${\mathcal{N}=(8,0)}$ AdS vacua of three-dimensional supergravity

We give a classification of fully supersymmetric chiral ${\cal N}=(8,0)$ AdS$_3$ vacua in general three-dimensional half-maximal gauged supergravities coupled to matter. These theories exhibit a wealth of supersymmetric vacua with background isometries given by the supergroups OSp$(8|2,\mathbb{R})$, F(4), SU$(4|1,1)$, and OSp$(4^*|4)$, respectively. We identify the associated embedding tensors and the structure of the associated gauge groups. We furthermore compute the mass spectra around these vacua. As an off-spin we include results for a number of ${\cal N}=(7,0)$ vacua with supergroups OSp$(7|2,\mathbb{R})$ and G$(3)$, respectively. We also comment on their possible higher-dimensional uplifts.


Introduction
Supersymmetric Anti-de Sitter backgrounds of string theory and supergravity are of central importance in the holographic AdS/CFT correspondence [1][2][3]. Within higher-dimensional supergravity, these correspond to supersymmetric solutions of the form AdS D × M, and may give rise after consistent truncation to a D-dimensional gauged supergravity with a stationary point in its scalar potential. The AdS D solution of the lower-dimensional theory with all scalars constant located at the stationary point then corresponds to the higher-dimensional AdS D × M solution.
A systematic approach to the classification of such backgrounds may start directly from a classification of supersymmetric AdS D backgrounds in D-dimensional gauged supergravity. These supergravities are determined by the choice of a constant embedding tensor which encodes the gauge structure and couplings of the theories [4][5][6]. Rather than searching AdS vacua within a given theory, one may in turn determine the most general embedding tensor such that the Tab. 1.1 Supergroups G with SL(2, R) factor and eight supercharges [22,23]. We also list their even part G even SL(2, R) × G R-sym and their R-symmetry group G R-sym . The supercharges are given in representations of G R-sym .
resulting theory admits a supersymmetric AdS D vacuum, thereby determining the relevant Ddimensional theories together with their solutions. For half-maximal supergravities in D ≥ 4 dimensions, such an analysis has been performed in [7][8][9][10][11], where the general gauging admitting a fully supersymmetric AdS vacuum has been determined and analyzed. AdS 3 vacua have so far escaped a similar classification. This is mostly due to the fact that the structure of gauged supergravity theories and their solutions in three space-time dimensions is very rich. Already the maximal (N = 16) gauged supergravity in three dimensions offers a plethora of fully supersymmetric AdS 3 vacua [12]. This is in marked contrast to higher dimensions, where there is a single maximally supersymmetric AdS vacuum in D = 7 [13] and D = 5 [14], together with a one-parameter family of maximally supersymmetric AdS 4 vacua [15,16]. Similarly, many AdS 3 vacua have been identified in theories with N = 9 and N = 10 supersymmetry [17,18]. The wealth of three-dimensional structures is based on the particular properties of three-dimensional gauge and gravitational theories. The gravitational (super-)multiplet in three dimensions is non-propagating which allows for the construction of a gravitational Chern-Simons action for any AdS 3 supergroup [19]. Further coupling to scalar matter offers ample possibilities due to the on-shell duality between scalar and gauge fields in three dimensions. Again, the possible structures are most conveniently encoded in terms of a properly constrained embedding tensor [4,20,21]. Finally, the AdS 3 isometry group SO(2, 2) ∼ SL(2, R) L × SL(2, R) R is not simple, but a product of two factors. Consequently, the supergroup of AdS 3 background isometries in general factors into a direct product of simple supergroups G L × G R for which there are various options [22,23]. The supercharges accordingly split into N = (p, q) charges transforming under G R and G L , respectively. As a result, there is an immense number of half-maximal N = 8 AdS vacua in three dimensions.
In this paper, we take a first step towards their classification, by determining all chiral N = (8, 0) AdS 3 vacua within half-maximal D = 3 gauged supergravity. For these vacua, the background isometries build a supergroup SL(2, R) L × G R , where the simple supergroup G R is chosen among the options listed in Tab. 1.1. Some such vacua have recently appeared in Ref. [24] as particular type IIA AdS 3 compactifications on a six sphere S 6 = SO(7)/SO(6) fibered over an interval, preserving the exceptional supergroup F (4). We give a classification of the halfmaximal D = 3 gauged supergravities admitting an N = (8, 0) AdS 3 vacuum. These theories couple the non-propagating N = 8 supergravity multiplet to n scalar multiplets, realizing different gauge groups embedded into the SO(8, n) isometry group of ungauged supergravity. We restrict to n ≤ 8 matter multiplets. For each of the supergroups of Tab. 1.1, we identify the possible three-dimensional supergravities, characterized by their gauge groups which we list in Tab. 1.2, together with the external global symmetry group G ext preserved by the vacuum. We also indicate in this table, which of these theories admit a free parameter, entering in particular  the structure constants and the scalar potential. In the main body of this paper, we perform the explicit analysis of consistency conditions on the embedding tensor, leading to this classification. We moreover compute for every vacuum the associated mass spectrum, organized into supermultiplets of SL(2, R) L × G R . As a by-product of our constructions, we also identify a number of AdS 3 vacua with N = (7, 1) and N = (7, 0) supersymmetry, respectively. Some of the latter with G(3) superisometry group have also emerged in the type IIA compactifications of Ref. [24] with fluxes responsible for the breaking of the R-symmetry group down to G 2 . We list our findings in Tab. 1.3. Let us recall that there is no three-dimensional supergravity theory with N = 7 local supersymmetries and non-trivial matter content [25]. As a consequence, N = (7, 0) vacua can only be realized within half-maximal N = 8 theories with 1/8 of supersymmetry spontaneously broken at the vacuum. Indeed, we find that both our N = (7, 0) vacua live in half-maximal theories which also admit a fully supersymmetric N = (8, 0) vacuum, c.f. Figs. 5.1 and 5.2 below.
The rest of this paper is organized as follows. In Sec. 2, we review the structure of halfmaximal D = 3 gauged supergravities, specifically their embedding tensors and the set of algebraic constraints imposed onto the embedding tensors in order to ensure consistency of the gauging and the existence of a supersymmetric AdS 3 vacuum. In the following two Secs. 3 and 4, we then turn to the analysis and solution of these constraints. The computation is organized by choice of supergroup from Tab. 1.1 together with the inequivalent embeddings of the desired AdS 3 R-symmetry group SO(p) × SO(q) into the isometry group SO(8, n) of ungauged D = 3 supergravity. For every solution, we list the explicit form of the embedding tensor, determine the gauge group of the D = 3 theory, and compute the mass spectrum of the N = (8, 0) vacuum. In Sec. 5, we collect some partial results on AdS 3 vacua with N = (7, 1) and N = (7, 0) supersymmetry. In Sec. 6, we raise and answer the question which of the AdS 3 vacua can in fact be further embedded as vacua in a maximal (N = 16) three-dimensional supergravity. This translates into a couple of additional algebraic constraints to be imposed onto the embedding tensor, which we check for all our vacua. Finally, in Sec. 7 we summarize our findings in Tabs. 7.1-7.3, and discuss possible generalizations, in particular the possible higher-dimensional origin of these vacua.

