Geometry and supersymmetry of heterotic warped flux AdS backgrounds

We classify the geometries of the most general warped, flux AdS backgrounds of heterotic supergravity up to two loop order in sigma model perturbation theory. We show under some mild assumptions that there are no $AdS_n$ backgrounds with $n\not=3$. Moreover the warp factor of AdS$_3$ backgrounds is constant, the geometry is a product $AdS_3\times M^7$ and such solutions preserve, 2, 4, 6 and 8 supersymmetries. The geometry of $M^7$ has been specified in all cases. For 2 supersymmetries, it has been found that $M^7$ admits a suitably restricted $G_2$ structure. For 4 supersymmetries, $M^7$ has an $SU(3)$ structure and can be described locally as a circle fibration over a 6-dimensional KT manifold. For 6 and 8 supersymmetries, $M^7$ has an $SU(2)$ structure and can be described locally as a $S^3$ fibration over a 4-dimensional manifold which either has an anti-self dual Weyl tensor or a hyper-K\"ahler structure, respectively. We also demonstrate a new Lichnerowicz type theorem in the presence of $\alpha'$ corrections.


Introduction
For over 30 years, AdS backgrounds have found widespread applications originally in supergravity [1] and more recently in string theory compactifications, and in AdS/CFT, see reviews [2,3,4]. Despite many developments, the geometry of all such backgrounds has not been specified and their classification remains an open problem, for some selected publications see [5]- [17]. Only recently, the fractions of supersymmetry preserved by flux, warped AdS backgrounds of 11-dimensional and type II supergravities have been specified in [18,19,20] without making any assumptions on the form of the fields and that of the Killing spinors apart from imposing the symmetries of AdS space on the former. The computation of these fractions has also uncovered a new class of Lichnerowicz type theorems.
To make progress towards the classification of AdS backgrounds, it is instructive to have a paradigm where all the issues that arise can be worked out in a simpler setting than that of the 11-dimensional and type II supergravities. Such theories are the common sector of type II supergravities and the heterotic supergravity. One simplification that occurs in the common sector and heterotic supergravities is that all solutions of the KSEs are known and the geometry of the backgrounds has been determined [21]. In particular, AdS 3 configurations arise naturally in the context of classification of supersymmetric heterotic backgrounds [21] and their geometry has also been examined 1 in [22]. The AdS backgrounds are also special cases of the heterotic horizons in [23] though the latter has been explored only for a closed 3-form field strength H. In addition, there is a nogo theorem for AdS 4 heterotic backgrounds that has been established in [24] without assuming supersymmetry but imposing smoothness on the fields and compactness of the transverse 6-dimensional space.
The α ′ corrections to heterotic backgrounds can be investigated in the perturbative or sigma model approach, see eg [25] and references within. In this, the field equations of the theory are the vanishing conditions of the sigma model beta functions which are expressed as a series in α ′ , the string tension. Furthermore, the Bianchi identity of the 3-form field strength gets corrected as part of a mechanism which cancels the chiral reparameterization and gauge anomalies of the 2-dimensional sigma model. This is referred to as the anomaly cancelation mechanism. As a result H is not closed any longer at one and higher loops. Instead, H at one-loop obeys the "anomalous Bianchi identity" in which dH is equal to the difference between the Pontryagin form of the gravitational sector and that of the gauge sector of the heterotic theory. After the modification of the Bianchi identity of H at one loop, consistency of the theory requires the inclusion of the two loop correction to the field equations. Generic heterotic backgrounds receive α ′ corrections to all loop orders.
There is also interest in the differential systems which arise from truncating the theory up to two loops in perturbation theory. There are at least two approaches to these. One is to consider the systems which arise from the KSEs and the anomalous Bianchi identity, and the other is to in addition consider the field equations up to and including two loops. The difference is that in the former case there are more options in the choice of connections 1 Our description of geometry differs from that of [22] specially for backgrounds preserving more than 2 supersymmetries. which enter in the Pontryagin forms of the anomalous Bianchi identity. In the latter case, consistency of the differential system requires that these connections satisfy an instanton type of condition. Solutions to such systems have been found sometime ago in [26] in connection with HKT geometry [27] on homogenous spaces. More recently attention has been centered on the solutions [28,29,30] of the Strominger system which is associated with R 3,1 compactifications of the heterotic string [31,32] and hermitian geometry.
AdS spaces can be seen as backgrounds in the perturbative approach to the heterotic string, and also as solutions to the differential system that arises after the theory is truncated to two loops. Because of this, both points of view will be explored, though the former is suitable for applications to the heterotic string. There are some differences in the analysis between the two cases. One difference is that in perturbation theory one begins with a background which satisfies dH = 0 and then corrects it order by order in α ′ . This is because one starts the perturbation theory from a world-sheet classical action which has a B-field coupling to the string. The existence of such zeroth order backgrounds puts a restriction on the theory. It is known for example that there are no-go theorems for such backgrounds that preserve some of the spacetime superymmetry [33]. If the zeroth order background exists and the perturbation theory can be set up, then typically the question that arises is whether there is a renormalzation scheme such that the symmetries of the zeroth order background get maintained to all orders in perturbation theory. Such computations for world-sheet supersymmetries can be found in [34] and for spacetime supersymmetries in [35,36], and more recently in [37].
On the other hand if the theory is truncated to two loops, α ′ is not treated as an expansion parameter but rather as a constant. As the anomalous Bianchi identity and the field equations do not have an explicit dependence on the B-field, the fields that are restricted by the differential system are the metric and the fundamental forms of the G-structure as typically H is expressed in terms of them. Therefore, the requirement of existence of dH = 0 configurations does not arise in the investigation of the differential system.
In this paper, we shall classify the geometry of supersymmetric warped, flux AdS backgrounds up to two loops in both heterotic perturbation theory and in the differential system that arises after truncating the higher than two loop contributions. We find in both cases that AdS 3 heterotic supergravity backgrounds admit 2, 4, 6 and 8 supersymmetries. We show that the geometry is always a product AdS 3 × M 7 , ie the warp factor is constant. Furthermore, the two spacetime supersymmetries restrict M 7 to have a conformally balanced G 2 structure compatible with a connection with skew-symmetric torsion. Four spacetime supersymmetries restrict M 7 to have a SU(3) structure compatible with a connection with skew-symmetric torsion. In addition M 7 is locally a circle fibration over a conformally balanced KT manifold with a U(3) structure and holomorphic canonical bundle. Six spacetime supersymmetries restrict M 7 to have a SU(2) structure. Moreover M 7 is locally a S 3 fibration over a 4-dimensional manifold B 4 which has vanishing selfdual Weyl tensor. The fibres S 3 twist over B 4 with an su(2) connection whose self-dual part is specified by a quaternion Kähler structure on B 4 and its anti-self-dual part is unrestricted. Eight spacetime supersymmetries restrict M 7 to have a SU(2) structure. Now M 7 is locally a S 3 fibration over a 4-dimensional hyper-Kähler manifold B 4 . The S 3 fiber is twisted with respect to an anti-self-dual su(2) connection.
We also show that there are no AdS n backgrounds for n = 3. For n > 3, this result has been established up to two loops in both perturbation theory and in the differential system approaches without making any further assumptions on the backgrounds apart from supersymmetry. For n = 2 this result has been established for backgrounds with dH = 0 and after assuming the application of the maximum principle 2 , and follows from the results of [39].
In addition we prove a new Lichnerowicz type theorem in the presence of α ′ corrections. This generalizes the theorem we have shown for dH = 0 configurations in [19] and [20]. We find that after choosing an appropriate Dirac operator that couples to fluxes on M 7 , all the zero modes of this Dirac operator are Killing spinors of the gravitino, dilatino and, in the perturbation theory, the gaugino KSEs. Furthermore, the validity of the theorem requires an appropriate choice for a connection of M 7 and that of the gauge sector which contribute to the higher order α ′ terms of field equations and anomalous Bianchi identity.
This paper is organized as follows. In section 2, we investigate the geometry of AdS 3 backgrounds provided dH = 0 and prove a Lichnerowicz type theorem. In section 3, we investigate the geometry of AdS 3 backgrounds provided dH = 0 and establish a Lichnerowicz type theorem for backgrounds with α ′ corrections. In section 4, we demonstrate that there are no AdS n backgrounds for n = 3. In appendices A and B, we summarize our notation and give some formulae needed for the proof of the Lichnerowicz type theorem, respectively. In appendix C, we derive the dilatino field equation from the other field equations of the theory. In appendix D, we examine the geometry of of AdS 3 backgrounds from another choice of Killing spinors and in appendix E, we give a component based proof on the non-existence of AdS n , n = 3, backgrounds.
2 AdS 3 backgrounds with dH = 0 The investigation of AdS 3 backgrounds will be separated into two cases depending on whether dH vanishes or not. For the common sector of type II supergravities as well as that for the heterotic string with the standard embedding which leads to the vanishing of the chiral anomaly, one has dH = 0. Furthermore dH = 0 at zeroth order in the α ′ expansion in the sigma model approach to the heterotic string. However, in the latter case dH = 0 to one and higher loops. For applications to the common sector, it is understood that we consider only one of the two chiral copies of the KSEs.

