Supersymmetric AdS$_{7}$ and AdS$_6$ vacua and their minimal consistent truncations from exceptional field theory

We show how to construct supersymmetric warped AdS$_7$ vacua of massive IIA and AdS$_6$ vacua of IIB supergravity, using"half-maximal structures"of exceptional field theory. We use this formalism to obtain the minimal consistent truncations around these AdS vacua.


Introduction
Finally, in 6, we derive the minimal consistent truncation around these vacua before concluding in section 7.
Note added: While finalising this manuscript, the paper [34] appeared which also constructs the minimal consistent truncation around supersymmetric AdS 6 vacua that we derive in section 6.2.

Half-maximal AdS vacua from ExFT
Supersymmetric AdS vacua can be naturally described in ExFT using the language of generalised Gstructures, analogous to the description of special holonomy spaces in Riemannian geometry. In particular, as was shown in [5][6][7], flux geometries of type II or 11-dimensional SUGRA admitting a half-maximal set of spinors can be described by a set of generalised tensors satisfying certain algebraic conditions. Here we will consider 10/11-dimensional geometries consisting of warped products M D × M int , where M D denotes the external spacetime, M int is the internal space and we will focus on D = 6, 7 in this paper.
In order for M D to be a half-maximal AdS vacuum, M int must admit a set of nowhere-vanishing d − 1 generalised vector fields J u ∈ Γ (R 1 ), where d = 11 − D and u = 1, . . . , d − 1, as well as a generalised tensor fieldK ∈ Γ (R D−4 ) satisfying (2.1) Here R i are different generalised tangent bundles whose fibre is the R i representation of E d(d) , listed in table 2, where also the representation R c is given. The ∧ products map ∧ : R i ⊗ R j −→ R i+j when i + j < D − 2 , where S denotes the space of scalar densities of weight 1 under generalised diffeomorphisms. The explicit expressions of ∧ in terms of E d(d) invariant is given in [7].
SL(5) 10 5 5 10 ∅ 6 Spin(5, 5) 16 10 16 N/R 1 Throughout, we will raise and lower the u, v = 1 . . . , d − 1 indices using δ uv . A set of generalised tensors J u ,K as above are called a half-maximal structure and are stabilised by a G half = SO(d − 1) subgroup of the exceptional group E d(d) given in table 2. The maximal commutant of SO(d − 1) ⊂ E d(d) is SO(d − 1) R and rotates the d − 1 J u 's amongst each other.
From the half-maximal structure one can also define the following generalised tensors that will be useful in the following: where K ∈ Γ (R 2 ), κ is a scalar density of weight 1 D−2 andĴ u ∈ Γ (R D−3 ). The explicit expressions for the above equations (2.1) and (2.3) in terms of E d(d) -invariants can be found in [7]. Furthermore, the BPS equations for the AdS D vacuum are equivalent to the following differential equations where Λ uvw = Λ [uvw] are completely antisymmetric, L denotes the generalised Lie derivative of ExFT [2,3,8] and d : is a certain nilpotent operator as defined in [5,7] and which also appears in studies of the tensor hierarchy of ExFT [35][36][37]. Table 2: G half structures and R-symmetry groups in 6 and 7 dimensions. In six dimensions, one can also have chiral half-maximal supersymmetry but we ignore this here since there are no chiral AdS6 vacua. The interested reader is referred to [7] for more information.
The differential conditions (2.4) encode the BPS conditions for the supersymmetric AdS D vacuum in a geometric language. As we will see the algebraic and differential conditions (2.1) and (2.4) can easily be solved in a variety of different cases, providing an efficient way of constructing supersymmetric AdS vacua of 10/11-dimensional supergravity. Moreover, once we have the half-maximal structure for an AdS vacuum, we can immediately construct a consistent truncations around the AdS vacuum to the Ddimensional half-maximal gauged SUGRA containing only the graviton supermultiplet [5,7], as we review in section 2.1. We will use this method to find the minimal consistent truncations around supersymmetric AdS 6 vacua of IIB SUGRA, as well as derive the consistent truncations around supersymmetric AdS 7 vacua of massive IIA SUGRA, where our expressions agree with [33].
The half-maximal structure encodes the 11-dimensional / type II supergravity fields, just like a complex and Kähler structure encode the metric. In ExFT, the supergravity fields parameterise the generalised metric M MN which lives in the coset space Before we proceed to discuss the AdS 6,7 cases in detail, we will first make some general observations about the differential conditions (2.4) and what they imply for J u andK. We see that the generalised Lie derivative of the J u 's generates an SO(d − 1) R rotation under which the J u 's transform in the vector representation whileK is invariant. However, as mentioned above, the generalised metric and hence the SUGRA fields are constructed from SO(d − 1) R -invariant combinations of the J u 's andK. Therefore, 6) and the J u are generalised Killing vector fields. Generalised vector fields are a formal sum of spacetime vector fields plus certain differential forms. Therefore, for J u to be generalised Killing, implies that either they consist of non-zero spacetime Killing vector fields with accompanying gauge transformations such that the gauge potentials are left invariant, or they have an identically vanishing spacetime vector field part and consist of trivial gauge transformations, i.e. exact differential forms. For such a "trivial" generalised Killing vector field V we would have acting on any tensor. We will make use of this general insight in sections 4 and 5 when constructing the AdS vacua.