Gauged supergravities in three dimensions
In this section, we recall some relevant facts about the three-dimensional half-maximal gauged supergravities. Their gauge structure is most conveniently encoded in a constant embedding tensor subject to a set of algebraic constraints. We spell out the conditions for supersymmetric AdS 3 vacua and give the general formulas for the mass spectra around these vacua.

Lagrangian
Half-maximal gauged supergravities in three dimensions have been constructed in Refs. [21,26] by deforming the half-maximal ungauged theory of Ref. [27]. This ungauged theory contains an N = 8 supergravity multiplet composed of a dreibein e α µ and eight Rarita-Schwinger fields ψ A µ , where µ and α denote respectively the curved and flat spacetime indices, and A is the index of the spinorial representation of the Minkowski R-symmetry SO (8). The matter fields combine into n copies of the N = 8 scalar multiplet, each one composed of eight scalars and eight spin-1 /2 fermions, transforming in the vectorial and cospinorial representations of SO(8), respectively. The scalar matter forms an SO(8, n)/(SO(8)×SO(n)) coset space sigma model. In the following, the indices I, J, . . . and r, s, . . . denote the vectorial indices of SO (8) and SO(n), respectively, which we combine into SO(8, n) vector indices M = {I, r}. The SO(8, n) invariant tenso r is defined as and the generators of so (8, n), in the chosen vectorial representation, are given by Then, {L IJ , L rs } form the generators of so(8) ⊕ so(n) while the coset is parametrized by the 8n scalar fields φ Ir through the SO(8, n) matrix Finally the spin-1 /2 fermions are denoted by χȦ r , with the indexȦ of the cospinorial representation of SO (8).
The gauging of the theory is described using the embedding tensor formalism [4,5]. We briefly review its main features in this context. Gauging amounts to promote a subgroup G 0 ⊂ SO(8, n) to a local symmetry, in such a way that the local supersymmetry remains preserved. The embedding of g 0 in so(8, n) is given by the embedding tensor Θ MN |PQ , so that the gauge group generators X MN are The embedding tensor Θ MN |PQ is antisymmetric in [MN ], and [PQ], moreover symmetric under exchange of the two pairs (in order to allow for an action principle of the gauged theory). It is thus contained in the symmetric tensor product of two adjoint representations of SO (8, n) and may accordingly be decomposed into its irreducible parts: where each box represents a vector representation 8 + n of SO (8, n). With this group-theoretical representation, the constraint on Θ MN |PQ that ensures supersymmetry of the gauged theory takes a simple form [21]: i.e. one has to project out the "Weyl-tensor" type representation. 1 This constraint is often called the "linear constraint". It can be explicitly solved by parametrizing Θ MN |PQ as where θ MN PQ is totally antisymmetric and θ MN is symmetric and traceless.
To ensure that this embedding defines a proper gauge group, the embedding tensor Θ MN |PQ must be invariant under transformations of g 0 itself. Explicitly, this reads Since X MN is defined in terms of the embedding tensor (2.4), this condition gives rise to a set of equations bilinear in Θ MN |PQ , referred to as the "quadratic constraints". For later use, we rewrite these constraints in the parametrization (2.7) after contraction with an antisymmetric parameter Λ RS In terms of SO(8, n) representations, the quadratic constraints (2.9a), (2.9b) can be shown to transform according to Any solution to these constraints defines a viable gauging. The gauging procedure then follows the standard scheme, introducing covariant derivatives with vectors A µ MN , and associated field-strengths F µν MN . These fields are not present in the ungauged Lagrangian, but may be defined on-shell upon dualizing the Noether currents of the global SO(8, n) symmetry. In the gauged theory they couple with a Chern-Simons term with the gauge parameter g, and do not carry propagating degrees of freedom. Their minimal coupling to scalars breaks supersymmetry, and necessitates the introduction of new terms to the Lagrangian, specifically fermionic mass terms, and a scalar potential. Before presenting the full Lagrangian, it is useful to introduce the so-called T -tensor, that encodes these additional terms in the Lagrangian. We define (2.11) 1 In Ref. [26] a stronger condition has been applied by projecting the embedding tensor on its totally antisymmetric and trace parts. This explains the extra pieces in our fermion mass terms, given in Eq. (2.13) below.
The full Lagrangian is then r χḂ s − e V. (2.12) Here, e = |det g µν |, R denotes the Ricci scalar, and f MN ,PQ KL describe the structure constants of so (8, n). We refer to App. A for further details, particularly for the supersymmetry transformations. The last four terms in Eq. (2.12) are the most relevant for the following as they carry the fermionic mass matrices and scalar potential characteristic for the given gauging. Explicitly, the fermionic mass tensors A 1,2,3 are given by as functions of the T -tensor (2.11) and products of the SO(8) Γ-matrices Γ I AȦ , while the scalar potential V is given by (2.14) We finally note that the quadratic constraints (2.9) imply the relation for the fermionic mass tensors, often referred to as a supersymmetric Ward identity.

