Generalized Connections, Spinors, and Integrability of Generalized

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1 Introduction 3

I First Part: Generalized connections and integrability 5

2 Preliminary material 5 2.1 Courantalgebroids ...... 6 2.2 E-connections and generalized connections ...... 7 2.3 Generalizedmetrics ...... 8 2.4 Generalized complex and hyper-complex structures ...... 9 2.5 Generalized K¨ahler and hyper-K¨ahler structures ...... 10 2.6 Intrinsic torsion of a generalized structure ...... 11 2.7 ApplicationtoBorngeometry ...... 13

3 Generalized almost complex structures 15 3.1 Intrinsic torsion of a generalized almost complex structure . . 15 3.2 Integrability using torsion-free generalized connections . . . . 16

4 Generalized almost hypercomplex structures 19 4.1 Intrinsic torsion of a generalized almost hypercomplex structure 19 4.2 Integrability using torsion-free generalized connections . . . . 20

5 Generalized almost Hermitian structures: integrability and torsion-free generalized connections 22

6 Generalized almost hyper-Hermitian structures: integrability and torsion-free generalized connections 23

II Second Part: Dirac generating operators 24

7 The space of local Dirac generating operators 24

8 The canonical Dirac generating operator 33

9 Standard form of the canonical Dirac generating operator 38

III Third Part: Applications 44

2 10 Generalized almost Hermitian structures: integrability and spinors 44

11 Appendix 49 11.1 Z2-gradedalgebrasandCliﬀordalgebras ...... 49 11.2 Integrability of generalized almost complex structures and spinors 50

1 Introduction

Generalized complex geometry is a well established field in present mathematics. It unifies complex and symplectic geometry and represents an important direction of current research in differential geometry and theoretical physics. In generalized geometry, the role of the tangent bundle T M of a manifold M is played by the generalized tangent bundle TM := T M T ∗M, ⊕ or, more generally, by a Courant algebroid. Many classical objects from differential geometry (including almost complex, almost Hermitian, almost hypercomplex and almost hyper-Hermitian structures) were defined and studied by several authors in this more general setting, see e.g. [16] (also [4, 5, 6]). While the integrability of such generalized structures is defined and un- derstood in terms of the Dorfmann bracket of the Courant algebroid, it seems that characterizations in terms of torsion-free generalized connections (analogous to the standard characterizations of integrability from classical geometry) are missing. In the first part of the present paper we fill this gap by answering this natural question. We shall do this by a careful analysis of the space of generalized connections which are ‘adapted’, i.e. preserve, a given generalized structure Q on a Courant algebroid E. In analogy with the classical case, we introduce the notion of intrinsic torsion tQ of Q, see Definition 15. This will play an important role in our treatment, when Q is a generalized almost complex structure or a generalized almost hypercomplex structure. We compute the intrinsic torsion for these structures and we prove that their integrability is equivalent to the existence of a torsion-free adapted generalized connection, see Theorem 23 and Theorem 32. In the same framework, we prove that a generalized almost Hermitian structure on a Courant algebroid is integrable if and only if it admits a torsion-free adapted generalized connection, see Theorem 36. A similar integrability criterion for generalized almost hyper- Hermitian structures is proved in Theorem 38. Our treatment shows that in generalized geometry the torsion-free condition on a generalized connection D adapted to a generalized structure Q does in general no longer determine D uniquely, even if the classical structure generalized by Q has a unique

3 torsion-free adapted connection. This was noticed already in [13, 10] for generalized Riemannian metrics: for a given generalized Riemannian metric G, there is an entire family of generalized connections which are torsion-free and preserve G. We will show that the same holds for the structures considered in this paper, for instance, for generalized hypercomplex structures: the Obata connection [22, 3] (see also [14]) of a hypercomplex structure is replaced in generalized geometry by an entire family of generalized connections. In more general terms, the theory developed here allows to decide whether a given generalized structure Q on a Courant algebroid admits an adapted generalized connection with prescribed torsion and to describe the space of all such generalized connections, see Proposition 17. A torsion-free generalized connection adapted to Q, for instance, exists if and only if the intrinsic torsion of Q vanishes and is unique if and only if the generalized first prolongation hh1i of the Lie algebra h of the structure group of Q is zero. As a further application of the generalized first prolongation we present an alternative proof for the uniqueness of the canonical connection of a Born structure defined in [9] (see Section 2.7). In the second part of the paper we present a self-contained treatment of the canonical Dirac generating operator on a regular Courant algebroid with scalar product of neutral signature. A Courant algebroid is regular if its anchor has constant rank. Regular Courant algebroids form an important class of Courant algebroids, which was studied systematically in [7]. A Dirac generating operator is a first order odd differential operator on a suitable irreducible spinor bundle which encodes the anchor and the Dorfman bracket of the Courant algebroid. We recover the crucial result of [1] (see also [15] and [8], [10]) namely that any regular Courant algebroid E with scalar product of neutral signature admits a canonical Dirac generating operator d/ and we express d/ in terms of a dissection of E. Such an expression for the canonical Dirac generating operator allows a better understanding of the relation between d/ and the structure of the regular Courant algebroid. The third part of the paper is devoted to applications. We assume that E is a regular Courant algebroid with scalar product of neutral signature. Owing to the failure of uniqueness of generalized connections adapted to various generalized structures Q on E it is natural to search for characterizations of the integrability of Q using the canonical Dirac generating operator of E. This was done in [1] when Q = is a generalized almost complex J structure. Here we present a criterion for a generalized almost Hermitian structure (G, ) on E to be generalized Kähler in terms of two canonical Dirac operatorsJ and the pure spinors associated to (where E are the J|E± ± subbundles of E determined by the generalized metric G), see Theorem 67. Our arguments use, besides the theory of Dirac generating operators, the

4 results on generalized connections adapted to a generalized almost Hermi- tian structure, developed in the first part of the paper. A similar spinorial characterization for generalized hyper-Kähler structures on regular Courant algebroids with scalar product of neutral signature is obtained in Corollary 70. In the appendix, intended for completeness of our exposition, we briefly recall basic facts we need on the theory of Z2-graded algebras and the integrability criterion in terms of pure spinors for generalized almost complex structures on regular Courant algebroids with scalar product of neutral signature, mentioned above. In Parts II and III of the paper the assumption that the scalar product of the Courant algebroid E has neutral signature plays a crucial role. Then the spinor bundles S on which the Dirac generating operators act are irreducible Z2-graded Cl(E)-bundles, with the essential property that any Cl(E)-bundle morphism f : S S is a multiple of the identity. (More → generally, considering signatures (t, s) with t s 0 or 2 (mod 8), would ensure the same property, where t stands for− the index≡ and the Clifford relation is v2 = v, v 1.) It would be interesting to extend the theory of Dirac h i generating operators to Courant algebroids of other signatures.

Part I

2 Preliminary material

We start by reviewing the basic definitions we need on Courant algebroids, generalized connections, their torsion (see [10]) and the definition of the generalized structures we shall consider in this paper. We prove a basic property of the Nijenhuis tensor of a generalized almost complex structure (see Lemma 9), we introduce the notion of intrinsic torsion of a generalized structure Q on a Courant algebroid and we describe the space of generalized connections adapted to Q, with prescribed torsion, as an affine space modeled on the space of sections of a vector bundle (see Proposition 17). The fibers of this vector bundle are isomorphic to the generalized first prolongation hh1i of the structure group H of Q, a notion which will be defined for any Lie subgroup H O(k,ℓ). ⊂

5 2.1 Courant algebroids Deﬁnition 1. A Courant algebroid on a manifold M is a vector bundle E M equipped with a non-degenerate symmetric bilinear form , Γ(Sym→ 2(E∗)) (called the scalar product), a bilinear operation [ , ] (calledh· ·i the ∈ · · Dorfman bracket) on the space of smooth sections Γ(E) of E and a homo- morphism of vector bundles π : E T M (called the anchor), such that the following conditions are satisﬁed: for→ all sections u,v,w Γ(E), ∈ C1) [u, [v,w]] = [[u, v],w] + [v, [u,w]]; C2) π([u, v]) = [π(u), π(v)]; C3) [u,fv]= π(u)(f)v + f[u, v]; C4) π(u) v,w = [u, v],w + v, [u,w] ; C5) 2[u,uh]= πi ∗d hu,u . i h i h i A Courant algebroid is called regular if the anchor π has constant rank.

Here π∗ : T ∗M E denotes the map obtained by dualizing π : E T M → → and identifying E∗ with E using the scalar product. Therefore C5) can be written in the equivalent way

2 [u,u], v = π(v) u,u . (1) h i h i Remark 2. As already pointed out in [24, 2], the axioms of a Courant algebroid can be reduced. One can show that axioms C2) and C3) from the definition of Courant algebroids follow from the other axioms. In fact, C2) can be checked by calculating [π(u), π(v)] w, e , for any w, e Γ(E), with the help of C4) and C1). Similarly, C3) canh bei checked by taking∈ the scalar product with a section w Γ(E) and evaluating the result with help of C4). ∈ Example 3. The fundamental example of a Courant algebroid is the generalized tangent bundle TM = T M T ∗M of a smooth manifold M with the canonical projection TM T M as⊕ anchor, the scalar product defined by X + ξ,Y + η = 1 (ξ(Y )+ η(→X)), and Dorfman bracket by h i 2 [X + ξ,Y + η] = [X,Y ]+ η i dξ + H(X,Y, ), (2) LX − Y · where H Ω3(M) is a closed 3-form and X,Y Γ(T M),ξ,η Γ(T ∗M). ∈ ∈ ∈ On the right-hand side, [X,Y ] stands for the usual Lie bracket of vector fields and X η for the Lie derivative of η in the direction of X. It is known that L ∗ every Courant algebroid for which the sequence 0 T ∗M π E π T M 0 is exact is isomorphic to a Courant algebroid of→ this form.→ Su→ch Courant→ algebroids are called exact.

6 2.2 E-connections and generalized connections Unless otherwise stated, E will denote a Courant algebroid, with anchor π : E T M, scalar product , and Dorfman bracket [ , ]. Let V M be a vector→ bundle. h· ·i · · →

Deﬁnition 4. i) An E-connection on V is an R-linear map

D : Γ(V ) Γ(E∗ V ), v Dv, → ⊗ 7→ which satisﬁes the following Leibniz rule

De(fv)= π(e)(f)v + fDev

∞ for all e Γ(E), v Γ(V ), f C (M), where Dev =(Dv)(e). ii) An∈E-connection∈ on E∈is called a generalized connection if it is compatible with the scalar product:

π(u) v,w = D v,w + v, D w (3) h i h u i h u i for all u,v,w Γ(E). ∈ Example 5. A connection on a vector bundle V M induces an E- connection D on V by the formula∇ →

D v := v, e ∇π(e) for all e Γ(E), v Γ(V ). If is a connection on E, then the induced E- ∈ ∈ ∇ connection is a generalized connection if and only if , is -parallel along π(E), that is , = 0 for all X π(E). Conversely,h· ·i ∇ if the Courant ∇X h· ·i ∈ algebroid E is regular, an E-connection D on V which satisfies De = 0 for all e ker π is induced by a connection on V and a similar statement holds for∈ generalized connections. In fact, an∇E-connection D on V such that 1 F De = 0 for all e ker π induces a partial connection : Γ(F ) Γ(V ) ∈ ∇ ×F → Γ(V ) along the distribution F = π(E) T M such that De = π(e) for all ⊂ ′ ∇ e E. Choosing a partial connection F : Γ(F ′) Γ(V ) Γ(V ) along a complementary∈ distribution F ′ T M,∇ we can define× a connection→ such ⊂ F F ′ ∇ ′ that De = π(e) for all e E by X+Y = X + Y for all X F , Y F . If V = E is∇ a regular Courant∈ algebroid∇ and∇ D ∇happens to be∈ a generalized∈ connection on E then the partial connection F is metric and we can choose ′ ∇ F (and hence ) to be metric as well. ∇ ∇ 1Recall that the notion of a partial connection is defined by the same properties as that of a connection, namely tensoriality in the first argument and Leibniz rule in the second.

7 Deﬁnition 6. The torsion T D Γ( 2E∗ E) of a generalized connection D is deﬁned by ∈ ∧ ⊗

T D(u, v)= D v D u [u, v]+(Du)∗v, u, v Γ(E), (4) u − v − ∀ ∈ where (Du)∗ is the metric adjoint of Du Γ(End E) with respect to , . ∈ h· ·i Note that the tensoriality of T D(u, v) in v is obvious since the operators Du and [u, ] satisfy the same Leibniz rule. The tensoriality in u is a consequence of the· skew-symmetry of T D, which follows from

[u, v] + [v,u],w = π(w) u, v (5) h i h i (polarization of (1)) and the compatibility of D with the scalar product. Moreover, one can check that T D(u,v,w) := T D(u, v),w is totally skew- symmetric. From (4), h i

[u, v],w = T D(u,v,w)+ D v D u,w + D u, v . (6) h i − h u − v i h w i We often identify E with E∗ using the scalar product , . By means h· ·i of this identiﬁcation, Λ2E End E is the subbundle of skew-symmetric endomorphisms of E, where ⊂

(e e )(e ) := e , e e e , e e , e E. 1 ∧ 2 3 h 1 3i 2 −h 2 3i 1 i ∈ With this convention, any two generalized connections are related by D˜ = D + η, for η Γ(E∗ Λ2E) and ∈ ⊗ T D˜ (u,v,w)= T D(u,v,w)+ η(u,v,w)+ η(w,u,v)+ η(v,w,u). (7)

The space of torsion-free generalized connections is non-empty, see [10]. If D and D˜ = D + η are torsion-free (or, more generally, have the same torsion), then η is of the form

η(u,v,w)= σ(u,v,w) σ(u,w,v), − for a section σ Γ(S2E E), see [10]. ∈ ⊗

2.3 Generalized metrics Let E be a Courant algebroid with anchor π : E T M, scalar product , → h· ·i and Dorfmann bracket [ , ]. · ·

8 Definition 7. A generalized metric on E is a vector subbundle E+ of E on which , is non-degenerate. h· ·i Let E := E⊥ be the orthogonal complement of E with respect to , . − + + h· ·i Then G := , E+×E+ , E−×E− is a non-degenerate metric on E. Al- ternatively,h· a generalized·i| − h· metric·i| on E can be defined as a non-degenerate symmetric bilinear form G on the vector bundle E such that the endomor- end end end 2 phism G , defined by G(u, v) = G u, v , satisfies (G ) = IdE. The h end i bundles E± are the 1-eigenbundles of G . Along the paper we denote by 1 end ± e± := 2 (Id G )e the E±-components of a vector e E in the decomposition E = E± E determined by a generalized metric∈ G. + ⊕ − Given a generalized metric G there is always a torsion-free generalized connection which preserves G (see [13, 10]). Such a connection is not unique if rank E > 1. It is called a Levi-Civita connection of G. The non-uniqueness is due to the fact that although the first prolongation so(k,ℓ)(1) is always trivial (which is responsible for the uniqueness of the Levi-Civita connection of a pseudo-Riemannian manifold), the generalized first prolongation so(k,ℓ)h1i (see Definition 16 below) is non-trivial if k + ℓ> 1.

