ITP-UH-17/10 Homogeneous heterotic supergravity solutions with linear dilaton Christoph N¨olle 1 1 Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover 0 Appelstraße 2, 30167 Hannover, Germany 2 email: [email protected] n a J Abstract 5 1 I construct solutions to the heterotic supergravity BPS-equations on products of Minkowski space with a non-symmetric coset. All of the ] h bosonic fields are homogeneous and non-vanishing, the dilaton being a t linear function on thenon-compact part of spacetime. - p e h [ Contents 2 v 1 Introduction 1 3 7 2 Heterotic supergravity 4 8 2 . 3 Homogeneous vector bundles 5 1 1 4 Spinors on cosets 10 0 1 5 Homogeneous Sasaki-Einstein manifolds 16 : v i X 6 Examples 17 r a 7 Conclusion 32 1 Introduction Theaimofthispaperistopresentsomehomogeneoussolutionstotheheterotic supergravity equations [8, 7] 1 ε= H γαβ ε=0 ∇−µ ∇µ− 8 µαβ (cid:0) 1 (cid:1) (1.1) γ dφ H ε=0 − 12 (cid:0) γ(F(cid:1))ε=0 1 and the Bianchi identity α dH = ′tr R+ R+ F F , (1.2) 4 ∧ − ∧ (cid:16) (cid:17) where ε is a spinor on a 10-dimensional Lorentzian manifold M, H is a three- form, φ a function, and F the curvature of a gauge field on M, and are ± ∇ two connections onthe tangent bundle TM,involvingH. In (1.1) γ is the map from forms to the Clifford algebra. The manifold M will be chosen of the form M = Rp,1 G/K for a non-symmetric, naturally reductive coset G/K, with × (mostly) simple compact Lie groups K G, equipped with the metric induced ⊂ by the Killing form. Thischoiceiscanonicalforthefollowingreasons. Firstofalleverysuchnon- symmetric coset carries a G-invariant (or homogeneous) three-form, which we willidentifywithH. UponpropernormalizationofH,thequestionwhetherthe gravitinoequation ε=0hasasolutionε,turnsintoasimplerepresentation- − ∇ theoretical problem. Furthermore, upon this choice of H we also get a solution forF,namelythecurvatureR oftheso-calledcanonicalconnection ,which − − ∇ also appears in the gravitino equation. It satisfies both γ(R )ε = 0 if ε solves − thegravitinoequation,anddH tr(R+ R+ R R ),leadingtoasolution − − ∼ ∧ − ∧ of all of the equations except the dilatino one, γ(dφ 1 H)ε=0. − 12 All this suggests that the heterotic supergravity equations are tailored to admit homogeneous solutions; in particular the Bianchi identity allowing for non-trivial dH is an important deviation from the standard supergravity rule dH =0,which wouldimmediately rule out these spaces. The situationchanges with the dilatino equation however. On a general coset G/K there are no homogeneous1-formswhichcouldserveasdφ, andifthey existthey tendto be non-exact. Therefore the only obvious choice would be to take dφ=0. This is not possible however, as we have γ(H)ε = 0, and there is also a simple no-go 6 theorem excluding this type of solutions. In [30] we proposed to circumvent this problem by allowing for non-trivial fermioncondensates,butinthis paperasolutionbasedonpurelybosonicback- grounds will be presented. The backdoorwe are going to use is to introduce an additional Rp factor to our spacetime M, with p the rank difference of G and K (or p=2 for equalrank groups). Then we can take φ to be a linear function on Rp, giving rise to a constant and thus homogeneous 1-form dφ. It will be shownbelow that upon this choice of φ it is often possible to solvealso the last equation. The amount of supersymmetry preserved in the space orthogonal to Rp G/K is then at least =2p p. For trivial K one gets a Wess-Zumino- × N − Witten (WZW) model coupled to a linear dilaton, and these models exist also intype II string theory. They wereconsideredalreadyin[16], asa certainlimit of NS5-branes. The BPS-equations (1.1) actually guarantee that our