7-dimensional ${\cal N}=2$ Consistent Truncations using $\mathrm{SL}(5)$ Exceptional Field Theory

We show how to construct seven-dimensional half-maximally supersymmetric consistent truncations of 11-/10-dimensional SUGRA using $\mathrm{SL}(5)$ exceptional field theory. Such truncations are defined on generalised $\mathrm{SU}(2)$-structure manifolds and give rise to seven-dimensional half-maximal gauged supergravities coupled to $n$ vector multiplets and thus with scalar coset space $\mathbb{R}^+ \times \mathrm{O}(3,n)/\mathrm{O}(3)\times\mathrm{O}(n)$. The consistency conditions for the truncation can be written in terms of the generalised Lie derivative and take a simple geometric form. We show that after imposing certain"doublet"and"closure"conditions, the embedding tensor of the gauged supergravity is given by the intrinsic torsion of generalised $\mathrm{SU}(2)$-connections and automatically satisfies the linear constraint of seven-dimensional half-maximal gauged supergravities, as well as the quadratic constraint when the section condition is satisfied.


Introduction
Finding consistent truncations of higher-dimensional supergravity to yield lower-dimensional theories is a notoriously difficult problem. By a consistent truncation we mean that solutions of the lower-dimensional equations of motion automatically satisfy those of the initial higherdimensional theory. Because of the non-linearity of the field equations such consistent truncation Ansätze are generically hard to find [1], unless the background has a lot of underlying symmetry. For example, Scherk and Schwarz [2] showed that consistent truncations can be defined on Lie groups, which are of course parallelisable manifolds. As a result the truncation has the same number of supersymmetries as the higher-dimensional theory.
With this set-up it has been possible to derive and study a variety of new consistent truncations on spheres and hyperboloids [23,34,35], on non-geometric backgrounds [20,36], on product manifolds [37] as well as to study the relationship between different consistent truncations [38,39]. 1 Because such generalised Scherk-Schwarz truncations are defined on generalised parallelisable spaces, they also preserve the same number of supersymmetries [13] and thus yield maximal gauged SUGRAs when used in EFT or half-maximal gauged SUGRAs when used in DFT, or their respective generalised geometry analogues. While it is possible to define a further truncation of the maximal gauged SUGRA to half-maximal ones, corresponding to the reduction of EFT to DFT, see e.g. [20,43], there are of course half-maximal gauged SUGRAs which cannot be obtained this way. Furthermore, there are half-maximal gauged SUGRAs which cannot be obtained by a consistent truncation of type II theories but require either the heterotic SUGRA or 11-dimensional SUGRA.
The purpose of this paper is to initiate the study of consistent truncations which break some amount of supersymmetry in exceptional field theory (and exceptional generalised geometry).
Here we will focus on the seven-dimensional case where we show how to obtain arbitrary half-maximal gauged SUGRAs coupled to n vector multiplets and thus with scalar coset space Our set-up allows us to capture consistent truncations to half-maximal gauged SUGRAs arising from either 11-dimensional or type II SUGRA, as well as the heterotic SUGRA, as shown in [44]. 2 In order to break half of the supersymmetry, the internal manifold must have generalised SU(2)-structure and we show how to define consistent truncations on such spaces. In particular, the embedding tensor is encoded in the generalised Lie derivative acting on the sections defining the truncation and automatically satisfies the linear constraint of half-maximal gauged SUGRA.
Exactly as in the maximal case [19], the section condition is sufficient for the gaugings to fulfil the quadratic constraint.
In order to understand how to obtain n = 3 vector multiplets it is important to distinguish between linear symmetry groups acting at each point in space and the symmetry group acting on the sections we keep in the truncation. The linear symmetry group is in the case of exceptional field theory just E d(d) , or in the case considered here SL (5). This is simply a consequence of the field content of the theory, and not a result of the backgrounds considered. 3 However, when we consider truncations on generalised parallelisable spaces then this also becomes the symmetry group acting on sections and this is why generalised Scherk-Schwarz reductions lead to gauged SUGRAs with global symmetry group E d(d) . On the other hand, when the background is not generalised parallelisable, as we will be considering in this paper, the group acting on the space of sections can be much larger because the number of independent sections can be larger. This is why the supergravities we obtain have global symmetry groups O(3, n) which are clearly not subgroups of SL (5).
To emphasise this point, let us consider the more familiar example of general relativity in d+4-dimensions on a product manifold so that its linear symmetry group is GL(d)×GL (4). When performing a truncation on T 4 , one obtains d-dimensional gravity minimally coupled to scalars parameterising a coset whose global symmetry group is indeed GL (4). However, when considering less supersymmetric truncations, for example on K3, one obtains duality groups which are not subgroups of GL(4). In the K3 example one obtains d-dimensional gravity minimally coupled to scalars parameterising the coset space O (3,19)/O(3) × O (9). The duality group O (3,19) acts of course on the space of sections defining the truncation on K3, i.e. the 22 harmonic forms. The linear symmetry group of the internal space, GL (4), which just tells us that at each point we have a 4-dimensional metric plays no direct role in the global symmetry group of the reduced 2 We should mention that the approach we take here differs from that in [45] which reduces the usual flux formulation of double field theory, which is only valid for paralellisable manifolds, on CY 3 to obtain a N = 2 scalar potential. Furthermore, our approach allows us to consider general flux backgrounds whereas [45] is restricted to fluxes which can be treated as small deviations to the Calabi-Yau background. 3 In this discussion we ignore the existence of the extra coordinates but we show how these fit into the picture in [44]. theory, O (3,19).
We begin by reviewing the SL(5) EFT relevant for truncations to 7-dimensional gauged SUGRAs in section 2 and introducing the tensors required to define a SU(2)-structure in section 3. Then we reformulate the theory in section 4 in a way that is more adapted to N = 2 SUSY. This involves rewriting the theory in terms of tensors defining the SU(2)-structure group rather than the generalised metric. That such a reformulation bypassing the generalised metric is necessary should not come as a surprise since as a particular example one could here consider the M-theory truncation on K3 for which the metric, and hence generalised metric, is not explicitly known. We show how to rewrite the supersymmetry variation of the gravitino as well as the scalar potential, kinetic terms and topological term in a way that is adapted to N = 2 SUSY. We next discuss how to perform a truncation of EFT on generalised SU(2)-structure manifolds in section 5. The conditions for a consistent truncation are compactly formulated in terms of the generalised Lie derivative. In particular, with the truncation Ansatz we present the internal coordinates can only appear in the action through the embedding tensor, which is defined by the generalised Lie derivative of the sections defining the truncation. Thus when the embedding tensor components are constant, the Ansatz guarantees that the action becomes independent of the internal coordinates and thus the truncation is consistent. Finally, we conclude in section 6 by discussing possible application and extensions of this work.
Summary of results Throughout this paper we are concerned with generalised SU(2)-structure manifolds. Such manifolds admit two linearly-independent, nowhere-vanishing spinors. This is equivalent to the manifold having the following nowhere-vanishing tensors under generalised diffeomorphisms: (κ, A a , A a , B u,ab ). Here a, b = 1, . . . , 5 denote SL(5) indices and u = 1, . . . , 3 are SU(2) R indices denoting the R-symmetry. Furthermore, κ is a scalar density of weight 1 5 and is related to the determinant of the external seven-dimensional metric. Additionally these structures satisfy This set of tensors reduce the USp(4)-structure group to SU(2) and thus define a generalised metric implicitly. The action can be rewritten completely in terms of the generalised SU(2)-structure, i.e. κ, A a , A a , B u,ab . To do so one introduces a generalised SU(2)-connection∇ which annihilates the SU(2)-structure∇ (5), where SU(2) S refers to the structure group while SU(2) R refers to the R-symmetry group. The intrinsic torsion can be used to rewrite the SUSY variations and the scalar potential. For example, the generalised Ricci scalar is given by