2.4 Generalized complex and hyper-complex structures Definition 8. A generalized almost complex structure on E is an endomor- 2 phism Γ(EndE) which satisfies = IdE and is orthogonal with respectJ to the∈ scalar product , of E.J We say− that is integrable (or is a generalized complex structureh·) if·i its Nijenhuis tensorJ N (u, v) := [ u, v] [u, v] ([ u, v] + [u, v]) (8) J J J − −J J J vanishes identically. The following simple lemma will be useful for us. Lemma 9. Let be a generalized almost complex structure on E. Then N (u,v,w) := NJ (u, v),w is a 3-form on E. J h J i Proof. The skew-symmetry of NJ (u,v,w) in the first two arguments follows from (5). In order to prove the skew-symmetry of NJ (u,v,w) in the last two arguments we compute

NJ (u,v,w)+ NJ (u,w,v) = [ u, v] [u, v],w + [ u, v] + [u, v], w h J J − i h J J J i + [ u, w] [u,w], v + [ u,w] + [u, w], v h J J − i h J J J i = [ u, v],w + v, [ u,w] ( [u, v],w + [u,w], v ) h J J i hJ J i − h i h i + [ u, v], w + [ u, w], v + [u, v], w + [u, w], v . h J J i h J J i h J J i h J J i

9 Using the axiom C4) from the deﬁnition of Courant algebroids we obtain

NJ (u,v,w)+ NJ (u,w,v) = π( u) v,w π(u) v,w + π( u) v, w + π(u) v, w =0. J hJ i − h i J h J i hJ J i

We have proved that NJ = NJ (u,v,w) is completely skew-symmetric. As it is C∞(M)-linear in w, it is C∞(M)-linear also in u, v.

Definition 10. A generalized almost hypercomplex structure on E is a triple ( , , ) of anti-commuting generalized almost complex structures such J1 J2 J3 that 3 = 1 2. We say that ( 1, 2, 3) is integrable (is a generalized hy- percomplexJ J structureJ ) if all areJ generalizedJ J (integrable) complex structures. Ji 2.5 Generalized Kähler and hyper-Kähler structures Definition 11. A generalized almost Hermitian structure on E is a pair (G, ) where G is a generalized metric and a generalized almost complex structureJ on E, such that G( u, v)= G(u,J v) for all u, v E. J J ∈ The endomorphism Gend of E determined by G (see Section 2.3) commutes with := and := Gend is a generalized almost complex J1 J J2 J structure. The generalized almost complex structures 1 and 2 preserve E (the 1-eigenbundles of Gend). Moreover, = Jon E . J ± ± J2 ±J1 ± Definition 12. A generalized almost Hermitian structure (G, ) is called a J generalized Kähler structure if = and = Gend are integrable. J1 J J2 J1 The argument from Proposition 2.17 of [17] works also in the setting of non-exact Courant algebroids and shows that (G, ) is a generalized Kähler J structure if and only if L (E±)C are closed under the Dorfmann bracket [ , ] of E, where L denotes the∩ (1, 0)-bundle of . · · J Definition 13. i) A generalized almost hyper-Hermitian structure is a generalized almost hypercomplex structure ( 1, 2, 3) together with a generalized metric G, such that (G, ) is a generalizedJ J almostJ Hermitian structure, for Ji all i =1, 2, 3. ii) A generalized almost hyper-Hermitian structure (G, , , ) is a J1 J2 J3 generalized hyper-Kähler structure if (G, i) is a generalized Kähler structure, for all i =1, 2, 3. J

10 2.6 Intrinsic torsion of a generalized structure Given a linear Lie group H O(k,ℓ), a generalized H-structure on a Courant algebroid E with scalar product⊂ , of signature (k,ℓ) is a reduction of the structure group of E from O(k,ℓh·) to·i H. 0 Rk+ℓ Let (Ti ) be a system of tensors on and Q =(Ti) a system of tensor 0 fields on E, such that Ti is pointwise linearly equivalent to Ti . Then we associate to Q the generalized H-structure defined as the principal bundle of standard orthonormal frames of E with respect to which each of the tensor 0 fields Ti takes the constant form Ti and the scalar product is represented by the matrix diag(1 , 1 ). Examples considered in this paper are: k − ℓ 1. generalized almost complex structures Q = , H = U( k , ℓ ), J 2 2

2. generalized almost hypercomplex structures Q = ( 1, 2, 3), H = k ℓ J J J Sp( 4 , 4 ), 3. generalized Riemannian metrics Q = G, H = O(k ,l ) O(k ,l ), + + × − − where , has signature (k ,ℓ ) (in particular, k + k = k and h· ·i|E± ± ± + − ℓ+ + ℓ− = ℓ), 4. generalized almost Hermitian structures Q =(G, ), H = U( k+ , ℓ+ ) J 2 2 × U( k− , ℓ− ), where , has signature (k ,ℓ ), and 2 2 h· ·i|E± ± ±

5. generalized almost hyper-Hermitian structures Q = (G, 1, 2, 3), k+ ℓ+ k− ℓ− J J J H = Sp( , ) Sp( , ). 4 4 × 4 4 We will refer to these simply as generalized H-structures Q. In analogy with the classical case (see e.g. [23]), given a generalized H- structure Q on E we may consider the space of generalized connections D which preserve, or are adapted, to Q (i.e. DQ = 0). A generalized connection adapted to a generalized almost complex (respectively, hypercomplex) structure Q = (respectively, Q = ( 1, 2, 3)) will be called complex (respectively, hypercomplexJ ). J J J

Lemma 14. Any generalized H-structure Q on a Courant algebroid E admits an adapted generalized connection.

Proof. Any connection on the principal H-bundle of frames which deﬁnes the generalized H-structure induces a connection on E for which Q and the scalar product , are parallel. This connection extends as in Example 5 to h· ·i a generalized connection adapted to Q.

11 The space of generalized connections adapted to a generalized H-structure Q is an affine space, modelled on the vector space of sections of the bundle ∗ 2 ∗ E adQ(E), where adQ(E) denotes the bundle of 2-forms α Λ E , adapted to ⊗Q. By the latter condition we mean that ad End( ∈), defined as the α ∈ TE natural extension of the action of α Λ2E∗ = so(E) End(E) to the tensor p ∗ q ∈ ∼ ⊂ bundle E = p,qE (E ) of E, annihilates the tensor fields from Q. When Q belongsT to the⊕ above⊗ list, we require that [α, ] = 0 and [α,Gend] = 0 when J ,G Q. These conditions are equivalent to ∗α = α and (Gend)∗α = α. J ∈ J Note that the fiber adQ(E)x, x M, of the bundle adQ(E) M is a Lie algebra isomorphic to the Lie algebra∈ h so(k,ℓ) of the structure→ group H. ⊂ The map

∗ 3 ∗ ∂Q : E adQ(E) Λ E , (∂Qη)(u,v,w) := η(u,v,w)+η(w,u,v)+η(v,w,u) ⊗ → (9) is called the algebraic torsion map. From (7), the image of T D Γ(Λ3E∗) 3 ∗ ∈ in the quotient bundle (Λ E )/im ∂Q is independent of the choice of adapted generalized connection D.

Deﬁnition 15. Let Q be a generalized H-structure on E and D an adapted D 3 ∗ generalized connection. The class of T in (Λ E )/im ∂Q is called the intrinsic torsion of Q.

We shall often consider the intrinsic torsion as a 3-form on E, by choosing 3 ∗ a suitable complement C(E) of im ∂Q in Λ E and identifying the quotient 3 ∗ (Λ E )/im ∂Q with C(E). Definition 16. The generalized first prolongation of a Lie algebra h so(k,ℓ) 2 ∗ Rk+ℓ ⊂ ∼= Λ V , V := , is the subspace hh1i := η V ∗ h ∂η =0 V ∗ Λ2V ∗, { ∈ ⊗ | } ⊂ ⊗ where ∂ : V ∗ Λ2V ∗ Λ3V ∗ is defined by: ⊗ → (∂η)(u,v,w) := η(u,v,w)+ η(w,u,v)+ η(v,w,u), u,v,w V. ∈ Proposition 17. Let Q be a generalized H-structure on a Courant algebroid D0 E and D0 an adapted generalized connection with torsion T . Given a section T Γ(Λ3E∗) there exists a generalized connection D adapted to Q ∈ D D0 with torsion T = T if and only if T T Γ(im ∂Q). The generalized − ∈ ∗ connection D is unique up to addition of a section of ker ∂Q E adQ(E). It is unique if and only if the generalized first prolongation⊂hh1i ⊗of the Lie algebra h so(k,ℓ) of the structure group H vanishes. ⊂

12 Proof. This is obtained by writing an arbitrary adapted generalized con- ∗ nection as D = D0 + η, where η Γ(E adQ(E)) and observing that D D0 ∈ ⊗ T = T + ∂Qη, see (7). The last statement follows from the fact that the h1i ﬁbres of the bundle ker ∂Q are isomorphic to h . The next corollary follows easily from Proposition 17. Corollary 18. In the setting of Proposition 17, Q admits an adapted torsion- free generalized connection D if and only if the intrinsic torsion of Q vanishes. The generalized connection D is unique if and only if hh1i =0.

2.7 Application to Born geometry The vanishing of the generalized first prolongation for certain Lie subalgebras of so(k,ℓ) can be used to prove the uniqueness of certain connections (rather than generalized connections). As an example we mention the canonical connection in Born geometry defined in [9], which is related to string compacti- fications and more specifically to double field theory. Its uniqueness, proven in [9], can be alternatively deduced from the vanishing of the generalized first prolongation of the diagonal so(n)-subalgebra

∆so = (A, A) so(n) so(n) A so(n) so(n, n). (n) { ∈ ⊕ | ∈ } ⊂ The corresponding diagonally embedded O(n)-subgroup ∆ O(n, n) is O(n) ⊂ precisely the automorphism group of the following data on V = R2n: 1. a scalar product η = , of neutral signature, h· ·i 2. a positive deﬁnite scalar product g and

3. a linear involution K, which satisfy the following compatibility conditions: 1. K is skew-symmetric with respect to η,

2. J = η−1g is an involution and

3. J anti-commutes with K. These properties imply that the triple (I := KJ, J, K) is a para-hypercomplex structure on V , i.e. I,J,K pairwise anti-commute, K = IJ and I2 = J 2 = K2 = Id. However, the structure is not para-hyper-Hermitian with− respect to η as only K is skew-symmetric, whereas I and J are symmetric with respect to η. A quadruple (η,I,J,K) with these properties is called a Born structure

13 on V if the symmetric bilinear form g = ηJ = η(J , ) is positive definite. So the data (η,g,K) with the above compatibility relations· · 1.-3. is equivalent to a Born structure (η,I,J,K) on V . Given smooth tensor fields (η,I,J,K) on a manifold M such that (ηp,Ip,Jp, K ) is a Born structure on T M for all p M, the data (η,I,J,K) is called a p p ∈ Born structure on M. It is proven in [9] that every Born structure (η,I,J,K) on a manifold M admits a canonical compatible connection , called the ∇ Born connection, with vanishing generalized torsion Ω3(M), where T∇ ∈ (X,Y,Z) := η( Y X [X,Y ]c +( X)∗Y,Z) T∇ ∇X −∇Y − ∇ is defined in terms of the so-called canonical D-bracket [X,Y ]c = c Y c X +( cX)∗Y. ∇X −∇Y ∇ Here B∗ denotes the η-adjoint of an endomorphism field B and c := η + 1 K ηK is the canonical connection compatible with the almost∇ para- ∇ 2 ∇ Hermitian structure (η,K), where η denotes the Levi-Civita connection of η. The Born connection is denoted∇ B and is defined by ∇ B Y := [X ,Y ]c + [X ,Y ]c +(K[X ,KY ]c) +(K[X ,KY ]c) , ∇X − + + + − − + + + − − − where stands for the projections onto the eigendistributions of J. It is ± clear that any two connections ′ and compatible with the same Born structure and having the same generalized∇ ∇ torsion differ by a section A ∈ Γ(T ∗M so(T M)) in the kernel of the total skew-symmetrization map ∂, ⊗ such that Ap belongs to the generalized first prolongation of the Lie algebra aut(TpM, ηp,Ip.Jp,Kp) ∼= ∆so(n). This shows, in particular, that the uniqueness of B follows from the next lemma. ∇ h1i Lemma 19. ∆so(n) =0. ′ ′ Proof. Let (e1,...,en, e1,...,en) be a basis of V , orthonormal with respect to both , and g, where h· ·i e , e = g(e , e )=1, e′ , e′ = g(e′ , e′ )= 1, h i ii i i h i ii − i i − ′ ′ ′ ′ such that K(ei) = ei, K(ei) = ei. Then J(ei) = ei and J(ei) = ei, from ′− ′ where we deduce that Av preserves span e1,...,en and span e1,...,en , ∗ { } { } for any A V ∆so(n) and v V (since Av commutes with J). We deduce ∈ ⊗ ∈ k ′k Rn that A is completely determined by two (1, 2)-tensors (Aij) and (A ij) on , k k ′ ′ where Aei ej = k Aijek and Aei ej = k A ijek, since (from AvK = KAv for ′ k ′ ′ ′k ′ any v) A e = A e and A ′ e = A e . From ∂A = 0 we obtain ei j P k ij k ei j P k ij k ′ ′ ′ ′ k 0=P Ae e , e + Ae′ e ,P ei + Ae′ ei, e = A h i j ki h j k i h k ji − ij ′k and similarly A ij = 0. This proves that A = 0.

