$\mathcal{N}=2$ consistent truncations from wrapped M5-branes

We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold $M$, preserving minimal $\mathcal{N}=2$ supersymmetry in five dimensions. These are based on $G_S \subseteq USp(6)$ structures for the generalised $E_{6(6)}$ tangent bundle on $M$, such that the intrinsic torsion is a constant $G_S$ singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS$_5 \times_{\rm w}M$ solutions arising from M5-branes wrapped on a Riemann surface. The generalised $U(1)$ structure associated with the $\mathcal{N}=2$ solution of Maldacena-Nu\~nez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and $SO(3)\times U(1)\times \mathbb{R}$ gauge group. The generalised structure associated with"BBBW"solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.


Introduction
Consistent Kaluza-Klein truncations are a precious tool for constructing compactifying solutions to ten or eleven-dimensional supergravity using a simpler lower-dimensional theory. Given a splitting of the higher-dimensional spacetime into an internal manifold M and an external spacetime X, a consistent truncation selects a finite subset of the KK modes of the higher-dimensional theory on M and provides an effective theory on X describing their non-linear dynamics. The selected KK modes must form a protected sector, in the sense that they do not mix in the equations of motion with the modes that have been truncated out. In this way all solutions of the lower-dimensional theory on X are guaranteed to also be solutions of the original higher-dimensional theory. For such non-trivial reduction to be possible, the internal manifold M should have a special geometric structure. The simplest case is when M admits a homogeneous action of a group G , that is M = G /H for some subgroup H ⊂ G . Then one can decompose all higher-dimensional fields into representations of G and truncate to the G -singlets. This G -invariant truncation is consistent, since the singlet fields can never source the nonsinglet fields. When in particular M is a group manifold, M = G , one has a conventional Scherk-Schwarz reduction [1]. Examples of such consistent truncations in the context of M-theory-which is our principal interest in this paper-can be found in [2][3][4][5].
As a step towards more general classes of truncations, it is convenient to think about reductions on homogeneous manifolds using the language of G-structures. Let us consider Scherk-Schwarz reductions for definiteness. A group manifold M = G admits a basis of globally defined left-invariant one-forms, {e a }, a = 1, . . . , dim M , that reduces the structure group to the identity (i.e. M is parallelisable). Furthermore, the group action implies that de a = 1 2 f bc a e b ∧ e c , where f bc a are the structure constants of the Lie algebra Lie G .
This means that the left-invariant identity structure has singlet, constant intrinsic torsion (singlet because de a is expressed in terms of the invariant {e a } basis, and constant because the coefficients of the expansion are constant). The truncation ansatz is defined by expanding all higher-dimensional fields in the basis of invariant tensors of the structure. When this is plugged into the equations of motion, we can again invoke the argument that only singlet tensors are generated and conclude that the truncation is consistent. Since the spin bundle is also trivialised, Scherk-Schwarz reductions preserve the full supersymmetry of the higher-dimensional theory. More generally, G -invariant consistent truncations on coset manifolds M = G /H are based on the existence of an H structure with constant, singlet intrinsic torsion, and preserve only a fraction of supersymmetry or none at all. Interestingly, the argument based on G-structure applies also to internal manifolds M that are not homogeneous. It is sufficient that M has a structure group G S with only constant, singlet intrinsic torsion; then the truncation to the G S -singlets is guaranteed to be consistent. This can preserve different fractions of supersymmetry, depending on how many G S -invariant spinors exist on M . In fact the G S structure data determine the full field content and gauge interactions of the truncated theory. Examples of this type in M-theory are the truncations based on Sasaki-Einstein and weak-G 2 holonomy manifolds of [6], and the tri-Sasakian reduction of [7].
However there are consistent truncations that are not captured by conventional Gstructures. Classic examples are the maximally supersymmetric consistent truncations on spheres, such as eleven-dimensional supergravity on S 7 [8] and S 4 [9]. M-theory truncations preserving less supersymmetry and containing warped AdS solutions can be found in [10][11][12][13]. 1 Building on the evidence emerging from these examples, a conjecture was formulated in [11] stating that for any warped, supersymmetric AdS D × w M solution to higherdimensional supergravity, there is a consistent truncation on M down to D-dimensional pure gauged supergravity with the same amount of supersymmetry.
Exceptional Generalised Geometry and Exceptional Field Theory offer an understanding of these more complicated examples that unifies them with the conventional ones. Exceptional Generalised Geometry uses an extension of the ordinary tangent bundle T M to a larger bundle E on M , whose fibres transform in a representation of the exceptional group E d (d) . In this way the diffeomorphism and gauge symmetries of higher-dimensional supergravity are unified as generalised diffeomorphisms on E. The notion of generalised G S structure, that is a G S structure of E, rather than of T M , leads to a new systematic approach to consistent truncations with different amounts of supersymmetry: one can argue that there is a consistent truncation any time a supergravity theory is reduced on a manifold M admitting a generalised G S structure with constant singlet intrinsic torsion [14]. In particular, all maximally supersymmetric truncations, both conventional Scherk-Schwarz reductions and sphere truncations, can be seen as generalised Scherk-Schwarz reductions on generalised parallelisable manifolds [15][16][17][18][19][20][21][22][23]. This also provides a connection to Poisson-Lie T-duality as described in [24] (see also [25]). Truncations preserving less supersymmetry are based on generalised structures larger than the identity, the half-maximal case having been explored rather extensively by now [14,[26][27][28][29][30]. Moreover, a proof of the conjecture of [11] was given in this framework [14,27], based on the fact that the conditions for a supersymmetric AdS D × w M vacuum can be rephrased as the requirement that M admits a generalised G S structure with vanishing non-singlet intrinsic torsion [31][32][33].
Although the general ideas were illustrated in [14] for any amount of supersymmetry, the Exceptional Generalised Geometry approach to consistent truncations has been developed just for maximal and half-maximal supersymmetry so far. In this paper we enlarge this framework and discuss in detail truncations of eleven-dimensional supergravity preserving minimal N = 2 supersymmetry in five dimensions.
While a strict USp(6) ⊂ E 6(6) generalised structure leads to a truncation to minimal N = 2 gauged supergravity in five dimensions, smaller G S ⊂ USp (6) structures lead to matter coupled supergravity. We show how the G S ⊂ USp(6) structure defines a continuous family of USp (6) structures, and identify the moduli space of this family with the vector multiplet and hypermultiplet scalar manifolds in the truncated five-dimensional theory. We also show how the generalised Lie derivative acting on the generalised tensors defining the G S structure specifies the isometries of the scalar manifold that are being gauged. This fully determines the truncated N = 2 supergravity theory.
We then derive general expressions that encode the uplift formulae for how the elevendimensional bosonic fields are encoded in terms of the moduli and the generalised tensors defining the G S structure. In order to make this truncation ansatz explicit we need to solve a number of technical issues. One is that, in contrast to the maximal and half-maximal case, the structure is not entirely characterised by the generalised vectors K I (i.e. sections of E, transforming in the fundamental of E 6(6) ) which control the vector multiplet sector of the truncated theory. We also need to consider generalised tensors J A belonging to the E 6(6) adjoint bundle, which eventually control the hypermultiplet sector. A related point, that is crucial to derive the scalar truncation ansatz, is the construction of the generalised metric on E, which receives contributions both from the K I and the J A . A significant advantage of the formalism however, is that the expressions are universal. The ansatz can be applied to any N = 2 background once one identifies the K I and J A singlets.
As application, we discuss M-theory truncations on geometries associated with M5branes wrapping a Riemann surface Σ. The near-horizon geometry of this brane configuration is given by a warped AdS 5 × w M solution to eleven-dimensional supergravity, where M is a fibration of a deformed S 4 over Σ [34,35]. The fibration corresponds to a topological twist in the dual superconformal field theory on the M5-branes, where the holonomy of the Riemann surface is cancelled by a U (1) in the SO (5) R-symmetry, which in the supergravity background is realised geometrically as the isometries of S 4 . Depending on which U (1) is chosen, one obtains different AdS 5 × w M solutions, and correspondingly different U (1) S generalised structures.
We start with the N = 2 background of Maldacena-Nuñez [34]: specifying its U (1) S generalised structure and discussing its singlet intrinsic torsion, we obtain a consistent truncation to five-dimensional N = 2 supergravity including four vector multiplets, one hypermultiplet, and a non-abelian SO (3) × U (1) × R gauging. This extends the abelian truncation of [36] (see also [10,37] for previous subtruncations) by adding SO(3) vector multiplets, which in the dual superconformal field theory source SO(3) flavour current multiplets. We also spell out the full bosonic truncation ansatz. The same construction also applies to the "BBBW" solutions [35,38], as the corresponding generalised structure is a simple deformation of the Maldacena-Nuñez one, controlled by a (discrete) parameter describing the choice of U (1) S in SO (5). The corresponding truncation features only two N = 2 vector multiplets, one hypermultiplet and an abelian gauging. We show that the Maldacena-Nuñez truncation admits a new non-supersymmetric AdS 5 solution when the Riemann surface is a sphere, which turns out to be perturbatively unstable. We also find new non-supersymmetric vacua in the BBBW truncations. Together with the consistent truncation including the N = 4 solution of [34], whose U (1) S generalised structure embeds in USp(4) ⊂ USp (6) and leads to half-maximal supergravity [14,39], the present study completes the landscape of what we believe are the most general consistent truncations that can be derived from eleven-dimensional supergravity on known smooth solutions associated with M5-branes wrapped over Riemann surfaces. 2 The rest of the paper is organised as follows. In Section 2 we characterise the generalised structure relevant for M-theory truncations on a six-dimensional manifold preserving N = 2 supersymmetry. In Section 3 we specify the truncation ansatz and discuss how the gauging is determined from the generalised structure. In Sections 4 and 5 we apply our formalism to consistent truncations associated with M5-branes wrapping a Riemann surface, first for Maldacena-Nuñez backgrounds and then for BBBW ones. We conclude in Section 6. The appendices contains a brief account of E 6(6) generalised geometry, a summary of five-dimensional N = 2 gauged supergravity and some technical details of our computations.

M-theory generalised structures and N = supersymmetry
In this section we first recall some basic notions of Exceptional Generalised Geometry for the case of interest here, namely eleven-dimensional supergravity on a six-dimensional manifold, and then we illustrate how the general procedure described in [14] applies to consistent truncations to five-dimensional N = 2 gauged supergravity. A more extended review of the relevant generalised geometry can be found in Appendix A.

