Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps

0.3 MB·English

by  Spiro Karigiannis

Hodge Theory for G -manifolds:
2
9 Intermediate Jacobians and Abel-Jacobi maps
0
0
2 Spiro Karigiannis

n Mathematical Institute
a
University of Oxford
J

7
2 Naichung Conan Leung

Institute of Mathematical Sciences and
]
G
Department of Mathematics
D Chinese University of Hong Kong
.
h
January 27, 2009
t
a
m
Abstract
[
WestudythemodulispaceMoftorsion-freeG2-structuresonafixedcompactmanifoldM7,
3 and define its associated universal intermediate Jacobian J. We define the Yukawa coupling
v
and relate it to a natural pseudo-K¨ahlerstructure on J.
7
We consider natural Chern-Simons type functionals, whose critical points give associative
8
andcoassociativecycles(calibratedsubmanifoldscoupledwithYang-Millsconnections),andalso
9
2 deformed Donaldson-Thomas connections. We show that themoduli spaces of thesestructures
. can be isotropically immersed in J by means of G2-analogues of Abel-Jacobi maps.
9
0
7 1 Introduction
0
:
v Compact manifolds with G holonomy play the same role in M-theory as Calabi-Yau threefolds do
2
i
X instringtheory,bothbeingRicci-flatandadmittingaparallelspinor. Infact,thereareanaloguesof
the mirror symmetry phenomenon [1, 11] that are expected to hold in the context of G -manifolds.
r 2
a These ideas are still not well understood mathematically.
TherearevariousspecialgeometricstructuresthatcanbeassociatedtoaG -manifoldandthere
2
arerelationshipsbetweenthesestructures. SpecificallywewishtoconsideranaloguesinG -geometry
2
of intermediate Jacobians and Abel-Jacobi maps which are familiar in algebraic geometry.
The moduli space of Calabi-Yau 3-folds is known to admit a special Kahler structure. Equiva-
lently, this means that the universal intermediate Jacobian J admits a hyperKa¨hler structure. The
images in J of the Abel-Jacobi maps are Lagrangian submanifolds. Some references for these facts
are[4,5, 8,10, 22]. Inthis paperwe proveanaloguesofthesestatements forG -manifolds. We now
2
describe the organization of the paper, and mention the main results of each section.
In Section 2 we briefly review some well-known facts about G -manifolds, which we will need
2
later, and discuss notation.
1
In Section 3 we begin with the description of the moduli space M of torsion-free G -structures
2
on a fixed G -manifold M, which has been studied by Joyce [16, 17] and Hitchin [15]. We define
2
a natural pseudo-Riemannian metric on M which is the Hessian of a superpotential function f.
Finally we define the Yukawa coupling Y on M, a fully symmetric cubic tensor, and relate this to
the superpotential function f.
Usingthesedata,inSection4wedefineintermediateG -Jacobiansandtheuniversalintermediate
2
G -JacobianJ ofthemodulispaceandshowthatitadmitsanaturalpseudo-K¨ahlerstructure,with
2
K¨ahler potential (essentially) given by f. Also, the projection J → M is a Lagrangian fibration,
and the associated cubic form to this fibration (as described in [8]) is exactly the Yukawa coupling
Y.
Section5discusseshowspecialgeometricstructuresassociatedtoaG -manifoldM canbeviewed
2
as critical points of certain natural functionals Φ of Chern-Simons type. Specifically we prove
k,ϕ
that the calibratedsubmanifolds ofM,together with specialconnectionsaresuchcriticalpoints, as
are deformed Donaldson-Thomas connections. We study each situation individually.
Additionally, we show that the moduli spaces of such structures can be immersed (via Abel-
Jacobi type mappings) into J isotropically with respect to the appropriate symplectic structure
on J. Therefore, these images are Lagrangian subspaces whenever they are half-dimensional. We
also discuss a topological number Ψ on the moduli spaces of these structures, and relate it to the
k,ϕ
critical points of Φ .
k,ϕ
Finally, in Section 6 we briefly consider the generalization of these results to the case when the
cohomology of the G -manifold M has torsion, which involves gerbes. We also discuss questions for
2
future study.