14 3 Generalized almost complex structures

3.1 Intrinsic torsion of a generalized almost complex structure

In this section we compute the intrinsic torsion tJ of a generalized almost complex structure on a Courant algebroid E and relate it to the Nijenhuis J tensor NJ . Before we need to recall basic facts on projectors. Recall that an endomorphism P Γ(EndV ) of a vector bundle V is a projector onto a ∈ subbundle V V if P 2 = P and im P = V . Then V decomposes as V = 0 ⊂ 0 V0 ker P and P (respectively Id P ) are the projections onto V0 (respectively ker⊕P ) along this decomposition.− In particular, there is a canonical choice of a complement of im P in V , namely, ker P. ∗ 1,1 ∗ 3 ∗ Consider now the algebraic torsion map ∂J : E ΛJ E Λ E of , 1,1 ∗ ⊗ → J deﬁned by (9), where adJ (E)=ΛJ E is the bundle of -invariant 2-forms on E. J

Lemma 20. i) The map Π Γ(End(Λ3E∗)) deﬁned by J ∈

(ΠJ α)(u,v,w) := 1 (α(u,v,w) α(u, v, w) α( u, v, w) α( u, v,w)) (10) 4 − J J − J J − J J is a projector onto the subbundle

Λ3 E∗ := α Λ3E∗, α( u,v,w)= α(u, v,w)= α(u, v, w) . J { ∈ J J J } ii) The equality im ∂J = ker ΠJ holds.

3 ∗ Proof. It is straightforward to check that im ΠJ ΛJ E . Also, for all 3 ∗ ⊂ 3 ∗ 2 α ΛJ E , ΠJ (α) = α. We obtain that im ΠJ = ΛJ E and ΠJ = ΠJ . Claim∈ i) is proved. To prove claim ii), we notice that Π ∂ = 0, i.e. J ◦ J im ∂ ker Π . Let J ⊂ J π˜ :Λ3E∗ E∗ Λ1,1E∗, (˜π α)(u,v,w) := α(u,v,w)+ α(u, v, w). J → ⊗ J J J J 1 ∗ It is straightforward to check that ΠJ = IdΛ3E 4 ∂J π˜J , which implies ker Π im ∂ . We proved that im ∂ = ker Π .− ◦ J ⊂ J J J The next corollary follows from Lemma 20 and our comments before this lemma.

Corollary 21. With the notation from Lemma 20, im ΠJ is a complement 3 ∗ of im ∂J in Λ E .

15 3 ∗ From Corollary 21, we can (and will) identify the quotient (Λ E )/(im ∂J ) with im Π = Λ3 E∗ and consider the intrinsic torsion t of as a section J J J J of Λ3 E∗. On the other hand, N ( u, v)= N (u, v) (easy check), which J J J −J J implies that NJ , considered as a 3-form (see Lemma 9), is also a section of 3 ∗ ΛJ E . Up to a constant, NJ and tJ coincide. More precisely, we have the following result.

Corollary 22. The torsion T D of a generalized connection D with D =0 J satisﬁes

D D D D T (u,v,w) T ( u, v, w) T (u, v, w) T ( u, v,w)= NJ (u,v,w), − J J − J J − J J (11) for all u,v,w Γ(E). In particular, t = 1 N (viewed as 3-forms on E). ∈ J 4 J Proof. Relation (11) follows from (8), together with

[u, v]= D v D u +(Du)∗v T D(u, v) (12) u − v − and D = 0. Relation (11) can be written as J 1 Π (T D)= N (13) J 4 J which implies the second statement.

3.2 Integrability using torsion-free generalized connections In this section we prove the following theorem.

Theorem 23. A generalized almost complex structure on E is integrable if and only if there is a torsion-free generalized connectionJD such that D =0. J Part of the statement of Theorem 23 follows from Corollary 22: if there is a torsion-free generalized connection D such that D = 0 then from relation J (11) is integrable. For the converse statement, let Γ(EndE) be a gen- eralizedJ almost complex structure. We will constructJ a ∈ generalized complex connection whose torsion equals the intrinsic torsion of . Such a gener- J alized connection will be torsion-free (and complex) when is integrable. This will conclude the proof of Theorem 23. The next remarkJ represents our motivation for the choice of generalized connection D˜ in Proposition 25.

16 Remark 24. Given an almost complex structure J and a torsion-free connection on a manifold M, the connection ∇ 1 1 ˜ Y = Y ( J)X,J Y J( J)Y, (14) ∇X ∇X − 4{ ∇ } − 2 ∇X where A, B := AB + BA denotes the anti-commutator of A and B and { } ( J)X Γ(End T M) is defined by Y ( Y J)X, is complex ( ˜ J = 0) ∇ ∈ ∇˜ 1 → ∇ ∇ and its torsion satisfies T (X,Y ) = 8 NJ (X,Y ) (see Theorem 3.4 of [20]; remark the difference by a multiplicative factor between our definition for the Nijenhuis tensor and that of [20]). In particular, if J is integrable, then ˜ is torsion-free (and complex). Now, for a generalized almost complex structure∇ and a generalized torsion-free connection D on the Courant algebroid E, Jwe may define the analogous expression 1 1 D˜ ′ v = D v (D )u, v (D )v. (15) u u − 4{ J J} − 2J uJ However, D˜ ′ defined by (15) is not a generalized connection (while D is J uJ skew-symmetric with respect to the scalar product , of E, the anticom- mutator (D )u, is not and D˜ ′ does not preserveh· ·i , , in general; we shall give{ moreJ detailsJ} on this argument in Lemma 26).h· In·i the next lemma we modify D˜ ′ in order to obtain a generalized connection. It will turn out that it has the required properties.

Proposition 25. Let be a generalized almost complex structure and D a torsion-free generalizedJ connection on E. Deﬁne 1 1 D˜ v = D v Asym, v (D )v, (16) u u − 4{ u J} − 2J uJ where A := (D )u and Asym is its , -symmetric part. Then D˜ is a u J u h· ·i generalized connection, which preserves . Its torsion is given by J 1 T D˜ (u,v,w)= N (u,v,w). (17) 4 J In particular, if is integrable, then D˜ is torsion-free (and complex). J We divide the proof of the above proposition into several lemmas.

Lemma 26. Equation (16) deﬁnes a generalized complex connection.

Proof. Note that Du is skew-symmetric with respect to the scalar product , of E (D isJ skew-symmetricJ and also D is skew-symmetric, being h· ·i uJ J uJ 17 the composition of two anti-commuting skew-symmetric endomorphisms). Similarly, Asym, is skew-symmetric, because Asym is symmetric and { u J} u J is skew-symmetric. We obtain that D˜ u and Du diﬀer by a skew-symmetric endomorphism, i.e. D˜ is a generalized connection. The generalized connection 1 D(1) := D D (18) u u − 2J uJ preserves . As Asym, commutes with , we obtain that J { u J} J 1 D˜ = D(1) Asym, (19) u u − 4{ u J} preserves as well. J In the next lemmas we prove relation (17). From (19), 1 D˜ v,w = D(1)v,w (η( v,u,w)+ η(w, u, v)) h u i h u i − 8 J J 1 + (η(v, u, w)+ η( w,u,v)) , (20) 8 J J where η = ηD,J is deﬁned by

η(u,v,w) := (D )v,w . (21) h uJ i Remark that η has the symmetries

η(u,v,w)= η(u,w,v), η(u, v,w)= η(u, v, w). (22) − J J Lemma 27. The torsion of D(1) is given by

(1) 1 T D (u,v,w)= η(u, v, w) (23) 2 J (u,v,wX) cyclic where the sum is over cyclic permutations on (u,v,w).

Proof. The claim follows from the torsion-free property of D together with relations (7) and (18).

Lemma 28. The following relation holds:

N (u,v,w)= (η(u, v, w)+ η( u,v,w)) . (24) J J J (u,v,wX) cyclic

18 Proof. Using relations (8), (12) and T D = 0, we obtain N (u, v)=(D )v (D )u (D )v J J uJ − J vJ −J uJ + (D )u +(D( u))∗ v (Du)∗v J vJ J J − (D( u))∗v (Du)∗ v. −J J −J J Taking the inner product of the above equality with w and using the symmetries (22) of η we obtain N (u,v,w)= N (u, v),w = η( u,v,w) η( v,u,w)+ η(w, u, v) J h J i J − J J η(v, u, w)+ η( w,u,v)+ η(u, v, w). (25) − J J J Taking in (25) cyclic permutations over u, v, w, using again the symmetries (22) of η and that NJ (u,v,w) is completely skew we obtain (24). The next lemma concludes the proof of Proposition 25 and Theorem 23. Lemma 29. The torsion of D˜ satisﬁes relation (17). Proof. From relations (20) and (7), we obtain

(1) T D˜ (u,v,w) T D (u,v,w)= − 1 (η( v,u,w)+ η(w, u, v) η(v, u, w) η( w,u,v)) − 8 J J − J − J (u,v,wX) cyclic 1 = (η( u,w,v)+ η(u, v, w)) , −4 J J (u,v,wX) cyclic where in the last equality we have used the symmetries (22) of η. From (23) we then obtain 1 T D˜ (u,v,w)= (η(u, v, w)+ η( u,v,w)) , (26) 4 J J (u,v,wX) cyclic which implies (17), from Lemma 28.

4 Generalized almost hypercomplex structures

4.1 Intrinsic torsion of a generalized almost hypercomplex structure

Let ( 1, 2, 3) be a generalized almost hypercomplex structure on a Courant J J J ∗ 1,1 ∗ 3 ∗ algebroid E and ∂H : E ΛH E Λ E the algebraic torsion map deﬁned by (9), where ⊗ → ad (E)=Λ1,1E∗ := α Λ2E∗, α( u, v)= α(u, v), u,v E, i =1, 2, 3 . (Ji) H { ∈ Ji Ji ∈ } 19 Lemma 30. The endomorphism P Γ(End(Λ3E∗)) deﬁned by ∈ 2 3 P := Π (27) 3 Ji i=1 X is a projector with ker P = im ∂H. In particular, im P is a complement of 3 ∗ im ∂H in Λ E .

Proof. For any generalized almost complex structure , deﬁne the endo- 0,2 0,2 J morphism Π of Λ2E∗ E by (Π α)(u, v),w = (Π α)(u,v,w), where J ⊗ h J i J α Λ2E∗ E∗, u,v,w E and (Π α)(u,v,w) is given by (10). Then Π0,2 ∈ ⊗ ∈ J J coincides with the operator deﬁned by relation (5) of [14]. We obtain that 2 3 0,2 2 ∗ (Pα)(u,v,w)= p(α)(u, v),w where p = 3 i=1 ΠJi Γ(End(Λ E E)) is the map fromh Lemma 1 of [14].i As p is a projector, ∈P is also a projector.⊗ P From Lemma 20 ii), im ∂H ker P. Let ⊂ 3 3 ∗ ∗ 1,1 ∗ π˜ :Λ E E ΛH E , (˜πα)(u,v,w) := α(u,v,w)+ α(u, v, w). → ⊗ Ji Ji i=1 X (28) It is straightforward to check that

∂H π˜ = 6(Id 3 ∗ P ), (29) ◦ Λ E − which implies ker P im ∂H and thus im ∂H = ker P. As P is a projector, ⊂ 3 ∗ im ∂H = ker P is a complement of im P in Λ E .

3 ∗ As in the previous section, we will identify (Λ E )/(im ∂H) with im P and consider the intrinsic torsion t of ( , , ) as a section of im P. (Ji) J1 J2 J3 Corollary 31. The intrinsic torsion t of ( , , ) is given by 1 3 N . (Ji) J1 J2 J3 6 i=1 Ji Proof. Let D be a hypercomplex connection. By Corollary 22 we haveP that D 1 D 1 3 ΠJi (T )= 4 NJi . So t(Ji) = P (T )= 6 i=1 NJi . P 4.2 Integrability using torsion-free generalized connections Our aim in this section is to prove the next theorem.

Theorem 32. A generalized almost hypercomplex structure ( , , ) on J1 J2 J3 a Courant algebroid E is integrable if and only if there is a torsion-free generalized connection D such that D =0 for all i =1, 2, 3. Ji

20 Part of the statement of Theorem 32 is obvious: if there is a torsion-free generalized connection which preserves all , then are integrable from Ji Ji Theorem 23. The converse statement is proved in the next proposition.

Proposition 33. Let ( 1, 2, 3) be a generalized almost hypercomplex structure and D a generalizedJ connectionJ J on E, such that D =0. Deﬁne J1 1 D(1) := D D (30) − 2J2 J2 and 1 (1) D˜ := D(1) π˜(T D ), (31) − 6 where π˜ is the map (28). Then D(1) and D˜ are generalized hypercomplex connections. Moreover,

1 3 T D˜ (u,v,w)= N (u,v,w). 6 Ji i=1 X

In particular, if ( 1, 2, 3) is a generalized hypercomplex structure, then D˜ is torsion-free (andJ hypercomplex).J J

Proof. The existence of a generalized connection D with D = 0 follows J1 from the proof of Lemma 26 (take any generalized connection, say D(0), (0) 1 (0) (1) and deﬁne D := D 2 1D 1). The same argument shows that D is a generalized connection− J whichJ preserves . As D = 0 and anti- J2 J1 J2 commutes with 1, we obtain that Du 2 anti-commutes with 1 as well, for any u E. ThenJ J J ∈ 1 1 D(1) = D [ D , ]= [ D , ]=0. u J1 uJ1 − 2 J2 uJ2 J1 −2 J2 uJ2 J1

(1) (1) (1) (1) As D 1 = D 2 = 0 also D 3 = 0 and D is hypercomplex. This proves theJ statementJ on D(1). J It remains to prove the statements on D.˜ Let η := 1 π˜(T D(1) ). With − 6 this notation, D˜ = D(1) + η. Since imπ ˜ E∗ Λ1,1E∗ and D(1) = 0, also ⊂ ⊗ H Ji D˜ i = 0 for all i =1, 2, 3, i.e. D˜ is hypercomplex. The torsion of D˜ is given byJ 3 D˜ D(1) 1 D(1) D(1) 1 T = T (∂H π˜)(T )= P (T )= N , (32) − 6 ◦ 6 Ji i=1 X where in the second equality we used (29) and in the third equality we used (13) (which holds since D(1) is hypercomplex).