Fields, Field Equations and Bianchi Identities
The most general metric and NS-NS 3-form flux of warped AdS 3 backgrounds which are invariant under the action of the sl(2, R) ⊕ sl(2, R) symmetry algebra of AdS 3 are where we have introduced the frame u, v, and z are the AdS 3 coordinates, ℓ is the AdS radius, and A is the warp factor. For more details on this parametrization of AdS 3 backgrounds see [40]. Furthermore, one finds that the dilaton, Φ, and the warp factor, A, and G depend only on the M 7 coordinates.
In addition X and A, and G are functions, and a 3-form on M 7 , respectively. The heterotic theory has in addition a 2-form gauge field F with gauge group a subgroup of E 8 × E 8 or SO(32)/Z 2 that is associated with the gauge sector. One way to impose the symmetries of AdS 3 on F is to take F to be the curvature of a connection on M 7 that depends only on the coordinates of M 7 . Alternatively, the gaugino 3 KSE for the backgrounds that we shall be considering implies that F vanishes along the AdS 3 directions and that the Lie derivative of F along the isometries of AdS 3 vanishes as well up to gauge transformations. These in particular imply that the associated Pontryagin forms vanish along AdS 3 and depend only on the coordinates of M 7 . Either results are sufficient for the analysis that will follow.
So far, we have not imposed the Bianchi identity on H and (2.1) applies equally to backgrounds regardless on whether dH vanishes or not. However imposing now the Bianchi identity, dH = 0, one finds The field equations for the dilatino and 2-form gauge potential can be expressed as where i, j, k = 1, . . . , 7. Moreover, the AdS component of the Einstein equation reads 5) and the M 7 components are where ∇ is the Levi-Civita connection on M 7 and R ij is its Ricci tensor. The Ricci scalar curvature of M 7 can be expressed This formula for A constant will be used later in the proof of a Lichnerowicz type theorem.