Minimal consistent truncation
Once we have constructed the half-maximal structure J u andK corresponding to a half-maximal AdS D vacuum, we can immediately construct a consistent truncation around this vacuum to a minimal halfmaximal D-dimensional SUGRA [7]. That such a consistent truncation should always exist for any warped supersymmetric AdS vacuum of 10-/11-dimensional SUGRA was conjectured in [28] and proven in the half-maximal case for D ≥ 4 in [7]. The truncation Ansatz is linear on the half-maximal structure and given as follows. We denote by Y M the internal coordinates on M int and by x µ the external coordinates on M D . Then, the truncation Ansatz (of the scalar sector) is given by [5,7] where X(x) is the scalar field of the D-dimensional half-maximal SUGRA. The consistency of the truncation Ansatz is guaranteed by the differential conditions (2.4) satisfied by the J u ,K as shown in [7]. Upon truncation, X(x) becomes the scalar field of the minimal half-maximal D-dimensional gauged SUGRA with embedding tensor given by Λ uvw in (2.4).
The consistent truncation can easily be extended to the other fields of the D-dimensional SUGRA as explained in [7]. However, since non-vanishing vacuum expectation values of these fields will typically break Lorentz symmetry, we will not include them in this paper.
3 Generalised metric from the half-maximal structure As we mentioned above, the half-maximal structure determines the supergravity fields which in ExFT are encoded in the generalised metric. We therefore need to find a way to compute the generalised metric from the half-maximal structure J u ,K. The generalised metric parameterises the coset space and hence must be an

Generalised metric in SO(5, 5) ExFT
In SO(5, 5) ExFT [39,40], the generalised metric is often used either in the R 1 = 16 representation or in the fundamental representation, R 2 = 10, of SO (5,5). The two are related by where η IJ is the constant SO(5, 5)-invariant matrix with which we raise/lower fundamental SO (5,5) indices. Furthermore, M IJ must satisfy We thus find the generalised metric and its inverse in the 16 are given by whereĴ u M and κ are defined in (2.3), and here given explicitly bŷ Similarly, the generalised metric in the 10 is (3.11) 4 AdS 7 vacua from massive IIA supergravity We will now show how to use this method to construct AdS 7 vacua of massive IIA SUGRA. First, we let Λ uvw = √ 2R −1 ǫ uvw so that the differential conditions (2.4) become We see that the J u 's generate SU(2) R rotations via the generalised Lie derivative. J u form triplets of SU(2) R , whileK are invariant. As we discussed in section 2 this implies that J u are generalised Killing vector fields. Since L Ju = 0, none of the J u are trivial generalised Killing vector fields and hence must contain spacetime Killing vectors. From (4.1) we see that these spacetime Killing vectors must generate an SU(2) R algebra and hence are related to an S 2 geometry. Therefore, we will consider the internal where I is an interval with coordinate z, where in principle we allow off-diagonal metrics between the S 2 × I (although we will see that supersymmetry does not allow these off-diagonal terms). We will parameterise S 2 by the three functions y u , u = 1, . . . , 3 satisfying y u y u = 1. Further details of our S 2 convention can be found in appendix B.

