Anomaly Corrected Heterotic Horizons
aa r X i v : . [ h e p - t h ] M a y DMUS–MP–16/07
Anomaly Corrected Heterotic Horizons
A. Fontanella , J. B. Gutowski and G. Papadopoulos Department of Mathematics, University of SurreyGuildford, GU2 7XH, UK.Email: [email protected]: [email protected] Department of Mathematics, King’s College LondonStrand, London WC2R 2LS, UK.E-mail: [email protected]
Abstract
We consider supersymmetric near-horizon geometries in heterotic supergravityup to two loop order in sigma model perturbation theory. We identify the conditionsfor the horizons to admit enhancement of supersymmetry. We show that solutionswhich undergo supersymmetry enhancement exhibit an sl (2 , R ) symmetry, and wedescribe the geometry of their horizon sections. We also prove a modified Lich-nerowicz type theorem, incorporating α ′ corrections, which relates Killing spinorsto zero modes of near-horizon Dirac operators. Furthermore, we demonstrate thatthere are no AdS solutions in heterotic supergravity up to second order in α ′ forwhich the fields are smooth and the internal space is smooth and compact withoutboundary. We investigate a class of nearly supersymmetric horizons, for which thegravitino Killing spinor equation is satisfied on the spatial cross sections but notthe dilatino one, and present a description of their geometry. Introduction
The effect of higher order corrections to supergravity solutions is of considerable interest,perhaps most notably for our understanding of quantum corrections to black holes. Thisis important in determining how string theory may resolve black hole singularities, as wellas the investigation of the properties of black holes away from the limit α ′ →
0. In higherdimensions the four dimensional uniqueness theorems [1, 2, 3, 4, 5, 6, 7] no longer hold,and there are exotic types of black hole solutions, such as the five dimensional black rings[8]. For ten and eleven dimensional supergravity, it is expected that there is a particu-larly rich structure of black objects, and the classification of these is ongoing. Progresshas recently been made in the classification of the near-horizon geometries of supersym-metric black holes. Near-horizon geometries of extremal black holes in supergravity areknown to generically undergo supersymmetry enhancement. This has been proven byanalysing the global properties of such solutions via generalized Lichnerowicz theorems[9, 10, 11, 12], and making use of index theory arguments [13]. One consequence of thesupersymmetry enhancement is that all such near-horizon geometries exhibit an sl (2 , R )symmetry. However, it is not apparent that these properties persist after including stringtheory corrections.There are several approaches to investigate how α ′ corrections can change the eventhorizons of black holes. Many black holes have AdS p × S q near-horizon geometries and asit is expected that the symmetries of such backgrounds persist in quantum theory, onlythe radii of the sphere and AdS receive α ′ corrections. However, we expect that exoticblack holes in higher dimensions need not necessarily have such near horizon geometries.Another approach, in the context of supersymmetric black holes in four and five dimen-sions, is to assume that the corrected near horizon geometries undergo an enhancement ofsupersymmetry in the near-horizon limit, which simplifies considerably the analysis of theKilling spinor equations. It is known that all supergravity D = 4 and D = 5 black holesundergo supersymmetry enhancement in the near-horizon limit [14, 15, 16]. In particular,the five dimensional BMPV black hole [17] undergoes supersymmetry enhancement from N = 4 to N = 8 (maximal supersymmetry) in the near-horizon limit [18]. Also, the super-symmetric asymptotically AdS black hole of [19] undergoes supersymmetry enhancementfrom N = 2 to N = 4 (half-maximal supersymmetry) in the near-horizon limit. Howeverit is not clear in general why one expects that the α ′ corrections should preserve thisproperty.The first systematic classification of supersymmetric near-horizon geometries in ahigher derivative theory in five dimensions [20] was done in [21], in which the only assump-tion made was that the solutions should preserve the minimal amount of supersymmetry.The five dimensional theory reduces to ungauged five-dimensional supergravity coupledto arbitrarily many vector multiplets when the higher derivative corrections are set tozero. In this limit, it is known that near-horizon geometries are maximally supersym-metric with constant scalars [22], which is consistent with the standard picture of theattractor mechanism. In contrast, when higher derivative terms are turned on, the listof near-horizon geometries determined in [21] includes not only the maximally supersym-metric geometries (which were classified in [23]), but also a set of regular non-maximallysupersymmetric solutions, on making use of a result of [24]. Although it is unclear if1hese particular near-horizon geometries can be extended to a full black hole solution,the existence of such a solution proves that for certain supergravity theories, the presenceof higher derivative terms can change how supersymmetry is enhanced for near-horizonsolutions.In this paper, we consider how higher derivative corrections to ten dimensional super-gravity affect the geometry and supersymmetry of near-horizon solutions. We shall chooseto begin this work by investigating heterotic theory which includes α ′ corrections up totwo loops in sigma model perturbation theory. This choice is motivated by two factors.Firstly, from the perspective of the standard supergravity, much more is known about thegeometric structure of generic supersymmetric solutions, and near-horizon geometries. Inparticular, as a consequence of the spinorial geometry classification techniques developedin [25, 26] which were then combined with a global analysis of near-horizon geometries in[27], there exists a full classification of all possible supersymmetric near-horizon geometriesin the heterotic supergravity. Secondly, the structure of higher derivative correction termsin the field equations, and in the Killing spinor equations, is significantly simpler for theheterotic theory when compared to the types of terms which arise in type II supergravity[29, 30, 31, 32], and associated references.The method we shall use to prove our results is that first we solve the Killing spinorequations in the near-horizon lightcone directions, and then simplify the remaining con-ditions as much as possible using both the local field equations and Bianchi identities, aswell as global analysis. For the global analysis, we shall assume that the spatial cross-section of the event horizon is smooth and compact, without boundary, and that allnear-horizon fields are also smooth. As a result of this analysis, we find that there are no AdS solutions (at zero and first order in α ′ ) to heterotic supergravity, which completesthe classification of heterotic AdS solutions in [33]. We also show that all of the conditionsof supersymmetry reduce to a pair of gravitino KSEs and a pair of algebraic KSEs on thespatial horizon sections. The latter are associated to the dilatino KSE. Throughout, weallow for all near-horizon data, including the spinors, to receive α ′ corrections.Using these conditions, we show that there is automatic supersymmetry enhancementat both zero and first order in α ′ in the case for which there exists negative light-conechirality Killing spinor η − up to O ( α ′ ) which does not vanish at zeroth order in α ′ . Inthis case the supersymmetry enhancement is obtained via the same mechanism as for thenear-horizon geometries considered in [27] without α ′ corrections, and the solution admitsan sl (2 , R ) symmetry. Such horizons admit 2, 4, 6 and 8 Killing spinors and their geometryis similar to that of horizons with vanishing anomaly contribution examined in [27]. Theremaining case, for which the negative light-cone chirality spinors vanish at zeroth order in α ′ remains open. We have investigated global aspects of these solutions by considering α ′ corrections to the global analysis carried out in [27], and also by constructing generalizedLichnerowicz theorems analogous to those proven in [9, 10, 11, 12], again incorporating α ′ corrections. However, in both cases, there is an undetermined sign in the O ( α ′ ) termsappearing, which precludes the extension of the maximum principle arguments to firstorder in α ′ .We also consider a class of near-horizon solutions which are “nearly” supersymmetric.These are not supersymmetric but some of their KSEs are satisfied. This is motivatedby the existence of WZW type of solutions to the heterotic theory with constant dilaton.2t is known that such solutions solve the gravitino KSE but not the dilatino one. In thepresent case, we consider horizons for which one of the gravitino KSEs is satisfied onthe spatial horizon section up order O ( α ′ ) but not the other and the algebraic KSEs.After some assumptions on the form of the fields, we give a complete description of thegeometry of such solutions.This paper is organized as follows. In section 2, we present the fields of heteroticnear horizon geometries and we integrate up the KSEs along the lightcone directions. Insections 3 and 4, we identify the independent KSEs by examining the various cases thatcan occur and in the process, prove that there are no AdS solutions. In section 5, wedetermine the conditions under which the horizons exhibit supersymmetry enhancement,and in section 6 we give the geometry of the horizon sections. In section 7, we generalizethe global analysis presented near-horizon geometries in [27] to include α ′ corrections.However because of a O ( α ′ ) sign ambiguity, it is not possible to prove that the horizonsection admits a G structure compatible with a connection with skew-symmetric torsion,as is the case at zeroth order in α ′ . We also generalize the Lichnerowicz type theorems tohigher orders in α ′ . Once again, O ( α ′ ) sign ambiguity means that it is not possible toprove that zero modes of the horizon Dirac equation (at zero and first order in α ′ ) satisfythe Killing spinor equations to the same order in α ′ , although the algebraic Killing spinorinvolving the 2-form gauge field is satisfied to the required order in α ′ . In sections 8 and9, we examine the geometry of nearly supersymmetric horizons focusing on those thatadmit a solution to the gravitino KSE on the horizon spatial section, and in section 10we give our conclusions.The paper contains several appendices. In appendix A, we summarize some key for-mulae that are used throughout in the computations of the paper and present the fieldequations of the theory. In appendix B, we provide the details of part of the proof toidentify the independent KSEs on the spatial horizon section. In section C, we presenta formula which relates the gravitino KSE to the gaugino KSE which is instrumentalin the investigation of the geometry of nearly supersymmetric horizons. In appendix D,we present further detail of the proof of the Lichnerowicz type theorem for the heterotictheory, and in Appendix E, we describe how AdS n +1 can be written as a warped productover AdS n , and describe how such constructions are inconsistent with our assumptionson the global structure and regularity of the solutions. The metric near a smooth killing horizon expressed in Gaussian null co-ordinates [34, 35]can be written as ds = 2 e + e − + δ ij e i e j , (2.1) Such solutions are not supersymmetric, and furthermore the spacetime gravitino KSE is not neces-sarily satisfied. e + = du , e − = dr + rh − r ∆ du , e i = e iJ dy J , (2.2) i, j = 1 , . . . , u, r are the lightcone coordinates, and the 1-form h , scalar ∆ and e i depend only on the coordinates y I , I = 1 , . . . ,
8, transverse to the lightcone. The blackhole stationary Killing vector field is identified with ∂ u . The induced metric on S is ds S = δ ij e i e j (2.3)and S is taken to be compact, connected without boundary. We denote the Levi-Civitaconnection of S by ˜ ∇ , and the Levi-Civita connection of the D=10 spacetime as ∇ .For the other heterotic fields, we assume that the dilaton Φ, and the real 3-form H ,and non-abelian gauge potential A admit well-defined near-horizon limits, and that ∂ u isa symmetry of the full solution: L ∂ u Φ = 0 , L ∂ u H = 0 , L ∂ u A = 0 . (2.4)In particular, this means that Φ = Φ( y ), and also H = e + ∧ e − ∧ N + r e + ∧ Y + W , (2.5)where N , Y and W are u, r -independent 1, 2 and 3-forms on S respectively, and we donot assume dH = 0. Moreover, A = r P e + + B (2.6)where P and B are a r, u -independent G -valued scalar and 1-form on S respectively. Thenon-abelian 2-form field strength F is given by F = dA + A ∧ A . (2.7)Our conventions for the heterotic theory including α ′ corrections are consistent with thoseof [29]. We assume that the near-horizon data admit a Taylor series expansion in α ′ . Wedenote this expansion by ∆ = ∆ [0] + α ′ ∆ [1] + O ( α ′ ) (2.8)and similarly for all near-horizon data, including spinors. For the supersymmetric solu-tions, we shall assume that that there is at least one zeroth order in α ′ Killing spinor, ǫ [0] = 0. In the previous treatments of heterotic near-horizon geometries [27], it was assumed thatthe anomaly vanishes and so the Bianchi identity dH = 0 was used to further simplifythe structure of the 3-form. Here, we shall not take dH = 0 as there is a non-trivial4ontribution from the heterotic anomaly, and so the 3-form takes the more general formgiven in (2.5).We remark that the KSE of heterotic supergravity have been solved in [25] and [26],and so, the solutions to the KSEs which we consider here correspond to a subclass of thesolutions in [25, 26]. However for horizons the global assumptions on the spatial section S , like compactness, allow the derivation of additional conditions on the spinors and onthe geometry. So it is particularly useful to re-solve the KSEs, decomposing the spinorsinto positive and negative lightcone chiralities adapted for the Gaussian null basis (2.2), ǫ = ǫ + + ǫ − , where Γ ± ǫ ± = 0 , Γ + − ǫ ± = ± ǫ ± . (2.9)We shall then extract from the KSEs the conditions imposed on ǫ ± that will be useful toapply the global conditions on S . We begin by considering the gravitino equationˆ ∇ M ǫ ≡ ∇ M ǫ − H MN N Γ N N ǫ = O ( α ′ ) . (2.10)First, on examining the M = − component of (2.10) we find that ǫ + = φ + + O ( α ′ ) , ǫ − = φ − + 14 r ( h − N ) i Γ − Γ i φ + + O ( α ′ ) , (2.11)where ∂ r φ ± = 0. Next, on examining the M = + component of (2.10), we find φ − = η − + O ( α ′ ) , φ + = η + + 14 u ( h + N ) i Γ + Γ i η − + O ( α ′ ) , (2.12)where ∂ r η ± = ∂ u η ± = 0. In additon, the M = + component of (2.10) implies a numberof algebraic conditions: (cid:18)