21 5 Generalized almost Hermitian structures: integrability and torsion-free generalized connections

In this section we characterize the integrability of generalized almost Hermi- tian structures using Levi-Civita connections (see Theorem 36 below). We begin with the following simple lemma. Lemma 34. Let (G, ) be a generalized K¨ahler structure on E. Then J [u, v] [u, v] =0, u Γ(E ), v Γ(E ). (33) J + −J + ∀ ∈ − ∈ + Above we denoted by e± the E±-components of a vector e E in the decomposition E = E E determined by G. Similarly, ∈ + ⊕ − [u, v] [u, v] =0, u Γ(E ), v Γ(E ). (34) J − −J − ∀ ∈ + ∈ − Proof. Relation (33) is equivalent to

[u, v] Γ((E )C L ), u Γ((E )C), v Γ((E )C L ), (35) + ∈ + ∩ 1 ∀ ∈ − ∈ + ∩ 1 where we have denoted by L1 the (1, 0)-bundle of 1 = . Remark that J J end (E )C L =(E )C L , where L is the (1, 0)-bundle of = G (since + ∩ 1 + ∩ 2 2 J2 J = on E ). In (35) we distinguish two cases: a) u Γ(L (E )C); b) J1 J2 + ∈ 1 ∩ − u Γ(L¯1 (E−)C). In case a), relation (35) follows from the integrability of ∈. In case∩ b), relation (35) follows from the integrability of . J J2 Remark 35. When the Courant algebroid is exact the above lemma can be proved using the Bismut connection for generalized Kähler structures, constructed in [18]. More precisely, for a generalized Kähler structure (G, ) on an exact Courant algebroid E, there is a unique generalized connectionJ D (called in [18] the Bismut connection), such that DG = 0, D = 0, and whose torsion is of type (2, 1)+ (1, 2) with respect to . The expressionJ of J D is given in Theorem 3.1 of [18]. Its mixed components Duv and Dvu, for u Γ(E+) and v Γ(E−), are Duv := [u, v]− and Dvu = [v,u]+. Relation (73)∈ follows from D∈ = 0. J Theorem 36. A generalized almost Hermitian structure (G, ) on a Courant algebroid E is generalized Kähler if and only if there is a Levi-CivitaJ connection of G which preserves . J Proof. In one direction the statement is obvious: if there is a Levi-Civita end connection D of G which preserves , then it preserves also 2 = G and we deduce that both and areJ integrable (from TheoremJ 23, becauseJ D J J2 is torsion free). We obtain that (G, ) is generalized Kähler. The converse statement follows from the next theorem.J

22 Theorem 37. Let (G, ) be a generalized Kähler structure on a Courant algebroid E and D a Levi-CivitaJ connection. Then the generalized connection 1 1 D˜ = D D Asym, , − 2J J − 4{ J} is a torsion-free generalized connection compatible with (G, ). J Proof. From Proposition 25 we know that D˜ is torsion-free and complex (D˜ = 0). So it suffices to show that DG˜ = 0. Since DG = 0 and is G- skew-symmetric,J the anticommuting endomorphisms and D (u JE) are J uJ ∈ both G-skew and, hence, Du as well. We conclude that the generalized 1 J J connection D 2 D preserves G. It remains− toJ checkJ that Asym, is G-skew-symmetric for all u E. { u J} ∈ We know that it is skew-symmetric with respect to , , since both D 1 ˜ h· ·i − 2 D and D are generalized connections. Therefore it suffices to check J J sym that Au , preserves the decomposition E = E+ E−. Since does, we { J} sym ⊕ J only need to check that Au E+, E− = 0. We check that AuE±, E∓ = 0, which implies the latter.h Let v Γ(i E ), w Γ(E ). Ifh u E ,i then ∈ ± ∈ ∓ ∈ ± A v,w = (D )u,w = 0, because D preserves the decomposition h u i h vJ i vJ E = E+ E− (for all v E). For u E∓, we use a) that D has zero torsion to express⊕ derivatives by∈ brackets with∈ the help of the previous equation (DE )E±, E∓ = 0 and the property that D preserves the subbundles E± andh b)J Lemmai 34:

a) b) A v,w = (D )u,w = D ( u) D u,w = [v, u] [v,u],w =0. h u i h vJ i h v J −J v i h J −J i

Note that the equations (DE± )E, E∓ = (DE± )E∓, E = 0, established in the proof, can be alsoh writtenJ as: i h J i

(D )E =0, (D )E E . (36) E± J ∓ E± J ± ⊂ ± 6 Generalized almost hyper-Hermitian structures: integrability and torsion-free generalized connections

Theorem 38. A generalized almost hyper-Hermitian structure (G, , , ) J1 J2 J3 on a Courant algebroid E is generalized hyper-K¨ahler if and only if there is a Levi-Civita connection D of G which satisﬁes D =0, for all i =1, 2, 3. Ji

23 Proof. If there is a Levi-Civita connection D of G which is hypercomplex, then (G, ) is generalized Kähler (see Theorem 36). Ji For the converse statement, let (G, 1, 2, 3) be a generalized hyper- Kähler structure and D a generalizedJ Levi-CivitaJ J connection of G with D = 0 (which exists, by Theorem 36). We will show that the gener- J1 alized connection D˜ constructed in Proposition 33, starting from D, is a Levi-Civita connection of G which preserves the . Define the generalized Ji connections 1 D(1) := D D (37) u u − 2J2 uJ2 and 1 D˜ := D(1) + η = D D + η , (38) u u u u − 2J2 uJ2 u where η := 1 π˜(T D(1) ). From Proposition 33, D(1) and D˜ are hypercomplex − 6 and D˜ is torsion-free. We claim that DG˜ = 0, which proves the theorem. (1) Note first that D G = 0, since DG = 0 and 2Du 2 is skew-symmetric with respect to G. So it suffices to show that η(Ju,v,wJ) = 0 for all u E, (1) ∈ v E , w E . Note that η = 1 π˜(T D ) has the following expression: ∈ ± ∈ ∓ − 6 η(u,v,w)= 1 (D )( w)+(D )(w) (D )( w)+ (D )(w),u 12h J1vJ2 J3 J2vJ2 − J3vJ2 J1 J2 vJ2 i 1 (D )(u) (D )(u) (D )(u) (D )(u), v , − 12hJ3 J1wJ2 − J2wJ2 −J1 J3wJ2 −J2 wJ2 i for any u,v,w Γ(E). For all u E, v E±, w E∓, each summand belongs either to∈ the set (D )E∈ , E or∈ to (D ∈ )E, E , which both h E± Jα ∓ i h E∓ Jα ±i reduce to zero by (36).

Part II

7 The space of local Dirac generating operators

Let E be an oriented Courant algebroid with anchor π : E T M, scalar → product , and Dorfman bracket [ , ]. We assume that , is of neutral signatureh· (n,·i n). We denote by Cl(E· )· the bundle of Cliﬀordh· ·i algebras over

24 (E, , ) with the Clifford relation e2 = e, e , e E. Let S M be a real vectorh· ·i bundle of irreducible Cl(E)-modules.h i We∈ will call S a→spinor bundle over Cl(E). The representation of Cl(E) on S, denoted by γ : Cl(E) End(S), a γ(a) := γ , (39) → 7→ a is an isomorphism of algebra bundles. To simplify notation, we shall some- times write as for the Clifford action γ s of a Cl(E) on s S. a ∈ ∈ Recall that the Clifford algebra bundle Cl(E) is Z2-graded. We denote i the subbundle of Cl(E) of degree i Z2 by Cl (E). Since , has neutral ∈ h· ·i 0 1 signature, the bundle S has a compatible Z2-grading denoted by S = S S , 0 1 1 1 ⊕ where S = 2 γ(1+ω)S and S = 2 γ(1−ω)S, with ω = e1 e2n the volume form of E, determined by a positive oriented orthonormal∧···∧ basis of E, and considered as an element of Cl(E) (see e.g. Proposition 3.6 of [21]). An argument analogous to the proof of Proposition 5.10 of [21] shows that the Cl0(E)-submodules S0 and S1 are pointwise inequivalent and irreducible. There is an induced Z2-grading on End(Γ(S)) and, in particular, on the algebra of differential operators on S, which includes Γ(End S) End(Γ(S)) ⊂ as the subalgebra of operators of 0-th order. We will denote by [A, B]= AB ( 1)deg A deg BBA − − the super commutator of two homogeneous elements A, B End(Γ(S)), where deg A stands for the degree of A. ∈ Definition 39. A first order odd differential operator d/ on a spinor bundle S over Cl(E) is called a Dirac generating operator for E if for all f C∞(M) ∈ and e, e , e Γ(E), 1 2 ∈ i) [[d,/ f],γe]] = π(e)(f),

ii) [[d,γ/ e1 ],γe2 ]= γ[e1,e2] and iii) d/ 2 C∞(M). ∈ Note that given (E, , ) and d/ one can reconstruct the full Courant algebroid structure fromh· i)·i and ii). This is why the operator d/ is called generating. Proposition 40. Suppose that there is a Dirac generating operator d/ for E on S. Then the set of Dirac generating operators for E on S has the structure of an aﬃne space modelled on the vector space V := e Γ(E) [d,γ/ ] C∞(M) . (40) d/ { ∈ | e ∈ }

In particular, Vd/ is independent of the choice of Dirac generating operator d/.

25 ′ Proof. We first check that d/ := d/ + γe0 is a Dirac generating operator for all e V . Since [γ , f] = 0 and [γ ,γ ]=2 e , e C∞(M) the properties 0 ∈ d/ e0 e0 e1 h 0 1i ∈ i) and ii) in Definition 39 for d/ immediately imply the same properties for ′ ′ 2 2 d/ . Finally, the equation (d/ ) = d/ + [d,γ/ e0 ]+ e0, e0 shows that property iii) holds for d/ ′ if it holds for d/ and e V . h i 0 ∈ d/ Conversely, we show that given Dirac generating operators d/ and d/ ′, there exists e V such that L := d/ ′ d/ = γ . We first observe that [L, f] is a 0-th 0 ∈ d/ − e0 order operator of odd degree for all f C∞(M). By property i) it satisfies ∈ [[L, f],γe] = 0 for all e Γ(E). This implies that [L, f] commutes with Cl0(E). Being of odd degree,∈ it interchanges S0 and S1 and we deduce that [L, f] = 0 since the irreducible Cl0(E)-modules S0 and S1 are inequivalent. This shows that the odd differential operator L is of 0-th order, that is L = γa for some a Γ(Cl1(E)). ∈ Next we consider the even 0-the order operator

′ L := [L, γe]= γ(ae + ea), e Γ(E). It commutes with Cl1(E), in virtue of property ii), and is hence a ∈ scalar operator (since , has neutral signature). We conclude that ae + ea h· ·i is a scalar in Cl(E), for any e E. This easily implies that a =: e0 Γ(E), by a straightforward computation∈ in which e runs through the elements∈ of an orthonormal frame. Now property iii) implies that e0 Vd/. The last claim is now obvious: if d/ and d/ ′ are two∈ Dirac generating ′ operators then d/ = d/ + γ for e V Γ(E) which implies V ′ = V . e ∈ d/ ⊂ d/ d/ The next theorem is our main result in this section.

Theorem 41 (Alekseev-Xu). Let E be a regular Courant algebroid with scalar product of neutral signature. Every spinor bundle S over Cl(E) admits locally a Dirac generating operator.

We divide the proof of Theorem 41 into several steps. Let D be a generalized connection on E. The existence of D is ensured by Example 5. The generalized connection D induces an E-connection in Cl(E), which we denote again by D. Next we choose an E-connection DS on S compatible with D in the sense that S S De (as)=(Dea)s + aDe s, for all e Γ(E), a Γ(Cl(E)), s Γ(S). The existence∈ of∈ such a connection∈ DS can be shown as follows. The bundle (E, , ) admits locally a spin structure. To this structure we can h· ·i associate a spinor bundle Σ over some domain U M. The E-connection D on E induces an E-connection DΣ on Σ. The connection⊂ form of DΣ with

26 respect to a local trivialization of the spin structure is one half of the connection form of D with respect to the corresponding local orthonormal frame of E. (Both forms can be considered as local sections of E∗ so(E), after identi- ⊗ fying spin(E) ∼= so(E) via the adjoint representation ad : spin(E) so(E), ad v := uv vu.) In more concrete terms, let (e ) be an orthonormal→ frame u − i of E and (σ ) a frame of Σ such that |U α β eiσα = Ciασβ, Xβ β where Ciα are constants. Let (ωij) be the (skew-symmetric) matrix of 1-forms deﬁned by

D (e )= ǫ ω (v)e =2 ω (v)(ǫ e∗ e ǫ e∗ e )(e ) v k − k jk j pj p p ⊗ j − j j ⊗ p k j j

tensor product with the connection DΣ. By considering an open covering (Ui) of M and a corresponding partition of unity, we can glue the E-connections DS,Ui to an E-connection DS. The E-connection DS gives rise to a ﬁrst order diﬀerential operator on S, which we call the Dirac operator:

1 2n D/ S = e˜ DS , (41) 2 i ei i=1 X where (ei) is any local frame of E and (˜ei) is the metrically dual frame, that is e , e˜ = δ . h i ji ij Lemma 42. For any generalized connection D and compatible E-connection DS, the operator 1 d/ = D/ S + γ (42) 4 T 3 ∗ satisfies conditions i) and ii) from Definition 39. Above T Γ( E ) ∼= Γ( 3E) Γ(Cl(E)) denotes the torsion of D. ∈ ∧ ∧ ⊂ 27 Proof. We compute [[d,/ f],γ ]] = [[D/ S, f],γ ] for f C∞(M) and v Γ(E). v v ∈ ∈ We find 1 [D/ S, f]= π(e )(f)γ , [[D/ S, f],γ ]= π(e )(f) e˜ , v = π(v)(f). 2 i e˜i v i h i i i i X X This shows that i) in Definition 39 is satisfied. Next we compute

S 1 S S 1 [D/ ,γv]= γe˜ γD v + D , [[D/ ,γv],γw]= [γe˜ γD v,γw]+ γD w, 2 i ei v 2 i ei v i i X X where w Γ(E). A simple calculation in the Cliﬀord algebra shows that for ∈ all u,v,w E: ∈ uvw wuv = 2 u,w v +2 v,w u. − − h i h i So 1 [γe˜ γD v,γw]= e˜i,w γD v+ De v,w γe˜ = γD v+ De v,w γe˜ 2 i ei − h i ei h i i i − w h i i i i i i i X X X X and using that D has torsion T we obtain

[[D/ S,γ ],γ ] = γ + D v,w γ v w Dvw−Dwv h ei i e˜i i X = γ + γ γ ∗ + D v,w γ T (v,w) [v,w] − (Dv) w h ei i e˜i i X = γT (v,w) + γ[v,w].

