DAMTP-2000-131 CTP TAMU-37/00 UPR-911-T MCTP-00-08 RUNHETC-2000-44 IHP-2000/07 hep-th/0012011 December, 2000 Ricci-flat Metrics, Harmonic Forms and Brane Resolutions M. Cvetiˇc , G.W. Gibbons♯, H. Lu¨⋆ and C.N. Pope † ‡ Department of Physics and Astronomy † University of Pennsylvania, Philadelphia, PA 19104 1 0 ♯DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, 0 2 Cambridge CB3 OWA, UK n a Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855 † J 6 ⋆Department of Physics 1 University of Michigan, Ann Arbor, Michigan 48109 2 v Center for Theoretical Physics 1 ‡ 1 Texas A&M University, College Station, TX 77843 0 2 1 ‡,†Institut Henri Poincar´e 0 0 11 rue Pierre et Marie Curie, F 75231 Paris Cedex 05 / h t ABSTRACT - p e h We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel : v metric, on the tangent bundle of Sn+1. We obtain explicit results for all the metrics, and i X show how they can be obtained from first-order equations derivable from a superpotential. r a Wethenprovideanexplicitconstructionfortheharmonicself-dual(p,q)-formsinthemiddle dimensionp+q = (n+1)fortheStenzelmetricsin2(n+1)dimensions. Onlythe(p,p)-forms are L2-normalisable, while for (p,q)-forms the degree of divergence grows with p q . We | − | alsoconstructasetofRicci-flatmetricswhoselevelsurfacesareU(1)bundlesoveraproduct of N Einstein-Ka¨hler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2- branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions. Contents 1 Introduction 2 2 Stenzel metrics 5 2.1 Geometrical and topological considerations . . . . . . . . . . . . . . . . . . 6 2.2 Detailed calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Covariantly-constant spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Ka¨hler form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Explicit solutions for Ricci-flat Stenzel metrics . . . . . . . . . . . . . . . . 13 3 Harmonic forms 14 3.1 Harmonic (p,q)-forms in 2(p+q) dimensions . . . . . . . . . . . . . . . . . 14 3.2 L2-normalisable harmonic (p,p)-forms in 4p dimensions . . . . . . . . . . . 15 3.3 Non-normalisable harmonic (p,q)-forms . . . . . . . . . . . . . . . . . . . . 16 3.4 Canonical form, and special Lagrangian submanifold . . . . . . . . . . . . . 17 4 Applications: resolved M2-branes and D3-branes 17 4.1 Fractional D3-brane using the 6-dimensional Stenzel metric . . . . . . . . . 18 4.2 Fractional M2-brane using the 8-dimensional Stenzel metric . . . . . . . . . 19 5 Ricci-flat K¨ahler metrics on Ck bundles 22 5.1 Curvature calculations, and superpotential . . . . . . . . . . . . . . . . . . . 24 5.2 Solving the first-order equations. . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 General results for N Einstein-Ka¨hler factors in the base space . . . . . . . 32 6 More fractional D3-branes and deformed M2-branes 35 6.1 The resolved fractional D3-brane . . . . . . . . . . . . . . . . . . . . . . . . 35 6.1.1 Harmonic 3-form on the C2 bundle over CP1 . . . . . . . . . . . . . 35 6.1.2 The issue of supersymmetry in the Pando Zayas-Tseytlin D3-brane . 37 6.2 Harmonic 4-form for C2/ZZ and C2 bundles over CP2, and smooth M2-branes 39 2 6.3 Harmonic 4-form for C2 bundle over CP1 CP1, and smooth M2-brane . . 42 × 6.4 Deformed M2-brane on the complex line bundle over CP3 . . . . . . . . . . 43 6.5 Deformed M2-brane on an 8-manifold of Spin(7) holonomy . . . . . . . . . 