supergravity vacua preserve some supersymmetry, and it was sometimes argued that they imply the usual equations of motions. Ivanov has proven that this is not the case if the equations of motion are truncated at order α as well, instead one would ′ havetoreplaceR+ intheBianchiidentityandtheequationofmotionbyR to − 2 ensure this [25]. The correct interpretation of this result seems to be that full compatibility between the supersymmetry equations (1.1) and the equations of motion requires the full tower of string corrections to both sets of equations, as explained in [7], based on results of [8]. Therefore, maybe one should take the solutions ofthe system(1.1) notas a proofbut rather asan indicationthat there exists a heterotic string theory on these backgrounds. Fromaphysicalperspectivethelineardilatoncertainlyrulesoutthesespaces as models for our universe; but see [27] for an intersecting brane scenario with chiral fermions. An interpretation of the type of solution considered here in terms of a decoupling limit of string theory is given in [3, 21]. On the other hand, a couple of homogeneous solutions to the above equa- tions havebeen presentedin recentyearswhere the dilatonis actually constant [18, 33]. The method used in these works is somewhat different from ours, as they take as starting point Strominger’s reformulation of the BPS equations [32]. Furthermore they avoid the above-mentioned no-go theorem by choosing M to be a non-semisimple Lie group (or a finite quotient thereof) equipped with a metric of negative scalar curvature, whereas in our models the metric always comes from the bi-invariant one on G and has positive scalar curvature. Although many of the spaces we discuss allow for other homogeneous metrics as well, the solution of the equations becomes more involved with these. One advantageofnotrelying onStrominger’sequationsis thatwe arenotrestricted to compactspacesof dimension six. In factwe will find solutions with compact spaces of arbitrary odd dimension, and also 6-dimensional ones. Afterintroducingthenecessarytoolsforhomogeneousspacesinsection3,we willinsection4developamethodwhichallowsusbothtoproveexistenceof - − ∇ parallelspinorsonmanyofthe consideredspaces,andtocalculatethe actionof γ(H)onthesespinors,thusenablingustodeterminethelineardilatonneededto satisfy also the dilatino equation. In section 5 we discuss homogeneous Sasaki- Einsteinmanifolds,whichareaparticularclassofspaceswherethismethodcan be applied. The last section 6 has some examples treated in detail, based on the cosets SU(n+1)/SU(n)=S2n+1 • Sp(n+1)/Sp(n)=S4n+1 • Sp(n)/SU(n) • SO(2n)/SU(n) • SO(n+1)/SO(n 1) • − Spin(7)/G =S7 2 • G /SU(3)=S6 2 • SU(3)/U(1) U(1) • × SO(5)/SO(3) max • It is intriguing that all of those spaces admit one of the following structures: 3 nearly K¨ahler (in 6D), • nearly parallel G (in 7D), 2 • Sasaki-Einstein(in odd dimension), • 3-Sasaki(in 4n+3 dimensions), • although the metric we use in most cases differs from the one defining this structure. These areexactlythe spaceswhoseconesadmitparallelspinors,and they play an important role in other types of string theory as well [1, 12]. The amountofsupersymmetrypreserveddependsonthegeometrictypeoftheman- ifold, nearly K¨ahler, G , and Sasaki-Einstein generically have = 1, whereas 2 N 3-Sasakianspaces preserve more supersymmetry. It should be mentioned that the method presented does not generalize to symmetric spaces, like Sn = SO(n+1)/SO(n) with its round metric; the equa- tion ε=0 does not have a solution there. ∇ 2 Heterotic supergravity The low-energy limit of heterotic string theory is given by 10D = 1 super- N gravity coupled to super Yang-Mills. The bosonic part of the