12 ∆ + 18 ( h − N ) −
18 ( dh + Y + h ∧ N ) ij Γ ij (cid:19) φ + = O ( α ′ ) , (2.13)and (cid:18) −
12 ∆ −
18 ( h − N ) −
18 ( dh + Y + h ∧ N ) ij Γ ij (cid:19) η − = O ( α ′ ) , (2.14)and (cid:18)
14 (∆ h i − ∂ i ∆)Γ i −
132 ( dh + Y ) ij Γ ij ( h − N ) k Γ k (cid:19) φ + = O ( α ′ ) . (2.15)We remark that (2.13) and (2.14) are equivalent to12 ∆ + 18 ( h − N ) = O ( α ′ ) , (2.16)5 dh + Y + h ∧ N ) ij Γ ij φ + = O ( α ′ ) , (2.17)and ( dh + Y + h ∧ N ) ij Γ ij η − = O ( α ′ ) , (2.18)respectively. Furthermore, using these conditions, (2.15) can also be rewritten as (cid:18)
14 (∆ h j − ∂ j ∆) −
18 ( h − N ) k (cid:0) dh + Y + 2 h ∧ N ) jk (cid:19) Γ j φ + = O ( α ′ ) . (2.19)Next, we consider the M = i components of (2.10). This implies˜ ∇ i φ + + (cid:18)
14 ( N − h ) i − W ijk Γ jk (cid:19) φ + = O ( α ′ ) , (2.20)and ˜ ∇ i η − + (cid:18)
14 ( h − N ) i − W ijk Γ jk (cid:19) η − = O ( α ′ ) , (2.21)together with the algebraic condition (cid:18) ˜ ∇ i ( h − N ) j + 12 ( h i N j − h j N i ) −
12 ( h i h j − N i N j ) − ( dh − Y ) ij − W ijk ( h − N ) k (cid:19) Γ j φ + = O ( α ′ ) . (2.22)These conditions exhaust the content of (2.10). Next again ignoring O ( α ′ ) terms we consider the dilatino KSE (cid:18) Γ M ∇ M Φ − H N N N Γ N N N (cid:19) ǫ = O ( α ′ ) . (2.23)On making use of the previous conditions, it is straightforward to show that the dilatinoKSE is equivalent to the following three conditions (cid:18) Γ i ˜ ∇ i Φ + 12 N i Γ i − W ijk Γ ijk (cid:19) φ + = O ( α ′ ) , (2.24)and (cid:18) Γ i ˜ ∇ i Φ − N i Γ i − W ijk Γ ijk (cid:19) η − = O ( α ′ ) , (2.25)and (cid:18)(cid:0) Γ i ˜ ∇ i Φ − N i Γ i − W ijk Γ ijk (cid:1) ( h − N ) ℓ Γ ℓ + Y ij Γ ij (cid:19) φ + = O ( α ′ ) . (2.26)6t remains to consider the gaugino KSE F MN Γ MN ǫ = O ( α ′ ) . (2.27)This implies the following conditions (cid:18) P + ˜ F ij Γ ij (cid:19) φ + = O ( α ′ ) , (2.28)and (cid:18) − P + ˜ F ij Γ ij (cid:19) η − = O ( α ′ ) , (2.29)and (cid:18) (cid:0) − P + ˜ F ij Γ ij (cid:1) ( h − N ) ℓ Γ ℓ + 2 (cid:0) h P + PB − BP − d P (cid:1) i Γ i (cid:19) φ + = O ( α ′ ) , (2.30)where ˜ F = d B + B ∧ B (2.31)The conditions (2.28) and (2.29) imply that P = O ( α ′ ) , (2.32)and so F = ˜ F + O ( α ′ ). Therefore (2.27) is equivalent to˜ F ij Γ ij φ + = O ( α ′ ) , (2.33)and ˜ F ij Γ ij η − = O ( α ′ ) , (2.34)and ˜ F ij Γ ij ( h − N ) ℓ Γ ℓ φ + = O ( α ′ ) . (2.35)In order to simplify these conditions further, we shall first consider the two cases forwhich either φ [0]+ ≡ φ [0]+ φ [0]+ ≡ Suppose that there exists a Killing spinor ǫ with ǫ [0]
0, but φ [0]+ ≡
0. Such a spinor musttherefore have η [0] −
0, and hence it follows that h [0] + N [0] = 0 . (3.1)7hen (2.21) implies that d k η [0] − k = − k η [0] − k h [0] . (3.2)In particular, this condition implies that if η [0] − vanishes at any point on the horizon section,then η [0] − = 0 everywhere. So, η [0] − must be everywhere non-vanishing.On taking the divergence of (3.2), and making use of the N = + , N = − componentof the 2-form gauge potential field equation (A.13), one obtains the following condition˜ ∇ [0] i ˜ ∇ [0] i k η [0] − k − (cid:0) ∇ i Φ [0] + k η [0] − k − ˜ ∇ [0] i k η [0] − k (cid:1) ˜ ∇ [0] i k η [0] − k = 0 . (3.3)As k η [0] − k is nowhere vanishing, an application of the maximum principle implies that k η [0] − k = const. , and hence (3.2) gives that h [0] = 0 , N [0] = 0 . (3.4)These conditions, together with (2.16), imply that∆ = O ( α ′ ) . (3.5)Then the dilaton field equation (A.15) implies that˜ ∇ i ˜ ∇ i ( e − ) = 16 e − W ijk W ijk + O ( α ′ ) , (3.6)and hence it follows that Φ [0] = const, W [0] = 0 . (3.7)Furthermore, this then implies that H = du ∧ dr ∧ N + rdu ∧ Y + W + O ( α ′ ) , (3.8)and hence dH = du ∧ dr ∧ ( dN − Y ) − rdu ∧ dY + dW + O ( α ′ ) . (3.9)As the ruij component on the RHS of the Bianchi identity is O ( α ′ ) this implies that Y = dN + O ( α ′ ) (3.10)and in particular, Y [0] = 0.Next consider the gauge equations. The + − component of the 2-form gauge potentialfield equations (A.13) is ˜ ∇ i N i = O ( α ′ ) . (3.11)Also, the u -dependent part of (4.18) implies that˜ ∇ i ( h + N ) j Γ j η − = O ( α ′ ) , (3.12)8hich gives that ˜ ∇ i ( h + N ) j = O ( α ′ ) . (3.13)Taking the trace of this expression, and using (3.14) yields˜ ∇ i h i = O ( α ′ ) . (3.14)Next, recall that the gravitino KSE (4.22) implies˜ ∇ i k η − k = −
12 ( h − N ) i k η − k + O ( α ′ ) (3.15)Taking the divergence yields, together with (3.11) and (3.14) the condition˜ ∇ i ˜ ∇ i k η − k = O ( α ′ ) (3.16)which implies that k η − k = const + O ( α ′ ). Substituting back into (3.15) gives thecondition N = h + O ( α ′ ), and hence (3.13) implies that˜ ∇ i h j = O ( α ′ ) . (3.17)So, to summarize, for this class of solutions, we have obtained the following conditionson the fields N = h + O ( α ′ ) , h [0] = 0 , Y = O ( α ′ ) , ˜ ∇ i h j = O ( α ′ ) , ∆ = O ( α ′ ) , H [0] = 0 , Φ [0] = const , (3.18)and it is straightforward to check that the generic conditions on φ + then simplify to˜ ∇ i φ + − W ijk Γ jk φ + = O ( α ′ ) , (3.19)and (cid:18) Γ i ˜ ∇ i Φ + 12 h i Γ i − W ijk Γ ijk (cid:19) φ + = O ( α ′ ) (3.20)and ˜ F ij Γ ij φ + = O ( α ′ ) . (3.21)The generic conditions on η − also simplify to˜ ∇ i η − − W ijk Γ jk η − = O ( α ′ ) (3.22)and (cid:18) Γ i ˜ ∇ i Φ − h i Γ i − W ijk Γ ijk (cid:19) η − = O ( α ′ ) (3.23)and ˜ F ij Γ ij η − = O ( α ′ ) . (3.24)In the next section, we shall consider the case for which there exists a Killing spinorwith φ [0]+
0. It will be shown that the conditions (3.18) on the bosonic fields and thesimplified KSEs listed above correspond to special cases of the corresponding conditionson the fields and simplified KSEs of φ [0]+
0. In particular, this will allow the KSEs for φ [0]+ ≡ φ [0]+ Solutions with φ [0]+ Suppose that there exists a Killing spinor ǫ , with ǫ [0] φ [0]+
0. Then consider(2.20); this implies that˜ ∇ i k φ + k = 12 ( h i − N i ) k φ + k + O ( α ′ ) , (4.1)and (2.22) gives that˜ ∇ i ( h − N ) j + 12 ( h i N j − h j N i ) −
12 ( h i h j − N i N j ) − ( dh − Y ) ij − W ijk ( h − N ) k = O ( α ′ ) . (4.2)Taking the divergence of (4.1), and using (2.20) together with the trace of (4.2), we findthat ˜ ∇ i ˜ ∇ i k φ + k − h i ˜ ∇ i k φ + k = O ( α ′ ) . (4.3)An application of the maximum principle (see e.g. [36]) then yields the condition˜ ∇ i k φ + k = O ( α ′ ) . (4.4)To see this, note that to zeroth order in α ′ , (4.3) implies that ˜ ∇ [0] i k φ [0]+ k = 0,on applying the maximum principle. Then (4.1) and (4.2) imply that N [0] = h [0] and Y [0] = dh [0] ; and from (2.16) we also have ∆ [0] = 0. Then it is useful to consider the fieldequations of the 2-form gauge potential (A.13), which imply that˜ ∇ i (cid:18) e − h i (cid:19) = O ( α ′ ) , (4.5)and e ˜ ∇ j (cid:0) e − dh ji (cid:1) + 12 W ijk dh jk + h j dh ji = O ( α ′ ) , (4.6)and the Einstein equations imply that˜ R ij + ˜ ∇ ( i h j ) − W imn W j mn + 2 ˜ ∇ i ˜ ∇ j Φ = O ( α ′ ) . (4.7)Using (4.5), (4.6) and (4.7) it follows that ˜ ∇ i ˜ ∇ i h + ( h − d Φ) j ˜ ∇ j h = 2 ˜ ∇ ( i h j ) ˜ ∇ ( i h j ) We remark that the condition (4.8) was also obtained in [27]. In that case, a bilinear matchingcondition was imposed in order to find N [0] = h [0] , Y [0] = dh [0] . Here we do not assume such a bilinearmatching condition, but nevertheless we find the same condition.
10 12 ( dh − i h W ) ij ( dh − i h W ) ij + O ( α ′ ) . (4.8)In particular, (4.8) implies that ˜ ∇ [0] i h [0] i = 0 on applying the maximum principle. Itfollows from (4.3) that˜ ∇ [0] i ˜ ∇ [0] i h φ [0]+ , φ [1]+ i − h [0] i ˜ ∇ [0] i h φ [0]+ , φ [1]+ i = 0 . (4.9)On multiplying this condition by h φ [0]+ , φ [1]+ i and integrating by parts, using ˜ ∇ [0] i h [0] i = 0,one finds that ˜ ∇ [0] i h φ [0]+ , φ [1]+ i = 0 as well. So, it follows that ˜ ∇ i k φ + k = O ( α ′ ).Then, (4.1) also implies that N = h + O ( α ′ ). Substituting these conditions back into(2.16), we find that ∆ [1] = 0 as well, so ∆ = O ( α ′ ). Also, (4.2) implies that Y − dh = O ( α ′ ) . (4.10)To summarize the conditions on the bosonic fields; we have shown that for solutionswith φ [0]+ = 0, we must have∆ = O ( α ′ ) , N = h + O ( α ′ ) , Y = dh + O ( α ′ ) (4.11)which implies that H = d ( e − ∧ e + ) + W + O ( α ′ ) . (4.12)The field equation (A.13) of the 2-form gauge potential can then be rewritten in terms ofthe near-horizon data as ˜ ∇ i (cid:0) e − h i (cid:1) = O ( α ′ ) , (4.13) e ˜ ∇ j (cid:0) e − dh ji (cid:1) + 12 W ijk dh jk + h j dh ji = O ( α ′ ) , (4.14)and e ˜ ∇ k (cid:0) e − W kij (cid:1) + dh ij − h k W kij = O ( α ′ ) . (4.15)In addition, P = O ( α ′ ) and so F = ˜ F + O ( α ′ ). The i, j component of the Einsteinequation then simplifies to ˜ R ij + ˜ ∇ ( i h j ) − W imn W j mn + 2 ˜ ∇ i ˜ ∇ j Φ+ α ′ (cid:18) − dh iℓ dh j ℓ + ˇ˜ R iℓ ,ℓ ℓ ˇ˜ R j ℓ ,ℓ ℓ − ˜ F iℓab ˜ F j ℓab (cid:19) = O ( α ′ ) . (4.16)Furthermore, dilaton field equation can be written as˜ ∇ i ˜ ∇ i Φ − h i ˜ ∇ i Φ − ∇ i Φ ˜ ∇ i Φ − h i h i + 112 W ijk W ijk + α ′ (cid:0) dh ij dh ij + ˜ F ij ab ˜ F ijab − ˇ˜ R ℓ ℓ ,ℓ ℓ ˇ˜ R ℓ ℓ ,ℓ ℓ (cid:1) = O ( α ′ ) . (4.17)11n making use of the conditions (4.11) on the bosonic fields, the KSEs on φ + thensimplify further to ˜ ∇ i φ + − W ijk Γ jk φ + = O ( α ′ ) , (4.18) dh ij Γ ij φ + = O ( α ′ ) , (4.19) (cid:18) Γ i ˜ ∇ i Φ + 12 h i Γ i − W ijk Γ ijk (cid:19) φ + = O ( α ′ ) , (4.20)and ˜ F ij Γ ij φ + = O ( α ′ ) . (4.21)Furthermore, KSEs on η − also simplify to˜ ∇ i η − − W ijk Γ jk η − = O ( α ′ ) , (4.22) dh ij Γ ij η − = O ( α ′ ) , (4.23) (cid:18) Γ i ˜ ∇ i Φ − h i Γ i − W ijk Γ ijk (cid:19) η − = O ( α ′ ) , (4.24)and ˜ F ij Γ ij η − = O ( α ′ ) . (4.25)In both cases above, (4.18) and (4.22) are a consequence of the gravitino KSE, (4.20)and (4.24) are associated to the dilatino KSE, while (4.21) and (4.25) are derived fromthe gaugino KSE. The two additional conditions (4.19) and (4.23) can be thought of asintegrability conditions. The KSEs we have stated in the previous sections (3.19)-(3.24) and (4.18)-(4.25) are notall independent. It turns out that the independent KSEs areˆ˜ ∇ η ± ≡ ˜ ∇ i η ± − W ijk Γ jk η ± = O ( α ′ ) (4.26)and (cid:18) Γ i ˜ ∇ i Φ ± h i Γ i − W ijk Γ ijk (cid:19) η ± = O ( α ′ ) . (4.27)This is the case irrespectively on whether φ [0]+ ≡ φ [0]+ = 0 though the conditions onthe bosonic fields are somewhat different. The proof of this independence of the KSEsrequires the use of field equations and Bianchi identities and it is rather involved. Thedetails can be found in appendix B. 12 Supersymmetry enhancement
A key ingredient in the investigation of heterotic horizons is that supersymmetry alwaysenhances. As a result horizons preserve 2, 4, 6 and 8 supersymmetries [27]. However thisis based on a global argument which we shall see does not necessarily apply to O ( α ′ ).As a result we shall seek some alternative conditions to guarantee that supersymmetryenhances. In particular we shall show that if there exists a Killing spinor ǫ = ǫ ( η + , η − ) upto O ( α ′ ), ie η − solves (4.26) and (4.27) up to O ( α ′ ), such that η [0] − = 0, and the horizonhas h [0] = 0, then there is automatic supersymmetry enhancement.To prove this, it suffices to demonstrate that h leaves all fields invariant and that itis covariantly constant with respect to the connection with torsion ˆ˜ ∇ on S . Indeed, firstnote that (B.15) implies thatˆ˜ ∇ i h j ≡ ˜ ∇ i h j − W ijk h k = O ( α ′ ) . (5.1)In particular, to both zeroth and first order in α ′ , h defines an isometry on S , with h = const + O ( α ′ ). Then the gauge equation (4.13) implies L h Φ = O ( α ′ ) . (5.2)Also, the u -dependent part of (4.21) implies( i h ˜ F ) i Γ i η − = O ( α ′ ) , (5.3)which implies that i h ˜ F = O ( α ′ ). So in the gauge for which i h B = 0, one has L h ˜ F = O ( α ′ ) . (5.4)Next we consider L h W , where L h W = − α ′ (cid:18) tr (cid:0) ( i h ˇ R ) ∧ ˇ R (cid:1)(cid:19) + O ( α ′ ) , (5.5)because dh = i h W + O ( α ′ ). To evaluate this expression, note first that the integrabilityconditions of ˆ˜ ∇ i η − = O ( α ′ ) , ˆ˜ ∇ i ( h ℓ Γ ℓ η − ) = O ( α ′ ) (5.6)are ˆ˜ R ijpq Γ pq η − = O ( α ′ ) , ˆ˜ R ijpq Γ pq ( h ℓ Γ ℓ η − ) = O ( α ′ ) (5.7)from which we obtain the condition h ℓ ˆ˜ R ijℓq = O ( α ′ ) , (5.8)and hence, as a consequence of (A.4), h ℓ ˇ˜ R ℓqij = O ( α ′ ) . (5.9)13oreover, h ℓ ˇ˜ R ℓq + − = h i ( dh ) iq = O ( α ′ ) . (5.10)It follows that the contribution of i h ˇ R to the RHS of (5.5) is of at least O ( α ′ ), and hence L h W = O ( α ′ ) . (5.11)So, we have shown that to both zero and first order in α ′ , the Lie derivative of the metricon S , as well as h, Φ and W with respect to h vanishes, and the Lie derivative of ˜ F withrespect to h vanishes to zeroth order in α ′ .Supersymmetry is therefore enhanced, because