A simple calculation in the Cliﬀord algebra shows that (for any 3-form T )

[[γ ,γ ],γ ]= 4γ . T v w − T (v,w) Thus we can conclude that 1 [[d,γ/ ],γ ] = [[D/ S + γ ,γ ],γ ]= γ . v w 4 T v w [v,w] So ii) in Definition 39 is also satisfied. In order to conclude the proof of Theorem 41 we therefore need to find a locally defined generalized connection D on E and a compatible E-connection DS on S such that condition iii) from Definition 39 holds as well. To ana- lyze condition iii) in Definition 39 we will use the following lemma, see [7]. (When , has arbitrary signature, the next lemma still holds with the only differenceh· ·i that the restriction of , to does not have neutral signature). h· ·i G 28 Lemma 43. ([7]) Let E be a regular Courant algebroid with scalar product , of neutral signature and anchor π : E TM. h· ·i → i) The bundle ker π E is a coisotropic subbundle of E, that is (ker π)⊥ ker π. ⊂ ⊂ ii) The bundle E decomposes as E = ker π where is isotropic. ⊕F F iii) The bundle ker π decomposes as ker π = (ker π)⊥ where is orthogonal to . ⊕ G G F iv) The decomposition E = (ker π)⊥ is orthogonal with respect to , . The restrictions of , to the two⊕F factors⊕ G (ker π)⊥ and have neutralh· ·i signature. h· ·i ⊕F G The above lemma implies that for any U M sufficiently small, the bundle E admits a frame (p , q ), a =1,...,n,⊂ such that (p ), a =1,...,n, |U a a a span a maximally isotropic subbundle of ker π, (pa), a = 1,...,s, span (ker π)⊥, (q ), a = 1,...,n, span a maximallyP isotropic subbundle of E, a Q qa ker π for any a s + 1, pa, qb = δab and [π(qa), π(qb)] = 0 for any a, b. For∈ the latter condition≥ weh are usingi that the image of π is an integrable distribution on M (by the axiom C2) in Definition 1). More precisely, using Lemma 43 iv), this basis can be constructed in the following way: start with any basis qa, a = 1, ,s, of such that [π(qa), π(qb)] = 0 for any a, b. Consider the basis p ,···a = 1, F,s, of (ker π)⊥ such that p , q = δ a ··· h a bi ab for any a, b. Finally, choose a basis pa, qa, a = s + 1, , n, of such that p ,p = q , q = 0 and p , q = δ for any a, b···= s +1,G , n. h a bi h a bi h a bi ab ··· The following inclusions summarize the properties of the two complementary maximally isotropic subbundles and of E: P Q (ker π)⊥ ker π, ⊥ = . ⊂ P ⊂ F⊂Q⊂F F ⊕ G The next corollary will be useful in the proof of Lemma 45 below. Corollary 44. For any σ Γ(ker π), n π(q ) σ, p =0. ∈ a=1 a h ai ⊥ Proof. Each term in the above sum vanishes:P if a s then pa Γ((ker π) ) and σ, p =0. If a s + 1, then q Γ(ker π) and≤ π(q )=0.∈ h ai ≥ a ∈ a Let be the connection on E U such that the frame (pa, qa) is parallel. Then ∇is flat, preserves the scalar| product , of E and S admits a flat ∇ h· ·i |U connection S compatible with . Then induces a generalized connection D on E and∇ S induces an E∇-connection∇ DS on S which is compatible |U ∇ |U with D. The next lemma concludes the proof of Theorem 41. Lemma 45. The operator (42) constructed using D and DS satisfies d/2 C∞(U) and is a Dirac generating operator. ∈

29 Proof. The Dirac operator D/ S has the expression 1 1 D/ S = p DS + q DS = p DS , 2 a qa a pa 2 a qa a a X X since π(pa) = 0. To see this it is sufficient to remark that the frame (˜qa, p˜b) dual to the frame (qa,pb) is precisely (pa, qb). Its square is given by 1 1 (D/ S)2 = p p DS DS = p ,p S S =0, (43) 4 a b qa qb 4 h a bi∇π(qa)∇π(qb) a a X X S S where we used pa = 0, the flatness of and [π(qa), π(qb)] = 0 for any a, b. ∇ ∇ S S S Next, we compute D/ γT + γT D/ = [D/ ,γT ]. We write the torsion T of D 1 ijk as T = 6 T eijk Cl(E), where (ei) is a D-parallel orthonormal frame ∈ ijk and eijk := eiejek. The coefficients T are given by P T ijk = T (˜e , e˜ , e˜ )= ǫ ǫ ǫ T (e , e , e )= ǫ ǫ ǫ [e , e ], e , (44) i j k i j k i j k − i j kh i j ki where (˜e ) is the frame of E metrically dual to (e ), i.e.e ˜ = ǫ e with i |U i i i i ǫ = e , e . Using the abbreviation γ = γ , we write i h i ii eiej ek ijk / S S ijk 12[D ,γT ]= [γe˜ℓ Deℓ , T γijk] i,j,k,ℓX ijk ijk ijk S = π(eℓ)(T )γe˜ℓ γijk + T γe˜ℓ γ(Deℓ eijk)+ T [γe˜ℓ ,γijk]Deℓ . i,j,k,ℓX i,j,k,ℓX =0 i,j,k,ℓX Note that, for any fixed ℓ, | {z } ijk ℓjk T [γe˜ℓ ,γijk]=6 T γjk. Xi,j,k Xj,k Hence T ijk[γ ,γ ]DS = 6 ǫ ǫ γ DS . e˜ℓ ijk eℓ − j k jk [ej ,ek] i,j,k,ℓX Xj,k To compute the last term we choose the orthonormal frame (ei) to be 1 1 (e ) = (p + q ), (p q ) , i i=1,...,2n √ a a √ a − a 2 2 a=1,...,n where (pa, qa) is the frame constructed above. Then π[ej, ek] = [πej, πek] = 0, because [π(q ), π(q )] = 0. This implies that DS = 0, showing that a b [ej ,ek] 1 [D/ S,γ ]= π(e )(T ijk)γ γ . (45) T 12 ℓ e˜ℓ ijk i,j,k,ℓX 30 From (43) and (45), we obtain

2 1 S 1 2 1 1 ijk 2 d/ = [D/ ,γT ]+ γT = π(eℓ)(T )γe˜ℓ γijk + γT . (46) 4 16 16 3 ! i,j,k,ℓX We compute

′ 2 1 ℓij ℓmn 1 ℓij ℓmn ijr 2 γT = ǫℓT T γijmn = ǫℓT T γijmn ǫiǫjǫr(T ) , 4 i,j,ℓ,m,n 4 i,j,ℓ,m,n − i,j,r X X X where the primed sum is only over pairwise distinct indices. Similarly,

′ ijk ijk ℓjk π(eℓ)(T )γe˜ γijk = π(eℓ)(T )γe˜ γijk +3 π(eℓ)(T )γjk. ℓ i,j,k,ℓ ℓ i,j,k,ℓX X Xj,k,ℓ On the other hand, for any j and k ﬁxed,

π(e )(T ℓjk) = ǫ ǫ π(e ) [e , e ], e ǫ ℓ − j k ℓ h j k ℓi ℓ Xℓ Xℓ = ǫ ǫ π(q ) [e , e ],p =0, − j k a h j k ai a X where we used (44) and Corollary 44 (with σ = [ej, ek], which is a section of ker π). Combining the above relations we obtain

′ ′ 2 1 1 ijk 1 ℓij ℓmn d/ = π(eℓ)(T )γe˜ γijk + ǫℓT T γijmn 16 3 i,j,k,ℓ ℓ 4 i,j,ℓ,m,n 1 X X ǫ ǫ ǫ (T ijr)2. − 16 i j r i,j,r X We aim to prove that 1 d/2 = ǫ ǫ ǫ (T ijk)2. (47) −16 i j k Xi,j,k For this, we need to show that

′ ′ 1 ijk 1 ℓij ℓmn ǫℓπ(eℓ)(T )γℓijk + ǫℓT T γijmn =0. (48) 3 i,j,k,ℓ 4 i,j,ℓ,m,n X X To prove (48) we use axiom C1) of Deﬁnition 1, where indices of tensor components are metrically raised and lowered according to standard conventions:

31 for any i, j, k ﬁxed,

0 = [e , [e , e ]] [[e , e ], e ] [e , [e , e ]] i j k − i j k − j i k = [e , T ℓe ] + [T ℓe , e ] + [e , T ℓe ] − i jk ℓ ij ℓ k j ik ℓ ℓ X = π(e )(T ℓ)+ π(e )(T ℓ) e + T ℓT m T ℓT m e − i jk j ik ℓ jk iℓ − ik jℓ m ℓ ℓ,m X X + [e , T ℓe ]) + π∗d T ℓe , e − k ij ℓ h ij ℓ ki ℓ X ∗ ℓ ℓ m π dTijk−Pℓ π(ek)(Tij )eℓ+Pℓ,m Tij Tkℓ em | {z } = π(e )(T )˜e π(e )(T ℓ)e T ℓT me , ℓ ijk ℓ − i jk ℓ − ij kℓ m m ! Xℓ (i,j,kX) cyclic Xℓ X ∗ ∞ where we have used that π df = ℓ π(eℓ)(f)˜eℓ for all f C (M). Therefore, for any i, j, k, ℓ ﬁxed, ∈ P

π(e )(T )˜e π(e )(T ℓ)e T sT ℓe =0. ℓ ijk ℓ − i jk ℓ − ij ks ℓ s ! (i,j,kX) cyclic X Taking now i, j, k, ℓ pairwise distinct, multiplying the above equality with γijk and summing over (pairwise distinct) i, j, k, ℓ, we obtain

′ ′ ℓijk ℓ mijk 4 π(eℓ)(Tijk)γ +3 Tij Tkℓmγ =0, i,j,k,ℓ i,j,k,ℓ,m X X which is precisely (48) after re-organising the indices. We proved relation (47) which implies in particular that d/2 C∞(U). From Lemma 42, d/ is a Dirac generating operator for E. ∈ Combining Proposition 40 with Theorem 41 we obtain:

Corollary 46. Let E be a regular Courant algebroid on a manifold M and S a spinor bundle over Cl(E). For any suﬃciently small open subset U M, ⊂ the set of Dirac generating operators for E U on S U has the structure of an aﬃne space modelled on the vector space | |

V := e Γ(E ), [d,γ/ ] C∞(U) , d/|U { ∈ |U e ∈ } where d/ is an arbitrarily chosen Dirac generating operator on S . |U

32 8 The canonical Dirac generating operator

Let E be a regular Courant algebroid with scalar product of neutral signature (n, n). In this section we construct a canonical Dirac generating operator d/S on a suitable spinor bundle S of E, of the form (42). Its deﬁnition will involve an arbitrary generalized connection D on E. By canonical we mean that d/S is independent of the choice of D. We start with an arbitrary spinor bundle S over Cl(E). Let D a generalized connection on E and DS a compatible E-connection on S. We begin S 1 S S / D / by analyzing the dependence of D + 4 γT on the data (D,D ), where D is the Dirac operator deﬁned by (41) Let D′ = D + A be another generalized connection on E, where A Γ(E∗ so(E)). ∈ ⊗ Proposition 47. The following holds.

(i) The torsions T ′ and T of D′ and D are related by:

T ′ = T + α, (49)

where α Γ( 3E∗) is given by ∈ ∧ α(u,v,w)= A v,w . (50) h u i (u,v,wX) cyclic (ii) The E-connection 1 (DS)′ := DS A − 2 is compatible with the generalized connection D′. Here A is considered 2 ∗ 2 as a map E E ∼= E Cl(E), so that Ae acts by Cliﬀord multiplication→ on ∧S for all ∧e E⊂. ∈ (iii) The Dirac operators D/ S and (D/ S)′ associated with (D,DS) and (D′, (DS)′) are related by

′ 1 1 (D/ S) = D/ S γ γ , (51) − 4 α − 4 vA where v = tr A = A e˜ Γ(E). A h·,·i ei i ∈ S 1 ′ S ′ 1 / /P / / ′ (iv) The operators dS = D + 4 γT and dS =(D ) + 4 γT are related by 1 d/′ = d/ γ . (52) S S − 4 vA

33 Proof. (i) is relation (7). (ii) To check the compatibility let e, v Γ(E) and s Γ(S): ∈ ∈ 1 1 (DS)′ (vs) = DS(vs) A (vs)= D (v)s + vDSs (A v)s e e − 2 e e e − 2 e 1 1 = D (v)s + vDSs [A , v]s vA s e e − 2 e − 2 e ′ S ′ = De(v)s + v(D )es.

In the fourth equality we used that the commutator [Ae, v] Γ(E) in the Cliﬀord algebra is related to the evaluation A (v) of A Γ(so∈(E)) on v by e e ∈ the formula 1 A (v)= [A , v]. (53) e −2 e

(iii) With respect to an orthonormal frame (ei) we write 1 1 A = Aijke (e e ), α = αijke e e , 2 i ⊗ j ∧ k 6 i ∧ j ∧ k Xi,j,k Xi,j,k ijk ijk ∗ where A := A(˜ei, e˜j, e˜k) and α := α(˜ei, e˜j, e˜k), where E and E are identi- 1 ijk fied with the help of the scalar product. In particular, Aei = 2 ǫi j,k A ej 1 ijk ∧ ek is identified with 2 ǫi A ejek in Cl(E). Similarly, α is identified with 1 ijk P α eiejek in Cl(E). With this notation, 6 P P 1 1 1 1 (D/ S)′ D/ S = e˜ A = e˜ A = Aijkγ − 2 i −2 ei −4 i ei −8 ijk i i X X Xj6=k ′ 1 ijk 1 jjk = A γijk A ǫjγk. −8 i,j,k − 4 X Xj,k Remark that

′ ′ ′ ijk 1 ijk 1 ijk A γijk = ( A γijk)= α γijk =2γα i,j,k 3 i,j,k 3 i,j,k X X (i,j,kX) cyclic X (where in the second equality we used (50)) and similarly

Ajjkǫ γ = A(e , e˜ ), e˜ γ = γ . j k h j j ki k vA Xj,k Xj,k Combining the above relations we obtain (51). (A formula equivalent to (51) is stated as equation (2.22) in [10].) (iv) follows from (49) and (51).