Solution of KSEs along AdS 3
The heterotic gravitino and dilatino KSEs are Therefore, the form of the two KSEs remains the same up to two and possibly higher loops. The gaugino KSE does not contribute in the investigation of backgrounds with dH = 0 and so it is not included. First let us focus on the gravitino KSE. The gravitino KSE along the AdS 3 directions reads and Γ ± ǫ ± = 0. Furthermore, using the relations we find that there is only one independent integrability condition As the Clifford algebra operator / ∂AΓ z does not have real eigenvalues, the above integrability condition for ℓ < ∞ can be satisfied provided that Thus the warp factor A is constant. The second equation above also implies that the component X of H along AdS 3 is constant. Furthermore, one can write where c 1 = ℓ 2 AX = ±1 as implied by (2.13). The KSEs (2.9) can be integrated to find provided that It is understood that the dependence of ǫ on the AdS 3 coordinates is given explicitly while τ ± and σ ± depend only on the coordinates of M 7 . It is clear from (2.16) that there are two solutions to the above conditions. If c 1 = 1, (2.16) implies that σ ± = 0. In turn the Killing spinor is Alternatively if c 1 = −1, (2.16) gives τ ± = 0 and the Killing spinor is Therefore depending on the sign of AX, which coincides in the sign of the contribution volume form of AdS 3 in H, there are two distinct cases to consider. In order to interpret the two cases that arise, note that AdS 3 can be identified, up to a discrete identification, with the group manifold SL(2, R). As such it is parallelizable with respect to either left or right actions of SL(2, R). The two associated connections differ by the sign of their torsion term which in turn is given by the structure constants of the sl(2, R). Of course the associated 3-form coincides with the bi-invariant volume form of AdS 3 .
To treat both cases symmetrically, we introduce B (±) which is equal to Ξ ± when it acts on σ ± and equal to Ξ ± ± 1 ℓ when it acts on τ ± . The integrability conditions are then succinctly expressed as B (±) χ ± = 0, χ ± = σ ± , τ ± , where with c 2 = 1 when χ ± = σ ± and with c 2 = −1 when χ ± = τ ± . The remaining KSEs on M 7 can now be expressed as is a metric connection with skew-symmetric torsion G associated with the gravitino KSE, and is associated with the dilatino KSE, and B (±) should be thought as a projector which restricts the first two equations on either σ ± or τ ± spinors. For the investigation of the geometry of these backgrounds it suffices to consider only the τ + or the σ + spinors. This is because, if χ − is a solution to the above KSEs, then χ + = A −1 Γ +z χ − is also a solution, and vice versa, if χ + is a solution, then χ − = AΓ −z χ + is also a solution. Incidentally, this also implies that the number of supersymmetries preserved by AdS 3 backgrounds is always even. Furthermore, it suffices to investigate the geometry of these backgrounds as described by the σ + spinors. As we have mentioned, the τ + spinors arise on choosing the other parallelization for AdS 3 and it can be treated symmetrically, see also appendix D.

Geometry
If the solution of the KSEs is determined by the σ ± spinors, the investigation of the geometry of M 7 can be done as a special case of that of heterotic horizons in [23] which utilized the classification results of [21]. To see this first note that h = − 2 ℓ dz and so the constant k which enters in the description of geometry for the heterotic horizons is (2.23) Next observe that as σ + and σ − are linearly independent, there are two Killing spinors given by Setting now σ − = AΓ −z σ + and after rescalling the second spinor with the non-vanishing constant −2ℓ −1 A −2 , we find that the two spinors can be rewritten as These are precisely the spinors that appear in the context of heterotic horizons, see [23] for a detailed description of the geometry of M 7 including the emergence of the (left) sl(2, R) symmetry of AdS 3 backgrounds as generated by the 1-form Killing spinor bilinears. Briefly, M 7 admits a G 2 structure compatible with a metric connection∇ with skew-symmetric torsion, Furthermore all field equations and KSEs are implied provided [23] that and ϕ is the fundamental form of the G 2 structure. The first condition in (2.26) is required for the existence of a G 2 structure on M 7 compatible with a metric connection with skew-symmetric torsion [41] and the second condition is the Bianchi identity (2.3). The dilatino KSE implies two conditions, one of which is that the G 2 structure on M 7 must be conformally balanced, θ ϕ = 2dΦ, both of which have been incorporated in the expression for G, where θ ϕ is the Lee form of ϕ. The conditions (2.26) are simpler than those that have appeared for heterotic horizons, because for AdS 3 backgrounds dh = 0.

Geometry of AdS 3 backgrounds with extended supersymmetry
We have shown that AdS 3 backgrounds always preserve an even number of supersymmetries. Furthermore, from the counting of supersymmetries for heterotic horizons [23], one concludes that AdS 3 backgrounds preserve 2, 4, 6 and 8 supersymmetries. In addition, AdS 3 backgrounds that preserve 8 supersymmetries and for which M 7 is compact are locally isometric to either AdS 3 × S 3 × T 4 or to AdS 3 × S 3 × K 3 . Again we shall not give the details of the proof for these results. However, we shall state the key formulae that arise in the investigation of the geometry for each case as they have some differences from those of the heterotic horizons.

Four supersymmetries
Let us first consider the AdS 3 backgrounds with 4 supersymmetries. The two additional spinors can be written as where σ 2 + is linearly independent from σ 1 + = σ + in (2.25). In fact it can be shown that the normal form for these spinors up to the action of Spin (7) can be chosen as σ 1 + = 1 + e 1234 and σ 2 The isotropy group of all four spinors is SU(3). Therefore M 7 is a Riemannian manifold equipped with metric ds 2 (7) and a 3-form G. Furthermore, the metric connection∇ with skew-symmetric torsion G is compatible with an SU(3) structure. The KSEs restrict this structure on M 7 further. In particular, the SU(3) structure on M 7 is associated with 1-form ξ, 2-form ω, and (3,0)-form χ spinor bilinears such that where ω and χ are the fundamental forms of an SU(3) structure in the directions transverse to ξ. All these forms are∇-parallel,∇ξ =∇ω =∇χ = 0. In particular∇ξ = 0 implies that ξ is Killing and that The dilaton Φ is also invariant under ξ. The full set of conditions on ξ, ω and χ can be found in [23]. The solution of these conditions 4 implies that M 7 can be locally constructed as a circle fibration on a conformally balanced 5 , θ ω = 2dΦ, KT manifold B 6 with Hermitian form ω, where the tangent space of the circle fibre is spanned by ξ. The canonical bundle of B 6 admits a connection 6 λ = k −1 w, such that

30)
i, j = 1, 2, . . . , 6, i.e. the canonical bundle is holomorphic and the connection satisfies the Hermitian-Einstein instanton condition, and in addition is the curvature of the canonical bundle induced from the connection with torsion G (6) = −i I dω on B 6 , and I is the complex structure of B 6 . The first condition is required for M 7 to admit an SU(3) structure compatible with the connection with skew-symmetric torsion G and the second condition is required by the Bianchi identity (2.3). Note that B 6 has a U(3) rather than an SU(3) structure compatible with a connection with skew-symmetric torsion. This is because the (3,0)-form χ is not invariant under the action of ξ (2.29). The metric and torsion on M 7 are given from those of B 6 as This summarizes the geometry for the AdS 3 backgrounds preserving four supersymmetries. Solutions can be constructed using the techniques developed in [23] to find solutions for heterotic horizons.