if η + satisfies (4.26) and (4.27), thenso does η ′− = Γ − h i Γ i η + . Conversely, if η − satisfies (4.26) and (4.27), then so does η ′ + =Γ + h i Γ i η − . The proof of this makes use of the conditions (5.1), together with (5.2) and(4.19) and (4.23), and the reasoning is identical to that used in [27]. This establishes a1-1 correspondence between spinors η + and η − satisfying (4.26) and (4.27), so the numberof supersymmetries preserved is always even.Next we wish to determine whether a similar supersymmetry enhancement argumentholds for η + spinors. In particular if there exists a solution to (4.26) and (4.27) with η [0]+ = 0 and h [0] = 0, does this imply that the number of η + solutions is equal to thenumber of η − solutions? This does not follow from a local analysis of (4.26) and (4.27),because there is no analogue of the condition (B.15) acting on η + . Nevertheless, in [27]a global analysis was used in order to establish such a correspondence, by computing theLaplacian of h and applying a maximum principle argument, in order to obtain (5.1) tozeroth order in α ′ . We shall revisit this analysis in section 7.1 including the α ′ corrections. It is a consequence of the results of [27], see also section 7.1, that horizons with non-trivialfluxes preserve an even number of supersymmetries up to O ( α ′ ). Furthermore we havealso demonstrated that such horizons with η − Killing spinors preserve an even numberof supersymmetries up to O ( α ′ ). It is straightforward to see that horizons with morethan 8 supersymmetries are trivial, ie the rotation h vanishes. Therefore, the heterotichorizons of interest preserve 2,4,6 and 8 supersymmetries.Up to O ( α ′ ), the investigation of geometry of all such horizons is identical to that givenin [27] for heterotic horizons with closed 3-form field strength. Here we shall describe thegeometry of the horizons that admit a η − Killing spinor up to O ( α ′ ). We have seen thatfor such horizons h is parallel with respect to the connection with torsion up to O ( α ′ ).Because of this, the geometry of such horizons is very similar to that of horizons withclosed 3-form flux. The only differences between the geometries of the two cases are solelylocated in the modified Bianchi identity for the 3-form flux. As the two cases are similar,the description of the geometry will be brief. G structure Such horizons admit two supersymmetries up to O ( α ′ ). In particular h satisfies (5.1). Thespacetime locally can be described as a (principal) SL (2 , R ) fibration over a 7-dimensional14anifold B which admits a metric d ˜ s and a 3-form ˜ H (7) such that the connection ˆ˜ ∇ (7) with torsion ˜ H (7) has holonomy contained in G . The spacetime metric and 3-form fluxcan be written as ds = η ab λ a λ b + d ˜ s + O ( α ′ ) ,H = CS ( λ ) + ˜ H (7) + O ( α ′ ) , (6.1)where CS ( λ ) is the Chern-Simons form of the principal bundle connection, λ − = e − , λ + = e + − k u e − − uh , λ = k − (cid:0) h + k u e − (cid:1) , (6.2) k = h is constant up to O ( α ′ ) and˜ H (7) = kϕ + e ⋆ d (cid:0) e − ϕ (cid:1) + O ( α ′ ) . (6.3)The 3-form ϕ is the fundamental G and it is related to the fundamental Spin (7) form ofthe η + Killing spinor via ϕ = k − i h φ + O ( α ′ ). The associated vector fields to λ − , λ + , λ satisfy a sl (2 , R ) algebra. The dilaton Φ depends only on the coordinates of B .To find solutions, one has to solve the remaining equations d [ e − ⋆ ϕ ] = O ( α ′ ) ,k − dh ∧ dh + d ˜ H (7) = − α ′ (cid:18) − dh ∧ dh + tr( ˇ R (8) ∧ ˇ R (8) − F ∧ F ) (cid:19) + O ( α ′ ) , ( dh ) ij = 12 ⋆ ϕ ijkl ( dh ) kl + O ( α ′ ) , F ij = 12 ⋆ ϕ ij kl F kl + O ( α ′ ) . (6.4)The first condition in (6.4) is required for B to admit a G structure compatible with aconnection with skew-symmetric torsion. The second condition is the anomalous Bianchiidentity of the 3-form field strength written in terms of B data. The curvature ˇ R (8) isthat of the near horizon section S with metric and skew symmetric torsion given by d ˜ s = k − h ⊗ h + d ˜ s + O ( α ′ ) , ˜ H (8) = k − h ∧ dh + ˜ H (7) + O ( α ′ ) . (6.5)As ˇ R (8) is invariant under h and i h ˇ R (8) = O ( α ′ ), it descends on B . Finally, the last twoequations in (6.4) imply that both dh and F are g instantons on B . SU (3) structure Such horizons preserve 4 supersymmetries. Locally the spacetime is a principal bundlewith fibre SL (2 , R ) × U (1) over a K¨ahler with torsion manifold (KT) B with Hermitianform ω (6) . The metric and 3-form field strength of the spacetime can be written as ds = η ab λ a λ b + d ˜ s + O ( α ′ ) , H = CS ( λ ) + ˜ H (6) + O ( α ′ ) , (6.6)where λ a , a = + , − , , a = + , − , λ = k − ℓ (6.7) Note that CS ( λ ) = du ∧ dr ∧ h + rdu ∧ dh + k − h ∧ dh . u (1) direction in the Lie algebra. h = k is constant up to O ( α ′ ).The curvature of the principal bundle connection λ a is expressed in terms of dh and dℓ which are 2-forms on B and it is required to satisfy that dh , = dℓ , = O ( α ′ ) , dh ij ω ij (6) = O ( α ′ ) , dℓ ij ω ij (6) = − k + O ( α ′ ) , (6.8)ie h is a su (3) instanton on B while ℓ is a u (3) instanton on B .The KT manifold B is in addition conformally balanced, ie θ ω (6) = 2 d Φ + O ( α ′ ) , (6.9)where θ is the Lee form and the torsion is˜ H (6) = − i I dω + O ( α ′ ) = e ⋆ d [ e − ω (6) ] + O ( α ′ ) . (6.10)The dilaton Φ depends only on the coordinates of B . The gauge connection is a su (3)instanton on B , i.e. F , = O ( α ′ ) , F ij ω ij (6) = O ( α ′ ) . (6.11)To find examples for such horizons two additional conditions should be satisfied. Oneis the restriction that ˆ˜ R (6) ij ω ij (6) = − k dℓ + O ( α ′ ) . (6.12)This arises from requirement that the U (3) structure on B lifts to a SU (3) structure onthe spacetime or equivalent the spatial horizon section S . The other is the anomalousBianchi identity which now reads k − dh ∧ dh + k − dℓ ∧ dℓ + d (cid:16) e ⋆ d [ e − ω ] (cid:17) = − α ′ (cid:18) − dh ∧ dh + tr( ˇ R (8) ∧ ˇ R (8) − F ∧ F ) (cid:19) + O ( α ′ ) , (6.13)where ˇ R (8) is the curvature of the connection with torsion on S which now its metric andtorsion are given by d ˜ s = k − ( h ⊗ h + ℓ ⊗ ℓ ) + d ˜ s + O ( α ′ ) , ˜ H = k − ( h ∧ dh + ℓ ∧ dℓ ) + ˜ H (6) + O ( α ′ ) . (6.14)Note that ˆ ∇ (8) has holonomy contained in SU (3) and so ˇ R (8) is a well defined form on B . SU (2) structure and 6 supersymmetries The spacetime is locally a SL (2 , R ) × SU (2) principal fibration over a 4-dimensional anti-self-dual Weyl Einstein manifold B with metric d ˚ s and quaternionic K¨ahler structure2-forms ω r ′ (4) . The spacetime metric and 3-form field strength can be expressed as ds = η ab λ a λ b + δ r ′ s ′ λ r ′ λ s ′ + e d ˚ s + O ( α ′ ) , H = CS ( λ ) + ˜ H (4) + O ( α ′ ) , (6.15)16here ˜ H (4) = − ˚ ⋆ de , the principal bundle connection λ a for a = + , − , λ r ′ = k − ℓ r ′ , (6.16)are the components along the su (2) subalgebra of the fibre. Furthermore the dilatondepends only on the coordinates of B , dh as well as the curvature ( F sd ) r ′ of λ r ′ are2-forms on B . In addition, we have that dh sd = O ( α ′ ) , ( F sd ) r ′ = k ω r ′ (4) + O ( α ′ ) , F sd = O ( α ′ ) (6.17)and dh ad , ( F ad ) r ′ and F ad are not restricted, where the self-dual and anti-self dual com-ponents are appropriately denoted. Geometrically, the set up is such that the SO (4) = SU (2) · SU (2) structure of B when lifted the 7-dimensional manifold which is the principalbundle with fibre SU (2) reduces to SU (2) as required from supersymmetry.The only remaining condition to find solutions is˚ ∇ e = −
12 ( F ad ) r ′ ij ( F ad ) ijr ′ − k − dh ij dh ij + 38 k e + α ′ (cid:18) − dh ij dh ij + tr( ˇ R (8) ij ˇ R (8) ij − F ij F ij ) (cid:19) + O ( α ′ ) . (6.18)Again ˇ R (8) is the curvature of the connection with torsion of the horizon section S whichhas metric and 3-form field strength d ˜ s = k − h ⊗ h + δ r ′ s ′ λ r ′ λ s ′ + e d ˚ s + O ( α ′ ) , ˜ H = k − h ∧ dh + CS ( λ r ′ ) + ˜ H (4) + O ( α ′ ) . (6.19)As ˆ R (8) has holonomy contained in SU (2), ˇ R (8) is a 2-form on B . For more details onthe geometry of heterotic backgrounds that preserve 6 supersymmetries and have SU (2)holonomy see [28, 27]. SU (2) structure and 8 supersymmetries This class of horizons have a similar geometry to those of the previous section that preserve6 supersymmetries. The differences are that( F sd ) r ′ = O ( α ′ ) , (6.20)so F r ′ is an anti-self dual instanton on B which now is a hyper-K¨ahler manifold withrespect to the metric d ˚ s . Furthermore the equation for the dilaton (6.18) now reads˚ ∇ e = − F r ′ ij F ijr ′ − k − dh ij dh ij + α ′ (cid:18) − dh ij dh ij + tr( ˇ R (8) ij ˇ R (8) ij − F ij F ij ) (cid:19) + O ( α ′ ) . (6.21)17herefore at zeroth order, a partial integration argument reveals that dh = O ( α ′ ) , F r ′ = O ( α ′ ) . (6.22)Thus B up to a local isometry is AdS × S × T or AdS × S × K and the dilaton isconstant. One does not expect additional α ′ corrections to the geometry in the case thatthe ˇ R (8) is identified with F . Though additional corrections are expected otherwise. Inthe absence of 5-branes, consistency requires that the Pontryagin number of the tangentbundle of B cancels that of the gauge bundle which is the vanishing condition for theglobal anomaly. h We shall revisit the global analysis of [27] by calculating the Laplacian of h , but includingalso α ′ correction terms. Then we shall examine the conditions imposed on the geometryby this expression. To avoid the trivial case when h = O ( α ′ ), we take h [0] = 0.Next we calculate the Laplacian of h to find that˜ ∇ i ˜ ∇ i h + ( h − d Φ) j ˜ ∇ j h = 2 ˜ ∇ ( i h j ) ˜ ∇ ( i h j ) + 12 ( dh − i h W ) ij ( dh − i h W ) ij − α ′ h i h j (cid:18) − dh iℓ dh jℓ + ˇ˜ R iℓ ℓ ℓ ˇ˜ R j ℓ ℓ ℓ − ˜ F iℓab ˜ F j ℓab (cid:19) + O ( α ′ ) . (7.1)In computing this expression, we made use of the Einstein equation (4.16) together withthe gauge field equations (4.13) and (4.14). We remark that the calculation proceedsin exactly the same way as in [27]; the α ′ terms in (7.1) originate from the α ′ terms in2 h i h j ˜ R ij . It should be noted that in order to fully control O ( α ′ ) terms in this expression,one would require to know the Einstein equations up to and including α ′ .To begin, we consider (7.1) to zeroth order in α ′ . We then re-obtain the conditionsfound in [27] via a maximum principle argument, i.e. h = const + O ( α ′ ) , ˜ ∇ ( i h j ) = O ( α ′ ) , dh − i h W = O ( α ′ ) (7.2)In particular, it follows from these conditions that i h dh = O ( α ′ ) , (7.3)and also L h Φ = O ( α ′ ) , L h W = O ( α ′ ) . (7.4)Furthermore, it also follows that if η + satisfies (4.26), then Γ − h i Γ i η + also satisfies (4.26)to zeroth order in α ′ . The integrability conditions therefore imply thatˆ˜ R ijmn h m Γ n φ + = O ( α ′ ) , (7.5)18nd hence ˇ˜ R mnij h m = O ( α ′ ) . (7.6)On substituting these conditions back into (7.1) one finds that the remaining contentof (7.1) is ˜ ∇ i (cid:18) e − ˜ ∇ i h (cid:19) + e − h j ˜ ∇ j h = α ′ e − h i h j ˜ F iℓab ˜ F j ℓab + O ( α ′ ) . (7.7)On integrating the O ( α ′ ) part of (7.7) over the zeroth order horizon section, one findsthat i h ˜ F = O ( α ′ ) , (7.8)and furthermore h = const + O ( α ′ ) . (7.9)It should be noted however that (7.1) does not in general imply (5.1). In particular, theconditions obtained from the analysis of the properties of h are not sufficient to implythat if η + , with η [0]+ = 0, satisfies (4.26) and (4.27), then η ′′− = Γ − h i Γ i η + also satisfies (4.26)and (4.27). Thus although (7.1) implies the horizons exhibit supersymmetry enhancementat O ( α ′ ), it does not imply the same at O ( α ′ ). Next we shall investigate whether it is possible to identify Killing spinors with the zeromodes of a suitable Dirac-like operator, by constructing a generalized Lichnerowicz typetheorem which incorporates the near-horizon fluxes. Such Lichnerowicz type theoremshave been established for near-horizon geometries in D=11 supergravity [9], type IIB[10] and type IIA supergravity (both massive and massless) [11, 12], as well as for
AdS geometries in ten and eleven dimensional supergravity [37, 38, 39, 33].To begin, let us first define the modified connection with torsion and the modifiedhorizon Dirac operator, respectively ∇ ( κ ) i ≡ ˆ˜ ∇ i + κ Γ i A , D ≡ Γ i ˆ˜ ∇ i + q A , (7.10)where κ, q ∈ R , and ˆ˜ ∇ i η ± = ˜ ∇ i η ± − W ijk Γ jk η ± , A = W ijk Γ ijk − i ˜ ∇ i Φ ∓ i h i . (7.11)It is clear that if η ± is a Killing spinor, i.e.ˆ˜ ∇ i η ± = O ( α ′ ) , and A η ± = O ( α ′ ) , (7.12)19hen D η ± = O ( α ′ ) also. Here we want to investigate the extent to which the converse istrue. We shall show that if D η ± = O ( α ′ ), thenˆ˜ ∇ i η ± = O ( α ′ ) , and A η ± = O ( α ′ ) , (7.13)and moreover dh ij Γ ij η ± = O ( α ′ ) , and ˜ F abij Γ ij η ± = O ( α ′ ) . (7.14)In order to obtain this result, we begin by considering the following functional I ≡ Z S e c Φ (cid:18) h∇ ( κ ) i η ± , ∇ ( κ ) i η ± i − hD η ± , D η ± i (cid:19) , (7.15)where c ∈ R , and we assume all the field equations. After some algebra, which is describedin appendix D, we find I = (cid:18) κ − κ (cid:19) Z S e − k A η ± k + Z S e − h η ± , Ψ D η ± i− α ′ Z S e − (cid:16) k /dh η ± k + k / ˜ F η ± k −h ˇ˜ R ℓ ℓ , ij Γ ℓ ℓ η ± , ˇ˜ R ijℓ ℓ , Γ ℓ ℓ η ± i (cid:17) + O ( α ′ ) , (7.16)which is true if and only if q = + O ( α ′ ) and c = − O ( α ′ ), and the Ψ is defined asfollows Ψ ≡ (cid:18) κ − (cid:19) A † − i ˜ ∇ i Φ −
16 Γ ℓ ℓ ℓ W ℓ ℓ ℓ + O ( α ′ ) . (7.17)The values of q and c are fixed by requiring that certain terms in the functional (7.15),which cannot be rewritten in terms of the Dirac operator D , or A † A , and which have nofixed sign, should vanish.The part of (7.16) which is of zeroth order in α ′ implies that if 0 < κ < , then D η ± = O ( α ′ ) = ⇒ (7 .
