Conformal structures with explicit ambient metrics and conformal G_2 holonomy

0.21 MB·English

by  Pawel Nurowski

CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT
G
7 2
METRICS AND CONFORMAL HOLONOMY
0
0
2
PAWEŠNUROWSKI

n
a 2 5
J Abstra
(3t,.2)Givenageneri
-plane(cid:28)eld[ogn]a -dimensionalmanifoldwe
on-
siderits -signature
onformalmetri
asde(cid:28)nedin[7℄. Every
onformal
1 [g]

lass obtained in this way has very spe
ial
onformal holonomy: it must
3 G2
be
ontained in the split-real-form of the ex
eptional group . In this note
2 5 [g]
weshowthatforspe
ial -plane(cid:28)eldson -manifoldsthe
onformal
lasses
]
G have the Fe(cid:27)erman-Graham ambient metri
s whi
h,
ontrary to the general
Fe(cid:27)erman-Graham metri
s given as a formal power series [2℄,
an be written
D
inanexpli
itform. Wepropose tostudytherelationsbetweenthe
onformal
G2 [g] G2
. -holonomyofmetri
s andthepossiblepseudo-Riemannian -holonomy
h
ofthe
orresponding ambientmetri
s.
t
a
m
[
(3,2)
1. The -signature
onformal metri
s
2
v Consider an equation
1 z′ =F(x,y,y′,y′′,z) with Fy′′y′′ =0,
9 (1.1) 6
8 y =y(x) z =z(x) x
fortworealfun
tions , ofonerealvariable . Tosimplifynotation
1 p=y q =y
′ ′′
0 introdu
e new symbols and . Equation (1.1) is totally en
oded in the
7 system of three 1-forms:
0 ω1 = dz F(x,y,p,q,z)dx
/ −
h ω2 = dy pdx
at (1.2) ω3 = dp−qdx,
m −
J (x,y,p,q,z)
: living on a 5-dimensional manifold parametrized by . In parti
ular,
v γ(t)=(x(t),y(t),p(t),q(t),z(t)) J
every solution to (1.1) is a
urve ⊂ on whi
h all
i ω1,ω2,ω3
X the forms identi
ally vanish.
r Weintrodu
eanequivalen
erelationbetweenequations(1.1)whi
hidenti(cid:28)esthe
a
equations having the same set of solutions. This leads to the following de(cid:28)nition:
z = F(x,y,y ,y ,z) z¯ = F¯(x¯,y¯,y¯,y¯ ,z¯)
′ ′ ′′ ′ ′ ′′
De(cid:28)nition 1.1. TJwo equJ¯ations and(x,y,p = y ,q = y ,z),
′ ′′
de(cid:28)ned on spa
es and parametrized, respe
tively, by
(x¯,y¯,p¯ = y¯,q¯ = y¯ ,z¯)
′ ′′
and φ:J, arJe¯said to be (lo
ally) equivalent, i(cid:27) there exists a
(lo
al) di(cid:27)eomorphism → transforming the
orresponding forms
ω1 =dz F(x,y,p,q,z)dx ω¯1 =dz¯ F¯(x¯,y¯,p¯,q¯,z¯)dx¯
− −
ω2 =dy pdx ω¯2 =dy¯ p¯dx¯
− and −
ω3 =dp qdx ω¯3 =dp¯ q¯dx¯
− −
Date:February 2,2008.
Thiswork wassupported inpart bythePolishMinisterstwoNauki iInformatyza
jigrant nr:
1P03B07529andtheUSInstitute forMathemati
sand ItsAppli
ationsinMinneapolis.
1
2 PAWEŠNUROWSKI
via:
φ (ω¯1) = αω1+βω2+γω3
∗
φ (ω¯2) = δω1+ǫω2+λω3 α,β,γ,δ,ǫ,λ,κ,ν J
∗
, with fun
tions on su
h that
φ (ω¯3) = κω1+µω2+νω3
∗
α β γ
 