Six supersymmetries
Next let us turn to AdS 3 backgrounds preserving 6 supersymmetries. For these M 7 admits an SU(2) structure compatible with∇. Furthermore, M 7 can be constructed locally as a SU(2) = S 3 fibration over a 4-dimensional manifold B 4 whose self-dual part of the Weyl tensor vanishes. SU (2) twists over B 4 with respect to a (principal bundle) connection λ which has curvature F r ′ such that the self-dual part satisfies where ω r ′ are the almost Hermitian forms of a quaternionic Kähler structure on B 4 . The anti-self dual part of F , F ad , is not restricted by the KSEs. The dilaton depends only on the coordinates of B 4 . The metric and G on M 7 are given by where CS is the Chern-Simons 7 form of λ. The only condition that remains to be solved to find solutions is∇ where the inner products are taken with respect to the ds 2 metric. For more details on the geometry of such backgrounds see [23].

Eight supersymmetries
Next let us turn to the AdS 3 backgrounds preserving 8 supersymmetries. The description of the geometry is as that of the backgrounds above preserving 6 supersymmetries. The only differences are that B 4 must be a hyper-Kähler manifold with respect to the ds 2 (B 4 ) metric, and that F sd = 0. The metric and 3-form G of M 7 are given as in (2.35) but now we have that∇ instead of (2.36). If B 4 is compact, a partial integration argument reveals tha F ad = 0 and so the only regular solutions, up to discrete identifications, are AdS 3 × S 3 × K 3 and AdS 3 × S 3 × T 4 . If B 4 is not compact, there are many smooth solutions, see [42].

Lichnerowicz type theorem on σ + , τ +
The Killing spinors of AdS 3 backgrounds (2.20) can be identified with the zero modes of a suitable Dirac-like operator coupled to fluxes on M 7 , and vice versa. This provides a new example of a Lichnerowicz type theorem for connections whose holonomy is not in a Spin group. This result is analogous to others that have been established for AdS backgrounds in 11-dimensional and type II supergravities [18,19,20]. However, there are some differences. One is that the spinor representation in the heterotic case is different from that of the previous mentioned theories. There are also some subtle issues associated with the modification of the Lichnerowicz type of theorem in the presence of α ′ corrections, which we shall consider in further detail in the next section.
To begin, let us first suppress the α ′ corrections, and take dH = 0. The Lichnerowicz type of theorem with α ′ corrections will be investigated later. We define the modified gravitino Killing spinor operator, for some q 1 , q 2 ∈ R. Observe that for q 1 , q 2 = 0, the holonomy of∇ (+,q 1 ,q 2 ) is not in Spin (7). Next define the modified Dirac-like operator It is clear that if χ + is a Killing spinor, ie satisfies (2.20), then it is a zero mode of D (+) .
Here we that prove the converse. In particular, we shall show that there is a choice of q 1 , q 2 such that all the zero modes of D (+) are Killing spinors. Thus we shall establish The proof relies on global properties of M 7 , which we assume to be smooth, and compact without boundary.
To prove the theorem, let us assume that D (+) χ + = 0 and consider the identity The first term on the right hand side can be further rewritten in terms of the differential operator∇ (+,q 1 ,q 2 ) by completing the square as while the second term can be rewritten using the identity / ∇ 2 = ∇ 2 − 1 4 R (7) , and D (+) χ + = 0, as Combining these, ∇ 2 χ + 2 can be rewritten as, Of the terms on the right hand side of (2.45), the first term is proportional to the gravitino Killing spinor equation squared, and so we expect that the remaining terms will be equal to some combination of the algebraic KSEs. The third term includes a derivative of χ + , however, and so we will attempt to write it in the form for some vector α and Clifford algebra element F that depend on the fields. In terms of the fields, the third term in the right hand side of (2.45) can be rewritten as Thus, we find that it can be separated as outlined above if and only if q 2 = − 1 7 . We will use this value of q 2 from here on. Then we find that and so, factoring out a Γ i on the right, and The F term part of the third term of (2.45) can be combined with the fourth term of (2.45) to give The last term on the right hand side of (2.45) is the only term involving derivatives of the fields other than Φ and the second derivative of Φ. However, we can use the Bianchi identity and the Φ field equation to rewrite this term as, and we can use the scalar part of the Einstein equation to rewrite the second term on the right hand side of (2.45) as is given by whereas It follows that This expression is suitable to apply the Hopf maximum principle on the scalar function χ + 2 on M 7 as for 0 < q 1 < 2 7 the right hand side of this equation is positive definite. Assuming that the conditions required for the maximum principle on the fields and M 7 apply, e.g. the fields are smooth and M 7 is compact without boundary, the only solutions to the above equation are that χ + 2 is constant, and that, Thus χ + is a Killing spinor which establishes the theorem.