44 7 Conclusions and comments on dual field theories 45 1 1 Introduction Fractional D3-branes have been extensively studied recently, since they can provide super- gravity solutions that are dual to four-dimensional N = 1 super-Yang-Mills theories in the infra-red regime [1, 2, 3, 4, 5, 6, 7, 8]. The idea is that by turning on fluxes for the R-R and NS-NS 3-form fields of the type IIB supergravity, in addition to the usual flux for the self- dual 5-form that supports the ordinary D3-brane, a deformed solution can be found that is freeoftheusualsmall-distancesingularbehaviourontheD3-branehorizon. Thisisachieved byfirstreplacingtheusualflat6-metric transversetotheD3-branebyanon-compact Ricci- flat Ka¨hler metric. It can then be shown that if there exists a suitable harmonic 3-form G satisfying a complex self-duality condition, then the type IIB equations of motion are (3) satisfied if the R-R and NS-NS fields are set equal to the real and imaginary parts of the harmonic 3-form, with the usual harmonic function H of the D3-brane solution now satis- fying the modified equation H = 1 m2 G 2 in the transverse space. A key feature of −12 | (3)| the type IIB equations that allows such a solution to arise is that there is a Chern-Simons or “transgression” modification in the Bianchi identity for the self-dual 5-form, bilinear in the R-R and NS-NS 3-forms. The construction can be extended to encompass other examples of p-brane solutions, and in [6] a variety of such cases were analysed. These included heterotic 5-branes, dyonic strings, M2-branes, D2-branes, D4-branes and type IIA and type IIB strings. The case of M2-branes was also discussed in [9]. In all these cases, the ability to construct deformed solutions depends again upon the existence of certain Chern-Simons or transgression terms in Bianchi identities or equations of motion. The additional field strength contribution that modifiesthestandardp-braneconfiguration thencomes froman appropriateharmonicform inthetransversespace. Oneagain replaces theusualflattransversespacebyamoregeneral complete non-compact Ricci-flat manifold. In order to get deformed solutions that are still supersymmetric, a necessary condition on this manifold is that it must have an appropriate special holonomy that admits the existence of covariantly-constant spinors. One can easily establish that if the harmonic form is L2-normalisable, then it is possible to choose integration constants in such a way that the deformed solution is completely non- singular[6]. Inparticular,itcanbearrangedthatthehorizoniscompletely eliminated, with the metric instead smoothly approaching a regular “endpoint” at small radial distances. At large distances, the metric then has the same type of asymptotic structure as in the undeformed case, with a well-defined ADM mass per unit spatial world-volume. If, on the other hand, the harmonic form in the transverse manifold is not L2-normalisable, then the 2 deformed solution will suffer from some kind of pathology. Usually, one chooses a harmonic form that is at least square-integrable in the small-radius regime, and this can be sufficient to allow a solution which gives a useful infra-red description of the dual super-Yang-Mills theory. If the harmonic form fails to be square-integrable at large radius, then this will lead to some degree of pathology in the asymptotic structure of the deformed solution in that region. For example, the deformed KS D3-brane solution [2] is based on a non-normalisable harmonic 3-form in the six-dimensional Ricci-flat Ka¨hler transverse space, for which the integral of G 2 diverges as the logarithm of the proper distance at large radius. This | (3)| leads to a deformed D3-brane metric that is complete and everywhere non-singular, and for which the harmonic function H has the asymptotic structure Q+m2 logρ H c + (1.1) ∼ 0 ρ4 at large proper distance ρ. Although the metric is still asymptotic to dxµdx +ds2, where µ c ds2 is the metric on the six-dimensional Ricci-flat conifold, the effect of the deformation c involving the logarithm is that the associated ADM mass per unit 3-volume is no longer well-defined. This is because the effect of the logρ term in H is to cause a slower fall- off at infinity than the normal ρ 4 dependence that picks up a finite and non-zero ADM − contribution.1 This change in the asymptotic structure implies that the solution may not admit an AdS region, even when the constant c in (1.1) goes to zero in a decoupling limit. 