Acknowledgements. ThefirstauthorwouldliketothankBobbyAcharyaforusefuldiscussions
concerning the physics of G -manifolds, and in particular for informing him about their use of
2
F = −log(f) as the superpotential function. The first author is also greatly indebted to Dominic
Joyce for pointing out some problems with an initial draft of this article, and for useful discussions
withChristopherLinandJohnLoftin. Theauthorsalsothanktherefereeforusefulcommentswhich
improved an earlier version of this article. The researchof the second author is partially supported
by a research RGC grant from the Hong Kong government.
2 Review of G -structures
2
In this section we briefly review some facts about G -structures. Some references for G -structures
2 2
are [3], [17], [18], [20], [27], and [32]. In addition, the first examples of compact irreducible G man-
2
ifolds are described in [16] and [17].
LetM7 beaclosedmanifoldwithaG -structure. TheG -structureisgivenbyapositive3-form
2 2
ϕ, which induces an orientation and a Riemannian metric g in a non-linear way via the formula
ϕ
(X ϕ)∧(Y ϕ)∧ϕ=−6g (X,Y)vol
ϕ ϕ
wherevol isthevolumeformcorrespondingtotheorientationandthemetricg . Hencethereisan
ϕ ϕ
induced Hodge star operator ∗ and the associated dual 4-form ψ =∗ ϕ. The 3-form has constant
ϕ ϕ
pointwise norm |ϕ|2 =7 with respect to g .
ϕ
The G -structure is called torsion-free if ϕ is parallel with respect to its induced metric g . In
2 ϕ
this case the Riemannian holonomy is contained in G , and M7 is called a G -manifold. It is well
2 2
2
knownthataG -structureϕistorsion-freeifandonlyifϕisbothclosedandco-closed(withrespect
2
to g .)
ϕ
ThespaceofformsΩ∗ onamanifoldwithG -structuredecomposesintoG -representations,and
2 2
thisdecompositiondescendstothecohomologyinthetorsion-freecase. Specifically,thecohomology
breaks up as
H2(M,R) = H2⊕H2
7 14
H3(M,R) = H3⊕H3⊕H3
1 7 27
H4(M,R) = H4⊕H4⊕H4
1 7 27
H5(M,R) = H5⊕H5
7 14
Wedefinebk =dim(Hk). Thenb3 =b4 =1,andbk =b (M),fork =2,3,4,5. Theactualholonomy
l l 1 1 7 1
groupHol(g )of(M,g )isdeterminedbythetopologyofM. Forexample,theholonomyisexactly
ϕ ϕ
G if and only if π (M) is finite, in which case b (M)=0 and all the bk =0. Such an M7 is called
2 1 1 7
irreducible.
A result which we will need repeatedly is the following.
Lemma 2.1. Let ϕ = ϕ +tη be a one-parameter family of G -structures for small t. Then we
t 0 2
have
∂ 4
(∗ ϕ )= ∗ π (η)+∗ π (η)−∗ π (η) (2.1)
∂t ϕt t 3 ϕ0 1 ϕ0 7 ϕ0 27
(cid:12)t=0
(cid:12)
where ∗ is the Hodge sta(cid:12)r induced from ϕ , and π is the orthogonal projection onto Ω3 (defined
ϕ0 (cid:12) 0 k k
with respect to ϕ .)
0
Proof. This is first mentioned in [16], and explicit proofs can be found in [15] and [20].
ItfollowsfromLemma2.1thatifψ =ψ +tθ isaone-parameterfamilyofpostive4-forms,then
t 0
∂ 3
(∗ ψ )= ∗ π (θ)+∗ π (θ)−∗ π (θ)
∂t ψt t 4 ϕ0 1 ϕ0 7 ϕ0 27
(cid:12)t=0
(cid:12)
(cid:12)
which motivates the follow(cid:12)ing definition.
