All timelike supersymmetric solutions of three-dimensional half-maximal supergravity

We first classify all supersymmetric solutions of the 3-dimensional half-maximal ungauged supergravity that possess a timelike Killing vector coming from the Killing spinor bilinear by considering their identification under the complexification of the local symmetry of the theory. It is found that only solutions that preserve $16/2^n, 1 \leq n \leq 3$ real supersymmetries are allowed. We then classify supersymmetric solutions under the real local symmetry of the theory and we are able to solve the equations of motion for all of them. It is shown that all such solutions can be expressed as a direct sum of solutions of the integrable Liouville and SU(3) Toda systems. This completes the construction of all supersymmetric solutions of the model since the null case has already been solved.


Introduction
Supersymmetric solutions are pivotal in the study of supergravity theories since they possess stability properties that survive quantum deformations. Assuming supersymmetry renders the solution space more tractable too. This is because studying the first-order Killing spinor equations is easier than the second-order equations of motion.
There are various related methods of attacking the problem of finding supersymmetric solutions. In the approach that is based on spinorial geometry one considers the reduction of the local symmetry of the theory, including the spacetime spin group, to the stability subgroup of Killing spinors. This method has been widely successful, especially so for maximally supersymmetric theories where the reduction of the spin bundle is straightforward (see for instance [1]), but also because the method can be applied to the reduction of the generalized (hidden) structure group of the theory (see for instance [2]). An equivalent approach is to study the various tensors formed by the Killing spinor bilinears as initiated by Tod in [3,4], a method successful in various dimensions and theories (see for instance [5]).
The latter approach was applied to study supersymmetric solutions of three-dimensional half-maximal supergravity in [6]. It follows from the algebra of supersymmetry variations that in any supergravity theory the vector formed by squaring a Killing spinor is at least Killing which is either null or timelike. For this model the null case has been completely solved in [6] and the most general solution is found to be a pp-wave. However, for the timelike case only few explicit solutions were obtained in [6]. In this paper our aim is to classify and solve for all supersymmetric timelike solutions of this model for which the metric is ds 2 = dt 2 − e 2ρ(x,y) dx 2 + dy 2 .
The scalar content of the theory parametrizes the coset and their Lie algebras as g = so(8, n) and k = so(8) ⊕ so(n), respectively. The coset representative is time independent, so the pull-back of the Maurer-Cartan form P + Q = V −1 dV only depends on the adapted coordinates x and y. Here P is the scalar current and Q is the SO(8) × SO(n) connection.
Recently a novel classification of supersymmetric backgrounds of the three-dimensional, maximally supersymmetric, ungauged supergravity was given in [7]. The motivation there was primarily the construction of interesting supersymmetric solutions with what is termed non-geometric monodromy. Rather than fixing a Killing spinor and thus reducing the symmetry of the theory, the authors instead fixed the element P under the action of some group. In a sense, the problem is turned on its head by asking which elements P admit one Killing spinor, two Killing spinors, etc. The general problem of fixing P this way is feasible. Moreover, the assumption of at least one supersymmetry implies that P has to be nilpotent in some Lie algebra. More precisely, by using the Zariski topology argument, the element which transforms under the local group of the theory K, is shown to be necessarily nilpotent as an element in the complexified version g C of g, where g is the Lie algebra of the global symmetry G. Note that the complexified version k C of the local algebra k acts on P z and preserves nilpotency in g C . What is then left is to classify nilpotent orbits of (g/k) C under K C , to which P z should belong. This is particularly attractive as nilpotent orbits are finite and can be classified for all classical groups. For the classification one then uses the Kostant-Segikuchi correspondence that asserts a one-to-one correspondence of nilpotent orbits in (g/k) C under K C to nilpotent orbits in g under G [7]. Although the method in [7] is applied to maximally supersymmetric ungauged supergravity in three dimensions, where the global symmetry G is E 8 and the local symmetry K is the maximally compact subgroup SO (16), their topology argument applies identically to the half-maximal ungauged supergravity as well.
Note that an element P z of a background that admits timelike supersymmetry is necessarily nilpotent in g C but the converse is not true. Therefore, after we obtain the nilpotent orbits in (g/k) C under K C we need to check for supersymmetry. This can be done by testing the element P z on the algebraic dilatino variation. We will show that this is sufficient as the integrability of the gravitino variation is indeed satisfied on-shell. The classification of nilpotent orbits under K C that admits supersymmetry is pretty concise to summarise. The supersymmetric orbits under K C to which such a P z belongs correspond to the partitioning of (8, n) into sums of (2, 2), (2,1), (1,0) and (0, 1). This decomposition can be thought of 1 as the decomposition of R 8,n into orthogonal subspaces R 2,2 , R 2,1 , R 1,0 and R 0,1 . The multiplicity µ of (2, 2) and multiplicity ν of (2, 1), and only these, determine the supersymmetry by the simple rule that each of them halve supersymmetry by a projection equation. Each class of elements, up to the action of K C , corresponds to a unique partition. We call the class N(µ, ν). That is, the classes are defined by A representative element for the class N(µ, ν) is called a normal form. They are useful as they allow us to work with a concrete element and are pretty easy to write down. However, note that the group used to identify the elements P z is the complexification K C of the symmetry of the theory K. Therefore, the orbits under K C may contain more than one, or even no solutions. For instance, a normal form under K C may not satisfy the equations of motion but some other representative that is K C -conjugate to it might do. That is, it does not make sense to use the normal form in order to start solving the equations of motion because the equations of motion are not covariant under K C . Therefore, we have to move on to classify the elements P z under the real local symmetry of the theory K in order to obtain exact solutions. This means that for each class N(µ, ν) and each element P ′ z ∈ N(µ, ν), we need to find all the elements P z that are distinct to P ′ z under the action of K but are identical to P ′ z under K C . We may call this space N(µ, ν)/K. The most general element P z ∈ N(µ, ν)/K is still easy to write and are given in (77). The equations of motion and in particular the integrability equations for P + Q = V −1 dV severely restrict the coefficients in P z . Consequently, the classification of the on-shell nilpotent elements that are in N(µ, ν) should be refined into spaces N(µ, ν r , ν c ), where ν = ν r + ν c . If P z ∈ N(µ, ν) and is indeed part of a solution, then P z ∈ N(µ, ν r , ν c ) . After this classification we analyze the field equations and integrability conditions and arrive at the following result: Main Result. The timelike supersymmetric backgrounds of the three-dimensional, half-maximal, ungauged supergravity are locally parametrized by µ + ν r + 2ν c meromorphic functions which are solutions to µ + ν r copies of Liouville's equation and ν c copies of an SU(3) Toda system. The µ and ν r copies of Liouville's equation are distinguished by their contribution to the coset space connection P + Q and to the spacetime curvature. Each ν, ν r and ν c copy is responsible for halving supersymmetry once.
We begin in section 2 with an introduction to the theory and set up our conventions for the timelike backgrounds. In section 3 we present the nilpotency classification. In section 4, we do not yet use the equations of motion but we present the elements P z in the classes up to the real symmetry. The restriction of P z due to the equations of motion and the solutions themselves are in section 5. We conclude in section 6 with some brief remarks. Most of the technical material is to be found in the appendices. In appendix A we review our spinorial conventions. We also give in appendix A some useful formulae for comparison with other methods in the literature. In appendix B we comment on a more direct matrix factorization of P z . Supersymmetry closure in the Zariski topology and construction of normal forms are explained in detail in appendices C and D, respectively.

