Geometry and supersymmetry of heterotic warped flux AdS backgrounds
aa r X i v : . [ h e p - t h ] J u l DMUS–MP–15/06
Geometry and supersymmetry of heterotic warpedflux AdS backgrounds
S. W. Beck , J. B. Gutowski and G. Papadopoulos Department of Mathematics, King’s College LondonStrand, London WC2R 2LS, UK.E-mail: [email protected]: [email protected] Department of Mathematics, University of SurreyGuildford, GU2 7XH, UK.Email: [email protected]
Abstract
We classify the geometries of the most general warped, flux AdS backgroundsof 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 = 3. Moreover the warp factor of AdS backgrounds is constant, the geometry is aproduct AdS × M and such solutions preserve, 2, 4, 6 and 8 supersymmetries. Thegeometry of M has been specified in all cases. For 2 supersymmetries, it has beenfound that M admits a suitably restricted G structure. For 4 supersymmetries, M 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 has an SU (2) structureand can be described locally as a S fibration over a 4-dimensional manifold whicheither has an anti-self dual Weyl tensor or a hyper-K¨ahler structure, respectively. Wealso demonstrate a new Lichnerowicz type theorem in the presence of α ′ corrections. Introduction
For over 30 years, AdS backgrounds have found widespread applications originally insupergravity [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 backgroundshas not been specified and their classification remains an open problem, for some selectedpublications see [5]-[17]. Only recently, the fractions of supersymmetry preserved byflux, warped AdS backgrounds of 11-dimensional and type II supergravities have beenspecified in [18, 19, 20] without making any assumptions on the form of the fields andthat 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 typetheorems.To make progress towards the classification of AdS backgrounds, it is instructive tohave a paradigm where all the issues that arise can be worked out in a simpler settingthan that of the 11-dimensional and type II supergravities. Such theories are the commonsector of type II supergravities and the heterotic supergravity. One simplification thatoccurs in the common sector and heterotic supergravities is that all solutions of the KSEsare known and the geometry of the backgrounds has been determined [21]. In particular,AdS configurations arise naturally in the context of classification of supersymmetricheterotic backgrounds [21] and their geometry has also been examined in [22]. TheAdS backgrounds are also special cases of the heterotic horizons in [23] though the latterhas been explored only for a closed 3-form field strength H . In addition, there is a no-go theorem for AdS heterotic backgrounds that has been established in [24] withoutassuming supersymmetry but imposing smoothness on the fields and compactness of thetransverse 6-dimensional space.The α ′ corrections to heterotic backgrounds can be investigated in the perturbative orsigma model approach, see eg [25] and references within. In this, the field equations of thetheory are the vanishing conditions of the sigma model beta functions which are expressedas a series in α ′ , the string tension. Furthermore, the Bianchi identity of the 3-form fieldstrength gets corrected as part of a mechanism which cancels the chiral reparameterizationand gauge anomalies of the 2-dimensional sigma model. This is referred to as the anomalycancelation 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 tothe difference between the Pontryagin form of the gravitational sector and that of thegauge 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 correctionto the field equations. Generic heterotic backgrounds receive α ′ corrections to all looporders.There is also interest in the differential systems which arise from truncating the theoryup to two loops in perturbation theory. There are at least two approaches to these. Oneis 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 Our description of geometry differs from that of [22] specially for backgrounds preserving more than2 supersymmetries. R , compactifications of the heterotic string [31, 32] and hermitian geometry.AdS spaces can be seen as backgrounds in the perturbative approach to the heteroticstring, and also as solutions to the differential system that arises after the theory istruncated to two loops. Because of this, both points of view will be explored, thoughthe former is suitable for applications to the heterotic string. There are some differencesin the analysis between the two cases. One difference is that in perturbation theory onebegins 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 actionwhich has a B-field coupling to the string. The existence of such zeroth order backgroundsputs a restriction on the theory. It is known for example that there are no-go theorems forsuch backgrounds that preserve some of the spacetime superymmetry [33]. If the zerothorder background exists and the perturbation theory can be set up, then typically thequestion that arises is whether there is a renormalzation scheme such that the symmetriesof the zeroth order background get maintained to all orders in perturbation theory. Suchcomputations for world-sheet supersymmetries can be found in [34] and for spacetimesupersymmetries 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 anexpansion parameter but rather as a constant. As the anomalous Bianchi identity andthe field equations do not have an explicit dependence on the B-field, the fields that arerestricted by the differential system are the metric and the fundamental forms of theG-structure as typically H is expressed in terms of them. Therefore, the requirement ofexistence of dH = 0 configurations does not arise in the investigation of the differentialsystem.In this paper, we shall classify the geometry of supersymmetric warped, flux AdSbackgrounds up to two loops in both heterotic perturbation theory and in the differentialsystem that arises after truncating the higher than two loop contributions. We find inboth cases that AdS heterotic supergravity backgrounds admit 2, 4, 6 and 8 supersym-metries. We show that the geometry is always a product AdS × M , ie the warp factor isconstant. Furthermore, the two spacetime supersymmetries restrict M to have a confor-mally balanced G structure compatible with a connection with skew-symmetric torsion.Four spacetime supersymmetries restrict M to have a SU (3) structure compatible with aconnection with skew-symmetric torsion. In addition M is locally a circle fibration overa conformally balanced KT manifold with a U (3) structure and holomorphic canonicalbundle. Six spacetime supersymmetries restrict M to have a SU (2) structure. Moreover M is locally a S fibration over a 4-dimensional manifold B which has vanishing self-dual Weyl tensor. The fibres S twist over B with an su (2) connection whose self-dualpart is specified by a quaternion K¨ahler structure on B and its anti-self-dual part isunrestricted. Eight spacetime supersymmetries restrict M to have a SU (2) structure.Now M is locally a S fibration over a 4-dimensional hyper-K¨ahler manifold B . The S fiber is twisted with respect to an anti-self-dual su (2) connection.2e also show that there are no AdS n backgrounds for n = 3. For n >
3, this resulthas been established up to two loops in both perturbation theory and in the differentialsystem approaches without making any further assumptions on the backgrounds apartfrom supersymmetry. For n = 2 this result has been established for backgrounds with dH = 0 and after assuming the application of the maximum principle , and follows fromthe 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 ,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 theoremrequires an appropriate choice for a connection of M and that of the gauge sector whichcontribute 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 backgrounds provided dH = 0 and prove a Lichnerowicz type theorem. In section 3, weinvestigate the geometry of AdS backgrounds provided dH = 0 and establish a Lich-nerowicz type theorem for backgrounds with α ′ corrections. In section 4, we demonstratethat there are no AdS n backgrounds for n = 3. In appendices A and B, we summarizeour notation and give some formulae needed for the proof of the Lichnerowicz type the-orem, respectively. In appendix C, we derive the dilatino field equation from the otherfield equations of the theory. In appendix D, we examine the geometry of of AdS back-grounds from another choice of Killing spinors and in appendix E, we give a componentbased proof on the non-existence of AdS n , n = 3, backgrounds. backgrounds with dH = 0 The investigation of AdS backgrounds will be separated into two cases depending onwhether dH vanishes or not. For the common sector of type II supergravities as well asthat for the heterotic string with the standard embedding which leads to the vanishingof 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 understoodthat we consider only one of the two chiral copies of the KSEs. The most general metric and NS-NS 3-form flux of warped AdS backgrounds which areinvariant under the action of the sl (2 , R ) ⊕ sl (2 , R ) symmetry algebra of AdS are ds = 2 e + e − + A dz + ds (cid:0) M (cid:1) ,H = AXe + ∧ e − ∧ dz + G, (2.1) AdS can be written as a warped product of AdS [38] and so there exist heterotic AdS backgrounds.However they do not satisfy the maximum principle. e + = du e − = dr − rℓ dz − rd ln A , (2.2) u, v, and z are the AdS coordinates, ℓ is the AdS radius, and A is the warp factor. Formore details on this parametrization of AdS backgrounds see [40]. Furthermore, one findsthat the dilaton, Φ, and the warp factor, A , and G depend only on the M coordinates.In addition X and A , and G are functions, and a 3-form on M , respectively.The heterotic theory has in addition a 2-form gauge field F with gauge group a sub-group of E × E or SO (32) / Z that is associated with the gauge sector. One way toimpose the symmetries of AdS on F is to take F to be the curvature of a connectionon M that depends only on the coordinates of M . Alternatively, the gaugino KSEfor the backgrounds that we shall be considering implies that F vanishes along the AdS directions and that the Lie derivative of F along the isometries of AdS vanishes as wellup to gauge transformations. These in particular imply that the associated Pontryaginforms vanish along AdS and depend only on the coordinates of M . Either results aresufficient for the analysis that will follow.So far, we have not imposed the Bianchi identity on H and (2.1) applies equallyto backgrounds regardless on whether dH vanishes or not. However imposing now theBianchi identity, dH = 0, one finds d ( A X ) = 0 , dG = 0 . (2.3)The field equations for the dilatino and 2-form gauge potential can be expressed as ∇ Φ = − A − ∂ i A∂ i Φ + 2( d Φ) − G + 12 X , ∇ k G ijk = − A − ∂ k AG ijk + 2 ∂ k Φ G ijk , (2.4)where i, j, k = 1 , . . . ,