Its intrinsic torsion lives in the representations
(1.5) where S, T are singlets of the intrinsic torsion, and S a , T a , U a are the (2, 2) of the intrinsic torsion. We perform a truncation by expanding the SU(2)-structure in terms of a finite basis of sections of the (1, 1), (1, 3) and (3, 1)-bundles of SU(2) S × SU(2) R . In particular because SU(2) S is nontrivially fibred over the manifold we use n sections of the (3, 1)-bundle and these will give rise to n vector multiplets in the reduced theory. We denote the sections by n a , n a and ω M,ab , where M = 1, . . . n + 3 collective denotes the sections of the (3, 1) and (1, 3)-bundles. These sections satisfy n a n a = 1 , where η MN is an O(3, n) metric whose signature depends on the number of (3, 1) sections.
The truncation Ansatz for the scalars is given by where we use to denote the truncation Ansatz, and ρ(Y ) is a density of weight In order to have a consistent truncation, the sections ρ, n a , n a and ω M,ab must satisfy three types of differential constraints. Firstly, any doublets must vanish, e.g.
where we defined the n + 3 generalised vectors (1.10) Secondly, the generalised Lie derivative of the sections ω M,ab must be expandable in a basis of the ω M,ab .
The embedding tensor of the half-maximal gauged supergravity is then given by the generalised Lie derivative of the sections defining the truncation. In particular, this satisfies the linear constraint of 7-d half-maximal gauged supergravities so that one can identify (1.11) By construction, closure of the algebra of generalised Lie derivatives (hence for example the section condition) is sufficient for the gaugings to satisfy the quadratic constraints of the gauged SUGRAs. Finally, the truncation is consistent when the embedding tensor (1.11) is constant.
Unlike in the construction of effective actions, the ω M,ab 's appearing here are not uniquely defined by the topology of the background. This is a reflection of the fact that a given background can admit multiple, different consistent truncations. Additionally, it is important to highlight that the consistent truncations defined here do not require the background to be a solution of the equations of motion. In this case, the gauged SUGRA will not have a vacuum at the origin of the scalar manifold, nor does it need to have a vacuum at all. Related to this, the fields in the truncated theory are not in general massless. In particular, the consistent truncation may have discarded some light modes but kept certain heavier modes. However, it does so in a manner in which any solutions can be uplifted to solutions of the full theory.