5 0 Of course this feature is itself of great interest, since it is associated with a breaking of conformal symmetry in the dual field theory picture. One might wonder whether there could be some other Ricci-flat Ka¨hler 6-manifold for which an L2-normalisable harmonic 3-form might exist. In fact rather general arguments establish that this is not possible, at least for the case where the 6-metric is asymptotically of the form of a cone, and the middle homology is one-dimensional.2 On the other hand, L2-normalisable harmonic forms can exist in non-compact Ricci-flat manifolds in other dimensions, and indeed some examples of fully resolved p-brane solutions based on such harmonic forms were obtained in [6]. We shall obtain further examples in this paper, using Ricci-flat Ka¨hler 8-manifolds to obtain smooth deformed M2-branes. Since the ADM mass is then well-defined, the asymptotic structure correspondingly may still allow an approach to AdS, if the constant term in the metric function H goes to zero, implying that the dual 1Forpracticalpurposes,theADMmassmeasuredrelativetothefiducialmetricdxµdxµ+ds2c isacertain constant times the limit of ρ5∂H/∂ρ as ρ goes to infinity. 2Weare grateful to Nigel Hitchin for extensivediscussions on this point. 3 field theory will still be a conformal one (three-dimensional in the case of M2-branes). In this paper, we explore some of these questions in greater detail. To begin, in section 2, we study the class of complete non-compact Ricci-flat Ka¨hler manifolds whose metrics were constructed by Stenzel [10]. These are asymptotically conical, with level surfaces that aredescribedbythecosetspaceSO(n+2)/SO(n), andtheyhaverealdimensiond = 2n+2. The n = 1 example is the Eguchi-Hanson instanton [11], and the n = 2 example is the six- dimensional “deformed conifold” found by Candelas and de la Ossa [13]. It is this example that is used in the fractional D3-brane KS solution in [2]. In section 2.1 we describe the geometry and topology of the general Stenzel manifolds, and then in section 2.2 we carry outdetailed calculations ofthecurvature,andshowhowRicci-flatsolutionscanbeobtained from a system of first-order equations derivable from a superpotential. In subsequent sub- sections we then obtain the explicit Ricci-flat Stenzel metrics and their Ka¨hler forms, and then we derive integrability conditions for the covariantly-constant spinors. In section 3 we obtain explicit results for harmonic forms in the middle dimension, that is to say, for harmonic (n+1)-forms in the 2(n+1)-dimensional Stenzel metrics.3 More precisely, we construct harmonic(p,q)-forms for all integers p and q satisfying p+q = n+1, where p and q count the number of holomorphic and antiholomorphic indices. We show that these are L2-normalisable if and only if p = q, which can, of course, occur only in dimensions d = 4p. Insection 4, wemakeuseof someoftheseresultsinordertoconstructdeformedp-brane solutions. Specifically, we firstreview the fractional D3-brane solution of [2]. Ourresults on harmonicformsallowustogiveanexplicitproofthattheirsolutionhasaharmonic3-formof type (2,1), which therefore ensures supersymmetry. We then construct a smooth deformed M2-brane, using the L2-normalisable (2,2)-form in the 8-dimensional Stenzel metric. This is also supersymmetric. In section 5 we construct another class of complete non-compact Ricci-flat Ka¨hler man- ifolds. These are again of the form of resolved cones, but in this case the level surfaces are themselves U(1) bundles over the product of N Einstein-Ka¨hler manifolds. Typical examples would be to take the base space to be = N CPmi, for an arbitrary set of M i=1 Q integers m . Infacttherequirements ofregularity of themetricmeanthatoneofthefactors i in the base space must be a complex projective space, but the others might be other M Einstein-Ka¨hler manifolds. Topologically, the total space is a Ck bundle over the remaining 3Nigel Hitchin has informed us that Daryl Noyce has independently constructed the unique harmonic form in the middledimension in the4N-dimensional Stenzelmanifolds. 