det δ ǫ λ =0.
6
κ µ ν
Itfollowsthatequation(1.1)
onsideredmoduloequivalen
erelationofDe(cid:28)nition
(3,2) [g ]
F
1.1 uniquely de(cid:28)nes a
onformal
lass of -signature metri
s on the spa
e
J (x,y,p,q,z)
. In
oordinates this
lass may be des
ribed as follows. Let
D =∂ +p∂ +q∂ +F∂
x y p z
J
be a total di(cid:27)erential asso
iated with equation (1.1) on . Then a representative
g [g ]
F F
of the
onformal
lass may be written as
g =[ DF2F2 +6DF DF F2 6DF F F2
F qq qq q qqq qq − qqq p qq −
3DDF F3 +9DF F3 9F F3 +
qq qq qp qq − pp qq
9DF F F3 18F F F3 +3DF F4
qz q qq − pz q qq z qq −
6DF F2F +6F F2F 8DF DF F F +
q qq qqp p qq qqp− q qq qq qqq
8DF F F F +3DDF F2F 3DF F2F
qq p qq qqq q qq qqq − p qq qqq −
3DF F F2F +4(DF )2F2 8DF F F2
z q qq qqq q qqq − q p qqq −
3(DF )2F F +4F2F2 +6DF F F F
q qq qqqq p qqq q p qq qqqq −
3F2F F 6DF F F2F +6F F F2F
p qq qqqq − q q qq qqz p q qq qqz −
3DF F3F +12F F3F +3F2F F
q qq qz p qq qz qq qqq y −
6DF F F2F +4DF F3F +6F F2F F +
qqq q qq z qq qq z q qq qqp z
8DF F F F F 4DF F2F F
(1.3) qq q qq qqq z − q qq qqq z −
9F F3F +F F2F F 8DF F F2 F +
qp qq z p qq qqq z − q q qqq z
8F F F2 F +6DF F F F F 6F F F F F +
p q qqq z q q qq qqqq z − p q qq qqqq z
18F3F +6F2F2F F +3F F3F F
qq qy q qq qqz z q qq qz z −
2F4F2+F F2F F2+4F2F2 F2
qq z q qq qqq z q qqq z −
3F2F F F2 9F2F3F ] (ω˜1)2+
q qq qqqq z − q qq zz
[ 6DF F2 6F2F 8DF F F +
qqq qq − qq qqp− qq qq qqq
8DF F2 8F F2 6DF F F +
q qqq − p qqq − q qq qqqq
6F F F 6F F2F +6F3F +
p qq qqqq − q qq qqz qq qz
2F2F F 8F F2 F +6F F F F ] ω˜1ω˜2+
qq qqq z − q qqq z q qq qqqq z
[ 10DF F3 10DF F2F +10F F2F
qq qq − q qq qqq p qq qqq −
10F4F +10F F2F F ] ω˜1ω˜3+
qq z q qq qqq z
G2
CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY3
30F4 ω˜1ω˜4+[ 30DF F3 30F F3 30F F3F ] ω˜1ω˜5+
qq q qq − p qq − q qq z
[ 4F2 3F F ] (ω˜2)2 10F2F ω˜2ω˜3+30F3 ω˜2ω˜5 20F4 (ω˜3)2
qqq − qq qqqq − qq qqq qq − qq
1
where
ω˜1 = dy pdx
−
ω˜2 = dz Fdx F (dp qdx)
q
− − −
ω˜3 = dp qdx
(1.4) −
ω˜4 = dq
ω˜5 = dx.
Itfollowsfromthe
onstru
tiondes
ribedinRef. [7℄thatwhentheequation(1.1)
φ g
F
undergoes a di(cid:27)eomorphism of De(cid:28)nition 1.1, the above metri
transforms

onformally.
[g ] (3,2)
F
The
onformal
lass of metri
s is veryspe
ialamongall the -signature

onformalmetri
sindimension5: theCartannormal
onformal
onne
tionforthis
so(4,3)

lass, instead of having values in full Lie algebra, has values in its
ertain
14-dimensionalsubalgebra. Thissubalgebraturnsouttobeisomorphi
tothesplit
g so(4,3)
2
real form of the ex
eptional Lie algebra ⊂ . Thus,
onformal metri
s
[g ]
F
provide an abundan
e of examples of metri
s with an ex
eptional
onformal
holonomy. This holonomy is always a subgroup of the non
ompa
t form of the
G F
2
ex
eptional Lie group . We strongly believe that randomly
hosen fun
tion ,
F = 0 [g ]
qq F
su
h that 6 , give rise to
onformal metri
s with
onformal holonomy
G
2
equal to .
[g ]
F
It is interesting to study the
onformal
lasses from the point of view of
F
the Fe(cid:27)erman-Graham ambient metri

onstru
tion [2℄. Sin
e for ea
h de(cid:28)ning
[g ]
F
equation(1.1)wehavea
onformal
lassofmetri
s indimension(cid:28)ve,thensin
e
(cid:28)ve is odd, Fe(cid:27)erman-Graham guarantees [2℄ that there is a unique formal power
(4,3) [g ]
F
series of a Ri