Overview of exceptional field theory
Let us begin by giving a brief review of the SL(5) exceptional field theory [8,9,46] with emphasis on the aspects needed for our purposes. We refer the interested reader to the reviews [47][48][49].
The SL(5) EFT can be viewed as a reformulation of 11-dimensional supergravity which makes the linear symmetry group SL(5) manifest. Thus, the starting point is 11-dimensional supergravity in a 7+4 split. Let us use x µ , µ = 1, . . . , 7, as coordinates for the "external" 7-d space and label yī, i = 1, . . . , 4 as the four "internal coordinates". These are part of 10 "extended coordinates", Y ab , forming the antisymmetric representation of SL(5), where we use a, b = 1, . . . , 5 as fundamental SL (5) indices. In the case where the internal geometry really is a torus, the extra six coordinates can be understood as being dual to wrapping modes of branes. However, the extra coordinates are always introduced, in a background-independent manner, and we will suggest a possible interpretation in the case where the four-dimensional part of the internal space is non-toroidal, e.g. a K3, in [44]. 4 We will always refer to the seven-dimensional space as external and the four-dimensional (or 10-dimensional if the extended viewpoint is taken) as "internal" although no truncation has been performed, i.e. all fields can depend on any of the (7 + 10) coordinates.
This coset can also be described by the generalised vielbein V a ij such that where i, j = 1, . . . , 4 are USp(4) indices which are raised/lowered by the symplectic invariant and for a scalar All other cases follow by linearity. Note that from the above considerations ∂ ab can be seen to carry weight − 1 5 under generalised diffeomorphisms. Furthermore, the parameter of generalised diffeomorphisms Λ ab is in the 10 of SL(5) and has weight 1 5 , so that under a generalised diffeomorphism it itself transforms as We will henceforth call any tensors in the 10 of SL(5) of weight 1 5 "generalised vectors", because they generate generalised diffeomorphisms.
For consistency the algebra of generalised diffeomorphisms must close, i.e. (2.7) Here the D-bracket just represents the action of a generalised Lie derivative, In order for (2.7) to hold one needs to impose the so-called section condition [13,51] ∂ where f and g denote any two objects of the SL(5) EFT. There are two inequivalent solutions to the section condition, one corresponding to 11-dimensional SUGRA while the other corresponds to type IIB [9,53,54]. Upon using a solution of the section condition, the generalised Lie derivative (2.4) generates the p-form gauge transformation and diffeomorphisms of the corresponding SUGRA. Similarly, the action that we are about to sketch reduces to the 11-dimensional SUGRA or IIB SUGRA action, upon imposing a solution of the section condition. However, one could also consider a set-up where there is not a globally well-defined solution to the section condition, in which case we obtain a non-geometric background. Given the generalised Lie derivative, one can introduce connections which give covariant derivatives with respect to these generalised diffeomorphisms. As usual one can also introduce a torsion as the tensorial part of a connection ∇. This can be conveniently defined via the generalised Lie derivative as where L ∇ Λ denotes the generalised Lie derivative (2.4) with all partial derivatives replaced by the covariant derivatives ∇ ab . It can be shown [13,55,56] that the torsion lives in the following irreps of SL (5) τ ab,c d ∈ 10 ⊕ 15 ⊕ 40 . (2.11) Using these concepts one can, for example introduce a generalised torsion-free USp(4) connection [9,13,24,55]. This connection is particularly useful for coupling fermions [57][58][59] and can also be used to derive a "generalised curvature scalar". We will make use of it throughout this paper and label it by ∇ ab . However, it is important to note that the torsion constraint does not fix the connection uniquely. Instead, only certain irreducible representations are uniquely fixed, see e.g. [13,24]. The generalised curvature scalar that can be derived in this way is in fact a scalar density under generalised diffeomorphisms which only involves derivatives with respect to the internal space of M ab and g µν . It is the EFT lift of the scalar potential of seven-dimensional gauged SUGRAs. Conversely, it reduces to the scalar potential of maximal seven-dimensional gauged SUGRAs upon imposing a Scherk-Schwarz Ansatz [22,23]. We should mention that there are also other geometric ways of constructing the generalised curvature scalar, e.g. [56,60]. In order to define the EFT on the full (7 + 10)-dimensional space one needs to introduce a seven-dimensional derivative which is covariant under generalised diffeomorphisms. This is given by the covariant external derivative and upon Scherk-Schwarz reduction this reduces to the gauge-covariant derivative of the gauged SUGRA.
The final ingredient required for constructing the EFT action are the field strengths of the vector fields, two-form and three-form potentials. We will label these as A µ ab , B µν,a , C µνρ a and D µνρσ,ab , which is the auxiliary 4-form potential appearing in the action without kinetic term [9,61,62]. These have weights 1 5 , 2 5 , 3 5 and 4 5 , respectively, under generalised diffeomorphisms. Following [61,62], their field strengths can be written in SL(5) index-free notation as where the E-bracket is the antisymmetrised generalised Lie derivative the • operation is defined as 15) and the (nilpotent) derivative∂ iŝ Note that the derivative∂ is a covariant derivative when acting on objects with the appropriate weight, i.e. when B a has weight 2 5 , C a has weight 3 5 and D ab has weight 4 5 . With all these ingredients one can construct the SL(5) EFT action [9,46,63] as Here L EH is the seven-dimensional modified Einstein-Hilbert term, where all ∂ µ are replaced by D µ [64], in order to be invariant under generalised diffeomorphisms. This is necessary because the seven-dimensional metric g µν is not a scalar but a density of weight 2 5 under generalised diffeomorphisms. The alternative is to use the vielbein formalism [54]. We define the modified Riemann tensor as The modified Einstein-Hilbert term is then Furthermore, where R is the generalised Ricci scalar [9,24] which involves only internal derivatives of the generalised metric. The topological term is best written as an integral over a 10-dimensional extended space and an eight-dimensional external spacetime, whose boundary is the seven-dimensional external spacetime we are considering [46,[61][62][63] (2.22) While each of these terms is individually a scalar (density) under generalised diffeomorphisms, it transforms anomalously under external diffeomorphisms. The various coefficients are fixed uniquely in order to ensure that the entire Lagrangian is invariant under external spacetime diffeomorphisms.

Spinor bilinears and SU(2) structure
In order to obtain a half-maximal theory in seven-dimensions, the internal space must admit two globally well-defined spinors, Θ 1 and Θ 2 . These two spinors form a SU(2) R doublet Θα, witḣ α = 1, 2, and are vectors of USp(4) ≃ Spin (5). The subscript R is used to emphasise that this SU(2) corresponds to the R-symmetry. Let us begin by fixing our spinor convention.

Spinor convention
The spinors transform as USp(4) vectors, i.e. we can write θα i for each spinor with i = 1, . . . , 4 the USp(4) index andα = 1, 2 the SU(2) R index as discussed above. For Spin(5) the charge conjugation matrix has to be antisymmetric [65]. The only invariant tensor we have is the symplectic tensor Ω ij and so we take this to be the charge conjugation matrix. In particular, it is also unitary because it satisfies Hence which shows that it is unitary, i.e. ΩΩ † = 1.
Because the charge conjugation matrix is antisymmetric we cannot define Majorana spinors.
Instead we can define symplectic Majorana spinors because we have extended SUSY. Thus we have Thus we will throughout use pseudo-real objects where both the USp(4) and SU(2) R indices are raised/lowered by complex conjugation.
Finally, let us normalise our spinors. We will take the EFT spinors to have weight −1/10, matching the usual EFT convention [57][58][59]. The full 11-d fermions are a product of the 7-d fermions and the internal spinors and should have no weight. Thus we take the internal spinors θα i to have weight 1/10 and impose the normalisation condition with κ a density of weight 1/5. This looks perhaps more natural if written as a positive definite product:

