On the supersymmetric solutions of the Heterotic Superstring effective action

We consider the effective action of the Heterotic Superstring to first order in α′ and derive the necessary and sufficient conditions that a field configuration has to satisfy in order to admit at least one Killing spinor using the spinor bilinear method in an arbitrary spinorial basis and corresponding arbitrary gamma matrices. As a previous step in this derivation, we compute the complete spinor bilinear algebra using the Fierz identities, obtaining as a by-product the algebra satisfied by the Spin(7) structure contained in the bilinears in an arbitrary basis. We find the off-shell relations existing between the bosonic equations of motion evaluated on supersymmetric field configurations using the Killing Spinor Identities instead of the (far more complicated) integrability conditions of the Killing Spinor Equations as it is common in the literature. We show how to include the Kalb-Ramond’s Bianchi identity in the Killing Spinor Identities.

JHEP06(2020)106 The construction and study of the classical solutions of a theory always provides a great deal of information about its properties and predictions. The fundamental rôle played by the Schwarzschild solution in the conceptual development of General Relativity, as well as in more mundane computations of testable predictions of this theory, is a very clear example that cannot be overstated. For these reasons, the construction and study of solutions of the Superstring Theory effective action (compactification backgrounds, pp-wave backgrounds, black holes, cosmological models) has been a very active area of research for almost 30 years and it is not surprising that some of the solutions found, such as the 3-charge black hole [1,2] used by Strominger and Vafa to compare the Bekenstein-Hawking entropy with that obtained by the first microstate counting in ref. [3], have also had a huge impact in the development of Superstring Theory. Near-horizon geometries, pp-waves and other Penrose limits of solutions provide further examples. The methods used to construct new solutions of Superstring Theory based on dualities and supersymmetry have probably been the most fruitful ones. Dualities transform solutions into solutions, sometimes with very different properties. The original solutions are required to satisfy only a minimal number of conditions such as the existence of isometries for T-duality. In contrast, supersymmetry methods can only be used to construct supersymmetric solutions, but, in general, they provide very general recipes that permit the construction of very general families of supersymmetric solutions such as all supersymmetric black holes of a given Superstring effective field theory (in the end, a supergravity theory). In Superstring Theory, supersymmetric solutions often describe the fields generated by non-perturbative extended objects such as Dp-branes and provide a way to learn more about them. Supersymmetric compactification backgrounds are an essential ingredient of many superstring phenomenological models as well. But supersymmetric solutions are also interesting in their own right because, often, they involve structures and enjoy properties of great physical and mathematical relevance. All this justifies the great deal of effort employed in the characterization and classification of all supersymmetric solutions of Superstring Theory via the characterization and classification of all supersymmetric solutions of all supergravity theories.
This effort started with the pioneering work of Gibbons, Hull and Tod [4,5] in pure N = 2, d = 4 supergravity. This theory is just the simplest of the very rich family of N = 2, d = 4 supergravities which have different matter contents (vector multiplets and hypermultiplets) and couplings (some of them associated to gaugings of their global symmetries). All of them have been studied from this point of view in a long series of papers [6][7][8][9][10][11][12][13] of increasing complexity using the "spinor bilinear" method of ref. [14], which we will also use and explain in this paper. The most general case, considered in ref. [13] has only been solved for the "timelike" case and the "null" supersymmetric solutions of theories with non-Abelian gaugings still have to be characterized. The supersymmetric solutions (both timelike and null) of the pure N = 4, d = 4 theory have also been characterized in refs. [15,16], but neither the matter-coupled nor gauged theories have been studied. 1

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Finally, since it is possible to treat all 4-dimensional supergravities with vector multiplets in a unified form, all their timelike supersymmetric solutions were characterized in a unified form in [17]. The null case and the gauged theories remain to be studied.
For the sake of making this short review of what has been accomplished in this field of research in d ≤ 6 complete, let us also mention the work done in maximal and half-maximal d = 3 supergravities in refs. [31][32][33][34] and also N = 1, d = 4 supergravity in refs. [35,36]. For a review on this topic and additional references on related work, see refs. [37,38].
In dimensions higher than six there are only supergravities with 16 or 32 supercharges. Many of them can be obtained via dimensional reduction from the 10-and 11-dimensional theories and, therefore, most of the work has been focused directly on these. Pure N = 1, d = 10 supergravity can be viewed as the effective field theory of the Heterotic or the Type I Superstrings (depending on the stringy interpretation of the supergravity fields) at lowest order in an expansion in terms of the Regge slope parameter α . One of the most important features of these theories is the presence of massless gauge vectors in their spectra. These occur at first order in α in the effective action, but there is no problem to accommodate them in N = 1, d = 10 supergravity as vector supermultiplets [39,40]. However, at this order in α , the Heterotic Superstring effective action contains more terms which can only be accommodated in N = 1, d = 10 supergravity if more terms are also included to preserve invariance under supersymmetry transformations at a given order [41,42]. The additional terms are of higher order in derivatives as well, which leads to very complicated equations of motion.
From the heterotic superstring effective action point of view, the neglect of the α corrections has to be justified in each particular solution. In general, this imposes constraints on the charges of the solutions or signals (typically high-curvature) regions of the solutions which cannot be trusted as good string theory solutions because they are bound to be modified once the neglected terms in the theory are reconsidered. For this reason, from the Superstring Theory point of view, it is important to take into account these possible complicated α corrections in the analysis from the onset.
However, even though the corrections to the equations of motion are very involved, it so happens that the Killing Spinor Equations (KSEs) only get "implicit" first-order α corrections and can be analyzed at zeroth or first order at the same time. For these reasons, in the first systematic studies of supersymmetric solutions refs. [43][44][45] the KSEs were solved at zeroth/first order but involving only the zeroth-order equations of motion. 3

