Strings on Semisymmetric Superspaces

Several string backgrounds which arise in the AdS/CFT correspondence are described by integrable sigma-models. Their target space is always a Z(4) supercoset (a semi-symmetric superspace). Here we list all semi-symmetric cosets which have zero beta function and central charge c<=26 at one loop in perturbation theory.


Introduction
A semi-symmetric superspace is a coset of a supergroup which possesses an additional 4 symmetry [1], thus generalizing the notion of ordinary, 2 invariant symmetric space. Sigma-models on semi-symmetric superspaces possess a number of interesting properties. Perhaps the main motivation to study them comes from the AdS/CFT duality. The holographic duals of superconformal field theories in diverse dimensions are string theories on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes. In many cases (and certainly in all maximally symmetric cases), the worldsheet sigma-models on such backgrounds are 4 cosets [2]. The best known example is the Green-Schwarz string action on AdS 5 × S 5 [3], which is a 4 coset of P SU(2, 2|4), the superconformal group in four dimensions 1 . One can define a Green-Schwarz-type sigma-model on any 4 coset. The 4 symmetry plays a crucial role in this construction by yielding the fermionic Wess-Zumino term in the sigma-model action [2].
A remarkable property of the Green-Schwarz-type 4 cosets is their classical integrability [5], which parallels integrability of bosonic symmetric-space sigma-models [6]. A Lax representation of the equations of motion in semi-symmetric cosets can be constructed using the 4 symmetry in a uniform, purely algebraic way 2 . Perhaps this is why integrability arises in the AdS/CFT correspondence.
All semi-symmetric superspaces are classified [1], and one can scan the list of the 4 cosets for potentially interesting integrable models, in particular for integrable string backgrounds. To be a string background, a 4 coset must satisfy two additional conditions: its beta function should vanish and it should have central charge c = 26.
After reviewing the construction of the Green-Schwarz-type sigma-model on a semisymmetric superspace, we will compute its beta function and central charge at one loop following [7,8,9]. Then we will list all cosets that satisfy the beta-function and the central charge constraints.

Sigma Model
A coset G/H 0 of a supergroup G is a semi-symmetric superspace if it is invariant under a 4 symmetry, generated by a linear automorphism Ω of the Lie algebra of G, Ω : g → g, Ω([X, Y ]) = [Ω(X), Ω(Y )], Ω 4 = id. The superalgebra g then admits a 4 decomposition: which is consistent with the (anti-)commutation relations: [h n , h m } ⊂ h (n+m) mod 4 . The subspace h n consists of the elements of g with the 4 charge n: The "supertrace" Str(· ·) denotes the G and 4 invariant bilinear form on g, and κ is the sigma-model coupling (κ 2 = 2πα ′ /R 2 , where R is the radius of G/H 0 ). The equations of motion for this action admit a Lax representation [5] making the world-sheet sigmamodel classically integrable. The expansion of the Lagrangian in (2.4) around g = 1 (the flat-space limit) has the form ∂X∂X +θ∂X∂θ typical for the Green-Schwarz superstring [10]. And indeed the Green-Schwarz action on many AdS backgrounds can be described as (2.4) for various 4 cosets [3,11,12,13,14,15,8,16,9,17,18,19,20]. Just like the ordinary Green-Schwarz action, (2.4) may possess local fermionic kappa-symmetries which, in effect, means that some of the fermion dimensions are unphysical and have to be removed by an appropriate gauge fixing prior to quantization. The rank of the kappa-symmetry depends on the structure of the coset and will be computed in sec. 4 for all sigma-models with the vanishing one-loop beta-function.
To illustrate these points and to set up the stage for the subsequent one-loop calculations, let us expand the action (2.4) around an arbitrary bosonic background 5ḡ (x), introducing the following notations for the background currents: (2.5) 3 These terms describe coupling to the metric and to the RR fields. In certain cases it should be possible to switch on the B-field (the theta-term in the sigma-model action) or its field strength (the bosonic Wess-Zumino term). For example, if the denominator of the coset contains a U (1) factor, it is possible to add a theta-term iϑε µν ∂ µ J U(1) ν 0 . 4 We consider the Euclidean worldsheet, which is why the second term in the Lagrangian is multiplied by i. After the Wick rotation the action becomes real. 5 The background-field calculations have been done for the Green-Schwarz-type cosets [8], as well as for many related pure-spinor type sigma-models [2,21,22,23,9,24].