i-(cid:29)at metri
of signature
orresponding to . Moreover,
F g
F
sin
e given the metri
is expli
itely determined by formula (1.3), we see that
F g
F
starting with real analyti
, the metri
is real analyti
. Thus, every analyti
F g
F
of (1.1) leads to analyti
and then, in turn, via Fe(cid:27)erman-Graham, leads
g˜ (4,3)
F
to a unique real analyti
ambient metri
of signature . Sin
e both the
g˜
F
Levi-Civita
onne
tion for and the Cartan normal
onformal
onne
tion forthe
g
F

orresponding5-dimensionalmetri
havevaluesin (possiblysubalgebrasof)the
so(4,3)
sameLiealgebra , it isinterestingto askabout the relationsbetweenthem.
We dis
uss these relations on examples.
2. The strategy for
onstru
ting expli
it examples of ambient
metri
s
We start with the Fe(cid:27)erman-Graham result [2℄ adapted to the 5-dimensional
[g ]
F
situation of
onformal metri
s .
g [g ] J
F F
Let be a repreJsentaRtive oRf the
onformal
lass (0de<(cid:28)nte,du)on Rby (1R.3).
+ +
Consider a manifold × × . Introdu
e
oordinates on × in
g
1 F
Note that formula for di(cid:27)ers from the one given in Ref. [7℄ by tilde signs over the all
g
F
omegas. In Ref. [7℄,when
opying the
al
ulated metri
, by mistake, we forgot to put these
g
F
tildesignsovertheomegas. Hen
e,inRef. [7℄,formulafor istrue,providedthatoneputsthe
tildesignsover theomegasand supplementsitbythede(cid:28)nitions(1.4)ofthetildedomegas.
4 PAWEŠNUROWSKI
J R R π : J R R J
+ +
× × . We have a naJturalJproRje
tionR × × → , whi
h enables
+
us to pullba
k forms from to × × . Ommiting the pulba
k sign in the
π (g )
∗ F
expressionslike we de(cid:28)ne a formal power series
∞
gˇ = 2dtdu+t2g utα+u2β+u3t 1γ+ ukt2 kµ .
(2.1) F − F − − X − k
k=4
α,β,γ,µ k =4,5,6,.... α,β,γ,µ
k k
HereJ J R, R gˇ,arepullba
ksofsymmetri
bJilinRearfoRrms
+ F +
from to × × . Thus isaformalbilinear form on × × . Thisformal
(4,3) u = 0
bilinear form has signature in some neighbourhood of . The following
theorem is due to Fe(cid:27)erman and Graham [2℄.
gˇ
F
Theorem 2.1. Among all the bilinear forms whi
h, via (2.1), are asso
iated
g g˜
F F
with metri
of (1.3) there is pre
isely one, say , satisfying the Ri

i (cid:29)atness
Ric(g˜ )=0.
F

ondition
g α,β,γ,µ g˜
F k F
Given ,allthebilinearforms in aretotallydetermined. Another
issueisto
al
ulatethemexpli
itely. Forexample,itisquitedi(cid:30)
ultto(cid:28)ndthegen-
µ
k
eral formulas for the higher order forms . Nevertherless the expli
it expressions
α,β,γ
for the forms are known [4, 5℄. We write them below in the form obtained
α β γ α = α dxidxj
ij ij ij ij
by C R Graham. We de(cid:28)ne the
oe(cid:30)
ients , and by ,
β = β dxidxj γ = γ dxidxj (xi) = (x,y,p,q,z) J
ij ij
, , where are
oordinates on .
α β γ
ij ij ij
Then Graham's expressionsfor , and are [4℄:
α =2 ,
ij ij
P
β = B + k ,
ij − ij Pi Pjk
3γ =B k 2W Bkl+4 B k 4 kB +4 klC
(2.2) ij ij;k − kijl Pk(i j) − Pk ij P (ij)k;l −
2CklC +C klC +2 k C l 2W k ml,
i ljk i jkl P k;l (ij) − kijlP mP
where
= 1(R 1Rg ),
Pij 3 ij − 8 Fij
g =g dxidxj
F Fij
is the S
houten tensor for the metri
,
W =R 2( g g )
ijkl ijkl i[k Fl]j j[k Fl]i
− P −P
is its Weyl tensor,
C =
ijk ij;k ik;j
P −P
is the Cotton tensor, and
B =C k klW
ij ijk; −P kijl
is the Ba
h tensor.
F
Of
ourse all the above quantities
an be expli
itely
al
ulated on
e , and in
g
F
turn the metri
, is
hosen.
F =F(x,y,p,q,z)
Intherestofthepaperwewill
hoseparti
ularfun
tions ,and
α,β,γ
we will
al
ulate the
orresponding forms for them. We will give examples
F γ
of 's for whi
h the bilinear form is identi
ally vanishing,
γ 0.
(2.3) ≡
F
Given su
h 's we will
onsider
g¯ = 2dtdu+t2g utα+u2β.
F F
− −
G2
CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY5
g¯ g˜
F F
Note that
oin
ides with the ambient metri
up to the terms quadrati
in
t,u g¯
F
the ambient
oordinates . If by
han
e the bilinear form satis(cid:28)es the Ri

i
(cid:29)atness
ondition
Ric(g¯ ) 0,
F
≡
g˜
F
then by the uniqueness of the ambient metri
stated in Theorem 2.1, it will
g˜
F

oi
ide with the ambient metri
:
g¯ g˜ .
F F
≡
g¯
F
The uniqueness result of Theorem 2.1, together with the Ri

i (cid:29)atness of , is
γ
powerfullenoughtoguaranteethatnotonlythe
oe(cid:30)
ient intheambientmetri
g˜ µ k =4,5,6,....,
F k
identi
ally vanishes, but that all the
oe(cid:30)
ients , vanish too!
g˜ g
F F
Thus the strategy of (cid:28)nding expli
it ambient metri
s for is as follows:
F = F(x,y,p,q,z) g
F
• (cid:28)nd for whi
h the
orresponding metri
has identi-
γ

ally vanishing form of (2.2);
g¯ F
F
•
al
ulate the approximate ambient metri
for su
h ;
Ric(g¯ ) g¯
F F
•
he
k if the Ri

i tensor of is identi
ally vanishing;
F g¯
F
• if you have with the aboveproperties then the approximatemetri
is
g˜ g
F F
the ambient metri
for .
3. Conformally Einstein example
g
F
As the (cid:28)rst example, following Ref. [7℄, we
al
ulate and its approximate
g¯
F
ambient metri
for a very simple equation:
z′ =F(y′′), with Fy′′y′′ =0.
6
[g ]
F 2
It was shown in Ref. [7℄ that the
onformal
lass may be represented by
15(F )10/3g =
′′ F
−
30(F )4 [ dqdy pdqdx ] + [ 4F(3)2 3F F(4) ] dz2+
′′ ′′
− −
2 [ 5(F′′)2F(3) 4F′F(3)2+3F′F′′F(4) ] dpdz+
− −
2 [15(F )3+5q(F )2F(3) 4FF(3)2+4qF F(3)2+3FF F(4)
′′ ′′ ′ ′′
− −
3qF F F(4) ] dxdz+
′ ′′
[ 20(F )4+10F (F )2F(3)+4(F )2F(3)2 3(F )2F F(4) ] dp2+
′′ ′ ′′ ′ ′ ′′
(3.1) − −
2 [ 15F (F )3+20q(F )4+5F(F )2F(3) 10qF (F )2F(3)+
′ ′′ ′′ ′′ ′ ′′
− −
4FF′F(3)2 4q(F′)2F(3)2 3FF′F′′F(4)+3q(F′)2F′′F(4) ] dpdx+
− −
[ 30F(F )3+30qF (F )3 20q2(F )4
′′ ′ ′′ ′′
− − −
10qF(F )2F(3)+10q2F (F )2F(3)+4F2F(3)2
′′ ′ ′′
−
8qFF F(3)2+4q2(F )2F(3)2 3F2F F(4)+
′ ′ ′′
−
6qFF F F(4) 3q2(F )2F F(4) ] dx2.
′ ′′ ′ ′′
−
gˆ = e2Υ(q)g
F F
As noted in Ref. [7℄ this metri
is
onformal to a Ri

i (cid:29)at metri
Υ=Υ(q)
with a
onformal s
ale satisfying se
ond order ODE:
90F 2(Υ Υ2) 60F F(3)Υ +3F F(4) 4F(3)2 =0.
′′ ′′ ′ ′′ ′ ′′
− − −
2
−15(TFh′e′)1m0/e3tri
presented here di(cid:27)ers from this of [7℄ by a
onvenient
onformal fa
tor equal to
.
6 PAWEŠNUROWSKI
F =F(q) [g ]
F
Thus,sin
eforea
h the
onformal
lass
ontainsaRi

i(cid:29)at metri
,
G
2
its
onformalholonomymust be a propersubgroup of the non
ompa
t form of .
Aninterestingfeatureofthis
onformal
lassisthatit isveryspe
ialamongallthe
g
F

onformal
lassesasso
iatedwithequation(1.1). Notonlyhas veryspe
ial
on-
formalholonomy,makingitverysimilartotheLorentzian4-dimensionalBrinkman
metri
s;moreover,sin
eitsWeyltensorhasessentiallyonlyonenonvanishing
om-
ponent(see Ref. [7℄ fordetails) it isnot weaklygeneri
(see Ref. [3℄ forde(cid:28)nition).
[g ] N
F
This makes analogousto the Lorentzian type metri
s in 4-dimensions,su
h
as for example, Fe(cid:27)erman metri
s.
g
F
Having of (3.1) we used the symboli

omputer
al
ulation program Math-
γ
emati
a to
al
ulate its asso
iated form of (2.2). We
he
ked that this form
identi
ally vanishes. We further used Mathemati
a to
al
ulate the
orresponding
g¯
F
approximate ambient metri
. On doing that we obsereved that, surprisingly,
β
the bilinear form is also identi
ally vanishing. The expli
it formula for the ap-
proximate ambient metri
is given below:
g¯ =t2g 2 dtdu 2tuF 4/3Pdq2,
F F ′′
(3.2) − −
with
4F(3)2 3F F(4)
P = − ′′ ,
90(F )10/3
′′
g g¯ J R R
F F +
and given by (3.1). The metri
is de(cid:28)ned lo
ally on × × with
(x,y,p,q,z,t,u) (4,3)

oordiantes . It obviously has signature . We also
he
ked,
Ric(g¯ ) 0
F
againusingMathemati
a,that ≡ . Thus,weful(cid:28)ledthestrategyoutlined
g¯
F
inSe
tion2. Thisenablesusto
on
ludethat of(3.2)
oin
ideswiththeambient
g˜ g
F F
metri
for . To give expressionsfor the Cartan normal
onformal
onne
tion
g g˜ =g¯
F F F
for andtheLevi-Civita
onne
tionfor we(cid:28)rstintrodu
eanonholonomi
(θ1,θ2,θ3,θ4,θ5) J

oframe on given by
θ1 =dy pdx
−
θ2 =dz Fdx F (dp qdx)
′
− − −
θ3 = 2 (F )1/3(dp qdx)
−√3 ′′ −
30(F )10/3θ4 = 3F F F(4) 4F F(3)2 10(F )2F(3) dp qdx +
′′ ′ ′′ ′ ′′
(cid:0) − − (cid:1)(cid:0) − (cid:1)
4F(3)2 3F′′F(4) dz Fdx +30(F′′)3dx
(cid:0) − (cid:1)(cid:0) − (cid:1)
θ5 = (F′′)2/3dq.
−
g
F
In this
oframe the metri
is simply:
g =2θ1θ5 2θ2θ4+(θ3)2.
F
−
By means of the
anoni
al proje
tion
π(x,y,p,q,z,t,u)=(x,y,p,q,z)
(θ1,θ2,θ3,θ4,θ5)
t(θh1e,
θo2f,rθa3m,θe4,θ5) J R
Ran be pulba
ked to (cid:28)ve linearly independent forms
+
on × × . They
an be suplemented by
θ0 =dt and θ6 =du
(θ0,θ1,θ2,θ3,θ4,θ5,θ6) J R R
+
to form a
oframe on the ambient spa
e × × .
G2
CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY7
J
The Cartan normal
onformal
onne
tion, when written on in the
oframe
(θ1,θ2,θ3,θ4,θ5)
reads:
0 0 0 0 0 Pθ5 0
 − 
θ1 0 Qθ2+ 9 Pθ3 1 θ4 1 θ3 0 Pθ5
 2√3 √3 −2√3 − 
 
 
θ2 0 0 1 θ5 0 1 θ3 0 
 √3 −2√3 
 
 
ω =θ3 0 2√3Pθ5 0 1 θ5 1 θ4 0 .
G2  − √3 −√3 
 
 
θ4 0 0 2√3Pθ5 0 Qθ2+ 9 Pθ3 0 
 − 2√3 
 
 
θ5 0 0 0 0 0 0 
 
 
 
0 θ5 θ4 θ3 θ2 θ1 0 
− −
Here:
40F(3)3 45F F(3)F(4)+9F 2F(5)
Q= − ′′ ′′ .
90F 5
′′
(θ0,θ1,θ2,θ3,θ4,θ5,θ6)
Nowweuse
oframe to writedownthe Levi-Civita
onne
-
g˜
F
tion for . We have
g˜ =g θiθj,
F ij
i,j =0,1,2,...6 g
ij
with the indi
es range: , and the matrix given by
0 0 0 0 0 0 1
 − 
0 0 0 0 0 t2 0
 0 0 0 0 t2 0 0 
 − 
g = 0 0 0 t2 0 0 0 .
ij  
 0 0 t2 0 0 0 0 
 − 
 0 t2 0 0 0 2tuP 0 
 − 
 1 0 0 0 0 0 0 
−
g˜ J R R
F +
The Levi-Civita
onne
tion for on × × , when written in the
oframe
(θ0,θ1,θ2,θ3,θ4,θ5,θ6)
reads:
0 0 0 0 0 tPθ5 0
 − 
1tθ1+ tu2Pθ5 1tθ0 Qθ2+ 2√93Pθ3 √13θ4 −2√13θ3 tu2Pθ0− 3utQθ5− 1tPθ6 −1tPθ5
 
 
 1θ2 0 1θ0 1 θ5 0 1 θ3 0 
 t t √3 −2√3 
 
 
ω = 1θ3 0 2√3Pθ5 1θ0 1 θ5 1 θ4 0 .
LC  t − t √3 −√3 
 
 
 1θ4 0 0 2√3Pθ5 1θ0 Qθ2+ 9 Pθ3 0 
 t − t 2√3 
 
 
 1θ5 0 0 0 0 1θ0 0 
 t t 
 
 
 0 tθ5 tθ4 tθ3 tθ2 tθ1 uPθ5 0 
− − −
8 PAWEŠNUROWSKI
Σ = (x,y,p,q,z,t,u): u = 0, t = 1 θ0 0 θ6
Note that on { } we trivially have ≡ ≡ .
ω Σ ω ω
Thus,restri
tingtωheformulafor LC to ,wesπeethat G2 ≡ω LC|Σ. O(cid:27)thissetthe
two
onne
tions: LC and the pullba
ked-by- -
onne
tion G2, di(cid:27)er signi(cid:28)
antly.
ω π (ω )
To see this it is enough to observe that
ontrary to LC, the
onne
tion ∗ G2
π (ω )
has torsion. Indeed writing the (cid:28)rst Cartan stru
ture equations for the ∗ G2 in
(θ0,θ1,θ2,θ3,θ4,θ5,θ6)
the
oframe we (cid:28)nd that the torsion is:
0
 
θ0 θ1 Pθ5 θ6
− ∧ − ∧
 θ0 θ2 
 − ∧ 
dθi+π∗(ωG2)ij ∧θj = −θ0∧θ3 .
 θ0 θ4 
 − ∧ 
 θ0 θ5 
 − ∧ 
 0 
Σ
The vanishing of this torsion on the initial hypersurfa
e
on(cid:28)rms our earlier
ω ω
statemant that the two
onne
tions G2 and LC
oin
ide there.
dω +ω ω
LC LC LC
It is interesting to note that the
urvature ∧ does not depend
t u ∂ ∂
t u
on , and isanihilated by and . Thus it
anbe
onsideredto be a2-formon
Σ dω +ω ω
. Assu
hitispre
iselyequaltothe
urvature G2 G2∧ G2 ofthe
onne
tion
ω
G2:
0 0 0 0 0 0 0
 
