The GS Type IIB Superstring Action on AdS(3) X S(3) X T(4)

24 Pages·0.29 MB·English

by  Igor Pesando

DFTT-58/98
hep-th/9809145
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AdS S T4
n 3 3
The GS Type IIB Superstring A
tion on × ×
a
J

3
1
I. Pesando 1

2
v
Dipartimento di Fisi
a Teori
a , Universitá di Torino, via P. Giuria 1, I-10125
5
Torino, Istituto Nazionale di Fisi
a Nu
leare (INFN)- sezione di Torino, Italy
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1
9
0
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t
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p
e
h
ABSTRACT
:
v
AdS S
i We des
ribe the type IIB supergravity ba
kground on 3⊗ 3⊗
X T4 AdS S T4
34 3
r using the potentials of A|dS⊗34 ⊗ and we uκse the super-
a solvable algebra asso
iated to | to
ompute the gauge (cid:28)xed
type IIB string a
tion. From the expli
it form of the a
tion we
an

learly see how passing from pure NSNS ba
kgrounds to pure RR
ba
kgrounds the WZW term disappears.
1e-mail: ipesandoto.infn.it,pesandoalf.nbi.dk
Work supported bytheEuropean CommissionTMRprogrammeERBFMRX-CT96-004
1 Introdu
tion.
TheAdS/CFT
onje
ture([1℄,[2℄,[3℄)hasre
entlyattra
tedalotofinterestand
work, one of its
onsequen
es is that the
lassi
al (i.e. no string loops) type
AdS S
5 5
IIB superstring propagating in a ⊗ ba
kground is the master(cid:28)eld of
= 4 D = 4 U(N)
the N SYM based on the gauge group. It has therefore
be
ome important to write the
lassi
al a
tion for the above mentioned type
IIB superstring and then to (cid:28)rst quantize it. The (cid:28)rst step has been a

om-
plishedinaseriesofworks([7℄,[8℄,[9℄)whi
h
ulminatedintwodi(cid:27)erentproposal
([4℄,[5℄) whi
h were shown to be equivalent ([6℄). Even if the resulting a
tion

an be very easily written down and we
an thrust it is the proper
onformal
a
tion ([7℄,[10℄), it is un
lear how to pro
eed to quantize it expli
itly, even in a
simpli(cid:28)ed but equivalent form ([11℄), the main reason being it des
ribes a RR
ba
kground whi
h re(cid:29)e
ts into the fa
t that the theory is not a usual WZW
model be
ause it has not the left/right symmetry([7℄). In view of the fa
t that
= 4 D = 4
expli
it
omputations of N SYM amplitudes in the leading order
λ λ = Ng2 >> 1
([12℄) and next to leading order ([13℄) in ( YM ) based on this

onje
ture have revealed the possible appearan
e of logarithm and even if the
results in this (cid:28)eld are far from be de(cid:28)nitive be
ause of the di(cid:30)
ulties of
om-
putingallthe
ontributionsexpli
itly, ithasbe
omeevenmoreimportanttobe
able to understand how to quantize this a
tion.
In order to get some hint on how to pro
eed to quantize su
h a kind of
a
tion we
ould start to look at some simpler example where one
an at least
partially
ompare with a well de(cid:28)ned NSR formulation: it is infa
t purpose
κ
of this paper to write the expli
it gauge (cid:28)xed a
tion for type IIB string
AdS S T4
3 3
propagatingon ⊗ ⊗ whoseNSR formulationhasbeen workedoutin
([14℄). This non dilatoni
ba
kground
an be pure NSNS, pure RR or a mixing
of the two sin
e it
an be thought as generated by a dioni
string in the type
D =6
IIB supergravitydimensionalredu
edto : unfortunatlyit turns outthat
the two ba
kgrounds are so di(cid:27)erent to make their
omparison quite di(cid:30)
ult.
The plan of the paper is the following.
In se
tion 2 we
ompute the supergravitational ba
kground
orresponding
AdS S T4
3 3
to a spontaneous
ompa
ti(cid:28)
ationon ⊗ ⊗ and weexpressthe ba
k-
ground (cid:28)elds, thought as (cid:28)elds living in a superspa
e, through the (cid:28)elds of the
AdS S T4
34 3
superspa
e | and the vielbein and spin-
onne
tion of ⊗ .
κ
In se
tion 3 we derive the form of the ba
kground (cid:28)elds when (cid:28)xing the
symmetry using the supersolvable approa
h.
In se
tion 4 we give the a
tion for the type IIB superstring propagating in
this ba
kground,we dis
uss and
omment the form of the a
tion.
In a series of appendi
eswe give more details on the
omputations and we
κ
prove that the proposed a
tion a
tually (cid:28)xes the symmetry as
laimed.
1
2 The supergravitational ba
kground.
We will now
onstru
t the supergravitational ba
kground expli
itly: this will
be performed in three steps
1. we solve the equationsof motion and we get the (cid:28)eld strengths whi
h are
AdS S T4
3 3
asso
iated with the spontaneous
ompa
ti(cid:28)
ation on ⊗ ⊗ ;
2. sin
e the fermions are set to zero we look at their supersymmetri
trans-
formationsand wedetermine the form and the number of Killing ve
tors.
Their form suggests the ansatz for the gravitino in the next point;
SU(1,12,0)
3. we(cid:28)nallyexpressallthe(cid:28)eldsthroughthegeneralizedpotentialsof | ⊗
SU(1,10,2) S T4
3
| andthevielbeinandspin
onne
tionof ⊗ : inthisway
AdS
34
all the (cid:28)elds be
ome de(cid:28)ned on the superspa
e | .
AdS S T4
3 3
2.1 The spontaneous
ompa
ti(cid:28)
ation on ⊗ ⊗ .
Sin
e we want a solution of the equations of motion whi
h has a spontaneous
AdS S T4
3 3