N = (p, q) supersymmetry
In the following, we will classify and analyze supersymmetric AdS vacua of the Lagrangian (2.12). As a first condition, the existence of an AdS vacuum necessitates an extremal point of the scalar potential (2.14), i.e. 16) The precise amount of preserved supersymmetry can be read off from the eigenvalues of the gravitino mass matrix A AB 1 . In units of the AdS length 2 = 2/|V 0 |, the condition for N = (p, q) supersymmetry takes the form For a vacuum with p + q = 8, preserving all supercharges, these conditions together with Eq. (2.13) further imply that T IJKr = T Ir = 0 at the vacuum. From this, it follows that the vacuum condition (2.16) is automatically satisfied. Moreover, for an N = (8, 0) vacuum, the tensor T IJKL has to be anti-selfdual at the vacuum.
Around a supersymmetric AdS vacuum, the matter content of the theory (2.12) organizes into supermultiplets of the associated supergroup that extends the spacetime isometry group. As the AdS 3 isometry group SO(2, 2) SL(2, R) L × SL(2, R) R is not simple, the corresponding supergroup in general is a direct product G L × G R of two simple supergroups, whose even parts are isomorphic to the products SL(2, R) L,R × G R-sym L,R , of the AdS factor SL(2, R) L,R with the respective R-symmetry groups G R-sym L,R . These supergroups have been classified in Ref. [22], and further analyzed in Ref. [23]. Supersymmetry in three dimensions thus is factorizable and admits the decomposition N = (p, q), where p and q are the number of fermionic generators of G R and G L respectively. In the following, we will mainly focus on chiral N = (8, 0) vacua, for which the even part of t he supergroup is of the form (2.19) The relevant simple supergroups have been given in Tab. 1.1 above, along with their even subgroups and the representations of supercharges. In particular, the different R-symmetry groups are of the form G R-sym SO(p) × SO(q), with p + q = 8 and p = 4. The supercharges originally transform in the spinor representation of SO (8)  For the following analysis of vacua it is convenient to discuss the two cases in (2.21) separately, and we will do so in Secs. 3 and 4, respectively.

Vacua and spectra
In this paper we will classify and analyze AdS 3 vacua preserving N = (8, 0) supersymmetry for general choice of the embedding tensor. Without loss of generality, we will search for vacua at the scalar origin V = 1, since any extremal point located at a different V 0 can be mapped into an extremal point at the scalar origin of the theory with embedding tensor rotated by V −1 0 [28,29]. We thus simultaneously solve the quadratic constraints (2.9a) and (2.9b) together with the extremality condition (2.16) (evaluated at T MN |PQ = Θ MN |PQ ) and the supersymmetry condition (2.17). For every AdS 3 vacuum, we then determine the associated gauge group and compute the mass spectrum of fluctuations.
The analysis is simplified by the symmetries of the desired vacuum. For a given choice of supergroup G R , we first parametrize Θ MN |PQ as a singlet of the R-symmetry group G R-sym R , c.f. Eq. (2.19). In general, this group may be embedded in different ways into the compact SO(8) × SO(n) ⊂ SO (8, n), such that the representation of the supercharges branches into the relevant representation collected in Tab. 1.1. All G R-sym R admit a chiral embedding into SO(8) ⊂ SO(8) × SO(n) according to either of the options from (2.21), while for sufficiently large n, the group G R-sym R or one of its factors may also admit a diagonal embedding into SO(8) × SO(n), as we will see in the following. With the proper parametrization of Θ MN |PQ , we then solve the Eqs. (2.9a), (2.9b), (2.16) and (2.17) to identify the vacua. The gauge group of the theory is read off from the algebra satisfied by the generators (2.4). At the vacuum, it is spontaneously broken down to its compact subgroup.