7. Moreover, the AdS component of the Einstein equation reads ∇ ln A = − ℓ A − − A − ( dA ) + 2 A − ∂ i A∂ i Φ + 12 X , (2.5)and the M components are R (7) ij = 3 ∇ i ∇ j ln A + 3 A − ∂ i A∂ j A + 14 G ik k G j k k − ∇ i ∇ j Φ , (2.6)where ∇ is the Levi-Civita connection on M and R (7) ij is its Ricci tensor. The Ricci scalarcurvature of M can be expressed R (7) = 3 ∇ ln A + 3 A − ( dA ) + 14 G − ∇ Φ= − ℓ A − − A − ( dA ) + 512 G + 12 X + 12 A − ∂ i A∂ i Φ − d Φ) . (2.7)This formula for A constant will be used later in the proof of a Lichnerowicz type theorem. From now on we assume that the gaugino KSE has the same Killing spinors as the gravitino KSE,see [21] for a justification. .2 Solution of KSEs along AdS The heterotic gravitino and dilatino KSEs are ∇ M ǫ − /H M ǫ = 0 + O ( α ′ ) , (cid:0) /∂ Φ − /H (cid:1) ǫ = 0 + O ( α ′ ) . (2.8)Therefore, the form of the two KSEs remains the same up to two and possibly higherloops. 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 directionsreads ∂ u ǫ ± + A − Γ + z (cid:0) ℓ − − Ξ − (cid:1) ǫ ∓ = 0 ,∂ r ǫ ± − A − Γ − z Ξ + ǫ ∓ = 0 ,∂ z ǫ ± − Ξ ± ǫ ± + 2 rℓ A − Γ − z Ξ + ǫ ∓ = 0 , (2.9)where Ξ ± = ∓ ℓ + 12 /∂A Γ z ∓ AX, (2.10)and Γ ± ǫ ± = 0. Furthermore, using the relationsΞ ± Γ z + + Γ z + Ξ ∓ = 0 , Ξ ± Γ z − + Γ z − Ξ ∓ = 0 , (2.11)we find that there is only one independent integrability condition (cid:18) Ξ ± ± ℓ Ξ ± (cid:19) ǫ ± = (cid:18) − ℓ −
14 ( dA ) ∓ AX /∂A Γ z + 116 A X (cid:19) ǫ ± = 0 . (2.12)As the Clifford algebra operator /∂A Γ z does not have real eigenvalues, the above integra-bility condition for ℓ < ∞ can be satisfied provided that dA = 0 , − ℓ + 116 A X = 0 . (2.13)Thus the warp factor A is constant. The second equation above also implies that thecomponent X of H along AdS is constant. Furthermore, one can writeΞ ± = ∓ c ℓ , (2.14)where c = ℓ AX = ± ǫ = ǫ + + ǫ − = σ + + e − zℓ τ + + σ − + e zℓ τ − − ℓ − uA − Γ + z σ − − ℓ − rA − e − zℓ Γ − z τ + , (2.15)provided that Ξ ± σ ± = 0 Ξ ± τ ± = ∓ ℓ τ ± . (2.16)5t is understood that the dependence of ǫ on the AdS coordinates is given explicitly while τ ± and σ ± depend only on the coordinates of M .It is clear from (2.16) that there are two solutions to the above conditions. If c = 1,(2.16) implies that σ ± = 0. In turn the Killing spinor is ǫ = ǫ + + ǫ − = e − zℓ τ + + e zℓ τ − − ℓ − rA − e − zℓ Γ − z τ + . (2.17)Alternatively if c = −
1, (2.16) gives τ ± = 0 and the Killing spinor is ǫ = ǫ + + ǫ − = σ + + σ − − ℓ − uA − Γ + z σ − . (2.18)Therefore depending on the sign of AX , which coincides in the sign of the contributionvolume form of AdS in H , there are two distinct cases to consider.In order to interpret the two cases that arise, note that AdS can be identified, up toa discrete identification, with the group manifold SL (2 , R ). As such it is parallelizablewith respect to either left or right actions of SL (2 , R ). The two associated connectionsdiffer by the sign of their torsion term which in turn is given by the structure constants ofthe sl (2 , R ). Of course the associated 3-form coincides with the bi-invariant volume formof AdS .To treat both cases symmetrically, we introduce B ( ± ) which is equal to Ξ ± when itacts on σ ± and equal to Ξ ± ± ℓ when it acts on τ ± . The integrability conditions are thensuccinctly expressed as B ( ± ) χ ± = 0, χ ± = σ ± , τ ± , where B ( ± ) = ∓ c + c ℓ , (2.19)with c = 1 when χ ± = σ ± and with c = − χ ± = τ ± .The remaining KSEs on M can now be expressed as ∇ ( ± ) i χ ± = 0 , A ( ± ) χ ± = 0 , B ( ± ) χ ± = 0 , (2.20)where ∇ ( ± ) i = ∇ i + Ψ ( ± ) i , Ψ ( ± ) i = − /G i , (2.21)is a metric connection with skew-symmetric torsion G associated with the gravitino KSE,and A ( ± ) = /∂ Φ ± c ℓ A − Γ z − /G , (2.22)is associated with the dilatino KSE, and B ( ± ) should be thought as a projector whichrestricts the first two equations on either σ ± or τ ± spinors.For the investigation of the geometry of these backgrounds it suffices to consider onlythe τ + or the σ + spinors. This is because, if χ − is a solution to the above KSEs, then χ + = A − Γ + z χ − is also a solution, and vice versa, if χ + is a solution, then χ − = A Γ − z χ + is also a solution. Incidentally, this also implies that the number of supersymmetriespreserved by AdS backgrounds is always even. Furthermore, it suffices to investigate thegeometry of these backgrounds as described by the σ + spinors. As we have mentioned,the τ + spinors arise on choosing the other parallelization for AdS and it can be treatedsymmetrically, see also appendix D. 6 .3 Geometry If the solution of the KSEs is determined by the σ ± spinors, the investigation of thegeometry of M can be done as a special case of that of heterotic horizons in [23] whichutilized the classification results of [21]. To see this first note that h = − ℓ dz and so theconstant k which enters in the description of geometry for the heterotic horizons is k = h = 4 A − ℓ − . (2.23)Next observe that as σ + and σ − are linearly independent, there are two Killing spinorsgiven by ǫ = σ + , ǫ = σ − − ℓ − uA − Γ + z σ − . (2.24)Setting now σ − = A Γ − z σ + and after rescalling the second spinor with the non-vanishingconstant − ℓ − A − , we find that the two spinors can be rewritten as ǫ = σ + , ǫ = − k uσ + + Γ − /hσ + . (2.25)These are precisely the spinors that appear in the context of heterotic horizons, see [23]for a detailed description of the geometry of M including the emergence of the (left) sl (2 , R ) symmetry of AdS backgrounds as generated by the 1-form Killing spinor bi-linears. Briefly, M admits a G structure compatible with a metric connection ˆ ∇ withskew-symmetric torsion, ˆ ∇ i X j = ∇ i X j + G j ik X k . Furthermore all field equations andKSEs are implied provided [23] that d (cid:0) e ⋆ ϕ (cid:1) = 0 , dG = 0 , (2.26)where G = kϕ + e ⋆ d ( e − ϕ ) , (2.27)and ϕ is the fundamental form of the G structure. The first condition in (2.26) isrequired for the existence of a G structure on M compatible with a metric connectionwith 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 structure on M must be conformally balanced, θ ϕ = 2 d Φ, both of which have been incorporated in theexpression for G , where θ ϕ is the Lee form of ϕ . The conditions (2.26) are simpler thanthose that have appeared for heterotic horizons, because for AdS backgrounds dh = 0. backgrounds with extended supersym-metry We have shown that AdS backgrounds always preserve an even number of supersymme-tries. Furthermore, from the counting of supersymmetries for heterotic horizons [23], oneconcludes that AdS backgrounds preserve 2, 4, 6 and 8 supersymmetries. In addition,AdS backgrounds that preserve 8 supersymmetries and for which M is compact arelocally isometric to either AdS × S × T or to AdS × S × K . Again we shall not givethe details of the proof for these results. However, we shall state the key formulae thatarise in the investigation of the geometry for each case as they have some differences fromthose of the heterotic horizons. 7 .4.1 Four supersymmetries Let us first consider the AdS backgrounds with 4 supersymmetries. The two additionalspinors can be written as ǫ = σ , ǫ = − k uσ + Γ − /hσ , (2.28)where σ is linearly independent from σ = σ + in (2.25). In fact it can be shown that thenormal form for these spinors up to the action of Spin (7) can be chosen as σ = 1 + e and σ = i (1 − e ). The isotropy group of all four spinors is SU (3). Therefore M is aRiemannian manifold equipped with metric ds and a 3-form G . Furthermore, the metricconnection ˆ ∇ with skew-symmetric torsion G is compatible with an SU (3) structure. TheKSEs restrict this structure on M further. In particular, the SU (3) structure on M isassociated with 1-form ξ , 2-form ω , and (3,0)-form χ spinor bilinears such that i ξ ω = 0 , L ξ ω = 0 , i ξ χ = 0 , L ξ χ = ikχ , (2.29)where ω and χ are the fundamental forms of an SU (3) structure in the directions trans-verse to ξ . All these forms are ˆ ∇ -parallel, ˆ ∇ ξ = ˆ ∇ ω = ˆ ∇ χ = 0. In particular ˆ ∇ ξ = 0implies that ξ is Killing and that i ξ G = k − dw , where w ( ξ ) = k . As G is closed L ξ G = 0.The dilaton Φ is also invariant under ξ . The full set of conditions on ξ , ω and χ can befound in [23].The solution of these conditions implies that M can be locally constructed as a circlefibration on a conformally balanced , θ ω = 2 d Φ, KT manifold B with Hermitian form ω ,where the tangent space of the circle fibre is spanned by ξ . The canonical bundle of B admits a connection λ = k − w , such that dw (2 , = 0 , dw ij ω ij = − k , (2.30) i, j = 1 , , . . . ,