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In all these works, the so-called spinorial geometry method was used [46]. In this method many computations are streamlined by the use of a privileged basis of spinors, and, correspondingly, of gamma matrices. The downside of this approach is that the results obtained concerning spinors are written in that particular basis and, sometimes, a basis-independent form could be more desirable for some purposes and could also give further insights into the structure and interpretation of the supersymmetric solutions, as we will discuss below.
Our goal in this paper is to repeat part of the analysis carried out in refs. [43][44][45] using the spinor bilinear formalism without making any choice of spinor basis, to first order in α . Thus, we expect to obtain 1. The algebra of spinor bilinears in an arbitrary basis (see section A.4). This algebra includes the relations satisfied by the Spin(7) structure 4-form Ω abcd .
2. The form of the Killing spinors in a general spinorial basis. In particular, we will obtain the supersymmetry projectors using arbitrary gamma matrices (section A.5).
The form of the projector eq. (A.45b) suggests that the solutions with only one Killing spinor can be viewed as a multiple intersection of S5-branes, with volume form of the 4-dimensional transverse space of each of them entering the Spin(7) structure 4-form Ω abcd . This projector can be compared with eq. (3.10) of ref. [43]. They are equivalent, but the later is expressed in a particular basis and its form will change under a general change of basis while the form of the projector eq. (A.45b) will not. Furthermore, the intersection interpretation is lost in the simplified form.
3. The necessary and sufficient conditions that a supersymmetric field configuration has to satisfy, originally found in refs. [43][44][45]. These will be obtained in essentially the same form.
4. The relations that hold between the first-order in α -corrected equations of motion when they are evaluated on supersymmetric configurations, via the Killing Spinor Identities (see below). Relations of this kind have been derived recently as integrability conditions of the Killing Spinor Equations in ref. [38]. We discuss the relation with ours in section 4.2.
We are just interested in the general characterization of the supersymmetric solutions (those admitting (at least) one Killing spinor), and we will not try to study case by case, what happens when the solution admits 2 or more Killing spinors, as it has been done in full detail in refs. [43][44][45].
There are several important difficulties which prevent us from making this analysis at higher orders in α : 1. Not all α corrections to the action and supersymmetry transformation rules are known. In ref. [42], which we will use here, they were determined to cubic order in α (quartic order in curvatures) by supersymmetrizing the first-order terms (specially the Chern-Simons terms in the Kalb-Ramond 3-form field strength) which had been JHEP06(2020)106 found by other means. This supersymmetrization leads to a recursive procedure for introducing the terms of next order in α in the Kalb-Ramond 3-form field strength H. H occurs in the action and in the supersymmetry transformation rules, and part of the α corrections in them are introduced implicitly through H. However, apart from these, there are other α corrections of increasing complexity both in the action and in the supersymmetry transformation rules and this will force us to work only at first order in α and analyze the KSEs only up to that order. It has been suggested that higher order corrections to the KSEs could be absorbed into redefinitions of the torsionful spin connection or the Kalb-Ramond 3-form field strength which would preserve their first-order form, though, [42,47]. If this was proven, the first-order analysis would be sufficient.
2. As we have just said, many (probably most) of the higher-order terms in the action cannot be constructed by recursion and they are only known explicitly up to cubic order. The equations of motion 4 have a very large number of complicated terms. At first order, though, it is known that many of them can be ignored because they are proportional to the zeroth-order equations of motion. It is not known if similar simplifications take place at higher orders and, if one wants to find explicit solutions at higher orders one will have to deal with very complicated equations.
3. In most of the literature on the characterization or classification of supersymmetric solutions the following fact is used: the integrability conditions of the KSEs (adequately treated) are combinations of the off-shell equations of motion. This leads to non-trivial relations between them which simplify the problem of finding supersymmetric solutions, because only a small number of equations of motion are independent and have to be solved. At higher orders in α , computing the integrability conditions of the KSEs and recognizing in them the complicated higher-order equations of motion can be very difficult.
The Killing Spinor Identities (KSIs) derived in ref. [48] offer an alternative path, because, based on the invariance of the theory under local supersymmetry transformations, they yield the same relations between equations of motion 5 even if the form of the latter is not known explicitly [49]. The main ingredients are the supersymmetry transformations of the bosonic fields of the theory, which, usually, are much simpler than those of the fermions.
4. However, even though, in most cases, the supersymmetry transformations of the bosonic fields are not modified when the supergravity theory is gauged or deformed in other ways, in this case the Kalb-Ramond 3-form field strength is modified at first order in α by the addition of the Lorentz-and Yang-Mills-Chern-Simons terms that 4 We only consider purely bosonic configurations and, therefore, we always refer, implicitly, to the equations of motion of the bosonic fields of the theory. 5 As we will explain in section 4.1, the equations of motion usually considered in the literature are combinations of those one obtains by direct variation of the action where terms proportional to the lowestorder equations of motion are eliminated. In order to use the KSIs, we need to take into account these facts.

