously still valid, even when P ∈ Pin(d, d) is coordinate dependent. However, one has to be more careful in discussing the commutation of the exterior derivative operator d and the action of P , as now d also acts on P . Let us first discuss the case when the Pin(d, d) matrix does in fact lie in the subgroup Spin + (d, d), P = S ∈ Spin + (d, d) (so that we can use the useful identity (3.41)): where ρ(S −1 ) = h and in passing to the second line, we have used (3.33). To calculate the second term in (3.40) we use an important identity that follows from the fact that the Lie algebras of SO(d, d) and Spin(d, d) are isomorphic: Here, f ABC are the fluxes associated with the matrix S (see [17,41] for the definition). Now, we again assume that the transformation matrix S is such that (3.35) is obeyed. We emphasize again that this condition is trivially satisfied if the field χ does not depend on the coordinates along which S and hence h acts nontrivially. This is indeed the case for NATD and is guaranteed by the fact that NATD acts along isometry directions. Then, under this assumption, we have where Let us now discuss what happens when P involves odd number of Λ i factors, so that P does not lie in the Spin(d, d) subgroup (if the number of Λ i factors is even, then S is still in Spin(d, d), albeit not in the subgroup Spin + (d, d) connected to identity). For simplicity, we assume that P is of the form P = C n S, where S ∈ Spin + (d, d) and C n is as in (C.7) with n odd. Equation (3.33) is valid for all Pin(d, d) elements, so we have  [37]). Rewriting the first line of (3.40) for P = C n S and using (3.44) we have where we have also used P −1 ∂ M P = S −1 ∂ M S for P = C n S. Using (3.41) again, one can see that we have Here, n is the number of Λ i factors that appear in the definition of P = C n S, S ∈ Spin + (d, d).
Before we move on to the next subsection, we would like note that the transformation of pure spinor equations under a non-constant O(d, d) transformation was also studied in [57] and [58]. They called such transformations twist transformations and also used them as solution generating transformations in Type II theory.

Invariance under NATD
As we discussed in section 2, the transformation under NATD of the fields in the NS-NS sector can be performed via the action of the matrix T NATD given in (2.1). Accordingly, the transformation of the RR fields can be performed via the projected element S NATD under the double covering homomorphism between Pin(d, d) and O(d, d), see equations (2.2)- (2.5). An important point that should be stressed here is that S NATD and T NATD act on the so-called untwisted fields g(X), B(X), φ(X), Φ(X) and F (X). These untwisted fields depend only on 10 − dimG coordinates, where dimG is the dimension of the non-Abelian isometry group G, 10 and is related to the background fields g(X, θ), B(X, θ),

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φ(X, θ), Φ(X, θ) and F (X, θ) exactly as in (2.2)-(2.5), where we replace the NATD coordinates ν with the space-time coordinates θ associated with the isometry directions 11 and the matrices T NATD and S NATD with L and S L , respectively with, For more details see [17]. We also assume that the pure spinors associated with the background respect the isometry so that (3.13) also holds for both pure spinors: 12 Now suppose that the background we start with preserves at least N = 1 supersymmetry so that the pure spinor equations (3.1), (3.2) are satisfied. According to the discussions in the previous subsection and the paragraph above, this means that the untwisted fields (which have no dependence on the isometry directions) satisfy the deformed pure spinor equations (3.48), (3.49), where the deformation is determined by the flux associated with the matrices L and S L . But this is just geometric flux with f k ij = C k ij , see [17]. Now we act on these untwisted fields with the NATD matrices (2.1), (2.3) as in (2.2)-(2.5) to generate the NAT dual background. The resulting fields satisfy the field equations of Type II supergravity as was shown in [17] by embedding these equations in DFT. To check supersymmetry of the dual background we also transform the untwisted pure spinors of the initial background (that is, the pure spinors Φ(X) in (3.51) rather than Φ(X, θ)) as in (3.13) with P = S NATD . Now we have to check whether these new pure spinors Φ(X, ν) still satisfy the supersymmetry equations (3.1), (3.2). As discussed in the previous subsection, this is equivalent to checking whether the untwisted pure spinors satisfy the deformed supersymmetry equations (3.48), (3.49), where the deformation is determined by the flux associated with the NATD matrix S NATD . As discussed above, due to the special form of the NATD matrix: S NATD = C n S θ , the associated flux can be computed by calculating the flux associated with S θ first (which gives the H-flux) and then raising one index with J 6 n . This yields geometric flux with f k ij = C k ij , and we already know that the untwisted pure spinors satisfy these deformed equations due to the existence of isometry respected by the initial background and the pure spinors associated with it. This completes the proof that a background that preserves N = 1 supersymmetry will still be supersymmetric after NATD. 11 We choose a coordinate system adapted to the isometries so that the fields can be written in this separated form. 12 Note that the second equality in (3.51) is valid due to the special form of L and SL, see equations (4.22)-(4.23) in [17].

