arXiv:hep-th/0412280 v1 23 Dec 2004
MPP-2004-170 hep-th/0412280
Generalised G
2-structures and type IIB superstrings
Claus Jescheka and Frederik Wittb
a Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, FRG
E-mail: jeschek@mppmu.mpg.de
b University of Oxford, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, U.K.
E-mail: witt@maths.ox.ac.uk
ABSTRACT
The recent mathematical literature introduces generalised geometries which are defined by a reduction from the structure group SO(d, d) of the vector bundle Td⊕Td∗ to a special subgroup. In this article we show that compactification of IIB superstring vacua on 7- manifolds with two covariantly constant spinors leads to a generalisedG2-structure associated with a reduction fromSO(7,7) toG2×G2. We also consider compactifications on 6-manifolds where analogously we obtain a generalised SU(3)-structure associated with SU(3)×SU(3), and show how these relate to generalised G2-structures.
1 Introduction
From a duality and a phenomenological point of view, the idea of compactifying superstring theories and M-theory is a rather appealing one. It also points to interesting geometrical issues as requiring a certain amount of supersymmetry to be preserved puts constraints on the internal background geometry and thus leads to special G-structures. For instance, compactification on a 7- or 6-manifold together with N = 1 supersymmetry1 leads to a dilaton and a Killing spinor equation which induces a G2- or SU(3)-structure with various non-trivial torsion classes.
For N = 2 supersymmetry we basically need two special G-structures inside a given metric structure [6]. This naturally leads one to consider the class of the so-called generalised geometries, a concept which goes back to [9]. Over a d-manifold Md, contraction on the vector bundleT ⊕T∗ defines an inner product of signature (d, d) and therefore induces an SO(d, d)-structure over Md. A generalised geometry is then described by the (topological) reduction to a special subgroup ofSO(d, d) together with a suitable integrability condition.
In the present article we investigate type IIB superstring vacua by compactifying on seven and six dimensional manifolds. We shall focus on the geometrical structure of the vacuum space admitting a certain amount of supersymmetry in the external space, which can be achieved by a spinorial formulation of the supersymmetry variations. An additional analysis of the equations of motion single out physical vacua.
As we will explain, considering the “doubled” vector bundle T ⊕T∗ accounts for N = 2 supersymmetry by reducing the structure group from SO(7,7) or SO(6,6) to G2 ×G2 or SU(3)×SU(3) in the same way as compactification withN = 1 supersymmetry is accounted for by a reduction from the structure groupSO(7) orSO(6) toG2 andSU(3). Our starting point is [8] where Hassan investigated T-duality issues along d directions. This naturally leads to the consideration of ”general” supersymmetry transformations invariant under the action of the so–called generalised T-duality orNarain group SO(d, d). Here, “generalised”
means that we treat the left- and right-moving sectors of the worldsheet independently under the supersymmetry variations. This is similar in spirit to the investigation of topologically twisted non-linear sigma models on target spaces admitting a generalised complex structure in the sense of Hitchin [10] and Gualtieri [7], see for instance [1], [4], [11], [12].
By taking Hassan’s supersymmetry variations, we see that preserving supersymmetry on the external space requires the variations of the gravitinos Ψ±X and dilatinos λ± to vanish, i.e.
δ±Ψ±X = 0 and δ±λ±= 0.
These equations were also derived by Gauntlett et al. [5] from the perspective of wrapped NS5-branes in IIB supergravity. There, the authors found a solution by assuming N = 2 supersymmetry in 3d from a classical SU(3)-structure point of view.