effective action is [7] 1 α S = Scalg+4dφ2 H 2+ ′tr R+ 2 F 2 Volg, (2.1) | | − 2| | 4 | | −| | ZM(cid:16) (cid:0) (cid:1)(cid:17) where we adopt the widely-used convention to denote by tr a positive-definite form on a Lie algebra, in fact always minus the ordinary trace over tangent space in our examples. It leads to the following field equations (to order α): ′ 1 α Ric +2( dφ) H H αβ + ′ R+ R+αβγ tr F F α =0, µν ∇ µν − 4 µαβ ν 4 µαβγ ν − µα ν Scal+4∆φ 4dφ2 h 1 H 2+ α′tr R+(cid:0) 2 F (cid:1)2i=0, − | | − 2| | 4 | | −| | e2φd e 2φF +A F Fh A+ H Fi=0, − ∗ ∧∗ −∗ ∧ ∗ ∧ d e−2φH =0. ∗ (2.2) The full action with fermions is invariant under supersymmetry, acting on the fermions as δψ = ε, µ ∇−µ 1 1 δλ= γ dφ H ε, (2.3) −2 − 12 1 (cid:0) (cid:1) δχ= γ(F)ε, −4 where ψ is the gravitino, λ the dilatino, and χ the gaugino. The quantization map γ is explicitly 1 γ ω dxµ1 dxµp =ω γµ1...γµp, (2.4) p! µ1...µp ∧···∧ µ1...µp (cid:16) (cid:17) 4 where we use the convention γµ,γν = 2gµν. The requirement that these { } variations vanish ensures that a background preserves supersymmetry, and is precisely the set of equations (1.1). Here the connections are related to the ± ∇ Levi-Civita connection of g via (Γ−)abc =Γabc+ 21Habc, (Γ+)abc =Γabc− 21Habc. (2.5) In addition to the equations of motion or supersymmetry equations, one has to impose the Bianchi identity α dH = ′tr R+ R+ F F . (2.6) 4 ∧ − ∧ (cid:16) (cid:17) Ithasbeenproposedtochoosethesameconnectioneverywhereintheequations, instead of + and , but here we stick to the usual convention, which seems − ∇ ∇ tobepreferredfromastringtheoreticalpointofview[7]. Wecannotexpectthe equations of motion (2.2) to be implied by the supersymmetry equations then, as this would require taking into account all α corrections. Those equations ′ however which do not involve the gauge field will be satisfied (and the Yang- Mills equation for F as well). They are the H-equation d e 2φH =0 and the − ∗ following combination of dilaton equation and trace of the Einstein equation: 1 Scal 8dφ2+6∆φ+ H 2 =0. (2.7) − | | 2| | Fromthiswecanderiveasimpleno-gotheorem. Supposethedilatonisconstant, then 1 Scal= H 2, (2.8) −2| | and the scalar curvature must be non-positive. It should be mentioned that there is also a constraint on the cohomology class of H: [H] H3 M, 4π2αZ , (2.9) ′ ∈ if dH = 0, which leads to the quant(cid:0)ization of t(cid:1)he level in WZW models for instance. For dH = 0 the requirement will be that a certain combination of H 6 and the Chern-Simons forms of R+ and F defines an integer cohomologyclass, but we will simply ignore this condition in what follows, as most of the spaces we consider have H3(M,Z)=0 anyway. As mentioned inthe introduction, in this paper we will solvethe BPSequa- tions (1.1) together with the Bianchiidentity (1.2), andignore the equations of motion completely. 3 Homogeneous vector bundles Let G be a connected compact simple Lie groupequipped with the bi-invariant Riemannian metric g (induced by minus the Killing form on its Lie algebra g), and K a naturally reductive subgroup. This means we have an orthogonal splitting of the Lie algebra g = k m, with ad(k)m m, so that m carries a ⊕ ⊂ representation of k. Let (V,ρ) be a representation of K, and E = G V the K × associated vector bundle over G/K, which consists of equivalence classes [g,v] 5 with g G and v V, and identification [g,v] = [gk 1,ρ(k)v] for all k K. − ∈ ∈ ∈ Its sections are in a 1-1 correspondence with maps f :G V satisfying → f(gk)=ρ(k)−1f(g), k K. (3.1) ∀ ∈ G acts on the space of sections Γ(G/K,E) through (g f)(h) = f(g 1h). The − · set of G-invariant sections (also called homogeneous sections) is thus given by