The HV structure
Consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M are based on E 6(6) × R + generalised geometry. This extends the tangent bundle T M to the generalised tangent bundle E on M , and the corresponding structure group GL (6) to E 6(6) . The group E 6(6) contains GL(6) as its geometrical subgroup, and we can use the latter to decompose the generalised tangent bundle as (2.1) Therefore the sections of E consist, locally, of the sum of a vector, a two-form and a five-form on M , These are called generalised vectors and transform in the 27 of E 6(6) . All geometric structures of conventional geometry on M , such as tensors, Lie derivative, connections etc, admit an extension to E [40][41][42]. In particular, generalised tensors are defined by considering bundles whose fibers transform in different representations of E 6(6) . We can define dual generalised vectors Z as the sections of the dual tangent bundle transforming in the 27 of E 6(6) . Locally the dual vectors are sums of a one-formv, a two-vectorω and a five-vectorσ, The adjoint bundle transforms in the 1 + 78 of E 6(6) and, in terms of GL(6) tensors, is defined as where, locally, l ∈ R, r ∈ End(T M ), a ∈ Λ 3 T * M is a three-form,ã ∈ Λ 6 T * M is a six-form and α ∈ Λ 3 T M andα ∈ Λ 6 T M . This bundle plays an important role as the components of the M-theory three-form and six-form gauge potentials are embedded in adF . As we will see, the bosonic fields of eleven-dimensional supergravity can be unified into generalised tensors. The supergravity spinors on the other hand arrange into representations transforming under USp (8), the double cover of the maximal compact subgroup USp(8)/Z 2 of E 6 (6) . For example the supersymmetry parameters are section of the generalsied spinor bundle S, transforming in the 8 of USp (8). It will be this compact USp (8) or more generally a subgroup of it, that determines the R-symmetry of the reduced fivedimensional theory.
The manifold M admits a generalised structure, G S ⊂ USp(8)/Z 2 , when the structure group E 6(6) is reduced to the subgroup G S . Typically this can be characterised by the existence of globally defined generalised tensors that are invariant under G S . The amount of supersymmetry of the eleven-dimensional theory that is preserved by the G S structure is given by the number of G S singlets in the spinor bundle, S. 3 In this paper we are interested in structures preserving N = 2 supersymmetry. The generic case is provided by what has been called an HV structure [33,43,44]. It consists of a triplet of globally defined tensors in the adjoint bundle, J α ∈ Γ(ad F ), with α = 1, 2, 3, satisfying together with a globally defined generalised vector K ∈ Γ(E) having positive norm with respect to the E 6(6) cubic invariant, c(K, K, K) := 6 κ 2 > 0 , (2.8) where κ is a section 4 of (det T * M ) 1/2 , and satisfying the compatibility condition where · denotes the adjoint action. 5 See Appendix A for a definition of the cubic invariant and the other generalised geometry operations appearing in these formulae.
The HV structure {J α , K} defines a reduction of the structure group to USp(6) ⊂ E 6(6) . Indeed the vector K is stabilised by F 4(4) ⊂ E 6(6) , while the J α are invariant under the subgroup SU * (6). The compatible K and J α have SU * (6) ∩ F 4(4) ≃ USp(6) as a common stabiliser. The globally defined vector K ∈ Γ(E) with positive norm is called a vectormultiplet structure, or V structure for short. A triplet of J α ∈ Γ(adF ) that define the highest root su 2 subalgebra of e 6(6) and satisfy the conditions (2.7) is called a hypermultiplet structure, or H structure. This justifies the name HV structure for the compatible pair {J α , K}.
It is easy to check that the amount of supersymmetry preserved by a HV structure is N = 2. Under the breaking (2.10) the spinorial representation decomposes as 8 = (6, 1) ⊕ (1, 2) and we see that the are only two USp(6) singlets. The SU (2) H factor in (2.10) is the R-symmetry of the reduced theory so that the two singlets form an R-symmetry doublet, as expected for N = 2 supersymmetry parameters. A strict USp(6) structure is not the only option to obtain N = 2 supersymmetry. In fact, any subgroup G S that embeds in USp (6) in such a way that there are no extra singlets in the decomposition of the spinorial representation of USp(8) does the job. Although the number spinor singlets is unchanged, when the structure group is smaller than USp (6) in general one finds more G S singlets in the decomposition of the 27 and the 78 representations. Let us denote by the set of independent generalised vectors corresponding to G S singlets in the 27, and by the set of independent sections of the adjoint bundle corresponding to G S singlets in the 78 that also satisfy the condition 6 J A · K I = 0 ∀ I and ∀ A . 14) with f AB C being the structure constants of H. The generalised structure G S ⊆ USp (6) is fully characterised as the group preserving the set We can always normalise such that the n V + 1 generalised vectors satisfy c(K I , K J , K K ) = 6 κ 2 C IJK , (2. 16) with C IJK a symmetric, constant tensor and κ is a section of (det T * M ) 1/2 fixed by the structure. In addition we can normalise the adjont singlets to satisfy where η AB is a diagonal matrix with −1 and +1 entries in correspondence with compact and non-compact generators of H, respectively. Any generalised structure has an associated intrinsic torsion [31], which is defined as follows. LetD be a generalised connection compatible with the G S -structure, that is, sastisfyingDQ i = 0 for all i, where Q i is the set of invariant generalised tensors that define the structure. Formally, the generalised torsion T ofD is defined by, acting on any generalised tensor α, where L is the generalised Lie derivative, LD is the generalised Lie derivative calculated usingD and · is the adjoint action on α. 7 The intrinsic torsion is the component of T that is independent of the choice of compatible connectionD, and hence is fixed only by the choice of generalised structure. In general, one can decompose the intrinsic torsion into representations of G S . In particular, for a consistent truncation we will be interested in the case where only the singlet representations are non-zero.

The generalised metric
An important ingredient to derive a consistent truncation is the generalised metric G on M . This is a positive-definite, symmetric rank-2 tensor on the generalised tangent bundle, that encodes the degrees of freedom of eleven-dimensional supergravity that correspond to scalars in the reduced theory. We provide the explicit relation between the generalised metric and the supergravity fields on M in Eq. (3.19). The generalised metric is defined in analogy to the ordinary metric: a metric g on M can be seen as an O(6) structure on T M that at each point on M parameterises the coset GL(6)/O (6). Similarly, at each point p ∈ M a choice of a generalised metric corresponds to an element of the coset Since a G S ⊂ USp(8)/Z 2 , the G S structure will determine a G S -invariant generalised metric, given in terms of the invariant tensors that are used to define the G S structure. The expression of G M N that is relevant for truncations preserving maximal and half-maximal supersymmetry was given in [14,15,22]. Here we will discuss the N = 2 case.
Consider first the case of a generic USp(6) structure. As discussed in the previous section this is specified by an invariant generalised vector, K, together with an su(2) triplet of sections of the adjoint bundle J α , α = 1, 2, 3. These objects define a USp(6)-invariant generalised metric through the formula This formula can be motivated as follows. The globally defined K induces the splitting of the 27 of E 6(6) into orthogonal subspaces in the singlet and 26 representation of F 4(4) ; correspondingly, the E 6(6) cubic invariant on the 26 reduces to the symmetric invariant form of F 4(4) This expression however is not positive definite, since the symmetric form of F 4(4) has signature (14,12) and overall (2.23) has signature (14,13). The first term in (2.21) contains the contribution from the singlet component V 0 and makes the metric positive definite in the singlet. To do the same in the 26 we need the full HV structure. Under SU (2) H × USp (6) the 27 decomposes as 24) and the action J α on V projects on the (2, 6) part, as the rest is an SU (2) H singlet. Then we can write the contribution to the metric in the (2, 6) as 25) and add it in (2.21) to make it positive definite. Note that (2.21) only contains one element of the triplet J α , that we chose to be J 3 . This is because, for each J α , where there is no sum over α and in the last equality we have used that (J α ) 2 = −1 in the V (2,6) subspace. We see that the action of each of the J α gives the same result. This reflects the fact that the generalised metric is independent of the action of the SU (2) H supergravity R-symmetry. For the purpose of constructing the truncation ansatz by comparing with (3.19), we will also need the inverse generalised metric. We can exploit the isomorphism between the generalised tangent bundle E and its dual E * provided by the generalised metric to construct a USp(6) singlet K * ∈ Γ(E * ) as K * (V ) = G(K, V ), where V is any generalised vector. Then, denoting by Z ∈ Γ(E * ) a generic dual vector, the inverse generalised metric is given by (2.27) where the action of the cubic invariant c * and of the adjoint elements J α on the dual generalised vectors can be found in Appendix A.

The HV structure moduli space and the intrinsic torsion
When the G S structure is a subgroup of USp(6) (and there is no supersymmetry enhancement), it determines an USp (6) structure and hence by definition defines a generalised metric. However, a given G S structure can determine several different USp(6) structures. Thus one gets a family of generalised metrics that can be obtained from the G S -invariant tensors, depending on which USp(6) structure one chooses. Concretely, we use the K I and J A tensors characterising the G S structure to construct a generalised vector K and a triplet of J α satisfying (2.7)-(2.9), which then we use to build the generalised metric as in (2.21). The parameterisation of K and J α in terms of K I and J A provides a set of deformations of a reference USp(6)-invariant metric, that correspond to acting on the structure with elements of E 6(6) that commute with G S , modulo elements of USp(8)/Z 2 that commute with G S . The resulting generalised metric thus parameterises the coset This is the moduli space of our G S structure, namely the space of deformations of the reference USp (6) structure that preserve the G S structure. For the N = 2 structures of interest in this paper, this splits in the product where M V is the V structure moduli space, corresponding to deformations of K that leave J α invariant, while M H is the H structure moduli space, which describes deformations of J α that leave K invariant. The fact that these deformations are independent follows from the requirement (2.13). When given a dependence on the external spacetime coordinates these deformations provide the scalar fields in the truncated theory, with M V and M H being identified with the vector multiplet and the hypermultiplet scalar manifolds, respectively. We next outline how to construct the V and H structure moduli spaces. The procedure will be further illustrated in Sections 4 and 5, where concrete examples will be discussed in detail.
The V structure moduli space. A family of V structures is obtained by parameterising the generalised vector K as the linear combination where h I , I = 0, . . . , n V , are real parameters, and imposing the property (2.8). Using (2.16), this is equivalent to showing that the n V + 1 parameters h I are constrained by one real relation and thus define an n V -dimensional hypersurface, This is our V structure moduli space. It will be identified with the vector multiplet scalar manifold in five-dimensional supergravity. The metric on M V is obtained by evaluating the generalised metric on the invariant generalised vectors, Using (2.21), it is straightforward to see that this gives where h I = C IKL h K h L . Then g ambient = 3 2 a IJ dh I dh J gives the metric on the ambient space, 8 and the metric on M V is obtained as the induced metric on the hypersurface.
The H structure moduli space. A family of H structures is obtained by parameterising the possible su 2 subalgebras of the algebra spanned by the J A . The fact that we only have two singlet spinors means that C USp(8)/Z 2 (G S ) must contain an SU (2) H factor (as in (2.10)) that acts on the two singlet spinors. Furthermore, the corresponding su 2 algebra must be generated by a highest root in e 6 (6) . The Lie algebra h = Lie H generated by the J A is the simple subalgebra of the Lie algebra of C E 6(6) (G S ) that contains the su 2 factor. Since h ⊂ e 6(6) the su 2 algebra is generated by a highest root in h.
The H structure moduli space is the space of choices of such highest root su 2 algebras in h, namely the symmetric space 9 Such spaces are known as "Wolf spaces" and are all quaternionic-Kähler, as expected from the fact that M H is going to be identified with hyperscalar manifold in five-dimensional supergravity. Points in M H can be parameterised by starting from a reference subalgebra j ≃ su 2 ⊂ h and then acting on a basis {j 1 , j 2 , j 3 } of j by the adjoint action of group elements h ∈ H, defined as Clearly, this action acts trivially on j if h ∈ SU (2) H ≃ exp(j), or if h belongs to the commutant of this SU (2) H in H, that is h ∈ C H (SU (2) H ). This way, we obtain a triplet of "dressed" generalised tensors J α , α = 1, 2, 3, which depend on the coset coordinates and parameterise our family of H structures.
The intrinsic torsion. This picture that the G S -structure defines a family of HV structures also allows us to give a characterisation of the intrinsic torsion. As discussed in [44], the intrinsic torsion of an HV structure is encoded in the three quantities where, given a generalised vector V ∈ Γ(E), one defines a triplet of functions 10 that formally are moment maps for the action of the generalised diffeomorphism group on the space of H structures. In general, if K G S is the space of G S -compatible connections, then the definition (2.18) defines a map τ : K G S → W where now we view the generalised torsion T as a section of W ⊂ E * ⊗ ad F . 11 The G S -intrinsic torsion is then an element of W G S int = W/W G S where W G S = Im τ . Now let p ∈ M be a particular point in the family of HV structures (2.28) and USp(6) p ⊂ E 6(6) be the corresponding structure group. By construction, G S is the common subgroup of all the USp(6) p subgroups. This means that that is, only a G S -compatible connection is compatible with every HV structure in the family. Hence W G S = p W USp(6)p and so In other words, knowing the intrinsic torsion of every HV structure in the family fixes the intrinsic torsion of the G S structure. Now, recall that each K in the family of HV structures is a linear combination of K I (with constant coefficients), while each J α is defined by the exponentiated adjoint action of a linear combination of J A (with constant coefficients) on a fixed reference su(2) algebra. Hence the intrinsic torsion components L K K and L K J α for the whole family are determined by knowing These also determine µ α (V ) when V has the form V = V I K I , even when the components V I are functions because of the condition (2.13). Thus the final components of the G S intrinsic torsion are determined by where we require c(K I , K J , W ) = 0, which defines a generalised vector that is orthogonal to those of the form V = V I K I . Note that the expressions (2.41) and (2.42) are in general not independent, but are sufficient to determine the intrinsic torsion.