Bianchi identities, field equations and KSEs
Let us first consider the modifications that occur in the Bianchi identity, field equations and KSEs of heterotic theory up to two loops in sigma model perturbation theory 8 . The anomaly cancelation mechanism requires the modification of the Bianchi identity for H as whereR is the curvature of a connection on the spacetime M which will not be specified at this stage, F is the curvature of the gauge sector connection of the heterotic theory and α ′ is the string tension which also has the role of the loop parameter. Thus dH is expressed as the difference of two Pontryagin forms, one is that of the tangent space of space-time and the other is that of the gauge sector bundle. Furthermore, global anomaly cancelation requires in addition that the form on the right-hand-side of the anomalous Binachi identity represents the trivial cohomology class in H 4 (M). This statement is modified upon the addition of NS5-brane sources but this will not be considered here.
In addition to the modification of the Bianchi identity, the field equations also get modified. In particular up to two loops in sigma model perturbation theory [43], the dilaton, 2-form gauge potential, and gauge sector connection field equations read 2) and the Einstein equation is 3) 8 We use the conventions and normalization of the field equations, Bianchi identities and KSEs of [35].
Furthermore, the KSEs [44] are In particular observe that the KSEs have the same form up to two loops in sigma model perturbation theory as that at the zeroth order. It is not known how these equations are modified at higher orders. The gauge indices of F have been suppressed. Before we proceed with the investigation of AdS 3 backgrounds, let us specifyR. In perturbative heterotic theory, the choice ofR is renormalization scheme dependent. In other words, one can choose asR the curvature of any connection on M. However in most applicationsR is chosen to be the curvatureŘ of the∇ = ∇ − 1 2 H connection on the spacetime. It is known that this choice has some key advantages. In particular it is required for the cancelation of world sheet supersymmetry anomaly [45] and also for the consistency of the anomalous Bianchi identity with the modified Einstein equations for supersymmetric backgrounds. This has been used in the calculations of [34,35] and recently emphasized [46]. The property ofŘ which is used to establish these is thatŘ satisfies instanton-like conditions, i.e. it satisfies the same conditions, to zeroth order in α ′ , as those implied on F by the gaugino KSE. To see this, consider the identitŷ The integrability condition of the gravitino KSE givesR M N,RS Γ RS ǫ = 0. As the righthand-side of the anomalous Bianchi identity is of order α ′ , it follows from (3.5) that, to zeroth order in α ′ ,Ř M N,RS Γ M N ǫ = 0 or equivalently/ Rǫ = 0 after suppressing the SO(9, 1) gauge indices. This is the same condition as that satisfied by the curvature of the gauge sector F in (3.4). To find solutions in the perturbative case, it is understood that the fields and Killing spinors are expanded in α ′ schematically as and similarly for the 3-form field strength, gauge potential and dilaton. Then the field equations and KSEs are solved order by order in α ′ to find the correction to the zeroth order fields. Next consider the case that the corrections to the heterotic theory are taken to be exact up to and including two loops. In such a case, α ′ is not an expansion parameter. The anomalous Bianchi identity (3.1), field equations, (3.2) (3.3), and KSEs (3.4) do not receive further corrections from the ones that have been explicitly stated. However consistency of the anomalous Bianchi identity with the field equations requires thatR satisfies the same conditions as those implied by the KSEs on the curvature F of the gauge connection, ie/ Rǫ = 0 after suppressing the gauge indices. It is not apparent that such a connection always exists but there are existence theorems in many cases of interest. Notice also the difference from the perturbation theory asR cannot be identified withŘ. This is because dH does not vanish in the right-hand-side of (3.5).

AdS 3 backgrounds in perturbation theory
Suppose that the symmetries of AdS 3 remain symmetries of the background after the α ′ corrections are taken into account. In such a case, the fields up to two loops in perturbation theory will decompose as This assumption is justified later. Furthermore, the field equations (3.2) and (3.3) read while component on M 7 is now, where i, j, k, ℓ = 1, 2, . . . , 7 and we have assumed thatŘ and F do not have components along the AdS 3 directions. As we shall see, this will follow from the KSEs. In addition, one finds that Similarly, the anomalous Bianchi identity of H reads As we shall see imposing the requirement that spacetime supersymmetry is preserved by the higher order corrections simplifies the above equations further.

Geometry of M 7 for backgrounds with two supersymmetries
In the perturbative approach to the heterotic string, one of the questions that arises is whether the higher order corrections preserve the spacetime supersymmetry of the zeroth order background. In other words, whether there is a renormalization scheme which preserves the spacetime supersymmetry order by order in perturbation theory. Here we shall not investigate the existence of such a scheme. Instead, we shall derive the conditions for such a scheme to exist. We have shown that for AdS 3 backgrounds admitting two spacetime supersymmetries at zero order in α ′ , M 7 has a G 2 structure compatible with a connection with skewsymmetric torsion. In particular at this order dH = 0, and A and X are constant and c 1 = ℓ 2 AX = ±1. The geometry of M 7 at this order is described in section 2.3. The contribution in the terms proportional to α ′ in the field equations, Bianchi identities and KSEs comes from the fields at zeroth order in α ′ . These depend onŘ and F . At zeroth order, the spacetime factorizes into a product AdS 3 × M 7 . Furthermore the choice of torsion on AdS 3 is such that∇| AdS 3 and∇| AdS 3 are either the left or right invariant parallelizing connection; AdS 3 is a group manifold. In either case,R| AdS 3 =Ř| AdS 3 = 0. Therefore the contribution in the α ′ terms of field equations, Bianchi identities and KSEs comes only from theŘ (7) curvature of M 7 . Furthermore, the KSEs imply that the gauge curvature F does not have components along AdS 3 and is invariant under the isometries of AdS 3 up to gauge transformations. As a result all gauge invariant tensors constructed from F are tensors on M 7 which do not depend on the coordinates of AdS 3 . These justify the choice ofŘ and F made in the previous section.
As the form of the gravitino KSE remains the same up to order α ′2 , this implies that A and X are constant up to that order and that again c 1 = ℓ 2 AX = ±1. Furthermore the metric and torsion of AdS 3 does not receive corrections at one loop, the form of the fields remains as in (3.7) up to order O(α ′2 ). The background remains factorized as AdS 3 × M 7 up to that order as well. Imposing all the above conditions on the fields, one finds that the anomalous Bianchi identity and field equations are simplified as in appendix B.
Next focusing on the geometry of M 7 , M 7 admits a G 2 structure compatible with a connection∇ with skew-symmetric torsion G. As a consequence of the gravitino and dilatino KSEs, G is as given in (2.27) up to order α ′2 . Moreover all the KSEs and field equations are satisfied provided that The first condition is required for the existence of a connection with skew-symmetric torsion which is compatible with the G 2 structure on M 7 while the second condition arises from the anomalous Bianchi identity. We have also assumed as in the dH = 0 case that all solutions ǫ of the gravitino KSEs are also solutions of the gaugino KSE, / F ǫ = 0. In this case, this implies that F is a G 2 instanton on M 7 , and so it satisfies where we have suppressed the gauge indices. This summarizes the geometry of M 7 up to order α ′2 .

Extended supersymmetry
Next let us investigate the geometry of AdS 3 backgrounds preserving 4, 6 and 8 supersymmetries up to order α ′2 . The geometry of the associated zeroth order backgrounds for which dH = 0 has already been described in section 2.4.