13) (7.18)and establishes the first part of the theorem. Next the integrability condition of ˆ˜ ∇ η ± = O ( α ′ ) is ˆ˜ R mn,ℓ ℓ Γ ℓ ℓ η ± = O ( α ′ ) , (7.19)which in turn implies that ˇ˜ R ℓ ℓ ,mn Γ ℓ ℓ η ± = O ( α ′ ) . (7.20)Hence we shall neglect the term in (7.16) which is quadratic in ˇ˜ R , as this term is O ( α ′ ).Then, assuming (7.18), the part of (7.16) which is first order in α ′ further implies (7.14).This completes the proof. 20 Nearly supersymmetric horizons
We have proven that for near horizon geometries the necessary and sufficient conditionsimposed by supersymmetry on the spinors can be reduced to (4.26) and (4.27). In thissection, we shall consider the case for which the supersymmetry is explicitly partiallybroken, in the sense that the gravitino KSE (4.26) admits solutions but not dilatino one(4.27). We also assume that the fields satisfy∆ = O ( α ′ ) , H = d ( e − ∧ e + ) + W + O ( α ′ ) . (8.1)These conditions were previously obtained via the supersymmetry analysis; here we shallassume them. In particular, all of the conditions obtained from the global analysis of theLaplacian of h in Section 7 remain true. As a consequence of this,ˆ˜ ∇ i h j = O ( α ′ ) . (8.2)However we do not assume that ˆ˜ ∇ h = O ( α ′ ).One consequence of these assumptions is that none of the spacetime Killing spinorequations are satisfied even at O ( α ′ ). In particular, the spacetime gravitino KSE requiresin addition the condition that dh ij Γ ij η + = O ( α ′ ) which is not one of our requirements.In what follows, we shall investigate the consequences of the above assumptions on thegeometry of the spatial horizon sections S . We shall also comment on the special casewhere ˆ˜ ∇ h = O ( α ′ ). A key property of backgrounds that satisfy the gravitino KSE but not the dilatino one isthe existence of additional parallel spinors, see also appendix C. In the present context toshow this focus on the spinor η + ; a similar analysis can be undertaken for the η − spinors.To proceed, it will be useful to define A = W ijk Γ ijk − i ˜ ∇ i Φ − h i Γ i , (8.3)so that the algebraic condition (4.27) on η + is equivalent to A η + = O ( α ′ ). We then notethe useful identity˜ ∇ i W ℓ ℓ ℓ Γ ℓ ℓ ℓ η + = ˜ ∇ i ( A η + ) − W iℓ ℓ Γ ℓ ℓ ( A η + )+ 3 W ℓ ℓ q W iℓ q Γ ℓ ℓ ℓ η + − (cid:0) ∇ m Φ + 3 h m (cid:1) W miℓ Γ ℓ η + + (cid:0) ℓ ˜ ∇ i ˜ ∇ ℓ Φ + 6 ˜ ∇ i h ℓ Γ ℓ (cid:1) η + . (8.4)The integrability conditions of (4.26) imply that16 (cid:18) ˜ ∇ i ( A η + ) − W iℓ ℓ Γ ℓ ℓ ( A η + ) (cid:19) − α ′ F iℓ ) ab Γ ℓ ( ˜ F q q ) ab Γ q q η + − α ′ dh iℓ Γ ℓ dh q q Γ q q η + = O ( α ′ ) , (8.5)21nd hence16 h η + , Γ i ˜ ∇ i ( A η + ) − W ℓ ℓ ℓ Γ ℓ ℓ ℓ ( A η + ) i + α ′ h (( ˜ F ℓ ℓ ) ab Γ ℓ ℓ η + , ( ˜ F q q ) ab Γ q q η + i + α ′ h dh ℓ ℓ Γ ℓ ℓ φ + , dh q q Γ q q η + i = O ( α ′ ) . (8.6)Integrating this expression over S yields the conditions˜ F ij Γ ij η + = O ( α ′ ) , dh ij Γ ij η + = O ( α ′ ) , (8.7)and substituting these conditions back into (8.5) then implies that˜ ∇ i ( A η + ) − W iℓ ℓ Γ ℓ ℓ ( A η + ) = O ( α ′ ) . (8.8)Therefore the spinor τ + = A η + is also ˆ˜ ∇ -parallel. As τ + has opposite chirality from η + cannot be identified as an additional Killing spinor within the heterotic theory. Never-theless it is instrumental in the description of the geometry of S . G holonomy Suppose that we consider solutions for which there exists a single solution η + to thegravitino KSE ˆ˜ ∇ η + = O ( α ′ ) , (8.9)for which (cid:0) A η + (cid:1) [0] = 0. This implies that the horizon section S [0] at zeroth order in α ′ admits a G structure.We begin by defining τ + = A η + , with τ [0]+ = 0. It will be particularly useful to define V i = h η , Γ i τ + i . (8.10)In what follows we shall show that V is a symmetry of all the fields of the spatial horizonsection.As τ [0]+ = 0, this implies that V [0] = 0. In addition, as η + and τ + satisfyˆ˜ ∇ η + = O ( α ′ ) , ˆ˜ ∇ τ + = O ( α ′ ) , (8.11)it follows that ˆ˜ ∇ V = O ( α ′ ) , (8.12)so that V = const. + O ( α ′ ), and V is an isometry of S to both zero and first order in α ′ . 22ext, we consider the relationship of V to h . In particular, the spinors h i Γ i A η + and V i Γ i A η + are both parallel with respect to ˆ˜ ∇ at zeroth order in α ′ . As we have assumedthat (8.9) admits only one solution, there must be a nonzero constant c such that V = ch + O ( α ′ ) . (8.13)In addition, we have L V W = i V dW + O ( α ′ ) , (8.14)because dV = i V W + O ( α ′ ). Also, as V = ch + O ( α ′ ) it follows that L V W = ci h dW + O ( α ′ ) . (8.15)As a consequence of (8.2), one has that i h dh = O ( α ′ ), and from the global analysis ofthe Laplacian of h , we find i h ˜ F = O ( α ′ ) as well as ˇ˜ R mnij h m = O ( α ′ ). These conditionsimply that L V W = O ( α ′ ) , (8.16)and so W is invariant.Next we consider L V Φ. As V = ch + O ( α ′ ) it follows that L V dh = c L h dh + O ( α ′ ) = O ( α ′ ) . (8.17)Also we have L V ˜ R ij,pq = O ( α ′ ) , (8.18)and (cid:0) L V ˜ F (cid:1) ijab ˜ F ijba = O ( α ′ ) , (8.19)which follows from L V ˜ F = c [ ˜ F , i h B ] + O ( α ′ ) . (8.20)Hence we have L V (cid:18) α ′ (cid:0) − dh ij dh ij + ˇ˜ R ij,pq ˇ˜ R ij,pq − ( ˜ F ij ) ab ( ˜ F ij ) ab (cid:1)(cid:19) = O ( α ′ ) . (8.21)So, on taking the Lie derivative of the trace of (4.16) with respect to V we find L V (cid:18) ˜ ∇ i h i + 2 ˜ ∇ i ˜ ∇ i Φ (cid:19) = O ( α ′ ) , (8.22)and hence, as a consequence of the field equation (4.13), we find L V (cid:18) h i ˜ ∇ i Φ + ˜ ∇ i ˜ ∇ i Φ (cid:19) = O ( α ′ ) . (8.23)23lso, on taking the Lie derivative of the dilaton field equation (4.17), we get L V (cid:18) − h i ˜ ∇ i Φ − ∇ i Φ ˜ ∇ i Φ + ˜ ∇ i ˜ ∇ i Φ (cid:19) = O ( α ′ ) (8.24)On taking the sum of (8.23) and (8.24), we find L V (cid:18) ˜ ∇ i ˜ ∇ i Φ − ˜ ∇ i Φ ˜ ∇ i Φ (cid:19) = O ( α ′ ) (8.25)and hence if f = L V Φ we have˜ ∇ i ˜ ∇ i f − ∇ i Φ ˜ ∇ i f = O ( α ′ ) . (8.26)We know L h Φ = O ( α ′ ) as a consequence of the analysis of the Laplacian of h , so f = α ′ f [1] + O ( α ′ ). Then, on integrating, (8.26) implies that Z S [0] e − [0] ˜ ∇ i f [1] ˜ ∇ i f [1] = 0 , (8.27)so f [1] = β for constant β , and so L V Φ = βα ′ + O ( α ′ ) . (8.28)As we require that Φ must attain a global maximum on S , at this point L V Φ = 0 to allorders in α ′ , for any V . This fixes β = 0, so L V Φ = O ( α ′ ) , (8.29)which proves the invariance of Φ.Next, we consider L V h . On taking the Lie derivative of the field equation of the 2-formgauge potential (4.15) we find d ( L V h ) ij − ( L V h ) k W ijk = O ( α ′ ) , (8.30)and on taking the Lie derivative of the Einstein equation (4.16) we get˜ ∇ ( i ( L V h ) j ) = O ( α ′ ) , (8.31)where we have used L h (cid:18) ˜ F iℓab ˜ F j ℓab (cid:19) = O ( α ′ ) . (8.32)It follows that ˆ˜ ∇ ( L V h ) j = O ( α ′ ) . (8.33)As V = ch + O ( α ′ ), it is convenient to write L V h = α ′ Λ + O ( α ′ ) , (8.34)24here ˆ˜ ∇ Λ = O ( α ′ ) . (8.35)As Λ j Γ j A η + and h j Γ j A η + are both parallel with respect to ˆ˜ ∇ at zeroth order in α ′ , itfollows as a consequence of (ii) that we must haveΛ = bh + O ( α ′ ) , (8.36)for constant b . It is also useful to compute h i ( L V h i ) = h i (cid:18) V j ˜ ∇ j h i + h j ˜ ∇ i V j (cid:19) = 12 L V h + h i h j ˜ ∇ i V j = O ( α ′ ) , (8.37)which follows because h = const + O ( α ′ ), and ˆ˜ ∇ V = O ( α ′ ). This implies that b = 0,and hence L V h = O ( α ′ ) . (8.38)So V is a symmetry of the full solution to both zeroth and first order in α ′ . We have shown that V is a symmetry of the backgrounds up O ( α ′ ). To investigatefurther the geometry of the horizon section S , let us first consider the consequences of theexistence of the η + Killing spinor. As the isotropy group of η + in Spin (8) is
Spin (7), thefundamental self-dual 4-form φ of Spin (7) on S is ˆ˜ ∇ -parallel. It is known that in sucha case, the torsion 3-form W can be uniquely determined in terms of φ and the metricwithout any additional conditions on the Spin (7) structure of S [40]. Next the conditionˆ˜ ∇ τ + = O ( α ′ ) with τ + = A η + is equivalent to requiring thatˆ˜ ∇ i (cid:0) (2 d Φ + h ) j − ( θ φ ) j (cid:1) = O ( α ′ ) , (8.39)where θ φ is the Lee form of φ , see [25]. As a result 2 d Φ + h − θ φ is a parallel 1-form. If itis not linearly dependent on V , it will give rise to an additional solution for the gravitinoKSE S . As we have assumed that there is strictly one parallel spinor of the same chiralityas η + , we have to require that2 d Φ + h − θ φ = λV + O ( α ′ ) , (8.40)for some non-zero constant λ ; for λ = 0 the dilatino KSE is satisfied as well.Let us next turn to investigate the G structure on S . As V is an isometry on S and i V W = dV , setting V = ℓ + O ( α ′ ) for ℓ constant, we can decompose the metric and3-form as d ˜ s = 1 ℓ V ⊗ V + ds + O ( α ′ ) , W = ℓ − V ∧ dV + W (7) + O ( α ′ ) , (8.41)25here ds is the metric on the space orthogonal to V and i V W (7) = 0. The data( ds , W (7) ) are thought (locally) as the metric torsion on the space of orbits M of V .For this observe that L V W (7) = 0 and as i V W (7) = 0, W (7) descends as a 3-form on thespace of orbits.The spatial horizon section S admits a G structure with fundamental form ϕ = ℓ − i V φ as ˆ˜ ∇ ϕ = O ( α ′ ). The question is whether this G structure descends on the space of orbitsof V . First observe that i V ϕ = 0. So it remains to investigate whether L V ϕ = O ( α ′ ). Forthis notice that under G representations dV decomposes as dV = dV + dV + O ( α ′ )because i V dV = O ( α ′ ). Then use (C.3) together with ˆ˜ ∇ ϕ = ˆ˜ ∇ V = O ( α ′ ) and i V dW = O ( α ′ ) to show that ˆ˜ ∇ dV = O ( α ′ ) . (8.42)As dV is a vector in S orthogonal to V , if it is not vanishing will generate an additionalˆ˜ ∇ -parallel spinor on S of the same chirality as η + . As we have restricted the numberof such spinors to one, we have to set dV = O ( α ′ ). It has been shown in [25] that aˆ˜ ∇ -parallel k-form α is invariant under the action of a ˆ˜ ∇ -parallel vector V , iff the rotation i V W leaves the form invariant. As i V W = dV + O ( α ′ ) and dV takes values in g , weconclude that L V ϕ = O ( α ′ ) . (8.43)and so M admits a G structure compatible with connection with skew-symmetric torsiongiven by the data ( ds , W (7) ). In such a case W (7) can be determined uniquely in termsof ϕ and ds provided a certain geometric constraint is satisfied [41].It remains to explore (8.40) from the perspective of M . Let us decompose h = V + h ⊥ ,where g ( V, h ⊥ ) = 0. Then (8.40) can be written as ℓ − g ( V, h ) −
16 ( W (7) ) ijk ϕ ijk = λℓ + O ( α ′ ) , d Φ + h ⊥ − θ ϕ = O ( α ′ ) , (8.44)where θ ϕ is the Lee form of ϕ on M . The former determines the singlet part of W (7) interms of V and h while the latter imposes the dilatino KSE on M . SU (3) holonomy Suppose there are exactly two linearly independent spinors η (1)+ , η (2)+ such thatˆ˜ ∇ η ( a )+ = O ( α ′ ) , a = 1 , , (9.1)26or which (cid:0) A η ( a )+ (cid:1) [0] = 0, ( a = 1 , S [0] admits a SU (3)structure at zeroth order in α ′ .We set τ ( a )+ = A η ( a )+ which are non-vanishing spinors that satisfyˆ˜ ∇ τ ( a )+ = O ( α ′ ) , a = 1 , . (9.2)Using these we define the 1-form and 2-form spinor bilinears V and ω by V i = h η (1)+ , Γ i τ (1)+ i , ω ij = h η (1)+ , Γ ij η (2)+ i , (9.3)and also let ˜ V = i V ω . (9.4)Observe that ˆ˜ ∇ V = O ( α ′ ) , ˆ˜ ∇ ω = O ( α ′ ) , ˆ˜ ∇ ˜ V = O ( α ′ ) . (9.5)We also define ˜ h by ˜ h = i h ω , (9.6)which satisfies ˆ˜ ∇ ˜ h = O ( α ′ ) . (9.7)The main task below is to show that both V and ˜ V leave invariant all the fields on S ,and that they generate a R ⊕ R lie algebra.As V and ˜ V are ˆ˜ ∇ -parallel, they are killing. Next consider the invariance of W . Thespinors V j Γ j A η ( a )+ , h j Γ j A η ( a )+ and ˜ h j Γ j A η ( a )+ are all parallel with respect to ˆ˜ ∇ to zerothorder in α ′ . In order for (9.1) to have exactly two solutions, we must have V = ch + ˜ c ˜ h + O ( α ′ ) , (9.8)for some constants c , ˜ c . Thus L V W = ci h dW + ˜ ci ˜ h dW + O ( α ′ ) . (9.9)To continue, since the two spinors η (1)+ and η (2)+ must satisfy (8.7), it follows that, at zerothorder in α ′ , ˜ F and dh are (1 ,