M-theory truncations to N = 2 supergravity in five dimensions
Any generalised G S structure on a manifold M with only constant, singlet intrinsic torsion gives rise to a consistent truncation of eleven-dimensional or type II supergravity on M . While the general ideas (along with the details for five-dimensional truncations preserving half-maximal supersymmetry) were given in [14], here we focus on the specific case of truncations of eleven-dimensional supergravity leading to N = 2 supergravity in five dimensions, based on G S ⊆ USp (6) structures. Although some of the formulae giving the truncation ansatz in terms of the structure are necessarily quite involved, a great advantage is that they are universal expressions good for any N = 2 consistent truncation. One does not have to search for the correct set of consistent modes on a case-by-case basis. All the particulars of the given truncations are encoded in terms of the given G S structure defined by the set of K I and J A singlets. For example, following the discussion in the previous section, the scalar matter content is determined by the commutant of G S in E 6(6) , giving n V vector multiplets and n H hypermultiplets, whose scalar manifolds are identified with the V structure and H structure moduli spaces, respectively. The gauge interactions of the truncated theory are determined by the torsion of the G S -structure, which in turn depends only the generalised Lie derivatives L K I K J and L K I J A . Together this data completely specifies the full five-dimensional supergravity.

The gauging
The gauge interactions of the truncated theory are determined by the intrinsic torsion of the generalised structure G S . As already emphasised, we assume that the intrinsic torsion takes values in the singlet representation of G S , with components that are constant on M . As explained in [14], this means that the generalised Lie derivative along the invariant vectors K I acting on any invariant tensor Q i , is given by where T int (K I ) is a G S singlet in the adjoint bundle. This means that T int (K I ) is in the Lie algebra of the commutant group G = C E d(d) (G S ). Thus −T int defines an "embedding tensor" [45,46], that is a linear map The image of this map defines the Lie algebra of the gauge group G gauge of the truncated theory and also how it embeds Lie G gauge = Im Θ ⊆ Lie G, thus giving G gauge as a subgroup of the commutant group For the structures of interest in the present paper, the relevant invariant tensors are the vectors K I and the adjoint bundle singlets J A that generate H ⊂ G. The former are the generators of the gauge algebra with structure constants f [IJ] L given by where (t A ) J L are the representations of the generators of Lie G acting on V. For the J A singlets we have where f AB C are the H structure constants, as in (2.14). For convenience we have also defined the linear combination of the J A with constant coefficients, so that the action of the generalised vector K I on J A is represented by the adjoint action of J (K I ) . Recall that generally the intrinsic torsion of the G S structure is captured by the expressions (2.41) and (2.42). The condition that one has singlet, constant intrinsic torsion is thus that (3.4) and (3.5) are satisfied with constant f IJ K and p IA B and in addition that where the generalised vector W satisfies c(K I , K J , W ) = 0. The condition on W implies it transforms non-trivially under G S and hence, since J A are singlets, the corresponding intrinsic torsion cannot be a singlet and so must vanish. Alternatively, recall from the discussion in Section 2. spectively, can be identified with the embedding tensor components that encode generic gaugings of five-dimensional N = 2 supergravity theories, including those involving vector fields that transform in non-adjoint representations of the gauge group, as well as antisymmetric rank-2 tensor fields. 12 For simplicity, here we just discuss the case where (3.4) define the structure constants of a Lie algebra, implying that it is not necessary to introduce antisymmetric rank-2 tensor fields. These determine the symmetries of the scalar manifold that are gauged, and hence all matter couplings of the N = 2 theory, completely fixing the five-dimensional Lagrangian (see Appendix B for a brief account of N = 2 supergravity in five dimensions). In particular, the vector multiplet scalar covariant derivatives and the gauge field strengths are given by The embedding tensor formalism is most commonly used to describe the gauging of maximal and half-maximal supergravity [45,46], see however [47,48] for its use in an N = 2 context.
where g is the gauge coupling constant and A I = A I µ dx µ are the five-dimensional gauge fields. In order to obtain the hyperscalar covariant derivatives, we need the Killing vectors on the H structure moduli space (2.35) that generate the gauged isometries. These can be constructed from (3.6) using the standard formalism of coset spaces, see e.g. [49]. Given the left-action of a generator J (K I ) on the coset representative L ∈ H, the corresponding Killing vector k I on M H is determined by the equation where the symbol ∼ means that the equality holds up to an element of the algebra one is modding out by, which in the present case is SU where q X denote the coordinates on M H , the hyperscalar covariant derivatives then read From the Killing vectors k I we can then compute the triholomorphic Killing prepotentials P α I , α = 1, 2, 3, that determine the fermionic shifts and the scalar potential of the N = 2 supergravity theory, see Appendix B for the relevant formulae. These Killing prepotentials are moment maps of the isometries being gauged, and as such can be nicely computed from the generalised geometry formalism. Recalling the definition of the moment map µ in (2.38), they are given by (3.12) In this formula, recall that the J α are the dressed triplet, hence the resulting moment maps are function of the H structure moduli. In the second line, we have used the fact that the singlet torsion components tr (J β (L K I J γ )) are constant on M and hence the integrals over

The truncation ansatz
Our conventions for eleven-dimensional supergravity are as in [42]. The eleven-dimensional bosonic action is (we denote by a hat the 11d quantities) whereF = dÂ andÂ is the three-form potential. The equations of motion arê (3.14) 13 A similar construction could be made for the Killing vectors that gauge isometries in the V structure moduli space, starting from the sections of the adjoint bundle that generate MV mentioned in Footnote 6. However, this will not be needed for our purposes.
The six-form potentialÂ dual to the three-formÂ may be introduced via the first-order relation * dÂ + 1 2Â ∧ dÂ = dÂ , (3.15) whose exterior derivative gives the Maxwell equation.
As first step of the truncation procedure, we arrange the eleven-dimensional bosonic fields into generalised tensors transforming in representations of GL(5, R) × E 6(6) , where GL(5, R) gives the tensorial structure of the fields in the five-dimensional theory obtained after reduction. Then we expand each E 6(6) representation in terms of the G S invariant tensors transforming in the same representation. We separate the eleven-dimensional coordinates in coordinates x µ , µ = 0, . . . , 4, on the external spacetime X, and z m , m = 1, . . . , 6, on the internal manifold M .
The bosonic fields of eleven-dimensional supergravity are decomposed aŝ where Dz m = dz m − h µ m dx µ , and all tensor field components may depend both on x µ and z m , except for the external metric, for which we assume a dependence on the external coordinates only, g µν = g µν (x).
The barred fields need to be redefined. In Appendix C we provide a justification for these redefinitions by studying the gauge transformations of the metric and three-form potential. For the three-form components we introduce the new fields A µνm , A µνρ viā Similar redefinitions apply to the six-form components with at least two external indices, however we will not discuss them in detail here.
The supergravity fields having all components on the internal manifold M arrange into the inverse generalised metric in the following way 14 19) where g m 1 m 2 , n 1 n 2 = g m 1 [n 1 g |m 2 |n 2 ] , and similarly for g m 1 ...m 5 , n 1 ...n 5 . Since the generalised metric is a scalar on the external spacetime, after imposing our truncation ansatz it will provide the scalar fields of the reduced five-dimensional theory.
The density κ introduced in Section 2.1 when defining the HV structure is related to the determinant of the generalised metric and is an E 6(6) invariant. For eleven-dimensional metrics of the form (3.16), this is given by [42,44] (3.20) The tensors with one external leg arrange into a generalised vector A µ on M , with components and will provide the gauge potentials of the reduced theory. The tensors with two antisymmetrised external indices define a weighted dual vector B µν on M , which is a section of det T * M ⊗ E * , with components and will give the two-form fields of the reduced theory. The last term in (3.22) is related to the dual graviton and we will not discuss it further here. The tensors with three antisymmetrised external indices arrange into the generalised tensor which is a section of (a sub-bundle of) the weighted adjoint bundle det T * M ⊗ ad F , whose components are labeled byα = 1, . . . , 57. See e.g. [50,51] for more details on this tensor hierarchy. As discussed in [14], the bosonic part of the truncation ansatz is obtained by imposing that the generalised tensors above are expanded in singlets of the G S structure. The generalised metric is obtained by constructing the K and J α parameterising a family of HV structures as detailed in Section 2.3, and plugging these generalised tensors in the formula (2.27). The resulting generalised metric depends on the H and V structure moduli; when given a dependence on the external coordinates x µ , these are then identified with the hyperscalar and vector multiplet scalar fields of the truncated N = 2 theory, respectively. Thus we have where L is the representative of the coset M H . Comparing the expression for the generalised metric with its general form (3.19), we obtain the truncation ansatz for ∆, g mn , A mnp (as well asÃ m 1 ...m 6 , whenever it is needed). Note that κ 2 given in (3.20) is independent of the scalar fields h I (x) and L(x), so it can be evaluated using any chosen representative of the family of HV structures defined by the G S structure. The gauge potentials A µ I (x) on the external space-time are defined by taking where span({K I }) ⊂ Γ(E) is the vector space spanned by the set of G S singlets K I , I = 0, 1, . . . , n V . Similarly the two-form fields are given by is the vector space spanned by the weighted dual basis vectors K I ♭ , the latter being defined by K I ♭ (K J ) = 3κ 2 δ I J . We also have is spanned by the G S singlets in the weighted adjoint bundle, here denoted by J ♭ A and given by J ♭ A = κ 2 J A . In Appendix C we show that these expressions, together with the field redefinitions (3.17), lead to the correct fivedimensional covariant objects, consistent with the expected gauge transformations.

N = 2 truncations on Maldacena-Nuñez geometries
We now apply the above formalism to consistent truncations of eleven-dimensional supergravity based on generalised structures arising from M5-branes wrapping a Riemann surface.
We start with the N = 2 AdS 5 × w M solution of Maldacena and Nuñez [34] and show that the manifold M admits a generalised U (1) structure with singlet intrinsic torsion, and therefore can be used to construct a consistent truncation. As we have stressed above, once we identify the singlet K I and J A tensors defining the structure it is straightforward to read off the form of the N = 2 supergravity.
We already observed in [14] that this process yields N = 2 supergravity with one hypermultiplet and four vector multiplets. Here we give the details of the construction and derive the gauging, which defines an SO(3) × U (1) × R gauge group. Our truncation includes as a subtruncation the reduction to N = 2 supergravity with one vector multiplet, one hypermultiplet and U (1) × R gauging recently obtained in [36].