Four supersymmetries
These backgrounds are a special case of those we have described in the previous section that preserve two supersymmetries. As a result up to order α ′2 , the geometry is a product AdS 3 × M 7 . The presence of two more supersymmetries restricts further the geometry of M 7 . As the form of the gravitino and dilatino KSEs remain the same as that of the zeroth order fields, the geometric restrictions on the geometry of M 7 are similar to those in section 2.4. The only difference here is that dH = 0. In particular, M 7 has an SU(3) structure compatible with a connection with skew-symmetric torsion. So it admits a Killing vector field ξ such that where w(ξ) = k. Moreover, M 7 can be locally described as a circle fibration of a conformally balanced, θ ω = 2dΦ, KT manifold B 6 with Hermitian form ω whose canonical bundle admits a connection k −1 w, such that 16) i.e. the canonical bundle is holomorphic and the connection satisfies the Hermitian-Einstein instanton condition, and in addition is the curvature of the canonical bundle induced from the connection with torsion G (6) = −i I dω on B 6 and I is the complex structure of B 6 . The first condition is required for M 7 to admit an SU(3) structure compatible with the connection with skew-symmetric torsion G and the second condition is required by the anomalous Bianchi identity (3.13), where now i, j, k, m = 1, 2, . . . , 6. It is understood that the expression in the right-hand-side of the second equation in (3.17) is evaluated at the zeroth order fields. The metric and torsion on M 7 are given from those of B 6 as in (2.33) but now of course the fields on B 6 obey the equations (3.17) above.

Six supersymmetries
The presence of additional supersymmetries restricts the geometry of M 7 further. In particular, the spacetime is still a product AdS 3 × M 7 up to order α ′2 . The geometry of the zeroth order configuration has already been described in section 2.4 and so M 7 is locally a S 3 fibration over a 4-dimensional manifold B 4 . As the gravitino and dilatino KSEs have the same form up to order α ′2 as the zeroth order equations, it is expected that M 7 admits three∇-parallel vector bilinears ξ r ′ , r ′ = 1, 2, 3. Thus ξ r ′ are isometries of the metric on M 7 and i ξ r ′ H = k −1 dw r ′ up to order α ′2 , where w r ′ (ξ s ′ ) = kδ r ′ s ′ . As the geometry of the spacetime is a product up to α ′2 , these commute with the isometries of AdS 3 . However, the gravitino and dilatino KSEs do not determine the Lie bracket algebra of ξ r ′ 's.
To determine [ξ r ′ , ξ s ′ ], first note that the commutator of two isometries is an isometry. Then using∇ξ r ′ = 0, we can establish the identities ). This follows from the fact that bothŘ and F contribute in dH via the zeroth order fields and so as a consequence of the gravitino and gaugino KSEs, i ξ r ′Ř = i ξ r ′ F = 0. In fact F has to be a anti-self-dual instanton in the directions transverse to AdS 3 and ξ r ′ . As a consequence, the commutator [ξ r ′ , ξ s ′ ] iŝ ∇-parallel up to order α ′2 . If [ξ r ′ , ξ s ′ ] is not expressed in terms of ξ r ′ , the holonomy of∇ is reduced to {1} implying that the zeroth order backgrounds are group manifolds. Such backgrounds preserve 8 supersymmetries and will be investigated below. Thus [ξ r ′ , ξ s ′ ] must close on ξ t ′ . Furthermore, one can use the Bianchi identitŷ 20) to show that dw r ′ restricted on the directions transverse to AdS 3 and ξ r ′ is∇-parallel. Then an analysis similar to that we have done for heterotic horizons [23] reveals that ξ r ′ close to a su(2) algebra. As a result, M 7 is locally a S 3 fibration over a 4-dimensional manifold B 4 . The geometry can be described exactly as in the zeroth order case but the various formulae are now valid up to order α ′2 . The only modification occurs in the equation for the dilaton which now reads where the inner products are taken with respect to the ds 2 metric. The additional α ′ contribution is due to the anomalous Bianchi identity of H.

Eight supersymmetries
The backgrounds with 8 supersymmeries can be investigated in a way similar to those with 6 supersymmetries described in the previous section. However there are some differences.
As we have already mentioned at zeroth order in α ′ , section 2.4, B 4 is a hyper-Kähler manifold and F sd = 0. Up to order α ′2 , the spacetime remains a product AdS 3 × M 7 .
The investigation of the closure properties of the three∇-parallel vector field ξ r ′ on M 7 is not necessary. This is because it is a consequence of the gravitino and dilatino KSEs that these vector fields close to a su(2) algebra [21]. The metric and torsion are given as in (2.35) but now the formulae are valid up to order α ′2 . The only modification from the zeroth order equations is that the dilaton equation now reads where the metric ds(B 4 ) is the zeroth order hyper-Kähler metric and the inner products have been taken with respect to it. For compact B 4 , at zeroth order F ad = 0, and in this case M 7 = S 3 × B 4 up to discrete identifications. As a consequence, the worldsheet action of the string factorizes into a sum of a WZW model on S 3 and a sigma model on the hyper-Kähler manifold B 4 . The latter has (4,0) worldsheet supersymmetry and as a result is ultraviolet finite [47]. However, in the presence of an anomaly, the couplings are corrected order by order in α ′ as a consequence of maintaining manifest (4,0) supersymmetry in perturbation theory [34].

Truncation to two loops
Suppose now that the theory up to two loops is exact. In such a case, the geometry of the solutions has to be re-examined as several arguments that have been applied in previous cases have been based on the closure of H either to all orders or at the zeroth order in perturbation theory. Moreover α ′ has been treated as an arbitrary parameter. None of these two assumptions are valid any longer. Nevertheless, there is a simplifying assumption. This is that the backgrounds have the symmetries of AdS 3 . In particular, the fields can be written as (2.1). The KSEs are We also assume that the gaugino KSE has as many Killing spinors as the gravitino KSE.