Here A µ is the background gauge field. We will denote by D µ the background covariant derivative: D µ = ∂ µ + [A µ , ·], and by F µν the background field strength: The currents A µ and K µ are assumed to satisfy the classical equations of motion: where ∇ µ K ν = D µ K ν + Γ ν µλ K λ and Γ ν µλ are the Christoffel symbols of the worldsheet metric. The first two equations are identities that follow from the flatness of the current g −1 ∂ µḡ . The equations of motion for the metric are the Virasoro constraints: where K ± µ are the chiral components of K µ : In order to expand around the classical backgroundḡ(x) we choose the coset representative in the form g =ḡ e κX , Under gauge transformation that also act on the background field:ḡ →ḡh, X transforms in the adjoin: X → h −1 Xh. It is straightforward to plug the coset representative (2.9) into the action and expand the latter in the powers of the coupling κ. The current (2.3) expands as where the long derivative D µ is defined by Unlike the covariant derivative D µ , which commutes with the 4 grading, the long derivative D µ does not have definite 4 charge. Thus, (D µ X) n = D µ X n for any n and (D µ X) 2 = D µ X 2 , but (D µ X) 1,3 = D µ X 1,3 + [K µ , X 3,1 ]. Plugging the expansion (2.10) into the action (2.4) and using the identities which follow from the equations of motion (2.6), one can bring the quadratic part of the action to the form where the chiral projections of a vector are defined in (2.8). In the conformal gauge (h µν = e φ δ µν ), the quadratic part of the Lagrangian becomes: where holomorphic and anti-holomorphic vector components are defined as D = D 1 +iD 2 , D = D 1 − iD 2 , and similarly for K. The fermion fluctuations of the worldsheet couple to the background currents, and if the currents vanish the fermion kinetic terms vanish too. Even if the background currents do not vanish, the Dirac operator may have zero modes, because the Lagrangian depends on X 1 (X 3 ) only in the combination [K, X 1 ] ([K, X 3 ]). IfK (K) has a nonempty commutant in h 1 (h 3 ), the Lagrangian degenerates and simply does not depend on the fermionic fluctuations in the corresponding directions. This is a manifestation of the κ-symmetry, a local fermion gauge invariance that has to be fixed in order to have well-defined perturbation theory.
The most simple and natural way to fix the kappa-gauge is to set to zero those components of X 1 and X 3 that drop out from the action anyway. These components are proportional to the Lie algebra generators from h 1 and h 3 which are annihilated by the adjoint action ofK or K. The rank of the κ-symmetry is the number of such generators: where K andK are sufficiently generic null elements of h 2 . The null condition follows from the Virasoro constraints The number of zero modes N κ or Nκ does not depend on the particular choice of K andK provided that they are sufficiently generic. For special (non-generic) classical solutions, the kappa-symmetry gauge condition may further degenerate. This is known to happen in AdS 4 × CP 3 [18]. However these degenerate cases occur on the surface of non-vanishing co-dimension in phase space. In the bulk of the phase space (for generic classical solutions) the rank of the kappa-symmetry is background independent, and is determined by the structure constants of the Lie superalgebra g.

Beta Function and Central Charge
To compute the central charge and the beta-function of the sigma-model, we integrate out X n , n = 1, 2, 3 in (2.12) and study the dependence of the effective action on the background currents and the 2d metric. The beta-function is determined by the logdivergent contribution to the unique dimension two operator: √ hh µν Str K µ K ν . The central charge is determined by the standard conformal anomaly. Since the beta function and the central charge are governed by different terms in the effective action, they can be computed separately. The beta function arises from the insertions of the mass operators K 2 X 2 2 and K 2 X 1 X 3 in the one-loop diagram and can be calculated in the conformal gauge. The central charge arises due to the short-distance anomaly in the fluctuation determinants and is insensitive to the masses. In computing the central charge the masses can thus be omitted, after which the Lagrangian (2.12) reduces to that of the Green-Schwarz string in flat space in the semi-light-cone gauge, the central charge for which was computed in [25].