The MN1 solution
We are interested in warped AdS 5 solutions to eleven-dimensional supergravity that describe the near-horizon region of M5-branes wrapping supersymmetric cycles in a Calabi-Yau geometry. The amount of supersymmetry of the solutions depends on how the cycle is embedded in the ambient geometry. This corresponds to a topological twist of the worldvolume (0, 2) theory on the M5-branes. The simplest examples are the solutions found by Maldacena and Nuñez [34] describing the near-horizon geometry of M5-branes wrapped on a Riemann surface Σ of negative constant curvature. The topological twist of the (0, 2) world-volume theory is realised by identifying the spin connection on Σ with a U (1) connection in the SO(5) R-symmetry group of the M5-brane theory. The theory preserves N = 2 or N = 1 superconformal symmetry in four dimensions, depending on how the U (1) is chosen inside SO (5). The corresponding supergravity solutions are warped products of AdS 5 times a six-dimensional manifold, M , which is the fibration of a deformed S 4 over Σ.
The SO(5) is realised via the action of the isometry group of the round S 4 . The structure of the fibration reflects the twist of the world-volume theory and determines the amount of supersymmetry of the solutions, which in five-dimensional language is either N = 4 or N = 2, respectively.
In this paper we focus on the N = 2 solution, which we call the "MN1 solution" in the following. The eleven-dimensional metric is 15 where g AdS 5 is the Anti de Sitter metric with radius ℓ = 3 2 R, R being the length scale of the internal space M . The metric on M takes the form Here, g Σ is the uniform metric on (a quotient of) the hyperbolic plane Σ = H 2 , with Ricci scalar curvature R Σ = −2, while υ is the spin connection on Σ satisfying with vol Σ the volume form on Σ. 16 The deformed S 4 is described as a foliation of a round S 3 over an interval, with the interval coordinate being ζ ∈ [0, π], while σ α , α = 1, 2, 3, are the standard SU (2) left -invariant forms on S 3 , expressed in terms of Euler angles {θ, φ, ψ}. 15 We present the solution in a form similar to the one given in [52,Sect. 5]. The precise dictionary with this reference is: where the variables on the left-hand side are those of [52] while the variables on the right-hand side are those used here. The length scale R that appears in our expressions is equal to the radius of S 4 in the related AdS7 × S 4 Freund-Rubin solution of eleven-dimensional supergravity. The four-formF in (4.5) has an overall opposite sign with respect to the one of [52],F = −FGMSW; this sign does not affect the equations of motion, it just modifies the projection condition satisfied by the supersymmetry spinor parameter. 16 Choosing local coordinates x, y on the hyperbolic plane, one can write gΣ = dx 2 +dy 2 y 2 , volΣ = dx∧dy y 2 , and υ = − dx y .
Their explicit expression can be found in Appendix D, together with more details on the parameterisation of S 4 . The warp factor is while the four-form readŝ Note that the invariant volume form (3.20) is given by where vol 4 is the volume form of the round S 4 of radius R. The solution has SU (2) left × U (1) right symmetry, which embeds in the SO(5) isometry group of a round S 4 as This symmetry is manifest as the solution is given in terms of the σ α . The globally-defined combination (σ 3 + υ) describes a fibration of S 4 over Σ, such that the U (1) right action on S 4 is used to cancel the U (1) holonomy of Σ.
The U (1) right factor provides the R-symmetry of the holographically dual N = 1 SCFT, while SU (2) left corresponds to a flavour symmetry. The dual N = 1 SCFT has been described in [53].

Generalised U (1) structure of the MN1 solution
The solution reviewed above admits a generalised U (1) S structure, which will be the basis for constructing our consistent truncation. In order to characterise it we proceed in two steps. The first is purely group theoretical: it consists in embedding the relevant U (1) S in E 6(6) , computing its commutant and the corresponding decompositions of the generalised tangent and adjoint bundles. To this end, it is convenient to decompose E 6(6) according to its maximal compact subgroup USp(8)/Z 2 . Since the usp(8) algebra can be given in terms of Cliff(6) gamma matrices (see Appendix E.1), this reduces the problem to gamma matrix algebra. The details of the derivation can be found in Appendix E; here we will just give the results. Once the relevant U (1) S singlets are identified, the second step is to express them in terms of the geometry of the six-dimensional manifold M .
The generalised U (1) S structure of the MN1 solution is the diagonal of the ordinary geometrical U (1) ≃ SO(2) ⊂ GL(2, R) structure on the Riemann surface and a U (1) factor in the SO(5) ⊂ SL(5, R) ≃ E 4(4) generalised structure for the generalised tangent space of the four-sphere. In terms of the isometry group decomposition (4.7) this can be identified with U (1) right . If we denote by 1 to 4 the directions in M along S 4 and by 5,6 those along Σ, the generator of U (1) S can be written as a usp(8) element as whereΓ m are six-dimensional gamma matrices. The first term corresponds to the U (1) holonomy of Σ while the second one is the U (1) right in SO (5). By computing the commutators of (4.8) in USp (8) we find that the U (1) S structure embeds in USp(8) as 17 where as above we distinguish the factor SU (2) H that gives the R-symmetry of the fivedimensional supergravity theory. Under this splitting, the spinorial representation of USp(8) decomposes as where the elements in the brackets denote the SU (2) × SU (2) H representations and the subscript gives the U (1) S charge. We then see that there are only two spinors that are singlets under U (1) S and that transform as a doublet of SU (2) H as required by N = 2 supersymmetry. The embedding of the U (1) S structure in the full E 6(6) is obtained in a similar way (see Appendix E.2 for details) where C E 6(6) (U (1) S ) is the commutant of U (1) S in E 6(6) . We can now determine how many generalised vectors and adjoint elements are U (1) S singlets. Under (4.11) the 27 decomposes as where the first subscript denotes the U (1) S charge and the second one the R + charge. We see that there are five singlets K I , I = 0, 1, . . . , 4, where is only charged under the R + , while form a vector of SO(3, 1). The singlets in the 78 adjoint representation are the generators of the commutant C E 6(6) (U (1) S ). However only the generators of the SU (2, 1) subgroup are relevant for 17 Here and below we give expressions ignoring subtleties involving the centres of each group; thus for instance we will not distinguish between embeddings in USp(8) and USp(8)/Z2. the structure. Indeed, (4.12) shows that the generators of R + × SO(3, 1) do not leave the singlet vectors invariant, and therefore, as discussed in Section 2.1, do not contribute to the truncation. As shown in (E.40) and (E.41), they can be obtained as products K I × ad K * J . We denote by J A , A = 1, . . . , 8, the elements of the adjoint bundle generating su 2,1 . Four of them are in the 36 of USp (8) and generate the compact subalgebra su 2 ⊕ u 1 , and four more are in the 42 of USp (8) and generate the rest of su 2,1 .
The U (1) S structure is then defined by The derivation of the explicit expressions for these generalised tensors relies on the way the solution of [34] is constructed by deforming the AdS 7 × S 4 background dual to flat M5-branes so as to describe their backreaction when wrapping a Riemann surface Σ. The world-volume theory on the wrapped M5-branes is made supersymmetric by a topological twist, where the spin connection on the Riemann surface is cancelled by switching on a background gauge field for a U (1) subgroup of the SO(5) R-symmetry. On the dual background the topological twist implies that M is an S 4 fibration over Σ The generalised tangent bundle for S 4 is given by and transforms under SL(5, R) ≃ E 4(4) . It is generalised parallelisable, meaning it admits a globally defined frame [15]. The idea is then to consider first the direct product Σ × S 4 , express the E 6(6) generalised tensors on this manifold in terms of the frame on Σ and the parallelisation on S 4 , and then implement the twist of S 4 over Σ so as to make globally well-defined objects. In the decomposition where GL(2, R) is the structure group of the conventional tangent bundle on Σ and SL(5, R) ≃ E 4(4) is the structure group of the generalised tangent bundle on S 4 , the E 6(6) generalised tangent bundle on Σ × S 4 decomposes as 19) and the adjoint bundle as (4.20) In the expressions above E 4 is the generalised tangent bundle on S 4 introduced in (4.17), ad F 4 is the adjoint bundle on S 4 , 21) and N 4 and N ′ 4 are the following bundles on S 4 , The bundles E 4 , N 4 and N ′ 4 admit the globally defined generalised frames see Appendix D for their expression in a coordinate basis and note that they include a contribution from the three-form gauge potential A S 4 of the flux on the S 4 . Geometrically this defines a generalised identity structure on S 4 . Given the way U (1) S is embedded in USp (8), we will find it useful to also introduce the following linear combinations of the generalised frame elements E ij on S 4 , Since their restriction to T M corresponds to the Killing vectors generating the SU (5) isometries of S 4 (again see Appendix D for their explicit expression), Ξ α and Ξ α , α = 1, 2, 3, may be seen as generalised Killing vectors generating the corresponding generalised isometries. As for the Riemann surface Σ, it can be (a quotient of) the hyperbolic plane H 2 as in the MN1 solution reviewed in Section 4.1, but we can also take a torus T 2 , or a sphere S 2 . We introduce orthonormal co-frame one-forms e 1 , e 2 on Σ, such that the constant curvature metric and the compatible volume form on Σ are given by The metric is normalised so that the Ricci scalar curvature is R Σ = 2κ, where κ = +1 for S 2 , κ = 0 for T 2 and κ = −1 for H 2 (and quotients thereof). We also define the U (1) spin connection, υ, on Σ as The decompositions (4.19) and (4.20) allow us to express the U (1) S invariant generalised tensors in terms of tensors on Σ and the S 4 generalised frames introduced above. We provide the derivation in Appendix E and here just present the resulting expressions. Let us first focus on the singlet generalised vectors K I . These can be written as where Υ is a section of the adjoint bundle implementing the twist of S 4 over Σ as in (4.16), ensuring that these are globally defined objects on the six-dimensional manifold. Recall that in the MN1 solution, the U (1) that is used to twist the four-sphere and compensate the spin connection υ on Σ is the Cartan of SU (2) right ⊂ SO (5). The E 6(6) twist element Υ is constructed in a way similar to the one used in [14], albeit with a different choice of U (1) in SO (5). We embed the connection one-form υ in a generalised dual vector, the Killing vector generating the Cartan of SU (2) right in the generalised vector Ξ 3 introduced above, and we project their product onto the adjoint of E 6(6) . That is, where × ad denotes the projection onto the adjoint and again R is the radius of S 4 . Evaluating its action in (4.27), we find that this is trivial for all the K I 's except for K 4 , and we obtain our final expressions A similar procedure applies to the singlets J A , A = 1, . . . , 8, in the adjoint bundle. In this way we obtain (see Appendix E for the derivation) where the superscript * denotes dual generalised vectors, transforming in the 27, and we introduced Ψ 1i = R e 1 ∧ E i and Ψ 2i = R e 2 ∧ E i . The adjoint action of e Υ is evaluated using the formula (A.21); we do not show the resulting expressions as they are rather lengthy. Evaluating the commutators [J A , J B ] using again (A.21), we checked that the J A satisfy precisely the SU (2, 1) commutation relations (see (F.4) for our choice of SU (2, 1) structure constants).

The V and H structure moduli spaces
We now construct the V structure and H structure moduli spaces. Applying the general discussion of Section 2.3 we have As we now show the first two factors give the V structure moduli space and the last factor the H structure moduli space.

The V structure
Evaluating (2.16) for the K I constructed above we obtain the constant, symmetric tensor C IJK . Using the invariant volume (4.6), we find that the non-vanishing components of C IJK are given by where η = diag(−1, −1, −1, 1) . (4.33) A family of V structures is then obtained by defining K as the linear combination (2.30) and imposing the condition (2.31). It follows that our V structure moduli space is the hypersurface so that while Σ (that we assume strictly positive) is a coordinate on R + , whose powers in (4.36) are dictated by the weight of the K I 's under the action of the R + that commutes with the generalised structure. The resulting V structure moduli space thus is and will determine n V = 4 vector multiplets in five-dimensional N = 2 supergravity. Note that by identifying SU (2) ≃ Spin(3) this matches the first two factors in (4.31). The isometry group is SO(3, 1) because the h I form a vector rather than a spinor representation of Spin(3, 1).

The H structure
We next turn to the H structure moduli space, again following the general discussion given in Section 2.3. Since the commutant of SU (2) H in SU (2, 1) is U (1), from (2.35) we obtain that the H structure moduli space is 18 .
This is a simple quaternionic-Kähler manifold of quaternionic dimension n H = 1. We will denote by {ϕ, ξ, θ 1 , θ 2 } (4.41) the coordinates on this space. In Appendix F we give the explicit parameterisation chosen for the coset space as well as the explicit form of the "dressed" su(2) elements J α , depending on {ϕ, ξ, θ 1 , θ 2 }, in terms of su(2, 1) elements. Below we will use this dressed triplet to construct the generalised metric. In Appendix F we also give the SU (2, 1) invariant metric on M H , which will provide the hyperscalar kinetic term in the five-dimensional theory.