(2.24)
Upon projecting out the Goldstone scalars and goldstini, the spectrum organizes into G supermultiplets. As the supergroup G = G L × G R is not simple, each ∆ decomposes itself as ∆ = ∆ L + ∆ R , with conformal dimensions ∆ L,R associated to the representations of G L,R . The spacetime spin s is identified as s = ∆ R − ∆ L .

Solutions with irreducible vector embedding (2.21a)
We are now in position to start analyzing the consequences of the algebraic equations (2.9a) and (2.9b). We will do this analysis separately for all the supergroups given in Tab. 1.1, with the two possible embeddings of the R-symmetry groups SO(p) × SO(q) according to Eq. (2.21). In this section, we will consider the case of an irreducible vector embedding (2.21a). We restrict to n ≤ 8 matter multiplets.

Constraining the embedding tensor
As explained above, upon implementing Eq. (2.18) in Eq. (2.13) for N = (8, 0), the remaining possibly non-vanishing components of the embedding tensor are a priori given by where θ − IJKL is anti-selfdual. The fact, that the embedding tensor is singlet under the respective R-symmetry group, embedded according to (2.21a), further restricts these components as with traceless ξȦḂ of signature (p, q) (only non-vanishing for p = 8). The first quadratic constraint (2.9a) with free indices chosen as (M, N ) = (I, J) is then identically satisfied. Choosing the free indices as (M, N ) = (I, r) gives rise to the equations Eq. (3.3a) determines the eigenvalues of the matrix θ rs to be Accordingly, we choose a basis in which θ rs is diagonal, split the indices r into {r + , r − } and denote by n ± the multiplicities of these eigenvalues. Tracelessness of θ MN implies that If we now set all θ MN PQ = 0, all remaining quadratic constraints are satisfied. Up to an arbitrary overall scaling factor, this yields an embedding tensor of the form respectively. These equations could be simultaneously solved by choosing κ = 2 λ . However, with Eq. (3.5) this choice implies that in fact κ = λ = 0, resulting in θ = 0 = θ MN . As a consequence, both tensors A AB 1 and A AȦr 2 from Eq. (2.13) vanish at the vacuum, inducing a vanishing potential (2.14) and thus a Minkowski vacuum, which is beyond the scope of the present analysis. In all the following we thus assume that κ = 2 λ. Eqs. (3.7) then imply that, after complete resolution of the first quadratic constraint (2.9a), the solution (3.6) can be extended to potentially non-vanishing components The remaining quadratic constraints restricting these components follow from evaluating Eq. (2.9b). For readability, we defer the full set of constraint equations to App. B, and in the following subsections treat each of the four possible supergroups separately.

Chiral embedding
The R-symmetry group in this case is SO (8). Let us first assume that it is chirally embedded into the first factor of SO(8) × SO(n) ⊂ SO(8, n). Since the embedding tensor must be singlet under this group, its possible non-vanishing components within θ MN PQ further reduce from Eq. (3.8) to The second quadratic constraint is then reduced to two non-trivial equations, given by Eqs. (B.4c) and (B.5a). They take the explicit form Following the discussion after Eq. (3.7), we restrict to the case κ = 2 λ, after which the first equation implies that the only non-vanishing components of θ pqrs is θ p − q − r − s − . Next, we solve the remaining equation (3.10b) by considering each value of n − ≥ 4 separately. Since θ pqrs does not enter in the mass formulas (2.22), (2.23), the spectra of all the resulting theories are still given by Tab. 3.1(a). But, we will find in the following that non-vanishing θ pqrs generically reduces the factor SO(n − ) of the gauge group to a subgroup K 0 , such that the n − in Tab. 3.1(a) is to be replaced by the corresponding representation of K 0 . n − = 4 Both sides of Eq. (3.10b) identically vanish, such that the general solution admits a non-vanishing The full solution then extends Eq. (3.6) to For |ξ| = 1 the gauge group is SO(8, n + ) × SO(4), as in the ξ = 0 case. On the other hand, when ξ takes a critical value ξ = ±1, the gauge group reduces to SO(8, is broken down to one of its chiral factors.
In particular, this implies that For κ = 2λ this implies that there is a basis such that 3 The full solution is then given by The gauge group in this case is SO(8, n + ) × U(3), i.e. due to the presence of a non-vanishing θ p − q − r − s − , the SO(6) factor is reduced to U(3) compared to solution (3.6).
with the constant Λ given by The full embedding tensor is then given by The gauge group is SO(8, n + ) × G 2 , i.e. compared to solution (3.6) the group SO (7) is reduced to G 2 .
n − = 8 In this case, Eq. (3.5) implies κ = 0 and Eq. (3.10b) is solved by a self-dual θ pqrs : with SO(8) Γ-matrices Γ pqrs ab and traceless ξ ab subject to the equation This implies that the eigenvalues of ξ ab are , (3.22) with multiplicity p and 8 − p respectively and p = 4. 5 The full embedding tensor then takes the form: There is an analogous solution for anti-selfdual choice of θ pqrs . The gauge group is SO . 4 We choose the normalisation of ω so that The case p = 4 implies κ = 2 λ thus again leading to a Minkowski vacuum. (8) Tab. 3.1 Mass spectra for the OSp(8|2, R) R solutions with (a) chiral and (b) diagonal embedding of the R-symmetry group. The spectrum organizes into multiplets of OSp(8|2, R), given in Eq. (5.3) and Tab. 7 of Ref. [23], respectively. For non-vanishing θ pqrs , the factor SO(n − ) in (a) reduces to a subgroup K 0 , and the representation n − is replaced by the corresponding representation of K 0 .