ompa
ti(cid:28)
ation on ⊗ ⊗ , it is natural to make the following ansatz
2
whi
h in the notation of [15℄ reads
Rab =r δab Rij =r δij Rrs =0
..cd 1 cd ..kl 2 kl ..tu (2.1)
G =g ǫ G =g ǫ
abc 1 abc ijk 2 ijk
(2.2)
Vα =const Q=0 P =0
a
± ⇒ (2.3)
F =0
a1...a5 (2.4)
ρ =0
ab (2.5)
λ=0 b b D λ=0
⇒ a (2.6)
bb
a,b,... 0,...,9 a,b,... 0,1,2 u,v,... 3,...,9 i,j,...
where ∈ { } , ∈ { }, b ∈ { }, ∈
3,4,5 r,s,... 6,...,9
G{abc =}bǫ,αbβV+αFaβbc∈.{ }( see appendix A for further
onventions ) and
It is then a simple matter to solve the equation of motions and to (cid:28)nd that
bbb bbb
r =r = e 2
1 2
− | | (2.7)
2 2
g = e g = eρ (ρ= 1)
1 2
3 3 ± (2.8)
e
where is an arbitrary
omplex number and to verify that the
omplex tensor
G DaG = 0
abc is
ovariantly
onstant in this ba
kground, i.e. abc . We noti
e
g2 = g2
that the
onstraint 1 2 is a
onsequen
e of the equbation of motion for the
s
bablbars. bbb
2 Withtheminormodi(cid:28)
ationsgivenbyψ→Ψ,ψ∗→Ψc≡CΨTand̟ab→Ωab.
b bb
2
2.2 The Killing ve
tors.
Asa(cid:28)rststepwewritethesupersymmetri
transformationrulesforthefermions
in this ba
kground
i
δλ = ΓabcǫG
−8 abc (2.9)
1
δΨ = Dǫ+bb3b2b 9bbΓbabVc+ΓabcdVd ǫcGabc (2.10)
(cid:16) (cid:17)
ǫ = C ǫT b bb b bbbb b ǫ
c b bbb
where is the
harge
onjugate of the in(cid:28)nitesimal parameters and
all the (cid:28)eld strengths are restri
ted to the spa
etime.
b bb bǫ
Sin
e the gravitino transformation rule involves both the parameters and
ǫ δΨ ǫ
c
its
harge
onjugate the only way to set to zero is to use a parameter
b
whi
hisasum oflinearlyindependentMajoranaspinors,thusbreakingat least
b b
half of the number of lo
al
harges,expli
itly:
0
ǫ= 16
eiϕN ǫN ηN (2.11)
(cid:18) ⊗ (cid:19)
ǫ = C ǫ bN N AdS
where N NN is a set of ( to be deteermined) 3 Majorana spinors
ηN =Cη N S T4
3
and is a set of ⊗ Majorana spinors.
The dilatino variation implies that
e ee
1 γ ηN =ρηN
2⊗ 5 (2.12)
ρ
where the sign is the same as the onee whi
heenters eq. (2.8).
The gravitinos variation yields more
onstraints and suggests how to ex-
pressthe gravitino(2.26), when thought as a (cid:28)eld on the superspa
e, using the
AdS S T4 Ψ
34 3
| ⊗ ⊗ generalized potentials . Expli
itly we get (we denote −the
Ψ
3
16 lower
omponents of the antiWeyl gravitino )
e
δΨ− = eiϕN DAdS3 ǫN +i2e−2iϕN γaǫN Va ⊗ηN
(cid:16) e (cid:17)
+ eiϕN ǫN ⊗ DS3⊗T4 ηN − 2ρe−2iϕN γi⊗14eηN Vi
(cid:16) (cid:17)
e e
whi
h
an be solved as
e
D ǫ is | | γ ǫ Ea = 0
AdS3 N − N 2 a N (2.13)
(cid:18) (cid:19)
e
D ηN +s | | γ ηN Ei = 0
S3⊗T4 N 2 i (2.14)
(cid:18) (cid:19)
e e e
where e e−2iϕN =sN = 1
− e ± (2.15)
| |
3 We also de(cid:28)ne DAdS3 ǫ = d− 14ωabγab ǫ and DS3⊗T4 η = d− 41ωijγij η where ω
θ
arethe
lassi
alspin-
onne
tions(cid:0),i.e. without(cid:1) dependen
e. (cid:0) (cid:1)
e e e
3
be
ause of the integrability
onditions of the two equations (2.13,2.14). Noti
e
that in eq.s (2.13,2.14) we have expli
itly used the fa
t that the restri
tion of
Va ′′θ′′=0 =Ea
the zehnbein to the spSa
etimTe4gives theηN
lassi
al zehnbein, i.e. | .
3
We will dsenote a s ⊗ηN spinor whi
h satis(cid:28)es eq. (2b.14) with thbe
substitution N → − as (s) in order to stress the relative sign of the two
e
terms of the equation (2.14).
e
Before
ountingthenumberofrealsupersymmetri