theconstantfunctions,andthereforeina1-1correspondencetotheK-invariant elements of V: Lemma 3.1. Let V carry a representation of K, then Γ(G/K,G V)G VK. K × ≃ We will adopt the following index convention. Basis elements of k will be denoted by I ,I ,..., those ofm by I ,I ,..., andthe full setof basiselements k l a b ofgbyI ,I ,.... Thedualbasisofleft-invariant1-formsonGisdenotedeµ,or µ ν ek andea forthosedualtoI andI . Thepull-backsoftheseformstoG/K will k a bedenotedbythe samesymbols,andthey satisfythe Maurer-Cartanequations 1 1 dek = fk el em fkea eb, −2 lm ∧ − 2 ab ∧ (3.2) 1 dea = faeb ec faeb ek, −2 bc ∧ − bk ∧ where fλ are the structure constants of g, defined by [I ,I ] = fλ I . Our µν µ ν µν λ metric on g will be minus the Killing form g(X,Y)=tr ad(X) ad(Y) , X,Y g, (3.3) g ◦ ∈ or in coordinates (cid:0) (cid:1) g = fd fc +2fkfc , g = fn fm+fb fa . (3.4) ab − ac bd ac bk kl − km ln ka lb g is g-invariant, th(cid:0)us also k-invarian(cid:1)t, and gives r(cid:0)ise to a homogen(cid:1)eous metric on G/K. The 3-form. Another important example of a homogeneous section is the following. Define H Λ3m through ∗ ∈ H(X,Y,Z)= g([X,Y],Z), X,Y,Z m, (3.5) − ∀ ∈ or in coordinates 1 H = f ea eb ec. (3.6) abc −6 ∧ ∧ Then H is K-invariant,andgives riseto a 3-formonG/K. Incase K is chosen trivial, H becomes a generator of H3(G,Z) = Z upon proper normalization of the metric g. In general H is a natural candidate for the 3-form of heterotic string theory. For the Bianchi identity we need to know the derivative of H, andforitsequationofmotiond H (usingthenotationeabcd =ea eb ec ed): ∗ ∧ ∧ ∧ Lemma 3.2. We have 1 1 dH = f fkeabcd = f fe eabcd, (3.7) −4 kab cd 4 abe cd whereas d H =0. ∗ 6 Proof. From the Maurer-Cartanequation we have 1 1 dH = f ff eabcd+ f fa edkbc. 4 abf cd 4 abc dk Consider the last term. It follows from the Jacobi identity that f fa splits abc dk into a partwhichis symmetric in bandd, andanother partsymmetric inc and d. Thereforethistermvanishes. UsingagainaJacobiidentity,weconcludethat f fe eabcd = f fkeabcd. abe cd − kab cd d H: We assume the I to forman othonormalbasis,suchthat f is totally µ µνλ ∗ antisymmetric. Furthermore we will not keep track of whether an index is up or down, but rather sum over any index appearing more than once. Then we have 1 H = εa1...anf ea4...an, ∗ −6(n 3)! a1a2a3 − with derivative 1 d H = εa1...anf fa4ebca5...an ∗ 12(n 4)! a1a2a3 bc − (3.8) 1 + εa1...anf fa4ebka5...an. 6(n 4)! a1a2a3 bk − The firsttermiseasily seento vanish: b andc onlyrunoverthe valuesofa ,a 1 2 and a , giving contributions of the type 3 εa1...anf f ea1a2a5...an, a1a2a3 a4a1a2 where the two f factors are symmetric in a and a , and thus vanish. Now let 3 4 us consider the second contribution in (3.8). We have 1 d H = εa1...anf fa4 ea3ka5...an ∗ 2(n 4)! a1a2a3 a3k − 1 = εa1...anf fa4ea3ka5...an, 2(n 3)! a1a2µ µk − which vanishes due to the Jacobi identity again. Connections. Due to the identification T (G/K) = G m , a connection ∗ K ∗ × on a homogeneous vector bundle G V can be considered as a map K × :C (G,V)K C (G,V m )K, ∞ ∞ ∗ ∇ → ⊗ satisfyingadditionalproperties. Thesimplestexampleistheso-calledcanonical connection , acting as − ∇ ∇−Xf =XL(f) ∀X ∈m, f ∈C∞(G,V)K. (3.9) Here X is the left-invariantvectorfield onG correspondingto X. In a trivial- L izationofT(G/K)inducedbyalocalmapG/K G,whichallowstopullback → theleft-invariant1-formsonGtolocally-defined1-formsonG/K,itsconnection form is given by Γ− =dρe(Ik)ek, (3.10) with dρ the differential of ρ : K Aut(V) at the identity. As is clear from e → thedefinition,theparallelsectionsof correspondtoconstantfunctions,and − ∇ thus to K-invariant elements of V: 7 Lemma 3.3. Let V carry a K representation, then