Set up 2.1 Theory
Half-maximal ungauged supergravity in three dimensions is described in the bosonic sector by a metric g on a three-dimensional spin manifold M and the coset map where the groups G and K are and their Lie algebras are g and k = so(8) ⊕ so(n). We pull-back and split the Maurer-Cartan form on the symmetric decomposition g = k ⊕ p, where p = g/k = R 8 ⊗ R n . The action of the model is where µ, ν = 0, 1, 2 are spacetime indices, I, A,Ȧ = 1, 2, . . . , 8 are respectively the vector, chiral and anti-chiral indices for Spin (8), and r, s = 1, . . . , n are SO(n) vector indices. Note that we use a mostly minus signature. The full theory was constructed already in [8]. The gaugings of the theory were classified in [9]. For other gauged three-dimensional supergravities with various amounts of supersymmetry see [10,11]. From the action we derive the equations of motion The integrability of P + Q = V −1 dV is dP + dQ + (P + Q) ∧ (P + Q) = 0, or explicitly dP Ir + Q IJ ∧ P Jr + Q rs ∧ P Is = 0 , The full theory has 16 real supersymmetries, which are locally given by ǫ A α but we usually suppress the spacetime spinor index α = 1, 2. With the gravitino ψ µ and dilatino χ put to zero, a Killing spinor should satisfy We will use {γ a , γ b } = −2η ab , where η ab has mostly minus signature, and {Γ I , Γ J } = −2δ IJ , so that all representations are real. We refer to appendix A for more details on our spinorial conventions.

Timelike backgrounds
Let us define the vector Since the derivative D in the gravitino variation (11) is in spin(1, 2)⊕spin(8), the vector V µ is easily shown to be parallel, i.e. ∇ µ V ν = 0. We may define the Killing spinor bilinear in order to derive via the Fierz identity which shows that V is either null or timelike: The null case was completely solved and few explicit solutions for the timelike case were obtained in [6]. In this paper we only consider the timelike case and so V µ is a timelike covariantly constant vector. It follows that we can find adapted coordinates (t, x, y) so that V = ∂ t and the metric is It is shown in [6] that ∂ t also leaves the coset representative invariant up to a local K transformation, so in particular we may choose a gauge where Q t = P t = 0.
The Einstein equations of motion for the metric (17) are only non-trivial in the (x, y) components, It thus follows that P Ir x P Ir x = P Ir y P Ir If we then define z = x + iy and the non-trivial components of the Einstein's equation are: and (26) that determine P and Q, the Einstein equation (23) that determines ρ, and finally the condition that P z admits Killing spinors via the algebraic equation (30). Therefore, when the equations of motion are satisfied, Killing spinors are characterized only by (30). Note that if ǫ A z is a Killing spinor, then so is i ǫ A z . We may thus assert the following: Theorem 1. Supersymmetric solutions with a timelike Killing vector admit an even amount of real supersymmetry and form a complex vector space.
We will see in Theorem 3 that not only is the amount of supersymmetry even, but it comes in powers of two: 16, 8, 4, 2.

Nilpotency
Our strategy in this section is to set aside the equations of motion for P , Q and ρ, and classify instead all elements P z that admit supersymmetry via equation (30). The classification is with respect to K C , the complexification of the local symmetry of the theory. That is, we identify all admissible P z up to the action of K C . The classes are parametrized by integers µ and ν and we call each class N(µ, ν).

Proof of nilpotency
We note that the symmetry of the dilatino supersymmetry equation (30) is SO(8) C × GL(n, C). Indeed, SO(8) C is the group that preserves the gamma matrices of the 8-dimensional Clifford algebra. For instance, take m ∈ so (8) and note that since 2 and all representations are real, we can complexify the Lie algebra element m.
On the other hand, the index r in (30) is a free index, whence the symmetry GL(n, C). Classifying P z up to the action of SO(8) C × GL(n, C) turns out to be too strong. However, it does prove that the algebraic supersymmetry equation (30) is a set of projection equations that halve the real supersymmetries according to 16, 8, 4, 2, we give the proof in appendix B. Instead, we classify the elements P z up to the action of Since it is a symmetry of the algebraic supersymmetry equation, we may consider the orbit space of the (g/k) C where P z belongs to, up to the action of K C : (g/k) C → (g/k) C . That is, since any other element in the same orbit admits the same amount of supersymmetry we may consider the orbit as a whole. The group K C is not a symmetry of the theory, in contrast to the group K, but one may hope to move from this classification to orbits under K once the first are obtained, which we do in section 4. Note also that relative to SO(8) C × GL(n, C), the orbit space of the action K C is more fine grained and thus perhaps more useful. In fact, it turns out that the orbit space under SO(8) C × GL(n, C) is labeled by the amount of supersymmetry. Similar to the case of maximal supergravity [7], we will now show that the element P z ∈ p C = (g/k) C is nilpotent in the adjoint representation of g C . That is, with the symmetric decomposition we will now show that (ad Pz ) p+1 = 0 for some positive integer p.
Our proof closely follows [7]. Consider an element P z ∈ p C . The Jordan-Chevalley decomposition tells us that it can be written as a sum of a semisimple element and a nilpotent element with P S , P N ∈ p C ⊂ so(8, n) C and [P S , P N ] = 0, see proposition 3 in [12]. Assume that (38) admitsñ > 0 algebraic Killing supersymmetries according to (30). Consider then the orbit O of P z under K C and assume P S = 0. The algebraic supersymmetry equation (30) implies that the closureŌ of the orbit O in the Zariski topology preserves at leastñ supersymmetries, a result of [7] that we review in appendix C. At the same time, it can be shown that in the Jordan-Chevalley decomposition, the semi-simple element P S is in the closure of the orbit, P S ∈Ō, see lemma 11 in [12]. Furthermore, any semi-simple element P S in p C is K C -conjugate to an element in the Cartan subalgebra in p C , by virtue of its semi-simplicity alone. In summary, if P z preservesñ supersymmetries and P S = 0, then there is an element in the Cartan subalgebra in p C that preserves at leastñ supersymmetries. Yet, it is easy to show that an element in the Cartan subalgebra in p C does not preserve any supersymmetry and hence P S has to vanish. In order to show this, assume first an orthonormal basis e I of R 8 and an orthonormal basisê r of R n . Then an element in the Cartan subalgebra in p C has to be diagonal and is expanded in this basis as P Ir S e I ⊗ê r = P 11 S e 1 ⊗ê 1 + P 22 The algebraic supersymmetry equation (30) for r = 1 becomes (if the component P 11 S is zero, take instead the first non-zero element) Since the gamma matrix Γ 1 squares to −1, this equation cannot admit a nonzero solution for ǫ A z . Hence, if P z admits some supersymmetry then P z = P N and the orbit O of P z under K C is nilpotent.
Our task is then to classify the nilpotent orbits in p C under K C , a space we may write as To this aid, we use the Kostant-Segikuchi correspondence, which is a correspondence between nilpotent elements in g up to the action of G and nilpotent elements in p C up to the action of K C : For more details, see appendix (D.1).