6, i.e. the canonical bundle is holomorphic and the connection satisfies theHermitian-Einstein instanton condition, and in additionˆ ρ (6) = dw , k − dw ∧ dw + dG (6) = 0 , (2.31)where ˆ ρ (6) ij = 12 ˆ R (6) ijkm I mk , (2.32)is the curvature of the canonical bundle induced from the connection with torsion G (6) = − i I dω on B , and I is the complex structure of B . The first condition is required for M 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 has a In fact with the data provided M admits a normal almost contact structure which however is furtherrestricted. θ ω is the Lee form of B . There some differences in the notation of this paper with that of [23]. For example w is denoted in[23] with ℓ . We have made this change because here we have denoted by ℓ the radius of AdS. (3) rather than an SU (3) structure compatible with a connection with skew-symmetrictorsion. This is because the (3,0)-form χ is not invariant under the action of ξ (2.29).The metric and torsion on M are given from those of B as ds ( M ) = k − w + ds ( B ) , G = k − w ∧ dw + G (6) . (2.33)This summarizes the geometry for the AdS backgrounds preserving four supersymme-tries. Solutions can be constructed using the techniques developed in [23] to find solutionsfor heterotic horizons. Next let us turn to AdS backgrounds preserving 6 supersymmetries. For these M admitsan SU (2) structure compatible with ˆ ∇ . Furthermore, M can be constructed locally as a SU (2) = S fibration over a 4-dimensional manifold B whose self-dual part of the Weyltensor vanishes. SU (2) twists over B with respect to a (principal bundle) connection λ which has curvature F r ′ such that the self-dual part satisfies( F sd ) r ′ = k ω r ′ , (2.34)where ω r ′ are the almost Hermitian forms of a quaternionic K¨ahler structure on B . Theanti-self dual part of F , F ad , is not restricted by the KSEs. The dilaton depends only onthe coordinates of B . The metric and G on M are given by ds ( M ) = δ r ′ s ′ λ r ′ λ s ′ + e d ˚ s ( B ) , G = CS ( λ ) − ˚ ⋆de , (2.35)where CS is the Chern-Simons form of λ . The only condition that remains to be solvedto find solutions is ˚ ∇ e = −
12 ( F ad ) + 38 k e , (2.36)where the inner products are taken with respect to the d ˚ s metric. For more details onthe geometry of such backgrounds see [23]. Next let us turn to the AdS backgrounds preserving 8 supersymmetries. The descriptionof the geometry is as that of the backgrounds above preserving 6 supersymmetries. Theonly differences are that B must be a hyper-K¨ahler manifold with respect to the d ˚ s ( B )metric, and that F sd = 0. The metric and 3-form G of M are given as in (2.35) but nowwe have that ˚ ∇ e = −
12 ( F ad ) , (2.37)instead of (2.36). If B is compact, a partial integration argument reveals tha F ad = 0and so the only regular solutions, up to discrete identifications, are AdS × S × K and AdS × S × T . If B is not compact, there are many smooth solutions, see [42]. Note that if F = 0, CS ( λ ) is proportional to the volume of S . .5 Lichnerowicz type theorem on σ + , τ + The Killing spinors of AdS backgrounds (2.20) can be identified with the zero modesof a suitable Dirac-like operator coupled to fluxes on M , and vice versa. This providesa new example of a Lichnerowicz type theorem for connections whose holonomy is notin a Spin group. This result is analogous to others that have been established for AdSbackgrounds in 11-dimensional and type II supergravities [18, 19, 20]. However, there aresome differences. One is that the spinor representation in the heterotic case is differentfrom that of the previous mentioned theories. There are also some subtle issues associatedwith 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 Lichnerowicztype of theorem with α ′ corrections will be investigated later. We define the modifiedgravitino Killing spinor operator,ˆ ∇ (+ ,q ,q ) i = ∇ (+) i + Γ i A (+ ,q ,q ) , (2.38)on the χ + spinors, where A (+ ,q ,q ) = − q A − Γ z B (+) + q A (+) , (2.39)for some q , q ∈ R . Observe that for q , q = 0, the holonomy of ˆ ∇ (+ ,q ,q ) is not in Spin (7). Next define the modified Dirac-like operator D (+) ≡ Γ i ˆ ∇ (+ ,q ,q ) i = Γ i ∇ i + Γ i Ψ (+) i + 7 A (+ ,q ,q ) . (2.40)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 , q such that all the zero modes of D (+) are Killing spinors. Thus we shall establish ∇ (+) i χ ± = 0 , A (+) χ ± = 0 , B (+) χ ± = 0 ⇐⇒ D (+) χ + = 0 . (2.41)The proof relies on global properties of M , which we assume to be smooth, and compactwithout boundary.To prove the theorem, let us assume that D (+) χ + = 0 and consider the identity ∇ k χ + k = 2 k∇ χ + k + 2 (cid:10) χ + , ∇ χ + (cid:11) . (2.42)The first term on the right hand side can be further rewritten in terms of the differentialoperator ˆ ∇ (+ ,q ,q ) by completing the square as2 k∇ χ + k = 2 (cid:13)(cid:13)(cid:13) ˆ ∇ (+ ,q ,q ) χ + (cid:13)(cid:13)(cid:13) − (cid:10) χ + , (cid:0) Ψ (+) i † + A (+ ,q ,q ) † Γ i (cid:1) ∇ i χ + (cid:11) − D χ + , (cid:0) Ψ (+) i † + A (+ ,q ,q ) † Γ i (cid:1) (cid:16) Ψ (+) i + Γ i A (+ ,q ,q ) (cid:17) χ + E = 2 (cid:13)(cid:13)(cid:13) ˆ ∇ (+ ,q ,q ) χ + (cid:13)(cid:13)(cid:13) − (cid:10) χ + , Ψ (+) i † ∇ i χ + (cid:11) − D χ + , (cid:0) Ψ (+) i † − A (+ ,q ,q ) † Γ i (cid:1) (cid:16) Ψ (+) i + Γ i A (+ ,q ,q ) (cid:17) χ + E , (2.43)10hile the second term can be rewritten using the identity / ∇ = ∇ − R (7) , and D (+) χ + =0, as 2 (cid:10) χ + , ∇ χ + (cid:11) = 2 (cid:10) χ + , Γ i ∇ i (cid:0) Γ j ∇ j χ + (cid:1)(cid:11) + 12 R (7) k χ + k = 12 R (7) k χ k − D χ + , ∇ i (cid:16) Γ i Γ j Ψ (+) j + 7Γ i A (+ ,q ,q ) (cid:17) χ + E − D χ + , (cid:16) Γ i Γ j Ψ (+) j + 7Γ i A (+ ,q ,q ) (cid:17) ∇ i χ + E . (2.44)Combining these, ∇ k χ + k can be rewritten as, ∇ k χ + k = 2 (cid:13)(cid:13)(cid:13) ˆ ∇ (+ ,q ,q ) χ + (cid:13)(cid:13)(cid:13) + 12 R (7) k χ + k + D χ + , h − (+) i † − i Γ j Ψ (+) j − q A − Γ zi B (+) − q Γ i A (+) i ∇ i χ + E + D χ + , − (cid:0) Ψ (+) i † − A (+ ,q ,q ) † Γ i (cid:1) (cid:16) Ψ (+) i + Γ i A (+ ,q ,q ) (cid:17) χ + E + D χ + , ∇ i h − i Γ j Ψ (+) j − q A − Γ zi B (+) − q Γ i A (+) i χ + E (2.45)where Ψ (+) † i = 18 /G i , B (+) † = − c + c ℓ , A (+) † = /∂ Φ + c ℓ A − Γ z + 112 /G . (2.46)Of the terms on the right hand side of (2.45), the first term is proportional to thegravitino Killing spinor equation squared, and so we expect that the remaining terms willbe equal to some combination of the algebraic KSEs. The third term includes a derivativeof χ + , however, and so we will attempt to write it in the form α i ∇ i k χ + k + (cid:10) χ + , F Γ i ∇ i χ + (cid:11) = α i ∇ i k χ + k − D χ + , F (cid:16) Γ i Ψ (+) i + 7 A (+ ,q ,q ) (cid:17) χ + E , (2.47)for some vector α and Clifford algebra element F that depend on the fields. In terms ofthe fields, the third term in the right hand side of (2.45) can be rewritten as D χ + , h − (+) i † − i Γ j Ψ (+) j − q A − Γ zi B (+) − q Γ i A (+) i ∇ i χ + E (2.48)= (cid:10) χ + , (cid:2) ℓ A − Γ zi ( q c + q c + 2 q c ) − q Γ i /∂ Φ+ q /G i + q Γ /G i i ∇ i χ + E . Thus, we find that it can be separated as outlined above if and only if q = − . We willuse this value of q from here on. Then we find that D χ + , h − (+) i † − i Γ j Ψ (+) j − q A − Γ zi B (+) + 2Γ i A i ∇ i χ + E = (cid:10) χ + , (cid:2) ℓ A − Γ zi (7 q c + 7 q c − c ) + 2Γ i /∂ Φ − /G i + Γ /G i i ∇ i χ + E , (2.49)and so, factoring out a Γ i on the right, F = 1 ℓ A − Γ z (7 q c + 7 q c − c ) − /∂ Φ − /G , (2.50)11nd α i = 2 ∂ i Φ.The F term part of the third term of (2.45) can be combined with the fourth term of(2.45) to give (cid:10) χ + , − (cid:0) Ψ (+) i † + q A − B (+) † Γ zi + A (+) † Γ i + F Γ i (cid:1)(cid:16) Ψ (+) i + q A − Γ zi B (+) − Γ i A (+) (cid:17) χ + E = D χ + , − h ℓ (cid:0) q c + 3 q c − c (cid:1) A − Γ zi − /∂ ΦΓ i + /G i + Γ /G i i(cid:2) − ℓ (cid:0) q c + q c − c (cid:1) A − Γ zi − Γ i /∂ Φ − /G i + Γ /G i (cid:3) χ + (cid:11) = D χ + , h − ℓ (7 q c + 7 q c − c ) A − − ( d Φ) − ∂ i ΦΓ /G i − ℓ (7 q c + 7 q c − c ) A − /G Γ z − /G /G − G (cid:3) χ + (cid:11) . (2.51)The last term on the right hand side of (2.45) is the only term involving derivatives ofthe fields other than Φ and the second derivative of Φ. However, we can use the Bianchiidentity and the Φ field equation to rewrite this term as, D χ + , ∇ i h − i Γ j Ψ (+) j − A − Γ zi B (+) + 2Γ i A i χ + E = (cid:10) χ + , (cid:2) ∇ Φ + /dG (cid:3) χ + (cid:11) = (cid:10) χ + , (cid:2) ℓ A − + 4( d Φ) − G (cid:3) χ + (cid:11) , (2.52)and we can use the scalar part of the Einstein equation to rewrite the second term on theright hand side of (2.45) as12 R (7) k χ + k = (cid:28) χ + , (cid:20) − ℓ A − − d Φ) + 524 G (cid:21) χ + (cid:29) . (2.53)Now we write the sum of (2.51), (2.52), and (2.53), as a linear combination of (cid:13)(cid:13) B (+) χ + (cid:13)(cid:13) , (cid:10) Γ z B (+) χ + , A (+) χ + (cid:11) , and (cid:13)(cid:13) A (+) χ + (cid:13)(cid:13) . In particular, the sum of (2.51), (2.52), and (2.53)is given by D χ + , h ℓ (cid:0) − q + 12 q + 12 q c c − q c c (cid:1) A − + ( d Φ) − ∂ i ΦΓ /G i − ℓ (7 q c + 7 q c − c ) A − /G Γ z − /G /G (cid:3) χ + (cid:11) , (2.54)whereas (cid:13)(cid:13) B (+) χ + (cid:13)(cid:13) = 1 + c c ℓ k χ + k , (cid:10) Γ z B (+) χ + , A (+) χ + (cid:11) = (cid:28) χ + , (cid:20) − ℓ (1 + c c ) A − − ℓ ( c + c ) /G Γ z (cid:21) χ + (cid:29) , (cid:13)(cid:13) A (+) χ + (cid:13)(cid:13) = h χ + , [( d Φ) + 1 ℓ A − − ∂ i ΦΓ /G i + c ℓ A − /G Γ z − /G /G ] χ + i . (2.55)It follows that ∇ k χ + k − ∂ i Φ ∇ i k χ + k = (cid:13)(cid:13)(cid:13) ˆ ∇ (+ ,q ,q ) χ + (cid:13)(cid:13)(cid:13) + 28 (cid:0) q − q (cid:1) A − (cid:13)(cid:13) B (+) χ + (cid:13)(cid:13)