Spinor bilinears
We can use these two spinors to construct a set of bilinears which define the SU(2) structure. In particular, we can form the following pseudo-real USp(4) tensors where u = 1, . . . , 3 and σ u are Pauli matrices. One can check that A ij ∈ 5 and B u ij ∈ 10. These tensors satisfy a set of compatibility conditions: Any set of such tensors of USp(4) have stabiliser SU(2) ⊂ USp(4) and thus define a generalised SU (2) structure. This can also be seen as follows. Consider decomposing USp(4) −→ SU(2) S × SU(2) R , where SU(2) S denotes the SU(2) structure group. The relevant representations then decompose as and A ij and B u ij correspond to the singlets under SU(2) S .
In order to define a reduction of the SL(5)× R + structure group to SU(2) we need to lift these objects to tensors of SL(5)× R + . We then have a SL(5) vector A a and three SL(5) antisymmetric tensors B u,ab , which satisfy Note that here we use the conventions that However, we see that it is impossible to impose all the compatibility conditions (3.7) as SL(5)×R + relations on A a and B u,ab . As a result, the objects (κ, A a , B u,ab ) are stabilised by SU(2) ⋉ R 4 ⊂ SL(5) × R + . Indeed, one can check that they define 18 degrees of freedom, which matches the dimension of the coset space The issue here is that by using spinor bilinears to construct invariant tensors, we are already taking the structure group to be a subgroup of USp(4). Thus the spinor bilinears can be used to further reduce the structure group from USp(4) to SU (2). However, to define a SU(2) structure of SL(5) × R + , without explicitly requiring the existence of spinors, we need to introduce another SL(5) covector A a satisfying The additional four degrees of freedom of A a are used to absorb the T 4 factor of the stabiliser and thus make the structure group a subgroup of USp(4). Indeed, the set parameterises the coset space and thus defines a SU(2) ⊂ SL(5) × R + structure. Because SU(2) ⊂ USp (4), this set of tensors also implicitly defines a generalised metric. To see that the stabiliser is indeed SU (2), note that up to a SL(5) × R + transformation we can take and thus This configuration is stabilised by SL(4) ⋉ T 4 but the T 4 degrees of freedom can be used to set Aī = 0. As a result, the stabiliser now becomes SL(4) ⊂ SL(5) × R + . The constraint implies that B u,ī5 = 0 and thus we are left to satisfy Three such antisymmetric rank-two tensors of SL(4) parameterise the coset space SL(4)/SU (2), see e.g. [66]. One should think of the objects A a , A a and B u,ab as the "exceptional generalisation" of the complex and Kähler structure on four-manifolds. We have already mentioned that they implicitly define a generalised metric, although there is no explicit formula relating the two. This is not surprising since, for example, the Kähler metric on K3 surfaces (which are an example of exceptional SU(2)-manifolds) is not known. However, by their definition we can identify A a and B u,ab with certain components of the coset representatives V a ij ∈ SL(5) USp (4) . This arises because the coset representatives define the map between SL(5) and USp(4) and thus

Properties of the spinor bilinears
Using A a we can actually "raise" the indices on B u,ab by defining the tensor Due to the compatibility conditions (3.14), it further satisfies The generalised vector of weight 1 5Ṽ u ab = κV u ab . (3.23) will play an important role in defining the intrinsic torsion.
Furthermore, using A a and A a we can project any SL(5) vector, Q a , onto a subspace parallel to A a and perpendicular to it by where we introduced the projector Note that this can also be expressed in terms of V u ab and B u ab as One can also form the following objects which are adjoint-valued: These satisfy the following algebra It is clear that these objects form a hyper-complex structure in the subspace perpendicular to A a . In the fluxless M-theory limit this reduces to the hyper-complex structure on 4-manifolds of SU(2)-structure.
Finally, we can also define a metric on the subspace perpendicular to A a using 4 Reformulating the SL(5) EFT We will now reformulate the SL(5) EFT in terms of the generalised SU(2)-structure, κ, A a , A a , B u,ab . This can be thought of as an SL (5) version of the rewriting in [67][68][69], but extended to the case where there are non-vanishing gauge fields. This will be necessary in order to obtain the full gauged SUGRAs after performing a consistent truncation and not just their vacua.
To perform the reformulation we need to introduce a generalised SU(2)-connection, which in general is not torsion-free. To motivate this, consider the case of maximal supersymmetry [56,70].
In that instance the consistent truncation is defined on a space with generalised identity-structure [24,28] and thus the compatible connection is uniquely given by the Weitzenböck connection. As showed in [56,70] the EFT scalar potential can be rewritten in terms of the torsion of this connection and upon truncation, the torsion becomes the embedding tensor of the maximal gauged SUGRA.
Here we perform the analogous construction in the case of generalised SU(2)-structures for which the connection is not unique. Nonetheless, the intrinsic SU(2)-torsion, which we define and discuss in 4.1, corresponds to generalised fluxes and can be used to reformulate the theory.
For example, the intrinsic torsion appears in the SUSY variations, as we show in 4.2, and in section 4.3 we show that we can express the scalar potential completely in terms of the intrinsic torsion. 5 In section 4.4 we rewrite the kinetic and topological terms in terms of the generalised SU(2)-structure.

Intrinsic SU(2)-torsion
We now introduce a SU(2)-connection whose intrinsic torsion will be identified with the embedding tensor of the half-maximal gauged supergravity obtained after truncating. A SU(2)-connection∇ ab is compatible with the tensors defining the SU(2)-structure, i.e.
This does not uniquely specify the connection, unlike in the maximally-supersymmetric case of an identity structure.
Recall from section 2 that the torsion of a connection ∇ is the tensor part of the connection and can be defined in terms of the generalised Lie derivative, by where U a has weight λ under generalised diffeomorphisms and τ ab is the trombone part of the embedding tensor. We know that for SL(5) the torsion has components only in the In the following discussion of the intrinsic SU(2)-torsion we essentially follow the general prescription outlined in [72]. The torsion map viewed as a map from the space of SU(2) connections to the space of torsions W may be neither injective nor surjective. In the first case, many SU (2) connections could have the same torsion, while in the latter, it is impossible to find a SU(2)-connection yielding an arbitrary torsion (the torsion map is not right-invertible on all of W ). The part of the torsion that is independent of the choice of SU (2) connection is called the intrinsic torsion, and is non-zero when the torsion map is not surjective.

Representations in the intrinsic SU(2)-torsion
To calculate the representations in which the intrinsic SU(2)-torsion transforms note that any two SU(2) connections must differ by an adjoint valued tensor in the 10, i.e. by Σ ∈ Γ(K SU (2) ) where K SU(2) = 10 ⊗ ad(P SU (2) ) . In terms of SU(2) S × SU(2) R representations 6 we have Now the torsion map is a map Thus we see that the image of the torsion map is at most and hence the set which is independent of the connection is given by Finally, for the sake of completeness let us mention that the kernel of the torsion map is at least although this will not concern us further.
In principle the image of τ could be smaller than the right-hand side of (4.8) in which case the intrinsic torsion is larger than the right-hand side (4.9). However, a direct calculation shows that this is not the case and so we find Because they are intrinsic these are the only components of the SU(2)-torsion that are physically relevant and we will see that these are related to the embedding tensor after truncation. In the following section we will show that the EFT can be rewritten entirely in terms of the SU(2)structure and its intrinsic torsion.

