Generalized structures of ten-dimensional supersymmetric solutions
aa r X i v : . [ h e p - t h ] N ov Generalized structures of ten-dimensionalsupersymmetric solutions
Alessandro TomasielloDipartimento di Fisica, Universit`a di Milano-Bicocca, I-20126 Milano, ItalyandINFN, sezione di Milano-Bicocca, I-20126 Milano, Italy
Abstract
Four-dimensional supersymmetric type II string theory vacua can be described ele-gantly in terms of pure spinors on the generalized tangent bundle T ⊕ T ∗ . In this paper,we apply the same techniques to any ten-dimensional supersymmetric solution (not neces-sarily involving a factor with an AdS or Minkowski metric) in type II theories. We finda system of differential equations in terms of a form describing a “generalized ISpin(7)structure”. This system is equivalent to unbroken supersymmetry, in both IIA and IIB.One of the equations reproduces in one fell swoop all the pure spinors equations for four-dimensional vacua. ontents ǫ . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Structure group from gamma matrices . . . . . . . . . . . . . . . . 62.1.3 Structure group from forms . . . . . . . . . . . . . . . . . . . . . . 82.2 Geometry defined by two spinors . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Structure groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Spinor bilinears versus metric . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Adding vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Summary of this section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.1 Structure of four-dimensional spinors . . . . . . . . . . . . . . . . . 254.1.2 Reproducing the pure spinor equations for four-dimensional vacua . 264.2 Minkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A Bispinors 30
A.1 The λ operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 Hodge star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 B Equivalence of (3.1) to supersymmetry 32
B.1 Deriving (3.1a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33B.2 Spinor basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3 Original supersymmetry equations in terms of intrinsic torsion . . . . . . . 35B.4 (3.1a), (3.1b) in terms of intrinsic torsion . . . . . . . . . . . . . . . . . . . 36B.5 The missing equations: (3.1c), (3.1d) . . . . . . . . . . . . . . . . . . . . . 37B.6 The missing equations and intrinsic torsion . . . . . . . . . . . . . . . . . . 39
Differential forms are in many respects easier to deal with than symmetric tensors. Grav-ity is usually described in terms of a symmetric tensor g MN . It is an old idea that itmight be understood better if written in terms of forms. Initially this was done in thehope that it might help with quantization; in four dimensions, one can express ordinarygeneral relativity in terms of self-dual two-forms [1–4]. More recently, it proved usefulin finding classical solutions to various supergravity theories. In this case, the forms areadditional data defined by the fermionic supersymmetry parameters of the supergravitytheory. Mathematically, they define a reduction of the structure group of the tangent bun-dle T to a certain group G , which is nothing but their little group (or stabilizer). Theseso-called G -structures have been used to reformulate the supersymmetry conditions moreefficiently, starting from [5, 6].In eleven-dimensional supergravity, there is only one supersymmetry parameter ǫ ,which can define two possible structure groups [5, 7]. In type II theories, each of the twosupersymmetry parameters ǫ , has its own stabilizer, and these can intersect in variousways. This gives rise to a variety of G -structures [8–11]. Moreover, for a supersymmetricsolution the stabilizer of ǫ , need not be the same everywhere: it can jump to a highergroup on some locus of spacetime. Because of all this, a complete classification quicklybecomes complicated.A possible reaction to this is to work on T ⊕ T ∗ , the direct sum of the tangent andcotangent bundles. This approach has been useful for four-dimensional “vacua” (namely,solutions of the form Minkowski × M or AdS × M , with M an arbitrary manifold).Here, the stabilizer in T of the two spinors ǫ , can be SU(2), SU(3), or it can be genericallySU(2) and jump to SU(3) on some loci. On T ⊕ T ∗ , however, the stabilizer is alwaysSU(3) × SU(3). This “generalized structure” allows then a unified treatment. It canbe described in an alternative way by two differential forms φ ± , sometimes called “purespinors”. The conditions for unbroken supersymmetry can then be summarized elegantlyin terms of the φ ± [12,13]; the system consists of three equations (see (4.16)). Interestingly,2f these, (4.16b) had been studied by mathematicians before the physical applicationbecame clear [14, 15].This success suggests one should be able to apply similar techniques to other typesof supersymmetric solutions, beyond four-dimensional vacua. There are several instanceswhere this has been attempted. For example, one can look at other dimensions: solutionsof the form Minkowski d × M − d or AdS d × M − d . A complete reformulation was achievedfor d = 3 in [16, 17], and for d = 6 in [18]; a set of necessary conditions was found for d = 1 in [19] and for all even d in [18]. Or, staying in four dimension, one can look forsolutions where the spacetime geometry is no longer Minkowski or AdS . For example,a particularly natural geometry to consider is that of a spherically symmetric black hole.This was considered in [20], with some Ansatz along the way.There are of course other classes of interesting supersymmetric solutions. It wouldbe interesting, for example, to have a classification of supersymmetric Lifschitz solutions(which are interesting as holographic duals to scale-invariant non-relativistic field theo-ries [21]), or to generalize the results of [20] to multi-center or asymptotically AdS blackholes.At present, however, every time one is interested in a new class of solutions one hasto start from scratch; one first needs to derive some differential equations on the relevantdifferential forms, then — much more painfully — one needs to prove that these equationsare equivalent to the conditions on ǫ , for preserved supersymmetry. It would be nice tohave a result which applies to any type of supersymmetric solution; this would combine theadvantages of the G -structure papers [5, 7–11], which make no Ansatz on the metric, withthe advantages of the generalized geometry approach [12, 13], which unifies the variouspossibilities for the stabilizers.In this paper, we find such a result. For type II supergravity, we give a systemof differential equations ((3.1) below) which is equivalent to supersymmetry, without anyAnsatz on the ten-dimensional metric or on any of the other field. The system is essentiallyidentical in IIA and IIB.The geometrical data appearing in the system are a single differential form Φ (evenin IIA, odd in IIB, but otherwise of mixed degree) and two sections e + · , · e + of T ⊕ T ∗ (our notation will be explained in section 2). The form Φ is not a pure spinor. Forfour-dimensional vacua, however, it does reduce to a certain sum of the pure spinors φ ± mentioned earlier (see (4.11) below). On T ⊕ T ∗ , Φ defines a complicated structuregroup, (2.35); this contains two copies of the “inhomogeneous Spin(7)” group, ISpin(7) ≡ Spin(7) ⋉ R , which is the structure group defined by a spinor in ten dimensions (see3or example [22]). Since the group in (2.35) is not a subgroup of Spin(9 , × Spin(9 , T ⊕ T ∗ .Among the differential equations, (3.1a) is particularly nice. When specialized tofour-dimensional vacuum solutions, it is easy to see that it implies all the pure spinorequations of [12] in one go. This equation is very similar to [19, Eq. (A.27)]; perhapsnot surprisingly, since their setup ( R × M ) is already very general. Deriving (3.1a) fromsupersymmetry is in fact even easier (see appendix B.1) than deriving the pure spinorequations for four-dimensional vacua (see [13, App. A]). A far harder task, however, isestablishing whether it is equivalent to, and not only implied by, supersymmetry. It is not;this is why we had to supplement (3.1a) by (3.1b), (3.1c), (3.1d). The latter two equationswere particularly hard to find, basically because the two sections e + · and · e + of T ⊕ T ∗ are not defined directly by the spinors ǫ i . It would be interesting to find alternative setsof equations complementary to (3.1a). The formalism set up in this paper will hopefullybe helpful in doing that. In section 2 we will look at the structure groups defined on T and T ⊕ T ∗ by the twosupersymmetry parameters ǫ , , and isolate the geometrical objects (Φ , e + · , · e + ) that willsummarize for us the data of the metric, of the B field and of the ǫ , . This long sectionis summarized in section 2.5. In section 3, we will describe the system (3.1) of differentialequations which reformulates the requirement of unbroken supersymmetry in terms of the(Φ , e + · , · e + ); its derivation is hidden in section B. In section 4 we will show how the sys-tem (3.1) reproduces earlier results about four- and three-dimensional Minkowski vacuumsolutions. In both cases, equation (3.1a) reproduces all the “pure spinor” equations. Theadditional equations (3.1c), (3.1d) are redundant for four-dimensional vacua, while forthree-dimensional vacua they reproduce a peculiar algebraic constraint that was foundin [16, 17]. In this section, we will describe how to encode the data of the metric, of the B field and ofthe supersymmetry parameters in a set of differential forms. A summary of these resultscan be found in section 2.5. These forms are the ones on which we will impose differentialequations in section 3. Some useful hints might also come from efforts towards reformulating type II supergravity usinggeneralized geometry; for a recent example, see [23]. .1 Geometry defined by one spinor Recall that the parameters for the supersymmetry transformations of type II supergravityare two ten-dimensional Majorana–Weyl spinors ǫ , ǫ . In type IIA ǫ has positive chiralityand ǫ has negative chirality. In type IIB both ǫ , have positive chirality.In this subsection, we will consider the geometry defined by one Majorana–Weyl spinor ǫ . We will work in a basis where all the γ M are real; the Majorana condition thensimply means that ǫ is real. In frame indices, γ is antisymmetric, whereas γ , . . . , γ aresymmetric. This can be summarized by saying γ tM = γ γ M γ ; (2.1)in other words, γ is the intertwiner between the representations { γ M } and { γ tM } of theClifford algebra. ǫ We will start with some preliminaries on how spinors are related to differential formsin ten dimensions. As usual, to a differential form we can associate a bispinor via theClifford map: C k ≡ k ! C M ...M k dx M ∧ . . . ∧ dx M k −→ (cid:0) C k ≡ k ! C M ...M k γ M ...M k . (2.2)Many formulas about bispinors are usefully summarized by this notation. For example,we will need in what follows: γ (cid:0) C k = ✭✭✭✭ ∗ λ ( C k ) , (2.3) γ M (cid:0) C k γ M = ( − ) k (10 − k ) (cid:0) C k (2.4)where γ = γ γ . . . γ is the chiral operator, and λ ( C k ) ≡ ( − ⌊ k ⌋ C k . The generalizationsof (2.3), (2.4) to any dimension (which we will need in section 4) can be found in (A.9) and(A.14) in appendix A. From now on, we will drop the slash, and freely confuse differentialforms with the associated bispinors. It is also useful to recall how wedges and contractionsare related to Clifford products: γ M C k = ( dx M ∧ + ι M ) C k , C k γ M = ( − ) k ( dx M ∧ − ι M ) C k (2.5)where ι M ≡ g MN ι N ≡ g MN ι ∂/∂x N . We will also sometimes use the notation → γ M = ( dx M + ι M ) , ← γ M = ( dx M − ι M )( − ) deg , (2.6)5here ( − ) deg C k ≡ ( − ) k C k , on a k -form C k .Consider now a Majorana–Weyl ǫ of chirality ±