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modify the gauge transformations of the Kalb-Ramond 2-form. The supersymmetry transformations of this field acquire additional terms related to these gauge transformations which results in a non-trivial modification of the KSIs at first order in α we have to deal with.
5. The main drawback of the KSIs is that, by construction, they never include the Bianchi identities satisfied by the field strengths in the relations obtained. The Bianchi identity of the Kalb-Ramond 3-form is typically one of the key equations to be solved, though. This deficiency of the KSI approach can be overcome by including the 6-form dual of the Kalb-Ramond 2-form in the derivation of the KSIs, as we will show in section 4.2. The equation of motion of the 6-form is, by definition, the Bianchi identity of the Kalb-Ramond 3-form field strength. We are not including the Bianchi identity of the Yang-Mills gauge field strength because in order to write it one needs to know the gauge connection, which completely determines and trivializes the Bianchi identity.
After considering all these difficulties and the solutions found for some of them, it is clear that, being conservative, we will have to content ourselves with carrying our program to first order in α only: we will determine necessary and sufficient conditions for unbroken supersymmetry valid to O(α ) and relations between equations of motion evaluated on supersymmetric configurations valid to the same order in α . We will not be able to simplify the independent equations of motion beyond O(α ), either, because it is not clear if they can be simplified. This paper is organized as follows: we start by reviewing section 2 the bosonic Heterotic Superstring effective action, equations of motion and supersymmetry transformation rules. Appendix A contains our conventions for gamma matrices and spinors, the definitions of the spinor bilinears used in the main text and the computation of the algebra satisfied by these bilinears using the Fierz identities in an arbitrary basis which we are going to use in section 3. 6 As a byproduct we will derive the algebra of the Spin(7) structure 4-form which is always present in the bilinear algebra. In section 3 we will determine the necessary conditions for a field configuration to admit one Killing spinor (summarized in 3.4) and we will show that they are also sufficient by solving explicitly the KSEs. We will make use of the spinor projector given explicitly in terms of the Spin(7) structure in eq. (A.45b). In section 4 we determine which independent equations have to be imposed on the supersymmetric field configurations determined in the previous section to ensure that they are solutions of all equations of motion of the theory. We explain how the equations of motion are obtained and simplified at first order in α and the derivation of the KSIs involving also the Bianchi identity of the Kalb-Ramond 3-form field strength. Finally, section 5 contains a brief discussion of the results obtained and their spinoffs.

JHEP06(2020)106 2 The Heterotic Superstring effective action
In this section we are going to review the Heterotic Superstring effective action was given in ref. [42], where it was constructed up to cubic order in α (quartic in derivatives) by demanding invariance of the action under supersymmetry (to that order). Here we will use the conventions of ref. [37]. 7 We start by defining recursively the 3-form field strength of the Kalb-Ramond 2-form B. The zeroth-order, it is given by (2.1) Using it as torsion, one can define the zeroth-order torsionful spin (or Lorentz) connections where ω a b is the (torsionless, metric-compatible) Levi-Civita spin connection 1-form. The curvature 2-forms of these connections and the (Lorentz-) Chern-Simons 3-forms are defined as 8 We will denote the gauge field 1-form by A A , where the indices A, B, C, . . . take values in the Lie algebra of the gauge group. The (Yang-Mills) gauge field strength and Chern-Simons 3-form are defined by where the Killing metric of the gauge group's Lie algebra in the relevant representation, K AB , assumed to be invertible and positive definite, has been used to lower the index of the structure constants f ABC ≡ f AB D K DB and of the indices of the gauge fields A A ≡ K AB A B .

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and using it as torsion, we obtain the first-order torsionful spin connections and their curvatures and Lorentz-Chern-Simons terms R (±) are obtained by plugging them into the above definitions. Then, the second-order Kalb-Ramond field strength is defined as etc. For many practical purposes it is advantageous to work with general H and Ω (±) and then restrict them to a given order when needed. This will allow us to work with the Killing spinor equations at an arbitrary order in α , for instance, because the only α corrections are contained in the definitions of H and Ω (+) . In the action, apart from the α corrections implicit in the definitions of H and Ω (−) , there are additional terms of higher order in curvatures that have to be included explicitly and which are known only to cubic order in α [42]. 10 It is understood that all terms above a certain order in α have to be ignored. Thus, with this understanding, we will omit the upper indices (n) from now on.
It is convenient to define an affine torsionful connection Γ (+) µν ρ via the Vielbein pos- Solving the above equation one finds that it is given by where ρ µν stands for the Christoffel symbols of the metric g µν = η ab e a µ e b ν .
It is also convenient to use the so-called "T -tensors" associated to the α corrections in the equations of motion and in the Bianchi identity of the Kalb-Ramond 3-form field strength. They are defined by (2.12) The Heterotic Superstring effective action, written in the string frame to cubic order in α in terms of the objects we have just defined tales the form (2.13) 10 And not completely: only the quartic terms that follow from the supersymmetrization of the Chern-Simons terms in H were determined, but there may be more [53][54][55].

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where R is the Ricci scalar of the string-frame metric g µν , φ is the dilaton field and its vacuum expectation value of e φ is the Heterotic Superstring coupling constant g s and where G (10) N is the 10-dimensional Newton constant. Observe that, to have all O(α 3 ) terms in the action, we have to use H (3) and Ω (3) (−) , disregarding all terms of higher order that arise from H 2 etc. in the above action. As explained in the Introduction, though, we will only work at first order in α .
Finally, let us consider the supersymmetry transformation rules of the bosons and of the fermionic fields (gravitino ψ µ , dilatino λ and gaugini χ A ) 11 for vanishing fermions and to first order in α . They can be written, respectively, as follows: where, at this order, with the gravitino field strength ψ µν defined as and

Supersymmetric configurations
The (purely bosonic) supersymmetric field configurations of this theory are those for which the Killing Spinor Equations (KSEs) δ ψ a = δ λ = δ χ A = 0 admit at least one solution called Killing spinor that we will denote by . Thus, if e a µ , B µν , φ describe a supersymmetric field configuration of this theory, there is an satisfying the KSEs

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When the spinor bilinears a and W a 1 ···a 5 defined in appendix A.2, are constructed with the Killing spinor that satisfies the above equations, they must satisfy certain some other equations apart from the algebraic relations found in appendix A. 4. In what follows, we are going to determine those equations and their immediate consequences, for each KSE.