Explicit expressions for the intrinsic SU(2)-torsion
We now wish to find explicit expressions for the intrinsic SU(2)-torsion which has irreducible components transforming in the representations (4.11). The fact that the intrinsic torsion is independent of the SU(2)-connection means that it can be expressed directly in terms of the SU(2)-structure and its partial derivatives with no connection appearing. Thus, the intrinsic torsion is given by SL(5) tensorial combinations of derivatives of the SU(2)-structure. For example, consider the combination It follows from the tensor hierarchy [61] that this is a tensor under generalised diffeomorphisms. Regardless we could have written it in terms of any connection∇ whereΓ ab,c d are the components of the connection∇. Because S is a tensor and the first term on the right-hand side of the above equation is a tensor, the final term must be a tensor too.
By definition, it is part of the torsion of∇. If we now specialise to the case where∇ is a SU(2)-connection we find that where as we said the right-hand side is part of the torsion. However, S was defined in (4.12) without referring to a specific SU(2)-connection and thus we see that it corresponds to the intrinsic torsion. In order to find expressions for the intrinsic torsion let us first define the projectors onto the representations appearing in (4.11). Firstly, note that A a define the singlets in the decomposition as SL(5) → SU(2) S × SU(2) R , and similarly A a for the 5 decomposition. Then the subspace perpendicular to A a in the 5 corresponds to the (2, 2). Thus, we use A a to project onto the (1, 1) and the projector we have met in section 3.3 for the (2, 2): and similarly of course for the conjugate SL (5) reps. For the 10 of SL(5) we have the decomposition as we break SL(5) → SU(2) S × SU(2) R . The three tensors V u ab project onto the (1, 3) representations, while A a can be used to project onto the (2, 2). Finally, we can use , (4.18) to project onto the (3, 1) since P ab cd B u,cd = 0 . Before giving the explicit expressions for the intrinsic torsion let us also define the projector We are now ready to give explicit expressions for the intrinsic torsion.

Singlets
(4.21) (1, 3) (4.22) (3, 1) (4.24) (4.25) 26) or more explicitly with Note that while one can think of other tensorial combinations transforming in the above representations they cannot be linearly independent from the expressions given above. For example, we can of course raise and lower the (2, 2) indices using the metric M ab defined in (3.29). We can also dualise the (3, 1) and (1, 3) indices using ǫ abcde A e . However, in this case it is clear that the resulting expressions are linearly dependent on the intrinsic torsion given above.

Intrinsic torsion in terms of spinors
In order to rewrite the supersymmetry variations it will be useful to express the intrinsic torsion in terms of the spinors θα ,i . We do this using the torsion-free USp(4) connection. For example, this allows us to write where ∇ ab is the torsion-free USp(4) connection as discussed in section 2. This can then be expressed in terms of the spinors θα ,i by the definition of A ij in equation (3.6). One finds that We see that we obtain a SU(2) R doublet of gravitini ψ µα as well as SU(2) S doublet. The gravitini forming a doublet of SU(2) S are responsible for enhancing the SUSY to N = 4 and thus we will ignore them here. Upon imposing the consistent truncation Ansatz they will correspond to massive gravitino multiplets of the gauged SUGRA and we ensure the truncation does not excite them.
By comparison with [57,58] and [50] one can see that the SUSY variation of the gravitini of the SL(5) EFT can be written (up to coefficients and γ-matrix orderings which are not important to us here) as (4.32) The N = 2 gravitini are embedded in the USp(4) ones via the internal spinors θ iα , hencẽ The N = 2 SUSY parameters are similarly embedded into the USp(4) ones as Hence we can write the variation of the N = 2 gravitini as In appendix B we show how one can rewrite this in terms of the SU(2)-structure and its intrinsic torsion. The result is (4.36) One could proceed similarly for the other fermions which do not form doublets under SU(2) S but we will not do so here as this is not necessary for our purposes.

Scalar potential
It is useful to write the scalar potential as Here R is the so-called generalised Ricci scalar [23] -although it is a density of weight − 2 5and contains only internal derivatives of the EFT scalars. It can also be written as the square of covariant derivatives of spinors [13,23,24], that is 1 16 Rǫ where∇ is the USp(4) connection without the seven-dimensional spin connection. This follows from the supersymmetry variation of the fermionic equations of motion which must be proportional to the bosonic equations of motion [24]. We show how to derive these coefficients in appendix A. We now write the spinor as ǫ i = θ iα ǫα in terms of a SU(2) R pair of spinors and use the fact that the right-hand side is linear in ǫα to find that We further integrate by parts to obtain We will show in 5.7 that this does reduce to the correct scalar potential of seven-dimensional half-maximal gauged SUGRAs. Now we are in a position to re-express the potential in terms of the spinor bilinears A a , A a and B u,ab via their intrinsic torsion (4.11). By expressing the intrinsic torsion in terms of the spinors θα i we find the generalised Ricci scalar to be
Finally, we claim that one can write where∇ ab is the SU(2)-connection and which acts on g µν as While this term vanishes when performing a consistent truncation as we are doing here, in [44] we show that this does reproduce the correct term in the heterotic DFT.

Kinetic terms
The kinetic terms of the scalar and gauge fields are usually written in terms of the generalised metric directly We need to rewrite these in terms of the SU(2) structures directly.
It is clear that the kinetic term for the scalars should be replaced by terms involving derivatives of A a , A a and B u,ab . Derivatives of κ are of course included in the Einstein-Hilbert term which needs no modification as it does not involve a generalised metric. There are only two such terms which are independent: However, as we are about to discuss in the next section 5.1, terms such as (4.50) necessarily vanish when we have an honest N = 2 theory and so we will not consider them. This possible omission is irrelevant for N = 2 theories which are the subject of this paper. We claim that the kinetic terms are given by (4.51) One may be able to derive the coefficients appearing here by requiring invariance under external diffeomorphisms. However, we have fixed the coefficients by comparison with gauged SUGRA.
As we will see in sections 5.8, (4.51) does reduce to the correct kinetic terms of seven-dimensional half-maximal gauged SUGRA. In [44] we also show that it reproduces the correct kinetic terms of the heterotic DFT.
5 Consistent truncations to half-maximal gauged supergravity

Decomposition of supergravity fields
The following discussion is the SL(5) EFT analogue of the discussion in section 2.2 of [67] and section 3 of [68] where they consider four-dimensional N = 2 truncations of 10-dimensional supergravity.
Let us begin by decomposing the EFT fields under SU(2) S × SU(2) R where the first factor labels the SU(2)-structure group and the second the R-symmetry group. We give the decompositions of the bosons in table 1 and that of the fermions in table 2. Now we can reorganise all these degrees of freedom into N = 2 supermultiplets. The singlets under the SU(2)-structure group form the graviton supermultiplet.
Those in the adjoint of the SU(2)-structure group form the vector multiplets (with A = 1, . . . , n).