4 Einstein-Ka¨hler factors.4 Having obtained generalresults forRicci-flat Ka¨hlermetrics inall thecases, wepresentsomemoredetailedexplicitformulaeforthree8-dimensionalexamples, corresponding to taking the base space to be S2 CP2, CP2 S2 and S2 S2 S2. We × × × × also discuss some well-known examples corresponding to complex line bundles over CPm. In section 6 we make use of our results for these Ricci-flat metrics, to obtain further ex- amplesofdeformedp-branesolutions. Webeginbyconsideringthecasewherethebasespace is = S2 S2 (i.e. m = m = 1), meaning that the level surfaces are the 5-dimensional 1 2 M × space known as T1,1 or Q(1,1), which is a U(1) bundle over S2 S2. Topologically, the × 6-dimensional manifold is a C2 bundle over CP1. Its Ricci-flat metric is present in [13], and it was discussed recently in [5], where it was used to provide an alternative resolution of the D3-brane. We construct the self-dual harmonic 3-form that was used in [5] in a complex basis, and by this means demonstrate that it contains both (2,1) and (1,2) pieces. This implies that the resolved D3-brane solution of [5] is not supersymmetric [6]. We also construct L2-normalisable harmonic 4-forms of type (2,2) in the 8-dimensional examples based on S2 CP2 and S2 S2 S2, and then use these in order to construct additional × × × deformed M2-branes, which are supersymmetric. A further smooth deformed M2-brane ex- ample, whichisnon-supersymmetric,resultsfromtakingthe8-dimensionaltransversespace to be the complex line bundle over CP3. We also include a discussion of a fifth completely smooth deformed M2-brane, which was obtained previously in [6]. This solution uses an 8-manifold of exceptional Spin(7) holonomy rather than a Ricci-flat Ka¨hler manifold. We give a simple proof of its supersymmetry. The paper ends with conclusions and discussions in section 7. 2 Stenzel metrics In this section we shall construct a sequence of complete non-singular Ricci-flat Ka¨hler metrics, oneforeachevendimension,ontheco-tangent bundleofthe(n+1)sphereT⋆Sn+1. Restricted to the base space Sn+1, the metric coincides with the standard round sphere metric. The sequence, which begins with the Eguchi-Hanson metric for n = 1, was first constructed in generality by Stenzel [10] following a method discussed in [14]. The case n = 2 was originally given, in rather different guise, by Candelas and de la Ossa [13] as a “deformation” of the conifold. The isometry group of these metrics is SO(n+2), acting in the obvious way on T⋆Sn+1. The principal (i.e. generic) orbits are of co-dimension one, 4Therearecertaintopological restrictionsonthepossible choicesfortheotherEinstein-K¨ahlerfactorsin the base space. For a detailed discussion, see [12]. 5 correspondingto the coset SO(n+2)/SO(n). Thereis a degenerate orbit (i.e. a generalized “bolt”)correspondingtothezerosection,i.e.tothebasespaceSn+1 SO(n+2)/SO(n+1). ≡ It is therefore possible to obtain the ordinary differential equations satisfied by the metric functions using coset techniques, and this we shall do shortly. Before doing so, however, we wish to make some comments about the geometry and topology of the metrics, which are intended to illuminate the subsequent calculations. 