Indecomposable types and their normal forms
Our goal is now to classify nilpotent elements of O(m, n) up to conjugacy. In particular we will construct normal forms, which are representatives in each class. Normal forms for elements in the classical linear groups have been described but not explicitly written in [13] (see also [14]). Consider the Lie algebra L(V, τ, σ) of a linear group that acts on a complex vector space V , preserves the bilinear τ and is compatible with the real or pseudoreal structure σ, where the latter is compatible with τ 3 . Let A ∈ L(V, τ, σ) and A ′ ∈ L(V ′ , τ ′ , σ ′ ). We take (A, V ) and (A ′ , V ′ ) as equivalent if there is an isomorphism φ such that: The equivalence class defines a so-called type ∆, that is (A, V ) ∈ ∆. If (A, V ) ∈ ∆ and A is reducible on the direct sum of τ -orthogonal, σinvariant subspaces V = V 1 ⊕ V 2 , then note that L(V i , τ |, σ|) is well-defined and we can write A ∈ L(V i , τ, σ) and (A, V i ) ∈ ∆ i for a type in the restricted linear algebra. In this case, we define the decomposition of types Note that we also have for the dimensions of the corresponding vector space decomposition. For the case of symmetric τ , the signature of the two types ∆ 1 and ∆ 2 should also add up to that of ∆, a property that we will use in our classification. The notion of decomposition of types in (44) lends to the definition of an indecomposable type. Finally, a decomposition into indecomposable types can be shown to be essentially unique. We give the indecomposable types ∆ of so(m, n) in table 1. The types in table 1 are denoted by ∆ p (ζ, · · · ), where p is the order of its nilpotent part N in the fundamental and in parentheses the (ζ, · · · ) are the eigenvalues of its semisimple part S. We also list the dimension and signature that any given type belongs to. Under a decomposition into indecomposables, see (46), the signatures add up as in (45). An algorithm to find the types of elements in so(m, n) is to partition the signature (m, n) into numbers (m i , n i ) that correspond to the indecomposable types in table 1.
From the table we see that if the indecomposable type is nilpotent, then there are only two possibilities: type ∆ ± p (0) and type ∆ p (0, 0). We construct normal forms for these types in appendix D.2, and in D.3 we give their corresponding Kostant-Segikuchi triples in so(m, n). Via the Kostant-Segikuchi correspondence, we thus arrive at the normal forms for the indecomposable nilpotent elements in p C up to the action of K C which we give in appendix D.4.

Supersymmetric nilpotency
In the previous subsection, we classified the complex nilpotent elements that P z necessarily belongs to. However, not all of them admit supersymmetry. We need to select those that admit a non-zero amount of supersymmetry according to the algebraic supersymmetry equation which leads us to: Theorem 2. Assume that P z admits some supersymmetry. If we decompose the element P z into nilpotent indecomposable types of SO(8, n), then the following hold a) Type ∆ p (0, 0) for p ≥ 3 does not appear in the decomposition, b) The multiplicity µ of ∆ 1 (0, 0) is responsible for projecting supersymmetry to a fraction (1/2) µ , c) Type ∆ p (0) for p ≥ 4 does not appear in the decomposition, d) The multiplicity of ∆ 0 (0) in the decomposition does not affect supersymmetry, e) Type ∆ + 2 (0) does not appear in the decomposition. The multiplicity ν of ∆ − 2 (0) is responsible for projecting supersymmetry to a fraction (1/2) ν . We give the proof of theorem 2 in appendix D.5. Types ∆ − 2 (0) and ∆ 1 (0, 0) are the only ones that determine supersymmetry, because type ∆ ± 0 (0) is represented by P z = 0. Assume a basis e I of R 8 andê r of R n related to SO(8, n) ungauged supergravity. Normal forms corresponding to each indecomposable type can be written in tensor product form (see (267) and (259) with some relabeling): If P z in its decomposition into ∆ − 2 (0) and ∆ 1 (0, 0) does not span the whole space p C , then one can use a parity transformation in the perpendicular directions and absorb the signs that appear in (47) and (48). If on the other hand the element P z spans the whole space, then all signs are again absorbed because the sign of the last type that appears in the decomposition is fixed to be one because of the chirality of Killing spinors. Indeed, the algebraic supersymmetry equation for each type in (47) and (48) is manifestly that of a BPS projection equation where Γ 1 corresponds to e 1 in (47) or (48) and Γ 2 corresponds to e 2 in (47) or (48). Note that the ǫ A z appearing in this equation is only K C -conjugate to the actual supergravity Killing spinor.
At this point we introduce the following notation: A supersymmetric element P z is said to belong to type N(µ, ν) if it decomposes into types as By using (47) and (48), a supersymmetric element P z ∈ N(µ, ν) is K Cconjugate to It follows from the signatures of the types in table 1 that each class N(µ, ν) corresponds to the partition of (8, n) into the sums of (1, 0), (0, 1), (2, 1) and (2,2), by using the convention with multiplicity µ of (2, 2) and multiplicity ν of (2, 1).
Note that having only one real supersymmetry is excluded because according to theorem 1 the vector space of Killing spinors is complex. In the case of µ + ν = 4 there are only three independent BPS projections due to chirality. Theorem 3 can also be shown in a more direct approach, which we do in appendix B.

Identification under K
The classification under K C is genuine and powerful. However, it is of little use if we cannot access the solutions. If P 1 ∈ N(µ, ν) is a normal form in the class but is not part of a solution, it does not follow that a conjugate element P 2 K C ∼ P 1 is also not a solution. If P 1 is indeed a solution, by using only P 1 we miss all other potential solutions that are related to P 1 by K C but not related to it by K, where the latter is the actual symmetry of the theory. Therefore, we should move from normal forms of a class N(µ, ν), that is under the identification of to all elements that are identified under We may think of starting with a specific normal form P 1 ∈ N(µ, ν) and act on it with all possible K C rotations, modulo its stabilizer that leaves the normal form invariant anyway, thus obtaining all elements in N(µ, ν). Subsequently, we should identify under K and obtain the space that we call N(µ, ν)/K. We are thus interested in the double quotient on the right-hand side of Note that the normal forms in (51) do not contain any coefficients so the spacetime variance of P z comes from the double coset alone. We will not parametrize the double quotient (59) directly. Instead, we will use the action of a real orthogonal group on the complexification of its associated vector space, which we describe in the next subsection. Then, we will be able to write the most general form of a P z ∈ N(µ, ν) after identifying the elements up to the real local symmetry of the theory.