(5.2)
Finally, all doublets of the SU(2)-structure group form a doublet of gravitino multiplets Gravitino multiplets: Let us first understand how we obtain n = 3 vector multiplets. A naive expectation would be to have three vector multiplets related to the (3, 1) representations. However, the generalised SU(2)-structure group is non-trivially fibred over the manifold. Thus, the number of sections of the (3, 1) bundle is in general n = 3 giving n = 3 vector multiplets. By contrast, the SU(2) R group is trivially fibred over the manifold and hence it contains exactly three sections. This is why, for example, we have exactly three vectors in the graviton multiplet and three scalars in each vector multiplet. Finally, the scalars in the vector multiplets, φαβ A , as well as the scalar in the graviton multiplet, φ, will correspond to deformations of the SU(2) structure A and B u that we have introduced in section 3. Now let us turn to the massive gravitino multiplets. These are associated to broken N = 4 SUSY. Indeed, one can only consistently couple these multiplets to seven-dimensional halfmaximal gauged SUGRA for n = 3 in which case we have a straightforward truncation of a N = 4 theory. Because we want an honest N = 2 theory, we do not want couplings to the gravitino multiplets in the truncated theory. This is ensured by not having any SU(2) S doublets in our Ansatz.
One can also understand the need for removing SU(2) S doublets in the truncation Ansatz differently. We want to have a generalised SU(2)-structure, not an identity structure. But from the discussion in 3 we see that a nowhere vanishing section in the doublet representation of the SU(2) S bundle would correspond to another pair of globally well-defined internal spinors. In this case it is clear that the structure group would be broken to an identity structure and we really have N = 4 SUSY. To avoid this, we project out all doublets of the SU(2)-structure group in our Ansatz.

Defining the truncation
We now wish to define a consistent truncation of the SL(5) EFT fields in order to obtain a sevendimensional half-maximal gauged SUGRA. For the scalar sector we expand the SU(2)-structure (κ, A a , A a , B u,ab ) in terms of a finite basis of sections which we are about to define.
In the analysis above we have seen that the SL(5) EFT degrees of freedom organise themselves into sections of the (1, 1), (1, 3) and (3, 1) bundles of SU(2) S × SU(2) R . Thus we choose a SL (5) density and a finite number of these sections, which we label by and where we have made it explicit that these objects only depend on the internal manifold. where I = 1, 2, 3 labels the SU(2) R adjoint sections and A = 1, . . . , n labels the SU(2) S adjoint sections.
We will throughout this paper use η MN to raise and lower (n + 3) vector indices. We can use these relations to introduce n + 3 sections of the ( Given these relationships we can further deduce the following identities which we will use copiously in this paper. In particular,ω M ab is a generalised vector and will be useful in formulating the consistency condition for our Ansatz. Before giving the truncation Ansatz, let us point out that in general we are not developing an effective theory because our truncation Ansatz may be keeping heavy modes, while discarding lower ones. 7 Instead we wish to perform a consistent truncation such that all solutions to the equations of motions of the lower-dimensional theory are also solutions to the equations of motions of the full exceptional field theory, and thus of 11-dimensional supergravity or type IIB. This allows us for example to perform a consistent truncation on a background that is not a solution of the equations of motion. Because we are only requiring a consistent truncation, not an effective one, the basis of sections which we use for the truncation are not in general analogues of harmonic forms. Indeed, they should not correspond to topological invariants of the background manifold on which we define the consistent truncation. This is because one manifold may admit several different consistent truncations for which different modes are kept, see for example the discussion in the case of maximal SUSY in [28]. Instead, we will require a weaker set of differential constraints on the sections which we discuss in subsection 5.4.

Scalar truncation Ansatz
We begin by expanding the generalised SU(2) structure in terms of the basis of sections defining the truncation. We let the coefficients in the expansion depend on x µ , the seven coordinates of the external space. These coefficients determine how the generalised SU(2)-structure, hence the geometry of the internal manifold, changes and they become scalars of the truncated sevendimensional theory.
We will denote the truncation Ansatz by angled brackets: . For the scalar fields it is given by

(5.12)
This implies that We must now check the compatibility conditions (3.14). The Ansatz (5.12) automatically satisfies A a A a = 1 2 . (5.14) However in order to satisfy We are left with 3n degrees of freedom which is the dimension of the coset space .
because of (5.16). Thus H MN is a symmetric element of O(3, n) and hence gives coordinates on the coset space M coset . It is the generalised metric of the seven-dimensional gauged supergravity. There are two further scalars d(x) and |ē|(x). These are related to the dilaton and the determinant of the seven-dimensional metricḡ 7 with |ē|= |ḡ 7 | 1/2 . In total we see that we obtain the scalar coset space where we are not counting |ē| as part of the scalar manifold because it forms part of the external metric.

Fermion, gauge field and external metric truncation Ansatz
Let us now give the truncation Ansätze for the fermions and gauge fields. Recall from the discussion in 4.2 that the N = 2 gravitini are embedded as USp(4) fermions by Furthermore, we have rewritten the SUSY variations in terms of ψ µα , χα, χ α,ij and θ iα . The truncation Ansatz for these objects is analogous to (5.12), e.g. for the gravitino it takes the form where Φ iα (Y ) is now an internal spinor with no weight under the generalised Lie derivative and is the fermionic analogoue of ω M ab , n a and n a , which can in turn be written as bilinears of Φ iα . Writing (5.22) in terms of the N = 2 gravitini and θ iα directly it becomes The SL(5) EFT has one-form, two-form and three-form gauge fields, as well as an auxiliary four-form valued in the 10, 5, 5 and 10 of SL(5) and of weight 1 5 , 2 5 , 3 5 and 4 5 respectively. This determines their truncation Ansätze to be The factor of −4 in the two-form Ansatz has been chosen to match the half-maximal gauged SUGRA conventions.
Similarly, the truncation Ansatz for the external metric is given by We have included the power of the dilaton in order to recover the string-frame action.