hargessurvivinginthis
N
ba
kgroundandbeingableηNtospe
ifyin whi
hrangetheTi4ndex S runs,weneed
3
to understand better how
an be deη
Nomposed w.r.t. and spinors. It is
not di(cid:30)
ult to see that the Majorana spinors
an be de
omposed as
e
η(Ns) =η((sn),m) =HpNq η(pse)⊗κq HN =σ2HN∗σ2 (2.16)
H ηN
where the
onstreaint oen N is due to the Majorana
ondition whi
h (s) have
to satisfy and to the fa
t that
e
κq q =1,2
• ( ) area
ouple of
onstant
hiralsymple
ti
-Majoranaspinors
on the 4-torus
γ κq =ρκq κ CκT =ǫ κp κ κq =δq
5 cq ≡ q qp p p (2.17)
ǫ =1
12
with and
ηp p=1,2 s
• (s) ( ) are (for any ) a
ouple of symple
ti
-Majoranaspinors on
the 3-sphere η CηT =ǫ ηq η ηq =δq
cp ≡ p pq p p (2.18)
whi
h satisfy the killing equation on the 3-sphere
e
D ηp s| |ργ ηp Ei =0
S3 (s)− 2 i (s) (2.19)
p q
It is now obvious that both and in (2.16)
an take two values and that
N = (n,m) 4
the index
an take values as we
an immediately (cid:28)nd 4 matri
es
H 1, iσ , iσ , iσ
1 2 3
whi
h satisfy the se
ond equation in eq. (2.16) : { } , hen
e
2 2 4=16
thenumber ofrealsupersymmetri

hargesis · · wherethe (cid:28)rst fa
tor
2 s
N
a

ounts for the two possible values of (2.15) and the se
ond fa
tor is the
ǫ
N
number of real
omponents of .
2.3 TAdhSe typeS IIBT4(cid:28)elds as (cid:28)elds living on the superspa
e
3|4 3
⊗ ⊗ .
We arenowreadyto writeouransatzon howto expressthe supergravity(cid:28)elds
leaving on superspa
e, to this purpose we use
AdS =
34
• the(cid:28)eldsofthesuperantideSitterin3Dand4
omplex
harges |
SU(1S,1O|2(,10,)2⊗)SUO((14,)1|0,2) ω(±)a(x,Θ(±)), A.(m±)n(x,Θ(±)), χn(±)(x,Θ(±)) (see
⊗
n o
4
appendi
esBandCfora
ompleteexplanationofthenotationsandderiva-
tion of these results ) whi
h satisfy
1
dω( )a+ ǫa ω( )bω( )c is( )χ( )nγaχ( ) = 0
± 2 .bc ± ± − ± ± n±
(2.20)
1
dA.(m±)n+A.(±p)nA.(±m)p−s(±) χ(±)nχm(±)− 2δmn χ(±)pχp(±) = 0
(cid:18) (cid:19)
(2.21)
1
dB(±)−s(±)2 χ(±)pχp(±) = 0
(2.22)
i
dχ( ) ω( )aγ χ( )+χ( )A( )m = 0
n± − 2 ± a n± m± .n±
(2.23)
x Θ( ) AdS
± 3
and depend on and whi
h are respe
tively the
oordinates
AdS
3
andasetof4
omplexGrassman
oordinateswhi
htransformastwo
AdS S T4
3 3