JHEP12(2021)071 4 Examples
In this section, we will demonstrate how the NATD transformation formulas (2.2)-(2.5) and (3.13) work by looking at a specific class of Type IIB backgrounds, which were first studied in [25]. The topology of the background we will study is R 1,3 × M 3 × S 3 so that there is an SU(2) isometry associated with S 3 , which can be utilized to perform NATD. The ansatz for the metric and the 5-form flux is 3 is the Hodge star operator on M 3 , and A is the warp factor. It is a function which has dependence only on the coordinates of M 3 . S 3 is assumed to be fibered over M 3 and hence the vielbeins e i on S 3 have the form Here, A i are 1-forms on M 3 and λ i are functions on M 3 . The forms σ i are left invariant 1forms for the isometry group SU(2) so that We also define (as in [25]) a set of undetermined frame fields h i so that Another assumption that is made in [25] is that this geometry preserves at least N = 1 supersymmetry in four dimensions in the form of an SU(3) structure characterized by the following 2-form J and 3-form Ω which are given by means of a vielbein e i and frame fields h i : As discussed in appendix A, SU(3) structure can be regarded as a special case of SU(3) × SU(3) structure with associated pure spinors of the form (A.4). In our case, setting Φ 1 = Φ − and Φ 2 = Φ + we have Due to assumption of preservation of supersymmetry, these pure spinors must satisfy the equations (3.1), (3.2). As shown in [25], this forces θ + = π 2 and A 1 = A 2 = 0. The possible values for θ − for different geometries is given in appendix B of [25]. Comparing (4.5) with (A.4) one can see that it is of the general form of a general SU(3) pure spinor with a = e iθ − /2 e iθ + /2 e A/2 and b = e iθ − /2 e −iθ + /2 e A/2 , which satisfy |a| 2 = |b| 2 = e A .
The ansatz (4.1) is general enough to cover many examples important for AdS/CFT duality, notably AdS 5 × T 1,1 , AdS 5 × Y p,q and AdS 5 × S 5 . The detailed description of how JHEP12(2021)071 these backgrounds fall within this general ansatz can be found in appendix B of [25]. For example, for T 1,1 background the required values are as follows: On the other hand, the required values for the AdS 5 × S 5 background are: Now, we perform the NATD transformation of the background described by the ansatz (4.1). We begin with the transformation of the metric and the B-field. For this we use (3.7) where T is obtained by embedding T NATD in (2.1) in O(6, 6) in the usual way. We will call this O(6, 6) matrix also T NATD . 13 Then we read off the transformed metric and the transformed B-field from the symmetric and antisymmetric parts of E , respectively: As mentioned before, this transformation is equivalent to what is given in (2.2). We refer to [17] for details. This gives These are the same as the results obtained in [25] (except for a sign difference in the B-field, see footnote 5). Next, we perform the NATD transformation of the RR flux F 5 from the transformation rule (2.4), with S NATD (2.6). To this end, it is convenient to write the spinor field F that packages the RR fluxes as a non-homogeneous differential form as follows (see [17,41]):

JHEP12(2021)071
where each p-form is decomposed according to how many legs it has along the SU (2) directions. This non-homogeneous differential form maps to a Clifford algebra element in the usual way where we identify the element σ i with the Clifford algebra element ψ i , for i = 1, 2, 3, see (3.19). It has the following form: Then, the spinorial action of ψ i on F is given by wedge product, whereas the spinorial action of ψ i is given by contraction [17,41]: Since there is no B-field, we will first calculate the action of CS θ on differential forms then apply e −B . The action of S θ in (2.6) on a non-homogeneous differential form α is as follows: On the other hand, the action of C given in (C.7) can be calculated by using (4.11). We calculate the following NATD transformed RR flux.
where the Hodge duality 3 is taken with respect to the transformed metric. This polyform packages all the RR fluxes of the NAT dual background, which we read off (after identifying ψ i with dν i ) to be: 14) These agree with the results obtained in [25] (up to sign differences in B , and the 6-and 8-forms due to differences in conventions, see footnote 5).
Finally, we will apply the NATD transformation rule (3.13) (with P = S NATD ) to the SU(3) pure spinors given in (4.5) (which are known to satisfy the supersymmetry equations (3.1), (3.2)) and obtain the NAT-dual pure spinors Φ + and Φ − . The explicit form of the transformed pure spinors are presented in appendix B. In obtaining the results there, we first calculate S θ Φ − : Applying √ G e −B C to S θ Φ − above, we obtain Φ − , whose explicit form is given in (B.2).

JHEP12(2021)071
Now we calculate S θ Φ + : Applying √ G e −B C to S θ Φ + we obtain Φ + , whose explicit form is given in (B.1). One can check by direct computation that the transformed pure spinors Φ − and Φ + can be written in the following form: where the complex 1-form z = v + iw, and the real and complex 2-forms j and ω are as given below Comparing (4.18), (4.19) with (A.6) one can see that they define an SU(2) structure, as can be seen by taking a = e iθ − /2 e iθ + /2 e A/2 and b = e −iθ − /2 e iθ + /2 e A/2 in (A.6). Note that |a| 2 = |b| 2 = e A , as needed. So, under NATD, a background with SU(3) structure is transformed to a background with SU(2) structure, as has been demonstrated many times in the literature previously, in particular in [25,26].
The results we present in (4.18)-(4.22) are in agreement with those obtained in [25]. 14 Whether these transformed pure spinors satisfy the supersymmetry equations (3.1), (3.2) was checked in [25] by direct computation. The results we obtained in section 3.3 make such a calculation redundant. Indeed, the pure spinors (4.18), (4.19) are obtained through the action of S NATD and we have proved that this transformation maps solutions of (3.1), (3.2) to new solutions.