12 4 q A − (cid:10) Γ z B (+) χ + , A χ + (cid:11) + 27 kA χ + k . (2.56)This expression is suitable to apply the Hopf maximum principle on the scalar function k χ + k on M as for 0 < q < the right hand side of this equation is positive definite.Assuming that the conditions required for the maximum principle on the fields and M apply, e.g. the fields are smooth and M is compact without boundary, the only solutionsto the above equation are that k χ + k is constant, and that, ∇ ( ± ) χ ± = 0 , A ( ± ) χ ± = 0 , B ( ± ) χ ± = 0 . (2.57)Thus χ + is a Killing spinor which establishes the theorem. backgrounds with dH = 0 Let us first consider the modifications that occur in the Bianchi identity, field equationsand KSEs of heterotic theory up to two loops in sigma model perturbation theory . Theanomaly cancelation mechanism requires the modification of the Bianchi identity for H as dH = − α ′ h tr( ˜ R ∧ ˜ R ) − tr( F ∧ F ) i + O ( α ′ ) , (3.1)where ˜ R is the curvature of a connection on the spacetime M which will not be specifiedat this stage, F is the curvature of the gauge sector connection of the heterotic theoryand α ′ is the string tension which also has the role of the loop parameter. Thus dH isexpressed as the difference of two Pontryagin forms, one is that of the tangent space ofspace-time and the other is that of the gauge sector bundle. Furthermore, global anomalycancelation requires in addition that the form on the right-hand-side of the anomalousBinachi identity represents the trivial cohomology class in H ( M ). This statement ismodified 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 getmodified. In particular up to two loops in sigma model perturbation theory [43], thedilaton, 2-form gauge potential, and gauge sector connection field equations read ∇ Φ = 2( d Φ) − H + α ′ h ˜ R MNST ˜ R MNST − F MNab F MNab i + O ( α ′ ) , ∇ R H MNR = 2 ∂ R Φ H MNR + O ( α ′ ) , ∇ N F MN + [ A N , F MN ] = 2 ∂ N Φ F MN + 12 H MNQ F NQ + O ( α ′ ) , (3.2)and the Einstein equation is R MN = 14 H MN − ∇ M ∇ N Φ − α ′ h ˜ R MLST ˜ R N LST − F MLab F N Lab i + O ( α ′ ) . (3.3) We use the conventions and normalization of the field equations, Bianchi identities and KSEs of [35]. ∇ M ǫ − /H M ǫ = 0 + O ( α ′ ) , (cid:0) /∂ Φ − /H (cid:1) ǫ = 0 + O ( α ′ ) ,/F ǫ = 0 + O ( α ′ ) . (3.4)In particular observe that the KSEs have the same form up to two loops in sigma modelperturbation theory as that at the zeroth order. It is not known how these equations aremodified at higher orders. The gauge indices of F have been suppressed.Before we proceed with the investigation of AdS backgrounds, let us specify ˜ R . Inperturbative heterotic theory, the choice of ˜ R is renormalization scheme dependent. Inother words, one can choose as ˜ R the curvature of any connection on M . However inmost applications ˜ R is chosen to be the curvature ˇ R of the ˇ ∇ = ∇ − H connection onthe spacetime. It is known that this choice has some key advantages. In particular itis required for the cancelation of world sheet supersymmetry anomaly [45] and also forthe consistency of the anomalous Bianchi identity with the modified Einstein equationsfor supersymmetric backgrounds. This has been used in the calculations of [34, 35] andrecently emphasized [46]. The property of ˇ R which is used to establish these is that ˇ R 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 identityˆ R MN,RS = ˇ R RS,MN − dH MNRS . (3.5)The integrability condition of the gravitino KSE gives ˆ R MN,RS Γ RS ǫ = 0. As the right-hand-side of the anomalous Bianchi identity is of order α ′ , it follows from (3.5) that, tozeroth order in α ′ , ˇ R MN,RS Γ MN ǫ = 0 or equivalently ˇ /Rǫ = 0 after suppressing the SO (9 , F in (3.4).To find solutions in the perturbative case, it is understood that the fields and Killingspinors are expanded in α ′ schematically as g = g + α ′ g + O ( α ′ ) , ǫ = ǫ + α ′ ǫ + O ( α ′ ) , (3.6)and similarly for the 3-form field strength, gauge potential and dilaton. Then the fieldequations and KSEs are solved order by order in α ′ to find the correction to the zerothorder fields.Next consider the case that the corrections to the heterotic theory are taken to beexact 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) donot receive further corrections from the ones that have been explicitly stated. Howeverconsistency of the anomalous Bianchi identity with the field equations requires that ˜ R satisfies the same conditions as those implied by the KSEs on the curvature F of thegauge connection, ie ˜ /Rǫ = 0 after suppressing the gauge indices. It is not apparent thatsuch a connection always exists but there are existence theorems in many cases of interest.Notice also the difference from the perturbation theory as ˜ R cannot be identified with ˇ R .This is because dH does not vanish in the right-hand-side of (3.5).14 .2 AdS backgrounds in perturbation theory Suppose that the symmetries of AdS remain symmetries of the background after the α ′ corrections are taken into account. In such a case, the fields up to two loops in perturbationtheory will decompose as ds = 2 e + e − + A dz + ds (cid:0) M (cid:1) + O ( α ′ ) ,H = AXe + ∧ e − ∧ dz + G + O ( α ′ ) . (3.7)This assumption is justified later. Furthermore, the field equations (3.2) and (3.3) read ∇ Φ = − A − ∂ i A∂ i Φ + 2( d Φ) − G + 12 X + α ′ (cid:2) ˇ R ij,kℓ ˇ R ij,kℓ − F ijab F ijab (cid:3) + O ( α ′ ) , ∇ k G ijk = − A − ∂ k AG ijk + 2 ∂ k Φ G ijk + O ( α ′ ) , (3.8)and the AdS component of the Einstein equation is unchanged, ∇ ln A = − ℓ A − − A − ( dA ) + 2 A − ∂ i A∂ i Φ + 12 X + O ( α ′ ) , (3.9)while component on M is now, R (7) ij = 3 ∇ i ∇ j ln A + 3 A − ∂ i A∂ j A + G ik k G j k k − ∇ i ∇ j Φ − α ′ (cid:2) ˇ R ik,st ˇ R jk,st − F ikab F j kab (cid:3) + O ( α ′ ) , (3.10)where i, j, k, ℓ = 1 , , . . . , R and F do not have componentsalong the AdS directions. As we shall see, this will follow from the KSEs.In addition, one finds that R (7) = 3 ∇ ln A + 3 A − ( dA ) + 14 G − ∇ Φ − α ′ (cid:2) ˇ R ij,kℓ ˇ R ij,kℓ − F ijab F ijab (cid:3) + O ( α ′ )= − ℓ A − − A − ( dA ) + 512 G + 12 X + 12 A − ∂ i A∂ i Φ − d Φ) − α ′ (cid:2) ˇ R ij,kℓ ˇ R ij,kℓ − F ijab F ijab (cid:3) + O ( α ′ ) . (3.11)Similarly, the anomalous Bianchi identity of H reads dG = − α ′ (cid:2) tr (cid:0) ˇ R ∧ ˇ R (cid:1) − tr( F ∧ F ) (cid:3) + O ( α ′ ) . (3.12)As we shall see imposing the requirement that spacetime supersymmetry is preserved bythe higher order corrections simplifies the above equations further.15 .3 Geometry of M for backgrounds with two supersymmetries In the perturbative approach to the heterotic string, one of the questions that arises iswhether the higher order corrections preserve the spacetime supersymmetry of the zerothorder background. In other words, whether there is a renormalization scheme whichpreserves the spacetime supersymmetry order by order in perturbation theory. Here weshall not investigate the existence of such a scheme. Instead, we shall derive the conditionsfor such a scheme to exist.We have shown that for AdS backgrounds admitting two spacetime supersymmetriesat zero order in α ′ , M has a G structure compatible with a connection with skew-symmetric torsion. In particular at this order dH = 0, and A and X are constant and c = ℓ AX = ±
1. The geometry of M at this order is described in section 2.3.The contribution in the terms proportional to α ′ in the field equations, Bianchi identi-ties and KSEs comes from the fields at zeroth order in α ′ . These depend on ˇ R and F . Atzeroth order, the spacetime factorizes into a product AdS × M . Furthermore the choiceof torsion on AdS is such that ˆ ∇| AdS and ˇ ∇| AdS are either the left or right invariantparallelizing connection; AdS is a group manifold. In either case, ˆ R | AdS = ˇ R | AdS = 0.Therefore the contribution in the α ′ terms of field equations, Bianchi identities and KSEscomes only from the ˇ R (7) curvature of M . Furthermore, the KSEs imply that the gaugecurvature F does not have components along AdS and is invariant under the isometriesof AdS up to gauge transformations. As a result all gauge invariant tensors constructedfrom F are tensors on M which do not depend on the coordinates of AdS . These justifythe choice of ˇ R and F made in the previous section.As the form of the gravitino KSE remains the same up to order α ′ , this implies that A and X are constant up to that order and that again c = ℓ AX = ±
1. Furthermore themetric and torsion of
AdS does not receive corrections at one loop, the form of the fieldsremains as in (3.7) up to order O ( α ′ ). The background remains factorized as AdS × M up to that order as well. Imposing all the above conditions on the fields, one finds thatthe anomalous Bianchi identity and field equations are simplified as in appendix B.Next focusing on the geometry of M , M admits a G structure compatible witha connection ˆ ∇ with skew-symmetric torsion G . As a consequence of the gravitino anddilatino KSEs, G is as given in (2.27) up to order α ′ . Moreover all the KSEs and fieldequations are satisfied provided that d (cid:0) e ⋆ ϕ (cid:1) = 0 + O ( α ′ ) , dG = − α ′ (cid:2) tr (cid:0) ˇ R (7) ∧ ˇ R (7) (cid:1) − tr( F ∧ F ) (cid:3) + O ( α ′ ) . (3.13)The first condition is required for the existence of a connection with skew-symmetrictorsion which is compatible with the G structure on M while the second conditionarises from the anomalous Bianchi identity. We have also assumed as in the dH = 0 casethat 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 instanton on M , and so it satisfies F ij = 12 ⋆ ϕ ijkm F km + O ( α ′ ) , (3.14)where we have suppressed the gauge indices. This summarizes the geometry of M up toorder α ′ . 