1) traceless with respect to the almost complex structureobtained from ω . This, together with the conditions i h dh = O ( α ′ ) and i h ˜ F = O ( α ′ ),which follow from the global analysis of the Laplacian of h , implies that i ˜ h dh = O ( α ′ ) , i ˜ h ˜ F = O ( α ′ ) , (9.10)and hence i V dh = O ( α ′ ) , i V ˜ F = O ( α ′ ) . (9.11)27t is also useful to consider the spinors η ( a )+ and ˜ h ℓ Γ ℓ η ( a )+ . The integrability conditions ofˆ˜ ∇ η ( a )+ = O ( α ′ ) , ˆ˜ ∇ (cid:16) ˜ h ℓ Γ ℓ η ( a )+ (cid:17) = O ( α ′ ) , (9.12)are ˆ˜ R ij,pq Γ pq η ( a )+ = O ( α ′ ) , ˆ˜ R ij,pq Γ pq (cid:16) ˜ h ℓ Γ ℓ η ( a )+ (cid:17) = O ( α ′ ) , (9.13)which imply ˜ h p ˇ˜ R pq,ij = O ( α ′ ) . (9.14)It follows that i h dW = O ( α ′ ) and i ˜ h dW = O ( α ′ ), as a consequence of the Bianchiidentity, and therefore i V dW = O ( α ′ ). Thus we have shown that L V W = O ( α ′ ) . (9.15)This proves the invariance of W .Next we consider L V Φ. It follows from (9.7) that i ˜ h d ˜ h = O ( α ′ ) , (9.16)and also L ˜ h W = O ( α ′ ) . (9.17)Since ˜ h is an isometry of S to zeroth order in α ′ , we also have L ˜ h ˜ R ij,pq = O ( α ′ ) . (9.18)On taking the Lie derivative of the trace of (4.16) with respect to ˜ h , we find L ˜ h (cid:16) ˜ ∇ i ˜ ∇ i Φ (cid:17) = O ( α ′ ) , (9.19)which is equivalent, if g = L ˜ h Φ, to˜ ∇ i ˜ ∇ i g = O ( α ′ ) . (9.20)On integrating the zeroth order of (9.20), we find Z S [0] ˜ ∇ i g [0] ˜ ∇ i g [0] = 0 , (9.21)so g [0] = γ , for constant γ . Thus L ˜ h Φ = γ + O ( α ′ ) . (9.22)Since Φ must attain a global maximum on S , at this point L ˜ h Φ = 0 to all orders in α ′ .This fixes the constant γ = 0, and so L ˜ h Φ = O ( α ′ ) , (9.23)28hich implies L V Φ = O ( α ′ ) . (9.24)As V = ch + ˜ c ˜ h + O ( α ′ ), it follows that L V dh = c L h dh + ˜ c L ˜ h dh + O ( α ′ ) = O ( α ′ ) . (9.25)Since V is an isometry of S to first order in α ′ , we have L V ˜ R ij,pq = O ( α ′ ) . (9.26)Also we have ( L V ˜ F ) ij ab ˜ F ij ba = O ( α ′ ) , (9.27)which follows from L V ˜ F = c [ ˜ F , i h B ] + ˜ c [ ˜ F , i ˜ h B ] + O ( α ′ ) . (9.28)Using the conditions (9.24), (9.25), (9.26) and (9.27), we follow the analysis for the G case of the previous section undertaken from the equation (8.22) to (8.29), and concludethat L V Φ = O ( α ′ ) , (9.29)which proves the invariance of the dilaton Φ.Next we consider L V h . Equations (8.30) and (8.31), which have been established inthe previous section, hold here as well after using in the addition that L ˜ h (cid:18) ˜ F iℓab ˜ F j ℓab (cid:19) = O ( α ′ ) . (9.30)Then it follows that ˆ˜ ∇ i ( L V h ) j = O ( α ′ ) . (9.31)Furthermore we notice that L ˜ h h = O ( α ′ ) . (9.32)As V = ch + ˜ c ˜ h + O ( α ′ ), it is convenient to write L V h = α ′ Ψ + O ( α ′ ) , (9.33)where ˆ˜ ∇ Ψ = O ( α ′ ) . (9.34)29hen it follows that the spinors Ψ j Γ j A η + , h j Γ j A η + and ˜ h j Γ j A η + are all parallel withrespect to ˆ˜ ∇ at zeroth order in α ′ . In order for (9.1) to admit exactly two solutions, wemust have Ψ = bh + ˜ b ˜ h + O ( α ′ ) , (9.35)for constants b and ˜ b . Then using i h L V h = O ( α ′ ), which has been computed in (8.37),and h = const. + O ( α ′ ), it follows that b = O ( α ′ ) and therefore L V h = α ′ ˜ b ˜ h + O ( α ′ ) . (9.36)Next we consider the symmetries generated by ˜ V . Since V = ch + ˜ c ˜ h + O ( α ′ ), thenwe have ˜ V = c ˜ h − ˜ ch + O ( α ′ ) . (9.37)Since V and ω are both parallel with respect to ˆ˜ ∇ to first order in α ′ , we also haveˆ˜ ∇ ˜ V = O ( α ′ ) . (9.38)Then the analysis undertaken for V holds as well for ˜ V , because the only properties of V used through the analysis are that V , at zeroth order in α ′ , is a linear combination of h and ˜ h with constant coefficients, and V is parallel with respect to ˆ˜ ∇ to first order in α ′ .Thus we argue in a similar way that L ˜ V W = O ( α ′ ) , L ˜ V Φ = O ( α ′ ) , L ˜ V h = α ′ ˜ q ˜ h + O ( α ′ ) , (9.39)for a constant ˜ q .Finally, the V and ˜ V commute up to O ( α ′ ). To see this observe that since i V ˜ V = 0and i V W = dV + O ( α ′ ), we have that L ˜ V V = i ˜ V i V W + O ( α ′ ) . (9.40)Using (C.3) adapted to S as well as i V dW = i ˜ V dW = O ( α ′ ), we conclude thatˆ˜ ∇ i ˜ V i V W = O ( α ′ ) . (9.41)Therefore the vector i ˜ V i V W is ˆ˜ ∇ -parallel and moreover is orthogonal to both V and ˜ V .So if it is non-zero, it will generate additional ˆ˜ ∇ -parallel η + spinors on S . As we haverestricted those to be strictly two, we conclude that i ˜ V i V W vanishes and so[ V, ˜ V ] = O ( α ′ ) . (9.42)In particular as i V ˜ V = 0, we have that i V d ˜ V = i ˜ V dV = O ( α ′ ) . (9.43)This concludes the examination of symmetries of S .30 .1.2 Geometry It is clear from the examination of the symmetries of the fields on S and in particular(9.5) and (9.43) that we can set d ˜ s = ℓ − V ⊗ V + ℓ − ˜ V ⊗ ˜ V + ds + O ( α ′ ) W = ℓ − V ∧ dV + ℓ − ˜ V ∧ d ˜ V + W (6) + O ( α ′ ) , (9.44)where V = ˜ V = ℓ + O ( α ′ ) and ℓ is constant, ds is the metric in the orthogonalcomplement of V and ˜ V and i V W (6) = i ˜ V W (6) = O ( α ′ ).From construction S admits an SU (3) structure. We shall now investigate whetherthis (locally) descends on the space of orbits M of V and ˜ V . First the data ( ds , W (6) )define a Riemannian geometry on M with skew-symmetric torsion. In particular for thetorsion this follows from i V W (6) = i ˜ V W (6) = O ( α ′ ) and L V W (6) = L ˜ V W (6) = O ( α ′ ).Next consider the reduction of the (almost) Hermitian form ω . Choosing without lossof generality V and ˜ V orthogonal, one can write ω = ℓ − V ∧ ˜ V + ω (6) + O ( α ′ ) , (9.45)where i V ω (6) = i ˜ V ω (6) = O ( α ′ ). For ω (6) to descend to a Hermitian structure on M , itmust be invariant under the action of both V and ˜ V . Observe that ˆ˜ ∇ ω (6) = O ( α ′ ) andalso ˆ˜ ∇ V = ˆ˜ ∇ ˜ V = O ( α ′ ). Thus ω (6) is invariant iff the rotations i V W = dV + O ( α ′ ) and i ˜ V W = d ˜ V + O ( α ′ ) leave ω (6) invariant [25]. In turn this implies that the (2,0) and (0,2)parts of the rotations which we denote with [ dV ] , and [ d ˜ V ] , , respectively, must vanish.Using (C.3), ˆ˜ ∇ ω (6) = O ( α ′ ) and i V dW = i ˜ V dW = O ( α ′ ), we find thatˆ˜ ∇ [ i V W ] , = ˆ˜ ∇ [ i ˜ V W ] , = O ( α ′ ) . (9.46)As S has an SU (3) structure compatible with ˆ˜ ∇ , contracting with the (3,0)-form both[ i ˜ V W ] , and [ i ˜ V W ] , give rise to vector fields in S orthogonal to both V and ˜ V whichare ˆ˜ ∇ -parallel. Thus the requirement of strictly two η + ˆ˜ ∇ -parallel spinors leads to setting[ i ˜ V W ] , = [ i ˜ V W ] , = O ( α ′ ) which in turn implies that L V ω (6) = L ˜ V ω (6) = O ( α ′ ) . (9.47)Thus M admits an almost Hermitian structure compatible with a connection ˆ˜ ∇ (6) withskew-symmetric torsion W (6) . It is well known that in such case W (6) is determined interms of the almost complex structure on M and the metric, see eg [42].To find whether M inherits a SU (3) structure as well, let investigate whether the(3,0) fundamental SU (3) form χ of S descends on M . It can always be arranged suchthat i V χ = i ˜ V χ = 0. So it remains to see whether χ is invariant under the action of V and ˜ V . For this a similar argument to that explained above for ω (6) leads to the assertionthat χ is invariant iff the ω -traces i ˜ V W · ω and i ˜ V W · ω of i ˜ V W and i ˜ V W , respectively,vanish. Furthermore, an application of (C.3) implies that both i ˜ V W · ω and i ˜ V W · ω are31onstant but not necessarily zero. Thus M has generically a U (3) structure instead ofan SU (3) one.It remains to investigate the rest of the content of the conditions ˆ˜ ∇ τ ( a )+ = O ( α ′ ).First consider the (3,0) part of W (6) denoted by W , . An application of (C.3) using that dW is a (2,2) form yields that ˆ˜ ∇ W , = O ( α ′ ) . (9.48)Thus W , is another globally defined ˆ˜ ∇ -parallel (3,0)-form on S and so it can either beset to zero or be identified with χ . In the former case, the complex structure on M isintegrable and so M is a KT manifold [43].Writing h = λ V + λ ˜ V + h ⊥ , where h ⊥ is orthogonal to both V and ˜ V and λ and λ are constants, we find using (C.3) thatˆ˜ ∇ (cid:0) d Φ + h ⊥ − θ ω (6) (cid:1) = O ( α ′ ) . (9.49)Now if 2 d Φ + h ⊥ − θ ω (6) in non-vanishing and since it is orthogonal to V and ˜ V will giverise to more than two η + ˆ˜ ∇ -parallel spinors on S . Since we have assumed that there arejust two, we set 2 d Φ + h ⊥ − θ ω (6) = O ( α ′ ) . (9.50)This concludes the investigation of geometry. SU (2) holonomy It is known that if one requires the existence of an additional ˆ˜ ∇ -parallel spinor η + tothose of the SU (3) backgrounds on S , then the isotropy algebra of the all the five spinorsreduces to su (2). As a result, S admits 8 ˆ˜ ∇ -parallel spinors and the holonomy groupreduces to a subgroup SU (2). To describe the geometry of backgrounds with exactly 8such spinors, we consider four linearly independent spinors η ( a )+ , and impose the conditionˆ˜ ∇ η ( a )+ = O ( α ′ ) , a = 0 , , , , (9.51)for which (cid:0) A η ( a )+ (cid:1) [0] = 0, ( a = 0 , , , S [0] admits a SU (2) structure at zeroth order in α ′ . We continue by setting τ ( a )+ = A η ( a )+ . These arenon-vanishing and satisfy ˆ˜ ∇ τ ( a )+ = O ( α ′ ) , a = 0 , , , . (9.52)Furthermore, we also define 1-form and 2-form spinor bilinears V ( a ) and ω r , respectively,by V i ≡ V (0) i = h η (0)+ , Γ i τ (0)+ i , ( ω r ) ij = h η (0)+ , Γ ij η ( r )+ i , r = 1 , , , (9.53)32nd also let ˜ V r = i V ω r . (9.54)In fact ω r together with the metric and W define an almost HKT structure [43] on S asˆ˜ ∇ V = O ( α ′ ) , ˆ˜ ∇ ω r = O ( α ′ ) , ˆ˜ ∇ ˜ V r = O ( α ′ ) , (9.55)and the almost complex structures associated to ω r satisfy the algebra of unit quaternions.These follow from (8.8) and the su (2) isotropy of the parallel spinors. It is clear from (9.55) that V ( a ) , V ( r ) = V r , generate isometries on S and that i a W = dV ( a ) + O ( α ′ ) , (9.56)where i a denotes inner-derivation with respect to V ( a ) . Without loss of generality wechoose g ( V ( a ) , V ( b ) ) = ℓ δ ab + O ( α ′ ) for ℓ constant. An investigation similar to the oneexplained in section 9.1.1 reveals that L a Φ = O ( α ′ ) , L a W = O ( α ′ ) , L a h = O ( α ′ ) , i a dh = O ( α ′ ) ,i a F = O ( α ′ ) . (9.57)Next let us consider the commutator [ V ( a ) , V ( b ) ] = i a i b W . An application of (C.3) togetherwith the conditions above reveal thatˆ˜ ∇ [ V ( a ) , V ( b ) ] = O ( α ′ ) . (9.58)Thus the commutator is either linear dependent on V ( a ) or it will lead to further reductionof the holonomy of ˆ˜ ∇ to { } . In the latter case, the horizon section S will admit morethan four η + ˆ˜ ∇ -parallel spinors violating our assumptions. Thus, we conclude that[ V ( a ) , V ( b ) ] = f abc V ( c ) + O ( α ′ ) , (9.59)for some constants f with ℓ f abc = i a i b i c W + O ( α ′ ). As f is skew-symmetric, the Liealgebra spanned by V ( a ) is a metric (compact) Lie algebra. As it has dimension 4, it iseither isomorphic to ⊕ u (1) or to u (1) ⊕ su (2).Therefore the horizon section S can be viewed locally as a fibration with fibre either × U (1) or U (1) × SU (2) over the space of orbits M of V ( a ) . We shall determine thegeometry of S by specifying the geometry of M . To simplify the analysis, we choose up to an so (4) rotation V to be along a u (1) directionin