the parallel sections of G V w.r.t. are in a 1-1 correspondence with K-invariant elements of K − × ∇ V, and by Lemma 3.1 are precisely the G-invariant sections. As the notation suggests, we will identify with the connection appear- − ∇ ing in the gravitino equation ε = 0, and thereby translate the problem of − ∇ solving this differential equation into a representation-theoretical one for the holonomy group K. Besides , we have some further homogeneous connec- − ∇ tions on T(G/K), the Levi-Civita connection of g, and the connection +, ∇ ∇ given by Γ c =Γc + 1Hc , −ab ab 2 ab (3.11) Γ+c =Γc 1Hc . ab ab− 2 ab Explicitly, one finds [30]: Γ= fa ek+ 1faec I eb , kb 2 cb a⊗ Γ =(cid:16)faek I eb , (cid:17)(cid:0) (cid:1) (3.12) − kb a⊗ Γ+ = fkabek(cid:0)+fcabec(cid:1) Ia⊗eb =fµabeµ Ia⊗eb . (cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1) The structure group for these connections is generically SO(m), but has − ∇ structure group K SO(m). Their curvatures, as elements of End(m) Λ2m ∗ ⊂ ⊗ and in coordinates, are R+ = ad(I ) π ad(I )ea eb, (R+)c =2fc fk , − a ◦ k◦ b ∧ dab k[a b]d 1 (3.13) R− =−2fakbadm(Ik)ea∧eb, (R−)cdab =−fakbfkcd, with π : g k the orthogonal projection. For the Bianchi identity we need to k → know tr(R+ R+) and possibly tr(R R ). These are given by − − ∧ ∧ 1 tr (R R )= I ,I fkfl eabcd, m −∧ − 4h k lim ab cd (3.14) 1 tr (R+ R+)= I ,I fkfl eabcd, m ∧ −4h k lik ab cd where we introduced the (negative) Killing form . of the subalgebra k, and k h··i similarly I ,I =tr ad(I ) ad(I ) . k l m m k l h i ◦ Using the result of Lemma 3.2 we conc(cid:0)lude that (cid:1) tr R+ R+ R− R− =dH, (3.15) ∧ − ∧ whichisalmosttheBianch(cid:0)iidentity. Itlookshow(cid:1)everasifweneedtoputα =4 ′ tosolvetheBianchiidentity,butthisisduetoourarbitrarynormalizationofthe metric on G/K. Note that the lhs. of (3.15) is completely scale-independent, whereas the rhs. scales with the same factor as the metric. Therefore the Bianchi identity really fixes the scale in terms of α. ′ 8 Spinors. The spin bundle on a homogeneous manifold is constructed as fol- lows. Ad-invariance of the Killing form implies that k acts orthogonally on m, giving rise to an embedding ad :k so(m), (3.16) m → which can be composed with the spin representation dS : so(m) spin(m), → to give ad := dS ad . We assume that this lifts to a representation of K, m ◦ a sufficient condition for this being that K is simply-connected. Denoting the spinor spface overm by S (also S(m) occasionally),we get an associatedbundle =G S, (3.17) K S × which is the spinor bundle over G/K. The connections we considered before give rise to connections on , and the parallelsections w.r.t. correspondto − S ∇ K-invariantelementsofS. Todeterminewhetherthereexistparallelspinorswe thus need to know whether the trivial representation of K (or k) occurs in the decomposition of the spinor representation S over m into irreducibles, which is a purely algebraic task. Suppose then that ε is parallel w.r.t. , so that also R ε = 0. Then it − − ∇ followsfromthe symmetry propertyRa−bcd =Rc−dab thatR− annihilatesεunder the Clifford action as well, γ(R )ε = 0, which makes R a candidate for F − − solving the gaugino equation γ(F)ε=0. We have seen before that it is also an excellent candidate to solve the Bianchi identity. The following commutation relation between the quantized 3-form and ele- ments of k acting on spinors over m will be useful: Lemma 3.4. For X k we have ∈ [γ(H),ad(X)]=0. (3.18) Proof. A simple calculation in the Clifford algebra shows that f [γ(H),ad(X)]= 3ad(X)a f γbcd, (3.19) − b acd but this is proportionalto γ(ad(X) H), where denotes the actionof so(m) on f · · Λ3m , and we know that H is invariant