16 .4 Extended supersymmetry Next let us investigate the geometry of AdS backgrounds preserving 4, 6 and 8 super-symmetries up to order α ′ . The geometry of the associated zeroth order backgrounds forwhich dH = 0 has already been described in section 2.4. These backgrounds are a special case of those we have described in the previous sectionthat preserve two supersymmetries. As a result up to order α ′ , the geometry is a product AdS × M . The presence of two more supersymmetries restricts further the geometryof M . As the form of the gravitino and dilatino KSEs remain the same as that of thezeroth order fields, the geometric restrictions on the geometry of M are similar to thosein section 2.4. The only difference here is that dH = 0. In particular, M has an SU (3)structure compatible with a connection with skew-symmetric torsion. So it admits aKilling vector field ξ such that i ξ G = k − dw + O ( α ′ ) , i ξ F = 0 + O ( α ′ ) , (3.15)where w ( ξ ) = k . Moreover, M can be locally described as a circle fibration of a con-formally balanced, θ ω = 2 d Φ, KT manifold B with Hermitian form ω whose canonicalbundle admits a connection k − w , such that dw (2 , = 0 + O ( α ′ ) , dw ij ω ij = − k + O ( α ′ ) , (3.16)i.e. the canonical bundle is holomorphic and the connection satisfies the Hermitian-Einstein instanton condition, and in additionˆ ρ (6) = dw + O ( α ′ ) , k − dw ∧ dw + dG (6) = − α ′ (cid:2) tr (cid:0) ˇ R (7) ∧ ˇ R (7) (cid:1) − tr( F ∧ F ) (cid:3) + O ( α ′ ) , (3.17)where ˆ ρ (6) ij = 12 ˆ R (6) ij km I mk + O ( α ′ ) , (3.18)is the curvature of the canonical bundle induced from the connection with torsion G (6) = − i I dω on B and I is the complex structure of B . The first condition is required for M 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), wherenow i, j, k, m = 1 , , . . . ,
6. It is understood that the expression in the right-hand-sideof the second equation in (3.17) is evaluated at the zeroth order fields. The metric andtorsion on M are given from those of B as in (2.33) but now of course the fields on B obey the equations (3.17) above. 17 .4.2 Six supersymmetries The presence of additional supersymmetries restricts the geometry of M further. Inparticular, the spacetime is still a product AdS × M up to order α ′ . The geometryof the zeroth order configuration has already been described in section 2.4 and so M islocally a S fibration over a 4-dimensional manifold B . As the gravitino and dilatinoKSEs have the same form up to order α ′ as the zeroth order equations, it is expectedthat M admits three ˆ ∇ -parallel vector bilinears ξ r ′ , r ′ = 1 , ,
3. Thus ξ r ′ are isometriesof the metric on M and i ξ r ′ H = k − dw r ′ up to order α ′ , where w r ′ ( ξ s ′ ) = kδ r ′ s ′ . As thegeometry of the spacetime is a product up to α ′ , these commute with the isometries ofAdS . However, the gravitino and dilatino KSEs do not determine the Lie bracket algebraof ξ r ′ ’s.To determine [ ξ r ′ , ξ s ′ ], first note that the commutator of two isometries is an isometry.Then using ˆ ∇ ξ r ′ = 0, we can establish the identities k − w [ ξ r ′ ,ξ s ′ ] = i ξ r ′ i ξ s ′ H , i [ ξ r ′ ,ξ s ′ ] H = k − dw [ ξ r ′ ,ξ s ′ ] + i ξ r ′ i ξ s ′ dH . (3.19)Next note that i ξ r ′ i ξ s ′ dH = 0 + O ( α ′ ). This follows from the fact that both ˇ R and F contribute in dH via the zeroth order fields and so as a consequence of the gravitinoand gaugino KSEs, i ξ r ′ ˇ R = i ξ r ′ F = 0. In fact F has to be a anti-self-dual instanton inthe directions transverse to AdS and ξ r ′ . As a consequence, the commutator [ ξ r ′ , ξ s ′ ] isˆ ∇ -parallel up to order α ′ . If [ ξ r ′ , ξ s ′ ] is not expressed in terms of ξ r ′ , the holonomy of ˆ ∇ is reduced to { } implying that the zeroth order backgrounds are group manifolds. Suchbackgrounds preserve 8 supersymmetries and will be investigated below. Thus [ ξ r ′ , ξ s ′ ]must close on ξ t ′ . Furthermore, one can use the Bianchi identityˆ R M [ N,P Q ] = −
13 ˆ ∇ M H NP Q + 16 dH MNP Q , (3.20)to show that dw r ′ restricted on the directions transverse to AdS and ξ r ′ is ˆ ∇ -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 is locally a S fibration over a 4-dimensionalmanifold B . The geometry can be described exactly as in the zeroth order case butthe various formulae are now valid up to order α ′ . The only modification occurs in theequation for the dilaton which now reads˚ ∇ e = −
12 ( F ad ) + 38 k e + α ′ (cid:0) ˜ R (4)2 − F (cid:1) + O ( α ′ ) , (3.21)where the inner products are taken with respect to the d ˚ s metric. The additional α ′ contribution is due to the anomalous Bianchi identity of H . The backgrounds with 8 supersymmeries can be investigated in a way similar to those with6 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 is a hyper-K¨ahlermanifold and F sd = 0. Up to order α ′ , the spacetime remains a product AdS × M .18he investigation of the closure properties of the three ˆ ∇ -parallel vector field ξ r ′ on M is not necessary. This is because it is a consequence of the gravitino and dilatino KSEsthat these vector fields close to a su (2) algebra [21]. The metric and torsion are given asin (2.35) but now the formulae are valid up to order α ′ . The only modification from thezeroth order equations is that the dilaton equation now reads˚ ∇ e = −
12 ( F ad ) + α ′ (cid:0) ˜ R (4)2 − F (cid:1) + O ( α ′ ) , (3.22)where the metric d ˚ s ( B ) is the zeroth order hyper-K¨ahler metric and the inner productshave been taken with respect to it.For compact B , at zeroth order F ad = 0, and in this case M = S × B up todiscrete identifications. As a consequence, the worldsheet action of the string factorizesinto a sum of a WZW model on S and a sigma model on the hyper-K¨ahler manifold B .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]. Suppose now that the theory up to two loops is exact. In such a case, the geometryof the solutions has to be re-examined as several arguments that have been applied inprevious cases have been based on the closure of H either to all orders or at the zerothorder 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 simplifyingassumption. This is that the backgrounds have the symmetries of AdS . In particular,the fields can be written as (2.1). The KSEs are ∇ M ǫ − /H M ǫ = 0 , (cid:0) /∂ Φ − /H (cid:1) ǫ = 0 , /F ǫ = 0 . (3.23)We also assume that the gaugino KSE has as many Killing spinors as the gravitino KSE. The G case is rather straightforward. As the form of the gravitino and dilatino KSEsin (3.23) is the same as that for dH = 0 backgrounds and the fields are invariant underthe symmetries of AdS , one finds that the gravitino KSE implies that A, X are constantand c = ℓ AX = ±
1. As a result, the geometry locally decomposes as
AdS × M . Thegeometry of M can now be described as in the perturbative case with the only differencethat now the equations are exact. In particular, M admits a G structure compatiblewith a connection with skew-symmetric torsion. This G structure is further restrictedby the KSEs, Bianchi identities and field equations as d (cid:0) e ⋆ ϕ (cid:1) = 0 , dG = − α ′ h tr (cid:16) ˜ R (7) ∧ ˜ R (7) (cid:17) − tr( F ∧ F ) i , (3.24)19here ϕ is the fundamental G G = kϕ + e ⋆ d ( e − ϕ ), and ˜ R and F are G instantons, ie ˜ R (7) ij,pq = 12 ⋆ ϕ ij km ˜ R (7) km,pq , F ij = 12 ⋆ ϕ ijkm F km . (3.25)The condition on F follows from the gaugino KSE. Observe that ˇ R , which is no longer a G instanton because of (3.5) and dH = 0, has now been replaced with ˜ R (7) . Moreover α ′ in (3.24) is a constant rather than a parameter. The geometry of these backgrounds also factorizes as
AdS × M . Moreover, M ad-mits a SU (3) structure compatible with a connection ˆ ∇ 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 . As these 4 vector bilinears are ˆ ∇ -parallel, their commutatoris [ ξ a , ξ b ] = i ξ a i ξ b H . Since the geometry factorizes as AdS × M , it turns out that thecommutator of the generators of sl (2 , R ) with the fourth vector bilinear vanishes, and sothe 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 k − dw ∧ dw + dG (6) = − α ′ h tr (cid:16) ˜ R (6) ∧ ˜ R (6) (cid:17) − tr( F ∧ F ) i , (3.26)where ˜ R (6) is a su (3) instanton on B , ie ˜ R (6) is a (1,1)-form and ω -traceless. Thiscondition is also satisfied by F because of the gaugino KSE. The geometry factorizes as
AdS × M and M admits an SU (2) structure compatiblewith a connection with skew-symmetric torsion ˆ ∇ . The spacetime admits 6 vector Killingspinor bilinears. Three of these span an sl (2 , R ) symmetry of AdS , and the other three ξ r ′ are ˆ ∇ -parallel on M and commute with those generating the sl (2 , R ). We shall arguethat for non-trivial backgrounds the commutator of these three vector field must close inthe set. To see this, consider the identities in (3.19). As ξ r ′ are Killing, their commutatoris 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 anomalousBianchi 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 ˆ ∇ -parallel, so is the commutator [ ξ r ′ , ξ s ′ ]. If the commutatordoes not close in the set ξ r ′ , the holonomy of ˆ ∇ will reduce to { } . As a result thecurvature of ˆ ∇ vanishes. If this is the case, the contribution to the anomalous Bianchiidentity 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 solutionsof the gaugino one. For such backgrounds backgrounds dH = 0 and so the spacetime isa group manifold which preserves 8 supersymmetries. Thus for backgrounds with strictly20ix supersymmetries, we shall take that [ ξ r ′ , ξ s ′ ] closes in the set ξ r ′ . Then it can be shownusing (3.20) that the symmetry group of the spacetime generated by the vector spinorbilinears is sl (2 , R ) ⊕ su (2).The rest of the investigation of the geometry is similar to that we have done in theperturbative case. The only difference is that now˚ ∇ e = −
12 ( F ad ) + 38 k e + α ′ (cid:0) ˜ R (4)2 − F (cid:1) , (3.27)where ˜ R (4) and F are anti-self-dual instantons on B and the inner products are takenwith respect to the d ˚ s metric. B is a 4-manifold with vanishing self-dual Weyl tensorand metric d ˚ s ( B ). The investigation of the geometry of these backgrounds is simpler than that described inthe previous section for backgrounds preserving 6 supersymmetries. First the geometryfactorizes as