Complex vectors
We begin with the action of O(m) on complex vectors in C m with inner product defined as A · B = I A I B I . We will later specialize for m = 8 and m = n. This subsection will eventually serve our goal to fix Let us first consider complex null vectors, for instance a vector v ∈ C m such that v · v = 0. Let us use an orthonormal basis {e I } of C m . It is clear that one may O(m)-rotate the real part of v to only have a component in e 1 and then rotate its imaginary part, by using the stabilizer O(m − 1), to have components in e 1 and e 2 . The condition v · v = 0 though implies that its expansion in components is in terms of some real v 1 that can be chosen positive. If we wish to fix v under the action of O(m) C instead, there is a hyperbolic element in SO(2) C ⊂ SO(m) C that scales v and so v 1 can be set to one. Assume now an ordered set of µ complex null vectors {v (i) } µ i=1 that are linearly independent and orthogonal to each other. We may fix the first vector v (1) as in (60), fix the second vector v (2) to only have components in e 1 , e 2 , e 3 , e 4 , etc., a modification of the QR decomposition. Since the vectors are orthogonal to each other, only half of their coefficients are independent, . . . and the diagonal coefficients are positive by linear independence. If we use O(m) C instead, the diagonal entries can be scaled to one. If we are not interested in fixing the vectors completely, we may expand with the Einstein summation over j = 1, . . . , µ and use a non-degenerate µ × µ matrix v j (i) . There is a manifest U(1) µ SO(µ) ⊂ SO(m) symmetry acting on the expansion in terms of v j (i) in (63). The U(1) factors are complex phase rotations and the SO(µ) rotates the e 2i−1 +ie 2i in the fundamental representation. The group product U(1) µ SO(µ) is not a direct product, it is the group generated by the groups U(1) µ and SO(µ) as subgroups of SO(m): the set of all possible multiplications between the group elements of the subgroups. As these two subgroups do not commute the multiplication generates U(µ), see lemma 1 in appendix B. Similarly, one may fix under O(m) C and the matrix v j (i) can be made equal to the identity matrix, see appendix B. Let us now turn to an ordered set of ν ≤ 4 linearly independent complex vectors {r (i) } ν i=1 that are mutually orthogonal among themselves and with the previous ordered set {v (i) } µ i=1 of complex null vectors, but such that the norm of each r (i) is equal to one. Since they are orthogonal to the {v (i) } µ i=1 , by using O(m) and the expansion in (63), we may expand the r (i) as where the R (i) do not contain components in the complex span of e 1 , . . . , e 2µ . We may use the remaining symmetry O(m − 2µ) to fix the R (i) . The first R (1) may be brought to the form and we may choose ζ 1 to be real. Continuing this way, in a QR decomposition, we may partially fix the R (i) to be expanded in a basis with an Einstein summation over j and where the matrix Σ (i) j is given by the upper-left ν × 2ν submatrix of the 4 × 8 matrix (ν ≤ 4) The η i might be fixed to be real or imaginary 4 and the ζ i are all real. It might seem that Σ is completely fixed and there is no remaining symmetry, but this is not true if Σ is degenerate. This happens when some of the parameters in Σ are zero. The matrix Σ has the orthonormal property ΣΣ T = I ν×ν .
We have now described in general how to fix two ordered sets of vectors that are orthogonal among themselves and each other, where the first are null and the latter unit norm, under the action of O(m). Under SO(m) there might be a sign ambiguity in one of the components when 2µ + 2ν = m. Indeed, for 2µ + 2ν < m one may use a SO(m) rotation that contains a parity transformation perpendicular to the basis, so the sign in the basis is restored. If 2µ + 2ν = m and ν = 0, we may allow η i to be negative in (68). If ν = 0 and 2µ = m then we may need to replace e 2i−1 + i e 2i with e 2i−1 − i e 2i for some i in (63). This sign ambiguity will not be present in what follows due to the chirality of Killing spinors.

Elements in
The most general K C transformation is such that P z should be expanded in terms of independent orthogonal complex null vectors where the w (i) and r (i) are also mutually orthogonal together: This follows by the form given in (51). Indeed, the action of SO(8) C × SO(n) C preserves the inner product among the vectors appearing in (51) or the corresponding ones appearing in (69). That is, in (69) we necessarily have Finally, the vectors in (69) should be linearly independent.
We define an orthonormal basis of an orthogonal subspace R 2µ ⊕ R 2ν ⊆ R 8 and an orthonormal basis of an orthogonal subspace R 2µ ⊕R 2ν ⊆ R n . We will use a basis of null vectors and a basis of null and orthonormal vectors in According to the discussion in subsection 4.1, the vectors appearing in the element in (69) There are two invariants of the element as written in (77) that identify it as belonging to N(µ, ν): • The rank µ + ν of P Ir z e I ⊗ θ r , and • The rank ν of P Ir z P Jr z e I ⊗ e J .
Note in particular that P Ir z P Jr z has the same rank as the square of the righthand side of (51).
The form of P z in (77) is the most general element in N(µ, ν) up to partial fixing under K = SO(8) × SO(n) × Z 2 for the following reason: Recall that most of the discussion in subsection 4.1 was by using O(m), here we have so far used O(8) × O(n). If we were to use K it might seem that (77) still holds up to sign ambiguities in the bases. The mixed parity rotation in Z 2 makes this relevant only for the null basis (75) in C 8 . That is, we might need to replace e (1) 2i−1 + i e (1) 2i or e (2) 2i−1 + i e (2) 2i with its conjugate for at most one i. If µ + ν < 4 then this is not necessary, as one may find an even parity transformation, with one inversion in some complement to the basis (75) we use, which renders the basis (75) still valid for expanding P z . Finally, if µ + ν = 4 then the chirality of spinors Γ 12345678 AB ǫ B z = ǫ A z guarantees that supersymmetric elements in this class are also necessarily of the form (77).
However, we still have a lot of freedom in fixing the element under K. We are allowed to use U(µ + ν) ⊂ SO(8) on the basis (75), and U(µ) × SO(2ν) ⊂ SO(n) on (76). These groups act on the form of P z in (77) mixing the various coefficients but not changing the basis. We will now proceed to fix P z in the basis of (75) and (76) by using these groups.

Matrix factorizations
We will use both Takagi's factorization and a singular value decomposition on certain coefficients of P z . Takagi's factorization allows the diagonalization of a symmetric matrix MM T into a diagonal matrix D via the action of a unitary matrix S by using D = SMM T S T [15]. Note that the transpose of S is taken instead of the Hermitian transpose. The diagonalization is thus different than the spectral decomposition or diagonalization by a unitary matrix of a diagonalizable matrix. Takagi's factorization is always possible for symmetric matrices. Furthermore, the diagonal elements of D are real, non-negative. On the other hand, the singular value decomposition is the diagonalization of a not necessarily square matrix N under the action of two unitary matrices S 1 and S 2 by using N → S 1 NS † 2 , and it is always possible. The diagonal elements are again real and non-negative.
Consider the square of P z as a symmetric complex (µ+ν)×(µ+ν) matrix in the basis of {e 2j−1 + i e (1) 2j We use Takagi's decomposition by using the action of SU(µ + ν) so that MM T = D , (diagonal, real and positive) (79) We may assert that D does not have zero components because the rank of P Ir z P Jr z should be preserved under K C -conjugation 5 and is equal to the invariant ν. After this arrangement, the diagonal form of P Ir z P Jr z is preserved by at least U(µ) L ⊂ U(µ + ν) ⊂ SO(8) that acts on the e (1) 2i−1 + i e (1) 2i . The group that preserves P Ir z P Jr z might in fact contain an extra unitary group if the diagonal elements in D are not all different, but it is not necessary to take this into consideration. After performing Takagi's factorization, the full remaining symmetry is at least We have labeled the unitary subgroups with L (left) and R (right) to distinguish how they act on P z , whereas SO(2ν) ⊂ SO(n) has not been adorned. The condition MM T = D can be solved by partially fixing SO(2ν). We write where Σ is a ν × 2ν matrix which satisfies 5 More precisely, Takagi's factorization determines here the split of the basis into {e 2i ′ } ν i ′ =1 , but we have already assumed that the split is full rank on the first set. and on which U(ν) L acts on the left in the dual representation and SO(2ν) acts on the right. However, we need to mod out by the action of the symmetry of the theory, which is precisely the orthogonal group SO(2ν) acting on the right of Σ. By using SO(2ν) and a Gram-Schmidt orthogonalization we can fix Σ so that it is the upper-left block of the 4 × 8 matrix (85) This is the same decomposition we described in subsection 4.1. If Σ is degenerate, for instance if some of the parameters are zero, there is remaining freedom in SO(2µ) to further fix its form. This will turn out to be the case when we consider in section 5 the scalar coset integrability relation. We will then be able to fix Σ completely.
We still have a U(µ) L freedom acting on the basis e 2i−1 +i e 2i and a U(µ) R acting on the basisê (1) 2j−1 + iê (1) 2j . Their action does not spoil the form of M = √ DΣ with Σ described by (85), since we may always use a complementary SO(2ν) transformation. We use the singular value decomposition on N, N → S 1 NS † 2 with (S 1 , S 2 ) ∈ U(µ) L × U(µ) R , in order to make N diagonal, real, non-negative. We split the basis so that N is non-zero on the first µ a components and zero on the rest of the µ b components. There is some remaining symmetry in K, an anti-diagonal U(1) µa generated by and a U(µ b ), both of which act on the matrices A and B. We will not fix A and B though, because the equations of motion will eventually force A = B = 0 and µ b = 0.
We have (partially) fixed the most general element P z ∈ N(µ, ν) under the action of K, which can be summarized as follows: The class N(µ, ν) of an element P z is characterised by the rank µ + ν of P z and the rank ν of P Ir z P Jr z , in which case the element is expanded as in (77) in an adapted basis. The coefficients M and A in its expansion should satisfy the Takagi relations (79)-(81) and N should be diagonal, real, non-negative. At this point, we cannot prove that N is strictly positive, as it will turn out to be. There is some remaining symmetry acting on A and B and possibly on M from the right that we do not take advantage of. The basis we are using is in C n , but we will eventually show that µ b = 0 (so µ = µ a ) and drop the label a on which N is diagonal, real and strictly positive.