In this article we shall proceed as follows. In Section 2 we introduce the generalised super- symmetry variations of type IIB following [8]. Neglecting the action of the RR-sector we use standard compactification methods in Section3to obtain the modelR1,2×M7together with
1Here,N denotes the number of covariantly constant spinors in the internal space.
the equations
γa∂aφ∓ 1
12γabcHabc
η±= 0,
∂a+1
4(ωabc∓1
2Habc)γbc
η±= 0 (1) for two spinors η± on the internal background. In section 4 we will discuss the notion of a generalised G2-structure [13] and give an equivalent formulation of equations (1) by means of differential forms of mixed degree. As a quick illustration we shall apply this formulation to the setup taken from [5] in Section 5. We then discuss compactifications on 6-manifolds which lead to generalised SU(3)-structures in Section6. Issues about lifting string theory to generalised topological M-theory admitting a generalised G2-structure (see for instance [3]) naturally rises the question about the relation between generalisedG2- andSU(3)-structures which will be addressed in Section 7.
The authors would like to acknowledge S.F. Hassan and R. Blumenhagen for useful discus- sions. Some ideas are based on the second author’s doctoral thesis who wishes to thank his supervisor Nigel Hitchin.
2 Preliminaries
We briefly set up the notation for the type IIB theory following Hassan [8]. Let us consider the gravitinos Ψ±, the dilatinos λ± and the supersymmetry parametersǫ±. They all arise as the real and imaginary part of the complex Weyl spinors
ΨX = Ψ+X +iΨ−X, X ∈T M1,9, λ=λ++iλ− and ǫ=ǫ++iǫ−.
The gravitinos (of spin 3/2) and the dilatinos (of spin 1/2) originate from the (R, N S)- and (N S, R)-sectors. We shall use an additional subscript +/− to indicate if the R-spin representation is induced by the left or right moving sector. For instance, Ψ+X comes from the (R, N S)-sector.
In the following we want to treat the left and right moving sector independently with respect to space-time supersymmetry and we therefore introduce two supersymmetry variations δ±. In particular, δ+ and δ− act only on the left and right moving sectors and interchange R ↔ N S. Both the supersymmetry parameters ǫ± and the gravitinos Ψ± are supposed to be of positive chirality, i.e.
Γ11ǫ± =ǫ± Γ11Ψ±= Ψ± while
Γ11λ±=−λ±,
that isλ±is of negative chirality. We first neglect the action of the RR-fields and consider only the closed NS-NS 3-form flux H, dH = 0 and the dilaton Φ. Therefore the supersymmetry variations of the (R, N S)- and (N S, R)-sector are given by
δ±Ψ±X =
∇X ∓14X H·
ǫ±, X ∈T M1,9 δ±λ± = 12
dφ∓12H
·ǫ±. (2)
The vanishing of the supersymmetry variations, i.e.
δ±Ψ±X = 0, and δ±λ±= 0 (3)
is necessary to characterise the background manifold M1,9 in the vacuum case. To find a solution to (3), we shall make specific assumptions which will occupy us next.
3 Compactification on M
7In this section, we compactify the theory on a 7-manifold M7, that is we consider the direct product modelR1,2×M7 where H and φtake non-trivial values only over M7. We want to determine the constraints on the underlying geometry of the internal space M7 imposed by the vanishing of the supersymmetry variations (3).
To that end, we decompose the supersymmetry parameters ǫ±∈∆±M1,9 accordingly, that is ǫ±=X
N
ξ±N⊗ηN± ⊗ 1
0
where ξ and η live in the irreducible spin representation ∆R1,2 and ∆M7 of Spin(1,2) and Spin(7) respectively, andN ≤dim ∆M7 = 8.
We fix the 10-dimensional space-time coordinates XM (M=0,. . . ,9) and assume the back- ground fields to be independent of Xµ (µ= 0,1,2). Coordinates on the internal space will be labeled by Xa fora= 3, . . . ,9. We use the convention
{ΓM,ΓN}= 2ηM NI32×32
with signature (−,+, . . . ,+). We choose the explicit gamma matrix representation ΓM =
( γµ⊗I8×8⊗σ2 : µ= 0, . . . ,2 I2×2⊗γa⊗σ1 : a= 3, . . . ,9
where the (8×8)-matricesγa are imaginary. TheSO(1,2) gamma matricesγµand the Pauli matrices σi are
γ0 = 0 −1
1 0
!