Intrinsic torsion and gauging
For the U (1) S structure constructed in the previous section to lead to a consistent truncation, it must be checked that its intrinsic torsion only contains U (1) S singlets, and that these are constant. In particular we need to show that equations (3.4), (3.5) and (3.7) hold.
For the first two conditions we evaluate the generalised Lie derivatives of the tensors K I and J A in (4.29) and (4.30), using the action of generalised Lie derivative on a generalised vector and on sections of the adjoint bundle given in Appendix A.
Consider first the algebra of the generalised vectors (4.29). Using the fact that, under the generalised Lie derivative, the S 4 frames E ij generate an so(5) algebra one can show that the only non-zero Lie derivatives are so that the vectors K α , α = 1, 2, 3, lead to an SO(3) factor in the gauge group in the fivedimensional supergravity. 19 This embeds in the SO(3, 1) factor of the global symmetry group of the ungauged theory in the obvious way. Hence (4.43) determines the components of the embedding tensor acting on the vector multiplet sector of the five-dimensional supergravity theory. We thus have that the non-vanishing structure constants are f αβγ = −2 ǫ αβγ and the gauge coupling constant is g = 1 R . Recalling (3.8), the non-trivial vector multiplet scalar covariant derivatives are 18 More precisely one has MH = SU (2, 1)/S(U (2) × U (1)). 19 For simplicity, we use the indices α, β = 1, 2, 3 both to label the generators of the SU (2)H entering in the definition of the H structure and the generators of the SU (2) in the V structure, although these are different subgroups of E 6(6) .
while the gauge field strengths read In order to determine the gauging in the hypersector we also need to compute the Lie derivative of the J A along the generalised vectors K I . We find that the J A are neutral under the action of the SO(3) generators K α , consistently with the fact that the gauging in the vector multiplet sector does not affect the hypersector. On the other hand, the remaining generalised vectors K 0 and K 4 act non-trivially on the J A , and determine an abelian gauging of the SU (2, 1) generators. In detail, the generalised Lie derivative of the J A along K 0 gives while the one along K 4 yields (4.48) The actions (4.47) and (4.48) can equivalently be expressed in terms of an adjoint action as where the sections of the adjoint bundle correspond to SU (2, 1) generators acting on the H-structure moduli space (4.40) as isometries. We denote by k 0 and k 4 the corresponding Killing vectors on M H . These can be calculated applying (3.10) to the coset representative L given in Appendix F, and read These Killing vectors specify the isometries of M H that are gauged in the five-dimensional supergravity. The hyperscalar covariant derivatives (3.11) are determined as The triholomorphic Killing prepotentials P α I obtained by evaluating the moment maps (3.12) read with P α 1 = P α 2 = P α 3 = 0. The information above completely characterises the five-dimensional N = 2 supergravity obtained upon reduction on M . This will be discussed in Section 4.6. Before coming to that, we provide the explicit bosonic truncation ansatz.

The truncation ansatz
The truncation ansatz is built following the general procedure described in Section 3.2. We compute the inverse generalised metric (2.27) out of the U (1) S invariant generalised tensors. This depends on the V structure moduli {Σ, H 1 , H 2 , H 3 , H 4 } and on the H structure moduli {ϕ, ξ, θ 1 , θ 2 }, which are now promoted to scalar fields in the external, five-dimensional spacetime. Then we evaluate the generalised tensors A µ , B µν , C µνρ using (3.25)- (3.27). Separating the components of these tensors as described in Section 3.2, we obtain the ansatz for the eleven-dimensional metricĝ and three-form potentialÂ.
We start from the covariantised differentials Dz m = dz m − h µ m dx µ of the coordinates on M , that appear in (3.16). From (3.21) and (3.25) we see that h µ = h µ m ∂ m is given by where K I | T M is the restriction of K I to the tangent bundle of M . Evaluating the right hand side using the explicit form (4.29) of the generalised vectors K I , we obtain where we recall that ξ α ,ξ α , α = 1, 2, 3, are the pull-back to T M of the SU (2) left -and SU (2) right -invariant vectors on S 3 , respectively, whose coordinate expression is given in (D.2) and (D.3). It follows that Dz m , and thus both the eleven-dimensional metric and three-form, contain the five-dimensional gauge potentials A α , A 4 , gauging the SU (2) left × U (1) right isometries of S 3 in M . Notice that A 0 does not appear in (4.55) as K 0 does not have a component in T M , hence it will not enter in the eleven-dimensional metric.
However K 0 will appear in the ansatz for the three-form, as it does have a component in In order to express our ansatz in a more compact way, it will be convenient to introduce new one-forms Ω α and Ω α , α = 1, 2, 3, adapted to the symmetries of the problem, that incorporate the covariantised differentials above but also include some more terms. Recall that we describe S 4 as a foliation of S 3 , parameterised by Euler angles {θ, φ, ψ}, over an interval, parameterised by ζ. We define Ω 1 = cos ψ Dθ + sin ψ sin θ Dφ , Ω 1 = cos φ Dθ + sin φ sin θ Dψ , which are analogous to the left-and right-invariant forms σ α ,σ α given in (D.6) and (D.7), but with the ordinary differential of the coordinates being replaced by the new covariantised differential D. This extends the differential D given above and is defined as with where the five-dimensional scalars θ 1 , θ 2 are two of the H structure moduli, and we recall that e 1 , e 2 are the vielbeine on the Riemann surface Σ while υ is the connection on Σ.
The local one-formsÃ α , A α gauge all the left-and right-isometries of S 3 , respectively, and would correspond to SO (4) ≃ SU (2) left × SU (2) right gauge potentials in the reduction of eleven-dimensional supergravity on S 4 down to seven-dimensional supergravity. However, in the further reduction on Σ of interest here only A α , A 4 become five-dimensional gauge fields, while the rest of (4.58) implements the twist on the Riemann surface and introduces the five-dimensional scalars θ 1 , θ 2 .
We are now in the position to give the truncation ansatz for the eleven-dimensional metricĝ = e 2∆ g µν dx µ dx ν + g mn Dz m Dz n . (4.59) The warp factor is e 2∆ =∆ 1/3 (e ϕ Σ) 4/5 , (4.60) while the part with at least one internal leg reads where ⊗ s is the symmetrised tensor product, defined as Ω ⊗ s Ω = 1 2 (Ω ⊗ Ω + Ω ⊗ Ω). In these expressions we introduced the function ∆ = e −2ϕ Σ 3 −4/5 cos 2 ζ + e −2ϕ Σ 3 1/5 H + sin 2 ζ , Note that in the last expression the exterior derivative acts on the internal coordinates and not on the scalars H I , which only depend on the external coordinates. We next come to the eleven-dimensional three-form potentialÂ. We first give our result and then make some comments. The ansatz forÂ readŝ where the five-dimensional gauge field strengths, F, and the covariant derivatives, D, of the five-dimensional scalars were given in (4.45) and (4.52), respectively. Equation (4.64) has been obtained by first computingÂ through the general procedure of Section 3.2, then implementing a gauge shift by an exact three-form so as to obtain a nicer expression (this is why derivatives of the external fields appear), and finally dualising away the five-dimensional two-and three-forms, so that the only five-dimensional degrees of freedom contained in the ansatz are scalar and vector fields, in addition to the metric g µν . Let us outline how this dualisation is performed. Evaluating (3.26) and (3.27), we find that only one external two-form B and one external three-form C appear in the ansatz forÂ. These are paired up with the generalised tensors E 5 and E ′ 5 on S 4 , which, as generalised tensors on M , are sections of det T * M ⊗ E * and det T * M ⊗ ad F , respectively. The combination entering inÂ is where the subscript on the left-hand side indicates the restriction to the three-form part, and the last term in the expression is removable via a gauge transformation ofÂ. Hence B and C only appear in the combination (C − dB). This means that the two-form gets eaten by the three-form via the Stuckelberg mechanism, giving it a mass. While a massless three-form in five-dimensions is dual to a scalar field, here we dualise the two-form at the same time and also obtain a vector field. The duality relation is obtained considering the duality between the eleven-dimensional three-formÂ and six-formÂ given in (3.15), and looking at the relevant terms with three external indices. In this way we find that We have used this expression to eliminate (C − dB) completely from the truncation ansatz. This explains the * 5 F 0 term appearing in (4.64).
Our truncation ansatz reproduces the Maldacena-Nuñez solution given in Section 4.1 upon taking κ = −1 and setting the scalars to The consistent truncation of [36] is recovered as a subtruncation that projects out the fields transforming under SU (2) left , that is settingÃ α = H α = 0, α = 1, 2, 3, which also implies H 4 = 1. 20 The further truncation to minimal gauged supergravity is obtained by setting the scalars to their AdS value (4.67) and taking A 0 = −A 4 .
One can obtain a slightly larger subtruncation by projecting out only the modes charged under U (1) left , rather than SU (2) left , namely settingÃ 1 =Ã 2 = H 1 = H 2 = 0. This leaves us with two vector multiplets, one hypermultiplet and just the abelian gauging generated by the Killing vectors (4.51), which is the same as the one in the truncation of [36]. 21 A notable generalisation of this subtruncation will be discussed in Section 5.
The truncation of [36] was obtained via a reduction of gauged seven-dimensional supergravity on the Riemann surface Σ. Similarly, we can obtain our truncation ansatz by combining the well-known truncation of eleven-dimensional supergravity on S 4 [9], leading to seven-dimensional maximal SO (5) supergravity, with a further truncation reducing the seven-dimensional theory on Σ. Starting from the convenient form of the bosonic truncation ansatz on S 4 given in [56], we have explicitly checked that this procedure works out as expected and reproduces the ansatz above.