Definition 2.2. Let ϕ be a fixed torsion-free G -structure. We define the map ⋆ : Ωk → Ω7−k
0 2 ϕ0
for k =3,4 by
4
for η ∈Ω3; ⋆ (η) = ∗ π (η)+∗ π (η)−∗ π (η) (2.2)
ϕ0 3 ϕ0 1 ϕ0 7 ϕ0 27
3
for θ ∈Ω4; ⋆ (θ) = ∗ π (θ)+∗ π (θ)−∗ π (θ) (2.3)
ϕ0 4 ϕ0 1 ϕ0 7 ϕ0 27
Notice that ⋆ agrees (up to a constant) with ∗ on each Ωk, but the constants are different on
ϕ0 ϕ0 l
different components. Also note that ⋆2 =1.
ϕ0
Finally we make some remarks about notation. We use hα,βi and |α|2 to denote the pointwise
innerproductandnormonformsinducedfromg . Weusehhα,βii= hα,βivol and||α||2 =hhα,αii
ϕ M ϕ
to denote the corresponding L2 inner product.
R
As M is always taken to be compact, we use Hodge theory throughout, so we will often identify
a cohomology class [α] with its unique harmonic representative α. We use g = g to denote the
ϕ
Riemannian metric on M7 associated to the G -structure ϕ, and reserve G for the metric on the
2
3
modulispaceMofG -manifolds,andG forthepseudo-K¨ahlermetricontheuniversalintermediate
2 J
JacobianJ. The summationconventionis usedthroughout,andthe dimensionof the moduli space
M is b =n+1 with indices running from 0 to n.
3
3 Special Geometry of the G -moduli space
2
3.1 The moduli space of G -manifolds
2
We denote by M the moduli space of torsion-free G -structures on M. Explicitly, M is defined as
2
M= ϕ∈Ω3(M):∇ ϕ=0 /Diff (M)
+ ϕ 0
where Ω3(M) denotes the space (cid:8)of G -structures on M(cid:9)(also called the positive 3-forms), and
+ 2
Diff (M) is the space of diffeomorphisms of M isotopic to the identity. This should more properly
0
be called the Teichmuller space of G -structures on M.
2
Remark 3.1. The quotient of the torsion-free G -structures by the full diffeomorphism group of M
2
(whichisequivalenttothequotientofMbythemappingclassgroupofM)isingeneralanorbifold.
As most of our discussions are local, we could equally consider this moduli space, at least where it
is smooth.
Dominic Joyce proved in [16, 17] that M is a smooth manifold of dimension b (M). Note that
3
on a compact G -manifold, b ≥ 1, since ϕ is a non-trivial harmonic 3-form. Therefore the moduli
2 3
spaceisalwaysatleastone-dimensional,andthosemodulicorrespondtoconstantscalingsofϕ,and
hence scalings of the total volume of M. In [16, 17], Joyce also proved the following local result.
Theorem 3.2 (Joyce [16, 17]). The period map
P : M→H3(M,R)
P(ϕ) = [ϕ]
is a local diffeomorphism. Here [ϕ] denotes the cohomology class of ϕ. Hence, given ϕo ∈M, there
exists ǫ>0 such that
Uϕo = ϕ∈Ω3+(M):||ϕ−ϕo||gϕo <ǫ,∇ϕϕ=0 /Diff0(M)
n o
is diffeomorphic to an open set in the vector space H3(M,R).
This result says M has a natural affine structure and a flat connection ∇, and P is an affine
map. To see this, let η ,...,η be a basis of H3(M,R), where n+1 = b = b3 +b3 +1. Let
0 n 3 7 27
xi be coordinates on U with respect to this basis. Thus η = xiη describes a point in U and
ϕ0 i ϕ0
η = ∂ . Since U is an open set in a vector space, it admits a flat connection ∇=d, and because
i ∂xi ϕ0
∇(dxi)=d(dxi)=0, we say xi are flat coordinates. We can thus cover M by such flat charts.