The one-loop effective action in the conformal gauge is Here we used that −D µ D µ + ad K µ ad K µ = −DD + ad K adK because of the identity [D µ , D ν ] = ad F µν = −[ad K µ , ad K ν ] satisfied by the background currents in virtue of the equations of motion (2.6). The prime in Sp ′ 1⊕3 means that the zero eigenvectors of adK (ad K) in h 1 (h 3 ) should be omitted. They are eliminated by fixing the kappa-symmetry gauge.
The log-divergent contribution to the beta-function comes from the two diagrams in fig. 1. The bosonic contribution is easy to compute: The fermion contribution requires more care because of the kappa-symmetry projection. The Dirac operator in (3.1) can be factorized as where we used that [D, adK] = 0 = [D, ad K] due to the equations of motion. The Dirac operator acts on h 1 ⊕ h 3 , so the factor 0 ad K adK 0 is just the kappa-symmetry projector, up to proportionality factor. The fermion contribution to the effective action thus is given by Expanding to the second order in ad K, adK, we find: The prime in the trace is omitted in the second line because the integrand is proportional to the kappa-symmetry projector. Now, so it might seem that fermions renormalize also the operator Σ ab ε µν K a µ K b ν , where Σ ab is an anti-symmetric invariant tensor on h 2 . However, this operator is a total derivative, its variation being proportional to ε µν D µ K ν = 0, and integrates to zero.
Adding together bosonic and fermionic contributions we find: Finally, recalling that K µ ∈ h 2 and thus ad K µ maps h 2 to h 0 and vice versa, we find: Hence we can replace 2 tr 2 − tr 1 − tr 3 in (3.4) by tr 0 + tr 2 − tr 1 − tr 3 = Str adj . If we denote the Hermitian generators of h 2 by T a and introduce the metric on the bosonic section of the coset: the one loop beta-function is where f A BC are the structure constants of g. The beta-function is thus proportional to the Killing form. The same one-loop beta-function arises in the pure-spinor-type cosets [2,22,9], the supergroup principal field [26,27], and in the 2 cosets of supergroups [28,29]. The condition for the one-loop beta-function to vanish is that the Killing form of g vanishes 6 .
The calculation of the central charge for the Green-Schwarz string requires careful regularization of the integration measure [31], and yields the following result [25,8]: the bosons have central charge 1; the left (right) moving fermions contribute 2 to the left (right) central charge. In our case, X 3 and X 1 are, respectively, left and right movers so, in total, The average central charge, c = (c L + c R )/2, is determined by the dimension of the coset and the full rank of the kappa-symmetry: The central charge is manifestly positive, in contradistinction to the non-unitary 2 supercosets, which can have negative central charge [29] 7 . By an explicit calculation we will find that in all conformal cosets c L = c R . We can thus make no distinction between c, c L and c R . We will be also interested in the case when an external CFT is added to the coset. At first sight, this cannot change the central charge counting, because the coset and the external CFT interact only via 2d metric which does not carry dynamical degrees of freedom and can be eliminated by fixing the conformal gauge. However, this is not quite true. Adding an external CFT can partially or completely break the kappa-symmetry. The kappa-symmetry transformations act on the 2d metric and since the latter enters the action of the external CFT, kappa-symmetry gets broken. In the conformal gauge, the kappa-symmetry breaking can be attributed to the violation of the null condition for the currents (2.15), which does not hold in the presence of another CFT with a non-trivial energy-momentum tensor 8 .
The ranks of the left-and right-moving kappa-symmetries with the null condition relaxed will be denoted byN κ ,Nκ. They are computed by the same formulas (2.14) where K andK are now the most general elements of h 2 , not necessarily null. We will denote the central charge of the sigma-model coupled to an external CFT (the extrinsic central charge) by 9ĉ : In the next section we will compute extrinsic and intrinsic central charges for all conformal 4 cosets.

