Anomaly corrected heterotic horizons

We consider supersymmetric near-horizon geometries in heterotic supergravity up to two loop order in sigma model perturbation theory. We identify the conditions for the horizons to admit enhancement of supersymmetry. We show that solutions which undergo supersymmetry enhancement exhibit an sl2,ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{s}\mathfrak{l}\left(2,\kern0.5em \mathbb{R}\right) $$\end{document} symmetry, and we describe the geometry of their horizon sections. We also prove a modified Lichnerowicz type theorem, incorporating α′ corrections, which relates Killing spinors to zero modes of near-horizon Dirac operators. Furthermore, we demonstrate that there are no AdS2 solutions in heterotic supergravity up to second order in α′ for which the fields are smooth and the internal space is smooth and compact without boundary. We investigate a class of nearly supersymmetric horizons, for which the gravitino Killing spinor equation is satisfied on the spatial cross sections but not the dilatino one, and present a description of their geometry.


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, (4.21), (4.23), (4.25) 38 B.2 Elimination of u-dependent parts of (4. 18) and (4.20) 40 1 Introduction The effect of higher order corrections to supergravity solutions is of considerable interest, perhaps most notably for our understanding of quantum corrections to black holes. This is important in determining how string theory may resolve black hole singularities, as well as the investigation of the properties of black holes away from the limit α ′ → 0. In higher dimensions the four dimensional uniqueness theorems [1][2][3][4][5][6][7] no longer hold, and there are exotic types of black hole solutions, such as the five dimensional black rings [8]. For ten and eleven dimensional supergravity, it is expected that there is a particularly rich structure of black objects, and the classification of these is ongoing. Progress has recently been made in the classification of the near-horizon geometries of supersymmetric black holes. Nearhorizon geometries of extremal black holes in supergravity are known to generically undergo supersymmetry enhancement. This has been proven by analysing the global properties of such solutions via generalized Lichnerowicz theorems [9][10][11][12], and making use of index theory arguments [13]. One consequence of the supersymmetry enhancement is that all such nearhorizon geometries exhibit an sl(2, R) symmetry. However, it is not apparent that these properties persist after including string theory corrections. There are several approaches to investigate how α ′ corrections can change the event horizons of black holes. Many black holes have AdS p × S q near-horizon geometries and as it is expected that the symmetries of such backgrounds persist in quantum theory, only the radii of the sphere and AdS receive α ′ corrections. However, we expect that exotic black holes in higher dimensions need not necessarily have such near-horizon geometries.
Another approach, in the context of supersymmetric black holes in four and five dimensions, is to assume that the corrected near horizon geometries undergo an enhancement of supersymmetry in the near-horizon limit, which simplifies considerably the analysis of the Killing spinor equations. It is known that all supergravity D = 4 and D = 5 black JHEP10(2016)121 holes undergo supersymmetry enhancement in the near-horizon limit [14][15][16]. In particular, the five dimensional BMPV black hole [17] undergoes supersymmetry enhancement from N = 4 to N = 8 (maximal supersymmetry) in the near-horizon limit [18]. Also, the supersymmetric asymptotically AdS 5 black hole of [19] undergoes supersymmetry enhancement from N = 2 to N = 4 (half-maximal supersymmetry) in the near-horizon limit. However it is not clear in general why one expects that the α ′ corrections should preserve this property.
The first systematic classification of supersymmetric near-horizon geometries in a higher derivative theory in five dimensions [20] was done in [21], in which the only assumption made was that the solutions should preserve the minimal amount of supersymmetry. The five dimensional theory reduces to ungauged five-dimensional supergravity coupled to arbitrarily many vector multiplets when the higher derivative corrections are set to zero. In this limit, it is known that near-horizon geometries are maximally supersymmetric with constant scalars [22], which is consistent with the standard picture of the attractor mechanism. In contrast, when higher derivative terms are turned on, the list of near-horizon geometries determined in [21] includes not only the maximally supersymmetric geometries (which were classified in [23]), but also a set of regular non-maximally supersymmetric solutions, on making use of a result of [24]. Although it is unclear if these particular near-horizon geometries can be extended to a full black hole solution, the existence of such a solution proves that for certain supergravity theories, the presence of higher derivative terms can change how supersymmetry is enhanced for near-horizon solutions.
In this paper, we consider how higher derivative corrections to ten dimensional supergravity affect the geometry and supersymmetry of near-horizon solutions. We shall choose to begin this work by investigating heterotic theory which includes α ′ corrections up to two loops in sigma model perturbation theory. This choice is motivated by two factors. Firstly, from the perspective of the standard supergravity, much more is known about the geometric structure of generic supersymmetric solutions, and near-horizon geometries. In particular, as a consequence of the spinorial geometry classification techniques developed in [25,26] which were then combined with a global analysis of near-horizon geometries in [27], there exists a full classification of all possible supersymmetric near-horizon geometries in the heterotic supergravity. Secondly, the structure of higher derivative correction terms in the field equations, and in the Killing spinor equations, is significantly simpler for the heterotic theory when compared to the types of terms which arise in type II supergravity [29][30][31][32], and associated references.
The method we shall use to prove our results is that first we solve the Killing spinor equations in the near-horizon lightcone directions, and then simplify the remaining conditions as much as possible using both the local field equations and Bianchi identities, as well as global analysis. For the global analysis, we shall assume that the spatial cross-section of the event horizon is smooth and compact, without boundary, and that all near-horizon fields are also smooth. As a result of this analysis, we find that there are no AdS 2 solutions (at zero and first order in α ′ ) to heterotic supergravity, which completes the classification of heterotic AdS solutions in [33]. We also show that all of the conditions of supersymmetry reduce to a pair of gravitino KSEs and a pair of algebraic KSEs on the spatial JHEP10(2016)121 horizon sections. The latter are associated to the dilatino KSE. Throughout, we allow for all near-horizon data, including the spinors, to receive α ′ corrections.
Using these conditions, we show that there is automatic supersymmetry enhancement at both zero and first order in α ′ in the case for which there exists negative light-cone chirality Killing spinor η − up to O(α ′2 ) which does not vanish at zeroth order in α ′ . In this case the supersymmetry enhancement is obtained via the same mechanism as for the near-horizon geometries considered in [27] without α ′ corrections, and the solution admits an sl(2, R) symmetry. Such horizons admit 2, 4, 6 and 8 Killing spinors and their geometry is similar to that of horizons with vanishing anomaly contribution examined in [27]. The remaining case, for which the negative light-cone chirality spinors vanish at zeroth order in α ′ remains open. We have investigated global aspects of these solutions by considering α ′ corrections to the global analysis carried out in [27], and also by constructing generalized Lichnerowicz theorems analogous to those proven in [9][10][11][12], again incorporating α ′ corrections. However, in both cases, there is an undetermined sign in the O(α ′2 ) terms appearing, which precludes the extension of the maximum principle arguments to first order in α ′ .
We also consider a class of near-horizon solutions which are "nearly" supersymmetric. These are not supersymmetric but some of their KSEs are satisfied. This is motivated by the existence of WZW type of solutions to the heterotic theory with constant dilaton. It is known that such solutions solve the gravitino KSE but not the dilatino one. In the present case, we consider horizons for which one of the gravitino KSEs is satisfied 1 on the spatial horizon section up order O(α ′2 ) but not the other and the algebraic KSEs. After some assumptions on the form of the fields, we give a complete description of the geometry of such solutions. This paper is organized as follows. In section 2, we present the fields of heterotic near-horizon geometries and we integrate up the KSEs along the lightcone directions. In sections 3 and 4, we identify the independent KSEs by examining the various cases that can occur and in the process, prove that there are no AdS 2 solutions. In section 5, we determine the conditions under which the horizons exhibit supersymmetry enhancement, and in section 6 we give the geometry of the horizon sections. In section 7, we generalize the global analysis presented near-horizon geometries in [27] to include α ′ corrections. However because of a O(α ′2 ) sign ambiguity, it is not possible to prove that the horizon section admits a G 2 structure compatible with a connection with skew-symmetric torsion, as is the case at zeroth order in α ′ . We also generalize the Lichnerowicz type theorems to higher orders in α ′ . Once again, O(α ′2 ) sign ambiguity means that it is not possible to prove that zero modes of the horizon Dirac equation (at zero and first order in α ′ ) satisfy the Killing spinor equations to the same order in α ′ , although the algebraic Killing spinor involving the 2-form gauge field is satisfied to the required order in α ′ . In sections 8 and 9, we examine the geometry of nearly supersymmetric horizons focusing on those that admit a solution to the gravitino KSE on the horizon spatial section, and in section 10 we give our conclusions.