On two overlapping flat charts, the coordinate systems x˜i and xi are affinely related: xi =
Pix˜j +Qi, where Pi and Qi are constants. Thus ∂ = Pi ∂ , and the transition functions are
j j ∂x˜j j∂xi
constant. Hence the flat connection ∇is a well-defined connection on M, since if ∇= d+A with
A=0inoneflatchart,thenA˜=P−1AP+P−1dP =0inalloverlappingflatcharts. Suchstructures
also arise in [12, 13, 26, 28], for example.
Thereisanaturalpseudo-RiemannianmetricGonthemodulispaceMfirstdefinedbyHitchin[15]
which we now describe.
4
Definition 3.3. We define a smooth function f :M→R as follows:
3
f(ϕ)= (ϕ∧∗ ϕ)=3 vol (3.1)
ϕ ϕ
7
ZM ZM
where ϕ is a point in M. Thus, up to a constant, f([ϕ]) is the total volume of M with respect to
the metric and orientation induced by ϕ. We call f the superpotential function for M, for reasons
that will soon be clear.
Remark 3.4. Our definition differs from Hitchin’s by a constant factor, which we choose for later
convenience.
In [15], Hitchin shows that when restricted to closed G -structures in a fixed cohomology class,
2
the torsion-free G -structures are precisely those which are critical points of f. We will say more
2
about this in Section 5.4.
Theorem 3.5 (Hitchin [15]). In a flat coordinate chart (U ,x0,...,xn), the Hessian f = ∂2f
ϕ0 ij ∂xi∂xj
defines a pseudo-Riemannian metric G on M by G = f . In the case when Hol(g ) = G , the
ij ij ϕ 2
metric G is Lorentzian.
ij
Proof. Let ϕ be a point in M. We want to differentiate f in the η direction. We identify the
j
cohomologyclass η with its unique harmonic representative(with respectto the metric g induced
j ϕ
from ϕ.) Now by Lemma 2.1, we have
∂f 3 ∂ 3 ∂
= ϕ ∧∗ ϕ+ ϕ∧ ∗ ϕ
∂xj 7 ∂xj ϕ 7 ∂xj ϕ
ZM(cid:18) (cid:19) ZM (cid:18) (cid:19)
3 3 4
= η ∧∗ ϕ+ ϕ∧ ∗ π (η )+∗ π (η )−∗ π (η )
j ϕ ϕ 1 j ϕ 7 j ϕ 27 j
7 7 3
ZM ZM (cid:18) (cid:19)
3 4
= π (η )∧∗ ϕ+ ϕ∧∗ π (η )
1 j ϕ ϕ 1 j
7 7
ZM ZM
= π (η )∧∗ ϕ= η ∧∗ ϕ
1 j ϕ j ϕ
ZM ZM
wherewehaveusedthefactthatthesplittingofΩ3 isorthogonalwithrespecttog . Tocomputethe
ϕ
second derivative, let ϕ(t) be a one-parameter family of torsion-free G -structures with ϕ(0) = ϕ,
2
satisfying ∂ ϕ(t)=η (t), where η (t) is the unique harmonic representativeof the cohomologyclass
∂t i i
η with respect to the metric g induced from ϕ(t). Then η (t) = η (0)+dβ(t) for some smooth
i t j j
family of 2-forms β(t). Hence ∂ η (t)=dβ′(t) is exact.