Conformal Sigma Models
The string sigma-model must be defined on a real superspace, so the symmetry algebra g should be a real Lie superalgebra. However, the one-loop beta-function and the central charge depend only on the structure constants of g and therefore are the same for all real forms of a given complex superalgebra. Dealing with complex Lie superalgebras is technically simpler, and subsequent analysis will be done as if g were complex. We will pick a particular real form in the very end. If we want to have a string interpretation of the sigma-model, the real form must be such that the metric (3.5) has the Minkowski signature (− + . . . +). In the cases when the requisite real form does not exist, we will keep in mind the compact form of the coset with the (+ . . . +) metric. The basic complex Lie superalgebras with vanishing Killing form form two infinite series: psu(n|n) and osp(2n + 2|2n) [30]. The one-parameter family of exceptional superalgebras d(2, 1; α), a continuous deformation of osp(4|2), also has vanishing Killing form. But since the deformation parameter appears only in the anti-commutator of supercharges, the central charge counting for d(2, 1; α) is the same as for osp(4|2) and we need not discuss d(2, 1; α) separately, just keeping in mind that any OSp(4|2) coset can be generalized to D(2, 1; α).
From the discussion above we see that there are two series of conformal sigma-models on semi-symmetric superspaces, those with P SU(n|n) and OSp(2n + 2|2n) symmetry, which we will call type-U and type-O models. All possible 4 automorphisms of psu(n|n) and osp(2n + 2|2n) and the corresponding cosets were classified by Serganova [1]. They fall into six separate classes, four type-U and two type-O, conveniently described with the help of the supermatrix representation of the su(n|n) and osp(2n+2|2n) superalgebras 10 : Here J is the 2n × 2n matrix We will also need the diagonal matrix 4) and the following supermatrix operations 11 : These three operations and the adjoint action of the matrices J and I p allow one to build all possible 4 automorphisms of su(n|n) (table 1) and osp(2n + 2|2n) (table 2) [1]. The 4 decomposition corresponding to these cosets is described in more detail in the appendix. Coset Coset Almost all of the cosets in tables 1 and 2 can be used to define the action of a sigmamodel, except for type-U3. Any element of h 2 in a type-U3 coset is null, at least in the usual supertrace metric (the explicit 4 decomposition is given in sec. A.3). The bosonic part of the action then vanishes identically. Although we will formally compute the central charge for this coset, we will not discuss this sigma-model any further.
In addition to the models based on simple Lie superalgebras one can also consider the cosets of product groups. Such cosets naturally arise in the AdS 3 /CF T 2 correspondence, because the conformal algebra in two dimensions is a direct sum of two Virasoro algebras acting independently on the left and right movers. Independently of the AdS/CF T connection, the product structure is quite natural from the point of view of the coset construction, as it generally admits a 4 action. If p is a superalgebra, we can define a One can easily check that Ω([X, Y ]) = [Ω(X), Ω(Y )] for any X, Y ∈ p ⊕ p. It is also obvious that Ω 2 = (−1) F and thus Ω 4 = id. The invariant subalgebra of the 4 action is the bosonic diagonal h 0 = {(X, X)|X ∈ p}. Consequently, the supercoset is P × P/H 0 , where H 0 is the bosonic subgroup of P diagonally embedded in P × P . The bosonic section is the group manifold of H 0 . We refer to the tensor-product semi-symmetric spaces as type-T u cosets if P = P SU(n|n) and type-T o cosets if P = OSp(2n + 2|2n). There are also interesting cosets of U(n|n) [33], which we will not consider here.