The five-dimensional theory
We now put together the ingredients defining the truncated five-dimensional theory and discuss it in more detail. This is an N = 2 gauged supergravity coupled to four vector multiplets and one hypermultiplet. The vector multiplet scalar manifold is while the hypermultiplet scalar manifold is . (4.69) As discussed before, these have a geometric origin as the V and H structure moduli spaces of the internal manifold. At the bosonic level, the vector multiplets are made of gauge fields A I and constrained scalar fields h I , I = 0, 1, . . . , 4, which we have parameterised 20 Then the one-forms Ωα essentially reduce to those in [36], up to slightly different conventions, while Ωα drop out of the ansatz. When comparing our truncation ansatz with the one given in Section 4.1 of [36], one should take into account thatÂ here = −Â FNR (this is seen from comparing our 11d Maxwell equation with the one in [54], which provides the 7d to 11d uplift formulae used in [36]). Moreover ζ here = ζ FNR + π/2 , 21 Curiously, this five-dimensional supergravity with two vector multiplets and one hypermultiplet looks closely related to the N = 2 "Betti-vector" model obtained in [55, Section 7] as a consistent truncation of IIB supergravity on T 1,1 . The two models are not the same though, as the details of the couplings between the fields are different.
in terms of Σ and H I , I = 1, . . . , 4, in (4.36). The latter scalars satisfy the constraint η IJ H I H J = 1, with η = diag(−1, −1, −1, 1). We have also found that the non-vanishing components of the symmetric tensor C IJK are given by The kinetic terms in the vector multiplet sector are controlled by the matrix a IJ , given by the general formula (2.34), which in our case reads The hypermultiplet comprises the scalars q X = {ϕ, ξ, θ 1 , θ 2 }, and the kinetic term is given by the quaternionic-Kähler metric on M H that we derived in Appendix F, The gauge group is SO(3) × U (1) × R. The symmetries being gauged are the SO(3) ⊂ SO (3, 1) isometries of M V and two abelian isometries in M H , generated by the Killing vectors (4.51). Note that the ∂ ξ term generates the non-compact R factor and the θ 2 ∂ θ 1 − θ 1 ∂ θ 2 term generates the compact U (1).
We recall for convenience the gauge field strengths and the covariant derivatives of the charged scalars, where the gauge coupling constant is given by the inverse S 4 radius, g = 1 R . The scalars Σ, H 4 and ϕ remain uncharged. The gauging in the hypersector is the same as in [36], while the gauging in the vector multiplet sector is a novel feature of our truncation.
Plugging these data in the general form of the N = 2 supergravity action given in Appendix B, we obtain the bosonic action for our model, where G 5 is the five-dimensional Newton constant. 22 The scalar potential V is obtained from the Killing prepotentials of the gauged isometries as summarised in Appendix B. The Killing prepotentials were already given in (4.53). We can check this expression by starting from the Killing vectors (4.51) and evaluating (B.15) using a standard parameterisation for the universal hypermultiplet; we have verified that indeed the same result is obtained. Then (B.17) gives for the scalar potential (4.76) The supersymmetric AdS vacuum conditions summarised in Eq. (B.18) are easily solved and give the scalar field values that is precisely the values (4.67) that reproduce the MN1 solution reviewed in Section 4.1.
The negative curvature κ = −1 for the Riemann surface arises as a positivity condition for the scalars Σ and e 2ϕ . The critical value of the scalar potential yields the cosmological constant Λ ≡ V = − 8 3R 2 , corresponding to an AdS 5 radius ℓ = 3 2 R, again in harmony with the solution in Section 4.1.
By extremising the scalar potential (4.76) we can search for further AdS 5 vacua within our truncation. Then, by analysing the mass matrix of the scalar field fluctuations around the extrema we can test their perturbative stability. In the following we discuss the outcome of this analysis for the three extrema that we have found.
• We recover the supersymmetric vacuum (4.77). Being supersymmetric, this is stable.
The supergravity field fluctuations source SU (2, 2|1) superconformal multiplets in the dual N = 1 SCFT [53], with the supergravity mass eigenvalues providing the conformal dimension ∆ of the operators in the multiplets. The field fluctuations that were also considered in [36] correspond to the energy-momentum multiplet (containing the energy-momentum tensor with ∆ = 4 and the R-current with ∆ = 3) and to a long vector multiplet of conformal dimension ∆ = 1 + √ 7 (see [36] for more details). The additional SO(3) vector multiplet included in this paper sources a conserved SO(3) flavour current multiplet in the dual SCFT. The three scalar operators in this multiplet have conformal dimension ∆ = 2 (once) and ∆ = 4 (twice), while the SO(3) flavour current has conformal dimension ∆ = 3, as required for a conserved current. Another piece of information about the dual SCFT is given by the Weyl anomaly coefficients; these are obtained from the five-dimensional Newton constant G 5 and the AdS 5 radius ℓ through the formula a = c = πℓ 3 8G 5 . 22 As discussed in [33], the five-dimensional Newton constant is given by (G5) −1 ∝ M e 3∆ vol6 = M κ 2 .
In the present case, M κ 2 = R 2 VolΣ Vol4, where VolΣ = 4π(1−g) κ is the standard volume of a Riemann surface of genus g and Vol4 = 8π 3 3 R 4 is the volume of a round S 4 with radius R.
• When κ = −1 we also recover the non-supersymmetric vacuum discussed in [36], that was originally found in [57]. The analysis of the scalar mass matrix shows that the fluctuation of H 4 has a mass squared m 2 ℓ 2 ≃ −4.46, which is below the Breitenlohner-Freedman bound ℓ 2 m 2 BF = −4. We thus establish that this vacuum is perturbatively unstable. Note that the unstable mode lies outside the truncation of [36].
• For κ = +1, we find a non-supersymmetric vacuum with non zero value of the Hscalars, given by where ℓ is the AdS radius. This appears to be a new solution. It represents an SO (3) worth of vacua really, since the scalars H α , α = 1, 2, 3, can take any value such that . We find that a linear combination of the fluctuations of Σ, ϕ and H 4 has mass squared m 2 ℓ 2 ≃ −5.86 < m 2 BF ℓ 2 , hence this vacuum is perturbatively unstable. Nevertheless, it allowed us to perform a nontrivial check of our truncation ansatz for non-vanishing H-fields, as we have verified that its uplift does satisfy the equations of motion of eleven-dimensional supergravity.

Truncations for more general wrapped M5-branes
The N = 2 and N = 4 Maldacena-Nuñez solutions are special cases of an infinite family of N = 2 solutions [35,38], 23 describing M5-branes wrapping a Riemann surface in a Calabi-Yau geometry. These solutions, which we will denote as BBBW solutions, have the same general features of the MN1 solution. In particular, they all admit a generalised U (1) S structure, which we use to derive the most general consistent truncation to N = 2 gauged supergravity in five dimensions associated with such backgrounds. As we will see, the truncated theory has two vector multiplets, one hypermultiplet and gauge group U (1) × R. It generalises the U (1) right invariant subtruncation of the truncation presented in the previous section: the matter content is the same and the gauging is deformed by one (discrete) parameter. Our systematic approach allows us to complete the consistent truncation derived from seven-dimensional maximal SO(5) supergravity on Σ previously presented in [37] by including all scalar fields in the hypermultiplet and directly deriving the gauging. 24

The BBBW solutions
The BBBW solutions describe M5-branes wrapped on a Riemann surface Σ, such that the (2, 0) theory on the branes has a twisting over Σ depending on two integer parameters p and q. The way the Riemann surface is embedded in the ambient space determines the local structure of the latter. The authors of [35,38] showed that there is an infinite family 23 See also [58], where a subset of the solutions was previously found. 24 We thank Nikolay Bobev and Alberto Zaffaroni for pointing out this reference. of allowed geometries, corresponding to the fibration L 1 ⊕ L 2 ֒→ Σ of two complex line bundles over the Riemann surface, so that the total space is Calabi-Yau. The degrees of these line bundles are identified with the integers that parameterise the twist of the M5 world-volume theory, p = deg L 1 and q = deg L 2 . By the Calabi-Yau condition p and q must satisfy p + q = 2g − 2, with g the genus of Σ. In this setup, the N = 1 and N = 2 twistings considered in [34] arise from setting p = q and q = 0 (or p = 0), respectively.
where ℓ is the AdS radius. The six-dimensional manifold M is still a fibration of a squashed four-sphere over the Riemann surface, with metric where the Riemann surface metric g Σ satisfies (4.25), (4.26), and the metric on the squashed and fibered S 4 is The angles ϕ 1 , ϕ 2 vary in [0, 2π], 25 and µ 0 = cos ζ , µ 1 = sin ζ cos θ 2 , µ 2 = sin ζ sin θ 2 , (5.5) with ζ, θ ∈ [0, π]. The two circles ϕ 1 and ϕ 2 are independently fibered over the Riemann surface, with connections where υ is again the connection on Σ and the discrete parameter z is related to the integers p and q as The warping function∆ and the constants f 0 , g 0 depend on z and on the curvature κ of the Riemann surface as The four-form flux is given bŷ where the Hodge star * 4 is computed using the metric (5.4). The solution has two U (1) isometries corresponding to shifts of the angles ϕ 1 , ϕ 2 that parameterise the two diagonal combinations of the U (1) right and U (1) left subgroups of SO (5). It turns out that neither of them corresponds to the superconformal R-symmetry of the dual N = 1 SCFT, which is given by a linear combination involving X 1 , X 2 [35,38].

Generalised U (1) S structure
The construction of the generalised structure associated to the BBBW solutions follows the same lines as for the MN1 solution. We first embed the ordinary U (1) structure in E 6(6) and then look for the invariant generalised tensors. The generalised U (1) S structure of the solutions is determined by the topological twist of the M5 world-volume theory, as a linear combination of the U (1) holonomy of Σ and the U (1) right and U (1) left subgroups of the SO(5) R-symmetry group This embeds in E 6(6) as an element of its compact subgroup USp(8) with generator whereΓ 56 is the usp 8 element generating U (1) Σ and 1 2 (Γ 12 ±Γ 34 ) generate U (1) left/right . When p = q we recover the U (1) S structure group of the MN1 solution, whereas q = 0 (or p = 0) gives the MN2 structure considered in [14]. Below we assume that p, q are generic, and do not fulfill these special conditions which as we have seen lead to a larger truncation.
By looking at the singlets under u(1) S in the 27 and 78 representations of E 6(6) , we find that the U (1) S structure is defined by eight J A , A = 1, . . . , 8, in the adjoint bundle and three generalised vectors K I , I = 0, 1, 2. The singlets in the adjoint bundle have the same form (4.30) as for the MN1 solution, while the three singlet generalised vectors take the same form as a subset of the MN1 generalised vectors, 26

(5.13)
However now the twisting element Υ has a more general form dictated by the embedding (5.12), that is This makes our generalised tensors globally well-defined. We emphasise that these depend on the integers p, q only through (5.14).

Features of the truncation
The number of U (1) S singlets in the 27 and 78 implies that the truncated supergravity theory contains two vector multiplets and one hypermultiplet. The H structure moduli space is the same as for the MN1 case, .
As before, this is parameterised by real coordinates q X = {ϕ, ξ, θ 1 , θ 2 } and the metric is given by Eq. (4.72). The V structure moduli space is determined again following our discussion in Section 2, and is a subspace of the one for the MN1 truncation. Evaluating the cubic invariant on the singlets K I as in (2.16), we obtain that the non-zero components of the C IJK tensor are for I, J = 1, 2 , while again Σ parameterises R + . Thus the V structure moduli space is (1, 1) .
The kinetic matrix a IJ then takes the same form (4.71), that is 26 Before acting with Υ, the singlets for the BBBW solutions are related to those used for the MN1 solutions as and to the structure of the MN2 solution in [14] as ) .
a 01 = a 02 = 0 , The gauging of the reduced theory is obtained from the generalised Lie derivative L K I acting on the K J and the J A . The Lie derivatives among vectors are now trivial, As discussed in Section 2, the Lie derivatives L K I J A are conveniently expressed as the adjoint action of SU (2, 1) generators, Evaluating the generalised Lie derivatives we find It follows that the covariant derivatives of the charged scalars are where again the inverse S 4 radius 1 R plays the role of the gauge coupling constant. The Killing prepotentials can be computed either from (3.12) or from (B.15), and read P α 0 = 0 , 0 , 1 4 e 2ϕ , P α 1 = 0 , 0 , 1 4 κ z e 2ϕ , P α 2 = √ 2 e ϕ θ 1 , √ 2 e ϕ θ 2 , −1 + 1 4 e 2ϕ 2θ 2 1 + 2θ 2 2 − κ . (5.26) Notice that for z = 0 (that is p = q), the quantities above reduce to those obtained for the MN1 structure in Section 4.4. The five-dimensional bosonic action is then determined to be where the scalar potential reads It is straightforward to analyse the supersymmetric AdS 5 vacuum conditions (B.18). The hyperino equation gives where we assume κ = ±1 (hence leaving aside the case κ = 0). The gaugino equation gives Plugging the Killing prepotentials (5.26) and using (5.29) we obtain Taking into account the allowed range of the scalar fields, the solution to these equations is For κ = 1, well-definiteness of the fields requires |z| > 1, as in [35], while z can be generic for κ = −1. The MN1 case z = 0 is recovered as a limiting case after fixing κ = −1. The critical value of the scalar potential determines the AdS radius ℓ as Although we do not present the uplift formulae for this truncation, we have checked that the supersymmetric vacuum identified above matches the BBBW solution summarised in Section 5.1. To do so, we have computed the inverse generalised metric G −1 associated with the U (1) S structure under consideration; this depends on the V structure and H structure parameters. From the generalised metric we have reconstructed the ordinary metric g 6 and the three-form potential on M , as well as the warp factor e 2∆ . Substituting the values for the scalars found above, we find agreement with the solution in Section 5.1 upon fixing the S 4 radius as R = 1 2 and implementing the following dictionary: with our AdS radius being given in terms of the quantities appearing there as By extremising the scalar potential 27 we recover the supersymmetric vacuum and also find new non-supersymmetric vacua, where the scalar field values are rather complicated functions of the parameter z. As an example, we give the numerical values for one chosen value of z, that we take z = 1 2 . When κ = −1 we find a new extremum of the potential at Σ ≃ 0.9388 , ϕ ≃ 0.1109 , H 2 ≃ 1.0217 , θ 1 = θ 2 = 0 , ℓ ≃ 1.5276 R , (5.36) while when κ = 1 we find an extremum at Σ ≃ 0.8631 , ϕ ≃ 0.2812 , H 2 ≃ 1.5506 , θ 1 = θ 2 = 0 , ℓ ≃ 1.0644 R , (5.37) and another one at Σ ≃ 1.1580 , ϕ ≃ 0.8455 , H 2 ≃ 1.9847 , θ 1 = θ 2 = 0 , ℓ ≃ 0.6198 R , (5.38) where for each solution we have also indicated the corresponding AdS radius ℓ.