34 S 1 / / D The operator dS = D + 4 γT , deﬁned using a generalized connection D, depends on the choice of E-connection DS compatible with D. In order to remove this freedom we deﬁne

1 ∗ := S det S rS , (54) S ⊗| | 1 ∗ 1 where r := rank S and det S rS denotes the bundle of -densities of S. S rS We call the bundle the|canonical| spinor bundle of S. S Remark 48. i) For a vector bundle V of rank n and s R, we denote ∈ by det V ∗ s the line bundle of s-densities on V , whose ﬁber over p M | | n ∈ consists of all maps ωp :Λ Vp 0 R (called s-densities) which satisfy s \{n } → ωp(λ~v)= λ ωp(~v), for any ~v Λ Vp 0 and λ R 0 . Note that, when s is an integer,| | det V ∗ s is canonically∈ \{ isomorphic} ∈ to det\{V}∗ ⊗s and det V ∗ 2s to (det V ∗)2s. Also,| | | | | |

det ΛV ∗ = det V ∗ N/2, det (V V )∗) s = det V ∗ sn2 det V ∗ sn1 , (55) | | ∼ | | | 1 ⊗ 2 | ∼ | 1 | ⊗| 2 | where N := rank ΛV , V are vector bundles of rank n and s R. i i ∈ ii) An E-connection D on V induces an E-connection on det V ∗ and on ∗ s det V , for any s R. The latter is deﬁned as follows: if Duω = α(u)ω, | | ∈ ∗ s s s where u Γ(E) and ω Γ(det V ), then Du( ω )= sα(u) ω , where ω is ∈ ∈ s | | s| | | | the s-density deﬁned by ω (v1, , vn) := ω(v1, , vn) . Remark that if D′ = D + A are two E-connections| | ··· on V then| the··· induced| E-connections on ∗ s ′ det V are related by Du = Du s trace Au. | iii)| Let S and S′ be two irreducible− spinor bundles of E. Since , has neutral signature, we can write S′ = S L, where L is a line bundle.h· ·i From ⊗ the second relation (55) we obtain that ′ = L det L∗ . If S and S′ are isomorphic, then L is trivializable, i.e. orientable.S S ⊗ ⊗| If L is moreover| oriented, ∗ ∗ ′ then L det L ∼= L L is canonically trivial. We deduce that and are then⊗| canonically| isomorphic.⊗ This is another reason why is calledS theS canonical spinor bundle. S

Lemma 49. Let D be a generalized connection on E with torsion T and DS an E-connection on S compatible with D. Then DS induces a connection DS on , which is compatible with D and depends only on D. In particular, d/ : Γ( S) Γ( ) depends only on D. S S → S Proof. It is clear that the E-connection DS on induced by DS is again S compatible with D. Any other E-connection on S compatible with D is of S ′ S ∗ the form (D ) = D + ϕ IdS for some section ϕ Γ(E ). From Remark 48 ii) DS and (DS)′ induce⊗ the same connection on ∈. S

35 In order to deﬁne a Dirac generating operator independent of D we consider as in [10] the following canonical weighted spinor bundle

∗ 1 S := L, L := det T M 2 . (56) S ⊗ | | The line bundle L carries an induced E-connection DL deﬁned by 1 DLµ := µ div (v)µ, (57) v Lπ(v) − 2 D where v Γ(E), µ denotes the Lie derivative of the densitity µ Γ(L) ∈ Lπ(v) ∈ with respect to the vector ﬁeld π(v) and divD(v) := tr Dv.

∗ 1/r Remark 50. The Lie derivative X µ of a density µ Γ( det T M ) in L 1 ∈ | | 2r the direction of X X(M) is defined by X µ = 2r α(X)µ, when X µ = 2r ∈ L L ∗ 2 α(X)µ . In the latter expression X denotes the Lie derivative on (det T M) (defined by (ω2)= 2( ω)ω whereL ω is the Lie derivative of the vol- LX LX LX ume form ω of M). If is a torsion-free connection on M, then it induces a connection (also denoted∇ by ) on det T ∗M 1/r and one can show that ∇ | | 1 µ = µ trace ( X)µ, µ Γ( det T ∗M 1/r), X X(M). (58) ∇X LX − r ∇ ∀ ∈ | | ∈ Lemma 51. If D is the generalized connection from Lemma 45, then the E-connection DL defined by (57) is induced by a usual connection on L.

Proof. We need to show that div (v) = 0 for any v Γ(ker π) (see Example D ∈ 5). Consider the frame (pa, qa) constructed after Lemma 43 and recall that it is parallel with respect to the connection = D on E. Its dual frame ∇π(e) e is (˜pa, q˜b)=(qa,pb) and div (v)= ( D (v), q + D (v),p )= (v),p = π(q ) v,p D h pa ai h qa ai h∇π(qa) ai a h ai a a a X X X where we used π(p ) = 0 and p = 0. From Corollary 44 we obtain a ∇ a divD(v) = 0 as needed. Theorem 52. [1] Let D be a generalized connection on E with torsion T , DS = DS DL the induced compatible E-connection on the canonical weighted ⊗ S spinor bundle S, and D/ the corresponding Dirac operator on S. Then

S 1 d/S = D/ + γ : Γ(S) Γ(S) (59) 4 T → is a Dirac generating operator, independent of D.

36 Proof. Replacing D by another generalized connection D′ = D + A changes S ′ S S 1 ′ S D to (D ) = D 2 A (see Proposition 47). This implies that (D ) = S 1 − D 2 A. Here we use that the Cliﬀord action of Au on S is trace-free (see the− next remark) and Remark 48 ii). From Proposition 47 again (applied to S 1 S ′ 1 the spinor bundle ) we obtain that D/ + γ changes to (D/ ) + γ ′ = S 4 T 4 T / S 1 1 (D + 4 γT ) 4 γvA . (See Proposition 47 for the deﬁnition of vA.) On the other− hand, on S = L, S ⊗ 1 D/(s l)=(D/ S s) l + e˜ s DL l, s Γ( ), l Γ(L) ⊗ ⊗ 2 i ⊗ ei ∈ S ∈ and a similar expression holds for the Dirac operator D/ ′ on S computed with the generalized connection D′. We deduce that 1 1 1 (D/ + γ )(s l)=(D/ S s + Ts) l + e˜ s DL l, 4 T ⊗ 4 ⊗ 2 i ⊗ ei ′ 1 S 1 1 1 ′ L (D/ + γ ′ )(s l)=(D/ s + Ts v s) l + e˜ s (D ) l. 4 T ⊗ 4 − 4 A ⊗ 2 i ⊗ ei

′ L L 1 But div ′ = div v , implies that (D ) l = D l + v , v l. We deduce D D −h A ·i v v 2 h A i that (59) is independent on D. It remains to show that d/S is a Dirac generating operator. As this is a local property, we need to show that for any sufficiently small open subset U M, ⊂ the restriction d/S U : Γ(S U ) Γ(S U ) is a Dirac generating operator. Choose U like in Lemma| 45, and| let→D be the| generalized connection on E defined |U by the flat metric connection used in that lemma. From Lemma 51, DS DL on S = ( L) is∇ induced by a usual connection compatible ⊗ |U S ⊗ |U with . Since d/S is independent on D, we deduce that on U it coincides with ∇ the operator constructed in Lemma 45. In particular, it is a Dirac generating operator.

Remark 53. In the proof we have used the fact that any representation of spin(k,ℓ) Cl(Rk,ℓ) induced by a representation of the Cliﬀord algebra ⊂ Cl(Rk,ℓ) is trace-free. This follows from the fact that the Lie algebra spin(k,ℓ) is spanned by commutators uv vu, where u, v Rk,ℓ. − ∈ Deﬁnition 54. The operator d/S : Γ(S) Γ(S) is called the canonical Dirac generating operator associated to E. →

Remark 55. Our canonical Dirac generating operator coincides with the Dirac generating operator constructed in Theorem 4.1 of [1]. This follows from formula (53) of [1], by noticing a diﬀerence of a minus sign between our deﬁnition for the torsion of a generalized connection and that from [1] (see

37 Section 3.2 of this reference). Our Dirac generating operator also coincides (up to a multiplicative constant factor) with the Dirac generating operator 1 from Proposition 5.12 of [15] (in formula (5.14) of [15] the term 2 γ(C∇) should have a minus sign).

9 Standard form of the canonical Dirac generating operator

In this section we provide an expression for the canonical Dirac generating operator d/S which uses the structure of regular Courant algebroids (see [7]). We consider the setting from the previous section. For simplicity, from now on d/S will be denoted by d/. From Lemma 43, there is a vector bundle isomorphism I : E F ∗ F, (60) → ⊕ G ⊕ where we recall that F = π(E) T M is an integrable distribution and ker π E is a subbundle. The isomorphism⊂ I maps the anchor π : E TG M ⊂ of E to⊂ the map ρ(ξ + r + X) = X and the scalar product of E to a→ scalar product 1 ξ + r + X, η + s + y = (ξ(Y )+ η(X))+(r, s)G, (61) h i 2 where ξ + r + X, η + s + Y F ∗ F , and the scalar product ( , )G on ∈ ⊕ G ⊕ · · is of neutral signature. The bundle , together with ( , )G, is canonically Gassociated to E. More precisely, ( , ( ,G)G) is isomorphic· to· (ker π)/(ker π)⊥ with scalar product induced by theG scalar· · product of E. Moreover, is a G bundle of Lie algebras, with Lie bracket [ , ]G induced from the Dorfman bracket of E. The scalar product ( , )G is· invariant· with respect to [ , ]G, · · · · that is the adjoint representation of the Lie algebra is by skew-symmetric endomorphisms. In fact, these properties follow immediately from ker π. An isomorphism (60) as above is called a dissection of E [7]. G ⊂ The Dorfman bracket [ , ] of F ∗ F induced from E via a dissection satisﬁes · · ⊕ G ⊕ Pr [r ,r ] = [r ,r ]G, r Γ( ), (62) G 1 2 1 2 i ∈ G ∗ where PrG is the natural projection from F F on (we shall use a ⊕ G ⊕ ∗ G similar notation PrF ∗ for the natural projection on F ). Therefore, we may (and will) assume that the given regular Courant algebroid is of the form E = F ∗ F , with anchor ρ(ξ + r + X) = X, metric given by (61) and Dorfman⊕ bracket G ⊕ [ , ] satisfying (62). As proved in · ·

38 Lemma 2.1 of [7], the Dorfman bracket of E is determined by its components

: Γ(F ) Γ( ) Γ( ), (r) := Pr [X,r], ∇ × G → G ∇X G R : Γ(F ) Γ(F ) Γ( ), R(X,Y ) := PrG[X,Y ] × → G ∞ : Γ(F ) Γ(F ) Γ(F ) C (M), (X,Y,Z) := (Pr ∗ [X,Y ])(Z). H × × → H F Note that here [X,Y ] stands for the Dorfman bracket

[X,Y ]= (X,Y, )+ R(X,Y )+ Y H · LX of X,Y as sections of F E, whereas, for the rest of this section, the Lie ⊂ bracket of vector fields will be always denoted by X Y . The map is a partial connection on , the map R is a 2-form on LF with values in ∇ and is a 3-form on F . TheG properties of the triple ( ,R, ) are describedG in H ∇ H Theorem 2.3 of [7]. The next lemma was proved in [7] and can be checked directly (we remark a difference of sign between our definition for the torsion of a generalized connection and that from [7]). By a torsion-free connection on F we mean a partial connection F : Γ(F ) Γ(F ) Γ(F ) which satisfies Y X = ∇ × → ∇X −∇Y Y for any X,Y Γ(F ). LX ∈ Lemma 56. ([7]) Let F be a torsion-free connection on F . Then ∇ 1 2 E (η + s + Y ) := ( F η (X,Y, ), s + [r, s]G, F Y ) (63) ∇ξ+r+X ∇X − 3H · ∇X 3 ∇X is a generalized connection on E with torsion given by

E T ∇ (u,v,w)= (X,Y,Z) (R(X,Y ), t)G (R(Y,Z),r)G (R(Z,X),s)G − H − − − + ([r, s]G, t)G, (64) for any u = ξ + r + X, v = η + s + Y , w = ζ + t + Z (where ξ,η,ζ F ∗, ∈ r, s, t , X,Y,Z F ). ∈ G ∈ Remark 57. For any regular Courant algebroid E with anchor π : E T M, the quotient := E/(ker π)⊥ inherits a Lie algebroid structure from→ A the Dorfman bracket of E. This Lie algebroid is called in [7] the ample Lie algebroid associated to E. A dissection induces a bundle isomorphism = F and a Lie algebroid structure on F (inherited from the ∼ E ALie algebroidG ⊕ structure of ). The restriction ofG the ⊕ 3-form Ω := T ∇ to F is closed with respectA to the Lie algebroid diﬀerential of − F and G ⊕ G ⊕ its cohomology class is independent on the chosen dissection. It is called the Severa class of E, as it coincides with the Severa class when E is exact [7].

39 Let ΛF ∗ be the bundle of forms over F . It is a spinor bundle over the bundle of Clifford algebras Cl(F ∗ F ), where F ∗ F has scalar product ⊕ ⊕ ξ + X, η + Y = 1 (η(X)+ ξ(Y )). The Clifford representation is h i 2 (X + ξ) α := ι α + ξ α, (65) · X ∧ where ιX α := α(X, ) denotes the interior product. Let SG be an irreducible spinor bundle over· Cl( ), where is considered with the scalar product G G G ( , ) . We assume that is oriented, so that SG is Z2-graded. The Z2- · · G ∗ graded tensor product S := ΛF ˆ SG is an irreducible spinor bundle over Cl(E) = Cl(F ∗ F ) ˆ Cl( ). (Basic⊗ facts concerning the Z -graded tensor ⊕ ⊗ G 2 product are reviewed in more detail in Appendix 11; in particular, see relation (88) for the Clifford action of E on S.)

Lemma 58. The canonical weighted spinor bundle of S is given by

S = ΛF ∗ det (Ann F ) 1/2 ˆ (66) ⊗| | ⊗SG where Ann F T ∗M is the annihilator of F and = S det S∗ 1/g is the ⊂ SG G ⊗| G| canonical spinor bundle of SG (with g := rank SG). Proof. The isomorphism (given by contraction) between (det T M) det F ∗ ⊗ and det ((Ann F )∗), induces a canonical isomorphism

det F 1/2 det T ∗M 1/2 = det (Ann F ) 1/2. (67) | | ⊗| | ∼ | | The claim follows from the deﬁnition of the canonical weighted spinor bundle S, recall (56) and (54), combined with relations (55) and (67). Recall the deﬁnition of the E-connection DS from Theorem 52 computed from a generalized connection D.