The gravitino KSE
Using the torsionful affine connection defined in eq. (2.11), the gravitino KSE eq. (3.1) immediately leads to these two differential equations: Using eq. (A.28), these two equations lead to another equation for the 4-form Ω a 1 ···a 4 where we are using the same convention as in the appendix: all indices with the same Latin letter (here b 1 · · · b 5 ) are assumed to be fully antisymmetrized.
The symmetric and antisymmetric parts of eq. (3.4) indicate that the null vector a is a Killing vector (∇ is the standard Levi-Civita connection) and that 8) or, in the language of differential forms It is customary to introduce an auxiliary null vector n, dual to Then, we can use the 1-forms as the first two elements of a Vielbein basis {e + , e − , e m } m = 1, . . . , 8 in which the metric takes the form Eq. (3.9) can be interpreted as the + component of the first Cartan structure equations (de a = ω a b ∧ e b in our conventions) and from it we find that

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On the other hand, from eq. (3.6) we get these two equations in the above basis: Since, in this basis, the 4-form Ω's only non-vanishing components are those with transverse indices m, n, p, . . ., eq. (3.14b) implies that which, together with eq. (3. 13) gives It is not difficult to see that, if one uses coordinates adapted to the null Killing vector µ ∂ µ = e − µ ∂ µ and, therefore, ∂ − e m µ = 0, we have the property ω −mn = −ω mn− which, combined with eq. (3.16) leads to which, in its turn, used in eq. (3.14a), sheds the result in agreement with the use of adapted coordinates. The a = +, m components of eq. (3.14a) still need to be analyzed, but this requires a more detailed choice of coordinates that we are going to make next.

The metric
All metrics characterized by the existence of a null, generically not covariantly-constant nor hypersurface-orthogonal, Killing vector can be written in a common way (see, e.g. ref. [30]: first of all, we introduce null coordinates v, u through f is a scalar function and both f and β are independent of v (but, in general, not of u nor of the remaining coordinates x m ). Next, we write e − in the form where ω = ω m dx m is another 1-form in the transverse 8-dimensional space and K is a scalar function which are also independent of v.

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Choosing the Vielbeins e m to only have non-vanishing v-independent components in the transverse directions, e m = e m n dx n , the metric takes the form where the metric in the transverse space is given by It is clear, then, that the transverse components of the spin connection ω mnp only depend on the transverse Vielbeins e m and the a = m components of eq. (3.14a) are those of an equation in transverse space. We can rewrite it as and multiplying it by Ω q n 2 n 3 n 4 and using eqs. (A.40g) and (A.40f) we find where we have used the projector acting on 3-forms defined in appendix A.5.
The the a = + components of eq. (3.14a), on the other hand, can be written in the form and using the same properties, we find Finally, observe that the components of the spin connection are determined by the objects that occur in the metric: the scalar functions f, K, the transverse 1-forms ω, β and the transverse metric h. Via eqs. (3.16) they also determine the H ab− components of the Kalb-Ramond field strength. These components are constrained by the dilatino KSE and the constraints become constraints on the objects that occur in the metric.

The dilatino KSE
Multiplying the dilatino KSE by¯ from the left, we get If we multiply by¯ Γ ab from the left, we get In terms of the 4-form Ω we arrive to

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The a = +, m components of this equation give a pair of non-trivial equations in transverse space 12 If we multiply the dilatino KSE by¯ Γ a 1 ···a 4 from the left, we get Using again eq. (A.28) we get another pair of equations in transverse space It turns out that these two equations are just combinations of eqs. (3.31a) and (3.31b): if we multiply eq. (3.33b) by the 4-form Ω, contracting just 3 of the 4 free indices and using eqs. (A.40g) and (A.40d) we obtain eq. (3.31b). The same happens with eq. (3.33a). Therefore, the only three independent equations one obtains from the dilatino KSE are eq. (3.28), (3.31a) and (3.31b).
Eq. (3.33a) can be rewritten in the form and, combining this result with eq. (3.25) we can solve for the components H mnp :

The gaugino KSE
Multiplying the gaugino KSE eq. (3.3) by¯ Γ a ,¯ Γ abc and¯ Γ a 1 ···a 5 from the left, we get respectively. Using eq. (3.36) and the decomposition of W in terms of and Ω eqs. (3.36b) and (3.36c) lead to

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Observe that these two equations for F A mn have exactly the same form as eqs. (3.31b) and (3.33b) for H mn− and, therefore, they are equivalent by virtue of the properties of the 4-form Ω. The components F A m+ remain undetermined. Eq. (3.37a) is the natural generalization of the standard self-duality condition of Yang-Mills instantons in 8 dimensions. As a matter of fact, eq. (3.37a) is the defining relation of the "octonionic instanton" constructed in ref. [56] and which was used as source for the "octonionic superstring soliton" solution of the Heterotic Superstring of ref. [57]. Since this equation is just a necessary condition to have at least one supersymmetry, we notice that all supersymmetric solutions of the Heterotic Superstring effective action must satisfy it. In particular, the gauge field of the "gauge 5-brane" solution of ref. [58] (a SU(2) BPST instanton [59]) must satisfy it and, indeed, the self-duality condition on the gauge field strength of the BPST instanton as just the result of imposing the condition eq. (3.37a) on a gauge field that lives on a 4-dimensional subspace. For gauge fields that live in subspaces of dimensions larger than 4 and smaller than 8, eq. (3.37a) defines Yang-Mills instantons of gauge groups related to the holonomy of the Killing spinors. An intermediate example between the octonionic (Spin (7)) one and the BPST one is provided by the G 2 instanton and its associated heterotic string solution [60].