0 0 A 0 0 0 0
5
0 0 0 0 0 0 0
 
dωG2 +ωG2 ∧ωG2 =dωLC +ωLC ∧ωLC =0 0 0 0 0 0 0θ2∧θ5,
0 0 0 0 0 A5 0
 
0 0 0 0 0 0 0
 
0 0 0 0 0 0 0
3
where
224F(3)4+336F F(3)2F(4) 51F 2F(4)2 80F 2F(3)F(5)+10F 3F(6)
A = − ′′ − ′′ − ′′ ′′ .
5 100F 20/3
′′
4. Non-
onformally Einstein example
[g ]
F
To get quite di(cid:27)erent example of we
onsider equation (1.1) in the form:
z =y 2+a y6+a y5+a y4+a y3+a y2+a y +a +bz,
′ ′′ 6 ′ 5 ′ 4 ′ 3 ′ 2 ′ 1 ′ 0
a ,i = 0,1,...,6, b
i
where and are real
onstants. This equation has the de(cid:28)ning
fun
tion
F =q2+a p6+a p5+a p4+a p3+a p2+a p+a +bz
6 5 4 3 2 1 0
[g ]
F
and, via (1.3), leads to a
onformal
lass represented by a metri
15(2) 2/3g =[9a +2b2+27a p+54a p2+90a p3+135a p4]dy2+
− F 2 3 4 5 6
[15a +2(b2 3a )p2 3a p3+9a p4+30a p5+60a p6
0 2 3 4 5 6
− − −
20bpq+5q2+15bz]dx2+
(4.1)
[15a +4(3a b2)p 9a p2 48a p3 105a p4 180a p5+
1 2 3 4 5 6
− − − − −
3 A5
Weusetheletter todenote thenonvanishing
omponentofthe
urvature tobeina

or-
dan
ewith[7℄and Cartan's paper [1℄. Notehowever that inorder toavoid
ollisionofnotations
A5 a5
betweenthepresentandthenextse
tions weuse
apital insteadof ofpaper [7℄.
G2
CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY9
20bq]dxdy+20dp2
−
10(bp+q)dpdx+10bdpdy 30dqdy 15dxdz+30pdqdx.
− −
This metri
is not
onformal to an Einstein metri
. The qui
kest way to
he
k
C W g
ijk ijkl F
this is the
al
ulation of the Cotton, , and the Weyl, , tensors for .
On
e these tensors are
al
ulated, it is easy to observe that they do not admit a
Ki C +KlW = 0
ijk lijk
ve
tor (cid:28)eld su
h that . As a
onsequen
e the metri
is not
a
onformal C-spa
e metri
. This proves our statement sin
e every
onformally
Einstein metri
is ne

essarily a
onformal C-spa
e metri
(see e.g. Ref. [3℄).
g
F
Re
all that of (4.1), as a member of the family of metri
s (1.3), de(cid:28)nes a
[g ] H
F

onformal
lass with
onformal holonomy redu
ed tothenon
ompa
tgroup
G
2
or to one of its subgroups. But sin
e the metri
(4.1) is not
onformal to an
H = G
2
Einstein metri
, we do not have an immediate reason to
on
lude that 6 .
H =G
2
We
onje
ture that here and try to prove it in a subsequent paper [6℄.
g˜ g
F F
It is remarkable that the ambient metri
for of (4.1) assumes a very

ompa
t form:
g˜ =t2g 2 dtdu
F F
− −
2 tu [ 1 ( 2a +4b2+3a p+6a p2 20a p3 120a p4)dx2
20 − 2 3 4 − 5 − 6 −
9 (a 10a p2 40a p3)dxdy 9 (a +5a p+15a p2)dy2 ] +
20 3− 5 − 6 − 10 4 5 6
u2 [ 3 (a 10a p+60a p2)dx2+ 9 (a 12a p)dxdy+ 81 a dy2 ].
20(2)2/3 4− 5 6 4(2)2/3 5− 6 4(2)2/3 6
Thisis
he
kedby applyingourstrategydes
ribed in Se
tion 2to the metri
(4.1).
As in the previous example, using Mathemati
a, we
al
ulated the bilinear form
γ γ 0 g¯
F
for (4.1). It turned out to be equal to zero, ≡ . Then we
al
ulated ,
g¯
F
and
he
ked that it is Ri

i (cid:29)at. Thus we
on
luded that
oin
ides with the
g˜ g˜ g¯
F F F
ambient metri
for . The above given formula for is therefore just , whi
h
we
al
ulated using (2.2).
g
F
We (cid:28)nd this example as a sort of mira
le. Apriori there is no reason for
to have the ambient metri
trun
ated at the se
ond order in terms of the ambient
t u
parameters and . We are intrigued by this fa
t.
Now, following the general pro
edure outlined in [7℄, we introdu
e a spe
ial
g
F