∂t j
Writing η (0)=η , we compute the second derivative at ϕ as
j j
∂2f ∂ ∂
= η (t) ∧∗ ϕ+ η ∧ ∗ ϕ(t)
∂xi∂xj ∂t j ϕ j ∂t ϕ(t)
ZM(cid:18) (cid:12)t=0 (cid:19) ZM (cid:18) (cid:12)t=0 (cid:19)
(cid:12) (cid:12)
= dβ′(0)∧∗ϕϕ+(cid:12)(cid:12) ηj ∧ 4 ∗ϕπ1(ηi)+∗ϕπ7(ηi)(cid:12)(cid:12)−∗ϕπ27(ηi)
3
ZM ZM (cid:18) (cid:19)
4
= π (η )∧∗ π (η )+π (η )∧∗ π (η )−π (η )∧∗ π (η )
1 i ϕ 1 j 7 i ϕ 7 j 27 i ϕ 27 j
3
ZM(cid:18) (cid:19)
4
= hhπ (η ),π (η )ii+hhπ (η ),π (η )ii−hhπ (η ),π (η )ii
1 i 1 j 7 i 7 j 27 i 27 j
3
5
where the first term in the second line above vanishes by Stokes’ theorem since ∗ ϕ is closed and
ϕ
M is compact.
The Laplace and Green’s operators commute with the projections when ϕ is torsion-free, so
each π (η ) is harmonic. Therefore we see that f is a pseudo-Riemannian metric of signature
k j ij
(b3+b3,b3 )=(1+b ,b3 ).
1 7 27 1 27
When M is irreducible (so the holonomy is exactly G ), we know b =0. Since H1 ∼=H3, there
2 1 7
are no harmonic Ω3-forms, and thus in this case
7
4
f = hhπ (η ),π (η )ii−hhπ (η ),π (η )ii (3.2)
ij 1 i 1 j 27 i 27 j
3
which is a Lorentzian metric of signature (1,b3 ).
27
Remark 3.6. We defined this metric using a particular flat coordinate chart in a neighbourhood of
each point. However, this metric G is well-defined globally on M. This is because with respect
to any other overlapping flat coordinate chart (x˜0,...x˜n), the two sets of coordinates are affinely
related: xi =Pix˜j+Qi. Therefore ∂ =Pi ∂ ,andalso ∂2f =PiPj ∂2f . Thisis exactlythe
j ∂x˜k k∂xi ∂x˜k∂x˜l k l ∂xi∂xj
statement that the metric G is well-defined.
Remark 3.7. In the physics literature, a slightly different notion of superpotential function is used:
they define F = −log(f) and use the Hessian of F to define a metric on M. The advantage of
this definition is that the metric F is actually Riemannian when M is irreducible, as opposed to
ij
Lorentzian. However, with this choice of potential function and metric, other results fail to hold.
See Remarks 3.13, 4.9, and 5.8 for more details. It is for these reasons that we prefer to use f and
its associated Lorentzian metric.
We close this section by noting that we can write the metric G on M more concisely using the
operator ⋆ defined in (2.2) as
ϕ
G(η ,η )= η ∧⋆ η (3.3)
1 2 1 ϕ 2
ZM
for η ∈ T M = H3(M,R). We stress again that ⋆ is not the same as ∗ , and thus G is not the
i ϕ ϕ ϕ
usual L2 metric on forms. This metric G is more natural than the usual L2 metric, because the
fact that it is of Hessian type allows us to construct a pseudo-K¨ahler structure on the universal
intermediate Jacobian in Section 4.2.
3.2 The Yukawa Coupling
In this section we define the Yukawa coupling Y, a symmetric cubic tensor on the G -moduli space
2
M. We relate this tensor to the superpotential function f and the metric G on M. We will need
the following identity, which can be found in Lemma A.12 of [20]:
ϕ ϕ gkc =g g −g g −ψ (3.4)
ijk abc ia jb ib ja ijab
We recall some facts about the space Ω3 = Ω3⊕Ω3⊕Ω3 of 3-forms on a G -manifold. We adopt
1 7 27 2
the notation of [20]. The elements η ∈ Ω3⊕Ω3 are in one-to-one correspondence with symmetric
1 27
2-tensors h ∈S2(T∗). The correspondence is given by
ij
η =h glmϕ +h glmϕ +h glmϕ (3.5)
ijk il mjk jl imk kl ijm
6
where Ω3 corresponds to multiples of the metric g and Ω3 corresponds to the traceless symmetric
1 27
tensors. Note that the 3-form ϕ itself corresponds to the symmetric tensor 1g .