γ1 = 0 1 1 0
!
γ2 = 1 0 0 −1
!
and
σ1= 0 1 1 0
!
σ2= 0 −i i 0
!
σ3= 1 0 0 −1
! . Furthermore, we note the relations
Y
µ
γµ=−I2×2 Y
a
γa=−iI8×8. The chirality operator Γ11 is therefore Γ11=I2×2⊗I8×8⊗σ3.
With these splittings at hand we want to carry out the supersymmetry variations (2). The external part of the dilatino variation trivially vanishes. For the internal part, we first note the useful identity
ΓM1M2M3HM1M2M3 = (I2×2⊗γabc⊗σ1)Habc
by means of which we immediately obtain the dilatino variations δ±λ±= 1
2 h
I2×2ξ±N⊗(γa∂aφ∓ 1
12γabcHabc)ηN± ⊗σ1
1 0
i.
The conditionδ±λ±= 0 is then equivalent to γa∂aφ∓ 1
12γabcHabc
η±N = 0. (4)
Next we focus on the variation of the gravitinos δ±Ψ±M. The flatness ofR1,2 implies
∇µξN± = 0
which solves the external part, and consequently we are left with δ±Ψ±a=I2×2ξ±N ⊗
∂a+1
4(ωabc∓ 1
2Habc)γbc ηN± ⊗
1 0
. Imposing the condition δ±Ψ±X = 0 finally implies
∂a+1
4(ωabc∓1
2Habc)γbc
ηN± = 0. (5)
In this article we shall deal with the case N = 1, i.e. with exactly two internal spinors η±. Hence a solution consists of the internal background data (M7, g, H, η±, φ), where g is a metric, H a closed 3-form,η± two spinors in the associated irreducible spin representation
∆ which we assume to be of unit length and the scalar dilaton field φsuch that (4) and (5) hold.
Note that the considerations above can be easily modified to tackle the case of non-chiral type IIA theory which results in similar geometric conditions.
4 Generalised G
2-structures
In this section we want to formulate the geometry of the internal spaceM7 in the language of generalised G2-structures. We merely outline their theory – details, further results and examples can be found in [13] where these structures where introduced.
Consider the data of the previous section, that is a spinnable Riemannian 7-manifold (M7, g) together with two unit spinors η+ and η− in ∆ = 8, the irreducible Spin(7)-representation associated with a fixed spin structure over M7. In terms of G-structures, it is a well-known fact that each spinor induces a reduction to a principalG2-fibre bundle inside this spin struc- ture. For sake of clarity, we denote the associated structure groups byG2±. A basic instance of such a structure was considered in [5] where the spinors are assumed to be orthogonal, that is we effectively have an SU(3)-structure over M7. There the authors constructed various forms of pure degree by fierzing the tensor productη+⊗η− and projecting onto ΛpT7∗. The differential conditions on the spinors then translate into differential conditions on the forms.
The present discussion generalises this procedure and covers two new features.
Firstly, our choice of the two unit spinors η+ and η− is perfectly general. In particular, although the spinors generically induce anSU(3) =G2+∩G2−-structure, these may coincide
over some subset of M7, that is, the SU(3)-fibre bundle becomes singular. Geometrically speaking this subset is the zero locus of a certain vector field we define below, so that its Hopf index effectively counts (with an appropriate sign convention) the number of points where this happens.
Secondly, by identifyingη+⊗η− with an even or odd formρev,od0 we can regard it as aspinor forTd⊕Td∗. To fully appreciate this point, we remark that contraction on the vector bundle Td⊕Td∗ over an arbitraryd-manifold defines a natural inner product of signature (d, d) and induces a spinnableSO(d, d)-structure. An element X⊕ξ ∈Td⊕Td∗ acts on a formτ ∈Λ∗ by
X⊕ξ•τ =X τ +ξ∧τ.