Lemma 59. The E-connection S on S computed from the generalized connection E deﬁned in Lemma∇ 56 has the following expression: for any ω Γ(ΛF∇∗), τ Γ( det (Ann F ) 1/2) and s Γ( ) we have ∈ ∈ | | ∈ SG S(ω τ s)=( F ω) τ s + ω ( τ) s ∇u ⊗ ⊗ ∇X ⊗ ⊗ ⊗ LX ⊗ 1 1 + (ι ) ω τ s + ω τ ( 0,SG s (ad )(s)). (68) 3 X H ∧ ⊗ ⊗ ⊗ ⊗ ∇X − 3 r

Above u = ξ + r + X, X τ denotes the Lie derivative of τ in the direction 0,SG L of X Γ(F ), is an F -connection on G , induced by an F -connection 0,SG ∈ ∇ S on SG compatible with the F -connection of and adr so( ) is considered∇ as a 2-form on , which acts by Cliﬀord∇ multiplicationG ∈ on G. G SG 40 Proof. We remark that E = F ∗+F + G, where F ∗+F and G are E- connections on F ∗ F and∇ respectively,∇ ∇ deﬁned by∇ ∇ ⊕ G ∗ 1 F +F (η + Y )= F η (X,Y, )+ F Y, ∇u ∇X − 3H · ∇X 2 G(s) := s + ad (s) ∇u ∇X 3 r where ad (s) = [r, s]G. The F -connection F induces an F -connection (also r ∇ denoted by F ) on ΛF ∗. A straightforward computation shows that the E-connection∇ 1 F, spin := F + (ι ) (69) ∇u ∇X 3 X H ∧ on ΛF ∗ is compatible with F ∗+F . On the other hand, like in the proof of Proposition 47 ii), the E-connection∇ 1 G, spin := 0,SG ad (70) ∇u ∇X − 3 r

G F, spin G, spin on SG is compatible with . We obtain that is an E- ∗ ∇ ∇E ⊗∇ E connection on S = ΛF ˆ SG compatible with . The deﬁnition of ⊗ F ∇ ∇ implies that div E (u) = trace( X) and thus ∇ ∇ 1 ( E)Lµ = µ trace( F X)µ, ∇ u LX − 2 ∇ for any section µ of L = det T ∗M 1/2. We obtain that the E-connection S | | ∇ on S = (ΛF ∗ det F 1/2) ˆ L is given by ⊗| | ⊗SG ⊗ S := F, spin G, spin ( E)L (71) ∇ ∇ ⊗∇ ⊗ ∇ where we preserve the same symbols F, spin and G, spin for the E-connections F, spin G, spin ∇ ∗ ∇1/2 induced by and on ΛF det F and G respectively. In order to compute∇ S we shall∇ compute ⊗|F, spin ( | E)L andS G, spin separately. We begin with∇ F, spin ( E)L.∇ This is⊗ an∇ E-connection∇ on ΛF ∗ ∇ ⊗ ∇ ⊗ det F 1/2 L. Relation (69) holds also on ΛF ∗ det F 1/2, as the endomor- |phism|ω ⊗ 1 (ι ) ω of ΛF ∗ is trace-free. Let ω⊗| Γ(ΛF| ∗), l Γ( det F 1/2) → 3 X H ∧ ∈ ∈ | | and µ Γ(L). Then ∈ ( F, spin ( E)L) (ω l µ)= F,spin(ω l) µ + ω l ( E)L µ ∇ ⊗ ∇ u ⊗ ⊗ ∇X ⊗ ⊗ ⊗ ⊗ ∇ X 1 1 =( F (ω l)+ (i ω) l) µ + ω l ( µ trace( F X)µ) ∇X ⊗ 3 X H ∧ ⊗ ⊗ ⊗ ⊗ LX − 2 ∇ 1 =( F ω) l µ + ω (l µ)+ (ι ω) l µ (72) ∇X ⊗ ⊗ ⊗LX ⊗ 3 X H ∧ ⊗ ⊗

41 where in the last equality we used

F (ω l)=( F ω) l + ω F l ∇X ⊗ ∇X ⊗ ⊗∇X 1 F l = l + trace( F X)l. (73) ∇X LX 2 ∇ (see (58) for the second relation (73)). G, spin 0,SG Next, we compute . Relation (70) holds also on G (with 0,SG ∇ 0,SGS ∇ replaced by , the E-connection on G induced by ). This follows ∇ S 2 ∇ from the fact that the Cliﬀord action by adr Γ(Λ ) on SG is trace-free. The latter is a consequence of Remark 53. So∈ we haveG proven: 1 G, spin = 0,SG ad on . (74) ∇u ∇X − 3 r SG Combining (71), (72) and (74), we obtain (68). Notation 60. For any τ Γ( det (Ann F ) 1/2), τ is C∞(M)-linear in X, ∈ | | LX when X Γ(F ) (as fX ω = f Xω, for any ω Γ(Ann F )). The map Γ(F ) X∈ τ isL a 1-form onLF with values in∈ det (Ann F ) 1/2, which ∋ → LX | | was denoted by (τ). L We arrive now at what we call the standard form for the canonical Dirac generating operator in terms of the data encoding the regular Courant algebroid. Theorem 61. Let E be a regular Courant algebroid with anchor π : E T M and scalar product of neutral signature. In terms of a dissection F ∗ → F of E, the canonical Dirac generating operator is given by ⊕ G ⊕

d/(ω τ s)=(dF ω) τ s + (τ) ω s + 0,SG (s) ω τ ⊗ ⊗ ⊗ ⊗ L ∧ ⊗ ∇ ∧ ⊗ 1 ( ω) τ s + ( 1)|ω|+1ω τ Cs − H ∧ ⊗ ⊗ 4 − ⊗ ⊗ +( 1)|ω|+1R¯(ω τ s), (75) − ⊗ ⊗ where ω Γ(ΛF ∗), τ Γ( det (AnnF ) 1/2) and s Γ( ). Above dF : ∈ ∈ | | ∈ SG Γ(Λ•F ∗) Γ(Λ•+1F ∗) is the exterior derivative along the integrable distribution F →= im π, C Γ(Λ3 ∗) is the Cartan form C(u,v,w) = ([u, v]G,w)G ∈ G of viewed as a section of Cl( ), Cs denotes its Cliﬀord action on s and G G 1 R¯(ω τ s)= (R(X ,X ),r )G(α α ω) τ r˜ s, (76) ⊗ ⊗ 2 i j k i ∧ j ∧ ⊗ ⊗ k Xi,j,k where (Xi) is a basis of F , (αi) is the dual basis, i.e. αi(Xj)= δij, (ri) is a basis of and (˜r ) is the dual basis with respect to ( , )G. G i · · 42 Proof. The bases of E

(ei) := (Xi,rj,αk), (˜ei) := (2αi, r˜j, 2Xk) (77) are dual with respect to the scalar product (61) of E. Using Lemma 59 and the definition of the Clifford action, we obtain e˜ S (ω τ s)= (2(α F ω) τ s + 2(α ω) τ s i∇ei ⊗ ⊗ i ∧∇Xi ⊗ ⊗ i ∧ ⊗LXi ⊗ i i X 2X + (α ι ω) τ s + 2(α ω) τ 0,SG s 3 i ∧ Xi H ∧ ⊗ ⊗ i ∧ ⊗ ⊗∇Xi 1 +( 1)|ω|+1 ω τ (˜r ad )(s)). − 3 ⊗ ⊗ i ri But α F ω = dF ω, α ι =3 , α τ = (τ), i ∧∇Xi i ∧ Xi H H i ⊗LXi L i i i X 1 X X r˜ ad = (ad (r ),r )Gr˜ r˜ r˜ =3C, (78) i ri 2 ri j k i j k i X Xi,j,k where in the first equality (78) we used that F is torsion-free and the last equality holds in the Clifford algebra Cl( ). We∇ obtain that G 1 e˜ S (ω τ s)=(dF ω) τ s + (τ) ω s 2 i∇ei ⊗ ⊗ ⊗ ⊗ L ∧ ⊗ i X 1 +( 0,SG s) ω τ +( ω) τ s + ( 1)|ω|+1ω τ Cs. (79) ∇ ∧ ⊗ H ∧ ⊗ ⊗ 2 − ⊗ ⊗ From (79) and the definition of d/ we obtain d/(ω τ s)=(dF ω) τ s + (τ) ω s +( 0,SG s) ω τ ⊗ ⊗ ⊗ ⊗ L ∧ ⊗ ∇ ∧ ⊗ 1 1 E +( ω) τ s + ( 1)|ω|+1ω τ Cs + T ∇ (ω τ s). (80) H ∧ ⊗ ⊗ 2 − ⊗ ⊗ 4 ⊗ ⊗ In order to conclude our proof we need to express T ∇E as a section of Λ3E E E and to compute T ∇ (ω τ s), where T ∇ Γ(Λ3E) Γ(Cl(E)) acts by Clifford multiplication on⊗ ω ⊗ τ s. Using the∈ bases (77),⊂ we write ⊗ ⊗ E 1 E T ∇ = T ∇ (e , e , e )˜e e˜ e˜ 6 i j k i ∧ j ∧ k Xi,j,k 4 E = T ∇ (X ,X ,X )α α α 3 i j k i ∧ j ∧ k i,j,k X E +2 T ∇ (X ,X ,r )α α r˜ + C, i j k i ∧ j ∧ k Xi,j,k 43 ∇E ∇E where we used that T (αi, , ) = 0 and T (ri,rj,Xk) = 0 from relation (64). Again from relation (64),· ·

E T ∇ (X ,X ,X )α α α = (X ,X ,X )α α α = 6 i j k i ∧ j ∧ k − H i j k i ∧ j ∧ k − H Xi,j,k Xi,j,k and

E T ∇ (X ,X ,r )α α r˜ = (R(X ,X ),r )Gα α r˜ . i j k i ∧ j ∧ k − i j k i ∧ j ∧ k We have proven that T ∇E , as a section of Λ3E, is given by

E T ∇ = 8 2 (R(X ,X ),r )Gα α r˜ + C. − H − i j k i ∧ j ∧ k Xi,j,k This implies

E T ∇ (ω τ s)= 8( ω) τ s +( 1)|ω|ω τ Cs ⊗ ⊗ − H ∧ ⊗ ⊗ − ⊗ ⊗ + 2( 1)|ω|+1 (R(X ,X ),r )G(α α ω) τ r˜ s. (81) − i j k i ∧ j ∧ ⊗ ⊗ k Xi,j,k We conclude by combining (80) with (81).

Example 62. Consider a regular Courant algebroid E with surjective anchor π : E T M such that ker π = (ker π)⊥. A dissection of E deﬁnes an → isomorphism between E and the Courant algebroid T M T M ∗ from Example 3 (see Lemma 2.1 of [1]). From Theorem 61, the canonical⊕ Dirac generating operator acts on Γ(ΛT ∗M) by d/(ω)= dω ω. We recover the expression −H ∧ of the Dirac generating operator for exact Courant algebroids, see e.g. [16].

Part III

10 Generalized almost Hermitian structures: integrability and spinors

In Theorem 6.4 of [1] an integrability criterion for a generalized almost complex structure on a regular Courant algebroid E with scalar product of neutral signatureJ (n, n), using the canonical Dirac generating operator d/ : Γ(S) Γ(S) of E and the pure spinor associated to , was developed. For→ completeness of our exposition we recall it in theJ appendix. As

44 an application of the theory from the previous sections, we now characterize the integrability of a generalized almost Hermitian structure (G, ) on E J in terms of suitably chosen Dirac operators and the pure spinors associated to E± , where E = E+ E− is the decomposition of E determined by G. RemarkJ| that rank E and⊕ rank E are even (as preserves E and E ). + − J + − We consider E endowed with the (non-degenerate) scalar products , ± h· ·i|E± and we denote by Cl(E±) the bundle of Clifford algebras over (E±, , E±). 2 h· ·i| We assume that E± are oriented and that ω± = 1, where ω± are volume forms determined by positive oriented bases of E±, viewed as elements of Cl(E±). We assume that there are given irreducible Cl(E±)-bundles S± with Schur algebra R. The latter condition means that any vector bundle morphism f± : S± S± which commutes with the Cl(E±)-action is a multiple of the identity.→ A quick inspection of Table 1 from [21] (see page 29) implies that either , E± have both neutral signature, or dim E+ = dim E− is a multiple of eight,h· ·i| and one of , is positive definite (while the other is h· ·i|E± negative definite). Moreover, S± are Z2-graded with gradation defined by 0 1 1 1 S± := 2 γ(1+ω±)S± and S± := 2 γ(1−ω±)S±. Since S are irreducible Z -graded Cl(E )-bundles, S ˆ S is an irre- ± 2 ± +⊗ − ducible Z2-graded Cl(E)-bundle, with Clifford action given by

γ (s s )= γ (s ) s +( 1)|s+|s γ (s ), (82) v +⊗ − v+ + ⊗ − − +⊗ v− − for any v = v + v E (see appendix for more details). Remark that + − ∈ = + ˆ − is the canonical spinor bundle of S+ ˆ S−, where ± := S± S S ⊗S1 ⊗ S ⊗ r det S∗ S± is the canonical spinor bundle of S (and r is the rank of S ). | ±| ± S± ± Lemma 63. If η Γ(( )C) are pure spinors associated to , then ± ∈ S± J|E± η η Γ( C) is a pure spinor associated to . +⊗ − ∈ S J Proof. Let L be the (1, 0)-bundle of and J (E )C L = L := v (E )C γ η =0 . ± ∩ η± { ∈ ± | v ± } Both L L and L are subbundles of (E )C (E )C = EC. From η+ ⊕ η− η+⊗η− + ⊕ − (82), L L L . Since L is an isotropic subbundle of EC, its η+ ⊕ η− ⊂ η+⊗η− η+⊗η− rank is at most n. By comparing ranks we obtain Lη+ Lη− = Lη+⊗η− . As L = L L we obtain L = L as needed. ⊕ η+ ⊕ η− η+⊗η− Deﬁnition 64. The pure spinors η± from the above lemma are called pure spinors associated to (G, ). J Let D be a generalized Levi-Civita connection of G and D± the E- S± connections on E± induced by D. Choose E-connections D on S± compatible with D±. Like in the proof of Theorem 41 one can show that DS± exist

45 S± 0 0 S± 1 and preserve the grading of S±, i.e. De Γ(S±) Γ(S±) and De Γ(S±) Γ(S1 ), for any e Γ(E). Since S have Schur⊂ algebra R, any other E⊂- ± ∈ ± connection compatible with D± is related to DS± by D˜ S± = DS± +λ± Id , ⊗ S± for λ± Γ(E∗). We denote by DS± the E-connections induced by DS± on ∈ . Let DS := DS+ DS− be their tensor product (independent of grada- S± ⊗ tions), which is an E-connection on = + ˆ − induced by the E-connection S+ S− S S ⊗S D D on S+ ˆ S−. Recall that we use the convention v±s± for the Clif- ford action⊗ γ s .⊗ Similarly, to simplify notation, we write v(s s ) instead v± ± +⊗ − of γ (s s ), for any v E. v +⊗ − ∈ Lemma 65. The E-connection DS is compatible with the Cliﬀord action of Cl(E) on . S Proof. Using (82) and that DS+ preserves the grading of , we obtain S+ DS(v(s s )) = D (v)(s s )+ vDS (s s ), e +⊗ − e + ⊗ − e +⊗ − for any e, v Γ(E) and s Γ( ), as needed. ∈ ± ∈ S± In the above setting, there are three Dirac operators to be considered: two Dirac operators D/ S± : Γ( ) Γ( ) computed using the E-connections S± → S± DS± of the Cl(E )-bundles , deﬁned by ± S±

S± 1 ± S± D/ (s±) := e˜i D ± s±, 2 ei i X ± ± ± where (ei ) is a basis of E± and (˜ei ) is the metric dual basis, i.e.e ˜i belong to ± ± S E± and ei , e˜j = δij. The third Dirac operator D/ is the one from Section 8, computedh usingi the E-connection DS on the Cl(E)-bundle . Using that (e+, e−) and (˜e+, e˜−) are bases of E dual with respect to , , weS obtain that i j i j h· ·i