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The paper contains several appendices. In appendix A, we summarize some key formulae that are used throughout in the computations of the paper and present the field equations of the theory. In appendix B, we provide the details of part of the proof to identify the independent KSEs on the spatial horizon section. In section C, we present a formula which relates the gravitino KSE to the gaugino KSE which is instrumental in the investigation of the geometry of nearly supersymmetric horizons. In appendix D, we present further detail of the proof of the Lichnerowicz type theorem for the heterotic theory, and in appendix E, we describe how AdS n+1 can be written as a warped product over AdS n , and describe how such constructions are inconsistent with our assumptions on the global structure and regularity of the solutions.

Near horizon fields
The metric near a smooth killing horizon expressed in Gaussian null co-ordinates [34,35] can be written as where we have used the frame and S is taken to be compact, connected without boundary. We denote the Levi-Civita connection of S by∇, and the Levi-Civita connection of the D=10 spacetime as ∇.
For the other heterotic fields, we assume that the dilaton Φ, and the real 3-form H, and non-abelian gauge potential A admit well-defined near-horizon limits, and that ∂ u is a symmetry of the full solution: In particular, this means that Φ = Φ(y), and also 5) where N , Y and W are u, r-independent 1, 2 and 3-forms on S respectively, and we do not assume dH = 0. Moreover, where P and B are a r, u-independent G-valued scalar and 1-form on S respectively. The non-abelian 2-form field strength F is given by

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Our conventions for the heterotic theory including α ′ corrections are consistent with those of [29]. We assume that the near-horizon data admit a Taylor series expansion in α ′ . We denote this expansion by 8) and similarly for all near-horizon data, including spinors. For the supersymmetric solutions, we shall assume that that there is at least one zeroth order in α ′ Killing spinor, ǫ [0] = 0.

Supersymmetry
In the previous treatments of heterotic near-horizon geometries [27], it was assumed that the anomaly vanishes and so the Bianchi identity dH = 0 was used to further simplify the structure of the 3-form. Here, we shall not take dH = 0 as there is a non-trivial contribution from the heterotic anomaly, and so the 3-form takes the more general form given in (2.5).
We remark that the KSE of heterotic supergravity have been solved in [25] and [26], and so, the solutions to the KSEs which we consider here correspond to a subclass of the solutions in [25,26]. However for horizons the global assumptions on the spatial section S, like compactness, allow the derivation of additional conditions on the spinors and on the geometry. So it is particularly useful to re-solve the KSEs, decomposing the spinors into positive and negative lightcone chiralities adapted for the Gaussian null basis (2.2), We shall then extract from the KSEs the conditions imposed on ǫ ± that will be useful to apply the global conditions on S.

The gravitino equation
We begin by considering the gravitino equation First, on examining the M = − component of (2.10) we find that where ∂ r φ ± = 0. Next, on examining the M = + component of (2.10), we find where ∂ r η ± = ∂ u η ± = 0. In additon, the M = + component of (2.10) implies a number of algebraic conditions: and We remark that (2.13) and (2.14) are equivalent to and respectively. Furthermore, using these conditions, (2.15) can also be rewritten as Next, we consider the M = i components of (2.10). This implies together with the algebraic condition These conditions exhaust the content of (2.10).

Dilatino and gaugino KSEs
Next again ignoring O(α ′2 ) terms we consider the dilatino KSE On making use of the previous conditions, it is straightforward to show that the dilatino KSE is equivalent to the following three conditions

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and It remains to consider the gaugino KSE This implies the following conditions and In order to simplify these conditions further, we shall first consider the two cases for which either φ  − ≡ 0, and hence it follows that In particular, this condition implies that if η As η These conditions, together with (2.16), imply that Then the dilaton field equation (A.15) implies that and hence it follows that Furthermore, this then implies that and As the ruij component on the r.h.s. of the Bianchi identity is O(α ′2 ) this implies that

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Also, the u-dependent part of (4.18) implies that which gives that∇ Taking the trace of this expression, and using (3.14) yields (3.14) Next, recall that the gravitino KSE (4.22) implies Taking the divergence yields, together with (3.11) and (3.14) the conditioñ So, to summarize, for this class of solutions, we have obtained the following conditions on the fields and it is straightforward to check that the generic conditions on φ + then simplify tõ and The generic conditions on η − also simplify tõ and In the next section, we shall consider the case for which there exists a Killing spinor with φ and (2.22) gives that Taking the divergence of (4.1), and using (2.20) together with the trace of (4.2), we find that An application of the maximum principle (see e.g. [36]) then yields the conditioñ To see this, note that to zeroth order in α ′ , (4.3) implies that∇ and e 2Φ∇j e −2Φ dh ji + 1 2 6) and the Einstein equations imply that Using (4.5), (4.6) and (4.7) it follows that 2 Then, (4.1) also implies that N = h + O(α ′2 ). Substituting these conditions back into (2.16), we find that ∆ [1] = 0 as well, so ∆ = O(α ′2 ). Also, (4.2) implies that (4.10) To summarize the conditions on the bosonic fields; we have shown that for solutions with φ [0] The field equation (A.13) of the 2-form gauge potential can then be rewritten in terms of the near-horizon data as∇ and Furthermore, dilaton field equation can be written as On making use of the conditions (4.11) on the bosonic fields, the KSEs on φ + then simplify further to∇