under this action of k. ∗ Thismeansthatγ(H)leavesthesetofinvariantelementsinS(m)invariant, so if there is only one invariant spinor, γ(H) maps it to a multiple of itself. The dilaton. In the supergravity equations only the differential dφ occurs, and if we impose homogeneity again it gives rise to a K-invariant element of m . Often there do existK-invariantelements in m if the rank of K is smaller ∗ ∗ thanthe rankofG,whichcorrespondtoCartangeneratorsorthogonaltok,but the associated1-formsonG/K arenotexact,andthereforenotsuitableforour purpose. We have to conclude that dφ = 0 is the only admissible solution for the dilaton. On the other hand we have seen that a vanishing dilaton is not compatible with positive scalar curvature, which is why we will have to introduce a linear dilaton on an additional Rp factor of the total manifold to obtain a solution of all the supergravity equations. 9 Symmetric spaces. Suppose G/K is symmetric, meaning that [m,m] k. ⊂ Then H = 0, and from the dilatino equation we also have that dφ = 0. The equation η = 0 then tells us that there is a parallel spinor, implying that M ∇ is Ricci-flat [2], which is impossible for symmetric spaces with G semisimple. Thus there are no solutions for symmetric spaces. On the other hand, it is far from obvious to me why the relation [m,m] k ⊂ implies that the trivial k-representation does not occur in S(m), and a purely Lie algebraic proof would be desirable. 4 Spinors on cosets Representation-theoretic method. Given a coset G/K we need to de- termine whether the spin representation over m = g/k contains the trivial k- representation as an irreducible component. Suppose for the moment that k is simple, and denote the set of weights of m by Ω(m), whereas Ω+(m) contains only the positive ones. Then the weights that appear in S(m) are of the form 1 Ω(S(m))= α , (4.1) 2 ± (cid:26) α∈XΩ+(m) (cid:27) whereallcombinationsofsignsappear(thiscanbeunderstoodfromLemma4.1 below). Nowonlythedominantweightscanbehighestweightsofanirreducible representation of S(m), so it is often enough to determine all the dominant weights in Ω(S(m)). Then a couple of situations can occur. If the zero weight is not in Ω(S(m)), then the trivial representation is not contained. If it is, one cansometimesconcludebydimensionalreasoningthatthetrivialrepresentation mustormustnotoccurasacomponent. Sometimesthesituationisevensimpler: Consider the coset SO(n+1)/SO(n) = Sn. In this case m is simply the fundamental representation of so(n), and it follows that S(m) is the (Dirac) spinor representation,which is often reducible, but does not have invariant ele- ments. Thus Sn with its standardroundmetric does notadmit a homogeneous solution, in accordance with our general result for symmetric spaces. Despite its elegancewewill notemploy the representation-theoreticmethod inthefollowing,butusethemoredown-to-earthapproachexplainedinthenext paragraph. Thereasonforthatisthatthelattermethodallowsustodetermine how the three-formH acts on invariantspinors,and thereby how to choose the dilaton appropriately to solve also the dilatino equation γ(dφ 1 H)ε = 0. A − 12 drawback is that the method is not always applicable, as explained below. Direct method for lower-rank subgroups. Recall that the action of k on m defines an embedding k so(m). Here we give an explicit construction ⊂ of the spinor space for the case that k su(m), for a well-chosen complex ⊂ structure on m. We assume for the time being that rk(k) <rk(g), although for certain maximal rank subgroups we will be able to generalize our construction. A particular example where this is possible is the case where G/K is a six- dimensional nearly K¨ahler manifold. 10