We will calculate the one-loop central charge for the eight types of semi-symmetric cosets introduced above (type-U1-4, type-O1,2 and type-T u,o). The central charge counts the number of degrees of freedom in the sigma-model and depends on the rank of the kappa-symmetry (3.7)-(3.9), which in turn is given by the dimension of the commutant of a generic element of h 2 (2.14). The calculations reduce to simple algebra, but have to be done case by case. The details are given in the appendix, here we just describe  Coset N κ type-U1 (p = 1, q = 1) n 2 − 4n + 6 type-U2 (n = 2) 2 type-U4 (n = 4) 8 type-O1 (n = 1, p = 1) 1 type-O2 (n = 2, p = 1) 4 type-T u (n = 2) 4 the general pattern that emerges: • The left-and right-moving kappa-symmetries, which are associated with the h 3 and h 1 subspaces, are identical in almost all the cases. One and only exception is the type-U3 coset, for which h 3 and h 1 are not isomorphic and have different dimensions. The kappa-symmetry compensates for this, such that even in this case the left-and right-moving central charges are equal.
• The extrinsic kappa-symmetries follow a regular pattern and depend uniformly on the dimensionalities of the superalgebra and the coset (table 3) 12 .
• There is no difference between intrinsic and extrinsic kappa-symmetries and central charges in most cases, but in low ranks there are exceptions listed in table 4. Imposing the Virasoro constraints then increases the rank of the kappa-symmetry and decreases the central charge.
Many of the cosets above have been discussed in the context of the AdS/CFT duality. The first coset, supplemented by an external S 1 , describes the Green-Schwarz string on AdS 3 ×S 3 ×S 3 ×S 1 with completely fixed kappa-symmetry [20]. The action of the second coset can be interpreted as the 6d Green-Schwarz action on AdS 3 ×S 3 [11,35,12,14,16,9] and as such admits rank-eight kappa-symmetry (table 4). However, coupling to an external T 4 , which is necessary to compensate for the central charge deficit, breaks kappasymmetry and changes the central charge counting. This coset plus four compact bosons describes the Green-Schwarz string on 13 AdS 3 ×S 3 ×T 4 with fully fixed kappa-symmetry [20]. The fifth coset yields the 4d Green-Schwarz action on AdS 2 × S 2 [13,2,9]. Again, its (four-parameter) kappa-symmetry is completely broken by coupling to an external c = 14 CFT. The models 3, 4, 6 and 7 are seemingly new. The last two models are similar to the OSp(1|2)/U(1) coset considered in [15] -they have AdS 2 as the bosonic target space and no physical degrees of freedom on shell. The latter is due to kappasymmetry. Coupling of these sigma-models to an external CFT breaks kappa-symmetry and revives their fermion degrees of freedom. Other

Conclusions
The list of semi-symmetric superspaces potentially consistent as string backgrounds is not very long. We should stress that we have computed the beta function and central charge only at the one loop level. There is no guarantee that higher-order corrections identically vanish, and the list of consistent string backgrounds with the 4 symmetry may be even shorter. It is instructive to look at what happens in the principal chiral models and 2 cosets of supergroups. In the case of the principal chiral field, it is possible to prove finiteness to all orders in perturbation theory for the cosets with the vanishing one-loop beta-function [27,29]. Many one-loop finite 2 cosets are two-loop finite as well [29], but the full set of conformal 2 cosets seems to be smaller than the set of 2 cosets with vanishing one-loop beta-function [37]. The semi-symmetric cosets with non-zero beta-function can also be interesting for the AdS/CFT duality, if they are asymptotically free. A sigma-model with the AdS target cannot develop a mass gap because of the non-compactness. The asymptotic freedom at weak coupling then suggests that the beta-function has a non-trivial zero, which can potentially be interpreted as string theory on the AdS space of fixed radius [8] 14 .
The consistent Minkowski backgrounds, critical (5.1) and non-critical (5.2), all involve an AdS factor and are potentially dual to CFTs in dimensions d 4. In all these cases the worldsheet sigma-model is integrable and thus potentially solvable by Bethe ansatz. For the string sigma-models on AdS 5 × S 5 and AdS 4 × CP 3 the classical algebraic curve [38], the worldsheet S-matrix [39] and the asymptotic quantum Bethe equations [40] are known. The finite-volume TBA/Y-system solution is now also available [41]. It would be interesting to derive a unifying Bethe-ansatz solution for a generic semi-symmetric coset.