2.1 Geometrical and topological considerations AnyKa¨hlermetricisnecessarilysymplectic,andinthepresentcasethesymplecticstructure coincides with the standard symplectic structure on T⋆Sn+1. The sphere Sn+1 is thus automatically a Lagrangian sub-manifold. In other words the Ka¨hler form restricted to the (n + 1)-sphere vanishes. The complex structure on T⋆Sn+1 is however non-obvious, and arises from the fact that we may view T⋆Sn+1 as a complex quadric in Cn+2, zaza = a2, (2.1) where a =1,2,...,n+2. Setting sinh( pbpb) za = cosh( pbpb)xa+i p , (2.2) p a p √pbpb one obtains xbxb = a2 and p xb = 0. These are the equations defining a point xb lying on b an (n+1)-sphere of radius a in En+1, and a cotangent vector p . Note that as the radius a b is sent to zero we obtain the conifold, which makes contact with the work of Candelas and de la Ossa [13]. The strategy of Stenzel [10] is now to assume that the Ka¨hler potential K depends only on the quantity τ =z¯aza = cosh(2√p p ). (2.3) b b Fromthisitisclearthattheprincipalorbitsoftheisometrygroupcorrespondtothesurfaces of constant energy H = 1p p on the phase space T⋆Sn+1. The stabliser of each point on 2 b b the orbit consists of rotations leaving fixed a point on Sn+1 and a tangent vector p . The b transitivity of the action is equally obvious. Thus √p p , or some function of it, it will b b serve as a radial variable. InfactthelevelsetsH = constantcanbeviewedascirclebundlesovertheGrassmannian SO(n + 2)/(SO(n) SO(2)). To see why, recall that the Hamiltonian H generates the × geodesic flow on T⋆Sn+1. Each such geodesic is a great circle consisting of the intersection of a two-plane through the origin of En+2 with the (n + 1)-sphere. The circle factor in 6 the denominator of the coset corresponds to the fact that geodesics or great circles are the orbits of a circle subgroup of the isometry group SO(n+2) of the (n+1)-sphere. ThusthecirclefibreofthecirclebundleisanorbitoftheisometrygroupoftheRicci-flat Ka¨hler metric. In terms of Ka¨hler geometry, the quotient of T⋆Sn+1 by the circle action corresponds to the Marsden-Weinstein or symplectic quotient, and gives at each radius a homogeneous Ka¨hler metric of two less dimensions. At large distances the Stenzel metric tends to a Ricci-flat cone over the Einstein-Sasaski manifold SO(n + 2)/SO(n). At small radius the orbits collapse to the zero-section of T⋆Sn+1. Thus it is clear that the (n + 1)-sphere Σ H (T⋆Sn+1) provides the only n+1 ∈ interesting homology cycle, and it is in the middle dimension. In the case that n is odd, its self-intersection number Σ Σ Z is, dependinguponorientation convention, 2, while if n is · ∈ evenitsself-intersection numbervanishes. ThisisequivalenttothestatementthattheEuler characteristic of the even-dimensional spheres is 2, while for the odd-dimensional spheres it vanishes. To see this equivalence, recall that the topology of the co-tangent bundle is the same as that of the tangent bundle. Now the Euler characteristic of any closed orientable manifold is given by the number of intersections, suitably counted, of the zero section with any other section of its tangent bundle. In other words it is the number of zeros, suitably counted, of a vector field on the manifold. We shall see that these facts have consequences for the cohomology. In the case of a closed(2n+2)-manifold (i.e. compact, withoutboundary),onemayusePoincar´e duality M to see that if α and β are closed middle-dimensional (n+1)-forms representing elements of Hn+1( ), then the cup product α β is an integer-valued bilinear form on Hn+1( ) M ∪ M given by α β. (2.4) Z ∧ M The cup productis symmetric or skew-symmetric dependinguponwhether n is odd or even respectively. Thus if n is even, α α =0. (2.5) Z ∧ M Moreover, the Hodge duality operator ⋆ acts on Hn+1( ), and M ⋆⋆= ( 1)n+1. (2.6) − Thus if n is odd, Hn+1( ) decomposes into real self-dual or anti-self dual (n+1) forms. M Any such closed form mustnecessarily beharmonic, andits L2 normwill beproportionalto the self-intersection number. The total number of linearly-independent harmonic middle- dimensional forms will depend only on the topology of the closed manifold . M 7 If n is even, we can find a complex basis of self-dual harmonic forms in L2, but there is no relation between their normalisability and the integral in (2.4). Our manifolds are non-compact, and the situation is therefore more complicated and we must proceed with caution. The usual one-to-one correspondence between harmonic forms and geometric cycles may break down. One generally expects at least as many L2 harmonic forms as topology requires, but there may be more (c.f. [15]). It is still true that L2 harmonic forms must be closed and co-closed [16]. However, the notion of exactness must be modified since we are interested in whether closed forms in L2 are the exterior derivatives of forms of one lower degree which are also in L2. For example, the Taub-NUT metricadmitsanexactharmonictwo-form inL2,butitistheexterior derivativeofaKilling 1-form which is not in L2. In the present case, if n is odd it seems reasonable to expect at least one harmonic form in the middle dimension, which is Poincar´e dual to the (n+1)-sphere. Because the Stenzel metric behaves like a cone near infinity, all the Killing vectors are of linear growth. It follows [17] that any harmonic form must be invariant under the action of the isometry group. In the case of the Taub-NUT and Schwarzschild metrics, this observation permits the complete determination of the L2 cohomology [17, 18]. We shall obtain an L2 harmonic form in the middle dimension for all the Stenzel manifolds with odd n. We obtain a general explicit construction of harmonic (p,q)-forms in all the Stenzel manifolds, where p + q = n + 1. These middle-dimension harmonic forms include (p,p) forms when n is odd, and these are the L2-normalisable examples mentioned above. All the others are non-normalisable, with a “degree of non-normalisability” that increases with p q at fixed p+q. In particular, this accords with the expectation that if n is even we | − | should not find any harmonic form in L2.5 2.2 Detailed calculations Let L be the left-invariant 1-forms on the group manifold SO(n+2). These satisfy AB dL = L L . (2.7) AB AC CB ∧ 5NigelHitchinandTamasHauselhavebothpointedouttousthatresultsofAtiyah,PatodiandSingeron asyptotically cylindricalmanifolds[19]andsomepropetiesofK¨ahlermanifoldsusedin[17]canbeextended to asymptotically conical metrics, and they imply that the L2 cohomology is toplogical, i.e. isomorphic to the compactly-supported cohomolgy in ordinary cohomolgy. The results reported here are consistent with those theorems. We thankthem for helpful communications. 8 We consider the SO(n) subgroup, by splitting the index as A = (1,2,i). The L are the ij left-invariant 1-forms for the SO(n) subgroup. We make the following definitions: σ L , σ˜ L , ν L . (2.8) i 1i i 2i 12 ≡ ≡ ≡ These are the 1-forms in the coset SO(n+2)/SO(n). We have dσ = ν σ˜ +L σ , dσ˜ = ν σ +L σ˜ , dν = σ σ˜ , i i ij j i i ij j i i ∧ ∧ − ∧ ∧ − ∧ dL = L L σ σ σ˜ σ˜ . (2.9) ij ik kj i j i j ∧ − ∧ − ∧ Note that the 1-forms L lie outside the coset, and so one finds that they do not appear ij eventually in the expressions for the curvature (see also [20]). We now consider the metric ds2 = dt2+a2σ2+b2σ˜2+c2ν2, (2.10) i i where a, b and c are functions of the radial coordinate t, and then we define the vielbeins e0 = dt, ei = aσ , e˜i = bσ˜ , e˜0 = cν. (2.11) i i Calculating the spin connection, we find a˙ b˙ c˙ ω = ei, ω = e˜i, ω = e˜0, 0i −a 0˜i −b 0˜0 −c ω = Be˜i, ω = Aei, ω = Cδ e˜0, ˜0i ˜0˜i − i˜j ij ω = L , ω = L , (2.12) ij − ij ˜i˜j − ij where a dot means d/dt, and (a2 b2 c2) (b2 c2 a2) (c2 a2 b2) A= − − , B = − − , C = − − . (2.13) 2abc 2abc 2abc From this, we obtain the curvature 2-forms a¨ a˙ Cb˙ Bc˙ Θ = e0 ei + + e˜0 e˜i, 0i −a ∧ −(cid:16)bc b c (cid:17) ∧ ¨b b˙ Ca˙ Ac˙ Θ = e0 e˜i+ + + e˜0 ei, 0˜i −b ∧ (cid:16)ac a c (cid:17) ∧ c¨ c˙ Ba˙ Ab˙ Θ = e0 e˜0+ + + ei e˜i, 0˜0 −c ∧ (cid:16)ab a b (cid:17) ∧ 1 a˙2 1 Θ = ei ej + B2 e˜i e˜j, ij (cid:16)a2 − a2(cid:17) ∧ (cid:16)b2 − (cid:17) ∧ 1 b˙2 1 Θ = e˜i e˜j + A2 ei ej, ˜i˜j (cid:16)b2 − b2(cid:17) ∧ (cid:16)a2 − (cid:17) ∧ 9