SL(5)
ExFT and IIA SUGRA Supersymmetric AdS 7 vacua are characterised by three generalised vector fields J u ∈ Γ (R 1 ) and a generalised tensor fieldK ∈ Γ (R 3 ). In IIA SUGRA these become formal sums of spacetime vector fields and differential forms, where V u , λ u , σ u and φ u are the vector, 1-form, 2-form and scalar parts of J u , while ω (p) are the p-forms appearing inK. For completeness' sake, a generalised tensor K ∈ Γ (R 2 ) becomes whereω (p) are p-forms.
In IIA SUGRA, the wedge products appearing in the algebraic conditions (2.1) become The quadratic algebraic constraint onK is automatically fulfilled for SL(5) ExFT [7]. The differential operators appearing in the differential conditions (4.1) become where we have included the Roman's mass m as in [24,41].

Half-maximal structure
Before continuing, we need to discuss the possible gauge potentials living in M int . In IIA SUGRA, we need to consider a 2-form and 3-form field strength in M int which must form SU(2) R -symmetry singlets.
The 2-form gauge potential can always be chosen to be an SU(2) R -symmetry singlet. However, the 1-form gauge potential A will necessarily violate the SU(2) R -symmetry. As we will use R-symmetry as a guiding principle, we will have to include the 1-form gauge potential A by hand as a "twist term", as e.g. in [19].
The most general J u we can construct that is compatible with the SU(2) R symmetry and that satisfies the algbraic conditions (2.1) is, up to generalised diffeomorphisms (i.e. gauge transformations and diffeomorphisms), given by where v u are Killing vectors and θ u certain 1-forms on S 2 (see appendix B), and g(z), q(z), h(z) are so far arbitrary functions of z. Furthermore, the most generalK constructed from R-symmetry singlets is, up to generalised diffeomorphisms, given bŷ The algebraic condition J u ∧ J u ∧K > 0 now becomes Allowing for the S 2 to shrink at the boundary of the interval parameterised by z, we have with equality at the boundary of I. Finally, as discussed R-symmetry implies that dA = R 2 l(z) vol 2 , (4.11) for some l(z). With (4.7), the differential conditions (4.6) reduce to (4.12) We can always redefine the coordinate z to make h(z) any functions we choose. A particular convenient choice is to take h(z) = q(z), whereupon the differential conditions (4.12) becomė Without loss of generality we can integrateġ = −1 to g = −z, absorbing any constant of integration by shifting z. Furthermore, we can express s and q in terms of derivatives of t which must satisfy and t ≥ 0 with equality at the boundary of I.
Altogether the half-maximal structure then becomes

The AdS 7 vacua
The SUGRA fields with legs on M int = S 2 × I can be read off from the generalised metric constructed from J u andK in (4.15). For this we use the parameterisation of generalised metric by IIA SUGRA fields given in [21]. The warp factor the AdS 7 metric is given by [5,7] where |g| is the determinant of the internal space in string frame and ψ is the IIA dilaton. Thus, we find the infinite family of supersymmetric AdS 7 vacua determined by the function t(z) satisfying (4.16).
where the metric is expressed in string frame and F 2 = dA − m B 2 is the Ramond-Ramond 2-form field strength of mIIA SUGRA. This is clearly the family of AdS 7 solutions found in [29] in the coordinate choice of [42], where our variables are related to theirs by the rescaling (4.20)