Two supersymmetries
The G 2 case is rather straightforward. As the form of the gravitino and dilatino KSEs in (3.23) is the same as that for dH = 0 backgrounds and the fields are invariant under the symmetries of AdS 3 , one finds that the gravitino KSE implies that A, X are constant and c 1 = ℓ 2 AX = ±1. As a result, the geometry locally decomposes as AdS 3 × M 7 . The geometry of M 7 can now be described as in the perturbative case with the only difference that now the equations are exact. In particular, M 7 admits a G 2 structure compatible with a connection with skew-symmetric torsion. This G 2 structure is further restricted by the KSEs, Bianchi identities and field equations as where ϕ is the fundamental G 2 3-form, G = kϕ + e 2Φ ⋆ 7 d(e −2Φ ϕ), andR and F are G 2 instantons, ieR (7) ij,pq = 1 2 ⋆ 7 ϕ ij kmR (7) km,pq , F ij = The condition on F follows from the gaugino KSE. Observe thatŘ, which is no longer a G 2 instanton because of (3.5) and dH = 0, has now been replaced withR (7) . Moreover α ′ in (3.24) is a constant rather than a parameter.

Four supersymmetries
The geometry of these backgrounds also factorizes as AdS 3 × M 7 . Moreover, M 7 admits a SU(3) structure compatible with a connection∇ with skew-symmetric torsion.
There are 4 vector spinor bilinears and there is a basis such that 3 of them generate an sl(2, R) symmetry of AdS 3 . As these 4 vector bilinears are∇-parallel, their commutator is [ξ a , ξ b ] = i ξa i ξ b H. Since the geometry factorizes as AdS 3 × M 7 , it turns out that the commutator of the generators of sl(2, R) with the fourth vector bilinear vanishes, and so the symmetry algebra of the spacetime is sl(2, R) ⊕ u(1).
The rest of the analysis is similar to that we have described for the perturbative case. In particular, the equations (3.15), (3.16), (3.17) and (3.18) are still valid but now exactly. The only modification is in the second equation in (3.17) which now reads whereR (6) is a su(3) instanton on B 6 , ieR (6) is a (1,1)-form and ω-traceless. This condition is also satisfied by F because of the gaugino KSE.

Six supersymmetries
The geometry factorizes as AdS 3 × M 7 and M 7 admits an SU(2) structure compatible with a connection with skew-symmetric torsion∇. The spacetime admits 6 vector Killing spinor bilinears. Three of these span an sl(2, R) symmetry of AdS 3 , and the other three ξ r ′ are∇-parallel on M 7 and commute with those generating the sl(2, R). We shall argue that for non-trivial backgrounds the commutator of these three vector field must close in the set. To see this, consider the identities in (3.19). As ξ r ′ are Killing, their commutator is also Killing. Furthermore, the term i ξ r ′ i ξ s ′ dH in the second equation in (3.19) vanishes. This is because we have assumed that the connections that contribute in the anomalous Bianchi identity are those that satisfy the gaugino KSE. For all these i ξ r ′ F = i ξ r ′R = 0. As a result, if ξ r ′ and ξ s ′ are∇-parallel, so is the commutator [ξ r ′ , ξ s ′ ]. If the commutator does not close in the set ξ r ′ , the holonomy of∇ will reduce to {1}. As a result the curvature of∇ vanishes. If this is the case, the contribution to the anomalous Bianchi identity must vanish as well as the connections that contribute to it have zero curvature. This is implied by our assumption that all solutions to the gravitino KSE are also solutions of the gaugino one. For such backgrounds backgrounds dH = 0 and so the spacetime is a group manifold which preserves 8 supersymmetries. Thus for backgrounds with strictly six supersymmetries, we shall take that [ξ r ′ , ξ s ′ ] closes in the set ξ r ′ . Then it can be shown using (3.20) that the symmetry group of the spacetime generated by the vector spinor bilinears is sl(2, R) ⊕ su (2). The rest of the investigation of the geometry is similar to that we have done in the perturbative case. The only difference is that noẘ whereR (4) and F are anti-self-dual instantons on B 4 and the inner products are taken with respect to the ds 2 metric. B 4 is a 4-manifold with vanishing self-dual Weyl tensor and metric ds(B 4 ).

Eight supersymmetries
The investigation of the geometry of these backgrounds is simpler than that described in the previous section for backgrounds preserving 6 supersymmetries. First the geometry factorizes as AdS 3 × M 7 and M 7 admits a connection with skew-symmetric torsion compatible with a SU(2) structure. As in the previous case, M 7 admits 3∇-parallel Killing spinor bilinears ξ r ′ which commute with another three which span an sl(2, R) symmetry of AdS 3 . Furthermore the gravitino and dilatino KSEs imply that the symmetry algebra of these backgrounds is sl(2, R) ⊕ su (2). The analysis of the geometry proceeds as in the perturbative case. In particular, M 7 is an S 3 fibration over a hyper-Kähler manifold B 4 with metric ds(B 4 ). The only difference from the perturbative case is that noẘ whereR (4) and F are anti-self-dual instantons on B 4 .