AdS × M and M admits a connection with skew-symmetric torsion com-patible with a SU (2) structure. As in the previous case, M admits 3 ˆ ∇ -parallel Killingspinor bilinears ξ r ′ which commute with another three which span an sl (2 , R ) symmetryof AdS . Furthermore the gravitino and dilatino KSEs imply that the symmetry algebraof these backgrounds is sl (2 , R ) ⊕ su (2). The analysis of the geometry proceeds as in theperturbative case. In particular, M is an S fibration over a hyper-K¨ahler manifold B with metric d ˚ s ( B ). The only difference from the perturbative case is that now˚ ∇ e = −
12 ( F ad ) + α ′ (cid:0) ˜ R (4)2 − F (cid:1) , (3.28)where ˜ R (4) and F are anti-self-dual instantons on B . σ + , τ + The Lichnerowicz type theorem has to be re-examined in the presence of α ′ correctionsand in the case that the theory is truncated to two loops. Again, we shall focus on M ,and define the modified Dirac-like operator as in (2.40) but now dG = 0. Furthermore, weassume the Bianchi identities and field equations of appendix B but now we shall includethe α ′ terms, replacing the ˇ R (7) terms with ˜ R , and replacing F with ˜ F where ˜ R and ˜ F are arbitrary curvatures of T M and the gauge sector bundle respectively. In particular˜ R and ˜ F are not restricted by the KSEs. For the truncated theory at two loops, we takethe equations in appendix B as exact but again with ˇ R (7) and F replaced with ˜ R and ˜ F .The derivation of (2.51) is unaffected, but (2.52) becomes D χ + , ∇ i h − i Γ j Ψ (+) j − A − Γ zi B (+) + 2Γ i A (+) i χ + E = (cid:10) χ + , (cid:2) ∇ Φ + /dG (cid:3) χ + (cid:11) = D χ + , h ℓ A − + 4( d Φ) − G + α ′ h ˜ R ij,kℓ ˜ R ij,kℓ − ˜ F ijab ˜ F ijab i + α ′ h ˜ R i i ,jk ˜ R i i ,jk − ˜ F i i ab ˜ F i i ab i Γ i i i i i χ + E , (3.29)21nd (2.53) also picks up an α ′ term;12 R (7) k χ + k = (cid:10) χ + , (cid:2) − ℓ A − − d Φ) + G − α ′ h ˜ R ij,kℓ ˜ R ij,kℓ − ˜ F ijab ˜ F ijab ii χ + E . (3.30)On combining these expressions we obtain ∇ k χ + k − ∂ i Φ ∇ i k χ + k = (cid:13)(cid:13)(cid:13) ˆ ∇ (+ ,q ,q ) χ + (cid:13)(cid:13)(cid:13) + 28 (cid:0) q − q (cid:1) A − (cid:13)(cid:13) B (+) χ + (cid:13)(cid:13) + 4 q A − (cid:10) Γ z B (+) χ + , A χ + (cid:11) + 27 kA χ + k + α ′ k / ˜ F χ + k − α ′ h ˜ R ℓ ℓ ,mn Γ ℓ ℓ χ + , ˜ R p p ,mn Γ p p χ + i , (3.31)where we have suppressed the gauge index contraction in the k / ˜ F χ + k term, and q = − .We shall first consider the case of perturbation theory, and set ˜ R = ˇ R (7) . We beginby systematically analysing the conditions imposed by (3.31) order by order in α ′ . To zeroth order in α ′ , one obtains (provided that 0 < q < ), the conditions ∇ (+) χ + = 0 + O ( α ′ ) , A (+) χ + = 0 + O ( α ′ ) , B (+) χ + = 0 + O ( α ′ ) . (3.32)The condition ∇ (+) χ + = 0 + O ( α ′ ) implies the integrability conditionˆ R (7) mn,ℓ ℓ Γ ℓ ℓ χ + = 0 + O ( α ′ ) . (3.33)This in turn implies that ˇ R (7) ℓ ℓ ,mn Γ ℓ ℓ χ + = 0 + O ( α ′ ) . (3.34)It follows that the final term in (3.31) is in fact at least of order α ′ , and so can be ignored.It remains to show that (3.31) implies the KSEs to linear order in α ′ . For this considerthe perturbative expansion in the fields as in (3.6). One can show that if one assumesthat the zeroth order KSEs are imposed, (3.31) does not have an α ′ correction apart fromthe gaugino term, which leads to the condition / ˜ F χ + = 0 + O ( α ′ ) . (3.35)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 ′ terms is required which is not available.Observe that the above theorem also implies that all solutions of the gravitino and dilatinoKSEs are also solutions of the gaugino one. This is because the modified Dirac-likeoperator D (+) is constructed from only the gravitino and dilatino KSEs but neverthelessthe above theorem implies that the gaugino KSE is implied as well. We remark that in perturbation theory, the RHS of (3.31) is explicitly determined only up to firstorder in α ′ . The α ′ terms are not known, as one would require the corresponding α ′ corrections to theDirac operator, as well as dG and R (7) and ∇ Φ, in order to fix the α ′ terms.
22n the truncated theory, one can again formulate a Lichnerowicz type of theoremprovided that one imposes by hand the condition˜ R ℓ ℓ ,mn Γ ℓ ℓ χ + = 0 . (3.36)This condition (taking 0 < q < ) is sufficient to ensure that the RHS of (3.31) can bewritten as a sum of positive definite terms, which must all vanish. n , n ≥ and n = 2 back-grounds There are no AdS n , n ≥ α ′ correctionsup to two loops in sigma model perturbation theory. This includes the case for which thetheory 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 areinvariant under the symmetries of AdS n . Then we take a basis for the spacetime as { e λ = A ¯ e λ , e i } where ¯ e λ is a basis for AdS n , and e i is a basis for the internal space M − n . We take H to be a 3-form on M − n . The components of H , and the conformalfactor A , depend only on the co-ordinates of M − n .To proceed, consider the gravitino KSE along the AdS n frame directions, see alsoappendix E. This has no contribution from the 3-form H , and can be rewritten as¯ ∇ λ ǫ −
12 Γ λ ∂ i A Γ i ǫ = 0 . (4.1)where ¯ ∇ denotes the Levi-Civita connection on AdS n . The integrability condition of thisequation implies that (cid:18) Γ λ ¯ R µλ + (1 − n )( dA ) Γ µ (cid:19) ǫ = 0 . (4.2)where ¯ R µν is the Ricci tensor of ¯ ∇ . However, for AdS n , ¯ R µν = κ ¯ g µν where ¯ g is the metricon AdS n , and κ is a negative constant. The integrability condition (4.2) is then equivalentto (cid:18) κ + (1 − n )( dA ) (cid:19) ǫ = 0 (4.3)which admits no solution as κ < n ≥ dH = 0, and so excludesthe existence of AdS n , n >
3, backgrounds for the common sector and the heterotic the-ory for which there is not an anomalous correction to the Bianchi identity. This result isalso valid for the AdS n , n > n , n >
3, backgrounds in perturbative heterotic theorywith an anomalous contribution to the Bianchi identity, dH = 0. In this case, the argu-ment above implies that at zeroth order in α ′ , there are no such solutions. Furthermore,23t also excludes the existence of AdS n , n >
3, solutions up and including two loops insigma model perturbation theory that preserve all the symmetries of AdS n . However suchsolutions cannot completely be excluded in higher orders as it is not known how the KSEsand field equations are corrected. There is the possibility that one can start from anotherbackground which is allowed at zeroth order which then gets corrected in perturbationtheory to an AdS n , n > O ( α ′ ) in perturbation theorythere are no AdS n , n >
3, solutions to heterotic theory.It remains to investigate the existence of AdS solutions. It is a consequence of theinvestigation of near horizon geometries in [23] that if dH = 0, there are no AdS solutions.It should be emphasized though that the exclusion of AdS backgrounds requires the useof the maximum principle. Otherwise heterotic AdS backgrounds do exist [38]. Thisresult extends up to order α ′ in perturbation theory as it is unlikely that one can startfrom a different zeroth order background and correct it at one-loop approximation to anAdS background-though we do not have a proof for this. The existence of AdS solutionsfor the truncated theories will be examined elsewhere. Acknowledgements
JG is supported by the STFC grant, ST/1004874/1. GP is partially supported by theSTFC rolling grant ST/J002798/1.
Appendix A Notation and conventions
Our form conventions are as follows. Let ω be a k-form, then ω = 1 k ! ω i ...i k dx i ∧ · · · ∧ dx i k , (A.1)and dω = 1 k ! ∂ i ω i ...i k +1 dx i ∧ · · · ∧ dx i k +1 , (A.2)leading to ( dω ) i ...i k +1 = ( k + 1) ∂ [ i ω i ...i k +1 ] . (A.3)Furthermore, we write ω = ω i ...i k ω i ...i k , ω i i = ω i j ...j k − ω i j ...j k − . (A.4)It is well-known that for every form ω , one can define a Clifford algebra element /ω given by /ω = ω i ...i k Γ i ...i k (A.5)where Γ i , i = 1 , . . . n , are the Dirac gamma matrices. In addition we introduce thenotation /ω i = ω i i ...i k Γ i ...i k , / Γ ω i = Γ i i ...i k +1 ω i ...i k +1 . (A.6)The rest of our spinor conventions can be found in [21]24 ppendix B Bianchi identities and field equationsof the perturbed AdS background Using the conditions derived from the KSEs, the anomalous Bianchi identity and the fieldequations up to order α ′ can be written as follows. The Bianchi identity is dG = − α ′ (cid:2) tr (cid:0) ˇ R (7) ∧ ˇ R (7) (cid:1) − tr( F ∧ F ) (cid:3) + O ( α ′ ) . (B.1)The field equations can also be expressed as, ∇ Φ = 2( d Φ) − G + 12 X + α ′ h ˇ R (7) ij,kℓ ˇ R (7) ij,kℓ − F ijab F ijab i + O ( α ′ ) , ∇ k G ijk = 2 ∂ k Φ G ijk + O ( α ′ ) , (B.2)and R (7) ij = 14 G ik k G j k k − ∇ i ∇ j Φ − α ′ h ˇ R (7) ik,st ˇ R (7) jk,st − F ikab F j kab i + O ( α ′ ) . (B.3)From the latter, we find that R (7) = − ℓ A − + 512 G + 12 X − d Φ) − α ′ h ˇ R (7) ij,kℓ ˇ R (7) ij,kℓ − F ijab F ijab i + O ( α ′ ) . (B.4)This is used in the proof of the Lichnerowicz-type theorem for backgrounds with α ′ cor-rections. Appendix C Dilaton field equation
There are two ways to derive the dilaton field equation. The first makes use only of thebosonic field equations and yields the covariant derivative of the dilaton field equation.As a result the dilaton field equation is determined from the other field equations up toa constant. The second method makes use of the integrability conditions of the dilatinoand gravitino KSE, together with the gaugino KSE and the bosonic field equations, andyields the dilaton field equation. In this Appendix we present both methods.