As this squares to minus the identity2, this yields an isomorphism between Cliff(Td⊕Td∗) and End(Λ∗). The irreducible spin representations of Spin(d, d) are then given by
S± = Λev,odTd∗.
Note that a 2-form b can be naturally identified with an element in the Lie algebra so(d, d) and therefore acts on S± through the exponential eb = 1 +b+b2/2 +. . ..
In presence of a spinnable metricg on Td we can identify vectors with their duals and let a vector field X act on a form τ – seen as a spinor for Td⊕Td∗ – byX τ +X∧τ. It also acts on 8 by the usual Clifford multiplication X·. In order to compare these two actions on 8⊗8 and S± we write
Cliff(Td, g)⊗bCliff(Td,−g) =Cliff(Td⊕Td∗) where the isomorphism is given by extension of the map
X⊗Yb 7→(X⊕ −X g)•(Y ⊕Y g),
where• also denotes multiplication inCliff(Td⊕Td∗). One can then show that (X·ϕ⊗ψ)ev,od = X⊗1b •(ϕ⊗ψ)od,ev
(ϕ⊗Y ·ψ)ev,od = ±1⊗Yb •(ϕ⊗ψ)od,ev (6)
for any ϕ, ψ ∈8. Hence theG2+×G2−-invariant tensor product η+⊗η− induces elements ρev,od0 ∈S± whose stabiliser insideSpin(7,7) is conjugated to G2×G2.
Conversely, aG2×G2-invariant spinorρinS+ orS− can be uniquely written (up to a sign) as
ρ=e−φeb∧(η+⊗η−)ev,od∈S±.
We call the pair (M7, ρ) a generalised G2-structure which means that the structure group SO(7,7) of T7 ⊕T7∗ reduces to G2 ×G2. Alternatively, this can be characterised by the data (M7, g, b, η±, φ). We refer to the induced 2-form b as the B-field of the generalised G2-structure. We usually assume ρ to be even and writeρb =eb∧(η+⊗η−)ev or simply ρ0
ifb= 0. Note that the
g: 28, b: 21, η+: 7, η− : 7, φ: 1
2We follow the usual convention in mathematics where unit elements in the Clifford algebra square to−1.
degrees of freedom sum to 64 = dimΛev,od, so that this data effectively parametrises the open orbit of a G2 ×G2-invariant form under the action of R>0 ×Spin(7,7). Following the language in [9] such a spinor is called stable. Stability allows one to consider a certain variational principle which gained some attraction in the recent physics literature [3].
To see how a classical G2-structure relates to a generalisedG2-structure we derive an explicit description ofρev,odin terms of the underlying G2+∩G2−-invariants. The coefficients of the form η+⊗η− can be computed by
g(η+⊗η−, eI) =q(eI·η+, η−),
where q denotes a suitably scaled Spin(7)-invariant inner product on 8. To that end, we decomposeη−= cos(a)η++ sin(a)η+⊥ whereη+and η⊥+ are orthogonal to each other anda=
∢(η+, η−) describes the angle between the spinorsη+and η−. SinceSpin(7) acts transitively on the set of pairs of orthonormal spinors, we may choose an orthonormal basis in 8 such that
η+= (1,0,0,0,0,0,0,0,0)tr, and η− = (cos(a),sin(a),0,0,0,0,0,0)tr.
If the spinorsη+andη−are linearly independent, their isotropy groupsG2+andG2−intersect inSU(3) which acts onT7, leaving invariant a 1-formα, a symplectic formωand two 3-forms ψ+ and ψ− which are the real and the imaginary part of the SU(3)-invariant holomorphic volume form Ω [2]. We then find
ρev0 = c+sω−c(ψ−∧α+ω2
2 ) +sψ+∧α−sω3 6 ) ρod0 = sα−c(ψ++ω∧α)−sψ−−sω2
2 ∧α+c volg,
wherecandsare shorthand for cos(a) and sin(a). The underlyingSU(3)-structure fluctuates with the angleaand breaks down whens= 0 which means that both spinors coincide. Since sα·η+ = sη+⊥ this happens precisely over the zero locus of the vector field dual to sα as mentioned earlier. Consequently, only the forms sα, sω etc. are globally defined over M7. Moreover, ifa≡0, that isη=η+=η−, then
ρev = 1−⋆ϕ ρod = −ϕ+vol,
whereϕ denotes the invariant 3-form of theG2-structure defined byη.