either ⊕ u (1) or u (1) ⊕ su (2). This in particular implies that i i r W = O ( α ′ ). Thenthe metric and torsion of S can be written as d ˜ s = ℓ − δ ab V ( a ) ⊗ V ( b ) + d ˜ s + O ( α ′ ) , W = ℓ − V ∧ dV + CS ( V r ) + W (4) + O ( α ′ )(9.60)33here V ( a ) is viewed as a principal bundle connection and CS ( V r ) is the Chern-Simonsform which for the ⊕ u (1) case is CS ( V r ) = ℓ − X r V r ∧ dV r . (9.61)The data ( ds , W (4) ) define a geometry on M with skew-symmetric torsion.First, let us investigate the reduction of the almost HKT structure of S on M . Forthis observe that ω r = ℓ − V ∧ V r + ℓ − ǫ rst V s ∧ V t + ω (4) r + O ( α ′ ) , (9.62)where i a ω (4) r = O ( α ′ ). Next consider L a ω (4) r . As both V ( a ) and ω (4) r are ˆ˜ ∇ -parallel, L a ω (4) r is specified by the properties of the rotation i a W . In particular if i a W is invariant under ω (4) r , the Lie derivative vanishes.Next let us investigate the two cases ⊕ u (1) and u (1) ⊕ su (2) separately. In the abeliancase, as i a i b W = O ( α ′ ), i a W is a 2-form on M . Furthermore L a ω (4) r vanishes iff theself-dual part, i a W sd , of i a W is zero. However in general this may not be the case. Anapplication of (C.3) implies that ˆ˜ ∇ i a W sd = O ( α ′ ) , (9.63)and so there exist some constants u such that i a W sd = u ar ω (4) r + O ( α ′ ) , (9.64)otherwise the holonomy of ˆ˜ ∇ will be reduced further and it will admit more than four η + parallel spinors. Then L a ω (4) r = 2 u as ǫ srt ω (4) t + O ( α ′ ) . (9.65)The identity [ L a , L b ] = L [ V ( a ) ,V ( b ) ] gives( u ra u sb − u rb u sa ) = O ( α ′ ) . (9.66)The covariant constancy condition on M now readsˆ˜ ∇ (4) ω (4) r = 2 ℓ − V ( a ) u sa ǫ srt ω (4) t + O ( α ′ ) , (9.67)where now V ( a ) should be thought as the pull back of the principal bundle connection V ( a ) with a local section. It is clear that the relevant connection that determines the geometryof M is Z s = V ( a ) u sa .If u ra = 0, M is a HKT manifold. It is easy to see this as ω r are covariantly constantwith respect to a connection with skew-symmetric torsion and all three almost complexstructures are integrable. The latter follows because of dimensional reasons. Otherwiseone of the 3-vectors u a must be non-zero. Without loss of generality take u = 0. In sucha case the above equation can be solved as ( u ra ) = ( u r , u r v s ), where v s = | u | − P r u rs u r .34sing these data, the covariant constancy condition of ω (4) r on M can be written asˆ˜ ∇ (4) ω (4) r = 2 ℓ − ( V + V p v p ) u s ǫ srt ω (4) t + O ( α ′ ) . (9.68)It is clear from this that M is a KT manifold with respect to the Hermitian form | u | − u r ω r . In fact M is an (almost) QKT manifold [44] for which the holonomy ofthe Sp (1) connection has been reduced to U (1).Next let us turn to examine the non-abelian u (1) ⊕ su (2) case. It is easy to see that( u ra u sb − u rb u sa ) = 12 f abc u tc ǫ trs + O ( α ′ ) . (9.69)If the 3-vector u = 0, then all the rest of the components of u vanish. In such acase, M is an KT manifold. This class of solutions includes the WZW type of solution AdS × S × M where M = S × S with the bi-invariant metric and constant dilaton.Such a horizon is not supersymmetric but it is nearly supersymmetric.It remains to consider the case u = 0. One can then show that det u = 0 and so( u rs ) is invertible. Thus Z s = V ( a ) u sa takes values in the sp (1) Lie algebra. M is a QKTmanifold, see also [28].To conclude we remark that in all HKT and KT cases, there is an analogue of thecondition (9.49) for every Hermitian form ω r that determines these structures. If theassociated 2 d Φ + h ⊥ − θ r forms do not vanish, then the holonomy of the connection withtorsion reduces to { } and the number of parallel spinors enhance to 16. The solutionsare the group manifolds. The solution AdS × S × S × S mentioned above belongs tothe class where the holonomy of the connection with torsion is { } .There is an analogue of this in the QKT case but in such a case the condition from theperspective of M twists with sp (1). If 2 d Φ + h ⊥ − θ r do not vanish, again the holonomyof the connection with torsion on S reduces to { } . However now some of the data likethe Hermitian forms are not (bi-)invariant under the action of the group. It would be ofinterest to explore his further to see whether there are actual solutions.We conclude the examination of the geometry of nearly supersymmetric backgroundsin the G , SU (3) and SU (2) cases by pointing out that they exhibit an sl (2 , R ) up toorder O ( α ′ ) but not up to order O ( α ′ ). For the latter, h must be a symmetry of thetheory up to the same order and so it can be identified with V . The description of thegeometry of this special class of nearly supersymmetric backgrounds is very similar to theone we have given above. The only difference is that now we can identify h with V .
10 Conclusions
We have investigated the supersymmetric near-horizon geometry of heterotic black holesup to and including two loops in sigma model perturbation theory. Using a combinationof local and global techniques, together with the bosonic field equations and Bianchiidentities, we have proven that the conditions obtained from the KSEs are equivalent to apair of gravitino equations (4.26) and a pair of algebraic conditions, related to the dilatino In the definition of QKT structure in [44] an additional integrability condition was considered. α ′ . In particular, wehave shown that the KSE related to the gaugino is implied by the other KSEs and fieldequations.In all cases, we have also shown that there are no regular AdS solutions with compactwithout boundary internal space by demonstrating that ∆ = O ( α ′ ). This is not incontradiction with the fact that one can locally write AdS as a warped product over AdS [45], see also appendix E. This is because our assumptions on the internal space of AdS are violated in such a case.Furthermore, we have demonstrated that horizons that admit a non-vanishing η − Killing spinor up to order O ( α ′ ), which does not vanish at zeroth order in α ′ , exhibitsupersymmetry enhancement via the same mechanism as described in [27], and so preserve2, 4, 6 and 8 supersymmetries. We have described the geometry of such horizons in allcases and this is similar to that presented in [27] for the horizons with dH = 0.We have also considered in some detail the global properties of our solutions. Theanalysis of the global properties of h proceeds in much the same way as in the heterotictheory with dH = 0. However in the presence of anomaly, the consequences of the globalrestrictions on the geometry of the horizons are somewhat weaker. For example, it is onlypossible to prove that h is an isometry of the horizon section to zeroth order in α ′ . So onecannot establish a direct algebraic relation between η + and η − spinors to order O ( α ′ ),and therefore it is not possible to directly show that there is supersymmetry enhancementvia this mechanism, as was done in [27] for the theory with dH = 0.We have also constructed generalized Lichnerowicz type theorems, which relate spinorswhich are parallel with respect to a certain type of near-horizon supercovariant derivative,to zero modes of near-horizon Dirac operators. We have shown that if η is a zero mode ofthe near-horizon Dirac operator to both zero and first order in α ′ , then the Lichnerowicztheorems imply that η only satisfies the KSE (4.26) and (4.27) to zero order in α ′ . Hence,the types of arguments used to show supersymmetry enhancement via Lichnerowicz typetheorems in [9, 10, 11, 12] also do not work to the required order in α ′ for the heterotictheory.Finally, we have examined a class of nearly supersymmetric horizons for which thegravitino KSE is allowed to admit solutions on the spatial horizon section but not therest of the KSEs. Such solutions in general do not admit any spacetime Killing spinorsincluding solutions of the gravitino KSE. Under some conditions on the fluxes, we investi-gate the geometry of the spatial horizon sections using a combination of local and globaltechniques as well as the field equations. We find that those with a G , SU (3) and SU (2)structure admit 1, 2 and 4 parallel vectors on the spatial horizon sections with respect tothe connection with torsion. The geometry on the orbit spaces of these isometries is fullyspecified.The spacetime of both supersymmetric, and nearly supersymmetry horizons consideredhere admits a SL (2 , R ) symmetry at zeroth order in α ′ . In the supersymmetric case forwhich there is a η − Killing spinor to order O ( α ′ ) such that η − does not vanish at zerothorder, η [0] − = 0, this symmetry persists at first order in α ′ . The nearly supersymmetrichorizons also admit an SL (2 , R ) symmetry provided that h is parallel with respect to theconnection with torsion up to O ( α ′ ).It is not apparent whether the properties of the heterotic horizons described here36re going to persist to higher than two loops in sigma model perturbation theory. It islikely though that the presence of an sl (2 , R ) symmetry will persist after perhaps a suitablechoice of a scheme in perturbation theory. There is no apparent reason to hypothesize thatsuch a symmetry can be anomalous at higher loops. What happens to global properties ofthe horizons, for example the Lichnerowicz type theorems, is less clear. We have alreadyseen that these theorems do not hold to the expected order in α ′ even at two loops. Thiscan be taken as an indication that additional higher order corrections may further weakenthe consequences of such theorems. Acknowledgements
AF is partially supported by the EPSRC grant FP/M506655. JG is supported by theSTFC grant, ST/1004874/1. GP is partially supported by the STFC rolling grant ST/J002798/1.
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No additional research data beyond the data presented and cited in this work are neededto validate the research findings in this work.37 ppendix A Useful formulae
A.1 Spin Connection and Curvature
In our conventions, the curvature of a connection Γ is given by R AB,C D = ∂ A Γ CBD − ∂ B Γ CAD + Γ
CAN Γ NBD − Γ CBN Γ NAD , (A.1)We define connections ˇ ∇ and ˆ ∇ as followsˇ ∇ M ξ N = ∇ M ξ N − H N ML ξ L , ˆ ∇ M ξ N = ∇ M ξ N + 12 H N ML ξ L (A.2)for vector field ξ , where ∇ is the Levi-Civita connection. In particular, the ˇ R curvaturetensor can be written asˇ R AB,CD = R ABCD − ∇ A H CBD + 12 ∇ B H CAD + 14 H CAN H N BD − H CBN H N AD , (A.3)where R is the Riemann curvature tensor. Also, note thatˇ R AB,CD − ˆ R CD,AB = 12 ( dH ) ABCD . (A.4)We also define connections ˇ˜ ∇ and ˆ˜ ∇ on the horizon section S viaˇ˜ ∇ i Y j = ˜ ∇ i Y j − W jik Y k , ˆ˜ ∇ i Y j = ˜ ∇ i Y j + 12 W j ik Y k , (A.5)for vector fields Y on S , and where ˜ ∇ is the Levi-Civita connection of S , and we denotethe curvatures of the connections ˜ ∇ , ˇ˜ ∇ and ˆ˜ ∇ by ˜ R , ˇ˜ R and ˆ˜ R respectively.The non-vanishing components of the spin connection in the frame basis (2.2) of thenear horizon metric (2.1) areΩ − , + i = − h i , Ω + , + − = − r ∆ , Ω + , + i = 12 r (∆ h i − ∂ i ∆) , Ω + , − i = − h i , Ω + ,ij = − rdh ij , Ω i, + − = 12 h i , Ω i, + j = − rdh ij , Ω i,jk = ˜Ω i,jk , (A.6)where ˜Ω denotes the spin-connection of the spatial horizon section S in the e i basis. If f is any function of spacetime, then frame derivatives are expressed in terms of co-ordinatederivatives as ∂ + f = ∂ u f + 12 r ∆ ∂ r f , ∂ − f = ∂ r f , ∂ i f = ˜ ∂ i f − r∂ r f h i . (A.7)The non-vanishing components of the Ricci tensor is the basis (2.2) are R + − = 12 ˜ ∇ i h i − ∆ − h , R ij = ˜ R ij + ˜ ∇ ( i h j ) − h i h j R ++ = r (cid:0)
12 ˜ ∇ ∆ − h i ˜ ∇ i ∆ −