C.1 Dilaton equation via bosonic conditions alone.
In the absence of an anomaly contribution, the dilaton field equation is implied by thefield equations of the other fields, up to an additive constant [48]. We shall re-examinethis in the presence of an anomalous contribution both in perturbation theory and in thecontext of the truncated theory.Taking the divergence of the Einstein equation (3.3), one has ∇ M R MN = 14 ∇ M H MN − ∇ ∇ N Φ − α ′ ∇ M h ˜ R MK,ST ˜ R N K,ST − F MKab F N Kab i . (C.1)25hen observe that − ∇ ∇ N Φ = − ∇ N ∇ Φ − R MN ∇ M Φ= − ∇ N ∇ Φ + 2 ∇ N ( d Φ) − H MN ∇ M Φ+ α ′ h ˜ R MK,ST ˜ R N K,ST − F MKab F N Kab i ∇ M Φ , (C.2)where the Einstein equation has been used again. Substituting this back into (C.1), ∇ M R MN = − ∇ N ∇ Φ + 2 ∇ N ( d Φ) + 14 e ∇ M ( e − H MN ) − α ′ ∇ M ( e − h ˜ R MK,ST ˜ R N K,ST − F MRab F N Rab i ) . (C.3)On the other hand, the divergence of the Ricci curvature is ∇ M R MN = 12 ∇ N R = ∇ N (cid:18) H − ∇ Φ − α ′ h ˜ R MN,ST ˜ R MN,ST − F MNab F MNab i(cid:19) (C.4)so, rearranging the Einstein equation ∇ N ∇ Φ = 2 ∇ N ( d Φ) + 14 e ∇ M ( e − H MN ) − ∇ N H − α ′ e ∇ M ( e − h ˜ R MK,ST ˜ R N K,ST − F MKab F N Kab i )+ α ′ ∇ N h ˜ R L L ,L L ˜ R L L ,L L − F L L ab F L L ab i . (C.5)From the Bianchi identity for H , H MKL dH MNKL ≡ H MKL ∇ M H NKL − ∇ N H = 3 α ′ H MKL h ˜ R MN,ST ˜ R KL,ST − F MN ab F KLab i . (C.6)Thus H MKL ∇ M H NKL = 16 ∇ N H + α ′ H MKL h ˜ R MN,ST ˜ R KL,ST − F MN ab F KLab i . (C.7)This together with the field equation for H gives14 e ∇ M ( e − H MN ) = 14 e ∇ M ( e − H MKL ) H N KL + 14 H MKL ∇ M H N KL = 124 ∇ N H + α ′ H MKL h ˜ R M N,ST ˜ R KL,ST − F M Nab F KLab i . (C.8)26ubstituting this back into (C.5), we find ∇ N ∇ Φ = 2 ∇ N ( d Φ) − ∇ N H + α ′ H MKL h ˜ R M N,ST ˜ R KL,ST − F M Nab F KLab i − α ′ e ∇ M ( e − h ˜ R MKST ˜ R N KST − F MKab F N Kab i )+ α ′ ∇ N h ˜ R L L L L ˜ R L L L L − F L L ab F L L ab i . (C.9)Clearly this implies the field equation of the dilaton up to a constant for theories that donot have an anomaly contribution.For the theories with an anomaly contribution, the derivation requires further exami-nation. First, let us focus on the quadratic terms in F in (C.9). From the Bianchi identityfor F ,0 = 32 F MK (cid:0) ∇ [ M F NK ] + [ A [ M , F NK ] ] (cid:1) = F MK (cid:0) ∇ M F NK + [ A M , F NK ] (cid:1) − ∇ N F , (C.10)where the gauge indices, which are traced over in all terms quadratic in F , have beensuppressed. Furthermore, using the field equation for F , one has α ′ e ∇ M ( e − F MK F N K ) = α ′ e ∇ M ( e − F MK ) F N K + α ′ F MK ∇ M F N K = α ′ H M K L F ML F N K + α ′ ∇ N F . (C.11)Therefore the terms quadratic in F sum in (C.9) to just − α ′ ∇ N F as expected.It remains to see what happens to the quadratic terms in ˜ R in (C.9). First, considerthe perturbative case, with ˜ R = ˇ R . The curvature terms can be most straightforwardlysimplified by making use of the appropriate curvature Bianchi identity, given byˇ ∇ [ K ˇ R MN ] ,SL = − H Q [ MN ˇ R K ] Q,SL (C.12)or equivalently, ∇ [ K ˇ R MN ] ,SL = 12 H QS [ K ˇ R MN ] ,SL − H QL [ K ˇ R MN ] ,QS . (C.13)On contracting the above identity over the K, S indices, we obtain the following usefulexpression, valid to zeroth order in α ′ : ∇ K (cid:18) e − ˇ R MN,KL (cid:19) − e − H QK [ M ˇ R N ] K,QL − e − H QKL ˇ R MN,QK = 0 (C.14)where we have made use of the identityˇ R NL ≡ ˇ R K N,KL = − ∇ N ∇ L Φ − ∂ Q Φ H QNL + O ( α ′ ) (C.15)27hich follows from the Einstein equations (3.3), and the gauge field equations for H . Onusing (C.13) and (C.14), it follows that18 H Q Q Q ˇ R Q N,L L ˇ R Q Q ,L L − e ∇ M (cid:18) e − ˇ R ML ,L L ˇ R N L ,L L (cid:19) = − ∇ N (cid:18) ˇ R L L ,L L ˇ R L L ,L L (cid:19) + O ( α ′ ) . (C.16)The above expression produces the appropriate curvature squared term in the dilatonequation at first order in α ′ . For the truncated theory, the appropriate curvature term inthe dilaton equation is obtained if (C.13) and (C.14) hold exactly, on replacing ˇ R with ˜ R . C.2 Dilaton equation via supersymmetry
Another way to obtain the dilaton field equation is to assume the KSEs, see also [21]. Inparticular, consider the condition / ∇ (cid:18)(cid:0) /∂ Φ − /H (cid:1) ǫ (cid:19) = 0 + O ( α ′ ) . (C.17)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: (cid:18) ∇ Φ − d Φ) + 112 H − /dH (cid:19) ǫ = 0 + O ( α ′ ) . (C.18)The dH bianchi identity term can then be rewritten to imply (cid:18) ∇ Φ − d Φ) + 112 H − α ′ (cid:0) ˜ R L L ,L L ˜ R L L ,L L − F L L ab F L L ab (cid:1) + α ′
32 ˜ R N N ,ST Γ N N ˜ R M M ,ST Γ M M − α ′ /F ab /F ab (cid:19) ǫ = 0 + O ( α ′ ) . (C.19)In the perturbative case, the final term on the LHS can be neglected as a consequenceof the gaugino KSE. The penultimate term can also be neglected because, if ˜ R = ˇ R then˜ R M M ,ST Γ M M ǫ = ˆ R ST,M M Γ M M ǫ + O ( α ′ ) = O ( α ′ ) (C.20)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 zerothorder 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 thegaugino KSE. The penultimate term vanishes if one assumes that˜ R M M ,ST Γ M M ǫ = 0 (C.21)and then (C.19) implies the dilaton equation.28 ppendix D Geometry of AdS backgrounds and τ spinors To investigate the conditions on the geometry of AdS backgrounds imposed by the τ ± Killing spinors, we note that there are two linearly independent spinors given by ǫ = e − zℓ τ + − ℓ − rA − e − zℓ Γ − z τ + , ǫ = e zℓ τ − . (D.1)Furthermore, as τ − = A Γ − z τ + , the two Killing spinors can be rewritten as ǫ = e − zℓ τ + − ℓ − rA − e − zℓ Γ − z τ + , ǫ = e zℓ A Γ − z τ + . (D.2)Next one can compute the 1-form Killing spinor bilinears which after an appropriatenormalization can be written as κ + = e zℓ e + , κ − = e − zℓ (cid:0) e − − ℓ − r A − e + + 2 ℓ − rdz (cid:1) ,κ z = dz − ℓ − rA − e + . (D.3)The Lie bracket algebra of the associated vector fields is[ κ + , κ z ] = − ℓ − A − κ + [ κ − , κ z ] = 2 ℓ − A − κ − [ κ + , κ − ] = 2 ℓ − κ z , (D.4)which is isomorphic to sl (2 , R ), where we have denoted the 1-forms and the associatedvector fields with the same symbols. Furthermore after some algebra one can show thatthe above vector fields commute with the vector fields constructed from the Killing spinorsdetermined in terms of the σ ’s. This confirms the statement in section 2.2 that the twochoices of Killing spinors for AdS backgrounds in terms of σ ’s and τ ’s correspond to thetwo different choices of parallelization of AdS as a group manifold in terms of the leftand right group actions. Appendix E The no-go theorem for AdS n , n = 3 backgrounds revisited The non-existence of AdS n , n ≥ α ′ corrections up to two loops in sigma model perturbation theory can be stated in a coordi-nate system similar to that used in the analysis of AdS backgrounds as follows. Supposethat the fields are invariant under the symmetries of AdS n . Then we have ds = 2 e + e − + A dz + A e − zℓ n − X a =1 ( dx a ) + ds (cid:0) M − n (cid:1) ,H = G, (E.1)The gravitino KSE along the AdS n , n >
3, directions reads ∂ u ǫ ± + A − Γ + z (cid:0) ℓ − − Ξ − (cid:1) ǫ ∓ = 0 ,∂ r ǫ ± − A − Γ − z Ξ + ǫ ∓ = 0 , z ǫ ± − Ξ ± ǫ ± + 2 rA − Γ − z Ξ + ǫ ∓ = 0 ,∂ a ǫ + + A − Γ za Ξ + ǫ + = 0 ,∂ a ǫ − + A − Γ za (cid:0) Ξ − − ℓ − (cid:1) ǫ − = 0 , (E.2)where Γ ± ǫ ± = 0 and Ξ ± = ∓ ℓ + 12 /∂A Γ z . (E.3)Using the relationsΞ ± Γ z + + Γ z + Ξ ∓ = 0 , Ξ ± Γ z − + Γ z − Ξ ∓ = 0 , Ξ ± Γ za + Γ za Ξ ± = ∓ ℓ − Γ za , (E.4)we find that there is only one AdS-AdS integrability condition which reads (cid:18) Ξ ± ± ℓ Ξ ± (cid:19) ǫ ± = 0 . (E.5)On the other hand Ξ ± = (cid:20) ∓ ℓ + 12 /∂A Γ z (cid:21) (cid:20) ∓ ℓ + 12 /∂A Γ z (cid:21) = 14 ℓ ∓ ℓ /∂A Γ z −
14 ( dA ) , (E.6)and so Ξ ± ± ℓ Ξ ± = − ℓ − ( dA ) . (E.7)Therefore the integrability condition cannot be satisfied for ℓ < ∞ . The rest of theanalysis is as in section 4. 30 eferences [1] P. G. O. Freund and M. A. Rubin, Dynamics of Dimensional Reduction,