Summary of the necessary conditions for unbroken supersymmetry
1. The metric has to admit a null Killing vector a . If v is the null coordinate adapted to this isometry, this means that the metric can be written in the form eq. (3.21) which we rewrite here (as we will do with other formulae) for the sake of convenience All objects in the metric are v-independent.
2. There exists a v-independent 4-form Ω satisfying the properties eqs. (A.40b)-(A.40h) and which satisfies 3. The following relations between certain components of the matter fields must be satisfied: 4. The torsionful spin connection Ω (+) satisfies the following conditions: These conditions relate certain components of the Levi-Civita spin connection (and, hence, some of the objects that occur in the metric) to certain components of the Kalb-Ramond 3-form.

Sufficiency
Let us now check that the necessary conditions for having the minimal amount of unbroken supersymmetry previously identified are also sufficient.

Gaugino KSE
Let us start with the gaugino KSE eq. (3.3). The necessary conditions that the gauge field strength has to satisfy are eq. (3.40b) (3.40c). Then, where we have used the property eq. (A.46a) and the spinor projector eq. (A.45b). This equation is solved by demanding 13 Observe that, when F A m+ = 0 the first condition seems to be unnecessary. However, Π − is only idempotent when that condition is satisfied.

Dilatino KSE
Using eq. (3.40a) and the spinor projector eq. (3.43a), this equation reduces to Now we do two things: 1. First use eq. (3.40e) into eq. (A.46b) to eliminate Π + H mnp from the latter, solve the resulting equation for H mnp and substitute the result in the above equation.

Gravitino KSE
The projection eq. (3.43a) and the supersymmetry condition eq. (3.41a) bring the gravitino KSE to the form

Supersymmetric solutions
In this section we are going to study under which conditions, the supersymmetric field configurations that we have identified in the previous section are also solutions of the equations of motion of the theory. We start by reviewing the equations of motion that follow from the action eq. (2.13) at first order in α and finding the relations with the simplified equations which are usually solved. Of course, nothing but the sheer difficulty prevents us from deriving higher-order equations of motion from eq. (2.13) because it is not known if the simplifications that occur at first order in α have an analogue at higher orders. JHEP06(2020)106

Equations of motion
The equations of motion that follow from the action eq. (2.13) are very complicated. If we stay at first order in α , though, there are important simplifications. Following ref. [61], we can separate the variations with respect to each field in the action (except for the dilaton and Yang-Mills fields) into those corresponding to occurrences via Ω (−) a b , that we will call implicit, 14 and the rest, that we will call explicit, as follows: 15 δB νρ δB νρ .  14 The dilaton does not occur in the torsionful spin connection, neither do the Yang-Mills fields to the order in α we are considering. 15 The complete equations of motion of all the fields and δS/δΩ (−) µ a b can be found in appendix B to first order in α . The proof of the lemma of ref. [61] mentioned below can also be found there.

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The non-trivial relation between the complete equations of motion that one obtains from the action eqs. (B.1)-(B.4) and the standard equations of motion considered in the literature eqs. (4.2)-(4.5) that we have just explained has to be kept in mind when using the Killing Spinor Identities (KSIs) [48,49]  If a solution is given in terms of the 3-form field strength, we also need to solve the Bianchi identity dH − T (4) = 0 , (4.6) as well. This identity can be rewritten as the equation of motion of the 6-formB µ 1 ···µ 6 dual to the Kalb-Ramond 2-form: Apart from the Kalb-Ramond 2-form, there are no other fields in the action which only occur through their field strengths. Thus, we cannot formulate the equations of motion only in terms of those field strengths and imposing a Bianchi identity on them is utterly unnecessary. Observe that, for instance, in order to write the Bianchi identity for the Yang-Mills field strength (or the Riemann curvature tensor) it is necessary to know the gauge field (the connection). Therefore, we will not need any more equations.

Killing Spinor Identities
The equations of motion of theories with local symmetries are related off-shell by the socalled Noether (or gauge) identities. In a supergravity theory the Noether identities relate the bosonic and fermionic equations of motion. 16 These identities are valid for any field configurations but, if we restrict ourselves to purely bosonic field configurations admitting Killing spinors, such as those we have characterized in the previous section, it can be shown that the equations of motion of the bosonic fields are related by the so-called Killing Spinor Identities, first derived in ref. [48].
As shown in ref. [49], the KSIs are essentially equivalent to the relations obtained from the integrability conditions of the Killing spinor equations, but, in general, they are much easier to derive because, usually, only algebraic operations are required. These relations between the bosonic equations of motion of supersymmetric configurations can be used to reduce the number of independent equations that need to be checked in order to prove that 16 The invariance of the action of N = 1, d = 4 supergravity was first proven analytically in ref. [62] precisely by checking that the supersymmetric Noether identity was satisfied off-shell.

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a given supersymmetric field configuration is also a solutions of all equations of motion. Our goal in this section is to find the KSIs and determine the independent equations of motion that need to be checked in the case of the Heterotic Superstring effective action.
An important point to be stressed is that the generic form of the KSIs only depends on the supersymmetry transformation laws of the bosonic fields. The equations of motion of a supergravity theory change when we gauge it, deform it with mass terms or, as it is the case here, when we add the α corrections, but, in many cases, the relations that hold between them when they are evaluated on supersymmetric configurations, the KSIs, do not, because they only depend on the supersymmetry transformation rules of the bosonic fields, which do not change. 17 In the present case, however, the supersymmetry transformation rules of the bosons do get α corrections and this generic property will not hold true. In particular, as explained in the Introduction, the Kalb-Ramond 2-form supersymmetry transformation rules acquire two new terms of first order in α (see eq. (2.15)) associated to the Nicolai-Townsend ("anomalous") gauge transformations that arise when the Lorentz-and Yang-Mills-Chern-Simons terms are included in the field strength. At higher orders, new terms have to be taken into account and, although it has been suggested that they may be absorbed in redefinitions of the fields, this has not been explored systematically and we will restrict our analysis to the first order in α at which the only modification of the bosons' supersymmetry transformation rules is the one we have just discussed.
A disadvantage of this approach that the proof of the KSIs in ref. [48] assumes the existence of the potentials in the field strengths (the equations of motion are the first variations of the action with respect to them) or, equivalently, that the Bianchi identities are satisfied and, sometimes, we would like not to assume this and solve a different set of equations. The Bianchi identities appear explicitly in the integrability conditions, but it is usually very hard to compute them.
There is, however, a simple way to make the Bianchi identities appear in the KSIs: we just have to view them as the equations of motion of the dual potentials (as long as their supersymmetry transformation laws are known). In the case at hands, this means that, if we want to find KSIs including the Bianchi identity of the Kalb-Ramond 3-form field strength, we must view it as the equation of motion of the dual 6-form potentialB µ 1 ···µ 6 and use the supersymmetry transformation law of this field, given in refs. [51,52] and which we have rewritten in our conventions in eq. (2.16). Observe that, if denote by EB µ 1 ···µ 6 the equation of motion of the dual 6-formB µ 1 ···µ 6 (see below), and we denote by B H µ 1 ···µ 4 the Bianchi identity of the Kalb-Ramond 3-form, 18 they are each other's Hodge dual: 17 This observation was used in ref. [63] to prove the exactness of the maximally supersymmetric solutions of 5-dimensional supergravity when higher-order corrections are included. 18 Up to global factors, the components of the Bianchi identity eq. (4.6) are

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Taking into account this last point, using the definitions and splitting the variations with respect to the fields into variations with respect to explicit occurrences and variations with respect to implicit occurrences via the torsionful spin connection, 19 the general recipe in ref. [48] takes the form Using the supersymmetry transformations of the bosons in eqs. (2.14)-(2.18), we find that the KSIs of the theory at hands are the spinorial equations where we have defined the combination Using eq. (B.12), 14) and, taking into account that the Killing spinor satisfies ∇ (0)

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The upshot is that, to O(α 2 ), the KSE eq. (4.12a), can be replaced bȳ where C is a complicated tensor that contains E (0) B and Γ (7) is the antisymmetrized product of 7 gamma matrices.
Furthermore, observe that the combination of equations of motion that occurs in eq. (4.12c) (4.19) and it is convenient to rewrite the KSI in the form Eqs. (4.12b), (4.18) and (4.20) should be identical to the relations between the bosonic equations of motion obtained from the KSEs' integrability conditions. These have been recently presented in ref. [38] (eqs. (205)) and should be compared to eqs. (4.12b), (4.18) and (4.20). Apart from some minor differences (the equation of motion of the dual 6-form is replaced by the Bianchi identity of the 3-form and the Einstein and Maxwell equations used in ref. [38] are the combinations of our equations of motion described in eqs. (4.2)-(4.5)) the authors of ref. [38] have used an expansion of the equations of motion and Bianchi identities in powers of α that, for instance, does not take into account the definition of the Kalb-Ramond field strength H in terms of potentials eq. (2.7) and the corresponding expansion in α . Instead, they just assume the existence of field strengths which are not expanded in powers of α , satisfying certain (anomaly-corrected) Bianchi identities. Using the explicit form of the Bianchi identity for H it is possible to rewrite their eqs. (205) in exactly the same form we have obtained.

Tensorial KSIs
Eqs. (4.12b), (4.18) and (4.20) are the off-shell relations between the bosonic equations of motion we were after but, in order to make use of them, we must transform them into tensorial equations. Let us start with the simplest of them, eq. (4.20): if we hit it with and Γ µ 1 µ 2 µ 3 µ 4 from the right we obtain, respectively, µÊA µ = 0 , (4.21a)  We conclude, that all components of the Yang-Mills equationsÊ A µ , except for n µÊA µ , are automatically satisfied by supersymmetric field configurations. Hitting now eq. (4.12b) with Γ µ from the right and contracting the result with n µ , we arrive at E φ = 60 µ n ν EB µν , (4.25) where we have defined 26) because this combination appears very often and plays an interesting rôle. Often, in the literature, the Bianchi identity is assumed to be solved from the beginning. In that case, the dilaton equation is automatically solved on supersymmetric field configurations, but the above KSI allows us to assume that the dilaton equation is solved from the beginning, which would imply that the component µ n ν EB µν of the Bianchi identity is automatically solved.
Observe that the relation eq. (4.9) implies that so the dilaton equation is related to a single combination of the transverse components of the Bianchi identity. Next, let us consider eq. (4.18). Hitting it with from the right, and taking into account that the last term will not contribute, we get  20 In what follows, in order to get rid of the complicated contributions of the last term in eq. (4.18) we will assume that the Bianchi identity is satisfied at zeroth order in α ,

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This is a complicated identity that we can simplify by hitting it with , n, Ω in different ways.
Hitting eq. (4.43) with g α 1 ν and using several of the identities derived above, we get another constraint on the Bianchi identity 20 g µν ρσ + To summarize, the components of the equations of motion of the Vierbein, the Kalb-Ramond 2-form and its dual implied by supersymmetry are: and those of the gauge fields and dilaton arê

Remaining equations for supersymmetric solutions
A possible choice of independent equations of motion to be checked is although some combinations of the components of the Bianchi identity are automatically satisfied for supersymmetric field configurations. It should be remembered that, in order to derive some of the relations, we had to assume that the Bianchi identity is satisfied at lowest order in α .

Discussion
In this paper we have re-analyzed the problem of characterizing all the supersymmetric solutions of the Heterotic Superstring effective action to first order in α working in a general spinorial basis, instead of working on a particular or privileged basis as in it is done in refs. [43][44][45]. Thus, we have first computed the algebra of bilinears in an arbitrary basis in section A.4. It is this computation that allowed us to obtain the conditions necessary for unbroken supersymmetry and to construct the supersymmetry projectors necessary to prove the sufficiency of the conditions in an arbitrary basis. As explained below eq. (A.45b), the JHEP06(2020)106 explicit form of the supersymmetry projector for the Killing spinors suggests a physical interpretation of minimally supersymmetric field configurations which is entirely lost if one works in a privileged spinor basis. The form of the projector eq. (A.45b), for instance, is general, basis-independent. One can always evaluate it in a particular basis such as the one used in refs. [43][44][45] if that is needed in a specific calculation.
Another important result is the derivation of the set of relations existing between the equations of motion evaluated on supersymmetric configurations to first order in α obtained in section 4.2 using the Killing Spinor Identities (KSIs). One of the novelties in this result is the procedure through which we have obtained it, using the KSIs in this, more complicated, context. We have also shown how to include the Bianchi identity of the 3-form Kalb-Ramond field strengths in the KSIs.
Finally, we have also re-derived the set of conditions necessary for unbroken supersymmetry, summarized in section 3.4. They are, of course, completely equivalent to those obtained in refs. [43][44][45].
The computation of the bilinear algebra in an arbitrary basis is a very useful result because the algebras of bilinears of half-maximal supergravities 22 in lower dimensions are exactly the same, up to relabeling of the components of the bilinear forms [64], which implies the existence of a Spin(7) structure hidden in any supersymmetric solution of any half-maximal theory. This observation deserves further discussion. 23 Let us consider the supersymmetry condition of the Yang-Mills fields eq. (3.40c) which we rewrite here for the sake of convenience: This equation can be seen as an 8-dimensional generalization of the 4-dimensional self-duality condition that characterizes 4-dimensional Yang-Mills instantons such as the BPST instanton with gauge group SU(2) ⊂ SO(4) [59]. This instanton is part of the gauge 5-brane solution of Heterotic Supergravity [58], sourcing the gravitational and dilaton field. Combined with the solitonic (or NS) 5-brane of ref. [66] as in ref. [67], in which the source is a magnetic Kalb-Ramond field, one can obtain the so-called symmetric 5-brane [67], which is considered an exact solution of the Heterotic Superstring effective action to all orders in α . It is clear that this solution should fit into our general result and that the gauge field satisfying eq. (5.2) should obey eq. (5.1) for some Spin(7) structure 4-form. As a mater of fact, one can view the Spin(7) structure 4-form as a collection of volume forms in 4-dimensional manifolds (hyper-planes in 8-dimensional Euclidean space, [68]) and, if we simply restrict 22 Actually, of any half-maximal supersymmetric theory, including global supersymmetry. 23 Observe that, ultimately, this is a property of the Clifford algebra itself that will hold whenever (1,9)dimensional Majorana-Weyl spinors are at play. For instance, in ref. [65] it has been shown that any 10-dimensional Lorentzian manifold admits a real chiral spinor if and only if it admits a null vector such that the associated metric in the corresponding "screen bundle" is Spin(7).

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eq. (5.1) to the 4-dimensional subspace in which the gauge field is defined to live, we just get (up to a sign) eq. (5.2). Therefore, eq. (5.2) is included as a particular case in eq. (5.1). A solution to eq. (5.1) that does not assume that the gauge field lives in less than 8 dimensions is the so-called octonionic instanton of Nicolai and Fubini [56], whose gauge group is Spin(7) ⊂ SO (8). Observe that the use of a generic basis for the Spin(7) structure 4-form and the knowledge of the algebra it satisfies plays an important rôle in the construction of the solution. The octonionic instanton has been used to construct the octonionic superstring soliton of ref. [57], which is, actually, a O(α ) solution of the Heterotic Superstring effective action preserving exactly one supersymmetry and, therefore, a very good example of the characterization discussed in this paper. It is, on the other hand, a solution closely related to that of the symmetric 5-brane mentioned above: both of them are sourced by Yang-Mills instanton fields satisfying eq. (5.1), the main difference being the number of transverse directions the gauge fields do not depend upon and the number of isometries of the metric (8 to 4) and the absence of a (known) solution of the same kind with no gauge fields, sourced only by the Kalb-Ramond field, in analogy with the solitonic 5-brane. 24 It should be possible to consider solutions to eq. (5.1) for cases in between the full 8-dimensional dependence of the octonionic instanton and the 4-dimensional dependence of the BPST instanton, with gauge groups that can be embedded in SO(n), 4 < n < 8 and which should coincide with (some of) the special holonomy groups found in refs. [43][44][45]. Let us consider, for instance, the n = 7 case, in which the gauge field lives in a 7-dimensional space or, alternatively, does not depend on one of the original 8 transverse coordinates, x 8 , say. it is not difficult to see that only the components Ω mnpq with m, n, p, q = 1, . . . , 7 occur in eq. (5.1) and, due to the 8-dimensional selfduality of the Spin(7) structure, they can be rewritten in terms of the 3-form Σ mnp ≡ Ω mnp8 which satisfies the algebra of a G 2 structure. Thus, it should not be surprising that one can construct G 2 instantons in R 7 using an ansatz similar to Nicolai and Fubini's (or 't Hooft's for the BPST instanton) and a complete solution of the Heterotic Superstring effective action sourced by such an instanton. As a matter of fact, both the instanton and the solution were constructed in ref. [60]. It should be possible to find more intermediate cases, with a number of isometries ranging between 2 (the string, Spin(7) case) and 6 (the fivebrane, SU(2) case), so they can be interpreted as (n + 1)-brane solitons. Their existence would greatly enhance the spectrum of non-perturbative extended solitons of the Heterotic Superstring. Work in this direction is in progress.

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partially supported by the Angelo della Riccia Foundation Fellowship and partially by the German Research Foundation (DFG) via the Emmy Noether program "Exact Results in Extended Holography". TO wishes to thank M.M. Fernández for her permanent support.
A d = 10 gamma matrices, spinors and the algebra of bilinears A.1 d = 10 gamma matrices and spinors In this appendix, Γ a , a, b, c, . . . = 0, · · · , 9 are the 10-dimensional gamma matrices, satisfying the Clifford algebra where (η ab ) = diag(+ − · · · −) is the 10-dimensional Minkowski metric. The chirality matrix Γ 11 is defined to satisfy the relations so that, in particular, The charge-conjugation and Dirac-conjugation matrices C and D are defined by the properties The particular matrices we have chosen are and satisfy Given a 10-dimensional spinor ψ, using these matrices, we define its Dirac and Majorana conjugates, respectivelyψ and ψ c , bȳ

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Majorana spinors are defined by the propertȳ With the particular choices of C and D that we have made, they are neither purely real nor purely imaginary, but this is the most convenient choice for reducing them to symplectic-Majorana spinors in d = 6 dimensions (which will be useful in a forthcoming work [64]). The supersymmetry parameter of Heterotic Supergravity, , is a Majorana-Weyl spinor. We choose the convention (A.10)

A.4 d = 10 bilinear algebra
It is now straightforward to compute the products of the bilinears a and W a 1 ···a 5 using the Fierz identities we just derived in the previous section. Each of the identities obtained has been checked to be consistent with the self-duality of the 5-form, which imposes strong constraints on the possible combinations that can occur in the right-hand side.
To start with, we find for the product of 1-forms and we observe that the right-hand-side is equal to itself when we replace W by its dual, in agreement with the invariance under duality of the left-hand side.
For the product of a 1-form and one self-dual 5-form we have obtained 25 where we have used the fact that the term which occurs in the right-hand side vanishes identically due to the self-duality of W . The expression in the right-hand side is self-dual in the five b i indices, just as the left-hand side. In order to express the product of two self-dual 5-forms we have defined, first, for the sake of convenience, the following products and contractions of the self-dual 5-form with itself: 25 Here and in what follows, it is assumed that all indices that have the same letter (b1, · · · , b5 in this case) are always antisymmetrized. The brackets of the antisymmetrizers have been suppressed to avoid the cluttering of the formulae.

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The rest of the terms quadratic in W that can occur in the right-hand side are linear combinations of them, as can be seen by replacing W by its dual. These relations are and they allow us to use A, B, C, D and E as a basis for these products. Then, using this notation, we find All terms in the right-hand side of this expression are a ↔ b symmetric, as the left-hand side. Furthermore, the left-hand side is self-dual in the a and b indices separately. It can be checked that the right-hand side has the same property: the combination of the first three terms is self-dual and the combination of terms in the fourth line is also self-dual, 26 as can be seen by using the properties

A.4.1 Consequences
The selfduality of W implies Then, eq. (A.18) implies that is null: 26 Actually, up to a global factor, it is the only self-dual combination.

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Lowering the upper index of eq. (A. 19) and antisymmetrizing it with the rest leads to 27) which implies that W a 1 ···a 5 = 5 a 1 Ω a 2 ···a 5 , (A.28) for a certain 4-form Ω. We will see that this 4-form, which was first found in the supergravity context in ref. [69], satisfies the relations of a Spin (7) structure. Plugging this result back into eq. (A. 19) and contracting now the upper index with one of the lower ones we so Ω lives in the 8-dimensional space transverse to the null vector . It is useful to introduce a null vector n dual to : n 2 = 0 , n a a = 1 , (A. 30) and define the metric induced in the 8-dimensional space transverse to as and a fully antisymmetric tensor in that space as well The selfduality of Ω in the 8-dimensional transverse space eq. (A.33a) together with eq. (A.34a) implies The last product, eq. (A.23), gives the following expression for the product of two 4-forms: (A.39) Further simplifications of this general formula are possible, but we will not try to obtain them here.
Summarizing, the main relations involving the product and contractions of two 4-forms that we will use are 27

A.5 Projectors
The 4-form Ω m 1 ···m 4 can be used to construct projectors acting on 2-and 3-forms in the 8-dimensional transverse space and on spinors. 27 Not all these equations are independent. We quote all of them for their usefulness.

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The projectors acting on spinors and forms satisfy several relations. The two relations that we will use are

B Equations of motion at first order in α
In this appendix we collect the explicit form of the complete equations of motion that follow from the action eq. (2.13) at first order in α and ignoring the factor g 2 s /(16πG where we have used the shorthand notations The expression in parenthesis is a combination of the zeroth-order equations of motion: This is the proof of the lemma of ref. [42] mentioned in section 4.1.
The particular combination of the zeroth-order equations of motion that appears in the above expression is, in the notation introduced in section 4.1 of the main text, and, therefore,

B.1 Noether identities
In order to understand better the structure of the equations of motion, it is convenient to study the Noether identities that relate them as a consequence of the gauge symmetries of the theory. Associated to the standard gauge transformations of the

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If this identity is true, then, this one is also true as well: Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.