3 ij
From now on we assume that M is irreducible, so H3(M,R) = H3 ⊕H3 . Each cohomology
1 27
class has a unique harmonic representative, so for a fixed torsion-free G -structure ϕ with metric
2
g , each class η ∈H3(M,R) corresponds to a symmetric 2-tensor h .
ϕ k k
Definition 3.8. The Yukawa coupling Y is a fully symmetric cubic tensor on the moduli space
M, defined as follows. Let ηi, ηj, ηk be elements of TϕM ∼= H3(M,R), with associated symmetric
tensors h , h , h , with respect to g . Then we define
i j k ϕ
Y(η ,η ,η )= (h )aα(h )bβ(h )cγφ φ vol (3.6)
i j k i j k abc αβγ
ZM
where g and vol are with respect to ϕ, and (h )aα means (h ) gbagβα, raising indices with g. It is
i i bβ
clear that Y is fully symmetric in its arguments.
In Section 4.2, we will see that the Yukawa coupling Y is essentially the natural cubic form
associated to a Lagrangianfibration.
Fix ϕ in M. Let η0,η1,...,ηn be a basis for TϕM∼=H3(M,R), where η0 =ϕ and ηi ∈H237 for
i6=0.
Proposition 3.9. The following relations between the Yukawa coupling Y, the superpotential func-
tion f, and the Hessian metric G =f hold:
ij ij
14
Y(ϕ,ϕ,ϕ) = f(ϕ) (3.7)
27
Y(ϕ,ϕ,η ) = 0 (3.8)
i
1
Y(ϕ,η ,η ) = G (ϕ) (3.9)
i j ij
6
where i,j =0,...,n, and i6=0 in (3.8).
Proof. First we show that (3.7) and (3.8) follow from (3.9). Equation (3.8) is automatic since
G = 0 for i 6= 0 by (3.2). Also (3.2) shows G = 4 ϕ ∧ ∗ϕ = 28f(ϕ). Then (3.9) says
0i 00 3 9
Y(ϕ,ϕ,ϕ)= 1G = 14f(ϕ).
6 00 27 R
To establish (3.9), consider h = 1g in (3.6) and use (3.4). We obtain:
k 3
1
Y(ϕ,η ,η ) = (h )aα(h )bβϕ ϕ gcγvol
i j i j abc αβγ
3
ZM
1
= (h )aα(h )bβ(g g −g g −ψ )vol
i j aα bβ aβ bα abαβ
3
ZM
1
= (Tr(h )Tr(h )−Tr(h h ))vol (3.10)
i j i j
3
ZM
where (h h ) = (h ) glm(h ) denotes matrix multiplication. The third term vanished by the
i j ab i al j mb
symmetry ofthe h’s and the skew-symmetryof ψ. Leth =λg+h0 andh =µg+h0 where h0 and
i i j j i
h0 are the trace-free parts. Substituting these into (3.10) gives
j
1
Y(ϕ,η ,η )= 42λµ−Tr(h0h0) vol (3.11)
i j 3 i j
ZM
(cid:0) (cid:1)
7
In Proposition 2.15 of [20], it is shown that for η ,η ∈Ω3⊕Ω3 , we have
i j 1 27
(η ∧∗η )vol = (Tr(h )Tr(h )+2Tr(h h ))vol
i j i j i j
ZM ZM
= 63λµ+2Tr(h0h0) vol (3.12)
i j
ZM
(cid:0) (cid:1)
By (3.2), the metric G =f is given by
ij ij
G = η ∧∗η¯ vol
ij i j
ZM
where η¯ corresponds to the symmetric tensor h¯ = 4µg−h0. Substituting µ→ 4µ and h0 →−h0
j j 3 j 3 j j
in (3.12) shows
G = 84λµ−2Tr(h0h0) vol (3.13)
ij i j
ZM
(cid:0) (cid:1)
Comparing (3.11) and (3.13), shows Y(ϕ,η ,η )= 1G .