omplex spinors and not as ⊗ ⊗ spinors; we noti
e moreover
s( )
±
the presen
e of the arbitrary sign ;
S Ei(y) ωij(y)
3
• the
lassi
al dreibein and spin-
onne
tion of ( , ) whi
h
satisfy
dEi ̟i Ej = 0
− .j (2.24)
dωij ωikω.j = e2EiEj
− k | | (2.25)
S y
3
and depend on the
oordinates ;
T4 Er =dzr zr T4
• the
lassi
al vierbein of where are the
oordinates.
When we take in a

ount the form of the killing spinors (2.11) the ansatz for
the gravitinoreads:
0
Ψ = κ3 eiϕ(+)χ((+n,)m)(x,Θ)⊗η((sn),m)+eiϕ(−)χ((n−,)m)(x,Θ)⊗η((tn),m) !=
0
= κ3 eiϕ(+) χ(n+)(x,Θ)⊗ηe(ns)c+χ(c+)n(x,Θ)⊗η(s)n !e+
(cid:16) 0 (cid:17)
+κ3 eiϕ(−) χn(−)(x,Θ)⊗eη(nt)c+χc(−)n(x,Θ)⊗eη(t)n ! (2.26)
(cid:16) (cid:17)
χ
n
where we have de(cid:28)ned impli
itly the Meajorana
omponenteof as
1
χ = χ +iχ
n √2 (n,1) (n,2) (2.27)
1 (cid:0) (cid:1)
χn = χ iχ
c √2 (n,1)− (n,2)
(cid:0) (cid:1)
5
η
and inversely we have de(cid:28)ned the
omplex spinors out of the Majorana ones
η
N
as 1
η = η +iη =(H )e ηp κq
e n √2 (n,1) (n,2) n pq ⊗ (2.28)
(cid:0)ηn (cid:1) η =
and their
harge
eonjugate ec . Moreeoverthe Majorana killing spinors (s)N
η
(s)(m,l)
(2.16) are normalized as
e e
η η =δ
e (s)N (s)M NM (2.29)
η
Now that we have a set ofe
ompelex C-number spinors n with the proper
index stru
ture we
an write the ansatz for the remaining (cid:28)elds:
e
1
Va = Ea(x,Θ)= ω(+)a ω( )a
−
2e − (2.30)
| |(cid:16) (cid:17)
Vi = Ei(y)+κ(+)ηm (y)γiη (y)A(+)n(x,Θ)
1 (s) (s)n .m
+κ1(−)ηm(t)(y)eγiη(t)ne(y)eA(.−m)n(x,Θ) (2.31)
Vr = dzr
e 1e e (2.32)
Ωab = ωab(x,Θ)= ǫabc ω(+)+ω( )
2 c c− (2.33)
Ωij = ωij(y)+κ(+)ηm ((cid:16)y)γijη (y)(cid:17)A(+)n(x,Θ)
2 (s) (s)n .m
+κ2(−)ηm(t)(y)γeijη(t)ne(y)eA(.−m)n(x,Θ) (2.34)
Ωrs = 0
e e e (2.35)
In order to (cid:28)nd the solution we have to (cid:28)x the
oe(cid:30)
ients κ1(±), κ2(±) , κ3
κ > 0 ϕ ϕ
3 (+) ( )
( sin
eHitNs phase is given by s ) andt the phases ± ; to determine
the matri
es (2.16) and the sign and ; to (cid:28)nd to whi
h form of the
SU(1,12) ω(±)a(x,Θ) A.(m±)n(x,Θ) χn(±)(x,Θ)
superalgebra | the , , belong, i.e.
s( )
±
whi
h sign has in the Maurer-Cartansystem (2.20-2.23).
ExaminingthepuregravitationalBian
hiidentitiesusingtheseparametriza-
tion (2.26, 2.30-2.35)we get that
1
κ1(±) = ±2e (2.36)
| |
1
κ2(±) = 2 (2.37)
1
κ =
3
2 e (2.38)
| |
s= t = 1
p
− − (2.39)
s( ) = 1
±
± (2.40)
ei(ϕ(−)−ϕ(+)) = i
(2.41)
0 1
H = H =
1 1 2 0 (2.42)
(cid:18) − (cid:19) (cid:18) (cid:19)
6
A
while the integrability of the equation requires
e
e2iϕ(+) =
− e (2.43)
| |
A
We do not derive the expli
it form of here be
ause of its
ompli
ation but
κ
we will (cid:28)nd its expli