Diagonal embedding
For n = 8 matter multiplets, the R-symmetry group SO (8) alternatively allows a diagonal embedding as Moreover, there are inequivalent diagonal embeddings according to possible triality rotations in the two SO (8) factors. In this case the condition of being singlet under the R-symmetry group reduces the possible components of the embedding tensor from Eq. (3.8) to An SO(8) diag singlet in θ IJrs can be parametrized as Eqs. (3.27) fix ρ = −2 κ, thus leading to an embedding tensor The gauge group induced by this tensor is GL (8). The spectrum around this vacuum is given in Tab. 3.1(b). Upon triality rotation of the second factor in Eq. (3.24), the singlet in θ IJrs would alternatively be given by This however does not lead to a non-trivial solution of Eqs. (3.27).

Chiral embedding
We now consider the supergroup F(4), with R-symmetry group SO (7). We first assume that SO (7) is entirely embedded into the first factor of SO(8) × SO(n) ⊂ SO(8, n) according to Eq. (2.21a), such that the potentially non-vanishing components of the embedding tensor reduce from Eq. (3.8) to Note that a non-vanishing ξȦḂ is required in order to realize the breaking of R-symmetry from SO (8) to SO (7), whereas for vanishing ξȦḂ we are back to the case OSp(8|2, R) discussed in Sec. 3.2 above. The remaining quadratic constraints for θ − IJKL and θ pqrs are not coupled. For θ − IJKL , they are given by Eqs. (B.1a) and (B.2b): The second line shows that non-vanishing ξȦḂ requires that n + = 0, thus We are left with Eq. (3.32a), which fixes the proportionality constant in ξȦḂ to be Setting θ pqrs = 0 solves all remaining equations, in which case the embedding tensor is given by up to an arbitrary overall scaling factor. The associated gauge group is SO(7) × SO(n). The spectrum is given in Tab. 3.2, organized into supermultiplets of F(4). Finally, we consider the possibility of non-vanishing θ pqrs . The equations for θ pqrs are the same as in Sec. 3.2 above, and so are the solutions, with the only difference that Eq. (3.33) restricts to vanishing n + . As in the OSp(8|2, R) case, a non-vanishing θ pqrs does not affect the spectrum of the theories which is still given by Tab. 3.2. Rather, it will restrict the SO(n) factor of the gauge group to some subgroup. Without repeating the details of the derivation, in the rest of this section, we simply list the different solutions for non-vanishing θ pqrs , organized by the different values for n.
where the G 2 invariant three-form ω uvw was introduced in Eq. (3.19). The gauge group is SO(7) × G 2 .

Diagonal embedding
Similar to Eq. (3.24), the R-symmetry group SO(7) could in principle be embedded diagonally into SO(8, n), but unlike for OSp(8|2, R) this does not give any new solution.

Chiral embedding
The supergroup SU(4|1, 1) has an R-symmetry group SO(6) × SO(2). We first consider its chiral embedding into the first factor of SO(8) × SO(n) ⊂ SO(8, n). The components of the embedding tensor are then given by Setting θ pqrs = 0 solves all remaining equations, in which case the full embedding tensor is given by up to an arbitrary overall scaling factor. The associated gauge group is SO(6) × SO(2) × SO(n). The spectrum is given in Tab. 3.3, organized into supermultiplets of SU(4|1, 1).
In the remainder of this section, in analogy to the OSp(8|2, R) case, we list the different solutions for non-vanishing θ pqrs , organized by the different values for n. The spectrum of these theories is still given by Tab. 3.3.
where σ was introduced in Eq.

Diagonal embedding
Alternatively, the R-symmetry group SO(6) × SO(2) or one of its factors can be diagonally embedded into SO(8) × SO(n) ⊂ SO(8, n) for n ≥ 2. The non-vanishing components of the embedding tensor are given by The set of non-trivial constraints which follow from the second quadratic constraint is the same as the one given for the chiral embedding. The new solutions are listed below. They are all defined in terms of the matrices defined in accordance with the embedding (2.21a). The 2 × 2 matrix σ has been introduced in Eq. (3.14).
The gauge group is SL(2) × Sp(4, R). The spectrum is given in Tab. 3.5.