12 ∆ ˜ ∇ i h i + ∆ h + 14 ( dh ) ij ( dh ) ij (cid:1) R + i = r (cid:0)
12 ˜ ∇ j ( dh ) ij − ( dh ) ij h j − ˜ ∇ i ∆ + ∆ h i (cid:1) , (A.8)38here ˜ R is the Ricci tensor of the horizon section S in the e i frame.We remark that the non-vanishing components of the Hessian of Φ, are given by ∇ + ∇ − Φ = − h i ˜ ∇ i Φ , ∇ + ∇ i Φ = − r ( dh ) ij ˜ ∇ j Φ , ∇ i ∇ j Φ = ˜ ∇ i ˜ ∇ j Φ , (A.9)where in the above expression, we have set ∆ = 0.The non-vanishing components of the ˇ R curvature tensor in the basis (2.2) areˇ R − i, + j = ˜ ∇ j h i + 12 h ℓ W ℓij , ˇ R ij, + − = dh ij , ˇ R ij, + k = r (cid:18) ˜ ∇ k dh ij − h k dh ij + 12 ( dh ) im W mjk −
12 ( dh ) jm W mik (cid:19) , ˇ R ij,kℓ = ˜ R ijkℓ −
12 ˜ ∇ i W kjℓ + 12 ˜ ∇ j W kiℓ + 14 W kim W mjℓ − W kjm W miℓ = ˇ˜ R ij,kℓ , (A.10)where in the above expression, we have set ∆ = 0, N = h and Y = dh . Note thatthe ˇ R − i, + j and ˇ R ij, + k terms give no contribution to the Bianchi identity of H or to theEinstein equations, because ˇ R MN, − i = 0 for all M, N . A.2 Bosonic Field Equations
The Bianchi identity associated with the 3-form is dH = − α ′ (cid:18) tr( ˇ R ∧ ˇ R ) − tr( F ∧ F ) (cid:19) + O ( α ′ ) (A.11)where tr( F ∧ F ) = F ab ∧ F ba ( a, b are gauge indices on F ).The Einstein equation is R MN − H ML L H N L L + 2 ∇ M ∇ N Φ+ α ′ (cid:18) ˇ R ML ,L L ˇ R N L ,L L − F MLab F N Lab (cid:19) = O ( α ′ ) . (A.12)The gauge field equations are ∇ M (cid:18) e − H MN N (cid:19) = O ( α ′ ) , (A.13)and ∇ M (cid:18) e − F MN (cid:19) + 12 e − H NL L F L L = O ( α ′ ) . (A.14)39he dilaton field equation is ∇ M ∇ M Φ = 2 ∇ M Φ ∇ M Φ − H N N N H N N N + α ′ (cid:18) ˇ R N N ,N N ˇ R N N ,N N − F N N ab F N N ab (cid:19) + O ( α ′ ) . (A.15)This completes the list of field equations. We have followed the conventions of [46]. Appendix B Further Simplification of the KSEs
Here we shall show that the independent KSEs are given in (4.26) and (4.27). We first notethat the conditions on the bosonic fields (3.18) (obtained from the case when φ [0]+ ≡ φ [0]+ φ [0]+
0, as the simplification of the KSEs in thecase φ [0]+ ≡ B.1 Elimination of conditions (4.19), (4.21), (4.23), (4.25)
Let us assume (4.18), (4.20), (4.22) and (4.24). Then acting on the algebraic conditions(4.20) and (4.24) with the Dirac operator Γ ℓ ˜ ∇ ℓ , one obtains (cid:18) ˜ ∇ i ˜ ∇ i Φ ∓ h i ˜ ∇ i Φ − ∇ i Φ ˜ ∇ i Φ − h i h i + 112 W ijk W ijk + 14 (1 ± dh ij Γ ij + 14 ( − ± h k W kij Γ ij − dW ℓ ℓ ℓ ℓ Γ ℓ ℓ ℓ ℓ (cid:19) φ ± = O ( α ′ ) , (B.1)where we have made use of the field equations (4.13) and (4.15), together with the al-gebraic conditions (4.20) and (4.24). Next, on substituting the dilaton equation and theBianchi identity into the above expression, one finds (cid:18)
12 (1 ∓
1) ˜ ∇ i h i + 14 (1 ± dh ij Γ ij + 14 ( − ± i h W ) ij Γ ij + α ′ dh ij Γ ij dh pq Γ pq + α ′
32 ˜ F ij ab Γ ij ˜ F pqab Γ pq − α ′
32 ˇ˜ R ij,mn Γ ij ˇ˜ R pq,mn Γ pq (cid:19) φ ± = O ( α ′ ) . (B.2)Further simplification can be obtained by noting that the integrability conditions of theKSE (4.18) and (4.22) are ˆ˜ R ij,pq Γ pq φ ± = O ( α ′ ) , (B.3)and hence ˇ˜ R pq,ij Γ pq φ ± = O ( α ′ ) , (B.4)40rom which it follows that the final term on the RHS of (B.2) is O ( α ′ ) and hence can beneglected. So, (B.2) is equivalent to (cid:18)
12 (1 ∓
1) ˜ ∇ i h i + 14 (1 ± dh ij Γ ij + 14 ( − ± i h W ) ij Γ ij + α ′ dh ij Γ ij dh pq Γ pq + α ′
32 ˜ F ij ab Γ ij ˜ F pqab Γ pq (cid:19) φ ± = O ( α ′ ) . (B.5)We begin by considering the condition which (B.5) imposes on φ + : (cid:18) dh ij Γ ij + α ′ dh ij Γ ij dh pq Γ pq + α ′
32 ˜ F ijab Γ ij ˜ F pqab Γ pq (cid:19) φ + = O ( α ′ ) (B.6)To zeroth order this gives dh ij Γ ij φ + = O ( α ′ ) , (B.7)which implies that the second term on the LHS of (B.6) is of O ( α ′ ), and hence can beneglected. Using this, (B.6) gives that α ′ h ˜ F ijab Γ ij φ + , ˜ F pqab Γ pq φ + i = O ( α ′ ) , (B.8)which implies that ˜ F ijab Γ ij φ + = O ( α ′ ) . (B.9)Using this the third term on the LHS of (B.6) is also of O ( α ′ ). So, the remaining contentof (B.6) is dh ij Γ ij φ + = O ( α ′ ) . (B.10)Hence, we have proven that the KSE (4.18) and (4.20) imply the algebraic KSE (4.19)and (4.21).Next, we consider the condition which (B.5) imposes on φ − , which is (cid:18) ˜ ∇ i h i −
12 ( i h W ) ij Γ ij + α ′ dh ij Γ ij dh pq Γ pq + α ′
32 ˜ F ij ab Γ ij ˜ F pqab Γ pq (cid:19) φ − = O ( α ′ ) . (B.11)However, note also that the u -dependent part of (4.18), with (4.22), implies that (cid:18) ˜ ∇ i h j − W ijk h k (cid:19) Γ j φ − = O ( α ′ ) . (B.12)On contracting this expression with Γ i , we find (cid:18) ˜ ∇ i h i + 12 dh ij Γ ij −
12 ( i h W ) ij Γ ij (cid:19) φ − = O ( α ′ ) , (B.13)and on substituting this expression into (B.11) we get (cid:18) − dh ij Γ ij + α ′ dh ij Γ ij dh pq Γ pq + α ′
32 ˜ F ij ab Γ ij ˜ F pqab Γ pq (cid:19) φ − = O ( α ′ ) . (B.14)41ence, we find from exactly the same reasoning which was used to analyse the conditionson φ + , that (4.22) and (4.24) imply (4.23) and (4.25).So, on making use of the field equations, it follows that the necessary and sufficientconditions for supersymmetry simplify to the conditions (4.18) and (4.20) on φ + , and to(4.22) and (4.24) on η − . We remark that the u -dependent parts of the conditions (4.18)and (4.20) also impose conditions on η − . We shall examine the conditions on η − furtherin the next section, and show how these may be simplified. B.2 Elimination of u -dependent parts of (4.18) and (4.20) We begin by considering the u -dependent parts of (4.18) and (4.20), assuming that (4.22)and (4.24) hold. The u -dependent part of the condition on φ + obtained from (4.18) is (cid:18) ˜ ∇ i h j − W ijk h k (cid:19) Γ j η − = O ( α ′ ) , (B.15)and the u -dependent part of the algebraic condition (4.20) is given by (cid:18) Γ i ˜ ∇ i Φ + 12 h i Γ i − W ijk Γ ijk (cid:19) h ℓ Γ ℓ η − = O ( α ′ ) . (B.16)On adding h ℓ Γ ℓ acting on (4.24) to the above expression, we find that (B.16) is equivalentto the condition (cid:18) ˜ ∇ i h i − h i W ijk Γ jk (cid:19) η − , = O ( α ′ ) (B.17)where we have also made use of the field equation (4.13). On contracting (B.15) with Γ i ,it then follows that (B.17) is equivalent to dh ij Γ ij η − = O ( α ′ ) . (B.18)However, as shown in the previous section, this condition is implied by (4.22) and (4.24)on making use of the field equations.So, it remains to consider the condition (B.15). First, recall that the integrabilityconditions of the gravitino equation of (4.22) is given byˆ˜ R ij,kℓ Γ kℓ η − = O ( α ′ ) . (B.19)On contracting with Γ j , one then obtains (cid:18)(cid:0) − R ij + 12 W imn W j mn − ∇ k Φ W kij + dh ij − h k W kij (cid:1) Γ j + (cid:0) −
16 ( dW ) ijkℓ −
13 ˜ ∇ i W jkℓ + 12 W ij m W kℓm (cid:1) Γ jkℓ (cid:19) η − = O ( α ′ ) , (B.20)42here we have used the gauge equation (4.15). Also, on taking the covariant derivativeof the algebraic condition (4.24), and using (4.22), one also finds the following mixedintegrability condition (cid:18)(cid:0) ˜ ∇ i ˜ ∇ j Φ −
12 ˜ ∇ i h j + 12 W ikj ˜ ∇ k Φ − W ikj h k (cid:1) Γ j +Γ jkℓ (cid:0) −
112 ˜ ∇ i W jkℓ + 18 W jkm W iℓm (cid:1)(cid:19) η − = O ( α ′ ) . (B.21)On eliminating the ˜ ∇ i W jkℓ Γ jkℓ terms between (B.20) and (B.21), one obtains the condition (cid:18)(cid:0) − R ij + 12 W imn W j mn + dh ij − h k W kij − ∇ i ˜ ∇ j Φ + 2 ˜ ∇ i h j (cid:1) Γ j − dW ijkℓ Γ jkℓ (cid:19) η − = O ( α ′ ) . (B.22)Next, we substitute the Einstein equation (4.16) in order to eliminate the Ricci tensor,and also use the Bianchi identity for dW . One then obtains, after some rearrangement ofterms, the following condition (cid:18)(cid:0) ∇ i h j − h k W kij (cid:1) Γ j + α ′ (cid:0) dh ij Γ j dh kℓ Γ kℓ + 14 ˜ F ijab Γ j ˜ F kℓab Γ kℓ −
14 ˇ˜ R ij,mn Γ j ˇ˜ R kℓ,mn Γ kℓ (cid:1)(cid:19) η − = O ( α ′ ) . (B.23)The α ′ terms in the above expression can be neglected, as they all give rise to terms whichare in fact O ( α ′ ). This is because of the conditions (4.23) and (4.25), which we havealready shown follow from (4.22) and (4.24), together with the bosonic conditions, as wellas the fact that ˇ˜ R kℓ,mn Γ kℓ η − = O ( α ′ ) , (B.24)which follows from the integrability condition of (4.22). It follows that (B.23) implies(B.15). Appendix C A consistency condition
Suppose that we consider the Bianchi identity associated with the 3-form as dH = − α ′ (cid:18) tr( R ∧ R ) − tr( F ∧ F ) (cid:19) + O ( α ′ ) , (C.1)where R is a spacetime curvature which will be specified later. Also observe that the2-form gauge potential and the Einstein equation can be written together asˆ R MN + 2 ˆ ∇ M ∇ N Φ + α ′ (cid:18) R ML ,L L R N L ,L L − F MLab F N Lab (cid:19) = O ( α ′ ) . (C.2)43hen one can establish by direct computation thatˆ R M [ N,P Q ] = −
13 ˆ ∇ M H NP Q − dH MNP Q . (C.3)Using this and the field equations of the theory, one can derive the relationˆ R MN,P Q Γ N Γ P Q ǫ = −
13 ˆ ∇ M (cid:0) H LP Q Γ LP Q − ∂ L ΦΓ L (cid:1) ǫ − α ′ R MN,EF R P Q,EF − F MNab F P Qab ]Γ N Γ P Q ǫ + O ( α ′ ) . (C.4)If ǫ satisfies that gravitino KSE, the left hand side of this relation vanishes. Furthermorethe right-hand-side vanishes as well provided that dilatino and gaugino KSEs are satisfied,and in addition R P Q,EF Γ P Q ǫ = O ( α ′ ) . (C.5)Of course in heterotic string perturbation theoryˇ R P Q,EF Γ P Q ǫ = O ( α ′ ) , (C.6)as a consequence of the gravitino KSE and the closure of H at that order. Thus one canset R = ˇ R and the identity (C.4) will hold up to order α ′ .One consequence of the identity (C.4) is that if the gravitino KSE and gaugino KSEsare satisfied as well as (C.5) but the dilatino is not, then the gravitino KSE admits anadditional parallel spinor of the opposite chirality. Such kind of identities have beenestablished before for special cases in [42]. Here we have shown that such a result isgeneric in the context of heterotic theory. Appendix D Lichnerowicz Theorem Computation
In this appendix, we present the details for the calculation of the functional I defined in(7.15), and show how the constants q and c are fixed by requiring that certain types ofterms which arise in the calculation should vanish. We begin by considering the calculationat zeroth order in α ′ , and then include the corrections at first order in α ′ . We remarkthat we shall retain terms of the type h i ˜ ∇ i Φ throughout. This is because although theseterms vanish at zeroth order in α ′ as a consequence of the analysis in Section 8, it doesnot follow from this analysis that L h Φ = O ( α ′ ). However, as we shall see, it turns outthat the coefficient multiplying the terms h i ˜ ∇ i Φ, which depends on the constants q and c , vanishes when one requires that several other terms in I vanish as well. So these termsdo not give any contribution to I at either zeroth or first order in α ′ . D.1 Computations at zeroth order in α ′ Throughout the following analysis, we assume Einstein equations, dilaton field equationand Bianchi identity at zeroth order in α ′ . To proceed, we expand out the definition of44 ( κ ) i and D in I , obtaining the following expression I = Z S e c Φ κ − q ) h Γ i A η ± , ˆ˜ ∇ i η ± i + e c Φ (8 κ − q ) h η ± , A † A η ± i− e c Φ h ˆ˜ ∇ i η ± , Γ ij ˆ˜ ∇ j η ± i . (D.1)Now, after writing ˆ˜ ∇ in terms of the Levi-Civita connection ˜ ∇ and after integratingby parts, the expression (D.1) decomposes into I = I + I + I , (D.2)where I = Z S e c Φ κ − q ) h η ± , A † D η ± i + e c Φ (8 κ − κq + q ) h η ± , A † A η ± i (D.3) − e c Φ h η ± , Γ ℓ ℓ Γ ij Γ ℓ ℓ W iℓ ℓ W jℓ ℓ η ± i , (D.4)and I = Z S ce c Φ h η ± , Γ ij ˜ ∇ j η ± i + 18 e c Φ h ˜ ∇ i η ± , Γ ij Γ ℓ ℓ W jℓ ℓ η ± i− e c Φ h η ± , Γ ℓ ℓ Γ ij W jℓ ℓ ˜ ∇ η ± i , (D.5)and I = Z S − e c Φ h ˜ ∇ i η ± , Γ ij ˜ ∇ j η ± i . (D.6)In