Solutions
In this section we impose the equations of motion on the scalar current P z whose form is now fixed in (77). We first show that the scalar connection Q z is also restricted in form because it has to act on P z and preserve the basis that we use for the latter. We may then turn to the coset integrability equations in order to show that the form of P z is further restricted, for instance it turns out that the matrices A and B must be zero. The equations of motion for P z and Q z reduce more and we finally arrive at our main result: All solutions decompose into solutions of Liouville and SU(3) Toda systems.

Restricting the connection
In order to restrict the possible values of Q z , we make a general analysis of the equation of motion of P z (26), which we rewrite using the notation '•': We will assume that the stabilizer of P z , is trivial. Hence we focus on those elements that act effectively on P z and enter (91). From (91) we calculate the equation of motion for D ∂z P Ir z P Jr Due to the Takagi decomposition, (93) involves only the diagonal, positive, real D and we may restrict The u(1) ν act on the e (2) and enter (93) in the form ∂zD + Qz| u(1) ν • D = 0, while the u(µ) acts on the e (1) 2i and do not enter (93). We turn to (91) again because, since we have restricted Qz| SO(8) to a unitary group as in (94), we may assert that The two factors of u(1) µa act on the positive components of the diagonal, real N e (1a) and the remaining u(µ b ) preserves the diagonal form of N in P z but acts on the A and B. Finally, there is a so(2ν) that acts on theê (2) r ′ and thus on M and A from the right. These are the most general subgroups that acting on P z should preserve the form of ∂zP z and should thus enter (91).
In summary, we have restricted the connection Q z to take values in Explicitly, we have the following expansion This may look intimidating at first, but we will soon show that µ b = 0 and the middle two lines are absent. The components in so(2ν) will also be restricted.

Integrability of the connection
Now we are ready to analyze the integrability equations for Q which will restrict P z even further. Calculating the right-hand side of the integrability equation (27) for Q IJ as (recall that N is diagonal, real and non-negative) − Im P Ir z P Jr 2j−1 + i e (1) 2j 2i−1 + i e (1) 2i 2i−1 + i e (1) 2i From the form of Q| SO (8) in (96) we deduce that Similarly, we calculate the right-hand side of the integrability equation (28) for Q rs as − Im P Ir z P Is From the form of Q| SO(n) in (97) we deduce that We first show that B = 0 and that A is further restricted. Multiplying (106) with M * from the left gives However, the Takagi relations (see (79) and (81)) are MA T = 0 and that MM T = D is invertible. Hence B = 0. We also have that N is invertible only in the first µ a diagonal components. With B = 0, (106) becomes A † N = 0, hence A ir ′ = 0 for i = 1, . . . , µ a . Using B = 0 in (104), one finds that and recall the Takagi condition (81) on A: We now turn to solving M, which will later lead us to A = 0. By using B = 0, (103) states that M * M T is diagonal, which after the Takagi relation M = √ DΣ becomes Recall that we have partly fixed Σ in (85) by (partly) using SO(2µ). The condition (110) is satisfied provided that the parameters in Σ sup (85) satisfy When these hold, Σ becomes degenerate and can be reduced to a nondegenerate block form by use of SO(2µ). In particular, we can reduce Σ to be of the form of a (ν r + ν c ) × (ν r + 2ν c ) matrix with values We may also write for the matrix D in which case M is now given by (116) We may now return to imposing both (108) and (109) with this particular M and we arrive at A = 0.
Let us summarise what we found: By using the general element P z ∈ N(µ, ν) given as (77), the form of Q z in (96) and (97), and the integrability equations for the connection Q, the right-hand side of which are in (102) and (105), the most general element P z up to the action of K is shown to be equal to where all the µ components N i are positive real and M i ′ r ′ is as in (116). Here, we have set µ b = 0 and dropped any label a from the basis e 2i . Indeed, with A = B = 0 the diagonal non-negative N should be strictly positive in order for P z to have rank µ+ν. The (ν r +ν c )×(ν r +2ν c ) matrix M has a special decomposed block diagonal form according to (116). We thus say that the element P z belongs to the refined class N(µ, ν r , ν c ), a filtering of the elements we were thus far considering in N(µ, ν).

Field equations and Toda blocks
The block form of M in (116) suggests that we refine the basis we are using. We split the basis as where by "rest" we mean those orthonormal basis vectors in R n that do not appear in P z . That is, we are now using the basis vectors e (1) in C n . The expansion of an element P z ∈ N(µ, ν r , ν c ) in this basis is As a matrix in the {e I ⊗ê r } basis of C 8 ⊗ C n , the element P Ir z e I ⊗ e r is block diagonal with µ, ν r and ν c blocks of (respectively) the type and trailing zeros 6 . The and − Im P Ir z P Is respectively. With this, we can now reduce the connection to be Any other component can be gauged to zero, because the curvature of the connection is non-trivial only in these components. We observe that the blocks decouple and we can solve the equations of motion separately for each sector N i , D i r and (D i c , ζ i ). We call each independent sector a Toda block. For the N functions we have from which we derive the Liouville equation We proceed by writing the equations of motion that involve D ∂zD i r + 2iq with integrability conditions These give the equations Finally, the equations of motion for the ζ i can be found from DzΣ = 0 and are while there is a remaining integrability equation,