This description also reveals how to relate the G2 ×G2-invariant forms ρev0 and ρod0 . For a p-form ξp, define σ(ξp) to be 1 for p ≡ 0,3 mod 4 and −1 for p ≡ 1,2 mod 4. A direct computation shows that
⋆σ(η+⊗η−)ev,od= (η+⊗η−)od,ev.
For a general G2×G2-form ρb we need to take the B-field into account. We introduce the generalised Hodge- or box operator 2g,b: Λev,od→Λod,ev defined by
2g,bρ=eb∧⋆σ(e−b∧ρ) so that
2g,bρev,od=ρod,ev.
Finally we want to translate the supersymmetry equations into the generalised picture. Recall that the twisted Dirac operator over 8⊗8 transforms into d+d⋆ under fierzing. A more general argument taking into account the possible action of theB-fieldbcan then be invoked to show that (M7, g, η±, φ, H), where H is a closed NS-NS 3-form flux, satisfies (4) and (5) if and only if the correspondingG2×G2-invariant spinor satisfies
dHe−φρ0 =de−φρ0+H∧e−φρ0 = 0, dH2g,0e−φρ0 =d2g,0e−φρ0+H∧2g,0e−φρ0 = 0.
IfH is globally exact, i.e. H =db this can be written in the more succinct form de−φρb = 0, de−φ2g,bρb = 0.
Note that this is precisely the integrability condition for a generalisedG2-structure to define a critical point for the variational principle mentioned above.
5 Recovering the classical SU (3) -case
Equations (4) and (5) were first derived by Gauntlett et al. [5] from a quite different point of view. Starting with IIB supergravity they studied wrapped NS5-branes over calibrated submanifolds inside an internal 7-manifold admitting anSU(3)-structure. As an illustration of the previous section, we reconsider their setup which turns out to be described by a “static”
generalised G2-structure witha≡π/2 that is, the structure group reduces to a fixedSU(3), with an additional closed 3-form, the NS-NS flux H.
Under this assumption the form ρ defining the generalised G2-structure becomes ρ0 =−ω+ω3
6 +α∧ψ+
with associated odd form
2g,0ρ0 =−α+α∧ω2 2 +ψ−. The supersymmetry equations are equivalent to
dHe−φρ0= 0 and dHe−φ2g,0ρ0 = 0 which written in homogeneous components can then be rephrased by
dω = dφ∧ω ,
dα∧ψ+ = dφ∧α∧ψ+−α∧dψ++H∧ω , 1
2dω∧ω2 = dφ∧ω3
6 −H∧α∧ψ+, and
dα = dφ∧α ,
dψ− = dφ∧ψ−+H∧α , α∧dω∧ω = −dφ∧α∧ω2
2 +dα∧ ω2
2 +H∧ψ−.
We finally conclude
d(e−φα) = 0, d(e−φω) = 0,
α∧dψ+ = H∧ω , 13dω∧ω = α∧H∧ψ+, d(e−φψ−) = H∧α , dω∧ω∧α = ψ−∧H .
(7)
The equations of motion are solved since H is closed, dH = 0, as proved in [5], and there- fore (7) characterises the physical vacua.