Conclusions
In this paper we have illustrated the Exceptional Generalised Geometry approach to N = 2 consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M . We have argued that for the truncation to go through, M must admit a generalised G S ⊆ USp(6) structure with constant singlet intrinsic torsion, and we have explained how this completely determines the resulting five-dimensional supergravity theory. We have also given an algorithm to construct the full bosonic truncation ansatz. This formalism provides a geometric understanding of the origin of the truncations, in particular those that are not based on invariants of conventional G-structures on the tangent bundle. It also sidesteps the need to reduce the equations of motion in order to uncover the matter content and couplings of the truncated theory. The main technically involved part of this formalism is deriving the truncation ansatz. However a significant advantage is that once this is done the relevant expressions can be used to derive the uplift formulae for any N = 2 consistent truncation. One does not have postulate the set of consistent modes case-by-case. Furthermore the structure of the resulting gauged supergravity is then simply determined by the generalised structure.
To demonstrate the concrete effectiveness of the formalism we worked out the full bosonic truncation ansatz on Maldacena-Nuñez geometries, leading to five-dimensional N = 2 supergravity with four vector multiplets, one hypermultiplet and a non-abelian gauging, having the N = 2 AdS 5 solution of [34] as a vacuum solution. This extends the truncation of [36] by SO(3) vector multiplets. For the BBBW geometries [35,38], we obtained a truncation featuring two vector multiplets, one hypermultiplet and an abelian gauging, completing the truncation obtained in [37]. This can be seen as a one-parameter deformation of the truncation obtained from the one on Maldacena-Nuñez geometry by imposing invariance under the Cartan of SO (3). Although in this case we did not give all details of the truncation ansatz, it should be clear that it can be obtained by following precisely the same steps presented for the case of Maldacena-Nuñez geometry. Since the generalised geometry tensors on S 4 used in these trucations are a subset of those appearing in the reduction of eleven-dimensional supergravity to maximal SO (5) supergravity in seven dimensions, it should also be clear that our consistent truncations can equivalently be obtained as truncations of maximal SO(5) supergravity on a Riemann surface.
Together with the half-maximal truncation presented in [14,39], which is based on the N = 4 solution of [34], this work provides the largest possible consistent truncations of eleven-dimensional supergravity that have as seed known AdS 5 × w M supersymmetric solutions describing M5-branes wrapped on a Riemann surface (larger truncations may be possible by including degrees of freedom that go beyond eleven-dimensional supergravity, such as membrane degrees of freedom).
It would be interesting to explore further the relatively simple five-dimensional supergravity models obtained in this paper and construct new solutions thereof. These would have an automatic uplift to eleven dimensions, and may have an interpretation in the dual SCFT. For the subtruncation with no SO (3) vector multiplet, solutions of holographic interest have been discussed in [36]. Our larger consistent truncation may offer the possibility to obtain solutions where non-abelian gauge fields are activated, which are quite rare in holography. For instance, constructing a supersymmetric, asymptotically AdS 5 black hole with non-abelian hair would represent a qualitatively new type of solutions.
It will be natural to adapt our construction to truncations of eleven-dimensional supergravity on a seven-dimensional manifold, leading to four-dimensional gauged N = 2 supergravity. This uses G S ⊆ SU (6) structures in E 7(7) generalised geometry, and would allow one to derive new consistent truncations based on the generalised structures underlying the AdS 4 × w M 7 solutions of [59,60], which in terms of ordinary G-structures only admit a local SU (2) structure. The solutions of [59] are the most general N = 2 AdS 4 solutions to eleven-dimensional supergravity supported by purely magnetic four-form flux; they represent the near-horizon region of M5-branes wrapping a special lagrangian threecycle in M 7 . The solutions of [60] have both electric and magnetic flux, and should arise from M2-M5 brane systems. Analysing the respective generalised structure it will become possible to enhance the truncation to minimal gauged supergravity obtained in [11] and [61] (for the solutions of [59] and [60], respectively) by adding matter multiplets. One example of this construction has been given in [62].
It will also be useful to extend our formalism to N = 2 truncations of type II supergravity. Minimally supersymmetric AdS 5 solutions of type IIB and massive type IIA supergravity were classified in [63] and [64], respectively. It would be useful to reformulate the classification of explicit solutions in terms of generalised G S ⊆ USp (6) structures; this would be a first step towards constructing consistent truncations to five-dimensional supergravity using our approach. One concrete application would be to check if the IIB solution of [65], given by a warped product of AdS 5 and a deformed S 5 , admits a consistent truncation to five-dimensional supergravity including (massive) KK modes that do not belong to the well-known IIB truncation leading to maximal SO (6) gauged supergravity. This would be somewhat analogous to the IIB consistent truncation on Sasaki-Einstein structures [66,67], where only a subset of the retained KK modes are also captured by SO (6) gauged supergravity.
A more challenging generalisation of our formalism would be the one to truncations preserving only N = 1 supersymmetry in four dimensions. Although a considerable amount of work remains to be done, it should be clear that the generalised structure approach to consistent truncations has the potential to classify all possible consistent truncations of higher-dimensional supergravity that preserve any given amount of supersymmetry.
We will denote by · the gl(6) action on tensors: given a frame {ê a } for T M and a co-frame {e a } for T * M , a = 1, . . . , 6, the action, for instance, on a vector and a two-form is For M-theory on a six-dimensional manifold we use E 6(6) × R + generalised geometry. The generalised tangent bundle E is where, as customary, we decompose the various bundles in representations of GL (6), the geometric subgroup of E 6(6) . The sections of E, the generalised vectors, transform in the 27 of E 6(6) and can be written as where v is an ordinary vector field, ω is a two-form and σ is a five-form. 28 28 The generalised tangent bundle E has a non-trivial structure that takes into account the non-trivial gauge potentials of M-theory. To be more precise the sections of E are defined as where A+Ã is an element of the adjoint bundle,V = v +ω +σ, with v ∈ Γ(T M ) are vectors, ω ∈ Γ(Λ 2 T * M ) and σ ∈ Γ(Λ 5 T * M ), and · defines the adjoint action defined in (A.23). The patching condition on the overlaps Uα ∩ U β is where Λ (αβ) andΛ (αβ) are a two-and five-form, respectively. This corresponds to the gauge-transformation of the three-and six-form potentials in (A.5) as The respective gauge-invariant field-strengths reproduce the supergravity ones: The dual bundle E * is defined as with sections Z =v +ω +σ , (A.10) wherev is one-form,ω is a two-vector andσ is a five-vector. Generalised vectors and dual generalised vectors have a natural pairing given by We will also need the bundle N ≃ det T * M ⊗ E * . In terms of GL (6) tensors, N decomposes as and correspondingly its sections Z ♭ decompose as The bundle N is obtained from the symmetric product of two generalised vectors via the map ⊗ N : (A.14) The E 6(6) cubic invariant is defined on E and E * as 29 The adjoint bundle is defined as where locally l ∈ R, r ∈ End(T M ), a ∈ Λ 3 T * M , etc. The e d(d) sub-algebra is obtained by fixing the factor l in terms of the trace of r as l = 1 3 tr r. This choice fixes the weight of the generalised tensors under the R + factor. In particular it implies that a scalar of weight k is a section of (det T * M ) k/3 : It is also useful to introduce the weighted adjoint bundle whose sections are locally given by the sum whereφ, φ and ψ are obtained from the adjoint elements where vol 6 is a reference volume form. The action of an adjoint element R on another adjoint element R ′ is given by the commutator, R ′′ = [R, R ′ ]. In components, R ′′ reads where · denotes the gl(6) action defined in (A.2).
The action of an adjoint element R on a generalised vector V ∈ Γ(E) and on a dual generalised vector Z is also denoted by · and is defined as where the components of V ′ are v ′ = lv + r · v + α ω −α σ , and those of Z ′ arev ′ = −lv + r ·v −ω a +σ ã , The e 6(6) Killing form on two elements of the adjoint bundle is given by The combination of diffeomorphisms and gauge transformations by the three-form and six-form potentials defines the generalised diffeomorphisms. The action of an infinitesimal generalised diffeomorphism is generated by the generalised Lie (or Dorfman) derivative along a generalised vector. The Lie derivative between two ordinary vectors v and v ′ on T M can be written in components as a gl (6) action where the symbol × denotes the projection onto the adjoint of the product of the fundamental and dual representation of GL (6). The generalised Lie derivative is defined in an analogous way; we introduce the operators ∂ M = ∂ m as sections of the dual tangent bundle and we define the generalised Lie derivative as where V M , M = 1, . . . , 27, are the components of V in a standard coordinate basis, and × ad is the projection onto the adjoint bundle, whose explicit expression can be found in [42,Eq. (C.13)]. In terms of GL(6) tensors, (A.27) becomes The action of the generalised Lie derivative on a section of the adjoint bundle (A.17) is We will also need the action of L V on the elements of the bundle N . Given a section Z ♭ = λ + ρ + τ of N , its Lie derivative along the generalised vector V = v + ω + σ is (A.31) Since Z ♭ = V ′ ⊗ N V ′′ , this is easily obtained by applying the Leibniz rule for L V .
It is also straightforward to verify that In this appendix we summarise some essential features of matter-coupled five-dimensional N = 2 gauged supergravity [68][69][70], following the conventions of [70]. 30 We limit ourselves 30 However, in order to match the normalisations defined by our truncation ansatz, we rescale the gauge fields appearing in [70] as A I here = − 2 3 A I there . Since we maintain the same form of the covariant derivatives, it follows that the gauge coupling constant g is rescaled as g here = − 3 2 g there . This implies that the expression for the scalar potential given in (B.13) below acquires a multiplicative 2/3 factor compared to the one in [70].
to the bosonic sector and only consider gaugings that do not require the introduction of two-form fields, as these are enough to describe our examples in Sections 4 and 5.
The fields of five-dimensional N = 2 supergravity arrange into the gravity multiplet, n V vector multiplets and n H hypermultiplets. The bosonic content consists of the vielbein e a µ , n V + 1 vector fields A I µ , I = 0, . . . , n V , together with n V vector multiplet scalars φ x , x = 1, . . . , n V , and 4n H hypermultiplet scalars q X , X = 1, . . . , 4n H . The φ x parameterise a 'very special real' manifold M V , with metric g xy , while the q X parameterise a quaternionic-Kähler manifold M H , with metric g XY . All together, the scalar manifold of the theory is the direct product A very special real manifold M V is a hypersurface that is conveniently described in terms of n V + 1 embedding functions h I (φ), I = 0, . . . , n V , satisfying the constraint where C IJK is a completely symmetric constant tensor. The metric on M V is given by and with the lower-index functions being The matrix a IJ is assumed invertible, and also controls the gauge kinetic terms. In the gauged theory, a subgroup of the isometries of the scalar manifold M, which are global symmetries of the Lagrangian, is promoted to a gauge group. The gauge generators t I satisfy [t I , t J ] = −f IJ K t K , with the structure constants f IJ K obeying f I(J H C KL)H = 0.
The gauge covariant derivatives of the scalars are given by where k x I (φ) and k X I (q) are the Killing vector fields generating the gauged isometries in the vector multiplet and hypermultiplet scalar manifolds, respectively. Equivalently, the vector multiplet scalar covariant derivatives can be expressed in terms of the embedding functions h I as One also has the gauge field-strengths We now have all the elements to write down the bosonic Lagrangian. This reads The vector multiplet scalar kinetic term can also be written in terms of the constrained scalars h I using the identity Using a differential form notation, the action reads The scalar potential V is given as a sum of squares as are the fermionic shifts, also appearing in the supersymmetry variations of the fermion fields: P is the gravitino shift, P x is the gaugino shift, and N iA is the hyperino shift.
Here, the arrow symbol denotes a triplet of the SU (2) H R-symmetry, and f iA X are the quaternionic vielbeins, satisfying f iA X f Y iA = g XY . The Killing prepotentials P I on M H are defined for n H = 0 by 4n where J X Y is the triplet of almost complex structures defined on any quaternionic-Kähler manifold. Plugging these expressions in (B.13) and using the identity (cf. [70, App. C]) we can express the scalar potential as Notice that the Killing vectors k x I on M V do not appear here, i.e. the gauging in the vector multiplet sector does not contribute to the scalar potential. This is true as long as we restrict to gaugings that do not require the introduction of two-form fields.
Supersymmetric AdS 5 vacua are obtained by setting all gauge fields to zero, all scalar fields to constant, and imposing that the gaugino and hyperino shifts vanish, Then the gravitino shift gives the AdS cosmological constant via