Consistency conditions and the embedding tensor
We have already listed a set of algebraic constraints which the truncation basis needs to satisfy.
These are given by equations (5.5) and (5.7). However, this is not enough to guarantee a consistent truncation. As we already mentioned, we are in general not truncating to the massless or lowest-lying excitations of a background. Thus, our sections are not some sort of "exceptional harmonic forms". Instead we require them to satisfy a weaker set of constraints which can be naturally formulated in terms of the generalised Lie derivative and ensures that we have a consistent truncation.

Doublet and closure conditions
First of all, we must ensure that we do not excite any doublets of SU(2) S , as we already discussed in 5.1. Thus, we require that any doublets generated by the tensorial combinations of derivatives vanish. In particular, we impose n a Lω MωN,ab = 0 , Lω M n a = n a n b Lω M n b , ∂ ab n b ρ 3 = ρ 3 n a n b ∂ bc n c . (5.26) The first equation is manifestly a tensor while it can be checked from [61] that the second equation is also a tensor. It is easy to see using (5.12) that these conditions ensure that the doublets of the intrinsic torsion (4.25) and (4.26) vanish.
Furthermore, we require the sections ω M,ab to form a closed set under the generalised Lie derivative, i.e.
In other words, the generalised Lie derivative of ω M,ab can be expanded in the basis of ω M,ab 's.

Using (5.26) one can see that this implies
so that the ω M ab 's also form a closed set under the generalised Lie derivative.
These conditions are analogous to the differential conditions encountered when studying consistent truncations of SU(3)-structure manifolds [67,68]. There one requires the sections used in the truncation Ansatz to form a closed set under the exterior derivative. In the case of consistent maximally supersymmetric truncations of EFT, which are governed by generalised identity-structures, these conditions are satisfied automatically and thus do not need to be imposed by hand. However, these conditions are not yet enough to guarantee a consistent truncations. The remaining consistency condition is best understood by using the terminology of the embedding tensor to which we turn next.

The half-maximal embedding tensor
It is easy to show that the conditions (5.7), (5.9) imply the following identities Lω M ω N ab ω P,ab − Lω M ω N,ab ω P ab = 4n a Lω M n a η N P , where the third equation follows from the first two.
We will now show that the object contains only the irreducible representations allowed by the linear constraint of half-maximal gauged SUGRA [20] and can thus be identified with the embedding tensor. Let us first define It follows immediately from (5.29) that This implies that g MN P ∈ (n + 3) × (adj + (n + 3)). Furthermore, one can use the fact that the torsion lies in the 15 ⊕ 40 ⊕ 10 of SL(5) to show that Thus we see that the only irreducible representations of g MN P are given by These are exactly the representations allowed for the embedding tensor by the linear constraint of the half-maximal gauged supergravity and we will see that indeed these objects f MN P are to be identified with the embedding tensor. Additionally, there is a singlet deformation allowed in seven-dimensional half-maximal gauged supergravity [73] which we identify with Θ = ρ n a ∂ ab n b . If we contract with η MN the left-hand side vanishes identically whereas the right-hand side gives which indeed reproduces a quadratic constraint for the vector fluxes of half-maximal gauged SUGRA, see e.g. [43] for the case where n = 3.
As we already discussed in 2, the algebra of generalised Lie derivatives closes when the section condition is fulfilled. Thus, when the background satisfies the section condition, the gaugings automatically satisfy the quadratic constraint. There may however, be examples where the section condition is violated but the quadratic constraint is not. Furthermore, exactly as in the maximal case [23], we require ξ M = 0 in order to have an action principle for the reduced theory. This can be seen, exactly as in [23] by requiring integration by parts to be valid. We want boundary terms to vanish where |e| is the determinant of the external vielbein and V ab has weight −1/5 under generalised Lie derivatives. However, we can write ∂ ab |e|V ab = 2LV |e| 5/7 , (5.40) whereV ab = |e| 2/7 V ab is a generalised vector of weight 1 5 . After imposing the truncation Ansatz we find and hence we find that integration by parts is only possible when ξ M = 0. ξ M is known as the trombone gauging and it is also known from the gauged SUGRA perspective that such a gauging prohibits an action principle [74]. By performing the truncation on the SUSY variation in section 5.9 and particularly scalar potential in section 5.7, we will obtain further evidence that the objects f MN P , f M , ξ M and Θ are to be identified with the embedding tensor. Furthermore, we will see that upon using the reduction Ansatz, all the ω M ab 's, n a 's andn a 's will drop out and the only possible dependence on Y ab in the action will appear through the embedding tensor components f MN P , f M , ξ M and Θ and an overall factor given by a power of the internal density ρ(Y ). Thus when the embedding tensor components are constant and obey the quadratic constraint, e.g. by requiring the section condition the internal space, we obtain a consistent truncation to a seven-dimensional gauged SUGRA.

Intrinsic torsion and the T -tensor
Let us now evaluate the intrinsic torsion (4.21) -(4.26) using the truncation Ansatz (5.12), the relations (5.7), (5.9), (5.26) Here we defined the left-moving and right-moving projectors P MN − and P MN + as well as the antisymmetric tensor H MN P as (5.43) Similar to the maximally supersymmetric case [23] we expect these expressions to correspond to the T-tensor of the seven-dimensional half-maximal gauged supergravity, some components of which are given in [43] for the case of three vector multiplets, i.e. n = 3.

Reducing the external covariant derivative
As a first check that we are obtaining a half-maximal gauged SUGRA let us consider the reduction of the external covariant derivative D µ . We can consider acting with it on any generalised vector, i.e. an object in the 10 of weight 1 5 , call it W ab with truncation Ansatz Then from equations (5.34) and (5.31) we find Similarly, if we have the external covariant derivative D µ acting on a SL(5) vector X a which is truncated as then the external covariant derivative reduces as Again D µ corresponds to the gauge-covariant derivative of the gauged supergravity.

Reducing the scalar potential
We will now take ξ M = 0 in order to have an action for the reduced theory. Recall that the potential is given by It is easy to see that ∇ ab g µν = 0 , (5.50) By writing out the left-moving and right-moving projectors explicitly and recalling that the This is precisely the scalar potential of seven-dimensional half-maximal gauged SUGRA coupled to n vector multiplets, with general embedding tensor satisfying the linear constraint and including the singlet deformation Θ, see e.g. [43,75]. A particularly interesting feature is that we here automatically obtain the term which vanishes in truncations of double field theory when the section condition is fulfilled by the background.

Scalar kinetic terms
Consider first the scalar kinetic terms. These were given by Let us begin with the kinetic terms of A a . From the reduction Ansatz we find where we defined the gauge-covariant derivative of the dilaton, and thus Similarly, for B u,ab we find But from (5.18) we find This is the correct kinetic term for the scalars of seven-dimensional half-maximal gauged SUGRA, see e.g. [43].

Gauge kinetic terms and topological term
Let us first of all consider the reduction of the field strength. We find where F µν M , H µνγ and J µνγσ are the reduced field strength of the gauged SUGRA Here [A µ , A ν ] M denotes the Lie bracket of the gauge group defined by the embedding tensor It is now straightforward to see that which is the correct kinetic term for the vector fields. Finally, let us reduce the kinetic term for the two-form potentials.

Reducing the SUSY variations
Finally, let us use the truncation Ansatz (5.22) to evaluate the SUSY variations (4.36). For the gravitino variation we find ǫα is the gauge-covariant derivative of ǫα andγ µ =ē µμ γμ.

Conclusions
In this paper we showed how to construct seven-dimensional half-maximal consistent truncations of 10-and 11-dimensional supergravity using exceptional field theory. To do this, we began by reformulating the SL(5) exceptional field theory in a way that is adapted to N = 2 SUSY. In particular, we rewrote the theory by replacing the generalised metric M ab with a set of well-defined tensors κ, A a , A a , B u,ab subject to a compatibility condition, which define the SU(2)-structure. We showed that the existence of these tensors is equivalent to there being two well-defined spinors on the internal space thus ensuring we have N = 2 SUSY. Furthermore, we introduced generalised SU (2) (5). The number of sections of the (3, 1)-bundle determines the number of vector multiplets in the gauged SUGRA. These sections were subject to a number of differential constraints, in particular a "doublet constraint" which ensured that all SU(2) S doublets vanished and hence that we obtained a N = 2 gauged SUGRA. In addition, we had to impose that the sections defining the truncation formed a closed set under the generalised Lie derivative. We saw that the embedding tensor of the half-maximal gauged SUGRA is now given a geometric definition in terms of the intrinsic SU(2) torsion of the background on which the truncation is performed. It can thus be written in terms of the generalised Lie derivative of the sections defining the truncation. Exactly as in the case of maximal SUSY, it automatically satisfies the linear constraint of gauged SUGRA and satisfies the quadratic constraint whenever the algebra of generalised diffeomorphisms closes, for example by imposing the section condition. The truncation was shown to be consistent when the embedding tensor is constant.
The framework introduced here can be used to find uplifts of half-maximal seven-dimensional gauged SUGRAs which cannot be obtained by simple truncations of a maximal seven-dimensional gauged SUGRA. These include gauged SUGRAs with non-zero singlet part of the embedding tensor, Θ. A particularly interesting example of these admits a fully stable deSitter vacuum [43].
It would also be interesting to generalise the procedure of this paper to lower-dimensions.
In this case the number of possible truncations increases because more fluxes are available. For example, it would be nice to study the consistent truncation of IIA on K3 where the full O (20,4) duality group should become visible directly in EFT. Another generalisation that is possible in lower dimensions is to consider a larger amount of broken supersymmetry. For example, in four dimensions one could consider N = 2 truncations, corresponding to truncations on generalised SU(6)-structure manifolds, which would include "exceptional Calabi-Yau"s and their AdS counterparts [76,77]. Finally, as we show in [44] one can use the set-up introduced here to obtain the heterotic DFT by a reduction of EFT. In this case, the extended space contains a SU(2)-structure manifold, but the coefficients in the truncation Ansatz are still allowed to depend on the extended space, albeit in a restricted fashion. This is reminiscent of the way massive IIA SUGRA can be obtained by a Scherk-Schwarz-like reduction of EFT [78]. In particular the duality between M-theory on K3 and the heterotic string on T 3 arises naturally from this picture.

Acknowledgements
The author would like to thank Daniel Thompson for initial collaborations on this project. In addition the author thanks David Berman, Iñaki García-Etxebarria, Severin Lüst, Diego Marqués, Carmen Nuñez, Erik Plauschinn, Felix Rudolph and Henning Samtleben for helpful discussions. The author would also like to thank IAFE Buenos Aires for hospitality while part of this work was completed. This work is supported by the ERC Advanced Grant "Strings and Gravity" (Grant No. 320045).

A Scalar potential in terms of spinors
As discussed in [13,23,24] one can write the generalised Ricci scalar in terms of covariant derivatives of spinors, i.e.
where∇ is the torsion-free USp(4) connection but without spin connection.
To fix the right-hand side, we follow [24] and note that the potential must only involve the determined parts of the torsion-free USp(4) connection. Also, we know that the right-hand side must be linear in ǫ i , and hence cannot have any double partial derivatives acting on ǫ i (as well as no single partial derivatives acting on ǫ i ). This knowledge is enough to fix the right-hand side.
The first observation lets us write 1 16 Rǫ i = 1 2∇ jk∇ ki ǫ j − 1 2∇ jk∇ jk ǫ i + α∇ ik∇ jk ǫ j , (A.2) as we will discuss in A.1, while the second implies that the potential must only make use of the combinations coming from the commutator of two covariant derivatives and their projection onto the 5 (since then the second order partial derivatives vanish by the section condition), i.e. 1 16 These two conditions uniquely fix the Ricci scalar, up to an overall coefficient, to be ik∇ jk ǫ j , (A. 4) and in particular the right-hand side is linear in ǫ i .

A.1 Determined connections
We must use the covariant derivatives which only depend on the determined part of the torsionfree USp(4) connection [13,24]. There are four different possible combination, depending on whether we act on a spinor in the 4 or 16 of USp(4). Let us denote a generic spinor in the 4 by ǫ and a generic spinor in the 16 by χ which thus satisfies The unique operators are given by with × 4 and × 16 being the projectors onto the 4 and 16 respectively. In particular, we need Now we can write The other allowed combination involves the projector onto the 5 since this gives the section condition for the terms involving only partial derivatives. We write and thus combining the two we have Equating the two allowed expressions gives the unique answer (4.40) (up to overall rescalings of the coupling constant).

B SUSY variations of the gravitino
We begin with Let us go through this term-by-term. We use the product rule to write the first term as We can further rewrite D µ θ iβ in terms of the intrinsic torsion. By definition (2.12) where we have introduced an SU(2) connection∇ and used the definition of the torsion via the generalised Lie derivative (4.2).