Chiral embedding
We now turn to the last possible supergroup. OSp(4 * |4) has R-symmetry group SO(5) × SO (3) and we first consider its chiral embedding according to Eq. (2.21a) into the first factor of SO(8)× SO(n) ⊂ SO(8, n). The potentially non-vanishing components of the embedding tensor are then given by Again, the second quadratic constraint (2.9b) implies that θ − IJKL is non-vanishing only if which we will assume in the following. The only difference with the case F(4) is the signature of ξȦḂ in θ − IJKL , which is fixed by the SO(5) × SO(3) invariance. Setting θ pqrs = 0 solves all remaining equations, in which case the full embedding tensor is

Diagonal embedding
Alternatively, for n ≥ 3 we may consider embedding of the R-symmetry group SO(5) × SO(3) of OSp(4 * |4) or one of its factors diagonally into SO(8) × SO(n) ⊂ SO(8, n). The potentially non-vanishing components of the embedding tensor are (3.61) The non vanishing equations given by the second quadratic constraint are the same as in the SU(4|1, 1) case. There is only one new solution, for n + = 8 and SO (5)  The only relevant supergroups are F(4), SU(4|1, 1) and OSp(4 * |4), as for OSp(8|2, R) both embeddings (2.21a), (2.21b) are equivalent. In this case, the potentially non-vanishing components of the embedding tensor are then of the form with anti-selfdual θ − IJKL . The fact, that the embedding tensor is singlet under the respective R-symmetry group will pose further constraints on these components that we shall evaluate case by case in the following.
With the parametrization (4.2), the first quadratic constraint (2.9a), with free index values (M, N ) = (i, α) and depending on the different Λ parameters, gives the following set of independent equations: Eq. (4.3a) leaves two options, imposing either of the two factors to vanish. Setting θ rr = 8 λ however implies that θ IJ = λ δ IJ , i.e. we go back to the case of an irreducible vector embedding (2.21a) with the parametrization (3.2) carried out in Sec. 3. 6 We thus set for the rest of this section As in Eq. (3.3a) above, Eq. (4.6a) implies that we can take θ rs to be a diagonal matrix with eigenvalues with multiplicities that, as above, we denote as n ± . Tracelessness of θ MN then implies that Finally, the component (M, N ) = (r, s) of the first quadratic constraint is unchanged compared to the previous section (see Eqs. (3.7)) and together with Eqs. (4.6) it implies that the only potentially non-vanishing components of the embedding tensor are In the following, we solve the second quadratic constraint (2.9b) in this parametrization for the different supergroups with chiral and diagonal embeddings, respectively. Again, for readability we defer the full set of constraint equations to App. B.

Chiral embedding
We first consider the supergroup F(4), for which p = 7 and q = 1, so that the index splitting (4.1) gives i ∈ [ [1,7]] and α = 8. We first assume that the R-symmetry group SO (7) is entirely embedded into the first factor of SO(8) × SO(n) ⊂ SO(8, n). The potentially non-vanishing components of the embedding tensor are Setting all θ MN PQ always give a solution to the remaining quadratic constraints. This yields an embedding tensor of the form The gauge group is SO(7, n + ) × SO(1, n − ). The spectrum is given in Tab which implies that θ pqrs has θ p − q − r − s − as only non-vanishing components. We are then left with the following set of independent equations:

Diagonal embedding
The R-symmetry group could also be embedded diagonally into SO(8) × SO(n) ⊂ SO(8, n) for n ≥ 7. There is only one new solution for n + = 7, given by The gauge group is GL(7) × SO(1, n − ) and the spectrum is given in Tab. 4.1(b).

Chiral embedding
For the supergroup SU(4|1, 1), p = 6 and q = 2 and the index splitting (4.1) gives i ∈ [ [1,6]] and α ∈ {7, 8}. The R-symmetry group SO(6)×SO(2) could first be embedded into the first factor of where Ω r − s − is an anti-symmetric matrix and σ has been introduced in Eq. (3.14). A first solution is given by setting all θ MN PQ = 0. The embedding tensor then has the form The gauge group is SO(6, n + ) × SO(2, n − ). The spectrum is given in Tab which implies once again that the only non-vanishing components of θ pqrs are θ p − q − r − s − . The other equations are then equivalent to the following set of independent equations: As Ω p − q − is antisymmetric, there exists a basis where it has the form with scalar functions f 1 , . . . , f m , and σ from Eq. (3.14). With this parametrization, Eqs. (4.22) imply that (4.24) if n − = 2m, whereas they imply that all f i vanish for n − = 2m + 1. Thus, for κ = 2 λ there are solutions with non-vanishing θ MN PQ only if n − = 2m is even, and they are given by (4.25) The gauge group is SO(6, n + ) × SO(2, 1) if n − = 2 and SO(6, n + ) × U(n − /2, 1) otherwise. The spectrum is given in Tab. 4

Chiral embedding
We now consider the last possible supergroup, OSp(4 * |4). Here p = 5, and q = 3, and the index splitting (4.1) is given by i ∈ [ [1,5]] and α ∈ {6, 7, 8}. Let us first assume that SO(5) × SO (3) is entirely embedded into the first factor of SO(8) × SO(n) ⊂ SO(8, n). The potentially nonvanishing components of the embedding tensor are then given by The second quadratic constraint (2.9b) reduces to Eqs. (B.4c) and (B.5a), of which the former implies (2λ − κ) θ pqru Thus, all the components of θ pqrs vanish. The solution of the embedding tensor then is given by
The gauge group is SO(5, n + ) × SO(3, n − ) and the spectrum is given in Tab. 4.6.