General solutions
We are interested in solving the equations of the Toda blocks in a punctured bounded domain of the complex plane. The solution to the positive Liouville modes N i in (130) is Similarly, the solution to the positive D i r is The complex functions f i (z) and g i (z) are allowed here to be meromorphic. However, only simple poles of the f i and g i give smooth solutions in (130) and (135). A concrete answer on the nature of the singularities can be given by requiring finite coset space charge, which we do not analyze here. The solutions we presented above are a rewriting of Liouville's general solution such that the modes are manifestly positive. As such, the domain of the solution should not contain roots of 1 − |f i (z)| 2 = 0 or 1 − |g i (z)| 2 = 0. In order to solve the SU(3) Toda system, we should write it canonically. In particular, we should diagonalize the first-order equations for D i c and cosh ζ i . Define Their gauge-invariant equations of motion are derived from (132) and (137): while (139) becomes The connection q (c)i z can thus be found from (134) once we solve the above three equations andq (c)i z can be found from (146) if we have a solution for the Φ i 1 and Φ i 2 . We gauge fix (Φ i 1 , Φ i 2 ) to be real and positive. We can then eliminateq (c)ī z from the three equations: This has the form of the SU(3) Toda field equations where C ab is the SU(3) Cartan matrix.
A simple form for the general solution of the SU(N) Toda equation in twodimensional Minkowski spacetime and with negative coupling constant that is reminiscent of the Liouville solution was derived in [16] from Kostant's solution. We amend that solution for N = 3, Euclidean signature and positive coupling constant: Note that we keep the index i of the ν c copies. For G i (z) = 0 the solution indeed matches Liouville's. Similarly to the Liouville solutions, we may allow the functions to be meromorphic but restrictions should be applied to ensure that the coset charge is finite.
The full connection Q z can always be solved from the Toda block solutions of this section. What finally remains is the Einstein equation. Recall that its non-trivial component is given by (23) and allows us to solve for the conformal factor in the metric given by an exponential of ρ. Not only can ρ be solved for each Toda block, the Einstein equation is linear in the block decomposition: We have presented the Toda block solutions in the form ∂ z ∂z(· · · ) for this reason: the Einstein equation is thence easily integrated. By using the explicit solutions (140), (141), (150) and (151), the solution up to boundary terms is given by With this, we have locally found the metric (17) of the most general timelike supersymmetric solution. The scalar curvature can then be computed from R = 2e −2ρ ∂ z ∂zρ.
If the meromorphic functions are defined at infinity, in which case there are necessarily singularities elsewhere on the Riemann sphere, the function ρ will also have a well-defined limit at infinity. As an example let us look at the simplest solution, namely N(1, 0, 0), for the metric, but a similar analysis applies to the N(0, 1, 0) solution. The metric is of the form If f (z) has a simple pole only at the origin of the Riemann sphere, then f (z) = a + c/z. If we further choose c > 0 and a = 0, then the metric becomes We may consider then the exterior of r = c and the metric is manifestly asymptotically flat. We leave a more thorough analysis of the properties of the solutions for future work. We note that the half-BPS solutions of SO(8, n) with n > 2 are always given by the Toda blocks N(1, 0, 0), N(0, 1, 0) and N(0, 0, 1). Other examples are given in the following:  N(0, 3, 0), N(1, 1, 0) and N(0, 1, 1) that preserve respectively 2, 4 and 4 real supersymmetries. In each case, we need to fit the Toda blocks (124) in a 8 × n matrix P Ir z .
Example 4. The supersymmetric solution presented in subsection 5.2 of [6] is restricted to n ≤ 4. Since it has Q rs z = 0, we identify it initially with the N(0, ν r , 0) class. Then P Ir z is taken proportional to a constant matrix U ir , P Ir z ∼ P Ii U ir , with U † U = I n×n where P Ii is the null basis {e 2i−1 + i e 2i } 4 i=1 , see also the discussion around (199) in appendix A.2. The matrix U is thus effectively proportional to the n × n identity matrix and we identify 7 the solution with N(0, n, 0) and with all Toda fields D i r equal, that is D i r = D r for i = 1, . . . , n.

Discussion and comments
In this article we have classified and explicitly obtained all timelike supersymmetric solutions of three-dimensional half-maximal ungauged supergravity. The structure of the supersymmetric solutions that we found, which are in blocks of Liouville and SU(3) Toda systems, is new and surprisingly simple. With the null supersymmetric waves having already been solved in [6], all supersymmetric solutions of the ungauged SO(8, n) theory are now known.
It may at first seem surprising that the supersymmetric solutions of halfmaximal D=3 supergravity have only been classified and solved for more than 30 years after its construction in [8]. It is therefore of importance to trace our method and pinpoint its novelty. The classification under K C , as introduced first in [7], characterizes classes uniquely by two invariants: the rank µ + ν of P z and the rank ν of P Ir z P Jr z . When we refine this classification with respect to the real symmetry of the theory, these two invariants are preserved. One could do away with the detour into the indecomposable types of the complex group and with some work arrive at the same classes N(µ, ν r , ν c ) provided one uses the same two invariants. 7 The reduced equations of motion of [6] match with ours, as they should, provided we identify the fields ζ and g that appear there according to e ρ ζ = 2 √ D r and q (r)i z = gζ 2 , but we solve them essentially differently. Note also that in [6] the local coset symmetry SO(8) breaks into SO(2) × SO(6), whereas here it is broken to µ + ν r + ν c ≤ 4 copies of SO (2).
Given the elements of P z in these classes, and in particular due to the invariant ν, we were naturally led to the use of Takagi's factorization. This is a rather uncommon method compared to the spectral or eigenvalue decomposition that does not preserve the invariant ν. Furthermore, an eigenvalue decomposition or singular value decomposition on P z would have been impossible unless one enlarged the symmetry of the theory, for instance one might consider SO(2ν r ) → SU(2ν r ) or SO(n) → SU(n). The subsequent factorization of N that we employed by using the singular value decomposition comes as a concession, in the sense that we are manifestly allowed to use it after Takagi's factorization. We finally enforced the equations of motion, which further reduced the possible form of the coset representative.
The success of our method seems promising in employing it perhaps to the maximally supersymmetric supergravity. The classification under the complex local symmetry was already achieved in [7] and perhaps finding all elements up to the real local symmetry is possible. Certainly though, the SO(8, n) representations appearing here are easier to work with. Another interesting extension of our work is to examine interesting monodromies, similar to the reasoning in [7]. One now has the advantage that all solutions are known and requiring single-center monodromies is straightforward.
More generally, one would like to have a more thorough analysis of solutions to the Toda blocks and their geometric analysis. We have already noted that if the holomorphic functions are well-defined at infinity, then under some conditions one can conformally compactify the space that is now asymptotically flat. The fundamental BPS states, that are the non-smooth single-center solutions, are particularly interesting also for quantum considerations. Even classically, the smooth solutions of the theory are the smearing of the fundamental solutions, and a careful analytic study of the Toda blocks is lacking in our work.
Let us briefly comment on one more extension of our work. The success of our method might imply that it has a place in the non-abelian gauged version of the theory [9], in which a subgroup of the global SO(8,n) is gauged by Chern-Simons gauge fields. Although the gauged theory upon imposing supersymmetry possesses a corresponding structure, the starting equation (30) is deformed in such a way that the nilpotency argument can no longer be applied. It would be interesting to find a solution to this problem.
we use the vielbein θ 0 = dt, θ 1 = e ρ dx and θ 2 = e ρ dy. From we find the non-zero spin coefficient The Riemann curvature has non-zero component, in flat coordinates, the non-zero components of the Ricci tensor is and the Ricci scalar is R = 2e −2ρ ∂ k ∂ k ρ. By using the complex coordinate z = x + i y and ∂ z = 1 2 (∂ x − i∂ y ), we define the complex components (φ z , φz) for a one-form with φ t = 0, For two such one-forms, we have with the antisymmetric ǫ 12 = 1. That is, both the inner product and the wedge of the two one-forms φ and χ are recovered from the Hermitian product of complex functions φzχ z . We use the three-dimensional gamma matrices These satisfy the Clifford algebra {γ a , γ b } = −2η ab , they are real and satisfy γ 012 = γ 012 = 1. The Levi-Civita connection acting on spinors is For a real chiral spinor, we define complex coefficients as The complex coefficients have the property that Clifford multiplication by a two-dimensional one-form corresponds to With the image of φ z in the Clifford algebra φ z (γ 1 + iγ 2 ), the generator L = − 1 2 γ 12 acts on φ z → −iφ z and ǫ z → − 1 2 iǫ z . Equation (166) preserves the action as it should. The Levi-Civita connection becomes ∇ t ǫ z = ∂ t ǫ z and If we define the antisymmetric inner product by Requiring that two spinors ǫ z and ǫ ′ z do not square to a two-dimensional one-form is thus equivalent to ǫzǫ ′z = 0. We now introduce our notation for chiral spinors in S 8+ of Spin (8). We define the Clifford algebra matrices AḂ are antisymmetric with respect to the spin inner product and all matrices can be chosen to be real. The representation is chiral with Γ 12345678 = 1 on the real eight-dimensional spinors ǫ A ∈ S 8+ .
By using these conventions we have the spin equivariant map from the square of real chiral spinors into the Clifford algebra However, we are interested in complex chiral spinors in S C 8+ that are isomorphic to the real tensor product of spacetime spinors with the spinors in In the above equation, ǫ A α for α = 1, 2 are the spin coefficients for each A = 1, · · · , 8 and our previous conventions apply.