A Rank of kappa symmetry
In this appendix we compute the rank of the kappa-symmetry for all conformal 4 cosets. According to (2.14), the rank is equal to the dimension of the commutant of a generic element K(orK) ∈ h 2 in h 1 and h 3 . In the supermatrix representation, Commuting this with an odd element of the superalgebra, we find: The commutator vanishes if The number of solutions to these equations determines the rank of the kappa-symmetry. The dimension of the solution space for generic A and B determines the rank of the extrinsic kappa-symmetryN κ ,Nκ. To compute the rank of the intrinsic kappa-symmetries N κ , Nκ, one should in addition impose the Virasoro constraints (2.15). The matrices A and B then satisfy Throughout the calculation we will use a number of simple algebraic facts, which we collect below. Consider an equation for an m × n matrix X: where M and N are given quadratic matrices, which we assume to be sufficiently generic.
Since a generic matrix can be diagonalized by a similarity transformation, without loss of generality we can assume that M and N are diagonal. Denoting their eigenvalues by µ i , i = 1, . . . , m and ν a , a = 1, . . . , n, we find In analyzing the spectrum of various matrices we will repeatedly use the Stenzel theorem [42], which states that the non-zero eigenvalues of a product of two anti-symmetric matrices are doubly degenerate. Namely, the spectrum of an n × n matrix M = A 1 A 2 , where A t i = −A i , consists of [n/2] pairs of eigenvalues µ 1 , µ 1 , . . . µ [n/2] , µ [n/2] and, if n is odd, an additional zero eigenvalue associated with the vector annihilated by A 2 .

A.1 Type-U 1
The 4 automorphism of the type-U1 coset acts on the supermatrices as where I p , I q are defined in (4.4). The 4 decomposition in the supermatrix representation is given by Let us assume that n − p p and n − q q. The non-zero eigenvalues of the matrix A form p pairs ±α 1 , . . . , ±α p , where α 2 i are the eigenvalues of the p × p matrix A 1 A 2 . In addition A, has n−2p zero modes built from (n−p)-dimensional vectors v i annihilated by A 1 . Analogously, the right action of B produces 2q non-zero eigenvalues ±β 1 , . . . , ±β q , whose squares β 2 j are eigenvalues of B 1 B 2 , and n − 2q zero modes made of left n − q dimensional null vectors of B 2 , which we denote by u j . There are (n − 2p)(n − 2q) pairs of zero eigenvalues of A and B. In h 1 , they correspond to (n − 2p)(n − 2q) solutions to (A.2) of the form In general, this condition is too weak and does not imply any degeneracies. The only exception is p = 1 = q, when each of the matrices A and B has only one pair of non-zero eigenvalues. The eq. (A.12) then implies that these eigenvalues coincide up to a sign. We thus find two extra solutions to (A.2) in both h 1 and h 3 . In h 1 , the solutions are and This solutions exist provided that A 1 A 2 = B 1 B 2 , which for p = 1 = q is a consequence of the Virasoro constraints. Hence, The 4 automorphism of the type-U2 coset is which gives the following 4 decomposition: To find the rank of the kappa-symmetry, we need to solve the equation

A.3 Type-U 3
The 4 automorphism in this case is which gives the following 4 decomposition: This case is rather special, because h 1 and h 3 have different dimensions: dim h 1 −dim h 3 = 2n, potentially leading to the mismatch of the central charges of left and right movers.
We will see that this mismatch is precisely compensated by the kappa-symmetry. The equation (A.2), that determines the rank of the kappa-symmetry, becomes: The solutions of this equation in symmetric matrices giveNκ, the number of solutions in anti-symmetric matrices determinesN κ . Without loss of generality we can assume that A is diagonal: A = diag(a 1 , . . . , a n ). Then, In general, all the eigenvalues are different, and the solutions are diagonal matrix elements Θ ii and Ψ ii . All of them belong to h 1 , thus giving: The kappa-symmetry eliminates the extra right-moving degrees of freedom and reinstalls the balance of central charges: c L = c R . The Virasoro condition does not impose any new constraints, since the equation tr A 2 = tr B 2 automatically holds for any element of h 2 .