i j 6 ij
From the above proposition, the natural question which arises is whether Y(η ,η ,η ) is related
i j k
to f . The main theorem of this section is the following, which shows that this is indeed the case.
ijk
Theorem 3.10. Let ηi,ηj,ηk be in TϕM∼=H3(M,R). Then
1 1 ∂3f
Y(η ,η ,η )= f = (3.14)
i j k 2 ijk 2 ∂xi∂xj∂xk
for i,j,k=0,...,n.
Beforewecanprovethistheorem,weneedacoupleofpreliminaryresults. Thereasonisthatitis
difficulttodirectlydifferentiatef = η ∧⋆ η becauseofthecomplicatednatureoftheoperator
ij M i ϕ j
⋆ . WecircumventthisdifficultybyconsideringtheauxiliaryfunctionF =−log(f)instead,because
ϕ
R
it will turn out that the third derivative of F is easily computable.
Lemma 3.11. Let F =−log(f). Then the Hessian F = ∂2F is given by
ij ∂xi∂xj
1 1
F = η ∧∗ η = hhη ,η ii (3.15)
ij i ϕ j i j
f f
ZM
when M is irreducible.
Proof. In this proof, subscripts on f or F always denote partial differentiation. We evaluate all
derivatives at a fixed point ϕ in M. As before fix a basis η ,...,η of H3(M,R) so that η = ϕ.
0 n 0
From the proof of Theorem 3.5, and our choice of basis, we see that at the point ϕ in M,
7
f = f f =0 for i6=0 (3.16)
0 i
3
and also that
28
f = f f =0 f =− hη ,η ivol for i,j 6=0 (3.17)
00 i0 ij i j
9
Z
8
From F =−log(f) we have F =−f−1f and
i i
1 1
F = f f − f (3.18)
ij f2 i j f ij
Substituting the above expressions, we see that at the point ϕ in M, we have
7 1 1
F = = ϕ∧∗ ϕ F =0 F =+ η ∧∗ η for i,j 6=0
00 ϕ i0 ij i ϕ j
3 f f
Z Z
and hence in all cases we get
1 1
F = η ∧∗ η = hhη ,η ii i,j =0,...,n
ij i ϕ j i j
f f
ZM
as claimed.
Lemma 3.12. Let η , η , and η be 3-forms in Ω3⊕Ω3 , with associated symmetric tensors h , h ,
1 2 3 1 27 1 2
and h , respectively. Then
3
(η ) (η ) (h )iagjbgkcvol = 2Y(η ,η ,η )
1 ijk 2 abc 3 1 2 3
ZM
+2 (Tr(h )Tr(h h )+Tr(h )Tr(h h )+Tr(h )Tr(h h ))vol (3.19)
1 2 3 2 3 1 3 1 2
ZM
Proof. We use (3.5) and exploit the symmetry of h , h , h , g, and the skew-symmetry of ϕ to
1 2 3
compute:
(η ) (η ) (h )iagjbgkc = (h ) glmϕ (η ) (h )iagjbgkc
1 ijk 2 abc 3 1 il mjk 2 abc 3
+2(h ) glmϕ (η ) (h )iagjbgkc
1 jl imk 2 abc 3
and then
(η ) (η ) (h )iagjbgkc = (h ) (h ) glmgαβϕ ϕ (h )iagjbgkc
1 ijk 2 abc 3 1 il 2 aα mjk βbc 3
+2(h ) (h ) glmgαβϕ ϕ (h )iagjbgkc
1 il 2 bα mjk aβc 3
+2(h ) (h ) glmgαβϕ ϕ (h )iagjbgkc
1 jl 2 aα imk βbc 3
+2(h ) (h ) glmgαβϕ ϕ (h )iagjbgkc
1 jl 2 bα imk aβc 3
+2(h ) (h ) glmgαβϕ ϕ (h )iagjbgkc
1 jl 2 cα imk abβ 3
We use the identity (3.4) repeatedly, and simplify. The right hand side becomes
6 Tr(h h h )+2(Tr(h )Tr(h h )−Tr(h h h ))
1 2 3 2 3 1 2 3 1
+2(Tr(h )Tr(h h )−Tr(h h h ))+2(Tr(h )Tr(h h )−Tr(h h h ))
1 2 3 1 2 3 3 1 2 3 1 2
+2(h )ia(h )jb(h )kcϕ ϕ
1 2 3 ijk abc
which, upon further simplification and integration over M, is exactly (3.19).
9
Proof of Theorem 3.10. We begin by differentiating (3.18) with respect to xk and rearranging to
obtain
2 1 1 1
f =−fF − f f f + f f + f f + f f (3.20)
ijk ijk f2 i j k f i jk f j ki f k ij
We need to compute F . Using Lemma 3.11, this is
ijk
1 1 ∂
F = − f η ∧∗ η + η ∧∗ η
ijk f2 k i ϕ j f ∂xk i ϕ j
ZM ZM(cid:18) (cid:19)
1 ∂ 1 ∂
+ η ∧∗ η + η ∧ ∗ η (3.21)
f ∂xk j ϕ i f i ∂xj ϕ j
ZM(cid:18) (cid:19) ZM (cid:18) (cid:19)
As in the proof of Theorem 3.5, we have ∂ η = dβ′ is exact. Since the η ’s are harmonic with
∂xk i i i
respectto g , wehavethat∗ η isclosed,andhence by Stokes’theoremthe secondandthirdterms
ϕ ϕ i
above both vanish. The fourth term of (3.21) is
1 ∂ 1 ∂
(g (η ,η ))vol+ g (η ,η ) vol (3.22)
f ∂xk ϕ i j f ϕ i j ∂xk
ZM ZM
where we regard η and η as constant. In local coordinates, we have
i j
1
g (η ,η )= (η ) (η ) gaαgbβgcγ
ϕ i j i abc j αβγ
6
We are differentiating in the xk direction, which means that we are considering a one-parameter
family ϕ(t) of torsion-free G -structures, such that ∂ ϕ(t)=η (t), which at t=0 corresponds to a
2 ∂t k
symmetric 2-tensor h . From Corollary 3.3 of [20], such an infinitesmal variation induces
k
∂ ∂
gab =−2(h )ab vol=Tr(h )vol
∂xk k ∂xk k
as the variations of gab and vol, respectively. Substituting these results and simplifying expres-
sion (3.22), equation (3.21) becomes
1 1
F = − f η ∧∗ η + Tr(h )η ∧∗ η
ijk f2 k i ϕ j f k i ϕ j
ZM ZM
1
+ −(η ) (η ) (h )aαgbβgcγ vol (3.23)
i abc j αβγ k
f
ZM
(cid:0) (cid:1)
Now there are two cases. If k 6=0, then Tr(h )=0 and f =0 from (3.16). Whereas if k =0, then
k k
h = 1g, so Tr(h ) = 7, and f = 7f by (3.16). Thus in all cases the combination of the first two
0 3 0 3 0 3
terms of (3.23) vanishes. Now applying Lemma 3.12, we finally obtain:
−fF = 2Y(η ,η ,η )
ijk 1 2 3
+2 (Tr(h )Tr(h h )+Tr(h )Tr(h h )+Tr(h )Tr(h h ))vol
i j k j k i k i j
ZM
Substituting this into (3.20) gives
2 1 1 1
f = 2Y(η ,η ,η )− f f f + f f + f f + f f (3.24)
ijk 1 2 3 f2 i j k f i jk f j ki f k ij
+2 (Tr(h )Tr(h h )+Tr(h )Tr(h h )+Tr(h )Tr(h h ))vol
i j k j k i k i j
ZM
10