1. We can build from it the bispinor ǫ ⊗ ǫ , where ǫ ≡ ǫ t γ . (2.7)The Fierz identities (see (A.12)) allow us to rewrite it as a sum of differential forms: ǫ ⊗ ǫ = X k k ! ( ǫ γ M k ...M ǫ ) γ M ...M k . (2.8)Actually, most of the bilinears ǫγ M ...M k ǫ vanish identically. First of all, γ γ M ...M k is anantisymmetric matrix for k = 0 , , , ,
8, so the corresponding bilinears vanish. Secondly,since ǫ has chirality opposite to ǫ , the bilinears vanish when k is even. That leaves uswith three cases: k = 1 , ,
9. The case k = 9 is ∗ -dual to the case k = 1, thanks to (2.3) ;so the independent bilinears are K M ≡ ǫγ M ǫ , Ω M ...M ≡ ǫγ M ...M ǫ , (2.9)and (2.8) reads ǫ ⊗ ǫ = K + Ω ± ∗ K = K (1 ∓ γ ) + Ω . (2.10)We can now compute, using (2.4), γ M ǫ ⊗ ǫ γ M = − K (1 ± γ ) ; (2.11)it follows that Kǫ = K M γ M ǫ = 132 γ M ǫ ǫγ M ǫ = − K (1 ± γ ) ǫ = − Kǫ ⇒ Kǫ = 0 . (2.12)(Recall that ǫ has chirality ±
1, as declared before (2.7).) Hitting this result from the leftby ǫ , we get K M K M = 0 . (2.13)So K M is a null vector. We will now determine the structure group defined by ǫ , which is the stabilizer (or isotropygroup, or little group) for the Spin(9 ,
1) action on it. We will show that the structure Notice also that ∗ Ω = ± Ω . ⋉ R , and that the orbit of the action is 16-dimensional. (A similarcomputation can be found in [24], and more explicitly in [22].)Given a ten-dimensional Majorana–Weyl spinor ǫ , we have seen in section 2 that thebilinear K M in (2.9) is a null vector. We will assume in what follows that ǫ has chirality+; the discussion is virtually the same for chirality − .We will choose a frame in which this vector is the vielbein e − : K = e − . (2.14)Since K is null, there are eight more vectors which are orthogonal to it; choose a basis e α for them, α = 1 , . . . ,
8. Finally, we have to pick one more direction, e + , which is not orthogonal to K ; we will take e ± · e ± = 0 , e − · e + = 12 , e ± · e α = 0 . (2.15)The gamma matrices in this frame are γ − = K · , γ + , γ α . So for example (2.12) reads γ + ǫ = 0 = γ − ǫ . (2.16)Our decomposition of indices suggests to pick a basis for these gamma matrices where γ ± = γ ± (2) ⊗ , γ α = ( γ + − (2) ) ⊗ γ α (8) , (2.17)where { γ ± (2) } and { γ α (8) } are bases for the two- and eight-dimensional Clifford algebrasrespectively. In fact, we will take γ (2) − = (cid:0) (cid:1) , γ (2) + = (cid:0) (cid:1) , so that, from K · ǫ = γ − ǫ = 0,it follows that ǫ = | ↑ i ⊗ η + , | ↑ i ≡ ! , (2.18)where η + is an eight-dimensional Majorana–Weyl spinor.We know from (2.16) that one gamma matrix annihilates ǫ ; a priori there could bemore. In the basis we have chosen, this could come from an eight-dimensional gammamatrix: γ α (8) η + ? = 0. But a Majorana–Weyl spinor in eight dimensions is not annihilatedby any linear combination of gamma matricess. So ǫ is not annihilated by any vector otherthan K = γ − . This makes it very different from a pure spinor, which would be annihilatedby five gamma matrices. A consequence of this will be that our “generalized” treatmentof the ten-dimensional supersymmetry conditions will not deal with pure spinors, unlikethe treatment of flux compactifications in [12, 13].We can now look at the infinitesimal action of a Lorentz transformation on ǫ : δǫ = ω AB γ AB ǫ . (2.19)7e have to ask which products of two gamma matrices annihilate ǫ . One obvious suchproduct is γ + γ α = γ + α , since already γ + annihilates ǫ . This time, however, we also finda contribution from the eight dimensional gamma matrices, since a Majorana–Weyl η + is annihilated by 21 out of 28 of the γ αβ (8) . Group theoretically, the representation ofSO(8) decomposes as ⊕ under its subgroup Spin(7). So we can write:stab( ǫ ) = span { ω αβ γ αβ , γ + α } , (2.20)where ω αβ is any two-form in the of Spin(7). Notice that this is the adjoint; so the ω αβ γ αβ generate the Lie algebra of Spin(7). Moreover, their commutation relations withthe γ + α are those of the semi-direct groupISpin(7) ≡ Spin(7) ⋉ R ; (2.21)we have introduced the notation ISpin, for “inhomogeneous Spin”, in analogy to thenotation ISO( d ) for inhomogeneous SO( d ) groups.We can now also look at the orbit of the Lorentz group action, which is given by allspinors that can be written as γ AB ǫ . We can already determine the dimension of thisspace as the dimension of Spin(9 ,
1) minus the dimension of the isotropy group (2.21):this gives 45-29=16. So we expect the orbit to be 16-dimensional. Let us see this moreexplicitly. The only subtle components are the purely eight-dimensional ones, γ αβ η + . Wehave just seen that the contribution from the vanishes; so only the contribution fromthe is non-zero: ω αβ γ αβ ǫ = | ↑ i ⊗ (Π αβ γδ ) γ γδ (8) η + , γ − α ǫ = 2 | ↓ i ⊗ γ α (8) η + , γ + − ǫ = 2 ǫ . (2.22)(We have used our normalization g + − = 2.) So we have 7 + 8 + 1 = 16 non-zero elementsthat can be obtained from the Lorentz infinitesimal action. This confirms that the orbitof ǫ is 16-dimensional.It can in fact be shown [24] that the action of the Lorentz group on the space Σ ± of Majorana–Weyl spinors of either chirality has only two orbits: the zero spinor, andeverything else. In other words, any two non-vanishing spinors of the same chirality canbe mapped to one another by a Lorentz transformation. All non-vanishing Weyl spinorshave then the same stabilizer. The structure group Spin(7) ⋉ R can also be understood from the point of view of theforms K , Ω defined in (2.9). Namely, one can find it as the stabilizer of these two formsfor the action of the Lorentz group SO(9 , K . Since K is null:stab( K ) = ISO(8) = SO(8) ⋉ R . (2.23)This is just the generalization of the familiar little group ISO(2) of a null vector in fourdimensions (see for example [25, Ch. 2.5]); in that case, the quantum number of the SO(2)part of ISO(2) is helicity.We now have to ask which subgroup of ISO(8) keeps also Ω invariant. Notice that(2.12) implies K ( ǫ ⊗ ǫ ) = 0 = ( ǫ ⊗ ǫ ) K . Using (2.5), this implies K ∧ ( ǫ ⊗ ǫ ) = ι K ( ǫ ⊗ ǫ ) = 0.Recalling (2.10), we get: K ∧ Ω = ι K Ω = 0 . (2.24)This implies Ω = K ∧ Ψ (2.25)for some four-form Ψ . The form Ψ can also be understood as follows: consider thenine-dimensional space K ⊥ of vectors orthogonal to K . Since K is null, K ∈ K ⊥ . Thenwe can define the quotient K ≡ K ⊥ / h K i (2.26)of vectors which are orthogonal to K , modulo vectors which are proportional to K . If werestrict our original spinor ǫ to K , we obtain a Majorana–Weyl spinor in eight dimensions;this is known to give rise to a Spin(7) structure. In fact Ψ in (2.25) is nothing but thefour-form that describes this Spin(7) structure. If in the little group of K , stab( K ) =ISO(8), we consider the transformations that also leave this Spin(7) structure invariant,we reduce the SO(8) factor to Spin(7). This gives an alternative understanding to thestabilizer (2.21).We can also now notice that the map ǫ K (2.27)is a Hopf fibration. The space of ǫ such that ǫ t ǫ = 1 is a sphere S . (2.27) maps this tothe space of K which are null and such that K = 1 /
32. This is a slice of the light cone,so it is a copy of S . The fibre of the map (2.27) is then the space of ǫ ’s that map to thesame K : this is Spin(8) / Spin(7) ∼ = S . All this can be made more transparent by usingan octonion basis for the gamma matrices, as for example in [26]; S is then understoodas the octonionic projective line OP . 9 .2 Geometry defined by two spinors We will now move on to considering what happens with two different spinors ǫ and ǫ ,which is what we need in type II theories. As we will see, there are various possibilitiesfor the structure group in T , whereas the structure group defined in T ⊕ T ∗ is universal.This is similar to what one finds for four-dimensional vacua [12]. In that case, one findsboth SU(2) and SU(3) structures on T , and SU(3) × SU(3) on T ⊕ T ∗ .The list of structure groups in section 2.2.1 in itself is a curiosity; it will be a usefulpreliminary, however, towards writing down the possible explicit expressions for Φ insection 2.2.2. Moreover, the list of the generators in the stabilizer will be useful when weask whether Φ determines a metric in section 2.3. We have seen in section 2.1 that a single ten-dimensional spinor defines an ISpin(7) ≡ Spin(7) ⋉ R structure. With two spinors, we have to consider the isotropy group inSO(9,1) of both ǫ , . This is the intersection of two copies of ISpin(7); there are variouspossibilities, which have been listed for example in [10, 11]. We will now give a quickdescription of the various cases; we will describe them in more detail in section 2.2.2.In IIA, ǫ and ǫ have opposite chiralities. If the two null vectors K and K definedby them are proportional, we can use the gamma matrix basis defined in section 2.1.2for both of them. We then reduce ourselves to considering two eight-dimensional spinorsof opposite chirality. The intersection of their stabilizers is G . So overall we havea G ⋉ R structure. If, on the other hand, the two null vectors K and K are notproportional, without loss of generality we can assume that they are respectively e + and e − (up to a rescaling). The two spinors can then be written as ǫ = | ↑ i ⊗ η = (cid:0) η (cid:1) and ǫ = | ↓ i ⊗ η = (cid:0) η (cid:1) . We are then reduced to the common stabilizer of the twoeight-dimensional spinors η and η , which have the same chiralities. This is genericallySU(4), but can get enhanced to Spin(7) if η and η are proportional. So we have foundthree possibilities: G ⋉ R , SU(4) , Spin(7) (on T ; in IIA) . (2.28)Before we move on to IIB, it is interesting to compare (2.28) with what happens [5,7,26] in eleven dimensions. There is a single supersymmetry parameter ǫ , which defines a vector K M = ǫ γ M ǫ . However, unlike our ten-dimensional K , , which are both always null, K can be either timelike or null; even if the component of K along x vanishes, there10s no contradiction, since its projection along the remaining ten dimensions is K + K ,and the sum of two null vectors can be either timelike or null. The little group of ǫ is SU(5) when K is timelike and (Spin(7) ⋉ R ) × R when ǫ is null. When K istimelike, K and K cannot be proportional, and we get SU(4) in ten dimensions. When K is null, K and K can be either proportional (in which case we get G ⋉ R in tendimensions) or not (in which case we get Spin(7)).Coming now to IIB, ǫ and ǫ have the same chirality. If the two null vectors K and K are proportional, again we can use the gamma matrix basis defined in section 2.1.2 forboth, and we can write ǫ i = | ↑ i ⊗ η i , where η i are eight-dimensional spinors of the samechirality. The intersection of the stabilizers of the η i is generically SU(4), but can getenhanced to Spin(7) if they are proportional. So we conclude that the common stabilizerof the ǫ i is generically SU(4) ⋉ R , and Spin(7) ⋉ R when ǫ and ǫ are proportional.When K and K are not proportional, again without loss of generality we can assumethat they are respectively e + and e − (up to a rescaling). The two spinors can then bewritten as ǫ = | ↑ i ⊗ η = (cid:0) η (cid:1) and ǫ = | ↓ i ⊗ η = (cid:0) η (cid:1) , where this time η and η have opposite chiralities. The common stabilizer of two eight-dimensional spinors withopposite chiralities is G . In conclusion, we have found three possibilities:SU(4) ⋉ R , Spin(7) ⋉ R , G (on T ; in IIB) . (2.29)The occurrence of all these cases is similar to the appearance of both SU(2) and SU(3)structures in the classification of type II vacua , namely solutions of the form R , × M orAdS × M . Using the differential geometry associated with the structure groups in (2.28)and (2.29) would be complicated, and it would give rise to a plethora of “intrinsic torsion”classes. Moreover, the stabilizer of the spinors ǫ i may change from a point to another,even for a single solution.In the case of vacua, the classification is more elegant [12] when one considers thestructure group in T ⊕ T ∗ : one obtains there an SU(3) × SU(3) structure. In the samespirit, we will now show that in all these cases one can define the same structure groupon T ⊕ T ∗ , using the single bispinor Φ = ǫ ⊗ ǫ . (2.30)One might also think of considering ǫ ⊗ ǫ or ǫ ⊗ ǫ . As we will show in section 3, however,considering Φ in (2.30) (along with some descendants that we will introduce shortly) isenough to recast the conditions for unbroken supersymmetry in geometrical language.Notice that Φ is not a pure spinor. To see this, we can use the gamma matrix basis11escribed in section 2.1.2. Notice, however, that that basis will in general be different forthe spinors ǫ and ǫ . To take care of that, we will add subscripts or to all indices(similarly to the notation used in [13, App. A.4] to distinguish the two almost complexstructures defined by a pure spinor pair). In this notation, the only sections of T ⊕ T ∗ that annihilate Φ are Ann(Φ) = span { → γ − , ← γ − } , (2.31)whereas a pure spinor would have an annihilator of dimension 10. So Φ is quite different incharacter from the forms φ ± that can be used to reformulate the supersymmetry conditionsfor four-dimensional vacua [12, 13].The bundle T ⊕ T ∗ has rank 20, and it has a natural metric I ≡ (cid:0) (cid:1) defined bycontracting one-forms with vectors. Its structure group is then SO(10 , , ω AB Γ AB , (2.32)where Γ A = { dx m ∧ , ι m } , which generate the Clifford algebra Cl(10,10). The computationis much easier if one changes basis, using (2.6), to the ordinary Cl(9,1) gamma matricesacting from the left and from the right on a bispinor. We get:stab(Φ) = span ω α β → γ α β , → γ − α , ω α β ← γ α β , ← γ − α , → γ + − + ← γ + − → γ − ← γ α , → γ − ← γ + , → γ α ← γ − , → γ + ← γ − , → γ − ← γ − . (2.33)We have again used the notation, introduced above (2.31), of adding an extra subscript and to indices relative to spinors ǫ and ǫ respectively. The first line in (2.33) containsthe stabilizers of ǫ and of ǫ . The last element of the first line comes about because γ + − ǫ = − ǫ (which just follows from Clifford algebra). The generators on the second linedo not correspond to acting on the spinors; as we will see in section 2.3, they correspondto acting on the metric and B field.Using the ordinary Cl(9,1) gamma matrix algebra, we can also determine the Liealgebra of stab(Φ). To perform this computation, it is actually best to decorate again ← γ M with a degree operator ( − ) deg , as in (2.6), so that { → γ M , ← γ N ( − ) deg } = 0 . (2.34)The → γ M and ← γ N ( − ) deg then generate two anticommuting copies of Cl(9,1). We now seethat the generators ω α β → γ α β and ω α β ← γ α β generate two copies of Spin(7). Another12ubalgebra is spanned by the three generators { → γ + − + ← γ + − , → γ − ← γ + ( − ) deg , → γ + ← γ − ( − ) deg } ; this is isomorphic to Sl(2 , R ). The remaining 33 generators satisfy the commu-tation relations of a Heisenberg algebra H , with → γ − α , → γ − ← γ α ( − ) deg playing the roleof the x I , the → γ α ← γ − ( − ) deg , ← γ − α , playing the role of the p I , and → γ − ← γ − ( − ) deg as thecentral element. Having divided the generators of stab(Φ) in three subalgebras, we haveto look at how these commute with each other; we find once again a semidirect product:(Spin(7) × Sl(2 , R )) ⋉ H (on T ⊕ T ∗ ; in IIA / IIB) . (2.35)This is the structure group defined by Φ on T ⊕ T ∗ . Since the group (2.35) is a bit of atongue-twister, we will simply say that Φ defines a generalized ISpin(7) structure . The fact that Φ defines a certain structure group on T ⊕ T ∗ will be important in reformu-lating the supersymmetry equations in terms of forms. However, in practice one also needsto know the possible explicit expressions for Φ. There are several cases, corresponding tothe various structure groups in T we found in (2.28) and (2.29).As a preliminary, we will deal with the bilinears one gets in eight Euclidean dimensions.For simplicity, we can use a real basis for the gamma matrices. (A particularly nice one,which we mentioned earlier, can be written in terms of octonions; see for example [26].)Our bilinears will then all be real.We start with two real spinors η , η of the same chirality. When η and η areproportional, the structure group is just the stabilizer of a real Weyl spinor in eightdimensions, which is Spin(7). In this case, the bilinear simply reads φ Spin(7) ≡ η ⊗ η t = A (1 + Ψ + vol ) , ( η = Aη ) (2.36)where Ψ is the four-form that defines the Spin(7) structure. Notice that the two-formand six-form parts are absent: this follows from the fact that our gamma matrices aresymmetric.When η and η are not proportional, they define generically an SU(4) structure, as wementioned in section 2.2.1. Let us review why. In general, an SU( d ) structure is definedin 2 d dimensions by a pure Weyl spinor. A Majorana spinor can never be pure, but wecan combine our two spinors in η ≡ η + iη , which is not Majorana, but still Weyl.However, in eight dimensions, not all Weyl spinors are pure: the space of pure spinors is C × SO(8) / U(4), which has real dimension 14. This is two less than 16, the real dimension13f the space of all Weyl spinors. However, the constraint for a Weyl spinor η to be pureis simply that η t η = 0 . (2.37)Now, any η and η which are not proportional can be parameterized as η = cos( ψ )˜ η + sin( ψ )˜ η , η = A (cos( ψ )˜ η − sin( ψ )˜ η ) , (2.38)where A and ψ are real, and ˜ η t ˜ η = 0 , ˜ η t ˜ η = ˜ η t ˜ η . (2.39)We can now see that η ≡ ˜ η + i ˜ η satisfies (2.37), and hence it is pure. This shows thatthe two original spinors η and η define an SU(4) structure.We can now use this information to write down the bilinear η ⊗ η t . Since η is pure,its bilinears are simply η ⊗ η t = 12 Ω , η ⊗ η † = 12 e iJ , (2.40)where Ω and J are simply the holomorphic and symplectic form associated to the SU(4)structure. From (2.40) we can extract the bilinears ˜ η i ⊗ ˜ η tj ; going back to (2.38) we obtain φ SU(4) ≡ η ⊗ η t = A Re (cid:16) Ω + e − iψ e iJ (cid:17) . (2.41)The case where η and η are proportional is recovered as ψ →
0. Indeed in this limitwe see that the two-form and six-form parts disappear, and we are left with (2.36), withΨ = ReΩ − J / η and η have opposite chirality. In this case,one can define a vector v m = η t γ m η . Applying (2.36) to the case η = η , one gets η η t = 1 + Ψ − vol (the minus sign in front of vol being due to the fact that η is nowof negative chirality). Using (A.14) one can now compute, similarly to (2.12): v · η = 116 γ m η η t γ m η = 12 (1 + γ ) η = η , (2.42) In general, purity is equivalent to the condition that all bilinears η t γ m ...m k η should be zero exceptwhen k is half the dimension of the space. In d = 8, the cases k = 1 , , k = 0, which is (2.37). We have normalized || η || = 32; this is no loss of generality for us, because we would be able in anycase to reabsorb || η || in the costant A in (2.41). φ G ≡ η ⊗ η t = v · η η t = v · (1+Ψ − vol ) = v + φ + v ∧ ˜ φ −∗ v , ( η chir . +; η chir . − )(2.43)where φ ≡ v x Ψ (along with its seven-dimensional dual ˜ φ ) defines a G structure.We can now use these eight-dimensional bilinears to compute the ten-dimensional ones.We can use the fact that the annihilator (2.31) of Φ is generated by → γ − = K ∧ + K x and ← γ − ( − ) deg = K ∧ − K x . Actually, it will be convenient to work in terms of K ≡
12 ( K + K ) , ˜ K ≡
12 ( K − K ) , (2.44)so that for example ( ˜ K ∧ + K x )Φ = 0 . (2.45)By construction, K · ˜ K = 0, but in general neither K nor ˜ K is null. Their norms are K = − ˜ K = K · K , which is related in turn to Φ by K · K = 132 ( − ) deg(Φ) (Φ , γ M Φ γ M ) , (2.46)where we have used (A.3), (A.9) and (A.11). γ M Φ γ M can be further evaluated using(A.14).When K = K , we see from (2.45) that Ann(Φ) contains K ∧ . So Φ should be of theform K ∧ ( . . . ). When K = K , Φ will be of the form exp h − K K ∧ ˜ K i ∧ ( . . . ). Theremaining parts ( . . . ) come from the eight-dimensional bilinears η η t which we studiedearlier. In IIA, the possible structure groups were listed in (2.28): we getΦ G ⋉R = K ∧ φ G , Φ SU(4) = exp (cid:20) − K K ∧ ˜ K (cid:21) ∧ φ SU(4) , (IIA)Φ Spin(7) = exp (cid:20) − K K ∧ ˜ K (cid:21) ∧ φ Spin(7) , (2.47)where φ G was given in (2.43), φ SU(4) in (2.41), and φ Spin(7) in (2.36). In IIB, the possiblestructure groups were listed in (2.29), and we getΦ
SU(4) ⋉R = K ∧ φ SU(4) , Φ Spin(7) ⋉R = K ∧ φ Spin(7) , (IIB)Φ G = exp (cid:20) − K K ∧ ˜ K (cid:21) ∧ φ G . (2.48)In both IIA and IIB, as we recalled earlier, it is possible for Φ to be of a certain type ata point, and of a different type at another point.15 .3 Spinor bilinears versus metric In order to reformulate the conditions for supersymmetry in terms of Φ, we need to knowwhether it determines a metric, partially or totally. In the case of the classification ofvacua [12], the two pure spinors φ ± do determine a metric, and hence the supersymmetryequations can be rewritten as conditions on them and on nothing else. For our ten-dimensional generalization, we will see that Φ alone is not enough to determine a metric,so that some extra data are necessary.The way to determine whether a certain G -structure determines a globally definedmetric g MN is to think of the latter as an O( d ) structure. If G is a subgroup of O( d ), thenthe G -structure determines a metric. In abstract terms, this is because if the transitionfunctions leave invariant the tensor ω defining the G -structure, they will also lie in O( d ),and leave a metric invariant. In other words, if the stabilizer of ω is contained in O( d ), itleaves invariant a quadratic form at every point; this quadratic form is g MN . On the otherhand, if G is not a subgroup of O( d ), then the stabilizer of ω contains an element thatlies outside O( d ), and that element can be used to change the metric without changing ω ; this shows that ω cannot determine a metric.We will now apply this general criterion to the case at hand; this will hopefully alsomake it clearer. Actually, it is more convenient to work on T ⊕ T ∗ and ask whether Φdetermines a metric and B field. This will spare us from having to go through all the casesin (2.28) and (2.29). We can use the fact that the data of g MN and B MN can be encodedin an O(9 , × O(9 ,
1) structure. This works as follows [15, Chap. 6]. An O(9 , × O(9 , C ± in T ⊕ T ∗ ; these can be singledout by a matrix G that is equal to ∓ on C ± and hermitian with respect to the naturalmetric I = (cid:0) (cid:1) on T ⊕ T ∗ . From G , one can write two orthogonal projections (1 ± G )on C ± . It can be shown that such a G can be written as G = E − − ! E = − g − B g − g − g − Bg − Bg − ! , E ≡ E − E t ! , E ≡ g + B , (2.49)for some g and B , which can be identified as the metric and B field. So the data of g and B are encoded in a O(9 , × O(9 ,
1) structure, and we now need to ask whether thestabilizer of Φ defines a subgroup of O(9 , × O(9 , Actually, if G is contained in a smaller orthogonal group O( d ′ ), d ′ < d , then it is a subgroup of morethan one copy of O( d ), and the tensor ω will define more than one globally defined metric. For example,when G is the trivial group (a so called “identity structure”), the tangent bundle is trivial, and there isa basis of globally defined vectors. C ± ⊂ T ⊕ T ∗ that define the O(9 , × O(9 ,
1) structure.To see this, consider first the case B = 0. We recognize that the matrix E in (2.49) givesthe change of basis in (2.6); in this basis, the fact that G = E (cid:0) − (cid:1) E − just tells us that G gives ∓ C + is the bundle of left-acting gamma matrices, and C − is the bundle of right-actinggamma matrices. When B is non-zero, the change of basis E identifies C ± as generatedby e B ∧ → γ M e − B ∧ = ι M + E MN dx N ∧ , e B ∧ ← γ M e − B ∧ ( − ) deg = − ι M + E NM dx N ∧ . (2.50)These again generate two copies of Cl(9,1).We see that for a bispinor of the form ǫ ⊗ ǫ , the infinitesimal generators of the Liealgebra of O(9 , × O(9 ,
1) should beso(9 , ⊕ so(9 ,
1) = { → γ MN , ← γ MN } . (2.51)(For a bispinor e B ∧ ǫ ⊗ ǫ , the gamma matrices in (2.51) should be conjugated by e B as in (2.50).) However, we see that stab(Φ) in (2.33) also contains elements of the type → γ M ← γ N , which are not of the form (2.51). This already tells us that stab(Φ) is not asubgroup of O(9 , × O(9 , B field.Let us try to get a more concrete understanding of why elements of the form → γ M ← γ N signal that Φ does not determine the metric and B field. From (2.6) we can compute dx M ∧ ι N = 12 (cid:16) − → γ ( M ← γ N ) ( − ) deg + g MN (cid:17) + 14 (cid:16) → γ MN + ← γ MN (cid:17) , (2.52a) dx M ∧ dx N ∧ = 12 → γ [ M ← γ N ] ( − ) deg + 14 (cid:16) → γ MN − ← γ MN (cid:17) . (2.52b)The symmetric part of (2.52a) gives dx ( M ∧ ι N ) = 12 (cid:16) − → γ ( M ← γ N ) ( − ) deg + g MN (cid:17) . (2.53)We can interpret this as the effect on a bispinor of a change in metric. This comes aboutbecause the Clifford map (2.2) depends on the metric, through the gamma matrices γ M .If we deform the vielbeine as δe AM = 12 β M N e AN , (2.54)17he inverse vielbein transforms as δe MA = − β M N e NA , the gamma matrices as δγ M = − β M N γ N , and the metric as δg MN = 2 e A ( M δe AN ) = β ( MN ) . (2.55)We can then take β MN to be symmetric. The Clifford map is deformed as δ (cid:0) C k = − β N M ✭✭✭✭✭✭ dx N ∧ ι M C k . (2.56)So the operator that takes into account the change in metric on a bispinor is − δg MN dx M ∧ ι N = − (cid:18) − → γ ( M ← γ N ) ( − ) deg + g MN (cid:19) δg MN . (2.57)A change in the B field, on the other hand, is simply given by Φ → e B ∧ Φ. In-finitesimally, (2.52b) shows us that this is given by → γ [ M ← γ N ] together with an action onthe spinors ǫ and ǫ in Φ = ǫ ⊗ ǫ .So we see that, if we have elements of the form → γ M ← γ N , they can be interpreted as achange in metric and B field that does not change Φ. Since our Φ does have such elementsin its stabilizer (2.33), it cannot determine by itself a metric and B field.It is interesting to compare this failure with the way a pair of pure spinors φ ± determinethe metric and B field for vacua in [12]. In that case, the common stabilizer of φ + = η η † + and φ − = η η †− is given by the union of the stabilizers of η and η , span { ω i ¯ j → γ i ¯ j ,ω i ¯ j ← γ i ¯ j } (for more details, see [13, App. A.4]). This stabilizer does not contain anyelements of the type → γ M ← γ N ; in fact, it is isomorphic to SU(3) × SU(3), which is a subgroupof SO(6) × SO(6). In this case, φ ± do determine a metric and B field. In section 2.3, we have found that a generalized ISpin(7) structure Φ does not determineuniquely a metric and B field. We will now try to add more degrees of freedom to Φ, soas to resolve this ambiguity.It is useful to start from a particular manifestation of the problem. Given an explicitform such as the ones we presented in section 2.2.2, it is easy to compute a two-dimensionalspace Ann(Φ) of sections of T ⊕ T ∗ that annihilate Φ. We noticed in (2.31) that this isgiven by the span of → γ − and ← γ − . With no additional information, however, we haveno means of telling which element in this two-dimensional space is → γ − and which one is ← γ − . 18o we should try to pick these two elements of Ann(Φ). It is actually best to declarewhich elements of T ⊕ T ∗ correspond to the creators: → γ + = e + · ( ) , ← γ + ( − ) deg = ( ) · e + . (2.58)Here we introduced an alternative notation that will be useful later. e + , e + are thevector parts of → γ + and ← γ + , and the symbol · denotes Clifford multiplication; (2.58) arethen two elements of T ⊕ T ∗ . To formalize the fact that they are two creators, we shoulddemand that ( e + · Φ · e + , Φ) = 0 , (2.59)where ( A, B ) ≡ ( A ∧ λ ( B )) (2.60)is the Chevalley–Mukai pairing ( denotes keeping the ten-form part only). The reasonto demand (2.59) is that, for both i = 1 ,
2, we have ǫ i γ + i ǫ i = 32 K i e + i = 32 g − i + i = 16,which is non-zero. (Once we determine a metric, (2.59) will be equal to 16 vol .)To see whether adding the data (2.58) to Φ allows us to determine a globally definedmetric, we have to compute the common stabilizer of Φ and of (2.58). An element ω AB Γ AB of the Lie algebra so(10,10) acts on T ⊕ T ∗ as a commutator:[ ω AB Γ AB , · ] . (2.61)We see now that all of the offending elements in stab(Φ), of the form → γ M ← γ N , containeither a → γ − or a ← γ − , and so they do not commute with one or both of the two elements(2.58). The resulting stabilizer isstab(Φ , → γ + , ← γ + ) = span { ω α β → γ α β , ω α β ← γ α β } . (2.62)This is now contained in the Lie algebra so(10,10) (it is in fact simply spin(7) ⊕ spin(7)).So the three data (Φ , → γ + , ← γ + ) (2.63)do determine a globally defined metric and B field.Notice, however, that we have been overzealous in adding → γ + and ← γ + to Φ: inaddition to the elements outside O(9 , × O(9 , → γ α − and ← γ α − ,which are in the Lie algebra of O(9 , × O(9 , g MN , B MN , ǫ , ǫ ) = span { ω α β → γ α β , → γ − α , ω α β ← γ α β , ← γ − α } , (2.64)19hich has the Lie algebra structure of the group ISpin(7) × ISpin(7). This means thatwe are parameterizing ( g, B, ǫ , ǫ ), but that part of the information in → γ + and ← γ + isspurious. This is not a big issue for our purposes: it gives rise to a potential topologicalsubtlety in using a Spin(7) × Spin(7) structure rather than an ISpin(7) × ISpin(7) structure,but this should be of very little importance. Moreover, these spurious data are quantifiablyfew, as we will see shortly.Since Spin(7) × Spin(7) is a subgroup of O(8) × O(8) and not just of O(9 , × O(9 , B field. Concretely, onecan extract E MN ≡ g MN + B MN as follows. First of all, we can identify → γ − as the elementof Ann(Φ) which anticommutes with ← γ + ( − ) deg . From the stabilizer (2.62) we can nowalso find eight more elements → γ α . This gives us a basis for C + , the bundle of left-actinggamma matrices. Now we can extract E MN from (2.50). (We can run the same procedurewith the right-acting gamma matrices ← γ M , with the same results.)Now that we have determined a choice of g and B , we can write any bispinor as e B P i =1 ǫ i ⊗ ǫ i in terms of some spinors ǫ i and ǫ i . But, if more than one of the ǫ i were non-zero, the elements of the type → γ MN in (2.62) would not have the Lie algebrastructure of Spin(7), but of a subgroup. So we see that only one of the ǫ i should benon-vanishing; and similarly, only one of the ǫ i . So we can write Φ = e B ǫ ⊗ ǫ for some ǫ , . Moreover, we can determine these two spinors from their stabilizers. In the basisintroduced in section 2.1.2, both ǫ i can be written as | ↑ i ⊗ η ,i ; both η ,i can then bedetermined from the elements → γ α β , ← γ α β in stab(Φ).So we have seen that from (Φ , → γ + , ← γ + ) we can reconstruct ( g, B, ǫ , ). Since Φ canbe written as e B ǫ ⊗ ǫ , and since the action of Spin(9,1) on the space of Weyl spinorsis transitive, we see that the action of so(10,10) on the space of generalized ISpin(7)structures is also transitive. We can then compute the dimension of this space as thedimension of so(10,10) minus the dimension of stab(Φ) in (2.33): this gives 190-78=112.We see that this is 20 less than the degrees of freedom of the data ( g, B, ǫ , ), which are10 + 2 ×
16 = 132. So we see once again that Φ does not contain enough data.On the other hand, → γ + and ← γ + are two elements of T ⊕ T ∗ , so each has 20 degreesof freedom. Together with the 112 degrees of freedom of Φ, this brings us to 152, whichis 20 more than we needed to match ( g, B, ǫ , ). This quantifies the redundancy in ourparameterization (Φ , → γ + , ← γ + ). 20 .5 Summary of this section We have shown that the degrees of freedom ( g MN , B MN , ǫ , ǫ ) can be reformulated interms of the data (Φ , → γ + , ← γ + ), where: • Φ is a “generalized ISpin(7) structure”, namely a form whose stabilizer in Spin(10,10)is the group (2.35); practically speaking, this just means that at every point Φ canbe written in one of the ways listed in (2.47) or (2.48). Φ has a two-dimensionalannihilator Ann(Φ) ⊂ T ⊕ T ∗ . This can be thought of as generated by left Cliffordaction by a null vector K , and right action by a vector K . But Φ alone does notdetermine a metric. Simplifying quite a bit our discussion in section 2.3, we cansay that it gives nine elements { e − = K , e α } α =1 ,..., of a “left” vielbein, and nineelements { e − = K , e α } α =1 ,..., of a “right” vielbein; both of these vielbeine areincomplete. • → γ + = e + · ( ) and ← γ + = ( ) · e + are two elements of T ⊕ T ∗ . We can think of e + and e + as completing the left and right vielbeine mentioned earlier.The fact that the e + i represent the missing vectors in the vielbeine defined by Φ is en-conded in the “compatibility condition” (2.59) among the data (Φ , → γ + , ← γ + ).Within Ann(Φ), a distinctive role will be played by the element that commutes with → γ + − ← γ + ( − ) deg , namely → γ − − ← γ − ( − ) deg , which we called ˜ K ∧ + K x (see (2.45)).In the next section, we will reformulate the conditions for unbroken supersymmetryon ( g MN , B MN , ǫ , ǫ ) as differential equations on (Φ , → γ + , ← γ + ).21 Differential equations
We will now see how the supersymmetry conditions look like in terms of the forms sum-marized in section 2.5.
We give here the differential equations, relegating their derivation to appendix B. In bothIIA and IIB , they read d H ( e − φ Φ) = − ( ˜ K ∧ + ι K ) F ; (3.1a) L K g = 0 , d ˜ K = ι K H ; (3.1b) (cid:18) e + · Φ · e + , γ MN (cid:20) ± d H ( e − φ Φ · e + ) + 12 e φ d † ( e − φ e + )Φ − F (cid:21)(cid:19) = 0 ; (3.1c) (cid:18) e + · Φ · e + , (cid:20) d H ( e − φ e + · Φ) − e φ d † ( e − φ e + )Φ − F (cid:21) γ MN (cid:19) = 0 . (3.1d)Here, g is the metric, φ is the dilaton, H is the NSNS three-form, d H ≡ d − H ∧ , and F is the “total” RR field strength F = P F k . In the spirit of the “democratic” approachchampioned in [27], the sum is from 0 to 10 in IIA and from 1 to 9 in IIB, and one cutsthe number of forms down by half with the self-duality constraint F = ∗ λ ( F ) . (3.2)Finally, ( , ) is the usual pairing on forms given in (2.60).Equations (3.1) are necessary and sufficient for supersymmetry to hold; we give somedetails of this computation in appendix B. To also solve the equations of motion, oneneeds to impose the Bianchi identities, which away from sources (branes and orientifolds)read dH = 0 , d H F = 0 . (3.3)It is then known (see [28] for IIA, [29] for IIB) that almost all of the equations of motionfor the metric and dilaton follow. To make (3.1a) identical in IIA and IIB, we have changed the conventions for IIA with respect to [27]by setting F here = λ ( F there ), and H here = − H there . (This differs by a sign from the redefinition usedin [13].) We still have a sign of difference among the two theories: the upper sign in (3.1c) is for IIA, thelower for IIB. ∇ + + H + ) ǫ , ( ∇ + − H + ) ǫ (3.4)are completely absent from d Φ. (3.1b) do contain some components of (3.4), but notall of them. So one has to look for a way of re-expressing the missing components interms of differential forms; in appendix B, we show that (3.1c) and (3.1d) do the job.They are perhaps not as nice as one might have wished, because they do contain themetric explicitly in the Clifford products. On a more positive note, they do not containany covariant derivative. Moreover, the system (3.1) is necessary and sufficient for bothIIA and IIB, and for any Φ among the many possibilities we gave in (2.47), (2.48), orinterpolating among those. It is of course possible that a better version will be found inthe future. In any case, for the vast majority of situations (3.1a) and (3.1b) will actuallybe enough, and (3.1c), (3.1d) will not contain any new information. For example, we willsee in section 4.1 that this is the case for four-dimensional vacua. This is because in thatsituation (3.4) are related by four-dimensional Lorentz symmetry to other components ∇ µ ǫ , whose equations are already implied by (3.1a) and (3.1b). On the other hand,sometimes (3.1c) and (3.1d) do contain some new information, as we will see in section4.2, where they will be seen to reproduce a constraint first noticed in [16, 17]. The first equation in (3.1b) tells us that K is an isometry. Notice that the second in (3.1b)also implies L K H = 0. We also show at the end of section B.4 that (3.1a), (3.1b) imply L K φ = 0. 23e will now also show that L K F = 0. Notice first that { d H , ˜ K ∧ + ι K } = { d, ˜ K ∧} − { H ∧ , ι K } + { d, ι K } = ( d ˜ K − ι K H ) ∧ + L K = L K , (3.5)where, in the last step, we have used (3.1b). Now, if one acts on (3.1a) with d H , the lefthand side vanishes since d H = 0. The right hand side then gives0 = d H (( ˜ K ∧ + ι K ) F ) = { d H , ( ˜ K ∧ + ι K ) } F − ( ˜ K ∧ + ι K ) d H F = L K F − ( ˜ K + ι K ) d H F ; (3.6)so if one imposes the Bianchi identities (3.3), one gets L K F = 0 , (3.7)as promised. So we see that K is not just an isometry, but a symmetry of the fullsolution [30].In fact, K is also a supersymmetric isometry: namely, it keeps Φ invariant. To seethis, act on (3.1a) with ˜ K ∧ + ι K . The left-hand side gives,( ˜ K ∧ + ι K ) d H ( e − φ Φ) = L K ( e − φ Φ) − d H ( e − φ ( ˜ K ∧ + ι K )Φ) = L K ( e − φ Φ) ; (3.8)we have used (3.5), and (2.45). The right-hand side gives −
12 ( ˜ K ∧ + ι K ) F = − ˜ K · KF = −
14 ( K − K ) F = 0 , (3.9)where we have used (2.44) and that K , are null (from (2.13) applied to both spinors ǫ , ). Recalling from (B.23) that L K φ = 0, we conclude L K Φ = 0 . (3.10) We will now see how (3.1) reproduces known equations in particular setups. We will start by considering four-dimensional Minkowski vacua, namely solutions of theform R , × M where all fields preserve the maximal symmetry of the four-dimensionalspace. We will recover the equations considered in [12]. As we will see, they all comefrom (3.1a). 24 .1.1 Structure of four-dimensional spinors We first need to understand the geometry of four-dimensional spinors, similarly to ourdiscussion in section 2 of Majorana–Weyl spinors in ten dimensions. As is well-known, infour dimensions one can impose either the Majorana or Weyl conditions, but not both.Moreover, one can map a Majorana spinor to a Weyl one, and viceversa. We will considera Weyl spinor ζ + of positive chirality. We will also define its conjugate ζ − = ( ζ + ) ∗ , whichhas negative chirality. (We will work in the basis where all γ µ are real.) There are twobispinors we can consider: ζ + ⊗ ζ + and ζ + ⊗ ζ − . We will investigate them in turn.We start by looking at ζ + ⊗ ζ + , using the Fierz identity (A.12) for d = 4. Since ζ + ischiral, we see that only bilinears with an odd number of γ µ ’s survive. That leaves us with ζ + ⊗ ζ + = (1 + γ ) v = v + i ∗ v , (4.1)where v µ ≡ ζ + γ µ ζ + is a real vector. A similar computation as (2.12) also reveals that v ζ + = 0 . (4.2)This also implies that v is null: v µ v µ = 0 . (4.3)Moving on to ζ + ⊗ ζ − , by chirality only bilinears with an even number of γ µ ’s survivein the Fierz identity (A.12). Moreover, the bilinear ζ − γ µ ...µ k ζ + = ζ t + γ γ µ ...µ k ζ + vanisheswhen γ γ µ ...µ k is antisymmetric. That leaves us with only k = 2. So ζ + ⊗ ζ − = ω + ; (4.4)from (A.9) (with d = 4; we pick the convention c = i ) and the fact that ζ + is chiral wesee that ω + = − i ∗ ω + . Also, (4.2) and (2.5) tells us v ∧ ω + = ι v ω + = 0 , (4.5)so in fact ω + = v ∧ w for some complex one-form w , which also satisfies ι v w = 0 (recallthat v is null, and that w is complex). So, summing up: ζ + ⊗ ζ − = v ∧ w , (4.6)with v the same as in (4.1), and v µ v µ = v µ w µ = 0.Although we will not quite need it in what follows, we can also now compute thestructure group of ζ + . We can compute it geometrically as the stabilizer of v and w under25he Lorentz action, similarly to section 2.1.3. Since v is null, its stabilizer is ISO(2) =SO(2) ⋉ R . As for w , we see that it plays the role of Ψ in section 2.1.3 (compare (2.25)and (4.6)). In ten dimensions, Ψ breaks from SO(8) to Spin(7); in four dimensions, w breaks SO(2) completely. We are thus left with a structure group isomorphic to R .This can actually be seen directly from the spinor perspective. If one uses thatSpin(3 , ∼ = Sl(2 , C ), one can see that the stabilizer of e.g. the spinor (cid:0) (cid:1) is the setof matrices of the form (cid:0) z (cid:1) , which is isomorphic to C ∼ = R .Finally, using the same logic as in (2.27), one can also see that the map ζ v is againa Hopf fibration; this time it reproduces the classic S → S with fibre S . For a four-dimensional vacuum solution, Poincar´e invariance fixes the metric to be ds = e A ds + ds . (4.7)Moreover, the flux F should be F = f + e A vol ∧ ∗ λf , (4.8)for f a form on the internal manifold M . H is constrained to be purely internal.We now proceed to splitting the ten-dimensional spinors ǫ , in terms of the four-dimensional spinors we just studied. For an N = 1 four-dimensional vacuum, this decom-position reads ǫ = ζ + η + ζ − η − ,ǫ = ζ + η ∓ + ζ − η ∓ ; (4.9)here and later, the upper sign is for IIA, the lower for IIB. By definition, η a − = ( η a + ) ∗ ,so that both ǫ , are Majorana–Weyl. Poincar´e invariance demands that the ǫ i in (4.9)be supersymmetric for any ζ + ; this will give a total of four supercharges, as appropriatefor an N = 1 vacuum. Using this decomposition, we can now specialize the ingredientsin (3.1a): namely the bispinor ǫ and its annihilators K ± . The ten-dimensional gammamatrices can be decomposed as γ (10) µ = e A γ (4) µ ⊗ , γ (10) m = γ ⊗ γ (6) µ , (4.10)where { γ ( d ) } is a basis of gamma matrices in d dimensions, and γ is the chiral operatorin four dimensions. 26he bispinor Φ = ǫ ⊗ ǫ can be evaluated using the ten-dimensional Fierz identi-ties (2.8), and repackaged using the ones (A.12) in four and six dimensions; we getΦ = ∓ ( ζ + ζ + ) ∧ ( η η †∓ ) + ( ζ − ζ − ) ∧ ( η − η †± ) + ( ζ + ζ − ) ∧ ( η η †± ) ± ( ζ − ζ + ) ∧ ( η − η †∓ )= ∓ e A v + ie A ∗ v ) ∧ φ ∓ ) + 2Re( e A v ∧ w ∧ φ ± ) . (4.11) φ ± = η ⊗ η †± are two six-dimensional pure spinors associated to the internal geometry;they define together an SU(3) × SU(3) structure.We can similarly evaluate K and K . First, recall that for a six-dimensional Weylspinor η + we have η †− γ m η + = 0, with m = 1 , . . . , K mi = 0.For the spacetime components, from the definition (2.9) and from the spinorial decompo-sition (4.9) we get K µi = 132 e − A (cid:0) ζ + γ µ ζ + || η i + || + ζ − γ µ ζ − || η i − || (cid:1) = 18 e − A v µ || η i || (4.12)Let us assume for simplicity || η || = || η || ≡ || η || . Then we have ˜ K = 0, K = K = K (recall (2.44)).Now we can start applying our differential equations (3.1). We start from (3.1b).This tells us that K = v ( e − A || η || ) should be a Killing vector. But by construction v isalready a Killing vector; so we get || η || = c + e A (4.13)for some constant c + . This was indeed the condition on the spinor norm found in [12] (inthe present simplifying assumption that || η || = || η || ). To summarize so far, we have˜ K = 0 , K = 18 c + v . (4.14)Now we can turn to (3.1a). We have already evaluated Φ in (4.11). So the left-hand side of (3.1a) is easy to evaluate: since everything is indepedent of four-dimensionalspacetime, d H only acts on M . As for the right-hand side of (3.1a), using (4.8) and(4.14): − ( ˜ K ∧ + ι K ) F = − c + ι v ( e A vol ∧ ∗ λf ) = 18 c + ∗ v ∧ e A ∗ λf . (4.15)We can now use (4.15) and (4.11) in (3.1a); we get d H Re( e A − φ φ ∓ ) = 0 , (4.16a) d H ( e A − φ φ ± ) = 0 , (4.16b) d H Im( e A − φ φ ∓ ) = ∓ c + e A ∗ λ f . (4.16c)27hese are the pure spinor equations found in [12] (in the simplifying assumption || η || = || η || ); the upper sign is for IIA, the lower for IIB. So we see that (3.1a) reproducesthem all in one go. Notice also that nothing in our computation depended on a particularchoice of v and w ; this means that we have found not just one solution to (3.1a), but four— as many as the number of Weyl spinors ζ + ; this corresponds to four supercharges, asappropriate for an N = 1 solution.We still have to look at (3.1c), (3.1d). We do not expect anything new from these, sincewe have already reproduced all the equations found in [12]. We will look at (3.1c); theanalysis of (3.1d) is similar. First, we have to choose e + · and · e + . Since K = K = K ,it also makes sense to take e + = e + ≡ e + . Moreover, since K ∝ v has only four-dimensional components, we will take this to be the case for e + as well. More specifically,to take care of the powers of the warping e A coming from (4.10), we will write → γ + = e A e + ∧ + e − A e + x , ← γ + ( − ) deg = e A e + ∧ − e − A e + x , (4.17)where now x refers to a contraction using the four-dimensional metric. Since e + has nointernal components, d † ( e − φ e + ) vanish. Also, from (4.17) we get { d, ← γ + ( − ) deg } = { dx M ∧ ∂ M , e A e + ∧ − e − A e + x } = e − A ∂ + + dA ∧ → γ + . (4.18)We now apply all this to (3.1c):( γ + Φ γ + , γ MN (cid:2) ± d H ( e − φ Φ γ + ) − F (cid:3) ) = ( γ + Φ γ + , γ MN [ dA ∧ γ + e − φ Φ − f ]) ; (4.19)we have used (4.18) and (4.8). Let us now consider the various possibilities for the indices M N . If M = m and N = n , both terms in (4.19) vanish because γ = 0, and the equationhas no content. For M = µ , N = ν , the dA ∧ γ + Φ term drops out, and we are left with( γ + Φ γ + , γ µν f ) = ∓ ǫ γ + γ µν f γ + ǫ = −
132 ( ζ + γ + γ µν γ + ζ − )( η † + f η ± + c . c . ) ; (4.20)we have used (B.33) and the spinor decomposition (4.9). This implies Tr( φ †± f ) = 0; or,in terms of the six-dimensional Chevalley–Mukai pairing,( λ ∗ f, φ ± ) = 0 . (4.21)This actually follows from (4.16c) and from ( dφ ± , φ ∓ ) = ( φ ± , dφ ∓ ).Finally, for M = m and N = ν , a computation similar to (4.20) gives (cid:16) dA ∧ φ ± ± i c + e A ∗ λf, γ m φ ± (cid:17) = 0 . (4.22)28his can also be shown to follow from (4.16). Using the “intrinsic torsions” introducedfor example in [13, Eq. (A.19)] and the expression of the pure spinors as a bispinor, φ ± = η ⊗ η †± , one derives that in generalized complex geometry ( dφ ∓ , γ m φ ∓ ); computingthe dφ ∓ from (4.16c), one then obtains (4.22).In conclusion, for four-dimensional Minkowski vacua, the system (3.1) reproduces theconditions found in [12]. Equation (3.1b) reduces to a condition about the norm of thespinors, while (3.1a) reproduces all of the pure spinor equations in [12]. (3.1c) and (3.1d)are, in this case, redundant. We will now consider solutions of the form R , × M . One application of these vacua isto the study three-dimensional RG flows holographically. In the context of this paper,this case will be an example where equations (3.1c), (3.1d) are not redundant; they willreproduce an algebraic constraint found in [16, 17].Most of our discussion is similar to the one in section 4.1, so we will be more schematichere.We will first discuss the forms of three-dimensional spinors. This is much simplerthan in section 4.1.1. We can work in a basis where all the gamma matrices are real (andsymmetric). It then follows that the “norm” ζ ζ = 0 for any spinor ζ (just as we found inten dimensions). Applying (A.13) in either the even or the odd version we find ζ ⊗ ζ = v = − ∗ v , (4.23)where as usual we are omitting the slash: v = (cid:0) v = v M γ M , and similarly for ∗ v .We now need to decompose our ten-dimensional spinor in terms of three- and seven-dimensional ones. The decomposition of gamma matrices is a bit harder than (4.10): γ (10) µ = e A σ ⊗ γ (3) µ ⊗ , γ (10) m = σ ⊗ ⊗ γ (7) m , (4.24)where σ i are Pauli matrices. Taking γ (3) µ all real and γ (7) m purely imaginary, all the γ (10) M are real. Our two ten-dimensional Majorana–Weyl spinor now can be decomposed as ǫ = (cid:18) − i (cid:19) ⊗ ζ ⊗ η , ǫ = (cid:18) i (cid:19) ⊗ ζ ⊗ η , (4.25)where ζ , η , η are all real. We can use the Fierz identities (2.8) in ten dimensions,together with the ones (A.13) in 3 and 7 dimensions, to obtainΦ = ( ζ ⊗ ζ ) + ∧ φ ± ∓ ( ζ ⊗ ζ ) − ∧ φ ∓ == − ∗ v ∧ φ ± + v ∧ φ ∓ , (4.26)29here φ = η ⊗ η † . The subscript () ± refers to the possibility of obtaining an even or oddform in (A.13); ( ζ ⊗ ζ ) ± have then been evaluated using (4.23). As for K and ˜ K , moreor less the same computation that led us to (4.14) now gives us˜ K = 0 , K = c + v , (4.27)again under the simplifying assumption || η || = || η || .We can now apply (3.1a), using the same steps as in section 4.1.2. We get d H ( e A − φ φ ∓ ) = 0 , (4.28a) d H ( e A − φ φ ± ) = c + e A ∗ λ f , (4.28b)which can indeed be found in [16, Eq. (2.5)] (see also [17]).Finally we look at (3.1c), (3.1d). The computation is again very similar to the onein section 4.1.2. For M = m , N = n , and for M = m , N = ν , we find nothing new.For M = µ , N = ν , instead of (4.21) we now find, in terms of the three-dimensionalChevalley–Mukai pairing, ( f, φ ∓ ) = 0 . (4.29)Unlike what happened in four dimensions, however, this equation cannot be derived fromthe pure spinor equations. It is possible to rewrite it as ( ∗ λf, φ ± ) = 0; but, if we try toderive this from (4.28b), we now get a term ( dφ ± , φ ± ), about which we cannot in generalsay anything. Indeed (4.29) was listed in [16, Eq. (2.6)] as a separate algebraic constraint.In conclusion, also for this class of solutions our system (3.1) reproduces the conditionsfor supersymmetry found in previous work — in this case [16, 17]. The differential, pure-spinor-like equations are again all reproduced by (3.1a); equations (3.1c), (3.1d) give analgebraic constraint on the flux that was also found in [16,17]. We can regard (3.1c), (3.1d)as a generalization of that constraint. Acknowledgments
I would like to thank L. Martucci, R. Minasian, M. Petrini and A. Zaffaroni for interestingdiscussions. I am supported in part by INFN.
A Bispinors
In this section we will collect a few facts about gamma matrices that we need in the maintext and in appendix B. There is very little of substance here; we are mostly going to take30are of a few annoying but unavoidable signs.
A.1 The λ operator The annoying sign par excellence is the operator λ , already encountered in the main text: λ ( C k ) ≡ ( − ) ⌊ k ⌋ C k , (A.1)where the floor function ⌊·⌋ denotes the integer part. In this subsection, we are going tofocus on d = 10 in Lorentzian signature, with real gamma matrices as in section 2.1. λ is related to transposition: from (2.1) we see that C tk = ( − ) k − γ λ ( C k ) γ . (A.2)In particular we have( − ) deg(Φ) λ (Φ) = − γ Φ t γ = − γ ( γ ) t ǫ ǫ t γ = − ǫ ǫ . (A.3)Moreover, since Φ is even (odd) when ǫ is odd (even), we get γλ (Φ) = − ( − ) deg(Φ) λ (Φ) . (A.4) λ does not commute with wedges and contractions: since ( − ) ⌊ k +12 ⌋ = ( − ) k ( − ) ⌊ k ⌋ , we have λ ( dx M ∧ C k ) = ( − ) k dx M ∧ λ ( C k ) , λ ( ι M C k ) = ( − ) k +1 ι M C k , (A.5)from which, remembering (2.5), λ ( γ M C ) = λ ( C ) γ M . (A.6) A.2 Hodge star
We will now consider gamma matrices in any dimension. We will first consider the caseof Euclidean signature. The chiral operator can be written as γ = c γ . . . γ d , (A.7)where c is a constant such that c = ( − ) ⌊ d ⌋ . (A.8)(When d is odd, we will take γ = 1.) γ is related to the Hodge star by γ C k = c ∗ λ C k , C k γ = c ( − ) ⌊ d ⌋ λ ∗ C k . (A.9)31n Lorentzian signature, there is an extra minus sign: γ = c γ . . . γ d − , c = − ( − ) ⌊ d ⌋ . (A.10)With this definition, (A.9) still holds.We also need the relation between the Chevalley–Mukai pairing (2.60) and the traceof bispinors: 12 ⌊ d ⌋ Tr( ∗ A B ) = ( − ) deg( A ) ( A, B ) . (A.11)Finally, there are two formulas that we will use several times in the main text. Oneis the Fierz identity. For even dimension, it consists in expanding any bispinor C on thebasis { γ M ...M k } dk =0 : C = d X k =0 d k ! Tr( Cγ M k ...M ) γ M ...M k ⇒ ζ ⊗ ζ = d X k =0 d k ! ( ζ γ M k ...M ζ ) γ M ...M k . (A.12)For odd dimension, the matrices { γ M ...M k } dk =0 are twice as many as the dimension of thespace of bispinor, 2 ⌊ d ⌋ × ⌊ d ⌋ . So there are in fact two bases: { γ M ...M k } dk =0 ( k even) and { γ M ...M k } dk =0 ( k odd) . So we have two possibilities: C = d X k =0 k even / odd ⌊ d ⌋ k ! Tr( Cγ M k ...M ) γ M ...M k ⇒ ζ ⊗ ζ = d X k =0 k even / odd ⌊ d ⌋ k ! ( ζ γ M k ...M ζ ) γ M ...M k . (A.13)So when the dimension is odd the Clifford map (2.2) is not injective: an odd form C − andan even form C + can correspond to the same bispinor, (cid:0) C − = (cid:0) C + . In fact, this happenswhen C + and C − are related by c ∗ λ , as one can see from (A.9) (recalling that, when d is odd, γ = 1).The last formula we need in the main text is valid in any number of dimensions: γ M C k γ M = ( − ) k ( d − k ) C k . (A.14) B Equivalence of (3.1) to supersymmetry
We will describe here the derivation of (3.1), and show that it is equivalent to the con-ditions for unbroken supersymmetry. We will work in IIB; the computations for IIA arevery similar. 32 .1 Deriving (3.1a)
The derivation of (3.1a) is similar to the derivation of the pure spinor equations in [13,App. A]. A notable difference is that one cannot use the “spacetime gravitino” variation δψ µ , since there is no distinction between spacetime and internal indices now. As we willsee, we can proceed anyway, thanks to the properties of ten-dimensional spinors that wereviewed in section 2.1.For supersymmetry to be unbroken, we want to set the supersymmetry variations tozero: (cid:18) D M − H M (cid:19) ǫ + e φ F γ M ǫ = 0 , (cid:18) D − H − ∂φ (cid:19) ǫ = 0 (B.1a) (cid:18) D M + 14 H M (cid:19) ǫ − e φ λ ( F ) γ M ǫ = 0 , (cid:18) D + 14 H − ∂φ (cid:19) ǫ = 0 , (B.1b)where, as usual, D ≡ γ M D M is the Dirac operator, H ≡ (cid:0) H = H MNP γ MNP , H M = H MNP γ NP .We start by computing ( d H Φ − dφ ∧ Φ). For d Φ = dx M ∧ ∂ M Φ and dφ ∧ = ∂ M φ dx M ∧ ,we can just derive 2 dx M ∧ = → γ M + ← γ M ( − ) deg from (2.6). As for H ∧ , it can be obtainedby applying the same expression to each of the three wedges in dx M ∧ dx N ∧ dx P ∧ : H ∧ = 18 · H MNP (cid:16) → γ MNP + ← γ MNP ( − ) deg + 3 → γ M ← γ NP +3 → γ NP ← γ M ( − ) deg (cid:17) == 18 (cid:18) → H + ← H ( − ) deg + → γ M ← H M + ← γ M → H M ( − ) deg (cid:19) . (B.2)So we can now write2 ( d H Φ − dφ ∧ Φ) = [ γ M , ∂ M Φ − ∂ M φ ∧ Φ] − H ∧ Φ= (cid:18) Dǫ − Hǫ − ∂φǫ (cid:19) ǫ + γ M ǫ (cid:18) D M ǫ − ǫ H M (cid:19) − (cid:18) D M ǫ − H M ǫ (cid:19) ǫ γ M − ǫ (cid:18) Dǫ − ǫ H − ǫ ∂φ (cid:19) . (B.3)Using (B.1), this becomes e φ (cid:0) γ M ǫ ǫ γ M F + F γ M ǫ ǫ γ M (cid:1) . (B.4)We can now use the property (2.11). Crucially, the right hand side of that relation onlyinvolves vectors and one-forms, so that we can massage (B.4) without needing any extraequation (such as the external gravitino variation in [13, App. A]):( B.
4) = − e φ ( K (1 + γ ) F + F (1 − γ ) K ) = − e φ (cid:16) ˜ K ∧ + ι K (cid:17) F . (B.5)33e have used that γF = F (B.6)(which follows from self-duality, (3.2), and from (2.3)), and the definition of K and ˜ K in (2.44). Comparing now (B.3) with (B.5), we obtain (3.1a).As for (3.1b), those two equations have been already derived in [8, 19, 30], so we willnot give their derivation here. B.2 Spinor basis
In section 2.1.2, we introduced a basis for ten-dimensional gamma matrices. As we notedthere, the spinors { γ AB ǫ } (B.7)span the whole space of spinors with the same chirality as ǫ . However, we will also needlater a basis for the space of spinors with the opposite chirality. For ease of discussion,we will pick ǫ to be of chirality +, as we did in section 2.1.2. So we need a basis for thespace Σ − of spinors of negative chirality.Obviously, γ M ǫ cannot be enough, since there are only 9 of them (recall γ + ǫ = 0,from (2.16)), and the space Σ − has dimension 16. The next natural possibility is to usespinors of the form γ MNP ǫ : γ αβγ ǫ = | ↑ i ⊗ γ αβγ (8) η + , γ − αβ ǫ = | ↓ i ⊗ γ αβ (8) η + , γ + − α ǫ = 2 | ↑ i ⊗ γ α (8) η + . (B.8)However, in eight dimensions we have γ αβγ η + = Ψ αβγδ γ δ η + , (B.9)where Ψ αβγδ is the Spin(7) four-form in eight dimensions. So the γ αβγ ǫ are in fact depen-dent on the γ + − α ǫ , of which there are 8. We also know that there are only 7 non-vanishing γ αβ η + ; so we are left with only 8 + 7 = 15 spinors. The one which we are missing is γ + ǫ = | ↓ i ⊗ η + . (B.10)So neither { γ M ǫ } nor { γ MNP ǫ } give a complete basis for Σ − . One possibility wouldbe to use them both, as a redundant basis.Another possibility, which we will adopt, is to pick a particular spinor with negativechirality, and act on this with the γ MN . Given our discussion in section 2.3, a naturalchoice for this spinor is γ + ǫ . Hence our basis for spinors with chirality opposite to ǫ is { γ AB γ + ǫ } . (B.11)34 .3 Original supersymmetry equations in terms of intrinsic tor-sion We will now count the independent components of the supersymmetry equations, so thatwe can compare them in section B.4 with (3.1a) and (3.1b).We can now use the bases (B.7) and (B.11) for both ǫ and ǫ . Just like in section 2.2,we need to take care to distinguish the index + relative to ǫ from the index + relativeto ǫ , which we will by adding an index or ; and likewise for the indices − and α . Sowe will have indices + , − , α , and + , − , α .We can now define (cid:18) D M − H M (cid:19) ǫ = Q MNP γ NP ǫ , (cid:18) D − H − ∂φ (cid:19) ǫ = T MN γ MN γ + ǫ (B.12a) (cid:18) D M + 14 H M (cid:19) ǫ = Q MNP γ NP ǫ , (cid:18) D + 14 H − ∂φ (cid:19) ǫ = T MN γ MN γ + ǫ . (B.12b)There is no assumption here: the left hand sides are spinors that can be expanded onour basis, and the Q ’s and T ’s are the coefficients of this expansion. They are the ten-dimensional analogue of the coefficients introduced in [13, App. A.4]. They can be thoughtof as parameterizing the intrinsic torsion of the generalized ISpin(7) structure defined byΦ. Notice that some of their components do not multiply anything, and can be assumedto vanish: Q aMα a + a = 0 , T aα a − a = 0 . (B.13)For the same reason, we can assume the Q to be antisymmetric in their last two indices,and the T to be antisymmetric.We also need a basis for forms. Our basis (B.7), (B.11) also produces for us a basisfor the space of bispinors: γ MN ǫ ⊗ ǫ γ P Q , γ MN γ + ǫ ⊗ ǫ γ + γ P Q ; γ MN γ + ǫ ⊗ ǫ γ P Q , γ MN ǫ ⊗ ǫ γ + γ P Q . (B.14)In IIB, which is our focus in this appendix, the first two sets of generators are oddforms, and the second two are even; in IIA, the opposite would be true. Via the Cliffordmap (2.2), the basis (B.14) can also be used as a basis for forms. In this section, wewill use it to expand F . In IIB, F is an odd form; moreover, it is self-dual, so that wehave (B.6): γF = F . This tells us that it will be a linear combination of the first set ofgenerators in (B.14): F = R MNP Q γ MN ǫ ⊗ ǫ γ P Q . (B.15)35t also follows that λ ( F ) = R MNP Q γ QP ǫ ⊗ ǫ γ NM .Expanding (B.1) in terms of the coefficients we just introduced in (B.12), we get Q MNP = 4 e φ R NP M − , T MN = 0 ,Q MNP = 4 e φ R − MNP , T MN = 0 . (B.16) B.4 (3.1a), (3.1b) in terms of intrinsic torsion
We will now rewrite the equations in (3.1) in the language of the intrinsic torsions Q and T , so as to compare them with (B.16).We start from (3.1a). Using (B.3) and (B.12), we can write2( d H Φ − dφ ∧ Φ) = T MN γ MN γ + Φ − Q MNP γ M Φ γ NP − Q MNP γ NP Φ γ M + T MN Φ γ + γ MN (B.17)and2( ˜ K ∧ + ι K ) F = ( K · F + F · K ) = R MNP Q ( K M γ N Φ γ P Q + γ MN Φ γ P K Q ) , (B.18)where we have used K R γ R ǫ = 0, ǫ γ R K R and [ γ R , γ MN ] = 4 δ [ MR γ N ] . So, comparing (B.17)and (B.18) with (3.1a), we get Q MNα = 4 e φ R Nα M − ( M = + ) , T α β = 0 , (B.19a) Q MNα = 4 e φ R − MNα ( M = + ) , T α β = 0 , (B.19b) Q α + − + T α + = 4 e φ R + − α − , T − = − Q − + − , (B.19c) Q α + − + T α + = 4 e φ R − α + − , T − = − Q − + − . (B.19d)All of these equations are implies by (B.16), as they should. But the converse is not true:we see, for example, that the components Q NP , Q NP (B.20)never appear anywhere in (B.19). This is because they would multiply γ NP ǫ ǫ γ + and γ + ǫ ǫ γ NP , which vanish because γ + ǫ = 0.We can now try to add more equations to (3.1a). An obvious choice is (3.1b), whichalready appeared in [8, 19, 30]. In terms of the coefficients Q , the first in (3.1b) says One also needs to use that, in the basis of section 2.1.2, γ α + γ + ǫ = 2 γ α ǫ , γ + − γ + ǫ = γ − ǫ , γ + − ǫ = ǫ . MN ) − + Q MN ) − = 0, while the second says Q MN ] − − Q MN ] − = 0. So together theygive Q MN − = − Q NM − . (B.21)Unfortunately, not even this helps us recover the supersymmetry conditions (B.16): forexample, the components Q α β and Q α β do not appear in either (B.19) or (B.21).We note in passing that already (B.19) and (B.21) are enough to show that L K φ = 0,as claimed in section 3.2. By simply using the definition of T in (B.12), we see that e φ d † ( e − φ K i ) = − T i + i − i , i = 1 , . (B.22)Now, we can use (B.21) to write Q − + − = − Q − − , which is zero because the two lastindices of Q are antisymmetric by definition. So (B.19) implies that T − = 0; we canargue in a similar way that T − = 0. Summing now (B.22) with i = 1 and i = 2, andusing the fact that K is Killing (from (3.1b)), we get L K φ = 0 , (B.23)as claimed. B.5 The missing equations: (3.1c), (3.1d)
In section B.4, we have expressed (3.1a) and (3.1b) in terms of the “intrinsic torsions” Q , T introduced in (B.12), and unfortunately we have concluded that they are implied bythe supersymmetry conditions (B.16), but not equivalent to them. We will now find somedifferential equations that, once expressed in terms of (B.12), will provide the missingcomponents of (B.16).The first thing to remark is that we could have expected a priori that (3.1a) and (3.1b)would not be equivalent to supersymmetry. In section 2, we found that Φ does not containby itself enough data to determine a metric and B field; we have to also provide theelements → γ + and ← γ + of T ⊕ T ∗ . So we expect the missing equations to contain them.Unfortunately, → γ + and ← γ + are not bilinears of ǫ and ǫ , and so there is no straight-forward procedure to compute their derivatives. However, one can proceed indirectly; letus focus on → γ + . First of all we notice that ǫ γ + γ M γ + ǫ = 32 e M + . (B.24)This allows us to consider e + as a bilinear of γ + ǫ . We need to compute, however, thecovariant derivatives of this spinor. For our purposes, it will be enough to know that (cid:26)(cid:18) D − H − ∂φ (cid:19) , v M γ M (cid:27) = 2 v N (cid:18) D M − H M (cid:19) + (cid:0) dv + e φ d † ( e − φ v ) . (B.25)37his implies (cid:18) D − H − ∂φ (cid:19) γ + ǫ = (cid:0) − γ + T MN γ MN γ + + 2 Q NP γ NP + de + · + e φ d † ( e − φ e + ) (cid:1) ǫ . (B.26)We can now compute32 e φ d † ( e − φ e + ) = ǫ γ + ( D − ∂φ )( γ + ǫ ) + (cid:0) D M ( ǫ γ + ) γ M − ǫ γ + ∂φ (cid:1) γ + ǫ = ǫ h Q NP [ γ + , γ NP ] + [ e + , de + ] + 2 e φ d † ( e − φ e + ) γ + i ǫ = 32 h Q + − + 2 ι K ι e +1 de + + e φ d † ( e − φ e + )) i . (B.27)So we get Q + − = − ι K ι e +1 de + ; (B.28)and, going back to (B.26):( ǫ γ + ) (cid:18) D − H − ∂φ (cid:19) ( γ + ǫ ) = 16 e φ d † ( e − φ e + ) . (B.29)We are now ready to derive (3.1d). We can use the same strategy as in (B.3) tocompute ( d H − dφ ∧ ); with some manipulations we get ǫ γ + h { d H − dφ ∧ , → γ + } ǫ ǫ i γ MN γ + ǫ = ǫ γ + (cid:20)(cid:18)(cid:26) D − H − ∂φ, γ + (cid:27) ǫ (cid:19) ǫ + { γ P , γ + } ǫ (cid:18) D P ǫ − ǫ H P (cid:19)(cid:21) γ MN γ + ǫ = 16 e φ d † ( e − φ e + ) · K [ M e N ]+ + 32 (cid:18) D + ǫ − ǫ H + (cid:19) γ MN γ + ǫ ; (B.30)we have used (B.29) and the normalization ǫ γ + ǫ = 16 (which follows from our conven-tions in section 2.1.2).We now want to reexpress the last line of (B.30) in terms of the pairing (2.60). Ingeneral, for any C , using (A.3) we have: ǫ γ + Cγ + ǫ = − ( − ) deg(Φ) Tr( γ + λ (Φ) γ + C ) . (B.31)We also have, using (2.3), (A.6) and (A.4): ∗ ( γ + Φ γ + ) = γλ ( γ + Φ γ + ) = γγ + λ (Φ) γ + = ( − ) deg(Φ) γ + λ (Φ) γ + . (B.32)38utting (B.31) together with (B.32), and using the relation (A.11) between the Chevalley–Mukai pairing and the spinorial trace, we obtain ǫ γ + C γ + ǫ = − − ) deg(Φ) ( γ + Φ γ + , C ) . (B.33)In particular, for C = Φ γ MN :( γ + Φ γ + , Φ γ MN ) = − K [ M e N ]+ ( − ) deg(Φ) . (B.34)We can now use the conjugate of the first equation (the gravitino variation) in (B.12b),to rewrite the second term in the last line of (B.30) in terms of F :32 (cid:18) D + ǫ − ǫ H + (cid:19) γ MN γ + ǫ = 2 e φ ǫ γ + F γ MN γ + ǫ . (B.35)Using also (B.33) and (B.34), we obtain (3.1d). B.6 The missing equations and intrinsic torsion
Finally, we want to express (3.1d), which we have derived in section B.5, in terms of the“intrinsic torsions” Q and T introduced in (B.12). Once we do that, we will be able tocheck whether the system (3.1) really is equivalent to the supersymmetry conditions (B.1).We need to go back to the last line of (B.30). The crucial term is the second one,which we have already expressed in terms of F in (B.35). We now follow a different route,and we express it in terms of Q using (B.12): − Q P Q ǫ γ P Q γ MN γ + ǫ ≡ − Q P Q ˜ P P QMN . (B.36)The tensor ˜ P is not quite a projector, but it is diagonal, if viewed as a matrix whosefirst index is the pair of indices P Q , and the second index is the pair of indices
M N .Specifically, we find that the only non-zero entries are (up to antisymmetry in
P Q and
M N separately): ˜ P αβγδ = P αβ γδ , ˜ P α − γ − = 14 δ γα , ˜ P + − + − = 18 , (B.37)where P is the projector in the of Spin(7). We have omitted the indices (writing α rather than α , and so on) to make the equation more readable, and because the sameequation is true for the spinor ǫ (and in fact this is needed for the analysis of (3.1c),which we are not giving here). 39f we now decompose F as (B.15) in (B.35), and we compare to (B.36), we get( Q P Q − e φ R − + P Q ) ˜ P P QMN = 0 . (B.38)But the only possible values for the indices P Q in both Q P Q and R − + P Q are α β , α − and + − . So, in fact, applying (B.37) to ˜ P (that is, reinstating the indices : α → α and so on) tells us that Q P Q = 4 e φ R − + P Q (B.39)These are precisely the components we were missing in (B.19), as we observed there. Asimilar analysis for (3.1c) also gives us Q P Q = 4 e φ R P Q + − . (B.40)Together with (B.19a) and (B.19b), these give us the supersymmetry conditions (B.16)for Q MNα and Q MNα . We still seem to face a problem: (B.19c) and (B.19d) do not quitetell us what Q α + − , Q − + − , Q α + − , Q − + − (B.41)are, because of the contributions from T . However, (B.21) relates these components to − Q α − , − Q − − = 0 , − Q α − , − Q − − = 0 . (B.42)The two non-zero components, Q α − and Q α − , have already been accounted forin (B.39) and (B.40). This allows us to derive all the missing equations in (B.16).In conclusion, we have shown that the system (3.1) is equivalent to the conditions (B.1)for unbroken supersymmetry. References [1] W. Israel,
Differential forms in general relativity . Dublin Institute For AdvancedStudies, 2nd ed ed., 1979.[2] J. F. Plebanski, “On the separation of Einsteinian substructures,”
J.Math.Phys. (1977) 2511–2520.[3] A. Ashtekar, “New Variables for Classical and Quantum Gravity,” Phys.Rev.Lett. (1986) 2244–2247. 404] R. Capovilla, J. Dell, T. Jacobson, and L. Mason, “Self-dual 2-forms and gravity,” Class.Quant.Grav. (1991) 41–57.[5] J. P. Gauntlett and S. Pakis, “The geometry of D = 11 Killing spinors,” JHEP (2003) 039, hep-th/0212008 .[6] J. P. Gauntlett, D. Martelli, S. Pakis, and D. Waldram, “G structures and wrappedNS5-branes,” Commun.Math.Phys. (2004) 421–445, hep-th/0205050 .[7] J. P. Gauntlett, J. B. Gutowski, and S. Pakis, “The geometry of D = 11 null Killingspinors,” JHEP (2003) 049, hep-th/0311112 .[8] E. J. Hackett-Jones and D. J. Smith, “Type IIB Killing spinors and calibrations,” JHEP (2004) 029, hep-th/0405098 .[9] P. Saffin, “Type IIA Killing spinors and calibrations,” Phys.Rev.
D71 (2005) 025018, hep-th/0407156 .[10] U. Gran, J. Gutowski, and G. Papadopoulos, “The Spinorial geometry of supersym-metric IIB backgrounds,”
Class.Quant.Grav. (2005) 2453–2492, hep-th/0501177 .[11] U. Gran, J. Gutowski, and G. Papadopoulos, “Invariant Killing spinors in 11D andtype II supergravities,” Class. Quant. Grav. (2009) 155004, .[12] M. Gra˜na, R. Minasian, M. Petrini, and A. Tomasiello, “Generalized structures of N = 1 vacua,” JHEP (2005) 020, hep-th/0505212 .[13] M. Gra˜na, R. Minasian, M. Petrini, and A. Tomasiello, “A scan for new N = 1 vacuaon twisted tori,” JHEP (2007) 031, hep-th/0609124 .[14] N. Hitchin, “Generalized Calabi–Yau manifolds,” Quart. J. Math. Oxford Ser. (2003) 281–308, math.dg/0209099 .[15] M. Gualtieri, “Generalized complex geometry,” math/0401221 . Ph.D. Thesis (Advi-sor: Nigel Hitchin).[16] M. Haack, D. Lust, L. Martucci, and A. Tomasiello, “Domain walls from ten dimen-sions,” JHEP (2009) 089, .[17] P. Smyth and S. Vaul`a, “Domain wall flow equations and SU(3) × SU(3) structurecompactifications,”
Nucl. Phys.
B828 (2010) 102–138, .4118] D. Lust, P. Patalong, and D. Tsimpis, “Generalized geometry, calibrations and su-persymmetry in diverse dimensions,”
JHEP (2011) 063, .[19] P. Koerber and L. Martucci, “D-branes on AdS flux compactifications,”
JHEP (2008) 047, .[20] J. P. Hsu, A. Maloney, and A. Tomasiello, “Black hole attractors and pure spinors,”
JHEP (2006) 048, hep-th/0602142 .[21] S. Kachru, X. Liu, and M. Mulligan, “Gravity Duals of Lifshitz-like Fixed Points,” Phys.Rev.
D78 (2008) 106005, .[22] J. M. Figueroa-O’Farrill, “Breaking the M-waves,”
Class. Quant. Grav. (2000)2925–2948, hep-th/9904124 .[23] A. Coimbra, C. Strickland-Constable, and D. Waldram, “Supergravity as GeneralisedGeometry I: Type II Theories,” .[24] R. L. Bryant, “Remarks on spinors in low dimension,” ∼ bryant/Spinors.pdf .[25] S. Weinberg, The Quantum Theory of Fields . Cambridge University Press, 1995.[26] R. L. Bryant, “Pseudo-Riemannian metrics with parallel spinor fields and vanishingRicci tensor,”
ArXiv Mathematics e-prints (Apr., 2000) arXiv:math/0004073 .[27] E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest, and A. Van Proeyen, “New Formu-lations of D = 10 Supersymmetry and D8–O8 Domain Walls,” Class. Quant. Grav. (2001) 3359–3382, hep-th/0103233 .[28] D. L¨ust and D. Tsimpis, “Supersymmetric AdS compactifications of IIA supergrav-ity,” JHEP (2005) 027, hep-th/0412250 .[29] J. P. Gauntlett, D. Martelli, J. Sparks, and D. Waldram, “Supersymmetric AdS solutions of type IIB supergravity,” Class. Quant. Grav. (2006) 4693–4718, hep-th/0510125 .[30] J. M. Figueroa-O’Farrill, E. Hackett-Jones, and G. Moutsopoulos, “The Killing su-peralgebra of ten-dimensional supergravity backgrounds,” Class. Quant. Grav. (2007) 3291–3308, hep-th/0703192hep-th/0703192