AdS 6 vacua from IIB supergravity
We next consider supersymmetric AdS 6 in IIB SUGRA. We begin by rewriting the differential conditions (2.4) by introducing the SO(4) R vector Λ u defined as (5.1) Then the differential conditions (2.4) become The SO(4) R vector Λ u encodes the AdS 6 radius and hence cosmological constant as follows. We can use a SO(4) R rotation to write, without loss of generality,  However, in contrast to the AdS 7 case, the equation J 4 ∝ dK implies that J 4 is a trivial generalised Killing vector field, containing no spacetime vector field but only exact differential forms. It therefore generates trivial gauge transformations of the gauge potentials. On the other hand, the three generalised vector fields J A necessarily contain non-vanishing spacetime vector field components, since L JA = 0.
These spacetime vector fields must be Killing vector fields that generate the SU(2) R algebra and hence are related to an S 2 geometry. Therefore, we will consider the internal space where Σ is a Riemann surface with coordinates x α , α = 1, 2. We will, in principle, allow for metrics on M int with off-diagonal components between S 2 and Σ although we will see that supersymmetry forbids such components.
The SL(2) S of the IIB S-duality acts naturally on the Riemann surface and we will raise/lower all SL(2) S indices using the alternating symbols ǫ αβ = ǫ αβ = ±1 with and following a Northwest-Southeast convention. The S 2 will once again be parameterised by three functions y A , A = 1, . . . , 3 with y A y A = 1. Further S 2 conventions are described in appendix B.

SO(5, 5) ExFT and IIB SUGRA
Supersymmetric AdS 6 vacua are characterised by four generalised vector fields J u ∈ Γ (R 1 ) and a generalised tensorK ∈ Γ (R 2 ). In IIB SUGRA these become formal sums of spacetime vector fields and differential forms as follows where V u , λ u α and σ u denote the vector, 1-form and 3-form parts of J u , while ω (p) are p-forms appearing inK.

Half-maximal structure
In contrast to AdS 7 vacua of mIIA, the gauge potentials of IIB SUGRA on M int = S 2 × Σ can always be chosen to respect the SU(2) R symmetry. Therefore, they will automatically be included in the halfmaximal structures we construct here.
The most general J A we can construct from SU(2) R -triplets that satisfies the algebraic conditions (2.1) is, up to generalised diffeomorphisms, where v A are Killing vectors and θ A certain 1-forms on S 2 (see appendix B), m α = m α β dx β are 1forms on Σ, where m α β depends only on the coordinates of Σ, k α are functions on Σ, and we defined |m| = 1 2 m αβ m αβ . Next, we constructK such that is an SU(2) R -invariant and satisfiesK ⊗K| Rc = 0 and J A ∧J A ∧K > 0. We find the unique combination, up to generalised diffeomorphisms, and hence J 4 The algebraic condition is satisfied iff r |m| ≥ 0 with equality at the boundary of Σ, while the algebraic conditions for J 4 impose m α ∧ dp α = 0 , m α ∧ m β = dp α ∧ dp β , dr + p α dk α = 0 .

(5.16)
Note that the final condition can be used to simplify the expression of J 4 Finally, we are left to solve the differential conditions (5.10). Here these simplify to which implies m α = −dk α .
Thus, we find that determined entirely by the two SL(2)-doublets of real functions k α and p α on Σ, which satisfy the differential conditions dk α ∧ dk β = dp α ∧ dp β , dk α ∧ dp α = 0 , (5.20) and positivity condition (5.14) r|dk| ≥ 0 with equality at ∂Σ , (5.21) where |dk| = ∂ α k β ∂ α k β , and r is defined up to an integration constant by At this stage, one might wonder how the quadratic differential conditions (5.20) can underly supersymmetric AdS vacua, which ought to be described by a first-order BPS equation. The answer is that we still have residual diffeomorphism symmetry on the Riemann surface Σ that can be used to turn (5.20) into first-order differential equations. We will show how to do this after calculating the supergravity fields from the structures.