S 1 + S 1 − S D/ (s+ s−)= e˜i D + (s+ s−)+ e˜i D − (s+ s−), ⊗ 2 ei ⊗ 2 ei ⊗ i i X X for any s Γ( ). The next lemma can be checked from deﬁnitions. ± ∈ S± Lemma 66. The operators D/ S, D/ S+ and D/ S− are related by

S S S D/ (s s )=(D/ + s ) s +( 1)|s+|s (D/ − s ) +⊗ − + ⊗ − − +⊗ − 1 + S− |s+| S+ − + e˜i s+ (D + s−)+( 1) (D − s+) e˜i s− , (83) 2 ⊗ ei − ei ⊗ i X where s Γ( ). ± ∈ S± 46 S± In the next theorem we use the notation De∓ (η±) η± (for e∓ E∓) S± ≡ ∈ if De∓ (η±) = fη± for a function f = f(e∓) which depends on e∓. By D/ S± η η we mean D/ S± η γ (η ). ± ≡ ± ± ∈ (E±)C ± Theorem 67. In the above setting, the generalized almost Hermitian structure (G, ) on the regular Courant algebroid E with scalar product of neutral J signature is generalized Kähler if and only if there is a Levi-Civita connection D of G such that S D/ ± η η ,DS± η η (84) ± ≡ ± e∓ ± ≡ ± for any e∓ Γ(E∓). Here E = E+ E− is the decomposition determined by G and η ∈Γ( ) are pure spinors⊕ associated to (G, ). ± ∈ S± J S / / 1 Proof. Let D be a Levi-Civita connection of G. From (59), d = D + 4 γT = S D/ since D is torsion-free and, using DS = DS DL, ⊗ S 1 D/ (s l)= D/ S(s) l + e˜ s DL l, s Γ( ), l Γ(L), ⊗ ⊗ 2 i ⊗ ei ∀ ∈ S ∀ ∈ i X where (e ) is a basis of E and (˜e ) the dual basis with respect to , . We i i h· ·i obtain that a pure spinor η l from S = L is projectively closed if and S ⊗ S ⊗ only if D/ η γEC η. ∈ S Assume now that relations (84) hold, with D/ ± and DS± computed starting with D. From (83), we deduce that the pure spinor η = η+ η− associated S ⊗ to satisfies D/ η γEC η, i.e. is integrable (see Corollary 72). In a sim- J ∈ Jend ilar way, we show that 2 = G is integrable. For this, we use the fact J J S− η+ η¯− is a pure spinor associated to 2 and D/ η¯− γ(E−)C η¯− (because S⊗− J ∈ D/ : Γ(( )C) Γ(( )C) is the complex linear extension of its restric- S− → S− tion to and hence commutes with the natural conjugation of ( )C). We S− S− obtain that (G, ) is generalized Kähler. Conversely, assumeJ now that is integrable and let D be a Levi-Civita J connection of G, with D = 0 (which exists from Theorem 36). The relation J DS+ (vη )=(D+v)η + vDS+ η , e Γ(E), (85) e + e + e + ∈

S+ together with the fact that D preserves Lη+ = L (E+)C imply that vDe η+ = 0, for any v Γ(L ), i.e. ∩ ∈ η+ DS+ η = λ(e)η , e Γ(E), (86) e + + ∀ ∈

∗ S+ for λ Γ(E ). Relation (86) with e := e− Γ(E−) implies that De− η+ η+. ∈ + ∈ ≡ On the other hand, letting e := ei in (86), applying the Cliﬀord action of

47 e˜+ and summing over i we obtain that D/ S+ η = λ η where λ := λ is i + + + + |E+ a section of E∗ = E (identified using , ) and λ η = γ η is the + ∼ + h· ·i|E+ + + λ+ + Clifford action of λ E on η . We proved D/ S+ η η as needed. The + ∈ + + + ≡ + same argument with + and − interchanged shows that all relations (84) are satisfied. S S Lemma 68. Relations (84) are independent of the choice of Levi-Civita connection. Proof. Let D be a Levi-Civita connection. Since D is torsion-free and preserves E , for any e Γ(E ) and v ,w Γ(E ), ± − ∈ − + + ∈ + 0= T D(e , v ,w )= D v D e [e , v ],w + v ,D e − + + h e− + − v+ − − − + +i h + w+ −i = D v [e , v ],w , h e− + − − + +i which implies De− v+ = [e−, v+]+ (see also Lemma 3.2 of [10]). In particular, D+ = D is independent of the choice of Levi-Civita connection D, e− e− |Γ(E+) for any e Γ(E ). We obtain that any two E-connections DS+ and D˜ S+ − ∈ − on , compatible with any two Levi-Civita connections of G, satisfy D˜ S+ = S+ e− DS+ +λ (e )Id , for λ E∗ . This implies that the condition DS+ η η e− − − S+ − ∈ − e− + ≡ + is independent of the choice of D. In a similar way we prove the statement for DS− η η . e+ − ≡ − Next, consider two Levi-Civita connections D and D˜ = D + A of G. The S arguments from Propositions 47 and 49 show that DS+ (hence, also D/ + ) depends only on D+ and

S ˜ + S+ 1 1 D/ = D/ γ + γ (87) − 4 α − 4 vA+ + 3 ∗ + where α Γ(Λ E+) is given by α (u,v,w) := (u,v,w) cyclic Auv,w , u,v,w ∈ n + + + h i ∈ E+ and vA+ := A + e˜i Γ(E+), where (ei )and(˜ei ) are , -dual bases i=1 ei ∈ P h· ·i ˜ + of E+. As D andPD are torsion-free, α = 0 and we obtain that S ˜ + S+ 1 D/ η = D/ η v + η . + + − 4 A +

S+ This implies that the condition D/ η+ η+ is independent of the choice of ≡ S D. In a similar way we prove the statement for D/ − η η . − ≡ − Remark 69. The Dirac operators D/ S± are in fact independent of the Levi- Civita connection of G, as long as we ﬁx the divergence of D. The statement S+ for D/ follows from relation (87), by noticing that if divD = divD˜ then S− vA+ = 0 (a similar argument holds for D/ ).

48 From Theorem 67 combined with Lemma 68 we obtain the following characterization for the integrability of generalized almost hyper-Hermitan structures.

Corollary 70. Let E be a regular Courant algebroid with scalar product , of neutral signature. Let E = E E be the decomposition of E determinedh· ·i + ⊕ − by a generalized metric G. Assume that , E± are either both of neutral signature, or rank E = rank E is a multipleh· ·i| of eight and one of , + − h· ·i|E± is positive definite (and the other negative definite). A generalized almost hyper-Hermitian structure (G, 1, 2, 3) is generalized hyper-Kähler if and only if conditions (84) from TheoremJ J 67J hold for a Levi-Civita connection D i of G and each of the pure spinors η± associated to (G, i), i = 1, 2, 3. The conditions are independent of the choice of D. J

11 Appendix

11.1 Z2-graded algebras and Clifford algebras 0 1 0 1 Recall that if A = A A and B = B B are Z2-graded vector spaces, then the tensor product⊕ A B inherits a⊕Z -gradation ⊗ 2 (A B)0 := A0 B0 + A1 B1, (A B)1 := A0 B1 + A1 B0. ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ We denote by A ˆ B the vector space A B together with this gradation. If, moreover, A and⊗B are Z -graded algebras,⊗ then A ˆ B inherits the structure 2 ⊗ of a Z2-graded algebra with multiplication on homogeneous elements defined by (a b)(ã ˜b) := ( 1)|b||a˜|aa˜ b˜b, ⊗ ⊗ − ⊗ where a , ˜b 0, 1 are the degrees of a and ˜b. We| say| | |∈{ that a Z} -graded vector space S = S0 S1 is a Z -graded A- 2 ⊕ 2 module if it is a representation space for A and the action of A on S is i j i+j compatible with gradations, i.e. A S S for any i, j Z2. ′ · ⊂ ∈ Finally, if S and S are Z2-graded A- and B-modules respectively, then ′ their graded tensor product S ˆ S is a Z2-graded A ˆ B-module with action given by ⊗ ⊗ γ (s s′) := ( 1)|b||s|a(s) b(s′), a⊗b ⊗ − ⊗ where s S, s′ S′, a A, b B, b := deg(b), s := deg(s) are the degrees of∈ the homogeneous∈ ∈ elements∈ b and| | s. | | We apply these facts to Clifford algebras and their representations. As- sume that (V+, q+) and (V−, q−) are two vector spaces with scalar products

49 and let (V := V+ V−, q := q+ + q−) be their direct sum. As Cl(V±) are Z -graded algebras⊕ we can consider Cl(V ) ˆ Cl(V ) which is a Z -graded al- 2 + ⊗ − 2 gebra, and as such is isomorphic to Cl(V ) (for the latter statement see e.g. Chapter I of [21]). The isomorphism between Cl(V ) and Cl(V+) ˆ Cl(V−) is obtained by extending the map V Cl(V ) ˆ Cl(V ) which assigns⊗ to any → + ⊗ − v = v + v V V the vector v 1+1 v . + − ∈ + ⊕ − +⊗ ⊗ 2 Let S± be Z2-graded Cl(V±)-modules. From above, the graded tensor product S ˆ S is a Z -graded Cl(V ) ˆ Cl(V )-module, hence also a Z - +⊗ − 2 + ⊗ − 2 graded Cl(V )-bundle. Any v = v+ + v− V+ V− Cl(V ) acts on S+ ˆ S− as ∈ ⊕ ⊂ ⊗ γ (s s )= γ (s ) s +( 1)|s+|s γ (s ). (88) v++v− +⊗ − v+ + ⊗ − − +⊗ v− − 11.2 Integrability of generalized almost complex structures and spinors Let E be a regular Courant algebroid with scalar product of signature (n, n), anchor π : E T M and canonical Dirac generating operator d/ : Γ(S) → → Γ(S). An almost Dirac structure of EC is an isotropic complex subbundle of EC of rank n. It is integrable (or a Dirac structure) if is is closed under the (complex linear extension of) the Dorfman bracket of E. For a non-vanishing section η Γ(SC) we deﬁne ∈ L := v EC γ η =0 . η { ∈ | v }

The spinor η is called pure if Lη is a vector bundle of rank n. Assume that η is a pure spinor. A simple computation shows that Lη is isotropic, and, being of rank n, Lη is an almost Dirac structure. It is called the null bundle of η. The assignment L [η] is a one-to-one correspondence between η → almost Dirac structures of EC and classes of projectively equivalent pure spinors of S (two pure spinors η and η deﬁned on an open set U M are 1 2 ⊂ projectively equivalent if η2 = fη1 for a non-vanishing function f on U). A pure spinor η Γ(SC) is called projectively closed if there is v Γ(EC) such ∈ ∈ that d/(η) = γvη. (In order to simplify notation, we use the same symbols d/ and γ for their complex linear extensions). Note that any pure spinor which is projectively equivalent to a projectively closed spinor is also projectively closed. This follows from d/(fη)= γπ∗(df)(η)+ fd/(η), for any η Γ(SC) and f C∞(M, C). ∈ ∈ Theorem 71. ([1]) An almost Dirac structure L of EC is a Dirac structure if and only if, locally, any pure spinor η associated to L is projectively closed.

50 Proof. Assume that η is projectively closed and let e Γ(EC) such that d/(η) = γ η. Let v,w Γ(L). Using condition ii) from∈ Deﬁnition 39 and e ∈ γvη = γwη = 0, we obtain

γ η = [[d,γ/ ],γ ]]η = γ γ d/(η)= γ γ γ η. (89) [v,w] v w − w v − w v e On the other hand,

γ γ γ = γ γ γ +2 v, e γ . (90) w v e − w e v h i w

Combining (89) with (90) and using γvη = γwη = 0, we obtain γ[v,w]η = 0. This proves that L is a Dirac structure. Conversely, assume that L is a Dirac structure. Then, for any v,w ∈ Γ(L), [v,w] Γ(L) and γ[v,w]η = 0. This implies, using condition ii) from Deﬁnition 39,∈ [[d,γ/ ],γ ]η = 0, or γ [d,γ/ ]η =0, w Γ(L). We obtain that v w w v ∀ ∈ [d,γ/ v]η, which is equal to γvd/(η), is a multiple of η, i.e. γvd/(η) = λ(v)η for λ(v) C∞(M, C). Remark that λ Γ(L∗). Extend λ to a (complex linear) ∈ ∈ 1-form on EC and let v Γ(EC), such that 2v is dual to this 1-form with 0 ∈ 0 respect to the complex linear extension of , . Then h· ·i λ(v)η =2 v, v η = γ γ η + γ γ η = γ γ η. h 0i v v0 v0 v v v0

The above computations show that γv(d/(η) γv0 η) = 0, for any v Γ(L), ∞− C ∈ which implies d/(η) γv0 η = gη for g C (M, ). As d/ and γv0 are odd operators and pure− spinors are chiral,∈ i.e. either even or odd, we conclude g = 0. This shows that d/(η)= γv0 η, i.e. η is projectively closed. Let be a generalized almost complex structure on E. The (1, 0)-bundle J L EC of is isotropic with respect to , and satisfies L L¯ = EC. In ⊂ J h· ·i ⊕ particular, rank L = n and L is an almost Dirac structure. A pure spinor η Γ(S) is called associated to if L = L . From Theorem 71 we obtain: ∈ J η Corollary 72. A generalized almost complex structure on a regular Courant algebroid is integrable if and only if, locally, one (equivaJ lently, any) pure spinor associated to is projectively closed. J Acknowledgements. We are grateful to Mario Garc´ıa-Fernández, for explaining to us his characterization of generalized Kähler structures on exact Courant algebroids [12], which we have generalized to regular Courant algebroids in Theorem 67. We are also grateful to Thomas Mohaupt for discussions about Born geometry and for pointing out the reference [9]. We thank Paul Gauduchon for sending us a copy of his paper [14], Carlos Shah- bazi for drawing our attention to reference [2], Roberto Rubio for drawing

51 our attention to references [13, 24] and Thomas Leistner for useful comments. Research of V.C. was partially funded by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 Quantum Universe – 390833306. L.D. was supported by a grant of the Ministry of Research and Innovation, project no PN-III- ID-P4-PCE-2016-0019 within PNCDI.

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V. Cort´es: [email protected] Department of Mathematics and Center for Mathematical Physics, Uni- versity of Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany.

L. David: [email protected] Institute of Mathematics Simion Stoilow of the Romanian Academy, Calea Grivitei no. 21, Sector 1, 010702, Bucharest, Romania.