6 Compactification on M
6and generalised SU (3) -structures
Following the procedure of Section3we can also compactify on a 6-dimensional manifoldM6. Recall that we haveSpin(6) =SU(4) and that the irreducible spin representations of positive and negative chirality ∆± are just the SU(4)-vector representation 4 and its conjugate 4. The supersymmetry equations compactified onM6 thus become
∇LCX η±± 1
4X H·η± = 0, (dφ±1
2H)·η± = 0 (8)
for two complex spinorsη±. Since we work in type IIB theory both η+ and η− are assumed to be of positive chirality. Similarly, we can consider type IIA theory by choosing the spinors to be non-chiral.
Recall that SU(4)/SU(3) = S7, hence the choice of two unit spinors η± ∈ 4 induces a reduction to twoSU(3)±-subbundles. TheSU(4)-representations ∆±decompose into3±⊕1± and3±⊕1±. Consequently, we can also consider the correspondingSU(3)±-invariant spinors η±∈4. We want to describe the data (M6, g, η±, φ) in the language of generalised geometry where it gives rise to a generalisedSU(3)-structure. This is completely analogous to Section4 and the proofs of [13] carries over without difficulty. Again we content ourselves with a brief outline of the corresponding results.
Rather than working with the complex spinors we will consider the real SU(4)-module S obtained by forgetting the complex structure on 4 or 4, that is the complexification of S is justSC =4⊕4. Note that the Riemannian volume elementvolg induces a complex structure on S and acts on4 and 4 by multiplication with iand −irespectively. We let
ϕ±= Re(η±), ϕb±= Im(η±), so that
volg·ϕ±=−ϕb±.
Since S carries an SU(4)-invariant Riemannian inner product, we can identify S ⊗S with Λ∗T6∗ through fierzing so that (6) holds. This yields two formsϕ+⊗ϕ−and ϕ+⊗ϕb− which we can interpret asSU(3)×SU(3)-invariant spinors and want to decompose into an even and an odd part. Note that under complexification of this isomorphism, the components4⊗4and 4⊗4get mapped onto odd complex forms, while the off-diagonal components4⊗4and4⊗4 become even since4and4are dual to each other. Writingϕ+⊗ϕ−= (η++η+)⊗(η−+η−)/4
etc. we obtain
τ0 = (ϕ+⊗ϕ−)ev =−(ϕb+⊗ϕb−)ev = 12Re(η+⊗η−) b
τ0 = (ϕb+⊗ϕ−)ev =−(ϕ+⊗ϕb−)ev = 12Im(η+⊗η−) υ0= (ϕb+⊗ϕb−)od=−(ϕ+⊗ϕ−)od=−12Re(η+⊗η−) b
υ0= (ϕ+⊗ϕb−)od= (ϕb+⊗ϕ−)od= 12Im(η+⊗η−).
To see how these forms relate to each other, we note that in dimension 6 the 2g,b-operator respects the parity of the forms and satisfies 22g,b = −Id, that is 2g,b induces a complex structure on Λ∗T∗ (it is effectively the ∧-operator introduced in [10]). We then have
2g,bτb=τbb, 2g,bυb=bυb, whereτb=eb∧τ0 etc..
As in Section 4 we can compute a normal form description which we can express in terms of the underlying SU(2) = SU(3)+∩SU(3)−-invariants if the unit spinors η± are linearly independent. Using again the complexified isomorphismSC⊗SC∼= Λ∗T6∗Cand decomposing η−=c1η++c2η⊥+ with two complex scalars c1, c2 ∈Cwe find
η+⊗η−=iZ¯∧(c1·Ω +c2·eiω1) (9) and
η+⊗η−=eiα∧β(¯c1·eiω1+ ¯c2·Ω) (10) where expressed in a suitable local orthonormal basise1, . . . , e6 we have the two real 1-forms α = e5, β = e6, the complex 1-form Z = e5 +ie6, the self-dual 2-forms ω1 = e12+e34, ω2 =e13−e24,ω3 =e14+e23 and the complex symplectic form Ω =ω2−iω3. The normal forms of η+⊗η− and η+⊗η− are obtained by complex conjugation in Λ∗T6∗C.