particular, we note the identityΓ ℓ ℓ Γ ij Γ ℓ ℓ W iℓ ℓ W jℓ ℓ = 8 W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ − W ijk W ijk , (D.7)which simplifies I . After integrating by parts the second term in I , we have I = Z S ce c Φ h η ± , Γ ij ˜ ∇ j η ± i − e c Φ h η ± , (cid:0) Γ ij Γ mn − Γ mn Γ ij (cid:1) W j mn ˜ ∇ i η ± i− c e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i − e c Φ h η ± , Γ ℓ ℓ ℓ ℓ ˜ ∇ ℓ W ℓ ℓ ℓ η ± i , (D.8)where the last term is order α ′ , so we shall neglect it. Now we shall focus on the secondterm of (D.8). First note that (cid:0) Γ ij Γ mn − Γ mn Γ ij (cid:1) W jmn = − mn W imn = 43 W ℓ ℓ ℓ (cid:0) Γ ℓ ℓ ℓ Γ i + Γ iℓ ℓ ℓ (cid:1) . (D.9)Then, after an integration by parts and after writing ˜ ∇ in terms of D , we have Z S − e c Φ h η ± , (cid:0) Γ ij Γ mn − Γ mn Γ ij (cid:1) W j mn ˜ ∇ i η ± i = Z S − e c Φ h η ± , W ℓ ℓ ℓ Γ ℓ ℓ ℓ D η ± i + q e c Φ h η ± , W ℓ ℓ ℓ Γ ℓ ℓ ℓ A η ± i − e c Φ h η ± , W ℓ ℓ ℓ Γ ℓ ℓ ℓ W ijk Γ ijk η ± i + c e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + 112 e c Φ h η ± , Γ ℓ ℓ ℓ ℓ ˜ ∇ ℓ W ℓ ℓ ℓ η ± i . (D.10)45he last term of (D.10) is order α ′ , so we shall neglect it. To proceed further, we shallsubstitute W ijk Γ ijk in terms of A , using its definition. This produces terms proportionalto the norm squared of A η ± , together with a number of counterterms. In detail, oneobtains Z S − e c Φ h η ± , (cid:0) Γ ij Γ mn − Γ mn Γ ij (cid:1) W jmn ˜ ∇ i η ± i = Z S − e c Φ h η ± , W ℓ ℓ ℓ Γ ℓ ℓ ℓ D η ± i + e c Φ (cid:18) − q (cid:19) h η ± , A † A η ± i + e c Φ (cid:18) − q (cid:19) h η ± , Γ i ˜ ∇ i Φ A η ± i± e c Φ (cid:18) − q (cid:19) h η ± , Γ i h i A η ± i + 3 e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i ± e c Φ h η ± , h i ˜ ∇ i Φ η ± i + 34 e c Φ h η ± , h i h i η ± i + c e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + O ( α ′ ) . (D.11)Let us focus now on the first term of (D.8). After writing Γ ij as Γ i Γ j − δ ij and afterintegrating by parts, we have Z S ce c Φ h η ± , Γ ij ˜ ∇ j η ± i = Z S ce c Φ h η ± , Γ ℓ ˜ ∇ ℓ ΦΓ i ˜ ∇ i η ± i + c e c Φ h η ± , ˜ ∇ i ˜ ∇ i Φ η ± i + c e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i . (D.12)The first term in the RHS of (D.12) can be rewritten in terms of the modified Diracoperator D after subtracting suitable terms. The second term on the RHS can be furthersimplified using the dilaton field equation at zeroth order in α ′ . On performing thesecalculations, we have Z S ce c Φ h η ± , Γ ij ˜ ∇ j η ± i = Z S ce c Φ h η ± , Γ ℓ ˜ ∇ ℓ Φ D η ± i − c e c Φ h η ± , W ijk W ijk η ± i + c (cid:18) − q (cid:19) e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + c e c Φ h η ± , h i h i η ± i + 12 c (cid:18)
112 + c
24 + q (cid:19) e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + 6 c (cid:18) ± q (cid:19) e c Φ h η ± , h i ˜ ∇ i Φ η ± i + O ( α ′ ) . (D.13)Let us now focus on I . Recall thatΓ ij ˜ ∇ i ˜ ∇ j η ± = −
14 ˜
R η ± . (D.14)Therefore after integrating by parts and using (D.14) neglecting α ′ corrections from Ein-stein equations, I becomes I = Z S − e c Φ h η ± , W ijk W ijk η ± i + e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + 14 e c Φ h η ± , h i h i η ± i + e c Φ h η ± , h i ˜ ∇ i Φ η ± i + O ( α ′ ) . (D.15)46ollecting together all terms and substituting h i h i by inverting the zeroth order in α ′ dilaton filed equation, one finally gets I = Z S e c Φ h η ± , (cid:18) c Γ ℓ ˜ ∇ ℓ Φ − W ℓ ℓ ℓ Γ ℓ ℓ ℓ + 2( κ − q ) A † (cid:19) D η ± i + (8 κ − κq − q
12 + q ) e c Φ h η ± , A † A η ± i + 34 (cid:18) q − (cid:19) e c Φ h η ± , W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ η ± i− c (cid:18) q − (cid:19) e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + 6 (cid:18)
112 + q + c (cid:19) e c Φ h η ± , ˜ ∇ i ˜ ∇ i Φ η ± i + 12 c (cid:16) q + c (cid:17) e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + (cid:18) − q ± q ( c + 2) (cid:19) e c Φ h η ± , h i ˜ ∇ i Φ η ± i + O ( α ′ ) . (D.16)In order to eliminate the term h η ± , W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ η ± i , which has no sign and cannotbe rewritten in terms of D or A † A , we must set q = 112 + O ( α ′ ) . (D.17)and then in order to eliminate the h η ± , ˜ ∇ i ˜ ∇ i Φ η ± i term we must further set c = − O ( α ′ ) . (D.18)Then (D.16) simplifies to I = Z S e − h η ± , Ψ D η ± i + (cid:16) κ − κ (cid:17) Z S e − k A η ± k + O ( α ′ ) , (D.19)where Ψ ≡ − ℓ ˜ ∇ ℓ Φ − W ℓ ℓ ℓ Γ ℓ ℓ ℓ + 2 (cid:18) κ − (cid:19) A † . (D.20) D.2 Computations at first order in α ′ In this section we shall consider corrections at first order in α ′ . I and I gain α ′ correc-tions from bosonic field equations and Bianchi identity, while I does not. Therefore wehave I = Z S e c Φ κ − q ) h η ± , A † D η ± i + e c Φ (8 κ − κq + q ) h η ± , A † A η ± i− e c Φ h η ± , W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ η ± i + 116 e c Φ h η ± , W ijk W ijk η ± i + O ( α ′ ) , (D.21)47nd I = Z S ce c Φ h η ± , (cid:18) Γ ℓ ˜ ∇ ℓ Φ − W ℓ ℓ ℓ Γ ℓ ℓ ℓ (cid:19) D η ± i − c e c Φ h η ± , W ijk W ijk η ± i + c (cid:18) − q (cid:19) e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + e c Φ (cid:18) − q (cid:19) h η ± , A † A η ± i + e c Φ (cid:18) − q (cid:19) h η ± , Γ i ˜ ∇ i Φ A η ± i ± e c Φ (cid:18) − q (cid:19) h η ± , Γ i h i A η ± i + (cid:18)
34 + c (cid:19) e c Φ h η ± , h i h i η ± i + (cid:18) c + c cq + 3 (cid:19) e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + (cid:16) c ± cq ± (cid:17) e c Φ h η ± , h i ˜ ∇ i Φ η ± i− e c Φ h η ± , Γ ℓ ℓ ℓ ℓ ˜ ∇ ℓ W ℓ ℓ ℓ η ± i + α ′ c e c Φ (cid:18) − h η ± , dh ij dh ij η ± i + h η ± , ˇ˜ R ℓ ℓ ,ℓ ℓ ˇ˜ R ℓ ℓ ,ℓ ℓ η ± i − h η ± , ˜ F ijab ˜ F ijab η ± i (cid:19) + O ( α ′ ) , (D.22)and I = Z S − e c Φ h η ± , W ijk W ijk η ± i + e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + 14 e c Φ h η ± , h i h i η ± i + e c Φ h η ± , h i ˜ ∇ i Φ η ± i + α ′ e c Φ (cid:18) − h η ± , dh ij dh ij η ± i + h η ± , ˇ˜ R ℓ ℓ ,ℓ ℓ ˇ˜ R ℓ ℓ ,ℓ ℓ η ± i − h η ± , ˜ F ijab ˜ F ijab η ± i (cid:19) + O ( α ′ ) . (D.23)48ombining all together and considering α ′ corrections from substituting h i h i by invertingthe dilaton field equations, we have I = Z S e c Φ h η ± , (cid:18) c Γ ℓ ˜ ∇ ℓ Φ − W ℓ ℓ ℓ Γ ℓ ℓ ℓ + 2( κ − q ) A † (cid:19) D η ± i + (8 κ − κq − q
12 + q ) e c Φ h η ± , A † A η ± i + 34 (cid:18) q − (cid:19) e c Φ h η ± , W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ η ± i− c (cid:18) q − (cid:19) e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + 12 c (cid:16) q + c (cid:17) e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + 6 (cid:18)
112 + q + c (cid:19) e c Φ h η ± , ˜ ∇ i ˜ ∇ i Φ η ± i + (cid:18) − q ± q ( c + 2) (cid:19) e c Φ h η ± , h i ˜ ∇ i Φ η ± i + α ′ e c Φ (cid:18) h η ± , Γ ℓ ℓ ℓ ℓ dh ℓ ℓ dh ℓ ℓ i − h η ± , Γ ℓ ℓ ℓ ℓ ˇ˜ R ℓ ℓ ,ij ˇ˜ R ℓ ℓ ,ij η ± i + h η ± , Γ ℓ ℓ ℓ ℓ ˜ F ℓ ℓ , ab ˜ F ℓ ℓ ab η ± i (cid:19) + α ′ (cid:18) − q (cid:19) e c Φ (cid:18) − h η ± , dh ij dh ij η ± i + h η ± , ˇ˜ R ℓ ℓ ,ℓ ℓ ˇ˜ R ℓ ℓ ,ℓ ℓ η ± i − h η ± , ˜ F ij ab ˜ F ijab η ± i (cid:19) + O ( α ′ ) . (D.24)To further simplify (D.24), we note the following identity h η ± , Γ ℓ ℓ ℓ ℓ dh ℓ ℓ dh ℓ ℓ η ± i = h η ± , Γ ℓ ℓ dh ℓ ℓ Γ ℓ ℓ dh ℓ ℓ η ± i + 2 h η ± , dh ij dh ij η ± i . (D.25)Identities analogous to (D.25) hold also for the terms which involve ˇ˜ R ij,kℓ and ˜ F ij ab . Thisleads to I = Z S e c Φ h η ± , (cid:18) c Γ ℓ ˜ ∇ ℓ Φ − W ℓ ℓ ℓ Γ ℓ ℓ ℓ + 2( κ − q ) A † (cid:19) D η ± i + (8 κ − κq − q
12 + q ) e c Φ h η ± , A † A η ± i + 34 (cid:18) q − (cid:19) e c Φ h η ± , W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ η ± i− c (cid:18) q − (cid:19) e c Φ h η ± , Γ iℓ ℓ ℓ ˜ ∇ i Φ W ℓ ℓ ℓ η ± i + 12 c (cid:16) q + c (cid:17) e c Φ h η ± , ˜ ∇ i Φ ˜ ∇ i Φ η ± i + 6 (cid:18)
112 + q + c (cid:19) e c Φ h η ± , ˜ ∇ i ˜ ∇ i Φ η ± i + (cid:18) − q ± q ( c + 2) (cid:19) e c Φ h η ± , h i ˜ ∇ i Φ η ± i + 38 α ′ ( q −
112 ) e c Φ (cid:18) dh ij dh ij + ˜ F ijab ˜ F ijab − ˇ˜ R ℓ ℓ ,ℓ ℓ ˇ˜ R ℓ ℓ ,ℓ ℓ (cid:19) k η ± k − α ′ e c Φ k /dh η ± k − α ′ e c Φ k / ˜ F η ± k + α ′ e c Φ h ˇ˜ R ℓ ℓ , ij Γ ℓ ℓ η ± , ˇ˜ R ijℓ ℓ , Γ ℓ ℓ η ± i + O ( α ′ ) .
49n order to eliminate the term h η ± , W iℓ ℓ W iℓ ℓ Γ ℓ ℓ ℓ ℓ η ± i , which has no sign and cannotbe rewritten in terms of D or A † A , we must set q = 112 + O ( α ′ ) . (D.26)and then in order to eliminate the h η ± , ˜ ∇ i ˜ ∇ i Φ η ± i term we must further set c = − O ( α ′ ) . (D.27)Then (D.28) is significantly simplified to I = (cid:18) κ − κ (cid:19) Z S e − k A η ± k + Z S e − h η ± , Ψ D η ± i− α ′ Z S e − (cid:16) k /dh η ± k + k / ˜ F η ± k −h ˇ˜ R ℓ ℓ , ij Γ ℓ ℓ η ± , ˇ˜ R ijℓ ℓ , Γ ℓ ℓ η ± i (cid:17) + O ( α ′ ) , (D.28)where Ψ is defined in (D.20). Appendix E
AdS n +1 as warped product over AdS n The
AdS n +1 space can be written as a warped product over AdS n . This has been observedbefore in [45] for AdS and elsewhere, see eg [47]. For this, we label all geometrical objectsdefined on AdS n +1 and AdS n by n + 1 and n respectively, e.g. ds n +1 is the metric on AdS n +1 and ds n is the metric on AdS n . In principle AdS n +1 and AdS n can have differentradii, which are indicated by ℓ n +1 and ℓ n respectively. Coordinates on AdS n +1 are takento be as follows x I = ( x , x i ) , x ≡ y , i = 1 , ..., n . (E.1)We shall begin with an Ans¨atz for the metric on AdS n +1 as a warped product over AdS n , i.e. ds n +1 = dy + f ( y ) ds n . (E.2)We want to determine the necessary and sufficient conditions to impose on f ( y ) in orderfor ds n +1 to be the metric on AdS n +1 . To succeed, we have to impose the fact AdS n +1 isa maximally symmetric space. Locally, the necessary and sufficient condition is that theRiemann tensor must assume the following form R ( n +1) IJKL = − ℓ n +1 (cid:16) g ( n +1) IK g ( n +1) JL − g ( n +1) JK g ( n +1) IL (cid:17) , (E.3)Equation (E.3) implies also that the metric (E.2) is Einstein and the curvature scalar isconstant and negative, i.e. R ( n +1) IJ = − nℓ n +1 g ( n +1) IJ , R ( n +1) = − ℓ n +1 n ( n + 1) . (E.4)50he non-vanishing Christoffel symbols of (E.2) are:Γ ( n +1) ki = f ′ ( y ) f ( y ) δ ki , Γ ( n +1) 0 i j = − f ( y ) f ′ ( y ) g ( n +1) ij , Γ ( n +1) ki j = Γ ( n ) ki j . (E.5)The non-vanishing Riemann tensor components are R ( n +1) i ,k = − f ′′ ( y ) f ( y ) δ ki ,R ( n +1) i , ℓ = f ( y ) f ′′ ( y ) g ( n ) iℓ ,R ( n +1) ij,kℓ = R ( n ) ij,kℓ + f ′ ( y ) (cid:16) δ kj g ( n ) iℓ − δ ki g ( n ) jℓ (cid:17) , (E.6)and R ( n +1) i ,k = − f ( y ) f ′′ ( y ) g ( n ) ik ,R ( n +1) ij,kl = f ( y ) R ( n ) ij,kl − f ( y ) f ′ ( y ) (cid:16) g ( n ) ik g ( n ) jl − g ( n ) jk g ( n ) il (cid:17) . (E.7)The non-vanishing Ricci tensor components are R ( n +1)00 = − n f ′′ ( y ) f ( y ) ,R ( n +1) ij = R ( n ) ij + (cid:2) f ′ ( y ) (1 − n ) − f ( y ) f ′′ ( y ) (cid:3) g ( n ) ij . (E.8)The Riemann tensor on AdS n must assume the following form R ( n ) ijkℓ = − ℓ n (cid:16) g ( n ) ik g ( n ) jℓ − g ( n ) jk g ( n ) iℓ (cid:17) . (E.9)Now we impose (E.3). The ( i , k f f ′′ ( y ) = 1 ℓ n +1 f ( y ) . (E.10)The ( ij, kl )-components provide the second ordinary differential equation for ff ′ ( y ) − ℓ n +1 f ( y ) + 1 ℓ n = 0 . (E.11)Since equations in (E.4) are derived from (E.3), they would imply again (E.10) and (E.11),so there is nothing further to be learned from those conditions. The general solution of(E.10) and (E.11) is f ( y ) = α cosh (cid:18) yℓ n +1 (cid:19) + β sinh (cid:18) yℓ n +1 (cid:19) , (E.12)where α and β are constants which satisfy α − β = ℓ n +1 ℓ n . (E.13)The solution (E.12) leads us to the following conclusions51. if y ∈ ( −∞ , + ∞ ), then locally the AdS metric can be written as AdS × w R .2. if y ∈ [0 , AdS metric can be written as is AdS × w [0 ,
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