C Gauge transformations
In this appendix, we study the reduction gauge transformations of eleven-dimensional supergravity to five dimensions. We first repackage them in terms of generalised geometric objects and then use our truncation ansatz to derive the gauge transformations of fivedimensional N = 2 supergravity. The infinitesimal gauge transformations of the eleven-dimensional metric and threeand six-form potentials are wherev is a vector field,λ a two-form andλ a five-form. The hat on the Lie and exterior derivative operators emphasises that the derivatives are taken with respect to all the elevendimensional coordinates. The fields g,Â andÂ are decomposed as in (3.16), while the gauge parameters are expanded aŝ where, in the last line we omitted the terms that are not relevant in what follows. We only consider internal diffeomorphisms, as the external ones have the standard action dictated by the tensorial structure of the field. That is why the vectorv has only components on M . As in (3.16), we do not impose any restriction on the dependence of the fields on the coordinates {x µ , z m }. In (C.2) and the rest of this section we use a notation that manifestly displays the external indices and always contracts the internal ones. For the metric components we define g = g mn dz m dz n , and for a generic p-form ω ω = 1 p! ω m 1 ...mp dz m 1 ...mp , We already mentioned in Section 3.2 that the barred components of the three-and six-form must be redefined as 31Ā and similar redefinitions of the six-form. An analogous redefinition for the barred gauge parameters will be given later. As discussed in Section 3.2, the components of the metric, warp factor, three and sixform potentials and the dual gravitong with the same number of external legs fit into E 6(6) representations where G M N is the inverse generalised metric, A µ ∈ E is a generalised vector, B µν ∈ N is a weighted dual vector and C µνρ is a section of the weighted E 6(6) adjoint bundle (det T * ) ⊗ adF . The same holds for the gauge parameter, which we arrange into a generalised vector Λ, a weighted dual vectorΞ µ and a section of a sub-bundle of the 78,Φ µν , Φ µν =λ µν +λ µν + . . . .

(C.11)
In (C.7)-(C.10) we introduced the dual graviton to give the full E 6(6) representation. However in this paper we will not discuss the dual graviton since it is not relevant for the truncation we are interested in.
We can now decompose the gauge transformations given above. We find that the fields with no or purely internal legs transform as 31 The contractions are defined as follows where the Lie derivative L and the exterior derivative d are taken with respect to the internal coordinates only, although the fields and gauge parameters depend on both the internal and external coordinates. When repackaging all the fields with no external legs into the inverse generalised metric, the transformations (C.12) become the action of the generalised Lie derivative along the generalised vector Λ, Consider now the fields with one external leg. Their gauge transformations are and it is straightforward to verify that they can be recast into By redefining the gauge parameters 32 , and using (A.33) to compute we bring the variation (C.15) to an appropriate form to compare with five-dimensional gauged supergravity The variations of the fields with two external legs are (C.21) 32 In components the redefinition (C.18) reads By a lengthy but straightforward computation (C.21) can be written as where we defined the field strength Applying the Leibniz rule for the generalised Lie derivative and (A.33) one can show that 24) and the variation of B µν can be written in a form compatible with five-dimensional gauged supergravity where we have made the following redefinition of the gauge parameters 33 Finally we should consider the variations of the fields with three external legs. To our purposes it is enough to study the three-form (C. 28) In generalised geometry (C.28) embeds in the lowest component of the variation of the tensor C µνρ in (C.10). We introduce the modified field strength for the three-form field C µνρ , and by manipulations similar to what we did previously we can recast the gauge variations as (C. 26) up to terms involving a four-form gauge parameter, which would continue the tensor hierarchy.
The five-dimensional gauge transformations are obtained by plugging the reduction ansatz in the variations (C.20), (C. 25) and (C.30). The fields A µ I (x) are expanded as in where K I are the generalised vectors that are singlets of the G S structure. In (3.26), the two-from fields are expanded on the weighted duals K I ♭ of the generalised vectors K I . These are elements of the bundle N and can also written as where the tensor D IJK satisfies D IKL C JKL = 1/2δ I J where C IJK is defined in (2.16). So the two-forms are expanded as The gauge parameters have a similar expansion With the ansatze (C.31) and (C.34) for A µ and the gauge parameters, the variations (C.20) of the one-forms become where we used the algebra of the vectors K I (3.4) and (A.33). The variations of two-forms are reduced in a similar way. We expand the field strength H J µν as in (C.31) and use again the generalised Lie derivative of vectors K I given in (3.4). In this way we obtain for the gauge variations of the five-dimensional two-forms B µν,I (x) This is in agreemement with five-dimensional supergravity. The variation δC µνρ reduces analogously.

D Parameterisation of S 4 and generalised frames
The six-dimensional geometry of interest in this paper is given by a four-sphere S 4 fibered over a Riemann surface Σ. In this appendix we describe S 4 as a foliation of S 3 over an interval and review the generalised frames on S 4 .

D.3 Generalised frames on S 4
In generalised geometry all spheres are generalised parallelisable as they admit globally defined frames on their exceptional tangent bundle [15]. In particular the generalised tangent bundle on S 4 is E 4 ≃ T S 4 ⊕ Λ 2 T * S 4 , (D. 16) and its fibres transform in the 10 of the structure group SL(5, R). We will also need the bundles (D.17) whose fibres transform in the 5 and 5 ′ representations, respectively. These bundles admit globally defined frames, which in constrained coordinates read where the Hodge star * 4 is computed using (D.10), and the three-form potential A S 4 must satisfy This is the flux relevant for the AdS 7 ×S 4 supersymmetric Freund-Rubin solution to elevendimensional supergravity; the twist over the Riemann surface discussed in the main text will modify it. The E ij are generalised Killing vectors generating the so 5 algebra via the action of the generalised Lie derivative, In the main text we will need the following linear combinations, Using the map (D.11), the frame elements (D.18) can equivalently be expressed in terms of angular coordinates on S 4 . In particular, choosing a gauge such that the potential 22) we find that the combinations (D.21) are expressed in terms of the SU (2) left and SU (2) right invariant tensors as The Ξ α can be seen as left-invariant generalised Killing vectors generating the SU (2) right ⊂ SO (4) ⊂ SO(5) generalised isometries, whileΞ α are right-invariant generalised Killing vectors generating the SU (2) left generalised isometries. We will also need the expressions for E i and E ′ i in terms of angular coordinates, in fact just for i = 5. These read E Details on the generalised U(1) structure of MN1 solution In this appendix we give the details of the construction of the U (1) structure discussed in Section 4.2. In order to identify the correct U (1) subgroup of E 6(6) ×R + and its commutant it is convenient to decompose E 6(6) under its maximal compact subgroup USp (8) and then express the USp (8) representations in terms of Cliff(6) gamma matrices. For the latter step we also need the decomposition of E 6(6) under SL (6)×SL (2). We first give a brief summary of the decomposition of E 6(6) under USp (8) and SL(6) × SL(2) and then we apply this to the construction of the U (1) structure, which reduces to simple gamma matrix algebra.
(E. 16) Then the 27 and 27 of USp (8) are given in terms of SL(6) × SL(2) representation by where Γ mn7 denotes the anti-symmetric product of two gamma's and Γ 7 . where ⊗ denotes the tensor product of two gamma's, µ m n is traceless and µî mnp are antisymmetric in the three lower indices.

E.2 The U (1) structure
We can now give the details of the construction of the U (1) S structure discussed in Section 4.2. The N = 2 solution of [34] has an U (1) S structure corresponding to the diagonal of the SO(2) holonomy on the Riemann surface Σ, and the U (1) right subgroup of the SO(5) isometry of the four-sphere, according to the embedding Seen as an element of E 6(6) , the U (1) S corresponds to a compact generator and therefore belongs to USp (8). Using the expression (E.18) for the generators of USp (8), and now taking the indices m = 5, 6 for the direction along the Riemann surface, the U (1) S generator can be written as whereΓ 56 is the generator of the SO(2) holonomy of Σ and i 2 (Γ 12 −Γ 34 ) generates U (1) right ⊂ SO (5).
To embed this generator in E 6(6) and determine the invariant generalised tensors it is convenient to decompose all E 6(6) representations into USp (8) one's and then use the parameterisation of USp (8) in terms of gamma matrices of Section E.1. In this way the computation of the commutant, C E 6(6) (U (1) S ), of U (1) S in E 6(6) and the determination of the U (1) S singlets reduce to simple gamma matrix algebra.
We first compute the commutators of U (1) S with the generic elements of the 36 and 42 in (E.18). This will allow to determine the number of U (1) S singlets in the 78 and the commutant C E 6(6) (U (1) S ). Using (E.6) we find that there are eight singlets in the 36. .

(E.35)
The compact singlets give the commutant of U (1) S into USp (6), (E.39) The singlets in the 27 and 78 are all we need to specify the generalised U (1) S structure. However, the generators of SO(3, 1) and R + in (E.31) do not leave the singlets generalised vectors invariant and hence do not belong to the U (1) S structure. Using (E.10), one can show that they are obtained as products of the singlets in the 27 and their duals J SO(3,1) α = 2i ǫ αβγ (K β × ad K * γ ) , K SO(3,1) α = −i (K α × ad K * 4 ) , α = 1, 2, 3 , (E. 40) and In summary the generalised U (1) S structure is defined by the five generalised vectors and the eight generators of SU (2, 1) The last step is to derive explicit expressions for these generalised tensors in terms of geometrical objects on the six-dimensional internal manifold M . We use the fact that, in our case, M is a fibration of the four sphere over a Riemann surface and that the four-sphere is generalised parallelisable as reviewed in Appendix D.3.
We decompose the six-dimensional bundles in representation of GL(2, R), the ordinary structure group on the Riemann surface, and SL(5, R), the exceptional structure group of where vol Σ = e 1 ∧ e 2 is the volume form on the Riemann surface, R is the S 4 radius, and E i and E ′ i are the sections of N 4 and N ′ 4 defined in Appendix D.3. The elements of the (1, 10) are the Ξ α , Ξ α , with α = 1, 2, 3, defined in (D. 21), and E i5 with i = 1, 2, 3, 4.
Comparing with (E.38) and (E.39), we see that K 0 ∈ (1, 5) ∼ Λ 2 T * Σ ⊗ N ′ 4 , K I ∈ (1, 10) ∼ E 4 , for I = 1, . . . , 4 , (E. 49) and can then be written as generalised vectors on M as where α = 1, 2, 3. To have the final expressions for these five generalised vectors we still have to implement the twist of S 4 as described in Section 4.2. The E 6(6) element implementing the twist is where P i j is the matrix It is then straightforward to check that only K 4 is modified by the twist, and the expressions (4.29) are obtained.
Finally we need the expressions for the singlets in the 78 generating SU (2, 1). Under E 6(6) ⊃ GL(2, R) × SL(5, R) as where ad F 4 is the adjoint bundle on S 4 The expressions for the singlets are easily obtained from (E.10) as products of the 27 and 27. In this way we obtain precisely the expressions given in Eq. (4.30), where the twisting by Υ can be evaluated with the aid of (E.52).

F Parameterisation of the H structure moduli space
We discuss here our parameterisation of the coset space M H = SU (2,1) SU (2)×U (1) that describes the hypermultiplet structure moduli space. We model the generators of SU (2, 1) on the matrices j A , A = 1, . . . , 8, defined as: