Generalised Geometries, Constrained Critical Points and Ramond–Ramond Fields

The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria

Claus Jeschek Frederik Witt

Vienna, Preprint ESI 1748 (2005) November 25, 2005

Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at MPP-2005-107

Generalised geometries, constrained critical points and Ramond-Ramond ﬁelds

Claus Jescheka and Frederik Wittb

a Max-Planck-Institut für Physik, Föhringer Ring 6, D-80805 München, FRG E-mail: [email protected]

b Freie Universit¨at Berlin, FB Mathematik und Informatik, Arnimallee 3, D-14195 Berlin, FRG E-mail: [email protected]

ABSTRACT

We describe the geometry of type II string compactifications to 6– and 7–dimensional backgrounds with Ramond–Ramond–fields in terms of principal fibre bundle as generalised SU(3)– and G2– structures. We endow these structures with an integrability condition which can be characterised as a constrained critical point for the generalised Hitchin functional and provides a solution to the supersymmetry variations. Moreover, we discuss topological and geometrical properties of generalised SU(3)–structures which we show to coincide with the notion of a generalised Calabi-Yau metric in dimension 6.

Subj. Class.: special Riemannian manifolds, generalised geometries, supergravity, string theory 2000 MSC: 53C10, 53C25, 58E15, 83E30, 83E50 1 Introduction 2

2 Generalised geometries 6

2.1 The geometry of T ⊕ T ∗ ...... 6

2.2 Generalised SU(3)– and G2–structures...... 7

2.3 Topological reductions to SU(3) × SU(3) and G2 × G2 ...... 15

3 BackgroundsinstringtheoryandRamond-Ramondﬁelds 18

3.1 TypeIItheories...... 19

3.2 Supersymmetryvariations ...... 21

3.3 Compactiﬁcation on M 6 ...... 23

4 Constrained critical points 25

4.1 An invariant functional and its ﬁrst variation ...... 25

4.2 Spinorial formulation of the variational problem ...... 28

4.3 T –duality ...... 38

A Appendix: A matrix representation of Spin(6) and Spin(7) 39

References 40

1 Introduction

Generalised geometries go back to Hitchin’s foundational article [15] and were further developed in [12], [27] and [4]. The basic idea is to enhance the tangent bundle T n to the “doubled” vector bundle T n ⊕T n∗, where contraction defines a natural inner product of split signature. It is therefore associated with a principal SO(n,n)–fibre bundle which is spinnable. Chiral spinors for T ⊕ T ∗ can be thought of as differential forms of even or odd degree, and as in classical geometry, there are special forms which induce further reductions of the structure group, sitting now inside Spin(n,n). These new geometries have found many applications in physics since they were introduced by Kapustin in his seminal work about T –duality in twisted topological models [18]. In the present paper, we propose generalised SU(3)– and G2–structures, which we introduced earlier [28], [17], as models for the internal background geometry of type II supergravity compactifications to 6 and 7 dimensions. Apart from the physical relevance, this also points to interesting geometrical objects

2 such as generalised structures with “torsion”. Generally speaking, a supersymmetric vacuum background in type II supergravity, the low energy limit of the corresponding superstring theory, consists of a 10–dimensional space–time (M 1,9,g) together with some bosonic field content. The fields come in two flavours; they are either NS-NS (Neveu-Schwarz) or R-R (Ramond–Ramond). To the former class belong the space–time metric g, a 2–form b, the so–called B–field, and a scalar field φ – the dilaton. In the latter class we find a collection of differential forms of degree p, the Ramond–Ramond fields F p which satisfy the duality relation F p = ⋆F\10−p, where ∧ is a sign changing operator depending on p. The space–time supersymmetry is accounted for by two (nowhere vanishing) spinors ε1 and ε2. The theory is then said to be of type IIA or IIB, if ε1,2 are of opposite or equal chirality. The spinors are required to satisfy additional conditions, namely the vanishing of the supersymmetry variations. Following the idea of the so–called democratic formulation of Bergshoeff et al. [2], the vanishing of these is equivalent to

1 1 φ ev,od ev,od 0 = ∇X + XxH ·P (ε1, ε2) + e F · X · P (ε1, ε2) 4 16 1 1 φ ev,od p ev,od 0 = (dφ · + H ·P)(ε1, ε2) ∓ e 5F − pF · P (ε1, ε2), (1.1) 2 8 p=Xev,od where H is the field strength of the B–field and P and Pev,od are certain projection operators. Note that in type IIA, only R-R–fields of even degree can occur, while in type IIB they are necessarily odd. Some references in physics where these equations have also been considered are [8], [9], [22]. The usual ansatz for finding a solution to (1.1) is a compactification on an n–dimensional internal background. For us, this means to take as 10–dimensional space–time the direct product between flat Minkowski space R1,9−n and a Riemannian spin manifold M n equipped with b, φ, F ev or F od, and two nowhere vanishing spinors η1,2 in ∆, the space of Spin(n)–spinors. In [17], we showed that in absence of R-R–fields, the geometry on an n = 6– or n = 7–dimensional internal background is ∗ described by a generalised SU(3)– or G2–structure whose defining T ⊕ T –spinors are closed with respect to the twisted differential dH = d+H∧. Roughly speaking, the link between the “form” and the “spinor picture” of generalised SU(3)– or G2–geometry is this. Using the standard Spin(n)– equivariant embedding ∆ ⊗ ∆ ֒→ Λ∗ (known as fierzing in the physics literature), we can think of ∗ the bispinor η1 ⊗η2 as an exterior form. It becomes a spinor if regarded as being acted on by T ⊕T , so some extra work is required to check this action to be compatible with the embedding [28]. The −φ spinor e exp(b) ∧ η1 ⊗ η2 then induces a generalised SU(3)– or G2–structure and conversely, any T ⊕ T ∗–spinor defining such a structure can be represented in this way. Finally, the vanishing of the internal supersymmetry variations governed by (1.1) without R-R–field terms translates under this embedding to −φ dH e η1 ⊗ η2 =0. (1.2)

While the NS-NS–ﬁelds appear naturally, the biggest stumbling block in taking R-R–ﬁelds into account was the problem of incorporating these into the generalised setup. In the language of G–

3 structures, generalised SU(3)– or G2–geometry is characterised, as we will show, by a topological reduction from a principal R>0 × Spin(n,n)–ﬁbre bundle, n = 6 or 7, to an SU(3) × SU(3)– or

G2 ×G2–bundle. This gives 1+66−16 =51 or1+91−28 = 64 degrees of freedom respectively which are already exhausted by the data (g,b,φ,η1,η2). A way out of this problem could be glanced from the democratic formulation above, which is appealing for at least two reasons. Firstly, it displays the high degree of symmetry between type IIA and IIB and treats these types on equal footing as suggested by T -duality. Secondly, this formulation is particularly apt for applications in generalised geometry as it treats the R-R–fields not as a collection of p–forms, but as an even or odd form. Though it is an honest differential form (i.e. an object of a Spin(n)–tensor representation), this suggests that the even or odd R-R–form should arise in an inhomogeneous version of (1.2), where we decompose η1 ⊗ η2 into chiral spinors, that is, we project on the even or odd part. Incidentally, such an equation has already been considered in [28] in the context of constrained critical points for the generalised Hitchin functional. We recall that generalised G2–structures are defined by generic spinors for which we can define a generalised Hitchin functional. As in Hodge theory, we restrict this functional to a fixed cohomology class and ask for critical points which yields the differential equation (1.2). Generalised SU(3)–structures do not fit into this setup per se as they are defined by a non–chiral spinor which is not generic. However, its chiral components are pure, that is, they induce a reduction to SU(3, 3) and give rise to a generalised Calabi-Yau manifold. As pure spinors form an open orbit, an SU(3, 3)–structure can be characterised as a critical point for Hitchin’s variational principle [15]. In this way, generalised SU(3)–structures come out of a twofold variational process where we have to impose additional algebraic constraints on the individual critical points in order to achieve a reduction to SU(3) × SU(3). Now any even or odd exact form F • induces a linear functional on exact forms of corresponding parity by

• QF • (dγ) = γ ∧ F . ZM Restricting the generalised Hitchin functional to the trivialc cohomology class and asking for critical points subject to the constraint QF = const yields as integrability condition

−φ • dH e η1 ⊗ η2 = F . (1.3)

In a way, this is reminiscent of Hodge theory. There, we need to fix the metric before we can carry out the variational process. On the other hand, the unconstrained variational principle does not require any further data, a feature which has recently found applications in topological M–theory [6], [21]. It is the critical point which induces the metric etc. as the result of the (topological) reduction to its stabiliser in the principal SO(n,n)–fibre bundle. As we have just observed, this is not the case for the R-R–fields which, like the metric in Hodge theory, need to be put in by hand in order to set up the constrained variational principle considered in this paper. Again we can translate (1.3) into the spinor picture of the generalised structure and find precisely the internal supersymmetry variations coming from (1.1). For the proof, we need to decompose

4 the R-R–field terms into SU(3) × SU(3)– and G2 × G2–irreducible components and as a result, we • see that F is required to take values in an SU(3) × SU(3)– and G2 × G2–invariant piece of (complex) dimension 9 and of (real) dimension 49. This is somehow similar to situations encountered 4 in classical geometry. For instance, the equations for the G2–invariant 3–form ϕ, namely dϕ = T and d ⋆ ϕ = T 5 for a 4– and 5–form T 4,5, lead to non–trivial algebraic relations among T 4 and T 5 and demanding for skew–torsion of the Levi–Civita–connection requires the vanishing of certain components. The variational principle also points to interesting mathematical aspects. It yields natural integrability conditions and gives rise to new geometries interesting for their own sake, but incorporates known geometries in disguise, too. For instance, we show that a generalised Calabi-Yau metric in dimension 6 [12] is essentially a generalised SU(3)–structure which satisfies the unconstrained critical point condition. On the other hand, relaxing the integrability condition for n = 7 and considering the constrained variational problem results in new geometries, for we know from [28] that the only unconstrained critical points are manifolds of holonomy G2. Using the supersymmetric formulation of integrability, one can easily compute the Ricci–tensor for vanishing R-R–fields. Moreover, this formulation yields a natural pair of connections ∇± = ∇LC ±H/2 (in this context, see also [16]). The fact that the NS-NS–sector is already contained in the topological data of a generalised structure suggests that R-R–fields play a similar rôle in the generalised setup as torsion 3–forms do in classical geometry, that is, the R-R–field terms could be interpreted as a substantiation of torsion if we had a meaningful notion of this concept in the generalised case. On the other hand, adopting the equivalent formulation of integrability (1.3) allows us to construct explicit examples by using the powerful 1 tool of T –duality. The idea is to start with an S –invariant generalised SU(3)– or G2–structure and to change the topology along the S1–fiber without destroying integrability. In particular, we can take a classical SU(3)– or G2–structure with torsion (which we view as a “generalised” structure in an obvious way) and obtain a non–classical T –dual structure. This paper is aimed as much to a mathematical as to a physical audience. Some mathematical arguments are therefore explained in finer detail than usual in order to render the material more accessible to physicists. Conversely, some time is spent to explain the physical motivation of this paper. The material is therefore organised as follows. In Section 2 we introduce the formalism of generalised geometry. We define generalised SU(3)– and G2–structures and give a much more refined version of the formalism in [28] which shows how to pass from the spinor picture to the form picture and vice versa. We also discuss these geometries from a G–structure viewpoint. In particular, we show that a generalised Calabi-Yau metric in dimension 6 is essentially a generalised SU(3)–structure. Section 3 starts with an informal review of type II string and supergravity theories, before we explicitly derive the internal supersymmetry variations for a compactification on R1,3 ×M 6 in type IIB following the democratic formulation (1.1). Section 4 contains the core results. We first setup the constrained variational problem and derive equation (1.3). We then translate this integrability condition into the spinor picture to make contact with the supersymmetry variations

5 of Section 3. As a further application, we compute the Ricci–tensor of an unconstrained critical point. Finally, we discuss T –duality in this setting and outline an explicit construction method for supersymmetric backgrounds. The authors would like to thank Stefano Chiantese, Florian Gmeiner, Dieter L¨ust and Dimitrios Tsimpis for useful discussions. The second author also would like to thank the Erwin Schr¨odinger Institute, Vienna, for hospitality during the preparation of this paper.

2 Generalised geometries

2.1 The geometry of T ⊕ T ∗

We brieﬂy sketch the algebraic setup following [15]. We consider the vector bundle T ⊕ T ∗ which comes equipped with a natural orientation and an inner product of split signature deﬁned for v ∈ T and ξ ∈ T ∗ by 1 (v + ξ, v + ξ) = − ξ(v). 2 Note that GL(n) 6 SO(n,n) and as a GL(n)-space, the Lie algebra of SO(n,n) decomposes into

so(n,n)=Λ2(T ⊕ T ∗) = End(T ) ⊕ Λ2T ∗ ⊕ Λ2T.

In particular, any 2-form b deﬁnes an element in the Lie algebra so(n,n). Exponentiated to SO(n,n), its action on T ⊕ T ∗ is exp(b)(v ⊕ ξ) = v ⊕ (vxb + ξ).

n Over a manifold M , this gives rise to a principal SO(n,n)-bundle PSO(n,n) which is always spinnable. To construct the spinor bundle, we consider the action of T ⊕ T ∗ on Λ∗T ∗ given by contraction and wedging, namely

(v + ξ) • τ = vxτ + ξ ∧ τ.

As this squares to minus the identity it induces an algebra isomorphism Cliﬀ(T ⊕T ∗) =∼ End(Λ∗T ∗). Moreover, the exterior algebra S =Λ∗T ∗ splits into the irreducible Spin(n,n)-representation spaces ± ev,od ∗ ± ev,od ∗ S =Λ T . Restricted to GL(n)+ 6 Spin(n,n), the representation becomes S = Λ T ⊗ n 1/2 (Λ T ) , where GL(n)+ acts on the factors as usual. Moreover, we can deﬁne the following Spin(n,n)-invariant bilinear form. Let ∧ denote the anti-automorphism given on any element of degree p by αp = ǫ(p)αp where ǫ(p)=1 for p ≡ 0, 3mod4 and −1 for p ≡ 1, 2mod4. The bilinear form c hα, βi =[α ∧ β]n, where the superscript n indicates taking the top degreeb component, is non–degenerate, symmetric if n ≡ 0, 3mod4 and skew if n ≡ 1, 2 mod4. Moreover, S+ and S− are non-degenerate and orthogonal

6 if n is even and totally isotropic if n is odd. The action of a B–ﬁeld b on a spinor τ is given by 1 exp(b) • τ = (1 + b + b ∧ b + ...) ∧ τ = eb ∧ τ. 2

As in “classical” geometry associated with the tangent bundle T , one is interested in additional algebraic objects conveniently described by the choice of a structure group which means that the principal ﬁbre bundle PSO(n,n) or its double cover PSpin(n,n) reduces to a principal ﬁbre bundle associated with the stabiliser of these objects. In this paper, we shall deal with reductions to

SU(3) × SU(3) and G2 × G2 over 6– and 7–dimensional manifolds respectively.

2.2 Generalised SU(3)– and G2–structures

Next we introduce the algebro-topological setup of generalised SU(3)– and G2–geometries which we later show to induce the aforementionned topological reductions to SU(3) × SU(3) and G2 × G2. Integrability conditions will be considered in Section 4. Here and in the sequel, n = 6 or 7, and the letter G will refer to either of the groups SU(3) or G2. From the viewpoint of generalised geometry, these structures are most naturally defined in terms of G × G–invariant T ⊕ T ∗–spinors. However, our main concern lies in application to physics where the algebraic data is typically given by a collection of fermions, ojects living in a spin representation of Spin(n) which is associated with the tangent bundle, and bosons – objects living in a tensor representation. We therefore choose an approach in terms of ”classical” T –spinors. We will elaborate on the physical aspects in further detail in Section 3. Before we give a precise definition of generalised G–structures, we recall some basic spin geometric facts to fix the notation. The real Clifford algebra Cliff(Rm,g) is multiplicatively generated by Rm under the relation x·y+y·x = −g(x, y)1 for x, y ∈ Rm. As a vector space, Cliff(Rm,g) is canonically ∗ isomorphic to Λ by sending x1 · ... · xp to x1 ∧ ... ∧ xp and thus carries a natural grading. If J denotes this isomorphism, then the algebra product transforms for x ∈ Rm and a homogenous element a ∈ Cliff(Rm,g) as

J(x · a) = x ∧ J(a) − xxJ(a), J(a · x) = (−1)deg(a) x ∧ J(a) + xxJ(a) . (2.4)

In the sequel we will also make intensive use of the canonical involutions ∼, ∧ and ∨ of Cliﬀ (Rm,g) [13], given on elements of degree p by p mod4 0 1 2 3 ∼ + − + − ∧ + − − + ∨ + + − −

Note that ∨ and ∧ induce anti-automorphisms, that is, (a · b)∨ = ˇb · ˇa and (a · b)∧ = b · a. Moreover, the involutions commute with each other and ∧◦∼= ∨. The canonical vector space isomorphism J b b 7 carries these involutions over to the exterior algebra and one easily checks that they satisfy the same multiplicative properties. In particular, we obtain the ∧–operation defined in the previous section. The complex Clifford algebra Cliff(Cm,gC) is the complexification of Cliff(Rm,g) and is either p p p isomorphic to End(C2 ) if m = 2p or the direct sum End(C2 ) ⊕ End(C2 ) if m = 2p +1. Fixing such an isomorphism induces a representation κ : Cliff(Cm,g) → EndC(∆m) with the convention p to project on the first summand if m is odd. The representation space ∆m = C2 is the so–called space of (Dirac) spinors and we usually write a · Ψ for κ(a)Ψ. The explicit matrix representation we use throughout this paper is given in the appendix. Restricting κ to Spin(m) yields the spin representation for this group which can be decomposed into the irreducible modules ∆+ and ∆− if m is even. A spinor is chiral if it lies in either of these irreducible components and in this case, we will indicate the chirality by the subscript ±. Moreover, any Spin(m)-representation space comes equipped with an invariant, positive-definite hermitian inner product q. In fact, if we extend ∧ complex anti–linearly to Cliff(Cm,gC), then for any element a of this algebra,

q(a · Ξ, Θ) = q(Ξ, a · Θ), (2.5) that is a∗ = a. b Next we discuss the dimensions 6 and 7 relevant to us. Starting with n = 6, we first note that Spin(6)b is isomorphic to SU(4). Under this isomorphism, the irreducible spin representations 4 4 ∆+ and ∆− are given by the complex vector representation C and its complex conjugate C . In ∗ ∼ particular, ∆+ = ∆− and the hermitian inner product stabilised by SU(4) induces on ∆± the hermitian inner product q alluded to above. The conjugation map interchanges the chirality of a spinor and defines a real isometry compatible with the Clifford action, i.e. for x ∈ R6 we have x · Θ = x · Θ and iΞ = −iΞ. The spin action is transitive on the sphere which is homeomorphic to

SU(4)/SU(3), that is, SU(3) is the isotropy group of a unit spinor, say Ξ+, and its conjugate spinor 6 Ξ− = Ξ+. Projecting down to SO(6) yields an almost complex structure J on R deﬁned through 6 3 3 3 (x − iJx) · Ξ+ = 0, or equivalently through (x + iJx) · Ξ− = 0. Then R ⊗ C = C ⊕ C , where C and C3 are spanned by x ∓ iJx, and we can embed the complexiﬁcation isometrically into the spin 3 3 representation by z ⊕ z 7→ (z ⊕ z) · (Ξ+ ⊕ Ξ−), that is ∆+ = CΞ+ ⊕ C and ∆− = CΞ− ⊕ C . Next we step one dimension up and consider Spin(7). Again, ∆ carries a real structure and we therefore identify ∆, as a vector space, with R8. The induced inner product which we still denote by q is now real and thus symmetric. In particular, ∆ is self-dual. Moreover, Spin(7) also acts transitively on the sphere which is isomorphic to Spin(7)/G2. The G2–invariant spinor Ψ induces an equivariant isometric embedding x 7→ x · Ψ and thus ∆ = RΨ ⊕ R7.

Deﬁnition 2.1. [17], [28] (i) Let M 6 be a 6-dimensional, spinnable manifold. A topological generalised SU(3)-structure is 6 deﬁned by the 6-tuple (M ,g,b,φ, Ξ+, Θ+), where g is a Riemannian metric, b a 2-form, φ a smooth 6 function on M , and Ξ+, Θ+ ∈ ∆+ are two complex unit spinors of positive chirality. We denote the stabiliser of Ξ± and Θ± by SU(3)l and SU(3)r respectively.

8 7 (ii) Let M be a 7-dimensional, spinnable manifold. A topological generalised G2-structure is 7 defined by a 6-tuple (M ,g,b,φ, Ψ+, Ψ−), where g is a Riemannian metric, b a 2-form, φ a smooth 7 function on M and Ψ+, Ψ− ∈ ∆ are two real unit spinors. We denote the stabiliser of Ψ+ and Ψ− by G2l and G2r respectively. (iii) We refer to the 2-form b as the B–field and to the scalar φ as the dilaton field of the generalised structure.

We usually drop the adjective ”topological”’, although it is important to bear in mind that for the moment, we only deal with topological data. Trivial instances of these generalised structures are provided by straight generalised G-structures, where in the definition we choose twice the same spinor associated with an honest classical G-structure. Nowhere vanishing spinor fields exist if and only if the Euler class of the spinor bundle vanishes. For dimension 6 and 7 the fibre is diffeomorphic to R8 in either case, so this obstruction vanishes trivially on spinnable manifolds.

n Proposition 2.1. A manifold M , n = 6, 7, admits a generalised SU(3)– or G2–structure if and only if the ﬁrst and second Stiefel-Whitney class of M n vanish.

Remark: Note that the existence of two prinicipal G–fibre bundles does not imply the reduction to a principal fibre bundle associated with the intersection of Gl and Gr, as, for instance, the defining spinors may conicide at some points. In the case of generalised G2–structures, this coincidence number can actually be used for a topological classification [28]. The same argument implies on dimensional grounds that for n = 6 there is – up to vertical homotopy equivalence – only the straight SU(3)–structure.

In order to make contact with T ⊕ T ∗–spinors, we want to use the classical Spin(n)–equivariant embedding of the tensor product ∆⊗∆ into the exterior algebra Λ∗ known as ﬁerzing in the physics literature. The key point here, however, is that we regard both ∆⊗∆ and Λ∗ as Spin(n)×Spin(n)- modules, where the action on the latter space is induced by a inclusion into Spin(n,n) depending on g and b. More concretely, we consider the isomorphisms

b πb± : x ∈ (T, ±g) 7→ e (−x ⊕ ±xxg) = −x ⊕ xx(±g − b) (2.6)

∗ and put V± = π±(T ), g± = ±π±∗g. Then T ⊕ T = V+ ⊕ V− is a decomposition into a maximally space- and timelike subspace which we refer to as a metric splitting. In terms of structure groups this is equivalent to a reduction from SO(n,n) to SO(n) × SO(n) which, on the other hand, is characterised precisely by the choice of a metric g and a B–field b. Alternatively, one can encorporate the B–field in a generalised tangent bundle [16], but for our purposes it is sufficient to work with the fixed principal PSO(n,n)–bundle and to keep the B–field as extra data. With a metric splitting ∗ at hand, we can decompose the Clifford algebra Cliff(T ⊕ T ) into a Z2-graded tensor product ∗ Cliff(V+)⊗ˆ Cliff(V−) =∼ Cliff(T ⊕ T ), where the isomorphism is given by extension of the map

9 ∗ v+⊗ˆ v− 7→ v+ • v−, • denoting the product in Cliﬀ(T ⊕ T ). The isometries π± induce isomorphisms

Cliﬀ(T, ±g) =∼ Cliﬀ(V±,g±) which map Spin(n) = Spin(T, ±g) isomorphically onto Spin(V±,g±). Consequently, the compounded algebra isomorphism

∗ ιb : x⊗ˆy ∈ Cliff(T,g)⊗ˆ Cliff(T, −g) 7→ πb+(x) • πb−(y) ∈ Cliff(T ⊕ T ) maps Spin(T,g) × Spin(T, −g) onto Spin(V+) × Spin(V− ) inside Spin(7, 7). We drop the subscript if b = 0. Furthermore, the invariance of q (2.5) enables us to identify the tensor product ∆ ⊗ ∆ with EndC(∆) and ultimately with complex forms in an equivariant way. To that end we consider the assignment

Ξ1 ⊙ Ξ2(Θ) = q(Θ, Ξ2)Ξ1 −1 C for Ξ1,Ξ2 and Θ in ∆. The algebra isomorphism κ maps this to Cliﬀ(T ) which in turn we identify with the exterior forms on T C via the canonical isomorphism J. We write

−1 [Θ1 ⊗ Θ2] = J(κ (Θ1 ⊙ Θ2)) for this form and use the superscripts ev and od if we project on the even or odd part. The following theorem states how the map [· , ·] behaves under the action of Cliﬀ(T ⊕ T ∗) on Λ∗. To keep the notation compact, we write [· , ·]b for exp(b/2) ∧ [· , ·]. Theorem 2.2. For any x ∈ T n, ˆ ˆ ^ [x · Ξ ⊗ Θ]b = ιb(x⊗1) • [Ξ ⊗ Θ]b, [Ξ ⊗ x · Θ]b = ιb(1⊗x) • [Ξ ⊗ Θ]b ^ ˆ ^ ^ ˆ [x · Ξ ⊗ Θ]b = −ιb(x⊗1) • [Ξ ⊗ Θ]b, [Ξ ⊗ x · Θ]b = −ιb(1⊗x) • [Ξ ⊗ Θ]b. Moreover, we have \ [Ξ ⊗ Θ]b =[Ξ ⊗ Θ]b and [Ξ ⊗ Θ]=[Θ ⊗ Ξ].

Proof: The proof of the ﬁrst statement directly carries over from arguments in [28]. Secondly, by decomposing the complex spinors Ξ and Θ into real and imaginary components, the induced complex conjugation map on the complexiﬁed exterior algebra is easily seen to commute with conjugation on ∆. Finally, we have \ \ [Ξ ⊗ Θ] = J(κ−1(Ξ ⊙ Θ)) = J(κ−1(Ξ ⊙ Θ)∗), for κ(a)∗ = κ(a) by (2.5). But

q((Ξ ⊙ Θ)∗Ξ , Θ ) = q(Ξ , (Ξ ⊙ Θ)Θ ) = q(Θ , Θ)q(Ξ , Ξ) = q(q(Ξ , Ξ)Θ, Θ ) = q((Θ ⊙ Ξ)Ξ , Θ ), 0b 0 0 0 0 0 0 0 0 0 ∗ hence (Ξ0 ⊙ Θ0) = (Θ0 ⊙ Ξ0), and the assertion follows.

Corollary 2.3. For any x ∈ T , 1 ^ x 1 ^ x ∧ [Ξ ⊗ Θ] = 2 [x · Ξ ⊗ Θ] + [Ξ ⊗ x · Θ] x [Ξ ⊗ Θ] = − 2 [x · Ξ ⊗ Θ] − [Ξ ⊗ x · Θ] ^ 1 ^ x ^ 1 ^ x ∧ [Ξ ⊗ Θ] = − 2 [x · Ξ ⊗ Θ]+[Ξ ⊗ x · Θ] x [Ξ ⊗ Θ] = 2 [x · Ξ ⊗ Θ] − [Ξ ⊗ x · Θ] . In particular, the map [·,·] is Spin(n)-equivariant, where Spin(n) acts on ∆ ⊗ ∆ through its spin and on Λ∗ through its vector representation.

10 Corollary 2.4. For any aev,od ∈ Cliﬀ(T n,g)ev,od,

ev ev ev ev [a · Ξ ⊗ Θ]b = ιb(a ⊗ˆ 1) • [Ξ ⊗ Θ]b, [Ξ ⊗ a · Θ]b = ιb(1⊗âˇ ) • [Ξ ⊗ Θ]b od od ˆ od ˆ od ^ [a · Ξ ⊗ Θ]b = ιb(a ⊗1) • [Ξ ⊗ Θ]b, [Ξ ⊗ a · Θ]b = ιb(1⊗aˇ ) • [Ξ ⊗ Θ]b ev^ ev ˆ ^ êv ˆ ev ^ [a · Ξ ⊗ Θ]b = ιb(a ⊗1) • [Ξ ⊗ Θ]b, [Ξ ⊗ a · Θ]b = ιb(1⊗aˆ ) • [Ξ ⊗ Θ]b od^ od ˆ ^ ôd ˆ od [a · Ξ ⊗ Θ]b = −ιb(a ⊗1) • [Ξ ⊗ Θ]b, [Ξ ⊗ a · Θ]b = ιb(1⊗aˆ ) • [Ξ ⊗ Θ]b In particular, we see that ev (i) as a ιb SU(3)l ×SU(3)r –module the image [∆+ ⊗∆+]b in Λ decomposes into the irreducible subspaces [∆+ ⊗ ∆+]b = 1 ⊕ Λ3l ⊕ Λ3r ⊕ Λ9

⊥ ⊥ of (complex) dimension 1, 3 and 9 spanned by the images of Ξ+ ⊗ Θ+, Ξ+ ⊗ Θ+, Ξ+ ⊗ Θ+ and ⊥ ⊥ Ξ+ ⊗ Θ+ respectively. We obtain analogous decompositions for ∆+ ⊗ ∆−, ∆− ⊗ ∆+ and ∆− ⊗ ∆−. The stabiliser of the pair of complex forms [Ξ+ ⊗Θ±]b or [Ξ± ⊗Θ+]b is conjugate to SU(3)×SU(3). The irreducible decomposition of the real SU(3) × SU(3)–modules Λev,od is given by

ev,od R ev,od ev,od ev,od Λ =2 ⊕ Λ6l ⊕ Λ6r ⊕ Λ18 .

ev Here the trivial representations in Λ are spanned by the real and imaginary part of [Ξ+ ⊗ Θ−], evC ⊥ ⊥ evC ⊥ ⊥ evC ⊥ ⊥ ⊥ ⊥ Λ6l = [Ξ+ ⊗ Θ−] ⊕ [Ξ− ⊗ Θ+], Λ6r = [Ξ− ⊗ Θ+] ⊕ [Ξ+ ⊗ Θ−] and Λ18 = [Ξ+ ⊗ Θ−] ⊕ [Ξ− ⊗ Θ+ ], odC and similarly for Λ = [∆+ ⊗ ∆+] ⊕ [∆− ⊗ ∆−]. ∗ (ii) as a ιb G2l × G2r –module the image [∆ ⊗ ∆]b in Λ decomposes into the irreducible subspaces [∆ ⊗ ∆]b = 1 ⊕ Λ7l ⊕ Λ7r ⊕ Λ49

⊥ ⊥ ⊥ ⊥ of dimension 1, 7 and 49, spanned by the images of Ψ+ ⊗ Ψ−, Ψ+ ⊗ Ψ−, Ψ+ ⊗ Ψ− and Ψ+ ⊗ Ψ− ev,od respectively. The stabiliser of [Ψ+ ⊗Ψ−]b is conjugate to G2 ×G2. The irreducible decomposition ev,od of the G2 ×G2–modules Λ is given by projection on the even and odd part of the exterior algebra

ev,od ev,od ev,od ev,od ev,od Λ = 1 ⊕ Λ7l ⊕ Λ7r ⊕ Λ49 .

Proof: We content ourselves to show that the stabiliser is conjugate to SU(3) × SU(3) or G2 × G2 as the remaining assertions are straightforward. Since the stabiliser inside Spin(V+ ) × Spin(V−) was already seen to be conjugate to these groups, we only need to show that we do not acquire any further elements in the stabiliser under the full action of Spin(n,n), i.e. we need to show that the dimension of the stabiliser inside so(n,n) is less or equal than 16 or 28. We have

2 ∗ 2 2 2 so(n,n)=Λ (T ⊕ T )=Λ (V+ ⊕ V−)=Λ V+ ⊕ V+ ⊗ V− ⊕ Λ V− and a basis for the transformations in V+ ⊗ V− =∼ Lin(V−, V+) is given by

Vij = − ei ⊕ eix(g − b)) ∧ (−ei ⊕ eix(−g − b) . 11 Hence, for a generic element A = Aij Vij we obtain 1 P 1 ^ A • [Ξ ⊗ Θ] = A ι (e ⊗ˆ 1) • ι (1⊗ˆ e ) • [Ξ ⊗ Θ] = − A [e · Ξ ⊗ e · Θ]. (2.7) 2 ij b i b j 2 ij i j X X In the generalised G2–case, we consider the real spinors Ξ = Ψ+ and Θ = Ψ− to see that (2.7) vanishes if and only if

Aijei · Ψ+ ⊗ ej · Ψ− =0. i,j X Since the bispinors ei · Ψ+ ⊗ ej · Ψ− are linearly independent, this implies Aij = 0 for all i and j. In the generalised SU(3)–case we get, introducing the shorthand notation i = i +3,

Aijei · Ξ+ ⊗ ej · Θ+ = Aij zi · Ξ+ ⊗ zj · Θ+ − Aijzi · Ξ+ ⊗ zj · Θ+ i,j i,j=1 X X

− iAij zi · Ξ+ ⊗ zj · Θ+ − iAij zi · Ξ+ ⊗ zj · Θ+ 3

Aij ei · Ξ+ ⊗ ej · Θ− = Aij zi · Ξ+ ⊗ zj · Θ− + Aij zi · Ξ+ ⊗ zj · Θ− i,j i,j=1 X X

− iAij zi · Ξ+ ⊗ zj · Θ− + iAij zi · Ξ+ ⊗ zj · Θ− , from which we deduce

Aij ∓ Aij =0, Aij ± Aij =0 and thus Aij = 0 for all i, j = 6. The same argument applies to the pair [Ξ± ⊗ Θ+].

We can give a canonical form for the spinors [Ξ+ ⊗ Θ+],[Ξ+ ⊗ Θ−] and [Ψ+ ⊗ Ψ−] in terms of an orthonormal basis for (T n,g). As a preparation, we prove the

n Lemma 2.5. Let e1,...,en be an orthonormal basis for (T ,g). Then for Ξ, Θ ∈ ∆ 1 [Ξ ⊗ Θ] = q(Ξ, e · Θ)e , n I I XI where I is a multi–index.

Proof: The trace operator Tr A∗B defines a positive-definite hermitian inner product on EndC(∆), C and if eI defines an orthonormal basis for g , then so does κ(eI ) for Tr. To see this, consider the linear complex functional Tr κ(a) on Cliff(T nC,gC) for which there exists an element c ∈ Cliff(T C) such that gC(c,a) = Tr κ(a). An easy computation shows that gC(c,a · b) = gC(a · c,b) = gC(c · b,a). From Tr AB = Tr BA we immediately conclude that c belongs to the center of Cliff(T C,gC) and is b therefore equal to n (put a = 1). We thus obtain b 1 gC(a,b) = gC(1,a · b) = Tr κ(a)κ(b)∗ n b 12 and in particular 1 C Tr κ(e ) ◦ κ(e )∗ = g (e , e ) = δ n I J I J IJ for any two multi-indices I and J. Consequently, using an orthonormal basis Ξ1,..., Ξ8 of ∆, 1 Ξ ⊙ Θ = Tr(Ξ ⊙ Θ ◦ κ(e )∗)κ(e ) n I I XI 1 = q(e · Ξ , (Θ ⊙ Ξ)(Ξ ))κ(e ) n I k k I XI,k 1 = q(eb · Ξ, Θ)κ(e ), n I I XI and applying J ◦ κ−1 yields the assertion. b

Now the action of Spin(6) and Spin(7) on the Stiefel varieties V2(∆±) and V2(∆), the set of pairs of orthonormal spinors, is transitive. As a result, we may assume, given the matrix representation

κ with respect to a ﬁxed orthonormal basis Ξ1,..., Ξ8, that Ξ+ =Ξ1 and Θ+ = cΞ1 + sΞ2 for two complex or real numbers c and s with |c|2 + |s|2 = 1. In the case of Spin(7), c and s are just the

(co-)sine of the angle between Ψ+ and Ψ−.

Corollary 2.6. There exists an orthonormal basis e1,...,en such that the SU(3) × SU(3)– and

G2 × G2–invariant spinors can be written as

[Ξ+ ⊗ Θ+] = se6 − ise5 + c(e136 + e145 + e235 − e246) + ci(−e135 + e146 + e236 + e245)

+s(e125 + e345) + si(e126 + e346) − se12346 + sie12345

[Ξ+ ⊗ Θ−] = c − ci(e12 + e34 + e56) + s(−e13 + e24) + is(e14 + e23))

+c(−e1234 − e1256 − e3456) + s(e1456 + e2356) + si(−e1356 + e2456) − cie123456, where c, s are complex numbers such that |c|2 + |s|2 =1, and

[Ψ+ ⊗ Ψ−] = c + se1 + s(−e23 − e45 + e67) + s(e247 − e256 − e346 − e356)

+c(e123 + e145 − e167 + e246 + e257 + e347 − e356)

+c(−e1247 + e1256 + e1346 + e1357 − e2345 + e2367 + e4567) +

+s(e1246 + e1257 + e1347 − e1356) + s(−e12345 + e12367 + e14567)

−se234567 + ce1234567, where c, s are real numbers such that c2 + s2 =1.

Remark: For a geometrical interpretation of the normal forms in terms of the underlying SU(3)l ∩

SU(3)r – or G2l ∩ G2r–invariants, see [17].

13 Figure 1 suggests that the knowledge of Re([Ξ+ ⊗ Θ±]) or Im([Ξ+ ⊗ Θ±)] and similarly, of either ev od [Ψ+ ⊗Ψ−] or [Ψ+ ⊗Ψ−] , is suﬃcient to reconstruct the full G×G–invariant spinor. The analysis of the relationship between these real spinors is our next task. To begin with, we give the following

∗C ∗ ∗ ∗C Deﬁnition 2.2. The generalised Hodge ⋆–operator 2b :Λ T → Λ T associated with the generalised metric (g,b) is deﬁned by

b/2 b/2 2bτ = 2bτb = (⋆gτ)b = e ∧ ⋆g(e ∧ τ).

In absence of a B–ﬁeld we stick to our generalb convention and writeb simply 2 for 20.

Λ∗C/Λ∗ iΛ∗/Λod

[∆+ ⊗ ∆+] ⊕ [∆− ⊗ ∆+]/ [∆ ⊗ ∆] i2ρ/ 2ρev

ρ/ρev Λ∗/Λev

[∆− ⊗ ∆−] ⊕ [∆+ ⊗ ∆−]/ ^ [∆ ⊗ ∆]

Figure 1: The embedding of ∆ ⊗ ∆ into Λ∗C/Λ∗ and the 2-operator for n =6/7

For subsequent computations it is convenient to have handy the commutation rules of the operators exp(b), ∧ and ⋆. The proof is straightforward.

Lemma 2.7. We have (i) (eb ∧ τ)∧ = e−b ∧ τ. ∓(⋆τ ev,od)∧ for n =6 (ii) ⋆(τ ev,od)∧ = . ( b(⋆τ ev,od)∧ for n =7 (iii) If αp is a p-form, then

2 p ev,od px2 ev,od 2 px ev,od p 2 ev,od b(α ∧ τb ) = (α τ )b, b(α τb ) = (α ∧ τ )b.

As drawn in Figure 1, the 2–operatore has a natural geometric interpretatione as a reﬂection along [∆ ⊗ ∆] in Λ∗C. We make this observation rigourous in the next lemma. Lemma 2.8. 2 ev,odC ev,odC 22 (i) In dimension 6 we have :Λ → Λ and b = −Id. The image of [∆• ⊗ ∆•] in ev,odC Λ is given by the ±i-eigenspaces of 2b: If Ξ±, Θ± ∈ ∆±, then

2b[Ξ+ ⊗ Θ−]b = i[Ξ+ ⊗ Θ−]b, 2b[Ξ− ⊗ Θ−]b = i[Ξ− ⊗ Θ−]b

2b[Ξ+ ⊗ Θ+]b = −i[Ξ+ ⊗ Θ+]b, 2b[Ξ− ⊗ Θ+]b = −i[Ξ− ⊗ Θ+]b.

14 In particular,

+ 2 ∗ [∆− ⊗ ∆−]b ⊕ [∆+ ⊗ ∆−]b = Eig2b (i) = {ρb = ρb − i bρb | ρ ∈ Λ } − 2 ∗ [∆+ ⊗ ∆+]b ⊕ [∆− ⊗ ∆+]b = Eig2b (−i) = {ρb = ρb + i bρb | ρ ∈ Λ }

2 ev,od od,ev 22 ^ (ii) In dimension 7 we have :Λ → Λ and b = Id. The image of [∆⊗∆]b and [∆ ⊗ ∆]b ∗ in Λ is given by the ±1-eigenspaces of 2b: If Ψ+, Ψ− ∈ ∆, then

2b[Ψ+ ⊗ Ψ−]b = [Ψ+ ⊗ Ψ−]b 2 ^ b[Ψ+ ⊗ Ψ−]b = −[Ψ+ ⊗ Ψ−]b.

In particular,

ev ev ev ev [∆ ⊗ ∆]b = {ρ + 2ρ | ρ ∈ Λ } ^ ev 2 ev ev ev [∆ ⊗ ∆]b = {ρ − ρ | ρ ∈ Λ }.

Proof: The matrix representation with respect to the splitting ∆ = ∆+ ⊕ ∆− can be written schematically as ∆ ⊗ ∆ ∆ ⊗ ∆ + − + + . ∆− ⊗ ∆− ∆− ⊗ ∆+ ! For n = 6, the Cliﬀord volume element vol acts then by the matrix

i Id 0 κ(vol) = . 0 −i Id !

Now in general, for any element a ∈ Cliﬀ(Cn,gC) the relation

⋆J(a) = J(a · vol), (2.8) holds, and consequently, b

−1 −1 −1 2bJ(κ (Ξ+ ⊙ Θ+))b = J(κ (Ξ+ ⊙ Θ+ ◦ κ(vol)))b = −iJ(κ (Ξ+ ⊙ Θ+))b.

The remaining assertions are straightforward.

As a result, the invariant spinors in Λ∗C or Λev,od which are induced by a generalised G-structure can be described by two real spinors ρev,od, 2ρev,od for G = SU(3) and by one real spinor ρev, 2ρev for G = G2. The existence of these spinors requires the principal bundle PSpin(n,n) to reduce to a principal G × G–bundle. This reduction process shall occupy us next.

15 2.3 Topological reductions to SU(3) × SU(3) and G2 × G2

Topological reductions from PSpin(n,n) to G × G–structures are in 1-1–correspondence with sections of the associated ﬁbre bundle whose typical ﬁbre Spin(n,n)/G × G is of dimension 50 (n = 6) and 63 (n = 7) respectively. This can be broken down into choices of the following set of data, namely a metric g, a 2-form b and two unit spinors Ξ+/Ψ+ and Θ+/Ψ− for which there are

g : 21/28 + b : 15/21 + Ξ+/Ψ+ :7 + Θ+/Ψ− :7 =50/63 degrees of freedom. The reduction process can be understood in two steps. As we pointed out in the

,−previous section, the inclusion G × G֒→ Spin(n) × Spin(n) ﬁrst yields a metric splitting V+ ⊕ V or equivalently, a metric g and a 2-form b. Secondly, this inclusion tells us that the vector bundles

V± are associated with principal G-ﬁbre bundles. Using the isometries π± (2.6), we can pull these G-structures back to the tangent bundle where they give rise to G–invariant T –spinors.

Proposition 2.9. The set of data (M n,g,b) and the choice of two unit spinors is equivalent to a reduction from the Spin(n,n)–principal ﬁbre bundle PSpin(n,n) to a principal G × G–ﬁbre bundle. Moreover, we obtain a G × G–invariant spinor.

In our situation, however, it is more natural to consider the action of the conformal spin group

R>0 × Spin(n,n), and to describe reductions inside the canonically extended principal ﬁbre bundle

PR>0×Spin(n,n). This additional degree of freedom is then accounted for by the dilaton φ. We ﬁrst ev,od ev,od examine the case of generalised G2–structures and denote by U ⊂ Λ the set of G2 × G2– invariant spinors. The conformal spin group, which is of dimension 92, acts transitively on the orbits U •, •∈{ev, od} [26], [28]. Hence, they are of dimension 64 and therefore open. Following the language of Hitchin, we call such spinors stable [15]. By abuse of notation let U • also be the open set of invariant sections in Ω•(M), and take an arbitrary section, say ρev ∈ U ev (the parity is of no importance here). This choice induces a reduction to a principal G2 × G2–bundle. In particular, we can split oﬀ a line bundle from Ωev(M) which is trivialised by ρev. On the other hand, we are ev given a trivialisation in the form of the invariant spinor [Ψ+ ⊗ Ψ−]b . Therefore, possibly after ev ev −φ taking −ρ , there exists a function φ so that ρ = e [Ψ+ ⊗ Ψ−]b. As a consequence, there is a

1-1-correspondence between generalised G2–structures, topological reductions from PR×Spin(7,7) to a principal G2 ×G2–bundle, and G2 ×G2–invariant spinors. The open orbit condition is the starting point for the variational problem we will consider in Section 4. To motivate the introduction of the conformal spin group in the case of generalised SU(3)– structures we need to make contact with Hitchin’s notion of a generalised Calabi-Yau manifold [15]. These are defined by a complex pure spinor τ ∈ Λev,odT ∗C such that q(τ, τ) 6= 0. Here, a complex spinor is said to be pure if the stabiliser under the Clifford action of (T 6 ⊕ T 6∗)C is maximally lightlike, i.e. has dimension 6. It induces a generalised complex structure J on T ⊕ T ∗, which means that J squares to minus the identity. Moreover, a complex T 6 ⊕ T 6∗–spinor τ = τ ev ⊕ τ od ∈ Λ∗C is said to define a generalised Calabi-Yau metric [12] if

16 • τ ev,od are pure

• the induced generalised complex structures J ev and J od commute with each other

ev,od ev od • J preserves a metric splitting (V±,g±) such that −J J = g+ ⊕ g−

• the induced complex structures on V± admit a nowhere vanishing holomorphic volume form

• the length condition q(τ ev, τ ev) = cq(τ od, τ od) holds for a real constant c. By a universal rescaling we may assume c = −1 (substituting, if necessary, τ ev by τ ev).

Remark: The first three properties define what is called a generalised Kähler structure. We chose a slightly different formulation from that in [12] where one requires −J evJ od to define a positive definite metric on T ⊕ T ∗. In view of our approach in terms of structure groups sitting inside Spin(n,n) it is more natural to consider a metric splitting compatible with (·, ·), the metric of split signature.

6 Proposition 2.10. Let (M ,g,b,φ, Ξ+, Θ+) be a generalised SU(3)–structure. The complex spinors

[Ξa ⊗ Θb], a, b = ±, are pure. Moreover,

q([Ξ+ ⊗ Θ±], [Ξ+ ⊗ Θ±]) = ∓8ivolg ,

∗ −φ −φ hence the pair of complex T ⊕ T –spinors e [Ξ+ ⊗ Θ±] or e [Ξ± ⊗ Θ−] deﬁnes a generalised Calabi-Yau metric.

Proof: First the length condition follows from a computation using the normal form. Next, let Jl,r denote the almost complex structure on T induced by SU(3)l,r , the stabilisers of Ξ+ and Θ+. We transport these to V± by the maps π±, that is

J+ιb(x⊗ˆ1) = ιb(Jlx⊗ˆ1), J−ιb(1⊗ˆy) = ιb(1⊗ˆJr y) deﬁne almost complex structurese on V± with nowheree vanishing holomorphic volume form. Now the generalised complex structures deﬁned with respect to the metric splitting V+ ⊕ V−,

−J 0 J 0 J ev = + , J od = + , 0 J− ! 0 J− ! e e e e commute and are compatible with the generalisede metric (g,b). Wee only need to show that they coincide with the generalised complex structures induced by [Ξ+ ⊗ Θ−] and [Ξ+ ⊗ Θ+], i.e. their respective annihilators are given by

ev ev ev N = {v + iJ v | (v + iJ v) • [Ξ+ ⊗ Θ−], v ∈ V+ ⊕ V−}, N od = {v + iJ odv | (v + iJ odv) • [Ξ ⊗ Θ ], v ∈ V ⊕ V }. e e + + + − e e 17 Indeed, we have

ev ev (v ⊕ iJ v) • [Ξ+ ⊗ Θ−] = ιb(x⊗ˆ1) + ιb(1⊗ˆy) + iJ (ιb(x⊗ˆ1) + ιb(1⊗ˆy)) • [Ξ+ ⊗ Θ−]

= ιb(x⊗ˆ1) + ιb(1⊗ˆy) − iιb(J+x⊗ˆ1) + iιb(1⊗ˆJ−y) • [Ξ+⊗ˆΘ−] e e ^ = [(x − iJ+x) · Ξ+ ⊗ Θ+] − [Ξ+ ⊗ (y + iJ−y) · Θ+], which vanishes by the deﬁnition of the complex structures J+ and J−. We can proceed similarly with the reamining spinors which proves the assertion.

The real parts of the complex spinors defining a generalised Calabi-Yau manifold are acted on transitively by R⋆ × Spin(6, 6) and form an open orbit U • – they are stabilised by SU(3, 3) [15]. Conversely, Hitchin shows that a stable spinor ρ• induces another stable spinor of same chirality ρ•∗ such that τ • = ρ• + iρ•∗ defines a generalised Calabi-Yau manifold. Moreover, ρ•∗∗ = −ρ•, so both the real and the imaginary part give rise to the same generalised structure. If τ ∈ Ω∗C(M 6) defines a generalised Calabi-Yau metric, we obtain a topological reduction to a principal SU(3) × SU(3)– ev,od bundle inside PSpin(6,6). Now we assume furthermore that τ are eigenspinors of the induced ev,od ev,od ev,od∗ ev,od 2b–operator, say with eigenvalue ±i. Then for ρ = Re τ we have ρ = ∓2bρ , that ev,od ev,od ev,od ev,od is τ = ρ ∓ i2bρ . Since ρ are sections of the line bundle spanned by Re [Ξ+ ⊗ Θ±]b ev,od ev,od −φev,od there are two scalar functions φ such that ρ = e Re[Ξ+ ⊗ Θ±]b. But now the length condition implies

−2φev −2φod e q Im[Ξ+ ⊗ Θ+]b, Re[Ξ+ ⊗ Θ+]b = −e q Im[Ξ+ ⊗ Θ−]b, Re[Ξ+ ⊗ Θ−]b , hence φev = φod . Summarising, we obtain the

Theorem 2.11. 6 (i) Generalised SU(3)-structures (M ,g,b,φ, Ξ+, Θ+) are in 1-1 correspondence with topological reductions from PR>0 ×Spin(6,6) to a principal SU(3) × SU(3)–bundle. This bundle can be characterised by an SU(3) × SU(3)–invariant spinor ρ = ρev ⊕ ρod such that

q(ρev, ρev∗) = −q(ρod , ρod∗).

In particular, generalised SU(3)–structures coincide with generalised Calabi-Yau metrics in (real) ev,od dimension 6 deﬁned by eigenspinors τ for the induced 2b–operator. The spinors can be uniquely written (modulo a simultaneous sign change for the spinors Ξ+ and Θ−) as

ev −φ od −φ ρ = e Re[Ξ+ ⊗ Θ−]b, ρ = e Re[Ξ+ ⊗ Θ+]b.

7 (ii) Generalised G2-structures (M ,g,b,φ, Ψ+, Ψ−) are in 1-1 correspondence with topological reductions from PR>0×Spin(7,7) to a principal G2 × G2–bundle. This bundle can be characterised by ev ev 7 a G2 × G2–invariant spinor ρ ∈ Ω (M ) which can be uniquely written (modulo a simultaneous sign change for Ψ+ and Ψ−) as ev −φ ev ρ = e [Ψ+ ⊗ Ψ−]b .

18 3 Backgrounds in string theory and Ramond-Ramond ﬁelds

In this section we derive the equations which govern the geometry of supergravity compactifications to a 6– or 7–dimensional background. The starting point are the 10–dimensional supersymmetry variations involving two kinds of fields, namely NS-NS– (Neveu-Schwarz) and R-R– (Ramond–Ramond) fields. Before we deal with the mathematical aspects we briefly sketch the theory of type II theories, following [10], [11], [20], [24], [25], [29].

3.1 Type II theories

Strings are 1–dimensional objects and therefore come in two flavours – they are open or closed if the parametrising intervall is open or closed. As strings evolve in a d-dimensional space-time M 1,d−1, they sweep out a 2-dimensional surface, the worldsheet Σ. Instead of thinking of Σ as an 1,d−1 embedded surface, one can interpret the embedding as a bosonic field XΣ :Σ → M . Moreover, Σ 1,d−1 we have two fermionic fields – the so–called left and right mover – ψL, ψR ∈ Γ(∆L,R ⊗ T M|Σ ), Σ where ∆L,R is a spin bundle associated with some spin structure on Σ (they are not necessarily the same). We can use these fields to define an action functional which determines the physical theory. After an appropriate quantisation process, it turns out that for a consistent theory the space-time needs to be of dimension 10. A key requirement for this action is to be invariant under supersymmetry transformations. These map the bosonic field content, which we recall we can think of in mathematical terms as elements in a Spin(1, 9)–tensor representation, in a 1-1–fashion onto the fermionic field content, that is, elements in a Spin(1, 9)–spin representation. The amount of supersymmetry one requires to be preserved depends on the theory. For the moment, we know five consistent superstring theories, namely type I, type IIA, type IIB, and two heterotic theories. They are non-trivially related to each other through duality maps. In this article we restrict ourselves to type II theories. Both types are defined by a closed string. In this case, Σ is diffeomorphic to the cylinder S1 ×R on which we introduce the complex coordinate w = σ1 + iσ2. Now there exist two spin structures on Σ which are determined by the Ramond or Neveu–Schwarz gluing condition

Ramond (R): ψL,R(w +2πσ1) = ψL,R(w),

Neveu–Schwarz (NS): ψL,R(w +2πσ1) = −ψL,R(w).

After quantisation the left and right movers become operators over Hilbert spaces which we accord- ingly label by NS and R. For these Hilbert spaces one can construct a discrete spectrum of states and a mass operator whose eigenvalues assign a “mass” to each of these states. For a meaningful theory we need a vacuum state, that is a massless ground state. This makes perfect sense from a phenomenological point of view, as the next level of the mass spectrum has Planck mass. However, particles of our “real” world are far too light and should be rather considered as perturbations of a

19 vacuum state. For the action of the zero modes, i.e. operators preserving the massless ground states, a careful analysis shows that the NS–ground states are non–degenerate and can be represented by a vector, while the zero modes acting on the (degenerated) R–ground states satisfy the Clifford algebra relations, reflecting the fact that R–ground states can be represented by a spinor. We single out the physically relevant part of the spectrum by a GSO–projection. For this we need to introduce the fermion operator F which labels every state with + or −. The combination NS-, however, is ruled out as the mass squared of these states is negative. In type II theories the various particles belong to sectors which are defined by the orderd pair of the zero mode ground states. For instance, the (NS+,R+)–sector indicates that the left moving ground state is NS+ while the right moving ground state is R+. To formulate the type II theories consistently we need to impose three selection rules, for instance modular invariance at 1-loop. For type IIA and IIB this forces the vacua to lie in the sectors

IIA: (NS+, NS+) ⊕ (R+, NS+) ⊕ (NS+,R−) ⊕ (R+,R−) IIB: (NS+, NS+) ⊕ (R+, NS+) ⊕ (NS+,R+) ⊕ (R+,R+).

The eﬀective dynamical degrees of freedom for the left and right mover are given by the normal 1,9 bundle NΣ of the world–sheet in T M . Consequently, the internal symmetry group for the degrees of freedom of the NS– and R–states is SO(8) and Spin(8). By the triality principle we can associate the vector representation 8V with NS+ and the two spin representations of positive and negative chirality, 8 and 8′, with R+ and R- respectively. In the sequel, we drop the label ± for sake of simplicity. The physical ﬁelds are then accounted for by elements in the irreducible components of the tensor product associated with a given pair. For instance, the NS-NS–sector is represented by

8V ⊗8V and is therefore bosonic. It can be decomposed into the trivial representation 1, the 2-forms 2 2 1,9 Λ 8V and the symmetric 2–tensors 8V. On the space–time M this gives rise to a dilaton field 2 1,9 φ, a B-field b ∈ Ω (M ) and a metricJ g respectively. The second bosonic sector is the R-R–sector. Here, the elements of the irreducible components induce the so–called Ramond-Ramond potentials. For type IIA one obtains C1 ∈ Λ1 and C3 ∈ Λ3 and for type IIB we have C0 ∈ Λ0, C2 ∈ Λ2 and the 4 4 self–dual 4–form C+ ∈ Λ+. In the fermionic sectors R-NS and R-NS we find the gravitino ΨX (which is of spin 3/2, X denoting the vector index) and the dilatino λ (of spin 1/2). The full description is given in Table 1. Note that the subscript of the gravitino distinguishes with which mover the R–sector is associated; + denotes the left and − the right mover. By considering only massless states and passing to the low energy limit of superstring theory, we effectively enter the realm of supergravity. Here, space–time supersymmetry can be achieved by introducing a globally well–defined supersymmetry parameter ǫ. This parameter is non–physical inasfar it is introduced for mathematical convenience. For type II theories, ε is built out of two space–time spinors ε1,2 (whence the name type II), which we rearrange as a 2–component vector

ε = (ε1, ε2). For type IIA, ε1,2 are of opposite and for type IIB of equal chirality. For the supergravity action to be invariant under the supersymmetry transformation induced by ε, the supersymmetry variations of the bosonic and fermionic ﬁelds have to vanish. It turns out that the supersymmetry

20 sector representation bosons/fermions dim 2 2 (NS+, NS+) 8V ⊗ 8V = 1 ⊕ Λ 8V ⊕ 8V φ ⊕ b ⊕ g bosonic 1+28+35 ′ 3 ′ (R+, NS+) 8 ⊗ 8V = 8 ⊕ Λ 8 J λ+ ⊕ ΨX+ fermionic 8+56 ′ 3 (R−, NS+) 8 ⊗ 8V = 8 ⊕ Λ 8 λ+ ⊕ ΨX+ fermionic 8+56 ′ 3 ′ (NS+,R+) 8V ⊗ 8 = 8 ⊕ Λ 8 λ− ⊕ ΨX− fermionic 8+56 ′ 3 (NS+,R−) 8V ⊗ 8 = 8 ⊕ Λ 8 λ− ⊕ ΨX− fermionic 8+56 2 4 0 2 4 (R+,R+) 8 ⊗ 8 = 1 ⊕ Λ 8V ⊕ Λ+8V C ⊕ C ⊕ C+ bosonic 1+28+35 ′ 3 1 3 (R+,R−) 8 ⊗ 8 = 8V ⊕ Λ 8V C ⊕ C bosonic 8+56

Table 1: The various sectors appearing in type II theories

variations of the bosonic fields depend only on the physical fermions (the gravitino and the dilatino) and the ε. Conversely, the variations of the fermionic fields depend only on the bosonic field content and ε. We make the usual assumption to deal with a vacuum background, that is there are no fermionic fields. This implis the vanishing of the bosonic variations, but of course we still need the vanishing of the fermionic variations,

δεΨX =0, δελ =0.

The gravitino variation comes down to a Killing spinor equation for the components ǫ1,2 of ε, where the spin connection is provided by the metric, while the dilatino variation involves the differential of the dilaton field dφ. In both equations, the B–field b and the R-R–potentials Cp−1 act through their differentials, i.e. their field strengths H ∈ Ω3(M 1,9) and F p ∈ Ωp(M 1,9). Now on the space– time we can generate forms of higher degree by taking, up to a sign, the Hodge dual of F p. This, of course, does not increase the number of degrees of freedom. For instance, consider the R-R– potential C2 in type IIB. It induces the field strength F 3 and thus yields a Hodge dual 7–form F 7 with corresponding R-R–potential C6. For type IIA/B, we thus obtain an odd or even R-R– potential Cod,ev with corresponding even or odd R-R–field strength F ev,od ∈ Ωev,od(M 1,9). For type IIA, this requires the introduction of an additional scalar field F 0 that describes the mass parameter in massive type IIA supergravity, but we do not enter into the details here. Summarising, the precise algebraic constraint on the R-R–field F ev,od is an anti–self–duality condition with respect to the space–time 2–operator, ev,od ev,od F = −2M 1,9 F . (3.9)

This point of view becomes particularly convenient when adopting the democratic formulation of Bergshoeﬀ et al. [2], which treats both type IIA and IIB on equal footing. We turn to this formulation next.

21 3.2 Supersymmetry variations

In this section we want to discuss the precise shape of the supersymmetry variations for type II theories in dimension 10, before we compactify down to a 6–dimensional background in the next section. Following the spirit of generalised geometry, and also the so–called democratic version of the supersymmetry variations [2], we regard R-R–ﬁelds as an even form F ev for type IIA and as an odd form F od for IIB.

For a vacuum background, the two supersymmetry variations – one for the gravitino ΨX , X ∈ Γ(T M 1,9), and one for the dilatino λ, are given by

1 1 φ ev,od ev,od δεΨX = ∇X + XxH ·P ε + e F · X · P ε, (3.10) 4 16 1 1 φ ev,od p ev,od δελ = (dφ · + H ·P)ε ∓ e 5F − pF · P ε. (3.11) 2 8 p=Xev,od

Here, ε = (ε1, ε2) is the 2–component vector containing the type II supersymmetry parameters ε1,2, and similarly for ΨX and λ. Note that for type IIB the parameters are spinors of equal chirality, that 11 11 is, using the standard notation Γ = vol10, we have Γ ε1,2 = ε1,2. The operator P is deﬁned to be 11 ev,od p Γ in type IIA and −σ3 in type IIB, while the components of P = p=ev,od P are determined according to the rule P p mod 4 01 2 3 p 11 P σ1 iσ2 Γ σ1 σ1 where σ1, σ2 and σ3 are the Pauli matrices

0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . 1 0 ! i 0 ! 0 −1 !

The supersymmetric space–time vacuum background is required to satisfy

δεΨX =0, δελ =0, but for our purposes it will be more convenient to consider the equivalent set of conditions

δεΨX =0, ek · δεΨek − δελ =0 X and to formulate these in terms of spinor ﬁeld equations. Let D denote the Dirac–operator which for a local orthonormal basis can be written as D =

ek ·∇ek . Substituting (3.10) in ek · δǫΨek gives for the NS-NS–part P 1 P 1 e · ∇ + e xH ·P ε = Dε + e ∧ (e xH) − e x(e xH) ·Pε k ek 4 k 4 k k k k X 3 X = D + HP· ε, 4

22 while using (2.4) for the R-R–part yields

ev,od ev,od ev,od ev,od ev,od ev,od ek · F · ek · P ε = ± ek · ek ∧ F · P ε + ekxF · P ε

X X ev,od ev,od ev,od = ∓ ekx(ek ∧ F ) − ek ∧ (ekxF ) · P ε

X ev,od p Xev,od = ∓ 10F − 2 pF · P ε. ev,odX Up to the factor eφ/8, this is precisely the R-R–part of (3.11) so we consider the expression 1 e · δΨ − δλ = D − dφ · + HP· ε, k ek 4 X known under the name of modiﬁed dilatino variation [8], [22]. In summary, the supersymmetry variations of IIA/B supergravity come down to the ﬁeld equations

1 1 φ ev,od ev,od ∇X + XxH ·P ε + e F · X · P ε =0 (3.12) 4 16 1 (D − dφ · + HP·)ε =0 (3.13) 4 for all X ∈ Γ(T M 1,9). To ﬁnd a solution of (3.12) and (3.13), we make a compactiﬁcation ansatz for the space M 1,9, that is we assume the space–time to be a direct product of the form R1,3 × M 6 or R1,2 × M 7.

3.3 Compactiﬁcation on M 6

We consider the compactification of type IIB supergravity on a 6–dimensional background in full detail; we will comment on the remaining cases at the end of this section, as certain problems arise in type IIA. On M 1,9 we choose a local coordinate system xK , K =0,..., 9 and label the external coordinates by xµ, µ = 0,..., 3 whereas the internal coordinates are labeled by Xa, a = 1,..., 6. We fix representations of Cliff(R1,9,g1,9), Cliff(R1,3,g1,3) and Cliff(R6,g6) and denote by ΓK , γµ and γa 1,9 1,9 the corresponding matrices. As in Section 2.2 we can write Cliff(R ,g )asa(Z2–graded) tensor product, so that e γ ⊗ I : µ =0,..., 3 ΓK = µ , 5 ( γ ⊗ γa : a =1,..., 6 5 7 e where we put γ = i vol4 and γ = −i vol6 (again, we follow the usual notation convention in the e 11 5 7 5 2 physics’ literature). In particular, we have Γ = vol10 = γ ⊗ γ . Note also that (γ ) = I. For the supersymmetrye parameters we choose the simple splitting e e ε1 = ζ+ ⊗ Θ+ + ζ− ⊗ Θ− ,

ε2 = ζ+ ⊗ Ξ+ + ζ− ⊗ Ξ− ,

23 1,3 ∼ C where ζ± ∈ ∆± and ζ− = ζ + (note that Spin0(1, 3) = SL(2, ), so we have, as for n = 6, a conjugation map on the space of spinors which interchanges the chirality), and the internal spinors 6 Θ±, Ξ± ∈ ∆± satisfy Θ− = Θ+. This simple ansatz is not only for computational convenience, it makes also sense from a physical point of view as we do not expect to observe supersymmetry directly. Instead, supersymmetry is supposed to be broken and therefore, it is only natural to start from a minimal supersymmetric vacuum background.

For the R-R–field F in 10 dimensions we introduce two sets of internal R-R–fields F1 and F2. We combine these to preserve 4–dimensional Poincaréinvariance, i.e.

F = vol4 ∧ F1 + F2. (3.14)

In this setting, the 10–dimensional Hodge duality constraint (3.9) can be rephrased as

F ev = 2F ev, F ev = −2F ev, 1 2 1 2 (3.15) od 2 od od 2 od F1 = − F2 , F1 = − F2 . d d Now we are in a position to explicitly carry out thed compactificationd on M 1,9 = R1,3 × M 6. We start with the gravitino variation (3.12) and focus on the external part first. We assume the H-field and the dilaton φ to live exclusively on the internal space, so that for the moment, we only need to take the R-R part into account. Then, for instance, we find for the R-R 1–form term

1 5 1 I 5 1 F · Γµ · P1 ε = (γ ⊗ F2 )(γµ ⊗ )iσ2ε = (−γµγ ⊗ F2 )iσ2ε.

The remaining terms follow analogously.e Appliede to ε1 we obtaine e

1 3 5 1 3 5 γµζ+ ⊗ F2 − F2 + F2 + i(F1 − F1 + F1 ) Θ+

1 3 5 1 3 5 + γeµζ− ⊗ − F2 + F2 − F2 + i(F1 − F1 + F1 ) Θ− = −γ ζ ⊗ F od + iF od Θ + γ ζ ⊗ F od − iF od Θ eµ + 2 1 + µ − 2 1 − (3.15) od 2 od od 2 od = −eγµζ+ ⊗ Fd2 − idF2 Θ+ +e γµζ− ⊗ dF2 +di F2 Θ−. Proceeding in the same waye with ε2,d the vanishingd of thee supersymmetryd variationd δεΨµ = 0 implies – in conjunction with (2.8) – the external algebraic constraints

od 2 od od od 2 od od (F2 − i F2 ) · Ξ+ = 2F2 · Ξ+ = 0, (F2 + i F2 ) · Ξ− = 2F2 · Ξ− = 0, od 2 od od od 2 od od (F2 − i F2 ) · Θ+ = 2F2 · Θ+ = 0, (F2 + i F2 ) · Θ− = 2F2 · Θ− = 0. (3.16) Nextd we investigated the internald part of the gravitinod variation.d For the NS-NS–partd we ﬁnd

1 bc 1 bc ζ+ ⊗ ∇a − Habcγ Θ+ + ζ− ⊗ ∇a − Habcγ Θ− 1 bc 4 4 I ⊗∇a + Habcγ (−σ3) ε = . 4  ζ ⊗∇ + 1 H γbcΞ + ζ ⊗ ∇ + 1 H γbcΞ  + a 4 abc + − a 4 abc −  

24 The internal R-R–part can be dealt with in a similar fashion as the external part. For instance we obtain for the R-R 5-form term

5 5 1 I 5 F Γa P5 ε = − iγ ⊗ F1 + ⊗ F2 (ε2, −ε1). Proceeding with the remaining terms in a similare fashion, we obtain for the entire R-R–part

od 2 od od 2 od ζ+ ⊗ F2 + i F2 γaΞ+ + ζ− ⊗ F2 − i F2 γaΞ− .  od 2 od od 2 od  ζ+ ⊗ F2 + i F2 γaΘ+ + ζ− ⊗ F2 − i F2 γaΘ−   Combining the NS-NS– andd R-R–part,d we ﬁnally derived the internald gravitino equation, namely

1 1 φ od ∇X Θ+ − (XxH) · Θ+ + e F · X · Ξ+ = 0 4 8 2 (3.17) 1 x 1 φ od ∇X Ξ+ + 4 (X H) · Ξ+ + 8 e F2 · X · Θ+ = 0 for all X ∈ Γ(T M 6). d For the modified dilatino equation (3.13) we note that there is no external contribution, for the spinors are parallel and the H–field and the dilaton φ are only defined on the internal space. Thus the modified dilatino variation boils down to 1 1 D − dφ − H · Θ =0, D − dφ + H · Ξ =0. (3.18) 4 + 4 + This procedure can also be carried out for type IIA, where we obtain the external constraint

(F ev + i2F ev) · Θ = 0, (F ev − i2F ev) · Θ = 0, 2 2 + 2 2 − (3.19) ev 2 ev ev 2 ev (F2 − i F2 ) · Ξ− = 0, (F2 + i F2 ) · Ξ+ = 0, the gravitino equationd d d d

∇ Θ + 1 (XxH) · Θ + 1 eφ(F ev + i2F ev) · X · Ξ = 0 X + 4 + 16 2 2 − (3.20) 1 x 1 φ ev 2 ev ∇X Ξ+ − 4 (X H)Ξ+ − 16 e (F2 + i F2 ) · X · Θ− = 0, d d and the modiﬁed dilatino variation

1 1 (D − dφ + 4 H) · Θ+ =0, (D − dφ − 4 H) · Ξ+ =0.

This set of equations was also derived in [22]. However, (2.8) implies the external constraint (3.19) to be trivially satisfied and the R-R–part in the gravitino variation (3.20) to vanish which is clearly unphysical and suggests a sign error in the democratic formulation in [2]. Finally, following [17], we briefly address compactifications of the form M 1,9 = R1,2 × M 7. Here, the situation is somewhat different as there are no chiral spinors in odd dimensions. We start with ev od an analogous ansatz as in (3.14), but we have to introduce an even F1 and an odd form F2 to ev incorporate the internal R-R–fields. However, the Hodge duality constraint (3.9) identifies F1 with od F2 , so that we only have one independent set of internal R-R–fields. In the same vein as above,

25 one then derives a similar set of conditions for the 7–dimensional background geometry. In summary, what we wish to solve are the equations (3.17) and (3.18) together with the algebraic constraint (3.16) for the data (M n,g,b,φ) plus two unit spinors, that is, we are given a topological generalised G–structure, together with an additional even or odd form F •. In the next section, we ﬁrst introduce the R-R–ﬁelds into the generalised setup, where they give rise to a constrained variational problem. Its spinorial formulation will then provide a solution to the internal supersymmetry variations.

4 Constrained critical points

4.1 An invariant functional and its ﬁrst variation

In this section we formulate a constrained variational problem which generalises the one considered in [28]. It is based on ideas from [14] and [15] which also have found applications in the recent mathematics and physics literature [16], [23], [9]. The starting point is the observation that we can associate with any stable spinor a volume form in a Spin(n,n)–invariant way. If the manifold is compact, then this volume form can be integrated and thus defines a smooth functional over the open set U • of stable spinors. As in classical Hodge theory, we consider the variation of this functional over a fixed cohomology class and ask for critical points. We first define the volume form. For generalised SU(3)–structures, the spinor ρ = ρev ⊕ ρod is not generic, but its chiral components ρev,od are stable. As pointed out above, any stable spinor ρ• gives rise to another stable spinor of same chirality, ρ•∗. On U •, we can then define

Q : ρ• ∈ U • ⊂ Λ• 7→ q(ρ•, ρ•∗) ∈ Λ6, which is smooth, Spin(6, 6)–invariant and homogeneous of degree 2. For n =7,a G2 × G2–invariant • •∗ • spinor ρ induces a stable of opposite chirality ρ = 2ρρ , where we wrote 2ρ to emphasise the dependence on ρ. We assign a volume form by the function

• • • 7 Q : ρ ∈ U ⊂ Λ 7→ q(ρ, 2ρρ) ∈ Λ which again is smooth, Spin(7, 7)–invariant and homogeneous of degree 2. In both cases, the deriva- tive of Q at ρ is given by [15], [28] ∗ DρQ(˙ρ) = q(˙ρ, ρ ). n • Next we look at an oriented, closed manifold M equipped with a stable form ρ . From a GL+(n)- point of view, ρ• is a section of a vector bundle with fibre Λ• ⊗(ΛnT )1/2 (cf. Section 2.1). Untwisting by the line bundle (ΛnT )1/2, we obtain a corresponding open set in the space of differential forms, • still denoted by U , on which we can consider the induced volume functional Q. It defines a GL(n)+- equivariant function since Q : U • ⊂ Λ•T ∗ → ΛnT ∗ as

Q(A∗ρ•) = (det A)−1Q(A • ρ•) = (det A)−1Q(ρ•).

26 n Associated with the GL+(n)–principal fibre bundle over M, Q thus takes values in Ω (M) and we obtain the volume functional V (ρ•) = Q(ρ•). ZM As stability is an open condition, we can differentiate this functional and consider its variation over a fixed cohomology class. Instead of working with ordinary cohomology only we will allow for an extra twist by a closed 3-form H. This means that we replace the differential operator d by the twisted operator dH = d + H∧. Closedness of H guarantees that dH still defines a differential complex. Moreover, we will also consider the constraint imposed by the following linear functional over H–exact forms. We first consider the case n = 6. Out of the Spin(6, 6)–invariant form q we obtain a non-degenerate pairing on Ωev,od(M) by

q(α, β) ZM for two forms α, β of equal parity. If α = dH γ is H-exact, then Stokes’ theorem implies that

q(dH γ, β) = − q(γ,dH β), ZM ZM \ for dβev,od = ∓(dβev,od)∧. Consequently, we can identify

ev,od 6 ∗ ∼ ev,od 6 ev,od 6 ΩH-exact(M ) = Ω (M )/ΩH-closed(M ).

od,ev 6 The twisted diﬀerential dH maps the latter space isomorphically onto ΩH-exact(M ) so that

ev,od 6 ∗ ∼ od,ev 6 ΩH-exact(M ) = ΩH-exact(M ).

ev,od od,ev 6 As a result, any exact form F induces a linear functional QF ev,od on ΩH-exact(M ) given by

QF (dH γ) = q(γ,F ). ZM In the same vein, we can identify the spaces

• 7 ∗ ∼ • 7 ΩH-exact(M ) = ΩH-exact(M ).

7 • 7 • over a 7-manifold M and obtain a linear functional on ΩH-exact(M ) for any form F which is exact. Theorem 4.1. ev,od ev,od An H–exact stable form ρ is a critical point subject to the constraint QF • (ρ ) = const, where F • ∈ Ω•(M n) is a form of suitable parity, if and only if there exists a real constant λ such that ev,od∗ • dH ρ = λF .

od,ev n Proof: We vary over the trivial cohomology class which we identify with dH Ω (M ). The ﬁrst ev,od variation of V at ρ = dH γ is

ev,od∗ ev,od∗ (δV )ρev,od (dH α) = Dρev,od (dH α) = q(dH α, ρ ) = ± q(α,dH ρ ), Z MZ MZ

27 ev,od whereas the diﬀerential of QF • at ρ is

• (δQF • )ρev,od (dH α) = q(α,F ). MZ

By Lagrange’s theorem, we need (δV )ρev,od = λ(δQF • )ρev,od for a critical point, or equivalently ev,od∗ • dH ρ = λF .

If we have two critical points ρev,od for the constrained variational problem defined by F = F ev ⊕ F od such that the spinor ρ = ρev ⊕ρod defines a generalised SU(3)–structure, then ρev,od∗ = ∓2ρev,od as in the generalised G2–case. We adopt the critical point condition as integrability condition for all G–generalised structures, whether or not M is compact or ρ• exact. Definition 4.1. Let d2 = 2 d 2. (i) Let (M 6, ρ) be a generalised SU(3)-structure, H a 3-form and F ∈ Ω∗(M 6). The structure is said to be integrable with respect to H and F if and only if

2ρ dH ρ =0, dH ρ = F.

ev ev od od Equivalently, we can write for τ = ρ − i2ρρ ⊕ ρ + i2ρρ more succinctly

dH τ = −i2τ F

7 • • • (ii) Let (M , ρ ) be a generalised G2-structure, H a 3-forme and F ∈ Ω (M). Then the structure is said to be integrable with respect to H and F ev,od if and only if

ev,od dH ρ = F ,

• • where ρ = ρ ⊕2ρρ . An integrable structure is said to be of even or odd type according to the parity of F •. (iii) An integrable generalised G–structure is parallel if F =0.

Remark: In [28] parallel structures were called strongly integrable. However, in view of the remark after Theorem 4.3 and 4.4 we shall stick to the new nomenclatura. Note also that part (ii) of the previous deﬁnition comprises the notion of a weakly integrable generalised G2-manifold [28] by • • • choosing F = λ2ρρ . In the present paper we put the emphasis on F which is why we use the adjectives even and odd in the opposite way.

4.2 Spinorial formulation of the variational problem

The central theorem of this paper is the reinterpretation of the integrability conditions in Deﬁni- tion 4.1 in terms of spinor ﬁeld equations for the underlying SU(3)l,r – and G2l,r–spinors. To this end, we shall make intensive use of the twisted Dirac operators D and D on ∆ ⊗ ∆, given locally by

D(Ξ ⊗ Θ) = ei ·∇ei Ξ ⊗ Θ + ei · Ξ ⊗∇ei Θ, D(Ξ ⊗ Θ) = ∇eibΞ ⊗ ei · Θ+Ξ ⊗ ei ·∇ei Θ X X b 28 for a local orthonormal basis e1,...,en. Here, ∇ designates the Levi-Civita-connection as well as its lift from the tangent bundle T to ∆, and d∗ = (−1)n(p+1)+1 ⋆d⋆ is the formal adjoint of d on p-forms. It is well-known that under the Spin(n)-equivariant identiﬁcation of ∆ ⊗ ∆ with p-forms, D and D transform into d + d∗ and (−1)p(d ± d∗) [19]. In the generalised context we obtain the following identiﬁcation. e 2b Proposition 4.2. Let d = 2b d 2b. We have

2 1 [D(Ξ ⊗ Θ)] = d[Ξ ⊗ Θ] − (−1)nd b [Ξ ⊗ Θ] + eb/2 ∧ dbx[Ξ ⊗ Θ] − db ∧ [Ξ ⊗ Θ] b b b 2 and ^ 2 ^ 1 ^ ^ [D(Ξ ⊗ Θ)] = −d[Ξ ⊗ Θ] − (−1)nd b [Ξ ⊗ Θ] + eb/2 ∧ dbx[Ξ ⊗ Θ] + db ∧ [Ξ ⊗ Θ] . b b b 2 Proof: eWe prove the identities for n = 6. Let us ﬁrst assume that b = 0, i.e. we have to show

2 ^ 2 ^ [D(Ξ ⊗ Θ)] = d[Ξ ⊗ Θ] − d [Ξ ⊗ Θ], [D(Ξ ⊗ Θ)] = −d[Ξ ⊗ Θ] − d [Ξ ⊗ Θ]. (4.21)

ev,od 2 ev,od But ⋆⋆ = (−1) Id, and the commutation rulese for ∧, ⋆ and d in Lemma 2.8 imply d ρ = ∗ ev,od −d ρ . According to Corollary 2.3, the isomorphism [·,· ]b is Spin(6)-invariant. Consequently, it commutes with the Levi-Civita connection. Since d and d∗ are obtained from skew-symmetrisation and minus the contraction of ∇ respectively, we get

[D(Ξ ⊗ Θ)] = ι(ek⊗ˆ 1) • [∇ek (Ξ ⊗ Θ)] X = ek ∧∇ek [Ξ ⊗ Θ] − ekx∇ek [Ξ ⊗ Θ] 2 = dX[Ξ ⊗ Θ] − d [Ξ ⊗ Θ].

Similarly, we ﬁnd [D(Ξ ⊗ Θ)] = −d[Ξ ⊗ Θ] − d2[Ξ ⊗ Θ]. Next, twisting with a B–ﬁeld yields 1 eeb/2 ∧ de−b/2[Ψ ⊗ Ψ ] = − db ∧ [Ψ ⊗ Ψ ] + d[Ψ ⊗ Ψ ] 1 2 b 2 1 2 b 1 2 b and again by Lemma 2.8,

b/2 2 −b/2 b/2 b/2 \ ∧ e ∧ d e [Ψ1 ⊗ Ψ2]b = e ∧ ⋆ d⋆(e ∧ [Ψ1 ⊗ Ψ2]b) b/2 −b/2 = 2b e ∧ d(e ∧ 2b[Ψ1 ⊗ Ψ2]b 1 2 = −2 ( db ∧ 2 [Ψ ⊗ Ψ ] ) + d b [Ψ ⊗ Ψ ] b 2 b 1 2 b 1 2 b 1 2 = −eb/2 ∧ ( dbx[Ψ ⊗ Ψ ]) + d b [Ψ ⊗ Ψ ] . 2 1 2 1 2 b Substituting this into the equations of (4.21) twisted with b proves the result.

As a second step, we need to understand the action on ∆ of the G × G–irreducible components of a bispinor in ∆ ⊗ ∆. Following Section 2.2, the form F = [Ξk ⊗ Θk] acts by

−1 F · Θ = κ(J [Ξk ⊗ Θk])Θ = Ξk ⊙PΘk(Θ) = q(Θ, Θk)Ξk X X X 29 which is just contraction of the right hand side with q(·, Θ), for

q(Θk, Θ)Ξk = q(Θ, Θk)Ξk = [Ξk ⊗ Θk] · Θ = F · Θ. (4.22) X X X Analogously, contraction on the left hand side gives F · Θ. For generalised SU(3)–structures, Corol- lary 2.4 asserts that we can decompose any form F ∈ [∆a⊗∆b], a, b = ±, into F = F1⊕F3l ⊕F3r ⊕F9, b regarding the left and right hand side of ∆ ⊗ ∆ as an SU(3)l and an SU(3)r –module respectively.

If, say, F ∈ [∆+ ⊗ ∆−], these pieces are realised as

F = λF Ξ+ ⊗ Θ− + αF · Ξ− ⊗ Θ− +Ξ+ ⊗ βF · Θ+ +ΓF (Ξ− ⊗ Θ+),

C C3 C3 with λF = q(F · Θ+, Ξ+) ∈ , αF ∈ and βF ∈ determined by

q(αF , Z) = q(F · Θ+, Z · Ξ−), q(Z, βF ) = q(F · Z · Θ−, Ξ+),

3 3 and the sesquilinear form ΓF ∈ C ⊗ C deﬁned as ΓF (W, Z) = q(F · W · Θ−, Z · Ξ−). We can also

ﬂip the representation spaces, that is, view the left and right hand side of ∆ ⊗ ∆ as an SU(3)r – and an SU(3)l–module, to obtain a further decomposition of F , namely

F = Fb1 ⊕ F3bl ⊕ F3br ⊕ Fb9

= λF Θ+ ⊗ Ξ− + αF · Θ− ⊗ Ξ− + Θ+ ⊗ βF · Ξ+ + ΓF (Θ− ⊗ Ξ+).

This links into the ﬁrstb decompositionb of the adjoint F by b b

tr b b b b b λF = λF , αF = βF , βF = αF , ΓF =ΓF for F ∈ ∆a ⊗ ∆b, a,b = ±. (4.23)

For a generalisedb G2–structure,b Corollaryb 2.4 impliesb similarly a decomposition into four pieces F = F1 ⊕ F7l ⊕ F7r ⊕ F49 which have an analogous description in terms of λF , αF , βF and ΓF . The details are left to the interested reader. We now possess all the necessary technical ingredients to prove the spinorial version of integrability.

6 Theorem 4.3. A generalised SU(3)–structure (M , τ) is integrable with respect to the forms H0 and F , i.e. 2 dH0 τ = −i τ F,

−φ if and only if for τ = e ([Ξ+ ⊗ Θ−]b ⊕ [Ξ+ ⊗ Θ+]b) ande H = db/2 + H0, the following conditions hold.

(i) The algebraic constraints: Seen as an endomorphism ∆ → ∆, F|∆a⊗∆b preserves the decomposition of ∆a,b into irreducible SU(3)l,r –modules, i.e. for all combinations a, b = ±, (F|∆a⊗∆b )3r,l = \ 0. Moreover, the 1– and 1–components of F ev,od and F ev,od couple via

ev od ev od bF · Θ± = F · Θ∓, F · Ξ∓ = −F · Ξ∓.

d d

30 (ii) The generalised Killing equations

1 x φ ev φ od ∇X Ξ+ + 4 (X H) · Ξ+ − e F−b · X · Θ− + e F−b · X · Θ+ = 0 1 x φ ev φ od ∇X Θ+ − 4 (X H) · Θ+ + e F−b · X · Ξ− + e F−b · X · Ξ+ = 0.

(iii) The dilatino equations d d

1 φ ev φ od DΞ+ − dφ · Ξ+ + 4 H · Ξ+ + e F−b · Θ− + e F−b · Θ+ = 0 1 φ ev φ od DΘ+ − dφ · Θ+ − 4 H · Θ+ − e F−b · Ξ− + e F−b · Ξ+ = 0.

7 od,ev Theorem 4.4. A generalised G2-structure (M , ρd ) is integrabled with respect to the forms H0 and F od,ev, i.e. ev,od dH0 ρ = F ev,od −φ ev,od ev,od 2 ev,od if and only if for ρ = e [Ψ+ ⊗Ψ−]b , H = db/2+H0 and F = F + F the following conditions hold. (i) The algebraic constraints: Seen as an endomorphism ∆ → ∆, F preserves the decomposition of ∆ into irreducible G2l,r–modules, i.e. F7l,r =0. (ii) The generalised Killing equations

1 x φ ∇X Ψ+ + 4 (X H) · Ψ+ ± e F−b · X · Ψ− = 0 1 x φ ∇X Ψ− − 4 (X H) · Ψ− − e F−b · X · Ψ+ = 0.

(iii) The dilatino equations b

1 φ DΨ+ − dφ · Ψ+ + 4 H · Ψ+ − e F · Ψ− = 0 1 φ DΨ− − dφ · Ψ− − 4 H · Ψ− ± e F · Ψ+ = 0.

Remark: If the generalised structure is parallel, then theb Theorems 4.3 and 4.4 assert the spinors

(Ξ+, Θ+) and (Ψ+, Ψ−) to be parallel with respect to the lift of Hitchin’s connections in Theorem 2 of [16]. From this point of view, F is most naturally interpreted as the “torsion” of these connections. As in the classical case, we obtain obstructions to integrability in the form of algebraic constraints on the “torsion” components.

Proof: We ﬁrst prove the case of a generalised SU(3)–structure in full detail before we sketch the case of a generalised G2–structure afterwards. 2 To ease the notation, we substitute F by Fb. The equation dH0 τ = −i bFb is equivalent to

de−φ[Ξ ⊗ Θ ] = −i2F ev − e−φH ∧ [Ξ ⊗ Θ ], + + + + e (4.24) −φ od −φ de [Ξ+ ⊗ Θ−] = i2F − e H ∧ [Ξ+ ⊗ Θ−] and thus by Corollary 2.8,

2 −φ ev −φ 2 −φ od −φ d e [Ξ+ ⊗ Θ+] = F − e Hx[Ξ+ ⊗ Θ+], d e [Ξ+ ⊗ Θ−] = F − e Hx[Ξ+ ⊗ Θ−].

31 From this we deduce

−φ ev− −φ −φ [D(e Ξ+ ⊗ Θ+)] = −F + e Hx[Ξ+ ⊗ Θ+] − e H ∧ [Ξ+ ⊗ Θ+] (4.25) −φ od− −φ −φ [D(e Ξ− ⊗ Θ+)] = −F + e Hx[Ξ− ⊗ Θ+] − e H ∧ [Ξ− ⊗ Θ+] (4.26) −φ ev+ −φ ^ −φ ^ [D(e Ξ+ ⊗ Θ+)] = F + e Hx[Ξ+ ⊗ Θ+] + e H ∧ [Ξ+ ⊗ Θ+] (4.27) [D(e−φΞ ⊗ Θ )] = −F od− + e−φHx[Ξ ^⊗ Θ ] + e−φH ∧ [Ξ ^⊗ Θ ], (4.28) e + − + − + − where we recalle our convention F •± = ρ• ∓ 2ρ• for a real form ρ• (cf. Lemma 2.8). Fixing an orthnormal basis e1,...,e6, we ﬁnd from a repeated application of Theorem 2.2 x 1 x 1 x ∼ H [Ξ ⊗ Θ] = 8 [ ek · Ξ ⊗ (ek H) · Θ − H · Ξ ⊗ Θ] + 8 [ (ek H) · Ξ ⊗ ek · Θ − Ξ ⊗ H · Θ] 1 x 1 x ∼ H ∧ [Ξ ⊗ Θ] = − 8 [P ek · Ξ ⊗ (ek H) · Θ − H · Ξ ⊗ Θ] + 8 [P(ek H) · Ξ ⊗ ek · Θ − Ξ ⊗ H · Θ] and P P x ^ 1 x 1 x ∼ H [Ξ ⊗ Θ] = − 8 [ (ek H) · Ξ ⊗ ek · Θ − Ξ ⊗ H · Θ] − 8 [ ek · Ξ ⊗ (ek H) · Θ − H · Ξ ⊗ Θ] ^ 1 x 1 x ∼ H ∧ [Ξ ⊗ Θ] = − 8 [P(ei H) · Ξ ⊗ ek · Θ − Ξ ⊗ H · Θ] + 8 [ Pek · Ξ ⊗ (ek H) · Θ − H · Ξ ⊗ Θ] .

As a result, contractingP (4.25) on the left-hand side with q(·, ΞP−) implies by (4.22) [ 1 q(De−φΞ , Ξ )Θ = F ev−(Ξ ) − e−φq(H · Ξ , Ξ )Θ . + − + + 4 + − + Consequently, [ev− pr ⊥ ◦ F (Ξ+) = α\ · Θ− =0 Θ+ F ev+ and thus β ev+ = 0 according to (4.23). Similarly, we see β od+ , α ev+ and α od− to vanish by F b F F F contracting (4.26), (4.27) and (4.28) with Ξ+, Θ− and Θ+. Taking the complex conjugate yields

βF ev+ = βF ev− = 0 etc. and proves the first statement of (i). Next we need the generalised Killing equations. For this it will be convenient to use an orthonormal l,r basis adapted to the almost complex structures Jl,r on T induced by SU(3)l,r , i.e. ek+3 = Jl,r ek for l,r l,r l,r l,r l,r l,r k =1, 2, 3. We then define the complex quantities zk = (ek − iJek )/2 and zk = (ek + iJek )/2. Recall that by definition of the complex structure Jr,l (cf. Section 2.2), we have for instance 1 zl · Ξ = (e + iJe ) · Ξ = e · Ξ = iJe · Ξ . m + 2 m m + m + m +

To keep the notation compact, we drop the reference to Jl,r and rely on the context to determine a suitable almost complex structure. Expressed in the complex basis zk, zk, (4.27) and (4.28) become 1 Ξ ⊗ D(e−φΘ )+2 ∇ Ξ ⊗ z · Θ = F ev+ + e−φΞ ⊗ H · Θ + + zk + k + 4 + + X 1 − e−φ (z xH) · Ξ ⊗ z · Θ (4.29) 2 k + k + X1 Ξ ⊗ D(e−φΘ )+2 ∇ Ξ ⊗ z · Θ = −F od− + e−φΞ ⊗ H · Θ + − zk + k − 4 + − X 1 − e−φ (z xH) · Ξ ⊗ z · Θ . (4.30) 2 k + k − X 32 We contract (4.29) and (4.30) on the right-hand side with q(·, zm · Θ+) and q(·, zm · Θ−) to get 1 q(De−φΘ , z · Θ )Ξ +2e−φ∇ Ξ = F ev+ · z · Θ + e−φq(H · Θ , z · Θ )Ξ + m + + zm + m − 4 + m + + 1 − e−φ(z xH) · Ξ 2 m + 1 q(De−φΘ , z · Θ )Ξ +2e−φ∇ Ξ = −F od− · z · Θ + e−φq(H · Θ , z · Θ )Ξ − m − + zm + m + 4 − m − + 1 − e−φ(z xH) · Ξ . 2 m + Adding and substracting this pair of equation yields

−φ −φ ev+ od− 2Re q(De Θ+, zm · Θ+)Ξ+ +2e ∇em Ξ+ = F · zm · Θ− − F · zm · Θ+ 1 + e−φRe q(H · Θ , z · Θ )Ξ 2 + m + + 1 − e−φ(e xH) · Ξ 2 m + −φ −φ ev+ od− 2Im q(De Θ+, zm · Θ+)Ξ+ − 2e ∇Jem Ξ+ = −iF · zm · Θ− − iF · zm · Θ+ 1 + e−φIm q(H · Θ , z · Θ )Ξ 2 + m + + 1 + e−φ(Je xH) · Ξ 2 m +

We contract the ﬁrst equation with q(·, Ξ+) and q(Ξ+, ·) and add the result. By (i),

ev+ q(F · z · Θ , Ξ ) = q(z , β ev+ ) = 0 m − + m F , od− q(F · zm · Θ+, Ξ+) = q(zm, βF od− ) = 0

and since q(∇em , Ξ+) + q(Ξ+, ∇em Ξ+) = 0, we see that

−φ −φ 4Re q(De Θ+, zm · Θ+) = e Re q(H · Θ+, zm · Θ+).

The same holds for the imaginary part, hence 1 1 1 ∇ Ξ = eφF ev+ · z · Θ − eφF od− · z · Θ − (e xH) · Ξ em + 2 m − 2 m + 4 m + i i 1 ∇ Ξ = eφF ev+ · z · Θ + eφF od− · z · Θ − (Je xH) · Ξ , Jem + 2 m − 2 m + 4 m + and ﬁnally, using F • = (F •+ ⊕ F •−)/2, 1 ∇ Ξ = − (XxH) · Ξ + eφF ev · X · Θ − eφF od · X · Θ X + 4 + − + for all X ∈ Γ(T M). In the same way, we derive 1 ∇ Θ = (XxH) · Θ − eφF ev · X · Ξ − eφF od · X · Ξ . X + 4 + − + d d

33 Next we derive the second part algebraic constraint and the dilatino equation by looking at the Spin(n)–Dirac operator, namely

−φ −φ De Ξ+ = e (−dφ · Ξ+ + DΞ+) 3 = −e−φdφ · Ξ − e−φH · Ξ + e · F ev · e · Θ − e · F od · e · Θ . + 4 + k k − k k + X X On the other hand side, contracting (4.25) with q(·, Θ+) yields 1 De−φΞ +e−φ q(∇ Θ , Θ )e ·Ξ = −F ev−·Θ + e−φ q((e xH)·Θ , Θ )e ·Ξ −H·Ξ . + ek + + k + − 4 k + + k + + X X Substituting the generalised Killing equation for ∇ek Θ+ ﬁnally implies 1 De−φΞ = −F ev− · Θ − e−φH · Ξ . + − 4 + In the same vein, we derive 1 De−φΞ = −F od+ · Θ − e−φH · Ξ , + + 4 + by contracting the conjugate of (4.26) by q(·, Θ−) and using the generalised Killing equation for

∇ek Θ−. Consequently, 1 F ev± · Θ = F od∓ · Θ , De−φΞ = −F ev · Θ − F od · Θ − e−φT · Ξ ± ∓ + − + 4 + and therefore 1 eφ e · F ev · e · Θ − eφ e · F od · e · Θ = −eφF ev · Θ − eφF od · Θ + dφ · Ξ + H · Ξ k k − k k + − + + 2 + X X From this we deduce 1 DΞ = dφ · Ξ − H · Ξ − eφF ev · Θ − eφF od · Θ . + + 4 + − + and by proceeding with (4.27) and (4.28) in a similar fashion, we ﬁnd 1 F ev ± · Ξ = −F od ∓ · Ξ , DΘ = dφ · Ξ + H · Ξ + eφF ev · Ξ − eφF od · Ξ , ∓ ± + + 4 + − + which showsd the necessityd of (i)–(iii). d d

Conversely, suppose that these conditions hold. Let e1,...,e6 be an orthonormal basis. Then by

34 using (ii) and (iii)

d[Ξ+ ⊗ Θ+] = ek ∧∇ek [Ξ+ ⊗ Θ+] 1X ^ 1 ^ = ([DΞ ⊗ Θ ] + [Ξ ⊗ DΘ ]) + [e · Ξ ⊗∇ Θ ] + [∇ Ξ ⊗ e · Θ ] 2 + + + + 2 k + ek + ek + k + 1 1 1 X 1 = [( dφ · Ξ − H · Ξ + eφF ev · Θ + eφF od · Θ ) ⊗ Θ ] 2 + 8 + 2 − 2 + + 1 1 1 1 +[Ξ ⊗ ( dφ · Θ + H · Θ + eφF ev · Ξ − eφF od · Ξ )] + 2 + 8 + 2 − 2 + 1 1 1 + [e · Ξ ⊗ ( (e xH) · Θ + eφdF ev · e · Ξ +deφF od · e · Ξ )] k + 8 k + 2 k − 2 k + X 1 1 1 + [(− (e xH) · Ξ + eφF ev · e ·dΘ − eφF od · e ·dΘ ) ⊗ e · Θ ] 8 k + 2 k − 2 k + k + = dφX∧ [Ξ+ ⊗ Θ+] − H ∧ [Ξ+ ⊗ Θ+] 1 + eφ[(F ev · Θ + F od · Θ ) ⊗ Θ +Ξ ⊗ (F ev · Ξ − F od · Ξ )] 2 − + + + − + 1 + eφ[ e · Ξ ⊗ (F ev · e · Ξ + F od · e ·dΞ )] d 2 k + k − k + 1 X + eφ[ (F ev · e · Θd− F od · e · Θd) ⊗ e · Θ ]. 2 k − k + k + X + − ev ev+ od od Now on hand, we have Ξ+ ⊗ F · Ξ− = (F )1/2 and F Ξ− = −F · Ξ+ etc., and on the other, d d d 6 3 ev [ev+ tr ev− ek · Ξ+ ⊗ F · ek · Ξ− = zk · Ξ+ ⊗ F · zk · Ξ− = Γ\ (Θ− ⊗ Ξ+) = F . F ev+ 9 k=3 k=1 X X d b Dealing in the same with the remaining terms, we ﬁnally obtain, using (i),

d[Ξ+ ⊗ Θ+] = dφ ∧ [Ξ+ ⊗ Θ+] − H ∧ [Ξ+ ⊗ Θ+] 1 + eφ F ev− + F ev+ + F ev− + F od− + F ev+ − F od− 2 1 9 9 9 9 1 = dφ ∧ [Ξ ⊗ Θ ] − H ∧ [Ξ ⊗ Θ ] + eφF ev. + + + + 2

Proceeding similarly with d[Ξ+ ⊗ Θ−] yields (4.24).

In the generalised G2–case, we start with the equations

−φ ev,od −φ de [Ψ+ ⊗ Ψ−] = F − e H ∧ [Ψ+ ⊗ Ψ−] 2 −φ ev,od −φ d e [Ψ+ ⊗ Ψ−] = 2F + e Hx[Ψ+ ⊗ Ψ−]. (4.31)

Put F = F ev,od + 2F ev,od. Then Proposition 4.2 implies

−φ −φ −φ [De Ψ+ ⊗ Ψ−] = F + e Hx[Ψ+ ⊗ Ψ−] − e H ∧ [Ψ+ ⊗ Ψ−] (4.32) −φ −φ ^ −φ ^ [De Ψ+ ⊗ Ψ−] = ±F + e Hx[Ψ+ ⊗ Ψ−] + e H ∧ [Ψ+ ⊗ Ψ−] (4.33)

e 35 with + for an even and − for an odd structure. By construction, F is self-dual with respect to 2 and can thus be written as a sum of bi-spinors. Contracting (4.32) and (4.33) with q(·, Ψ+) and q(·, Ψ−) on the left and right hand side respectively yields 1 q(De−φΨ , Ψ )Ψ = F · Ψ − e−φq(H · Ψ , Ψ )Ψ + + − + 4 + + − 1 q(De−φΨ , Ψ )Ψ = ±Fb · Ψ + e−φq(H · Ψ , Ψ )Ψ . − − + − 4 − − +

From this we deduce as above that αF = βF = 0, i.e. F7l = F7r = 0. The gravitino equation is obtained by contracting (4.32) and 4.33 with q(·, em · Ψ+) and q(·, em · Ψ−) on the left and right hand side. Then 1 q(De−φΨ , e · Ψ )Ψ + e−φ∇ Ψ = F · e · Ψ − e−φq(H · Ψ , e · Ψ )Ψ + m + − em − m + 4 + m + − 1 + be−φ(e xH) · Ψ 4 m − 1 q(De−φΨ , e · Ψ )Ψ + e−φ∇ Ψ = ±F · e · Ψ + e−φq(H · Ψ , e · Ψ )Ψ − m − + em + m − 4 − m − + 1 − e−φ(e xH) · Ψ . 4 m + and therefore 1 ∇ Ψ + (XxH) · Ψ ∓ eφF · X · Ψ = 0 X + 4 + − 1 ∇ Ψ − (XxH) · Ψ − eφF · X · Ψ = 0. X − 4 − + We evaluate the Dirac-operator to get b 3 DΨ = − H · Ψ ± eφ e · F · e · Ψ . + 4 + k k − X On the other hand, contracting (4.32) with q(·, Ψ−) and comparing both terms ﬁnally gives 1 DΨ = dφ · Ψ − H · Ψ + eφF · Ψ . + + 4 + − Similarly, we derive the second modiﬁed dilatino equation 1 DΨ = dφ + H · Ψ ± eφF · Ψ . − 4 + + Conversely, assume that (i)–(iii) of Theorem 4.4 hold. As aboveb we substitute the generalised Killing and dilatino equations to obtain

d[Ψ+ ⊗ Ψ−] = dΦ ∧ [Ψ+ ⊗ Ψ−] − H ∧ [Ψ+ ⊗ Ψ−] 1 + eφ[F · Ψ ⊗ Ψ + e · Ψ ⊗ F · e · Ψ ] 2 − − k + k + Xk 1 b ± eφ[Ψ ⊗ F · Ψ + F · e · Ψ ⊗ e · Ψ ]∼ 2 + + k − k − Xk b 1 = dΦ ∧ [Ψ ⊗ Ψ ] − H ∧ [Ψ ⊗ Ψ ] + eφ(F + F + F + F ) + − + − 2 1 49 1 49 . e e

36 ev,od 1 By hypothesis F7l,r = 0, and thus F = 2 (F1 + F49 + F1 + F49) which implies (4.31).

e e Remark: (i) In the situation of Section 3.3, only even or odd R-R–fields are present. Therefore Theo- rem 4.3 yields precisely, after some trivial adjustments, the exterior constraint (3.16) as well as equations (3.17) and (3.18) for type IIB. Assuming a sign error in the democratic formulation, the same holds for type IIA. (ii) Reformulated in terms of SU(3)×SU(3)–representation theory, the algebraic constraint (3.16) says that F • has no singlet component. Thus, as a result of Theorem 4.3, only the 9–component of F • is present. (iii) A analogous statement holds for compactifications on 7-manifolds, i.e. we can characterise the vacuum background of an internal 7-fold by a generalised G2-structure which is integrable in the sense of Definition 4.1.

As a geometrical application of the spinorial formulation we computate the Ricci tensor of a parallel generalised G–structure, i.e. F • = 0. Let ∇± = ∇ ± H/2 denote the metric connections with skew–symmetric torsion ±H which by Theorem 4.3 and 4.4 preserve the Gl,r–structure, i.e ± ± Ξ+, Θ+ and Ψ+, Ψ− are parallel with respect to ∇ . By, Ric and Ric we denote the associated Ricci tensors. Generally speaking, the Ricci tensor Ric associated with a G-invariant spinor Ξ and a G-preserving, metric linear connection ∇ with skew torsion H is determined by the formula [7] g 1 Ric(X) · Ξ =e (∇ H) · Ξ + (XxdeH) · Ξ X 2 and relates to the metric Riccig tensor throughe e e 1 1 Ric(X, Y ) = Ric(X, Y ) + d∗H(X, Y ) + g(XxH, Y xH). 2 4 Applied to our situation, this givesg e e e

Proposition 4.5. If g is a metric underlying a parallel generalised G-structure, then its Ricci tensor is given by 1 Ric(X, Y ) = −2Hφ(X, Y ) + g(XxH, Y xH), 4 φ where H (X, Y ) = X.Y.φ−∇XY.φ is the Hessian of the dilaton φ. It follows that the scalar curvature of the metric g is 3 Scal = Tr(Ric) = 2∆φ + kHk2, 4 (·) where ∆(·) = −TrgH is the Riemannian Laplacian.

Proof: For sake of concreteness, we consider the case of a parallel generalised SU(3)–structure.

The G2–case was already dealt with in [28], but our new computation corrects a slight error in the

37 coeﬃcients there. According to the previous formulae we obtain 1 1 Ric(X, Y ) = Ric+(X, Y )+Ric−(X, Y ) + g(XxH, Y xH). 2 16 We thus need to compute the Ricci tensors Ric±. Now the dilatino equation of a parallel structure can be rewritten as 1 1 (dφ + H) · Ξ =0, (dφ − H) · Θ =0. 2 + 2 + ± Since ∇ preserves the SU(3)l,r -structure, we deduce

+ + − − Ric (X) · Ξ+ = −2(∇X dφ) · Ξ+, Ric (X) · Θ+ = −2(∇X dφ) · Θ+.

± Now consider an orthonormal frame that satisﬁes ∇ei ej = 0 at a ﬁxed point, or equivalently, ∇ei ej = 1 ± 2 k Hijkek. Then

P ± ± ± Ric (ei, ej) = −2g(∇ei dφ,ej) = −2ei.g(dφ,ej) + g(dφ, ∇eiej ) = −2ei.ej.φ ± Hijkek.φ. Xk φ The Hessian evaluated in this basis is just H (ei, ej) = ei.ej.φ, whence the result.

4.3 T –duality

In this section we discuss the device of T –duality as introduced in [28], following ideas from [3]. 1 n−1 Consider a principal S –ﬁbre bundle p : PS1 → M over a compact, n − 1–dimensional base n−1 1 manifold M , and assume that PS1 carries an S –invariant generalised G–structure deﬁned by a

G × G–invariant spinor ρ. We choose a connection form θ and denote by Xθ the corresponding dual vertical vector field, i.e. Xθxθ = 1, and by ω the induced curvature, regarded as a 2-form on M, so dθ = p∗ω. Moreover, assume to be given a closed, integral and S1-invariant 3-form H such that the 2-form ωt defined by p∗ωt = XxH is also integral. In practice, we will assume H = 0 so that this condition is automatically fulfilled. Integrality of ωt ensures the existence of another principal 1 t t t ∗ t S -bundle PS1 , the T -dual of P defined by the choice of a connection form θ with dθ = −p ω . Writing H = θ ∧ ωt + H where H∈ Ω3(M), we define the T -dual of H by

Ht = −θt ∧ ω + H.

The pull-back p∗ was dropped to ease notation. As the G×G–invariant spinor ρ is also S1–invariant, we can decompose ρ into forms ρ0, ρ1 living on the base manifold, ρ = θ ∧ ρ0 + ρ1. The T -dual of ρ is then deﬁned to be t t ρ = θ ∧ ρ1 + ρ0, so T -duality is enacted by multiplication with the element Xθ ⊕ θ ∈ P in(n,n) on ρ followed by the substitution θ → θt. In particular, it is parity–reversing. The crucial feature of this operation is to preserve the P in(n,n)–orbit structure on Λev,odTP ∗. In particular, ρt is also G × G–invariant.

38 t t Consequently, 2ρt ρ and (2ρρ) are both G × G–invariant spinors which are therefore equal up to a universal scalar which we henceforth ignore. In the same vein, we decompose an S1–invariant t t form of mixed degree F = θ ∧ F0 + F1 and deﬁne the induced dual form to be F = θ ∧ F1 + F0. Integrability translates then as follows.

Proposition 4.6. Let ρ and F be S1–invariant. Then

t t dH ρ = F ⇐⇒ dHt ρ = −F .

The proof is a straightforward computation using the S1–invariance and the deﬁnition of the T –duals.

Corollary 4.7. If ρ defines an integrable SU(3)–structure with respect to H and F , then so does t t t • ρ with respect to H and −F . Similarly, if ρ defines an integrable G2–structure of odd or even type with respect to H and F od,ev, then ρ• t defines an integrable structure of even or odd type with respect to Ht and −F od,ev t.

Note that compacticiy of M is only needed for the existence of the T –dual bundle. For non– compact M, this machinery can still be put into action as follows. Assume we are given a straight generalised G–structure, that is G = Gl = Gr, where G comes from an invariant G–structure deﬁned over a prinicpal S1–bundle. Then let H = 0 which forces the T -dual bundle to be ﬂat, so that we can choose P t = M × S1.

Example: In absence of an R-R–field F , any S1–invariant G–structure whose holonomy is contained in G can thus be T –dualised to give further examples of parallel generalised G–structures. Note that we acquire non–trivial torsion Ht = −dt∧ω if the initial S1–bundle is not flat. Moreover, additional torsion of the underlying G–structures can sometimes be absorbed into non–trivial R-R–fields. For 1 instance, consider a co–calibrated, S –invariant G2–structure (see, for example, [5]). If ϕ denotes ev the induced stable form, this means d⋆ϕ ϕ = 0, but in general dϕ = F 6= 0. The G2 ×G2–invariant ev od spinors of the corresponding straight generalised G2–structure are ρ =1 − ⋆ϕ and ρ = −ϕ + vol (cf. Corollary 2.6), hence dρev = 0 and dρod = F ev. Applying T –duality yields a new structure with a non–trivial H–field. This is in sharp contrast to the case of compact parallel G2–structures for which necessarily H = 0 [28].

A Appendix: A matrix representation of Spin(6) and Spin(7)

In general one constructs the spin representations out of an explicit identiﬁcation κ of Cliﬀ(Cm,gC) ± with a complex matrix algebra, for example by using the isomorphism provided in [1]. If κ (ei) m denotes the matrix acting ∆± → ∆∓ associated with an orthonormal basis e1,...,em of (R ,g),

39 then we obtain for m =6

0 00 −i 0 001

±  0 0 i 0  ±  0 010  κ (e1) = ± , κ (e2) = ,  0 −i 0 0   0 −1 0 0       i 00 0   −1 000          0 0 −i 0 001 0

±  00 0 −i  ±  0 0 0 −1  κ (e3) = ± , κ (e4) = ,  i 0 0 0   −100 0       0 i 0 0   010 0          0 −i 0 0 01 00

±  i 0 00  ±  −10 00  κ (e5) = ± , κ (e6) = .  0 0 0 i   00 01       0 0 −i 0   0 0 −1 0          With respect to this basis, the conjugation map C = :∆± → ∆∓ is given by the matrix

0 Id C = . Id 0 !

7 For Cliﬀ(R ,g) we use a more “geometric” representation [13]. Fix an orthonormal basis e1,...,e7 7 and identify R with the imaginary octonions Im O by e1 = i,...,e7 = e · k. Deﬁne

7 Ru 0 u ∈ R =∼ Im O 7→ ∈ EndR(O) ⊕ EndR(O), 0 −Ru ! where Ru denotes right multiplication by u in O. By the universal property, this extends to a Cliﬀord algebra isomorphism and projection on the ﬁrst summand EndR(O) yields an explicit matrix repre- 8 sentation. In terms of the standard basis of skew-symmetric matrices, Eij = (δjkδil − δikδjl)k,l=1, i < j, it is given by

κ(e1) = E1,2 − E3,4 − E5,6 + E7,8,

κ(e2) = E1,3 + E2,4 − E5,7 − E6,8,

κ(e3) = E1,4 − E2,3 − E5,8 + E6,7,

κ(e4) = E1,5 + E2,6 + E3,7 + E4,8,

κ(e5) = E1,6 − E2,5 + E3,8 − E4,7,

κ(e6) = E1,7 − E2,8 − E3,5 + E4,6,

κ(e7) = E1,8 + E2,7 − E3,6 − E4,5.

40 References

[1] H. Baum, T. Friedrich, R. Grunewald, and I. Kath. Twistors and Killing spinors on Rie- mannian manifolds. 124 in Teubner-Texte zur Mathematik, B.G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991.

[2] E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest, and A. Van Proeyen. New formula- tions of D = 10 supersymmetry and D8 - O8 domain walls. Class. Quant. Grav., 18:3359–3382, 2001, hep-th/0103233.

[3] P. Bouwknegt, J. Evslin, and V. Mathai. T-Duality: Topology Change from H-ﬂux. Commun. Math. Phys., 249(2):383–415, 2004, hep-th/0306062.

[4] G. R. Cavalcanti. New aspects of the ddc-lemma, DPhil thesis, University of Oxford 2004, math.DG/0501406.

[5] S. Chiossi and S. Salamon. The intrinsic torsion of SU(3) and G2 structures. In Dif- ferential geometry Valencia 2001, 115–133, World Sci. Publishing, River Edge, NJ, 2002, math.DG/0202282.

[6] R. Dijkgraaf, S. Gukov, A. Neitzke and C. Vafa. Topological M-theory as Uniﬁcation of Form Theories of Gravity, hep-th/0411073.

[7] T. Friedrich and S. Ivanov. Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math., 6(2):303–335, 2002, math.DG/0102142.

[8] M. Grana, R. Minasian, M. Petrini, and A. Tomasiello. Generalized structures of N=1 vacua, hep-th/0505212.

[9] M. Grana, J. Louis, and D. Waldram. Hitchin Functionals in N=2 Supergravity, hep- th/0505212.

[10] M. B. Green, J. H. Schwarz, and E. Witten. Superstring theory. Vol. 1: Introduction. Univ. Pr., Cambridge, UK, 1987.

[11] M. B. Green, J. H. Schwarz, and E. Witten. Superstring theory. vol. 2: Loop amplitudes, anomalies and phenomenology. Univ. Pr., Cambridge, UK, 1987.

[12] M. Gualtieri. Generalized complex geometry, DPhil thesis, University of Oxford 2003, math.DG/0401221.

[13] F. R. Harvey. Spinors and Calibrations, 9 of Perspectives in Mathematics. Academic Press, Boston, MA, 1990.

41 [14] N. J. Hitchin. Stable forms and special metrics. In Global diﬀerential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000) edited by M. Fernandez´ and J. A. Wolf, 288 in Contemp. Math., pages 70–89, 2001, math.DG/0107101.

[15] N. J. Hitchin. Generalized Calabi-Yau manifolds. Q. J. Math., 54(3):281–308, 2003, math.DG/0209099.

[16] N. J. Hitchin. Brackets, forms and invariant functionals, math.DG/0508618.

[17] C. Jeschek and F. Witt. Generalised G2-structures and type IIB superstrings. JHEP 03 054, 2005, hep-th/0412280.

[18] A. Kapustin. Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys., 1:49–81, 2004, hep-th/0310057.

[19] H.B. Lawson, Jr. and M.-L. Michelsohn. Spin geometry, 38 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1989.

[20] D. Lust¨ and S. Theisen. Lectures on string theory. Lect. Notes Phys., 346:1–346, 1989.

[21] N. Nekrasov. A la recherche de la M–Th´eorie perdue. Z Theory: Chasing M/F Theory, hep-th/0412021.

[22] L. Martucci and P. Smyth. Supersymmetric D-branes and calibrations on general N=1 backgrounds, hep-th/0507099.

[23] V. Pestun and E. Witten. The Hitchin functionals and the topological B-model at one loop, hep-th/0503083.

[24] J. Polchinski. String theory. Vol. 1: An introduction to the bosonic string. Univ. Pr., Cambridge, UK, 1998.

[25] J. Polchinski. String theory. Vol. 2: Superstring theory and beyond. Univ. Pr., Cambridge, UK, 1998.

[26] M. Sato and T. Kimura. A classiﬁcation of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J., 65:1–155, 1977.

[27] F. Witt. Closed forms and special metrics, DPhil thesis, University of Oxford 2004, math.DG/0502443.

[28] F. Witt. Generalised G2-manifolds, math.DG/0411642.

[29] B. Zwiebach. A ﬁrst course in string theory. Cambridge, UK: Univ. Pr. (2004).