Finally we wish to state the supersymmetry equations (8) in terms of the SU(3)×SU(3)- invariant formsτ0,bτ0,υ and υ. The real version of (8) is given byb
∇LCX ϕ±±1
4X H·ϕ±= 0, (dφ± 1
2H)·ϕ±= 0 and
∇LCX ϕb±±1
4X H·ϕb± = 0, (dφ±1
2H)·ϕb±= 0.
and the same computation as in the generalised G2-case shows that this is equivalent to dHe−φτ0 =dHe−φτb0= 0, dHe−φυ0=dHe−φυb0 = 0,
that is
dHe−φη+⊗η−= 0, dHe−φη+⊗η−= 0.
IfH is globally exact, that isH =db, we can write these equations more succinctly as de−φτb=de−φ2g,bτb= 0, de−φυb=de−φ2g,bυb = 0.
7 Dimension 7 vs. 6
The inclusion SU(3) ⊂ G2 allows one to pass from an SU(3)-structure in 6 dimensions to a G2-structure in 7 dimensions. In the same vein, the inclusion SU(3)×SU(3)⊂ G2×G2
relates generalisedSU(3)- to generalisedG2-structures. In this final section we want to render this link explicit in both the spinorial and the form picture of a generalised structure. We first discuss the algebraic setup before we turn to integrability issues.
To start with, assume that we are given a generalised G2-structure (g, b, η±, φ) over the 7- dimensional vector spaceT7 =T together with a preferred unit vectorα. We want to induce a generalisedSU(3)-structure onT6 =Tedefined byT =Te⊕Rα. Sinceα·α =−1 the choice of such a vector induces a complex structure on the irreducibleSpin(7)-module ∆ =8 which is compatible with the spin-invariant Riemannian inner product. Hence the complexification of ∆ is
8⊗C= ∆1,0⊕∆0,1, where
∆1,0/0,1 ={η∓iα·η|η∈∆}.
The choice ofαalso induces a reduction fromSO(7) toSO(6) which is covered bySpin(6) = SU(4), and as anSU(4)-module we have ∆1,0=4 and ∆0,1 =4. We define
ψ±=η±−iα·η±
and let eg=g|Te andeb=α (α∧b). Then a generalised SU(3)-structure over Te is given by (T ,e g,eeb, ψ±, φ). Moreover, we get a (possibly zero) 1-form β=α b∈Λ1Te∗. It is clear that we can reverse this construction by defining a metric g=eg+α⊗α, b=eb+α∧β and two spinors η±∈8through η±= Re(ψ±).
To see what happens in the form picture, we start with the special G2×G2-invariant form ρ0 = (η+⊗η−)ev =f0+α∧f1,
wheref0 ∈ΛevTe∗ and f1 ∈ΛodTe∗. It follows from (6) that α∧(η+⊗η−)ev,od = 1
2(α·η+⊗η−∓η+⊗α·η−)od,ev α (η+⊗η−)ev,od = 1
2(−α·η+⊗η−∓η+⊗α·η−)od,ev. Therefore the formsf0 and f1 can be expressed by
f0 =α (α∧ρ0) = 1
2(η+⊗η−+α·η+⊗α·η−)ev and
f1 =α ρ0 =−1
2(α·η+⊗η−+η+⊗α·η−)od. Using the spinorsψ± as defined above we find
ψ+⊗ψ− = (η+⊗η−−α·η+⊗α·η−)−i(α·η+⊗η−+η+⊗α·η−) ψ+⊗ψ− = (η+⊗η−+α·η+⊗α·η−) +i(−α·η+⊗η−+η+⊗α·η−).
Consequently, we have
f0 = 1
2Re(ψ+⊗ψ−) =τ0
and
f1 = 1
2Im(ψ+⊗ψ−) =υb0. In the same vein, decomposing 2g,0ρ0 =g1+α∧g0 yields
g0 = 1
2Im(ψ+⊗ψ−) =bτ0
and
g1 = 1
2Re(ψ+⊗ψ−) =−υ0.