oframe for given by:
θ1 =dy pdx
−
θ2 =dz Fdx 2q(dp qdx)
− − −
θ3 = 24/3(dp qdx)
− √3 −
θ4 =2 1/3dx
−
15(2)1/3θ5 =(9a +2b2+27a p+54a p2+90a p3+135a p4)(dy pdx)+
2 3 4 5 6
−
10b(dp qdx) 30dq+
− −
15(a +2a p+3a p2+4a p3+5a p4+6a p5+2bq)dx.
1 2 3 4 5 6
g
F
In this
oframe the metri
is:
g =2θ1θ5 2θ2θ4+(θ3)2.
F
−
As in the previous se
tion, we use the
anoni
al proje
tion
π(x,y,p,q,z,t,u)=(x,y,p,q,z)
10 PAWEŠNUROWSKI
(θ1,θ2,θ3,θ4,θ5) (θ1,θ2,θ3,θ4,θ5)
topJullbRa
kthRe
oframe to(cid:28)velinearlyindependentforms
+
on × × , whi
h are further suplemented by
θ0 =dt and θ6 =du
(θ0,θ1,θ2,θ3,θ4,θ5,θ6) J R R
+
to form a
oframe b = 0 on thJe ambiJent sRpa
e R× × .
+
It turns out that if the
oframes on and × × de(cid:28)ned in this
way are suitable to analyze the relations between the Cartan normal
onformal
ω [g ] ω g˜ b = 0

onne
tion G2 for F and the Levi-Civita
onne
tion LC for F. If 6
ω (θ1,θ2,θ3,θ4,θ5) ω
the
one
tion G2 in the
oframe and the
onne
tion LC in the
(θ0,θ1,θ2,θ3,θ4,θ5,θ6) t = 1 u = 0

oframe do not
oin
ide on , . We will not
analyze this
ase here.
Restri
ting to the
b=0

ase we (cid:28)nd the following:
ω (θ1,θ2,θ3,θ4,θ5)
• the
onne
tions G2 in the
oframe and the
onne
tion
ω (θ0,θ1,θ2,θ3,θ4,θ5,θ6) t=1 u=0
LC
in the
oframe
oin
ide on , .
π (ω ) (θ0,θ1,θ2,θ3,θ4,θ5,θ6)
• the torsionof ∗ G2 in the
oframe is nonvanish-
t=1 u=0
ing o(cid:27) the set ,
dω +ω ω
LC LC LC
• unliketheexampleofthepreviousse
tionthe
urvature ∧
t u
sigini(cid:28)
antly depends on and .
t = 1 u = 0 dω +ω ω
• even on , , the
urvature G2 G2 ∧ G2 and the restri
tion
dω +ω ω
LC LC LC
of ∧ do not
oin
ide.
5. A
knowledgements
IamverygratefultoTPBranson,MEastwoodandWMillerJr,theorganizers
of the 2006 IMA Summer Program (cid:16)Symmetries and Overdetermined Systems of
Partial Di(cid:27)erential Equations(cid:17), for invitating me to Minneapolis to parti
pate in
this very fruitful event. The topi

overed by this note is inspired by the talk of C
R Graham whi
h I heard in Minneapolis during the program. In parti
ular, I am
very obliged to C R Graham for sending me the formulas (2.2), whi
h I used to
prepare the examples in
luded in this note.
Referen
es
[1℄ Cartan E, (cid:16)LessystemesdePfa(cid:27)a
inq variables et les equations aux derivees partielles du
se
onde ordre(cid:17) Ann. S
. Norm. Sup.27109-192 (1910)
[2℄ Fe(cid:27)erman C, Graham C R, (cid:16)Conformal invariants(cid:17), in Elie Cartan et mathematiques
d'aujourd'hui,Asterisque,horsserie(So
iete MathematiquedeFran
e, Paris)95-116 (1985)
[3℄ Gover A R, Nurowski P, "Obstru
tions to
onformally Einstein metri
s in n dimensions"
Journ. Geom. Phys.56450-484 (2006)
[4℄ GrahamCR,private
ommuni
ations,unpublished
[5℄ deHaroS,SkenderisK,SolodukhinSN,(cid:16)Holographi
re
onstru
tionofspa
etimeandrenor-
malization in the AdS/CFT
orresponden
e(cid:17), Comm. Math. Phys. 217 (2001), 594(cid:21)622,
hep-th/0002230
[6℄ LeistnerTh,NurowskiP,inpreparation
[7℄ NurowskiP,"Di(cid:27)erentialequationsand
onformalstru
tures" Journ.Geom.Phys.5519-49
(2005)
Instytut Fizyki Teorety
znej, Uniwersytet Warszawski, ul. Ho»a 69, Warszawa,
Poland
E-mail address: nurowskifuw.edu.pl