Diagonal embedding
Finally, for n ≥ 3, there are the solutions when the entire R-symmetry group SO(5) × SO(3) or one of its factors is embedded diagonally in SO(8) × SO(n) ⊂ SO(8, n). The potentially non-vanishing components of the embedding tensor are Evaluating the quadratic constraints with this parametrization gives rise to a number of new solutions for the different embeddings which we list in the following. (3) is embedded diagonally, there is one new solution for n − = 3 and n + = 5. It is given by

OSp(7|2, R)
For OSp(7|2, R), a pair of N = (7, 0) vacua is given by the embedding tensor with n = 8 . The gauge group is SO(8) × SO (7), which at the vacuum is spontaneously broken down to a diagonal SO (7). The spectrum is given in Tab. 5.1. (7) 3 /2 Tab. 5.1 Mass spectrum for the OSp(7|2, R) solution with N = (7, 0) supersymmetries. The gauge group is SO(8) × SO (7). Closer inspection shows that these embedding tensors may be related to the embedding tensor of Eq. (3.23) (with selfdual θ pqrs ) by an SO (8,8) rotation of the form In view of our discussion in Sec. 2.3, we have thus identified three vacua which all belong to the same three-dimensional theory. To illustrate this structure, we evaluate the scalar potential (2.14) on the 1-scalar truncation (5.5) to SO(7) singlets, which takes the form that is sketched in Fig. 5.1. It exhibits the fully symmetric N = (8, 0) vacuum at the scalar origin φ = 0, together with the two N = (7, 0) vacua at φ = φ ± from Eq. (5.6). This potential may be cast into the following form Tab. 5.2 Mass spectrum for the G(3) solution with N = (7, 1) supersymmetry. The gauge group is SO(n, 1) × G 2 .
in terms of a real superpotential which shares its stationary points with V (φ).
The embedding tensors (5.11) turn out to be related to the previously found N = (8, 0) solution (3.38) by an SO(8, 7) rotation of the form Starting from Eq. (3.38) (up to a change of basis) at the origin φ = 0, the tensors (5.11) are obtained at φ ± = ln 3 ± 2 √ 2 . In view of our discussion in Sec. 2.3, we have thus identified three vacua which all belong to the same three-dimensional theory. To illustrate this structure, we evaluate the scalar potential (2.14) on the 1-scalar truncation (5.12) to G 2 singlets, which takes the form that is sketched in Fig. 5.2. It exhibits the fully symmetric N = (8, 0) vacuum at the scalar origin φ = 0, together with the two N = (7, 0) vacua at φ ± = ln 3 ± 2 √ 2 . Let us finally note, that the scalar potential (5.13) admits a "fake" superpotential W (φ) = − 5 32 33 + 84 cosh(φ) − 56 cosh(2 φ) + 28 cosh(3 φ) + 7 cosh(4 φ) ) , (5.14) in terms of which it may be written as However, the N = (7, 0) vacua are no stationary points of W (φ).  It is an interesting question to ask, which of the vacua identified in this analysis can actually be embedded into a maximally supersymmetric (N = 16) three-dimensional supergravity, thus spontaneously breaking half of the supersymmetries of the theory. For D = 4 supergravities, the analogous question has been addressed in Ref. [36].
To answer this question, we first recall some of the basic structures of maximal D = 3 supergravities [4,20]. In this case, the scalar sector describes an E 8(8) /SO (16) coset space sigma model and the embedding tensor which defines the gauge group generators within E 8(8) in analogy to Eq. (2.4) transforms in the 1 ⊕ 3875 representation of E 8 (8) . The maximal theory can be truncated to a half-maximal subsector upon truncating the coset space (8)) . (6.1) Under SO (8,8), the embedding tensor of the maximal theory decomposes as into SO (8,8) and its orthogonal complement (transforming in the spinor representation 128 s of SO (8,8)), an embedding tensor of the maximal theory triggered by the first three terms in Eq. (6.2) takes the form in the maximal theory, the gauge group is thus given by an extension of the gauge group of the half-maximal theory by the additional generators X A = Θ A|B Y B . We may now address the following question: given an embedding tensor (2.7) of the half-maximal SO (8,8) theory, satisfying the quadratic constraints (2.9a), (2.9b), does the associated embedding tensor (6.4) satisfy the quadratic constraints of the maximal theory and thereby define a consistent gauging of the maximal theory? By consistency of the truncation, the N = (8, 0) vacuum of the halfmaximal theory then turns into a vacuum of some maximal gauged supergravity, breaking half of the supersymmetries spontaneously.
To answer this question, we recall that the quadratic constraints of the maximal theory transform in the 3875 ⊕ 147250 (6.6) under E 8 (8) . Breaking these under SO (8,8) and restricting to the representations that can actually appear in the symmetric tensor product of two half-maximal embedding tensors shows that in order to define a maximal N = 16 gauging, the components of the embedding tensor (6.4) must satisfy additional constraints transforming as 35 ⊕ 6435 c , i.e.
where the last term refers to the anti-selfdual contribution in the 8-fold antisymmetric tensor. These additional conditions may be worked out explicitly in analogy to Eq. (2.8) and take the following form Let us note that the first equation has the same representation content as the constraint obtained from proper contraction of the constraints (2.9b) of the N = 8 theory. Furthermore, the choice of anti-selfduality (vs. selfduality) in Eq. (6.7) is a pure convention here, depending on the embedding of SO (8,8) into E 8 (8) .
To summarize, an embedding tensor of the half-maximal theory which in addition to the quadratic constraints (2.9a), (2.9b) of the half-maximal theory satisfies the additional constraints (6.8) defines a consistent maximal three-dimensional supergravity. The half-maximal theory is recovered upon truncation (6.1). Vacua of the half-maximal theory then give rise to vacua within the maximal theory. It is a straightforward task to check the additional constraints (6.8) for all the vacua we have identified in this paper. For theories with a free parameter these constraints single out specific values of the parameter. We collect the result on the possible embedding into the maximal theory for every vacuum in the summarizing Tabs. 7.1-7.3.

Summary and outlook
In this paper, we have presented a classification of N = (8, 0) AdS 3 vacua in half-maximal D = 3 supergravities. Analyzing the consistency constraints on the embedding tensor, we have determined the full set of possible gauge groups embedded in the SO(8, n) isometry group of ungauged D = 3 supergravity, for n ≤ 8. There are four classes of such vacua according to the different superisometry groups OSp(8|2, R), F(4), SU(4|1, 1), and OSp(4 * |4), respectively. For each of the vacua, we have determined the explicit embedding tensor, the gauge group embedded into the SO(8, n) isometry group of ungauged D = 3 supergravity, and the physical mass spectrum, organized in terms of supermultiplets. We summarize our results in Tabs. 7.1, 7.2. For all the vacua identified, we have furthermore determined if and under which conditions on the free parameters, the half-maximal theories admit an embedding into a maximal (N = 16) supergravity, with the gauge group enhanced by additional generators according to Eqs. (6.4), (6.5), above.
In Tab. 7.3, we collect our findings of N = (7, 1) and N = (7, 0) AdS 3 vacua. We have shown that the latter vacua are realized in half-maximal theories that also admit fully supersymmetric N = (8, 0) AdS 3 vacua as different stationary points in their scalar potential, c.f. Figs. 5.1, 5.2 above. In particular, this indicates the existence of domain wall solutions interpolating between N = (8, 0) and N = (7, 0) AdS 3 vacua. It would be very interesting to generalize our findings to a classification of general fully supersymmetric N = (p, 8 − p) vacua of the half-maximal theories. In this case, the classification will be organized by products of smaller supergroups, which presumably leaves even more possibilities for the potential embeddings of their bosonic parts into SO (8, n).
An immediate question to address for the vacua identified in this paper is the existence and the structure of their moduli spaces as submanifolds of the scalar target space SO(8, n)/(SO(8)× SO(n)). In the holographic context, this relates to the exactly marginal operators of the putative holographically dual conformal field theories.
Probably the most outstanding issue about these vacua and theories concerns their possible higher-dimensional origin. More precisely, it would be very interesting to embed the halfmaximal D = 3 supergravities identified in this paper as consistent truncations into ten-or eleven-dimensional supergravities, such that in particular the AdS 3 vacua would uplift to full D = 11 or IIB solutions, subject to the constraints from higher-dimensional classifications and no-go results [37][38][39]. For example, it would be interesting to see if the type IIA compactifications with exceptional supersymmetry of Ref. [24] can be embedded into consistent truncations to D = 3 supergravities constructed in this paper. In particular, for the solution with F(4) supersymmetry, Tab. 7.2 offers several candidates with gauge group factors SO(7) potentially realized as the isometry group of the round six sphere S 6 = SO(7)/SO (6) .
A systematic approach for higher-dimensional uplifts builds on the reformulation of the higher-dimensional supergravities as exceptional field theories based on the group SO(8, n) [40]. In this framework, consistent truncations are described as generalized Scherk-Schwarz reductions, leaving the task of solving the consistency equations for suitable Scherk-Schwarz twist matrices. An apparent obstacle to a standard geometrical uplift of many of the theories collected in Tabs. 7.1-7.3 is the rank and the size of their gauge groups which do not admit a geometric realization as the isometry group of a 7-or 8-dimensional internal manifold. It remains to be seen if the rich structure of three-dimensional supergravity hints at some more general reduction mechanisms specific to three-dimensional theories. In this context, it may be advantageous to exploit the possibility of embedding the D = 3 half-maximal theory into maximal higher-dimensional supergravities via their formulation as an E 8(8) exceptional field theory [41] upon suitable generalization of the methods developed in Ref. [42]. Alternatively, one may entertain the possibility of embedding some factor of the large D = 3 gauge groups into a non-abelian gauge structure of the heterotic theory in ten dimensions.

A Notations and supersymmetry transformation
We list here the conventions for the gauging and the supersymmetry transformations in the Lagrangian (2.12), following Refs. [20,26]. The coset representative V(φ) defined in Eq. (2.3) transforms as with g ∈ SO(8, n) and h(φ) ∈ SO(8) × SO(n). For the ungauged theory, its coupling to fermions is described in terms of the Cartan-Maurer form Once the theory is gauged, the covariantization gives The full covariant derivatives of the fermions then read with the spin-connection ω µ αβ , the three-dimensional γ matrices γ α in flat spacetime and γ αβ = γ [α γ β] . Finally, the full supersymmetry variations are with supersymmetry parameter ε A and γ µ = e µ α γ α .

B Second quadratic constraint
We list here the full set of independent equations contained in the second quadratic constraint