A.4 Type-U 4
The type-U4 cosets are defined for P SU(n|n) with even n. We can also assume that n > 2, as h 2 is empty for 15 n = 2. The 4 symmetry acts as .26) and leads to the 4 decomposition The second equation in (A.2) is a consequence of the first for this coset. To find the rank of the kappa-symmetry, we just need to solve AΘ = ΘB. (A.29) Non-trivial solutions to this equation correspond to pairs of equal eigenvalues of A and B.
A matrix that satisfies the condition (A.27) can be represented as a product of two anti-symmetric matrices: A = −(AJ)J, and consequently has doubly-degenerate spectrum, by Stenzel theorem. In addition, A and B are traceless and so have n/2 − 1 independent eigenvalues. In general these eigenvalues have no reasons to coincide. Consequently, the coset has no extrinsic kappa-symmetries: the other will automatically follow. The 4 automorphism of the type-O1 cosets in the supermatrix representation (4.2) acts as follows: The associated 4 decomposition is The kappa-symmetry condition (A.32) in h 1/3 reduces to Then Θ 2 = −A t 1 Θ 1 B −1 ± , and we are left with the equation for the p × n matrix Θ 1 . In general this equation has no solutions and, consequently, there will be no kappa-symmetries:N κ =Nκ = 0, (A. 38) because −A 1 A t 1 and B ∓ B ± have different eigenvalues for generic matrices A 1 , B 1 , B 2 . The null condition (A.3) relates the sums of the eigenvalues, because This is still insufficient for the eigenvalues to coincide, except for the special case of p = 1, n = 1. Then both −A 1 A t 1 and B ∓ B ± are numbers rather than matrices, which must coincide once the trace condition (A.3) is imposed. We thus find one solution in h 1 and one in h 3 : There is also an extremely degenerate case of p = 0, n = 1. The target space then is AdS 2 , without any extra factors. There are no propagating bosonic degrees of freedom. The number of kappa-symmetries, and consequently the number of fermionic degrees of freedom, depends on whether the string is left-or right-moving in the target space. In one case, there is no kappa-symmetries, and in the other case the kappa-symmetry removes all the fermions: N κ = 4 = Nκ. The string then is purely topological.

A.6 Type-O2
The 4 automorphism in this case acts as (A.41) It is convenient to work in the basis in which . (A.42) The 4 decomposition in this basis takes the form (we assume that n − p p): The zero-mode equation (A.32) in the h 1/3 subspace boils down to a system of two equations for matrices Θ 1 , Θ 2 : We need to distinguish even and odd n. Consider first odd n. The anti-symmetric matrices A ± are then non-degenerate and we can express Θ 2 through Θ 1 : Θ 2 = A −1 ± Θ 1 B 1 , substitute the result into the first equation and get: Both A ± A ∓ and (B 1 JB t 1 )J are products of two anti-symmetric matrices, their spectra are thus degenerate and contain, respectively, (n + 1)/2 and p different eigenvalues: α 1 , α 1 , . . . , α (n+1)/2 , α (n+1)/2 and β 1 , β 1 , . . . β p , β p . These eigenvalues are in general different. Hence there are no non-trivial solutions for Θ 1 , and there are no kappa-symmetries: The only exception is the degenerate case of n = 1, p = 0, which is completely analogous to the type-O1 coset with n = 1, p = 0, discussed at the end of sec. A.5. If n is even, the matrix A ± has one zero eigenvalue: The solutions due to the null eigenvalues are: Potentially, there may also be solutions due to coincident non-zero eigenvalues of A and B, which are given by the equation (A.45). This requires coincidence of some eigenvalues of A ± A ± , α 1 , α 1 , . . . , α n/2 , α n/2 , 0, and B 1 JB t 1 J, β 1 , β 1 , . . . , β p , β p . In general, this does not happen, and thusN and there are no exceptional cases.