The AdS 6 vacua
We will now compute the supergravity background corresponding to the half-maximal structures (5.19).
The supergravity fields are encoded in the generalised metric (3.9), (3.11) as summarised in appendix A. Moreover, the 6-D metric is warped by the factor [7] where |g| is the determinant of the internal four-dimensional space. From this, we find the following background in Einstein frame As we mentioned previously, we can use diffeomorphisms to turn the differential equations for k α and p α into first-order PDEs. In particular, we can always use diffeomorphisms to make the metric on Σ conformally flat. From (5.24) we see that this would impose Together with (5.20), and requiring (5.21) the differential conditions become the Cauchy-Riemann equations dk α = I · dp α , (5.27) where I β α = δ αγ ǫ γβ is a complex structure on Σ. This implies that p α and k α are the real and imaginary parts of two holomorphic functions on Σ We now immediately see that our solutions match those of [30] upon identifying our holomorphic functions with the A ± of [30] as follows

AdS 6
We can similarly use (2.8) to find the minimal consistent truncation corresponding to the supersymmetric AdS 6 vacua of IIB SUGRA we described here and which were previously constructed in [30]. We find in Einstein frame and k α , p α and r satisfy (5.20), (5.21), (5.22). Upon truncation, X becomes the scalar field of the minimal 6-dimensional gauged SUGRA, the so-called F (4) gauged SUGRA [44]. All these AdS vacua correspond to the same vacuum of the 6-dimensional gauged SUGRA.

Conclusions
In this paper we showed how ExFT can be used to efficiently construct supersymmetric AdS vacua. We focused on supersymmetric AdS 7,6 vacua of mIIA and IIB SUGRA, respectively, and found the class of infinite solutions desribed in [29] and [30]. Our method allowed us to immediately derive the minimal consistent truncation around these AdS vacua. We rederived the consistent truncation around AdS 7 vacua of mIIA given in [33], and found the minimal consistent truncation around the AdS 6 vacua of IIB SUGRA. These consistent truncations are a useful tool in studying the AdS vacua, for example by finding RG flows between different AdS vacua. It would be interesting to explore whether one can keep vector multiplets in the consistent truncation, using the procedure discussed in [7], allowing us to differentiate between the various AdS vacua in the lower-dimensional theory.
The method presented here can be generalised to lower dimensions and different amounts of SUSY [7,11], where it may yield new AdS vacua of 10-/11-dimensional SUGRA. Perhaps, it can even be used to provide a classification of supersymmetric AdS vacua of 10-/11-dimensional SUGRA.
Moreover, this formalism is clearly suited to studying the moduli of AdS vacua. For example, [12] showed that in the absence of isometries beyond the U(1) R , all infinitesimal moduli of N = 2 AdS 5 vacua can be exponentiated to finite deformations. One could therefore attempt to compute the deformations of the SUGRA backgrounds corresponding to moduli of the AdS vacua. The finite deformations are holographically dual to exactly marginal deformations of the SCFTs and thus by computing the metric on the AdS moduli space, we could holographically determine the Zamolodchikov metric on the conformal manifold.

B S 2 conventions
We describe the S 2 by three functions y u , u = 1, . . . , 3 satisfying y u y u = 1 . (B.1) In terms of these functions, the round metric on S 2 and its volume form are given by The Killing vectors of the round S 2 are given by v i u = g ij ǫ uvw y v ∂ j y w , (B .3) where i, j = 1, 2 denote a local coordinate basis and g ij is the inverse metric of the round S 2 . Alternatively, the Killing vectors can be defined as in [19]. We also make repeated use of the 1-forms which form a "dual span" of the T * (S 2 ) to the Killing vectors, i.e.
Note that the 1-forms dy u , θ u and Killing vectors v u satisfy y u dy u = y u θ u = y u v u = 0 . (B.6) All the objects we introduced above transform naturally under the SU(2) R symmetry generated by the Killing vector fields.

C Matching different AdS 6 conventions
Upon imposing the Cauchy-Riemann equations (5.27) and identifying the holomorphic functions as in (5.29), we find the following match between our objects and those of [30]. To differentiate our κ and our parameter R from the objects denoted by the same symbols in [30] we will denote theirs by an underline, κ and R. Using (5.29), we find To compare our two-forms and axio-dilaton with those of [30], it is important to translate our SL(2) representations into their SU(1, 1) ones. These are mapped via C = −C where C denotes the 2-form and B the axio-dilaton of [30]. The latter encodes the complex axio-dilaton τ = e ψ + i C 0 as