Phys. Lett.B (1980) 233.[2] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza-Klein Supergravity,
Phys. Rept. (1986) 1.[3] M. Grana,
Flux compactifications in string theory: A Comprehensive review,
Phys.Rept. (2006) 91; [hep-th/0509003].[4] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz,
Large N fieldtheories, string theory and gravity, Phys. Rept. (2000) 183; [hep-th/9905111].[5] L. Castellani, L. J. Romans and N. P. Warner,
A Classification of CompactifyingSolutions for d = 11 Supergravity,
Nucl. Phys. B (1984) 429.L. J. Romans,
New Compactifications of Chiral N = 2 d = 10 Supergravity,
Phys.Lett. B (1985) 392.[6] C. N. Pope and N. P. Warner,
Two New Classes of Compactifications of d = 11 Supergravity,
Class. Quant. Grav. (1985) L1.[7] G. W. Gibbons and P. K. Townsend, Vacuum interpolation in supergravity via superp-branes, Phys. Rev. Lett. (1993) 3754; [hep-th/9307049].[8] I. R. Klebanov and E. Witten, Superconformal field theory on three-branes at aCalabi-Yau singularity,
Nucl. Phys. B (1998) 199; [hep-th/9807080].[9] B. S. Acharya, J. M. Figueroa-O’Farrill, C. M. Hull and B. J. Spence,
Branesat conical singularities and holography,
Adv. Theor. Math. Phys. (1999) 1249;[hep-th/9808014].[10] M. Cvetic, H. Lu, C. N. Pope and J. F. Vazquez-Poritz, AdS in warped space-times,
Phys. Rev. D (2000) 122003; [hep-th/0005246].[11] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Super symmetric AdS(5)solutions of M theory,
Class. Quant. Grav. (2004) 4335; [hep-th/0402153].[12] D. Lust and D. Tsimpis, Supersymmetric AdS(4) compactifications of IIA supergrav-ity,
JHEP (2005) 027; [hep-th/0412250].D. Lust and D. Tsimpis, New supersymmetric AdS(4) type II vacua,
JHEP (2009)098; [arXiv:0906.2561 [hep-th]].[13] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS(5) solu-tions of type IIB supergravity,
Class. Quant. Grav. (2006) 4693; [hep-th/0510125].[14] N. Kim and J. -D. Park, Comments on AdS(2) solutions of D=11 supergravity, JHEP (2006) 041; [hep-th/0607093].J. P. Gauntlett, N. Kim and D. Waldram, Supersymmetric AdS(3), AdS(2) andBubble Solutions,
JHEP (2007) 005; [hep-th/0612253].3115] M. Gabella, D. Martelli, A. Passias and J. Sparks, N = 2 supersymmetric AdS solutions of M-theory, Commun. Math. Phys. (2014) 487; [arXiv:1207.3082 [hep-th]].[16] F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello,
AdS solutions of typeII supergravity, JHEP (2014) 099; [arXiv:1406.0852 [hep-th]].F. Apruzzi, M. Fazzi, D. Rosa and A. Tomasiello, All
AdS solutions of type IIsupergravity, JHEP (2014) 064; [arXiv:1309.2949 [hep-th]].[17] N. T. Macpherson, C. Nunez, L. A. Pando Zayas, V. G. J. Rodgers and C. A. Whit-ing, Type IIB Supergravity Solutions with AdS From Abelian and Non-Abelian TDualities,
JHEP (2015) 040; [arXiv:1410.2650 [hep-th]].[18] J. B. Gutowski and G. Papadopoulos, Supersymmetry of AdS and flat backgroundsin M-theory,
JHEP (2015) 145; [arXiv:1407.5652 [hep-th]].[19] S. W. Beck, J. B. Gutowski and G. Papadopoulos, Supersymmetry of AdS and flatIIB backgrounds,
JHEP (2015) 020; [arXiv:1410.3431 [hep-th]].[20] S. Beck, J. B. Gutowski and G. Papadopoulos, Supersymmetry of IIA warped fluxAdS and flat backgrounds, [arXiv:1501.07620 [hep-th]].[21] U. Gran, P. Lohrmann and G. Papadopoulos,
The Spinorial geometry of supersym-metric heterotic string backgrounds,
JHEP (2006) 063; [hep-th/0510176].U. Gran, G. Papadopoulos, D. Roest and P. Sloane, Geometry of all supersymmetrictype I backgrounds,
JHEP (2007) 074; [hep-th/0703143 [HEP-TH]].G. Papadopoulos, Heterotic supersymmetric backgrounds with compact holonomy re-visited,
Class. Quant. Grav. (2010) 125008 [arXiv:0909.2870 [hep-th]].[22] H. Kunitomo and M. Ohta, Supersymmetric AdS(3) solutions in Heterotic Super-gravity,
Prog. Theor. Phys. (2009) 631; [arXiv:0902.0655 [hep-th]].[23] J. Gutowski and G. Papadopoulos,
Heterotic Black Horizons,
JHEP (2010) 011;[arXiv:0912.3472 [hep-th]].J. Gutowski and G. Papadopoulos, Heterotic horizons, Monge-Ampere equation anddel Pezzo surfaces,
JHEP (2010) 084; [arXiv:1003.2864 [hep-th]].[24] S. R. Green, E. J. Martinec, C. Quigley and S. Sethi, Constraints on String Cosmol-ogy,
Class. Quant. Grav. (2012) 075006 [arXiv:1110.0545 [hep-th]].F. F. Gautason, D. Junghans and M. Zagermann, On Cosmological Constants fromalpha’-Corrections,
JHEP (2012) 029 [arXiv:1204.0807 [hep-th]].[25] M. B. Green, J. H. Schwarz and E. Witten,
Superstring Theory,
Vols. 1 and 2,Cambridge, Uk: Univ. Pr. ( 1987) 596 P. ( Cambridge Monographs On MathematicalPhysics) 3226] A. Opfermann and G. Papadopoulos,
Homogeneous HKT and QKT manifolds; [math-ph/9807026].[27] P. S. Howe and G. Papadopoulos,
Twistor spaces for HKT manifolds,
Phys. Lett. B (1996) 80; [hep-th/9602108].[28] K. Becker, M. Becker, J. X. Fu, L. S. Tseng and S. T. Yau,
Anomaly cancellationand smooth non-Kahler solutions in heterotic string theory,
Nucl. Phys. B (2006)108; [hep-th/0604137].[29] J. X. Fu and S. T. Yau,
The Theory of superstring with flux on non-Kahler man-ifolds and the complex Monge-Ampere equation,
J. Diff. Geom. (2009) 369;[hep-th/0604063].[30] M. Fernandez, S. Ivanov, L. Ugarte and R. Villacampa, Non-Kaehler Heterotic StringCompactifications with non-zero fluxes and constant dilaton,
Commun. Math. Phys. (2009) 677; [arXiv:0804.1648 [math.DG]].[31] A. Strominger and E. Witten,
New Manifolds for Superstring Compactification,
Com-mun. Math. Phys. (1985) 341.[32] C. M. Hull,
Compactifications of the Heterotic Superstring,
Phys. Lett. B (1986)357.[33] S. Ivanov and G. Papadopoulos,
A No go theorem for string warped compactifications,
Phys. Lett. B (2001) 309; [hep-th/0008232].S. Ivanov and G. Papadopoulos,
Vanishing theorems and string backgrounds,
Class.Quant. Grav. (2001) 1089; [math/0010038 [math-dg]].[34] P. S. Howe and G. Papadopoulos, Finiteness and anomalies in (4,0) supersymmetricsigma models,
Nucl. Phys. B (1992) 360; [hep-th/9203070].[35] J. Gillard, G. Papadopoulos and D. Tsimpis,
Anomaly, fluxes and (2,0) heteroticstring compactifications,
JHEP (2003) 035; [hep-th/0304126].[36] H. Lu, C. N. Pope, K. S. Stelle and P. K. Townsend, Supersymmetric deformationsof G(2) manifolds from higher order corrections to string and M theory,
JHEP (2004) 019; [hep-th/0312002].[37] K. Becker, D. Robbins and E. Witten, The α ′ Expansion On A Compact ManifoldOf Exceptional Holonomy,
JHEP (2014) 051; [arXiv:1404.2460 [hep-th]].[38] A. Strominger, AdS(2) quantum gravity and string theory,” JHEP (1999) 007;[hep-th/9809027].[39] U. Gran, J. Gutowski and G. Papadopoulos, Index theory and dynamical symmetryenhancement near IIB horizons,
JHEP (2013) 104; [arXiv:1306.5765 [hep-th]].[40] U. Gran, J. Gutowski and G. Papadopoulos, AdS backgrounds from black hole hori-zons,
Class. Quant. Grav. (2013) 055014; [arXiv:1110.0479 [hep-th]].3341] T. Friedrich, S. Ivanov, Killing spinor equations in dimension 7 and geometry ofintegrable G -manifolds J. Geom. Phys. 48 (2003), 1-11; [math.DG/0112201].[42] G. Papadopoulos,
New half supersymmetric solutions of the heterotic string,
Class.Quant. Grav. (2009) 135001; [arXiv:0809.1156 [hep-th]].[43] C. M. Hull and P. K. Townsend, The Two Loop Beta Function For Sigma ModelsWith Torsion,
Phys. Lett. B (1987) 115.[44] E. A. Bergshoeff and M. de Roo,
The Quartic Effective Action Of The HeteroticString And Supersymmetry,
Nucl. Phys. B , 439 (1989).[45] P. S. Howe and G. Papadopoulos,
Anomalies in Two-dimensional SupersymmetricNonlinear σ Models,
Class. Quant. Grav. (1987) 1749.[46] D. Martelli and J. Sparks, Non-Kahler heterotic rotations,
Adv. Theor. Math. Phys. (2011) 131; [arXiv:1010.4031 [hep-th]].[47] P. S. Howe and G. Papadopoulos, Ultraviolet Behavior of Two-dimensional Super-symmetric Nonlinear σ Models,
Nucl. Phys. B (1987) 264.[48] A. A. Tseytlin, σ Model Weyl Invariance Conditions and String Equations of Motion,
Nucl. Phys. B294