In presence of a non-trivial B-field b ∈ Λ2T∗ we write b = eb+α ∧β. Since eeb+α∧β = eeb∧(1 +α∧β) we obtain for the general case the expressions
ρb =e−φτeb+α∧(e−φυbeb+β∧e−φτeb) (11) and
2g,bρb =−e−φυeb+α∧(e−φτbeb−β∧e−φυeb). (12) Conversely, if (T, g, b, ρ0, φ) defines a generalised G2-structure and α ∈ T is a unit vector, then the forms eb = α b, τ0 = α (α ∧ρ0) and υ0 = −α 2g,0ρ0 define a generalised SU(3)-structure (T ,e eg,eb, τ0, υ0, φ) with eg=g|T.
To see how integrability conditions relate to each other over the manifolds M7 = M and M6 = Mf, consider a smooth family (g(t),e eb(t), τ0(t), υ0(t), φ(t)) of metrics eg(t), of 2-forms eb(t), of even and odd forms τ0(t) andυ0(t) and of scalar functions φ(t) which we assume to define a generalised SU(3)-structure for any t lying in some open interval I, together with 1-formsβ(t)∈Ω1(Mf). In order to obtain an integrable generalisedG2-structure over Mf×I defined by (Mf×I, g, b, ρ0, φ) whereg=egt⊕dt⊗dt,b=eb(t) +dt∧β(t),ρ0 =τ0(t) +dt∧υb0(t) and φ=φ(t), we need to solve the equations
dρ= 0, d2g,bρ= 0.
We decompose the exterior differential dover M =Mf×I into d|Ωev,od· →d|Ωev,od·=de|Ωev,od· ±∂t|Ωev,od· ∧dt,
where deis the exterior differential on Mf. From (11) we conclude the first equation to be equivalent to
dρ=dee−φτeb+dt∧(∂te−φτeb−dee−φbυeb−d(βe ∧e−φτeb)) so that
dee−φτeb = 0, ∂te−φτeb=dee−φυbeb+dβe ∧e−φτeb. (13) By (12) the second equation reads
d2ρ=−dee−φτeb+dt∧(−∂te−φυeb−dee−φbτeb−d(βe ∧e−φυeb))
and therefore yields
dee−φυeb= 0, ∂te−φυeb =−dee−φτbeb+dβe ∧e−φυeb. (14) If we let ¯β(t) = Rs
0 β(s)ds we can bring (13) and (14) into Hamiltonian form, that is the generalised G2-structure is integrable if and only if
d(ee −φedeβ¯∧υeb) = 0, deυb = ∂t(e−φedeβ¯∧τeb),
d(ee −φedeβ¯∧τeb) = 0, debτ = −∂t(e−φedeβ¯∧υeb). (15) We illustrate the previous discussion by considering a classical SU(3)-structure defined by a unit spinor η and taking the generalised SU(3)-structure given by (M6, g, η) with trivial B-field and vanishing dilaton, b= 0 andφ= 0. We can compute the formsτ0 etc. by using the normal form description (9) and (10) wherec1 = 1 andc2= 0. Using the notation of [2]
we obtain
υ0 = −12ψ−, υb0 = 12ψ+, τ0 = 12(1−ω22), τb0 = 12(ω−ω63), and equations (15) become the Hitchin flow equations
dψe − = 0, dψe + = −∂tω∧ω , dωe ∧ω = 0, dωe = ∂tψ−,
which appeared in [2] and go back to [9]. Note that although equations (15) are, like the Hitchin flow equations, in Hamiltonian form we have not shown yet that if the data (eg(t),eb(t), τ0(t), υ0(t), φ(t)) defines a generalised SU(3)-structure att=t0 and satisfies (15), then it automatically defines a generalised SU(3)-structure for t > t0, as it is the case for classical SU(3)-structures evolving along the Hitchin flow.
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