A.2 Basis of timelike spinors
In finding supersymmetric solutions of a theory, the form of the Killing spinor is usually fixed by using the symmetry of the theory. For our model this was done in [6]. In this work we have instead used the symmetry K to fix P Ir z . Furthermore, we do not need to explicitly solve for the Killing spinors because the integrability of the gravitino variation is guaranteed in our analysis. Here we present a few complementary details on the Killing spinors once we have fixed P z to a certain form.
We choose a representation of the Γ IJ AB matrices, such that the generators of the Cartan subalgebra Γ 12 AB , Γ 34 AB , Γ 56 AB , Γ 78 AB are block diagonal and proportional to (178) This follows from Darboux's theorem, or equivalently because a two-form in SO(8) decomposes into a sum of ∆ 0 (σ i , −σ i ) and ∆ − 0 (0) in table 1. Since they need to square to −1, the σ i are all signs. The choice of which of the commuting Γ IJ AB correspond to which of the Γ ±±±± is restricted by the following rule: Any two products should trace to zero and the product of the four should be proportional to the identity.
Up to reflections, there are only two choices for the signs σ i . This is to be expected since the two chiral algebras are not isomorphic. We freely choose 8 The condition iΓ 12 In fact, we can define a basis ǫ (±±±) by The basis satisfies manifestly the condition ǫ A z ǫ A z = 0 so squaring any two timelike spinors, (ǫ A , γ µ ǫ ′A ) will be zero for components in the µ = 1, 2 directions, see (171).
If a timelike background allows 8 real supersymmetries, the Killing spinors span the timelike spinor basis and we can fix a basis ǫ A (i)z such that each basis Killing spinor is proportional to one and only one of the ǫ A (+σ 1 σ 2 ) . The most general N = 8 Killing spinor is where F σ 1 σ 2 are functions of z. If a timelike background allows 4 real supersymmetries, this arises from the algebraic supersymmetry equation (30) imposing both iΓ 12 ǫ z = ǫ z and iΓ 34 ǫ z = ǫ z . The Killing spinor is now in the span of ǫ ++± and we can choose a basis of Killing spinors proportional to ǫ ++± , where F ± are functions of z. Finally 2 real supersymmetries mean that there is a single basis Killing spinor proportional to ǫ +++ . Forñ = 1, 2, 4 complex supersymmetries there is an action of SU(ñ) on the Killing spinors in the R-linear span of the ǫ A (i)z , which we now describe. First note that the matrix is diagonal and constant. We can use a constant GL(ñ, C) action ∆ → M∆M T in order to make it proportional to the identity. The matrices have some interesting properties.
Indeed, the right hand side is real and antisymmetric in A, B. The group SU(ñ) acts on the Killing spinor basis via spin rotations Under the SU(ñ) we can essentially bring any timelike Killing spinor to be proportional to ǫ +++ . The SU(ñ) action is important because we can make precise contact with other formulations. For instance, the so(8) element F IJ , which was called Ω IJ in [6], is given by the square of ǫ +++ : F = e 1 ∧ e 2 + e 3 ∧ e 4 + e 5 ∧ e 6 + e 7 ∧ e 8 . (199) One can then find the eigenstates of F IJ , which were called P Ii in [6], and are simply the null basis e 1 + i e 2 , e 3 + i e 4 , e 5 + i e 6 , e 7 + i e 8 .
We thus understand that the result of [6], that P z should be expanded in P Ii , is equivalent to our complex null basis e 2i−1 + i e 2i of the main text.

B Direct matrix factorizations of P z
We give here a direct analysis of how the form of P z can be fixed if we use K C or the maximal symmetry of (30), namely SO(8) C × GL(n, C), without the nilpotency argument. This gives an alternative proof that the real supersymmetries come in powers of two. We begin with two useful lemmas.
Proof. Consider a complex orthonormal basis {e i } µ i=1 of C µ with respect to the Hermitian inner product on C µ and its Hermitian dual and the u(1) µ generators are Their commutator is All of the generators of SU(µ) are thus generated from the group product U(1) µ SO(µ). On the other hand, the group generated preserves the Hermitian inner product on C µ so it cannot be larger than U(µ). Finally, we can assert that the group contains the non-special unitary U(1) and is fully U(µ).
Proof. The scaling R + is given by i L i , where we use L i and L ij of lemma 1. Clearly, all matrices in GL(µ, C) can now be generated (symmetric and antisymmetric, real and imaginary) similarly to (203).
Assume an element P z that admits some supersymmetry according to the algebraic supersymmetry equation (30). Multiplying the equation with P Is z Γ I and symmetrizing over (r, s) we derive P Ir z P Is z = 0 .
By using SO(8) C , we fix it as P Ir z e I ⊗ θ r with the group R + × U(1) 4 · SO(4) C = GL(4, C) acting on the left. The equality in (206) follows from lemma 2. The factors of U(1) come from the rotation e 2i−1 + i e 2i → i (e 2i−1 + i e 2i ), the SO(4) C is manifestly a subgroup of SO(8) C , and the scalings R + are the complex SO(2) C rotations that are not in SO (2). The element P z is represented by a 4 × n matrixP ir in (205) and inherited from SO(8) C × SO(n) C is the group GL(4, C) × SO(n) C acting onP ir by left/right multiplication. Similarly, the group inherited from SO(8) C × GL(n, C) acting onP ir is GL(4, C) × GL(n, C). The rank ofP ir is not necessarily full.
The action of GL(4, C) and permutations in SO(n) C can be used to rotate theP ir to one of the following forms 1 0 0 0 * · · · * 0 1 0 0 * · · · * 0 0 1 0 * · · · * 0 0 0 1 * · · · *     (207) for respectively 16, 8, 4, 2, 2 real supersymmetries. Stars signify here possibly non-zero elements. The reason why the upper-left square block is the identity matrix rather than a triangular matrix is because of the action of the stabilizers of one, two, three and four complex vectors in GL(4, C): In particular, the p copies of R 4−p (for p = 1, 2, 3) are translations that set the first p components of the next column to be fixed (the (p + 1)'th column) equal to zero.
The matrices in (207) describe the QR decomposition ofP ir with respect to GL(4, C) acting on the left. On the other hand, the group SO(8) C × GL(n, C) is such that all stars in (207) may be fixed to zero. The classification under SO(8) C × GL(n, C), the maximal symmetry of the supersymmetry equation, thus describes finite classes with each class representing uniquely a certain fraction of supersymmetry. Whichever of these two groups we use, or indeed if we use the factorization of P z under K with a similar method to the above, the real supersymmetry can be shown to come in powers of two.