Lichnerowicz type Theorem on σ + , τ +
The Lichnerowicz type theorem has to be re-examined in the presence of α ′ corrections and in the case that the theory is truncated to two loops. Again, we shall focus on M 7 , and define the modified Dirac-like operator as in (2.40) but now dG = 0. Furthermore, we assume the Bianchi identities and field equations of appendix B but now we shall include the α ′ terms, replacing theŘ (7) terms withR, and replacing F withF whereR andF are arbitrary curvatures of T M 7 and the gauge sector bundle respectively. In particular R andF are not restricted by the KSEs. For the truncated theory at two loops, we take the equations in appendix B as exact but again withŘ (7) and F replaced withR andF . The derivation of (2.51) is unaffected, but (2.52) becomes and (2.53) also picks up an α ′ term; On combining these expressions we obtain where we have suppressed the gauge index contraction in the / F χ + 2 term, and q 2 = − 1 7 . We shall first consider the case of perturbation theory, and setR =Ř (7) . We begin by systematically analysing the conditions imposed by (3.31) order by order in α ′ . 9 To zeroth order in α ′ , one obtains (provided that 0 < q 1 < 2 7 ), the conditions The condition ∇ (+) χ + = 0 + O(α ′ ) implies the integrability condition R mn,ℓ 1 ℓ 2 Γ ℓ 1 ℓ 2 χ + = 0 + O(α ′ ) . (3.33) This in turn implies thatŘ (7) ℓ 1 ℓ 2 ,mn Γ ℓ 1 ℓ 2 χ + = 0 + O(α ′ ) . It follows that the final term in (3.31) is in fact at least of order α ′3 , and so can be ignored. It remains to show that (3.31) implies the KSEs to linear order in α ′ . For this consider the perturbative expansion in the fields as in (3.6). One can show that if one assumes that the zeroth order KSEs are imposed, (3.31) does not have an α ′ correction apart from the gaugino term, which leads to the condition So we cannot conclude that the KSEs, apart from the gaugino, are implied from (3.31) to order α ′ . For this some control over the α ′2 terms is required which is not available. Observe that the above theorem also implies that all solutions of the gravitino and dilatino KSEs are also solutions of the gaugino one. This is because the modified Dirac-like operator D (+) is constructed from only the gravitino and dilatino KSEs but nevertheless the above theorem implies that the gaugino KSE is implied as well.
In the truncated theory, one can again formulate a Lichnerowicz type of theorem provided that one imposes by hand the conditioñ R ℓ 1 ℓ 2 ,mn Γ ℓ 1 ℓ 2 χ + = 0 . (3.36) This condition (taking 0 < q 1 < 2 7 ) is sufficient to ensure that the RHS of (3.31) can be written as a sum of positive definite terms, which must all vanish. 4 A no-go theorem for AdS n , n ≥ 4 and n = 2 backgrounds There are no AdS n , n ≥ 4 backgrounds in heterotic theory with or without α ′ corrections up to two loops in sigma model perturbation theory. This includes the case for which the theory is treated as exact up to and including two loops.
The proof of this relies on the solution of the KSEs. Suppose that the fields are invariant under the symmetries of AdS n . Then we take a basis for the spacetime as {e λ = Aē λ , e i } whereē λ is a basis for AdS n , and e i is a basis for the internal space M 10−n . We take H to be a 3-form on M 10−n . The components of H, and the conformal factor A, depend only on the co-ordinates of M 10−n .
To proceed, consider the gravitino KSE along the AdS n frame directions, see also appendix E. This has no contribution from the 3-form H, and can be rewritten as where∇ denotes the Levi-Civita connection on AdS n . The integrability condition of this equation implies that whereR µν is the Ricci tensor of∇. However, for AdS n ,R µν = κḡ µν whereḡ is the metric on AdS n , and κ is a negative constant. The integrability condition (4.2) is then equivalent to which admits no solution as κ < 0 and n ≥ 4. The above argument clearly applies for all backgrounds with dH = 0, and so excludes the existence of AdS n , n > 3, backgrounds for the common sector and the heterotic theory for which there is not an anomalous correction to the Bianchi identity. This result is also valid for the AdS n , n > 3 solutions of the truncated theory as well. It remains to investigate the existence of AdS n , n > 3, backgrounds in perturbative heterotic theory with an anomalous contribution to the Bianchi identity, dH = 0. In this case, the argument above implies that at zeroth order in α ′ , there are no such solutions. Furthermore, it also excludes the existence of AdS n , n > 3, solutions up and including two loops in sigma model perturbation theory that preserve all the symmetries of AdS n . However such solutions cannot completely be excluded in higher orders as it is not known how the KSEs and field equations are corrected. There is the possibility that one can start from another background which is allowed at zeroth order which then gets corrected in perturbation theory to an AdS n , n > 3 solution. Although this cannot be excluded, it may be a rather remote possibility. We conclude therefore that up to order O(α ′2 ) in perturbation theory there are no AdS n , n > 3, solutions to heterotic theory.
It remains to investigate the existence of AdS 2 solutions. It is a consequence of the investigation of near horizon geometries in [23] that if dH = 0, there are no AdS 2 solutions. It should be emphasized though that the exclusion of AdS 2 backgrounds requires the use of the maximum principle. Otherwise heterotic AdS 2 backgrounds do exist [38]. This result extends up to order α ′2 in perturbation theory as it is unlikely that one can start from a different zeroth order background and correct it at one-loop approximation to an AdS 2 background-though we do not have a proof for this. The existence of AdS 2 solutions for the truncated theories will be examined elsewhere.
Then observe that where the Einstein equation has been used again. Substituting this back into (C.1), On the other hand, the divergence of the Ricci curvature is Substituting this back into (C.5), we find Clearly this implies the field equation of the dilaton up to a constant for theories that do not have an anomaly contribution. For the theories with an anomaly contribution, the derivation requires further examination. First, let us focus on the quadratic terms in F in (C.9). From the Bianchi identity for F , where the gauge indices, which are traced over in all terms quadratic in F , have been suppressed. Furthermore, using the field equation for F , one has Therefore the terms quadratic in F sum in (C.9) to just − α ′ 16 ∇ N F 2 as expected. It remains to see what happens to the quadratic terms inR in (C.9). First, consider the perturbative case, withR =Ř. The curvature terms can be most straightforwardly simplified by making use of the appropriate curvature Bianchi identity, given by On contracting the above identity over the K, S indices, we obtain the following useful expression, valid to zeroth order in α ′ : which follows from the Einstein equations (3.3), and the gauge field equations for H. On using (C. 13) and (C.14), it follows that The above expression produces the appropriate curvature squared term in the dilaton equation at first order in α ′ . For the truncated theory, the appropriate curvature term in the dilaton equation is obtained if (C.13) and (C.14) hold exactly, on replacingŘ withR.

C.2 Dilaton equation via supersymmetry
Another way to obtain the dilaton field equation is to assume the KSEs, see also [21]. In particular, consider the condition On expanding out the LHS, one obtains, after making use of the H gauge field equation, and the gravitino and dilatino KSE the following expression: The dH bianchi identity term can then be rewritten to imply In the perturbative case, the final term on the LHS can be neglected as a consequence of the gaugino KSE. The penultimate term can also be neglected because, ifR =Ř theñ as a consequence of the integrability conditions of the gravitino KSE. The first line of (C.19) then implies the dilaton equation to the required order, provided that the zeroth order in α ′ part of the Killing spinor ǫ is nonzero.
In the truncated case, again the final term on the LHS of (C.19) vanishes due to the gaugino KSE. The penultimate term vanishes if one assumes that and then (C.19) implies the dilaton equation.