Conclusions and outlook
In this paper, we studied how the pure spinor equations (3.1), (3.2) transform under NATD. These are equations to be satisfied for preservation of N = 1 supersymmetry in compactifications of Type II string theory to four dimensions. Our approach in analyzing supersymmetry under NATD is different from those in the literature in that we exploit the recently discovered fact that NATD can be described as an O(d, d)/Pin(d, d) (in the NS-NS/RR sectors) transformation. Although this is a coordinate dependent transformation we start in section 3.1 by considering constant Pin(d, d) transformations. Writing the equations (3.1), (3.2) in terms of DFT fields makes it easy to show that they are Pin(d, d) covariant. This then means that solutions of these equations will be mapped to new solutions under Pin(d, d). This analysis can be regarded as a generalization of those carried out in [34] and [56], where the behavior of pure spinor equations under Abelian T-duality (a certain type of O(1, 1)/Pin(1, 1) transformation) and LM deformations (a certain type of O(2, 2)/Pin(2, 2) transformation) was studied, respectively.
Since the NATD matrix is coordinate-dependent, further analysis is needed to see whether solutions are mapped to solutions under NATD. This is done in section 3.2. We show in that section that this is indeed the case, due to the simple fact that the fluxes generated by the NATD matrix (regarded as a twist matrix within the formalism of GDFT) is the same as the geometric flux associated with the isometry group that is used to perform NATD. This idea of 'preservation of flux' has been used before to analyze field equations of supergravity under NATD in [17], under YB deformations in [42][43][44][45][46] and under U-duality transformations in [47].
As we emphasized before, our approach here in analyzing supersymmetry equations under NATD is novel, as it implements NATD as an O(d, d)/Pin(d, d) transformation. We believe that this starting point is quite useful, as it has been realized in various works recently that there are other interesting O(d, d) transformations that can be utilized to generate new supergravity backgrounds, notably related with integrable deformations of string sigma models [15,43,45], [60]- [63]. The approach taken here would also be useful to analyze supersymmetry of such backgrounds. Also, viewing NATD as a Pin(d, d) transformation makes it easier to apply it to other backgrounds, which fall outside the ansatz considered in section 4 with different isometry groups and supporting a generic SU(3) × SU(3) structure [36,64,65]. The methods we employed here are well suited to analyze the supersymmetry and structure group of the resulting backgrounds. We plan to consider these issues in future work.

A SU(3) and SU(2) structures and pure spinors
The structure group of the generalized tangent bundle T M ⊕ T * M of the six dimensional internal manifold M reduces to SU(3) × SU(3) if there exists two globally defined SU(3) × SU(3) pure spinors Φ 1 and Φ 2 of non-vanishing norm, [35,36]. Adopting the conventions JHEP12(2021)071 of [56], the explicit form of these pure spinors can be given as below: where c 1 , c 2 , c 3 , c 4 are complex functions on M . For a background of the form (3.3) requirement of existence of supersymmetric branes imposes that |c 1 | 2 + |c 2 | 2 = |c 3 | 2 + |c 4 | 2 = e A [56]. 15 Here z = v + iw is a complex 1-form, j is a real 2-form and ω is a complex 2-form.
Reduction of the structure group of the tangent bundle T M to SU(3) is equivalent to existence on M of an invariant real 2-form J and a complex 3-form Ω satisfying the following compatibility conditions, [66][67][68][69].
On the other hand, SU(2) structure on M is characterized by the existence of a complex 1-form z = v + iw, a real 2-form j and a complex 2-form ω satisfying the following compatibility conditions [66][67][68][69]: Again, SU(2) structure can be regarded as a special case of SU(3) × SU(3) structure [28][29][30] and the form of the pure spinor describing the SU(2) structure is a special case of (A.1) and (A.2). The corresponding pure spinors are [35,36,56]:

JHEP12(2021)071 B NAT-dual pure spinors
Transformation of the SU(3) pure spinors given in (4.5) under the NATD transformation yields the pure spinors presented below: Similarly, we transform Φ − under NATD and obtain Note that after the transformation ψ i are identified with dν i . It can be checked by direct computation that the spinors Φ + and Φ − can be written as in (4.18)-(4.22).

C Mukai pairing
Mukai pairing is the natural inner product on the Clifford module ∧ • T * and described as follows.

This then means that ρ(Λ
(C.14) The charge conjugation matrix which appears in the definition of K ∈ Spin(d, d) is C d for even d, whereas it is C + d for odd d, as explained in [21].