C Supersymmetry in the Zariski topology
We review here result 2(a) of [7]. The proof is identical with minor changes. It states that if an orbit O of an element P z admitsñ supersymmetries, then elements in the closureŌ in the Zariski topology preserve at leastñ supersymmetries.
Let us first consider all elements P z that preserve at leastñ complex supersymmetries. The condition is that for at leastñ linearly independent spinors ǫ A z . Via the rank-nullity theorem the rank of the 8×8 matrix P Ir z Γ I AȦ is at most (8−ñ). All (9−ñ)×(9−ñ) submatrices of P Ir z Γ I AȦ should thus have vanishing determinant. The elements we are considering are roots of a finite number of homogeneous polynomial equations that define the set Cñ. The topology's closed sets are defined by such homogeneous polynomials and the sets Cñ are thus closed in the topology. Determinant theory implies that Cñ ⊂ Cñ′ if and only ifñ ≥ñ ′ . Take then an orbit O ⊂ Cñ of an element P z ∈ Cñ. Due to closure we haveŌ ⊂ Cñ and so elements inŌ and their orbits preserve at leastñ supersymmetries.

D Constructing normal forms D.1 Kostant-Segikuchi correspondence
Let us define θ C the Cartan involution of a real Lie algebra g. That is, the algebra decomposes as where k is the maximally compact subalgebra. We will eventually take g = so(8, n) and k = so(8) ⊕ so(n).
A standard triple {E, F, H} is an ordered set of elements in g or g C (depending on the context) that generate sl 2 and with canonical relations We define a Kostant-Segikuchi triple {E, F, H} in g to be a standard triple such that From this it also follows that θ C H = −H. We also define a Kostant-Segikuchi triple {e, f, h} in g C to be a standard triple such that From this it also follows that θ C h = h. The Kostant-Segikuchi correspondence establishes the correspondence between Kostant-Segikuchi triples in g up to the action of G and Kostant-Segikuchi triples in p C up to the action of K C . By an adaptation of the Jacobson-Morozov theorem, this is a correspondence between nilpotent elements in g up to the action of G and nilpotent elements in p C up to the action of K C : Explicitly, the correspondence is given by

D.2 Normal forms in g
Indecomposable types can be classified as follows: Let be the Jordan-Chevalley decomposition corresponding to an indecomposable element A ∈ L(V, τ, σ) and (A, V ) ∈ ∆. The definition of ∆ was given in section 3.2. Let the order of the nilpotent part N be p, that is N p+1 = 0 in the fundamental representation 9 . By proposition 3 in [13], it is true that KerN m = NV . We define the non-degenerate formτ onV = V /NV as which has symmetry |τ |(−1) p where |τ | is the symmetry of τ . By proposition 3 again, the restrictionĀ of S acting onV is well-defined, semisimple and (Ā,V ) ∈∆ is an indecomposable type of L(V ,τ ,σ). Proposition 2 in [13] asserts that Theorem 4. An indecomposable type ∆ is completely determined by p and ∆.
According to theorem 4, in order to classify indecomposable types ∆, what remains is to classify the indecomposable semisimple types∆ of certain linear algebras L(V ,τ ,σ). The semisimple types are labeled by their eigenvalues (ζ, · · · ) onV . We refer to [13] for further details and for a proof of the multiplicity of the eigenvalues. For the problem at hand, we have listed the indecomposable types of O(m, n) in table 1. In particular, we are interested in the nilpotent elements given in the last two rows of the table.
Although [13] does not list explicit normal forms, these can be easily constructed based on the proof of proposition 2 in [13] that extends lemma 2 in [13]: Theorem 5. Suppose A ∈ L(V, τ, σ) is such that its nilpotent part N has order p and NV = kerN p . Then there exists an S-invariant and σ-invariant subspace W such that is a sum of mutually disjoint subspaces with the following properties The conditions of theorem 5 are met for elements of an indecomposable type. It follows from theorem 5 that for two elements We remind the reader that the symmetry ofτ now also depends on p mod 2.
Assume then that we have identified the space W ⊂ V and that we specify the irreducible type∆ ofĀ that is S acting on W =V as an operator in L(W,τ , σ). The normal form of A = S + N acting on V can be constructed as follows: 1. the operator N is the ladder operation on V = ⊕ i N i W . It is left undetermined up to scalings of each ladder-step operation.
2. The normal form of S is given by extending 10 the action ofĀ from W to V . This is essentially the method we will use to write normal forms. That is, we identify W in an explicit basis V and construct S and N accordingly. We define appropriately a basis of V with the requisite signature of table 1 and inspect the left-hand side of (220). This allows us to identify the subspace W such that τ (φ(·), φ(·)) is non-zero only for τ (N i u, N p−i v) with u, v ∈ W . (221) All elements of V are of the form N iṽ , i = 0, 1, · · · , p. The inner product is τ (N kṽ , N p−kṽ ) = ±(−1) k k = 0, 1, · · · , p 2 − 1 We choose the null basis {η k ,η k , θ} with η k = N kṽ ,η k = ±(−1) k N p−kṽ , θ = N p 2ṽ . The inner product is thus non-zero on One can construct N using the fact that is a ladder operator The coefficient can be scaled freely. We will later choose appropriately so that N belongs to a KS triple.
With η k = N kṽ ,η k = (−1) k N p−kw , θ k = N kw ,θ k = (−1) k+1 N p−kṽ . The non-zero inner product is N is the ladder operator where the a k , b k and c are constants that can be scaled freely.

D.3 Kostant-Segikuchi triples in g
In order to construct normal forms for elements in p C up to the action of k C , we will use the Kostant-Segikuchi correspondence. We are thus interested in Kostant-Segikuchi triples in so (8, n). That is, we need to construct triples of the form {E, F, H} that satisfy the condition F = −θ C E. The construction in appendix D.2 used the simplest coefficients for a nilpotent part of an element N. By using boosts, we amend the normal form of a nilpotent element E such that E, F = −θ C E and indeed satisfy the standard sl 2 relations. This can always be done and it fixes the scalings of the ladder operators completely. We give the normal form of Kostant-Segikuchi triples here and using the Kostant-Segikuchi correspondence we give the corresponding nilpotent element in p C in the subsection D.4. There are two nilpotent complex types in p C : One inherited from the indecomposable type ∆ ± p (0) of signature ±(−1) p 2 ( p 2 + 1, p 2 ) with p even, and one from the indecomposable ∆ p (0, 0) of signature (p + 1, p + 1) with p odd.

D.5 Proof of theorem 2
In this section we prove theorem 2 on page 15. In order to facilitate our calculations, let us use the notation of Clifford multiplication vǫ of a vector v in Cl(8, 0) acting on a spinor ǫ of the Clifford module, and similarly for a higher-degree form.
where again e 1 and e 0 square to −1, while a 2 0 = p = p 2 . There is thus no supersymmetry allowed.
Proof of (d). Type ∆ 0 (0) ± corresponding to a spacelike or timelike R ⊂ R 8,n is such that e = 0. It imposes no supersymmetry restriction itself from the algebraic supersymmetry equation (30).
The sign here is that of b. If we choose the direction of r corresponding to the timelikeẽ, we arrive at the equation with solution ǫ = 0. If we consider ∆ − 2 (0) instead and use the orthonormal basis {e 0 ,ẽ,ê 0 } of R 2,1 , (265) is replaced by e = (∓ẽ ⊗ê 0 + ie 0 ⊗ê 0 ) .
The sign in this equation is again that of b. This is a single projection equation of the form i e 0 ∧ẽǫ = ±ǫ .
Each appearance of ∆ − 2 (0) implies a single projection equation that halves supersymmetry.