Supersymmetry enhancement
A key ingredient in the investigation of heterotic horizons is that supersymmetry always enhances. As a result horizons preserve 2, 4, 6 and 8 supersymmetries [27]. However this is based on a global argument which we shall see does not necessarily apply to O(α ′2 ).
As a consequence we shall seek some alternative conditions to guarantee that supersymmetry enhances. In particular we shall show that if there exists a Killing spinor To prove this, it suffices to demonstrate that h leaves all fields invariant and that it is covariantly constant with respect to the connection with torsion∇ on S. Indeed, first note that (B.15) implies that

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In particular, to both zeroth and first order in α ′ , h defines an isometry on S, with h 2 = const + O(α ′2 ). Then the gauge equation (4.13) implies Also, the u-dependent part of (4.21) implies To evaluate this expression, note first that the integrability conditions of∇ from which we obtain the condition and hence, as a consequence of (A.4), It follows that the contribution of i hŘ to the r.h.s. of (5.5) is of at least O(α ′ ), and hence So, we have shown that to both zero and first order in α ′ , the Lie derivative of the metric on S, as well as h, Φ and W with respect to h vanishes, and the Lie derivative ofF with respect to h vanishes to zeroth order in α ′ . Supersymmetry is therefore enhanced, because if η + satisfies (4.26) and (4.27), then so does Conversely, if η − satisfies (4.26) and (4.27), then so does The proof of this makes use of the conditions (5.1), together with (5.2) and (4.19) and (4.23), and the reasoning is identical to that used in [27]. This establishes a 1-1 correspondence between spinors η + and η − satisfying (4.26) and (4.27), so the number of supersymmetries preserved is always even.

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Next we wish to determine whether a similar supersymmetry enhancement argument holds for η + spinors. In particular if there exists a solution to (4.26) and (4.27) with η [0] + = 0 and h [0] = 0, does this imply that the number of η + solutions is equal to the number of η − solutions? This does not follow from a local analysis of (4.26) and (4.27), because there is no analogue of the condition (B.15) acting on η + . Nevertheless, in [27] a global analysis was used in order to establish such a correspondence, by computing the Laplacian of h 2 and applying a maximum principle argument, in order to obtain (5.1) to zeroth order in α ′ . We shall revisit this analysis in section 7.1 including the α ′ corrections.

Geometry
It is a consequence of the results of [27], see also section 7.1, that horizons with non-trivial fluxes preserve an even number of supersymmetries up to O(α ′ ). Furthermore we have also demonstrated that such horizons with η − Killing spinors preserve an even number of supersymmetries up to O(α ′2 ). It is straightforward to see that horizons with more than 8 supersymmetries are trivial, ie the rotation h vanishes. Therefore, the heterotic horizons of interest preserve 2, 4, 6 and 8 supersymmetries.
Up to O(α ′ ), the investigation of the geometry of all such horizons is identical to that given in [27] for heterotic horizons with closed 3-form field strength. Here we shall describe the geometry of the horizons that admit a η − Killing spinor up to O(α ′2 ). We have seen that for such horizons h is parallel with respect to the connection with torsion up to O(α ′2 ). Because of this, the geometry of such horizons is very similar to that of horizons with closed 3-form flux. The only differences between the geometries of the two cases are solely located in the modified Bianchi identity for the 3-form flux. As the two cases are similar, the description of the geometry will be brief.

Horizons with G 2 structure
Such horizons admit two supersymmetries up to O(α ′2 ). In particular h satisfies (5.1). The spacetime locally can be described as a (principal) SL(2, R) fibration over a 7-dimensional manifold B 7 which admits a metric ds 2 (7) and a 3-formH (7) such that the connection∇ (7) with torsionH (7) has holonomy contained in G 2 . The spacetime metric and 3-form flux can be written as where CS(λ) is the Chern-Simons form 3 of the principal bundle connection, and
To find solutions, one has to solve the remaining equations The first condition in (6.4) is required for B 7 to admit a G 2 structure compatible with a connection with skew-symmetric torsion. The second condition is the anomalous Bianchi identity of the 3-form field strength written in terms of B 7 data. The curvatureŘ (8) is that of the near horizon section S with metric and skew symmetric torsion given by Finally, the last two equations in (6.4) imply that both dh and F are g 2 instantons on B 7 .

Horizons with SU(3) structure
Such horizons preserve 4 supersymmetries. Locally the spacetime is a principal bundle with fibre SL(2, R) × U(1) over a Kähler with torsion manifold (KT) B 6 with Hermitian form ω (6) . The metric and 3-form field strength of the spacetime can be written as where λ a , a = +, −, 1, 6 is the principal bundle connections whose a = +, −, 1 components are as in (6.2) and which is along the u(1) direction in the Lie algebra. h 2 = k 2 is constant up to O(α ′2 ). The curvature of the principal bundle connection λ a is expressed in terms of dh and dℓ which are 2-forms on B 6 and it is required to satisfy that The KT manifold B 6 is in addition conformally balanced, ie where θ is the Lee form and the torsion is

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The dilaton Φ depends only on the coordinates of B 6 . The gauge connection is a su (3) instanton on B 6 , i.e.
To find examples for such horizons two additional conditions should be satisfied. One is the restriction thatR This arises from requirement that the U(3) structure on B 6 lifts to a SU(3) structure on the spacetime or equivalent the spatial horizon section S. The other is the anomalous Bianchi identity which now reads whereŘ (8) is the curvature of the connection with torsion on S for which the metric and torsion are given by (6.14) Note that∇ (8) has holonomy contained in SU(3) and soŘ (8) is a well defined form on B 6 .

Horizons with SU(2) structure and 6 supersymmetries
The spacetime is locally a SL(2, R) × SU(2) principal fibration over a 4-dimensional antiself-dual Weyl Einstein manifold B 4 with metric ds 2 (4) and quaternionic Kähler structure 2-forms ω r ′ (4) . The spacetime metric and 3-form field strength can be expressed as whereH (4) = −⋆ 4 de 2Φ , the principal bundle connection λ a for a = +, −, 1 coincides with that of (6.2) while are the components along the su(2) subalgebra of the fibre. Furthermore the dilaton depends only on the coordinates of B 4 , dh as well as the curvature (F sd ) r ′ of λ r ′ are 2-forms on B 4 . In addition, we have that and dh ad , (F ad ) r ′ and F ad are not restricted, where the self-dual and anti-self dual components are appropriately denoted. Geometrically, the set up is such that the SO(4) = SU(2) · SU(2) structure of B 4 when lifted the 7-dimensional manifold which is the principal bundle with fibre SU(2) reduces to SU(2) as required from supersymmetry.

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The only remaining condition to find solutions is AgainŘ (8) is the curvature of the connection with torsion of the horizon section S which has metric and 3-form field strength AsR (8) has holonomy contained in SU (2),Ř (8) is a 2-form on B 4 . For more details on the geometry of heterotic backgrounds that preserve 6 supersymmetries and have SU (2) holonomy see [27,28].

Horizons with SU(2) structure and 8 supersymmetries
This class of horizons have a similar geometry to those of the previous section that preserve 6 supersymmetries. The differences are that so F r ′ is an anti-self dual instanton on B 4 which now is a hyper-Kähler manifold with respect to the metric ds 2 (4) . Furthermore the equation for the dilaton (6.18) now reads Therefore at zeroth order, a partial integration argument reveals that Thus B 4 up to a local isometry is AdS 3 × S 3 × T 4 or AdS 3 × S 3 × K 3 and the dilaton is constant. One does not expect additional α ′ corrections to the geometry in the case that theŘ (8) is identified with F . Though additional corrections are expected otherwise. In the absence of 5-branes, consistency requires that the Pontryagin number of the tangent bundle of B 4 cancels that of the gauge bundle which is the vanishing condition for the global anomaly. We shall revisit the global analysis of [27] by calculating the Laplacian of h 2 , but including also α ′ correction terms. Then we shall examine the conditions imposed on the geometry by this expression. To avoid the trivial case when

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Next we calculate the Laplacian of h 2 to find that In computing this expression, we made use of the Einstein equation (4.16) together with the gauge field equations (4.13) and (4.14). We remark that the calculation proceeds in exactly the same way as in [27]; the α ′ terms in (7.1) originate from the α ′ terms in 2h i h jR ij . It should be noted that in order to fully control O(α ′2 ) terms in this expression, one would require to know the Einstein equations up to and including α ′2 .
To begin, we consider (7.1) to zeroth order in α ′ . We then re-obtain the conditions found in [27] via a maximum principle argument, i.e.
In particular, it follows from these conditions that and also Furthermore, it also follows that if η + satisfies (4.26), then Γ − h i Γ i η + also satisfies (4.26) to zeroth order in α ′ . The integrability conditions therefore imply that and henceŘ On substituting these conditions back into (7.1) one finds that the remaining content of (7.1) is∇ On integrating the O(α ′ ) part of (7.7) over the zeroth order horizon section, one finds that and furthermore It should be noted however that (7.1) does not in general imply (5.1). In particular, the conditions obtained from the analysis of the properties of h 2 are not sufficient to imply that if η + , with η

Lichnerowicz type theorems
Next we shall investigate whether it is possible to identify Killing spinors with the zero modes of a suitable Dirac-like operator, by constructing a generalized Lichnerowicz type theorem which incorporates the near-horizon fluxes. Such Lichnerowicz type theorems have been established for near-horizon geometries in D=11 supergravity [9], type IIB [10] and type IIA supergravity (both massive and massless) [11,12], as well as for AdS geometries in ten and eleven dimensional supergravity [33,[37][38][39].
To begin, let us first define the modified connection with torsion and the modified horizon Dirac operator, respectively It is clear that if η ± is a Killing spinor, i.e.
Here we want to investigate the extent to which the converse is true. We shall show that if and moreover dh ij Γ ij η ± = O(α ′ ), andF ab ij Γ ij η ± = O(α ′ ) . (7.14) In order to obtain this result, we begin by considering the following functional where c ∈ R, and we assume all the field equations. After some algebra, which is described in appendix D, we find which is true if and only if q = 1 12 + O(α ′2 ) and c = −2 + O(α ′2 ), and the Ψ is defined as follows

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The values of q and c are fixed by requiring that certain terms in the functional (7.15), which cannot be rewritten in terms of the Dirac operator D, or A † A, and which have no fixed sign, should vanish. The part of (7.16) which is of zeroth order in α ′ implies that if 0 < κ < 1 48 , then and establishes the first part of the theorem. Next the integrability condition of which in turn implies thatŘ Hence we shall neglect the term in (7.16) which is quadratic inŘ, as this term is O(α ′3 ). Then, assuming (7.18), the part of (7.16) which is first order in α ′ further implies (7.14). This completes the proof.

Description of the backgrounds
We have proven that for near horizon geometries the necessary and sufficient conditions imposed by supersymmetry on the spinors can be reduced to (4.26) and (4.27). In this section, we shall consider the case for which the supersymmetry is explicitly partially broken, in the sense that the gravitino KSE (4.26) admits solutions but not dilatino one (4.27).
We also assume that the fields satisfy These conditions were previously obtained via the supersymmetry analysis; here we shall assume them. In particular, all of the conditions obtained from the global analysis of the Laplacian of h 2 in section 7 remain true. As a consequence of this, However we do not assume that∇h = O(α ′2 ). One consequence of these assumptions is that none of the spacetime Killing spinor equations are satisfied even at O(α ′ ). In particular, the spacetime gravitino KSE requires in addition the condition that dh ij Γ ij η + = O(α ′ ) which is not one of our requirements.
In what follows, we shall investigate the consequences of the above assumptions on the geometry of the spatial horizon sections S. We shall also comment on the special case where∇h = O(α ′2 ).

Additional parallel spinors
A key property of backgrounds that satisfy the gravitino KSE but not the dilatino one is the existence of additional parallel spinors, see also appendix C. In the present context to show this focus on the spinor η + ; a similar analysis can be undertaken for the η − spinors. To proceed, it will be useful to define so that the algebraic condition (4.27) on η + is equivalent to Aη + = O(α ′2 ). We then note the useful identitỹ The integrability conditions of (4.26) imply that and hence Integrating this expression over S yields the conditions and substituting these conditions back into (8.5) then implies that Therefore the spinor τ + = Aη + is also∇-parallel. As τ + has opposite chirality from η + cannot be identified as an additional Killing spinor within the heterotic theory. Nevertheless it is instrumental in the description of the geometry of S.
In what follows we shall show that V is a symmetry of all the fields of the spatial horizon section.
As τ [0] + = 0, this implies that V [0] = 0. In addition, as η + and τ + satisfŷ so that V 2 = const. + O(α ′2 ), and V is an isometry of S to both zero and first order in α ′ . Next, we consider the relationship of V to h. In particular, the spinors h i Γ i Aη + and V i Γ i Aη + are both parallel with respect to∇ at zeroth order in α ′ . As we have assumed that (8.9) admits only one solution, there must be a nonzero constant c such that In addition, we have As a consequence of (8.2), one has that i h dh = O(α ′ ), and from the global analysis of the Laplacian of h 2 , we find i hF = O(α ′ ) as well asŘ mnij h m = O(α ′ ). These conditions imply that and so W is invariant.

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Hence we have So, on taking the Lie derivative of the trace of (4.16) with respect to V we find and hence, as a consequence of the field equation (4.13), we find Also, on taking the Lie derivative of the dilaton field equation (4.17), we get On taking the sum of (8.23) and (8.24), we find and We know L h Φ = O(α ′ ) as a consequence of the analysis of the Laplacian of h 2 , so f = α ′ f [1] + O(α ′2 ). Then, on integrating, (8.26) implies that so f [1] = β for constant β, and so As we require that Φ must attain a global maximum on S, at this point L V Φ = 0 to all orders in α ′ , for any V . This fixes β = 0, so which proves the invariance of Φ. Next, we consider L V h. On taking the Lie derivative of the field equation of the 2-form gauge potential (4.15) we find

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where we have used As Λ j Γ j Aη + and h j Γ j Aη + are both parallel with respect to∇ at zeroth order in α ′ , it follows as a consequence of (ii) that we must have for constant b. It is also useful to compute which follows because h 2 = const + O(α ′2 ), and∇V = O(α ′2 ). This implies that b = 0, and hence So V is a symmetry of the full solution to both zeroth and first order in α ′ .

Geometry
We have shown that V is a symmetry of the backgrounds up O(α ′2 ). To investigate further the geometry of the horizon section S, let us first consider the consequences of the existence of the η + Killing spinor. As the isotropy group of η + in Spin (8) is Spin (7), the fundamental self-dual 4-form φ of Spin(7) on S is∇-parallel. It is known that in such a case, the torsion 3-form W can be uniquely determined in terms of φ and the metric without any additional conditions on the Spin(7) structure of S [40]. Next the condition∇τ + = O(α ′2 ) with τ + = Aη + is equivalent to requiring that where θ φ is the Lee form of φ, see [25]. As a result 2dΦ + h − θ φ is a parallel 1-form. If it is not linearly dependent on V , it will give rise to an additional solution for the gravitino KSE S. As we have assumed that there is strictly one parallel spinor of the same chirality as η + , we have to require that for some non-zero constant λ; for λ = 0 the dilatino KSE is satisfied as well.

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Let us next turn to investigate the G 2 structure on S. As V is an isometry on S and i V W = dV , setting V 2 = ℓ 2 + O(α ′2 ) for ℓ constant, we can decompose the metric and 3-form as where ds 2 (7) is the metric on the space orthogonal to V and i V W (7) = 0. The data (ds 2 (7) , W (7) ) are thought (locally) as the metric torsion on the space of orbits M 7 of V . For this observe that L V W (7) = 0 and as i V W (7) = 0, W (7) descends as a 3-form on the space of orbits.
The spatial horizon section S admits a G 2 structure with fundamental form ϕ = ℓ −1 i V φ as∇ϕ = O(α ′2 ). The question is whether this G 2 structure descends on the space of orbits of V . First observe that i V ϕ = 0. So it remains to investigate whether L V ϕ = O(α ′2 ). For this notice that under G 2 representations dV decomposes as dV = dV 7 + dV 14 As dV 7 is a vector in S orthogonal to V , if it is not vanishing will generate an additional ∇-parallel spinor on S of the same chirality as η + . As we have restricted the number of such spinors to one, we have to set dV 7 = O(α ′2 ). It has been shown in [25] that a∇-parallel k-form α is invariant under the action of a∇-parallel vector V , iff the rotation i V W leaves the form invariant. As i V W = dV + O(α ′2 ) and dV takes values in g 2 , we conclude that and so M 7 admits a G 2 structure compatible with connection with skew-symmetric torsion given by the data (ds 2 (7) , W (7) ). In such a case W (7) can be determined uniquely in terms of ϕ and ds 2 (7) provided a certain geometric constraint is satisfied [41]. It remains to explore (8.40) from the perspective of M 7 . Let us decompose h = V +h ⊥ , where g(V, h ⊥ ) = 0. Then (8.40) can be written as where θ ϕ is the Lee form of ϕ on M 7 . The former determines the singlet part of W (7) in terms of V and h while the latter imposes the dilatino KSE on M 7 .
9 Nearly supersymmetric horizons with additional parallel spinors Using these we define the 1-form and 2-form spinor bilinears V and ω by and also letṼ = i V ω .
We also defineh byh = i h ω , The main task below is to show that both V andṼ leave invariant all the fields on S, and that they generate a R ⊕ R lie algebra. As V andṼ are∇-parallel, they are Killing. Next consider the invariance of W . The spinors V j Γ j Aη + must satisfy (8.7), it follows that, at zeroth order in α ′ ,F and dh are (1, 1) traceless with respect to the almost complex structure obtained from ω. This, together with the conditions i h dh = O(α ′ ) and i hF = O(α ′ ), which follow from the global analysis of the Laplacian of h 2 , implies that and hence It is also useful to consider the spinors η It follows that i h dW = O(α ′2 ) and ihdW = O(α ′2 ), as a consequence of the Bianchi identity, and therefore i V dW = O(α ′2 ). Thus we have shown that This proves the invariance of W . Next we consider L V Φ. It follows from (9.7) that and also LhW = O(α ′ ) .

(9.17)
Sinceh is an isometry of S to zeroth order in α ′ , we also have On taking the Lie derivative of the trace of (4.16) with respect toh, we find which is equivalent, if g = LhΦ, to∇

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Since V is an isometry of S to first order in α ′ , we have which follows from Using the conditions (9.24), (9.25), (9.26) and (9.27), we follow the analysis for the G 2 case of the previous section undertaken from the equation (8.22) to (8.29), and conclude that which proves the invariance of the dilaton Φ.
Next we consider L V h. Equations (8.30) and (8.31), which have been established in the previous section, hold here as well after using in the addition that Then it follows that∇ Furthermore we notice that Lhh = O(α ′ ) . (9.32) As V = ch +ch + O(α ′ ), it is convenient to write Then it follows that the spinors Ψ j Γ j Aη + , h j Γ j Aη + andh j Γ j Aη + are all parallel with respect to∇ at zeroth order in α ′ . In order for (9.1) to admit exactly two solutions, we must have for constants b andb. Then using i h L V h = O(α ′2 ), which has been computed in (8.37), and h 2 = const. + O(α ′2 ), it follows that b = O(α ′ ) and therefore Since V and ω are both parallel with respect to∇ to first order in α ′ , we also havễ Then the analysis undertaken for V holds as well forṼ , because the only properties of V used through the analysis are that V , at zeroth order in α ′ , is a linear combination of h andh with constant coefficients, and V is parallel with respect to∇ to first order in α ′ . Thus we argue in a similar way that for a constantq. Finally, the V andṼ commute up to O(α ′2 ). To see this observe that since i VṼ = 0 and i V W = dV + O(α ′2 ), we have that (9.40)

Using (C.3) adapted to S as well as
Therefore the vector iṼ i V W is∇-parallel and moreover is orthogonal to both V andṼ .
So if it is non-zero, it will generate additional∇-parallel η + spinors on S. As we have restricted those to be strictly two, we conclude that iṼ i V W vanishes and so In particular as i VṼ = 0, we have that This concludes the examination of the symmetries of S.

Geometry
It is clear from the examination of the symmetries of the fields on S and in particular (9.5) and (9.43) that we can set where V 2 =Ṽ 2 = ℓ 2 + O(α ′2 ) and ℓ is constant, ds 2 (6) is the metric in the orthogonal complement of V andṼ and i V W (6) = iṼ W (6) From construction S admits an SU(3) structure. We shall now investigate whether this (locally) descends on the space of orbits M 6 of V andṼ . First the data (ds 2 (6) , W (6) ) define a Riemannian geometry on M 6 with skew-symmetric torsion. In particular for the torsion this follows from i V W (6) = iṼ W (6) = O(α ′2 ) and L V W (6) = LṼ W (6) = O(α ′2 ).
Thus M 6 admits an almost Hermitian structure compatible with a connection∇ (6) with skew-symmetric torsion W (6) . It is well known that in this case W (6) is determined in terms of the almost complex structure on M 6 and the metric, see eg [42]. To find whether M 6 inherits a SU(3) structure as well, let investigate whether the (3,0) fundamental SU(3) form χ of S descends on M 6 . It can always be arranged such that i V χ = iṼ χ = 0. So it remains to see whether χ is invariant under the action of V andṼ . For this a similar argument to that explained above for ω (6) leads to the assertion that χ is invariant iff the ω-traces iṼ W · ω and iṼ W · ω of iṼ W and iṼ W , respectively, vanish. Furthermore, an application of (C.3) implies that both iṼ W · ω and iṼ W · ω are constant but not necessarily zero. Thus M 6 has generically a U(3) structure instead of an SU(3) one.
It remains to investigate the rest of the content of the conditions∇τ (a) + = O(α ′2 ). First consider the (3,0) part of W (6) denoted by W 3,0 (6) . An application of (C.3) using that dW is a (2,2) form yields that∇ Thus W 3,0 (6) is another globally defined∇-parallel (3,0)-form on S and so it can either be set to zero or be identified with χ. In the former case, the complex structure on M 6 is integrable and so M 6 is a KT manifold [43].

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Now if 2dΦ + h ⊥ − θ ω (6) in non-vanishing and since it is orthogonal to V andṼ will give rise to more than two η +∇ -parallel spinors on S. Since we have assumed that there are just two, we set 2dΦ This concludes the investigation of the geometry.

Assumptions and definitions
It is known that if one requires the existence of an additional∇-parallel spinor η + to those of the SU(3) backgrounds on S, then the isotropy algebra of the all the five spinors reduces to su (2). As a result, S admits 8∇-parallel spinors and the holonomy group reduces to a subgroup SU (2). To describe the geometry of backgrounds with exactly 8 such spinors, we consider four linearly independent spinors η Furthermore, we also define 1-form and 2-form spinor bilinears V (a) and ω r , respectively, by and the almost complex structures associated to ω r satisfy the algebra of unit quaternions. These follow from (8.8) and the su(2) isotropy of the parallel spinors.

Symmetries of the horizon section
It is clear from (9.55) that V (a) , V (r) = V r , generate isometries on S and that where i a denotes inner-derivation with respect to V (a) . Without loss of generality we choose g(V (a) , V (b) ) = ℓ 2 δ ab + O(α ′2 ) for ℓ constant. An investigation similar to the one explained in section 9.1.1 reveals that Thus the commutator is either linear dependent on V (a) or it will lead to further reduction of the holonomy of∇ to {1}. In the latter case, the horizon section S will admit more than four η +∇ -parallel spinors violating our assumptions. Thus, we conclude that for some constants f with ). As f is skew-symmetric, the Lie algebra spanned by V (a) is a metric (compact) Lie algebra. As it has dimension 4, it is either isomorphic to ⊕ 4 u(1) or to u(1) ⊕ su (2). Therefore the horizon section S can be viewed locally as a fibration with fibre either × 4 U(1) or U(1) × SU(2) over the space of orbits M 4 of V (a) . We shall determine the geometry of S by specifying the geometry of M 4 .

Geometry
To simplify the analysis, we choose up to an so(4) rotation V to be along a u(1) direction in either ⊕ 4 u(1) or u(1) ⊕ su (2). This in particular implies that i 0 i r W = O(α ′2 ). Then the metric and torsion of S can be written as where V (a) is viewed as a principal bundle connection and CS(V r ) is the Chern-Simons form which for the ⊕ 4 u(1) case is The data (ds 2 (4) , W (4) ) define a geometry on M 4 with skew-symmetric torsion. First, let us investigate the reduction of the almost HKT structure of S on M 4 . For this observe that where i a ω (4) r . As both V (a) and ω r , the Lie derivative vanishes. Next let us investigate the two cases ⊕ 4 u(1) and u(1) ⊕ su(2) separately. In the abelian case, as i a i b W = O(α ′2 ), i a W is a 2-form on M 4 . Furthermore L a ω (4) r vanishes iff the self-dual part, i a W sd , of i a W is zero. However in general this may not be the case. An application of (C.3) implies that∇ and so there exist some constants u such that i a W sd = u a r ω (4) r + O(α ′2 ) , (9.64) otherwise the holonomy of∇ will be reduced further and it will admit more than four η + parallel spinors. Then L a ω (4) r = 2u a s ǫ sr t ω The covariant constancy condition on M 4 now readŝ where now V (a) should be thought as the pull back of the principal bundle connection V (a) with a local section. It is clear that the relevant connection that determines the geometry of M 4 is Z s = V (a) u s a . If u r a = 0, M 4 is a HKT manifold. It is easy to see this as ω r are covariantly constant with respect to a connection with skew-symmetric torsion and all three almost complex structures are integrable. The latter follows because of dimensional reasons. Otherwise one of the 3-vectors u a must be non-zero. Without loss of generality take u 0 = 0. In such a case the above equation can be solved as (u r a ) = (u r 0 , u r 0 v s ), where v s = |u 0 | −2 r u r s u r 0 . Using these data, the covariant constancy condition of ω (4) r on M 4 can be written aŝ It is clear from this that M 4 is a KT manifold with respect to the Hermitian form |u 0 | −1 u r 0 ω r . In fact M 4 is an (almost) 4 QKT manifold [44] for which the holonomy of the Sp(1) connection has been reduced to U(1).
Next let us turn to examine the non-abelian u(1) ⊕ su(2) case. It is easy to see that where M 4 = S 1 × S 3 with the bi-invariant metric and constant dilaton. Such a horizon is not supersymmetric but it is nearly supersymmetric. It remains to consider the case u 0 = 0. One can then show that det u = 0 and so (u r s ) is invertible. Thus Z s = V (a) u s a takes values in the sp(1) Lie algebra. M 4 is a QKT manifold, see also [28].
To conclude we remark that in all HKT and KT cases, there is an analogue of the condition (9.49) for every Hermitian form ω r that determines these structures. If the JHEP10(2016)121 associated 2dΦ + h ⊥ − θ r forms do not vanish, then the holonomy of the connection with torsion reduces to {1} and the number of parallel spinors enhance to 16. The solutions are the group manifolds. The solution AdS 3 × S 3 × S 3 × S 1 mentioned above belongs to the class where the holonomy of the connection with torsion is {1}.
There is an analogue of this in the QKT case but in such a case the condition from the perspective of M 4 twists with sp(1). If 2dΦ + h ⊥ − θ r do not vanish, again the holonomy of the connection with torsion on S reduces to {1}. However now some of the data like the Hermitian forms are not (bi-)invariant under the action of the group. It would be of interest to explore his further to see whether there are actual solutions.
We conclude the examination of the geometry of nearly supersymmetric backgrounds in the G 2 , SU(3) and SU(2) cases by pointing out that they exhibit an sl(2, R) up to order O(α ′ ) but not up to order O(α ′2 ). For the latter, h must be a symmetry of the theory up to the same order and so it can be identified with V . The description of the geometry of this special class of nearly supersymmetric backgrounds is very similar to the one we have given above. The only difference is that now we can identify h with V .

Conclusions
We have investigated the supersymmetric near-horizon geometry of heterotic black holes up to and including two loops in sigma model perturbation theory. Using a combination of local and global techniques, together with the bosonic field equations and Bianchi identities, we have proven that the conditions obtained from the KSEs are equivalent to a pair of gravitino equations (4.26) and a pair of algebraic conditions, related to the dilatino KSE, (4.27), which are required to hold at zeroth and first order in α ′ . In particular, we have shown that the KSE related to the gaugino is implied by the other KSEs and field equations.
In all cases, we have also shown that there are no regular AdS 2 solutions with compact without boundary internal space by demonstrating that ∆ = O(α ′2 ). This is not in contradiction with the fact that one can locally write AdS 3 as a warped product over AdS 2 [45], see also appendix E. This is because our assumptions on the internal space of AdS 2 are violated in such a case.
Furthermore, we have demonstrated that horizons that admit a non-vanishing η − Killing spinor up to order O(α ′2 ), which does not vanish at zeroth order in α ′ , exhibit supersymmetry enhancement via the same mechanism as described in [27], and so preserve 2, 4, 6 and 8 supersymmetries. We have described the geometry of such horizons in all cases and this is similar to that presented in [27] for the horizons with dH = 0.
We have also considered in some detail the global properties of our solutions. The analysis of the global properties of h 2 proceeds in much the same way as in the heterotic theory with dH = 0. However in the presence of anomaly, the consequences of the global restrictions on the geometry of the horizons are somewhat weaker. For example, it is only possible to prove that h is an isometry of the horizon section to zeroth order in α ′ . So one cannot establish a direct algebraic relation between η + and η − spinors to order O(α ′2 ), and JHEP10(2016)121 therefore it is not possible to directly show that there is supersymmetry enhancement via this mechanism, as was done in [27] for the theory with dH = 0.
We have also constructed generalized Lichnerowicz type theorems, which relate spinors which are parallel with respect to a certain type of near-horizon supercovariant derivative, to zero modes of near-horizon Dirac operators. We have shown that if η is a zero mode of the near-horizon Dirac operator to both zero and first order in α ′ , then the Lichnerowicz theorems imply that η only satisfies the KSE (4.26) and (4.27) to zero order in α ′ . Hence, the types of arguments used to show supersymmetry enhancement via Lichnerowicz type theorems in [9][10][11][12] also do not work to the required order in α ′ for the heterotic theory.
Finally, we have examined a class of nearly supersymmetric horizons for which the gravitino KSE is allowed to admit solutions on the spatial horizon section but not the rest of the KSEs. Such solutions in general do not admit any spacetime Killing spinors including solutions of the gravitino KSE. Under some conditions on the fluxes, we investigate the geometry of the spatial horizon sections using a combination of local and global techniques as well as the field equations. We find that those with a G 2 , SU(3) and SU(2) structure admit 1, 2 and 4 parallel vectors on the spatial horizon sections with respect to the connection with torsion. The geometry on the orbit spaces of these isometries is fully specified.
The spacetime of both supersymmetric, and nearly supersymmetry horizons considered here admits a SL(2, R) symmetry at zeroth order in α ′ . In the supersymmetric case for which there is a η − Killing spinor to order O(α ′2 ) such that η − does not vanish at zeroth order, η It is not apparent whether the properties of the heterotic horizons described here are going to persist to higher than two loops in sigma model perturbation theory. It is likely though that the presence of an sl(2, R) symmetry will persist after perhaps a suitable choice of a scheme in perturbation theory. There is no apparent reason to hypothesize that such a symmetry can be anomalous at higher loops. What happens to global properties of the horizons, for example the Lichnerowicz type theorems, is less clear. We have already seen that these theorems do not hold to the expected order in α ′ even at two loops. This can be taken as an indication that additional higher order corrections may further weaken the consequences of such theorems.

A.1 Spin connection and curvature
In our conventions, the curvature of a connection Γ is given by We define connections∇ and∇ as followš for vector field ξ, where ∇ is the Levi-Civita connection. In particular, theŘ curvature tensor can be written aš where R is the Riemann curvature tensor. Also, note thať We also define connections∇ and∇ on the horizon section S viǎ for vector fields Y on S, and where∇ is the Levi-Civita connection of S, and we denote the curvatures of the connections∇,∇ and∇ byR,Ř andR respectively. The non-vanishing components of the spin connection in the frame basis (2.2) of the near horizon metric (2.1) are , whereΩ denotes the spin-connection of the spatial horizon section S in the e i basis. If f is any function of spacetime, then frame derivatives are expressed in terms of co-ordinate derivatives as The non-vanishing components of the Ricci tensor is the basis (2.2) are whereR is the Ricci tensor of the horizon section S in the e i frame.

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We remark that the non-vanishing components of the Hessian of Φ, are given by where in the above expression, we have set ∆ = 0. The non-vanishing components of theŘ curvature tensor in the basis (2.2) arě where in the above expression, we have set ∆ = 0, N = h and Y = dh. Note that thě R −i,+j andŘ ij,+k terms give no contribution to the Bianchi identity of H or to the Einstein equations, becauseŘ M N,−i = 0 for all M, N .

A.2 Bosonic field equations
The Bianchi identity associated with the 3-form is where tr(F ∧ F ) = F a b ∧ F b a (a, b are gauge indices on F ). The Einstein equation is The gauge field equations are and The dilaton field equation is This completes the list of field equations. We have followed the conventions of [46].

JHEP10(2016)121 B Further simplification of the KSEs
Here we shall show that the independent KSEs are given in (4.26) and (4.27). We first note that the conditions on the bosonic fields (3.18) (obtained from the case when φ [0] + ≡ 0) actually imply those of (4.11) (corresponding to the φ where we have made use of the field equations (4.13) and (4.15), together with the algebraic conditions (4.20) and (4.24). Next, on substituting the dilaton equation and the Bianchi identity into the above expression, one finds We begin by considering the condition which (B.5) imposes on φ + :

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To zeroth order this gives which implies that the second term on the l.h.s. of (B.6) is of O(α ′2 ), and hence can be neglected. Using this, (B.6) gives that Using this the third term on the l.h.s. of (B.6) is also of O(α ′2 ). So, the remaining content of (B.6) is Hence, we have proven that the KSE (4.18) and (4.20) imply the algebraic KSE (4.19) and (4.21). Next, we consider the condition which (B.5) imposes on φ − , which is However, note also that the u-dependent part of (4.18), with (4.22), implies that On contracting this expression with Γ i , we find and on substituting this expression into (B.11) we get Hence, we find from exactly the same reasoning which was used to analyse the conditions on φ + , that (4.22) and (4.24) imply (4.23) and (4.25). So, on making use of the field equations, it follows that the necessary and sufficient conditions for supersymmetry simplify to the conditions (4.18) and (4.20) on φ + , and to (4.22) and (4.24) on η − . We remark that the u-dependent parts of the conditions (4.18) and (4.20) also impose conditions on η − . We shall examine the conditions on η − further in the next section, and show how these may be simplified.

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B.2 Elimination of u-dependent parts of (4.18) and (4.20) We begin by considering the u-dependent parts of (4.18) and (4.20), assuming that (4.22) and (4.24) hold. The u-dependent part of the condition on φ + obtained from (4.18) is and the u-dependent part of the algebraic condition (4.20) is given by On adding h ℓ Γ ℓ acting on (4.24) to the above expression, we find that (B.16) is equivalent to the condition where we have also made use of the field equation (4.13). On contracting (B.15) with Γ i , it then follows that (B.17) is equivalent to However, as shown in the previous section, this condition is implied by (4.22) and (4.24) on making use of the field equations. So, it remains to consider the condition (B.15). First, recall that the integrability conditions of the gravitino equation of (4.22) is given bŷ On contracting with Γ j , one then obtains where we have used the gauge equation (4.15). Also, on taking the covariant derivative of the algebraic condition (4.24), and using (4.22), one also finds the following mixed integrability condition On eliminating the∇ i W jkℓ Γ jkℓ terms between (B.20) and (B.21), one obtains the condition

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Next, we substitute the Einstein equation (4.16) in order to eliminate the Ricci tensor, and also use the Bianchi identity for dW . One then obtains, after some rearrangement of terms, the following condition The α ′ terms in the above expression can be neglected, as they all give rise to terms which are in fact O(α ′2 ). This is because of the conditions (4.23) and (4.25), which we have already shown follow from (4.22) and (4.24), together with the bosonic conditions, as well as the fact thatŘ which follows from the integrability condition of (4.22). It follows that (B.23) implies (B.15).

C A consistency condition
Suppose that we consider the Bianchi identity associated with the 3-form as where R is a spacetime curvature which will be specified later. Also observe that the 2-form gauge potential and the Einstein equation can be written together aŝ Then one can establish by direct computation that Using this and the field equations of the theory, one can derive the relation If ǫ satisfies that gravitino KSE, the left hand side of this relation vanishes. Furthermore the right-hand-side vanishes as well provided that dilatino and gaugino KSEs are satisfied, and in addition Of course in heterotic string perturbation theory

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as a consequence of the gravitino KSE and the closure of H at that order. Thus one can set R =Ř and the identity (C.4) will hold up to order α ′2 . One consequence of the identity (C.4) is that if the gravitino KSE and gaugino KSEs are satisfied as well as (C.5) but the dilatino is not, then the gravitino KSE admits an additional parallel spinor of the opposite chirality. Such kind of identities have been established before for special cases in [42]. Here we have shown that such a result is generic in the context of heterotic theory.

D Lichnerowicz theorem computation
In this appendix, we present the details for the calculation of the functional I defined in (7.15), and show how the constants q and c are fixed by requiring that certain types of terms which arise in the calculation should vanish. We begin by considering the calculation at zeroth order in α ′ , and then include the corrections at first order in α ′ . We remark that we shall retain terms of the type h i∇ i Φ throughout. This is because although these terms vanish at zeroth order in α ′ as a consequence of the analysis in section 8, it does not follow from this analysis that L h Φ = O(α ′2 ). However, as we shall see, it turns out that the coefficient multiplying the terms h i∇ i Φ, which depends on the constants q and c, vanishes when one requires that several other terms in I vanish as well. So these terms do not give any contribution to I at either zeroth or first order in α ′ .

D.1 Computations at zeroth order in α ′
Throughout the following analysis, we assume Einstein equations, dilaton field equation and Bianchi identity at zeroth order in α ′ . To proceed, we expand out the definition of ∇ (κ) i and D in I, obtaining the following expression Now, after writing∇ in terms of the Levi-Civita connection∇ and after integrating by parts, the expression (D.1) decomposes into where and
E AdS n+1 as warped product over AdS n The AdS n+1 space can be written as a warped product over AdS n . This has been observed before in [45] for AdS 3 and elsewhere, see eg [47]. For this, we label all geometrical objects defined on AdS n+1 and AdS n by n + 1 and n respectively, e.g. ds 2 n+1 is the metric on AdS n+1 and ds 2 n is the metric on AdS n . In principle AdS n+1 and AdS n can have different radii, which are indicated by ℓ n+1 and ℓ n respectively. Coordinates on AdS n+1 are taken to be as follows x I = (x 0 , x i ) , x 0 ≡ y , i = 1, . . . , n . (E.1) We shall begin with an Ansätz for the metric on AdS n+1 as a warped product over AdS n , i.e. ds 2 n+1 = dy 2 + f (y) 2 ds 2 n .

(E.2)
We want to determine the necessary and sufficient conditions to impose on f (y) in order for ds 2 n+1 to be the metric on AdS n+1 . To succeed, we have to impose the fact AdS n+1 is a maximally symmetric space. Locally, the necessary and sufficient condition is that the Riemann tensor must assume the following form