it expression after we have the (cid:28)xed expression for the
zehnbein and the gravitini in the next se
tion .
κ
3 Fixing the symmetry using the supersolvable
κ
and the gauge (cid:28)xed expression for the super-
gravity ba
kground (cid:28)elds.
ISnUo(1rd,1er2)to (cid:28)nd the supersolvable algebras asso
iated withAdtShe t=woSfOo(r2m,2)s o=f
| , we look at the solvable algebra asso
iated to 3 SO(1,2)
SO(1S,2O)⊗(1S,2O)(1,2) ; from ([16℄) we see that the generators of AdS3
an be
hosen
to be
solv(AdS )= M ,M ,M
3 23 +0 +1
{ }
M SO(2,2) M = 1 (M +M )
where µν are the generators and +µ √2 2µ 3µ (see ap-
pendix B for further details and more pre
ise notations). We
an now see how
thesegenerators
anbewrittenintermsofthegeneratorsJc(±) =ǫcab41 Mab± 12ǫabµν Mµν
SO(1,2)
of the two (cid:0) (cid:1)
solv(AdS3)= J2(+)−J2(−), −J+(+)±J−(−) (3.1)
n o
Thisequation(3.1)
anbereadin thedualspa
ebysayingthatthebosoni
ω(+)2 ω( )2, ω(+)+ ω( )
− − −
partofthesupersolvablealgebrahastobegeneratedby − ± .
SU(1,12)
We writedownthe Maurer-Cartanequationsforthe tw(cid:8)o | and weask (cid:9)
ω(+) ω( )+
− −
for
onsisten
ywhen we set either or . If we generi
allydenote the
χ
two
omponents of the fermioni
1-form as
ζ
χ=
ξ
(cid:18) (cid:19)
(+)
then it easy to (cid:28)nd that the algebra reads
dω(+)2 = 0
(3.2)
dω(+)++ω(+)+ω(+)2 is(+)√2ξ∗nξn = 0
− (3.3)
1
dξ(+) ω(+)2ξ(+) = 0
n − 2 n (3.4)
( )
and the − algebra reads
dω( )2 = 0
−
(3.5)
7
dω( ) ω( ) ω( )2 is( )√2ζ nζ = 0
− − − − − − ∗ n
− − (3.6)
1
dζn(−)+ 2ω(−)2ζn(−) = 0 (3.7)
From the knowledgeofwhi
h1-formmakethe supersolvablealgebrawe
an
dedu
e the generi
element of the supersolvable algebra (see appendix for C
notations); for example for the (+) algebrawe get
1α(+)2
2
g(+) = −√i2α+ −21α(+)2 τm 
s(+)τ n
∗
−
 
it is now an easytask to derivethe expli
it dependen
e on these
oordinates of
G(+) =exp g(+) exp g(+) exp g(+)
the(+)1-formsfromthegeneri
groupelement 2 + τ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
:
ω(+)2 = dα(+)2
i
ω(+)+ = dα+ α+dα(+)2 s(+) τ∗n −→d ←−d τn
− − √2 −
(cid:16) (cid:16) (cid:17) (cid:17)
1
ξ(+) = dτ τ dα(+)2
n n− 2 n
Similarly for the (-) algebrawe get
ω(−)2 = dα(−)2
i
ω(−)− = dα−+α−dα(−)2 s(−) θ∗n −→d ←−d θn
− √2 −
(cid:16) (cid:16) (cid:17) (cid:17)
1
ζ( ) = dθ + θ dα( )2
n− n 2 n −
κ
Inorderto(cid:28)ndthe gauge(cid:28)xedexpressionforthedreibeinandthegravitini
AdS SO(1,2)
3
we have to relate the
oset
oordinates with those of the two :
baPa = ebaJa(+) ebaJa(−) α(+)aJa(+)+α(−)aJa(−)
thisisdone
omparing | | −| | with .
α(+)2 = α( )2 = eb2 α(+)+ = eb+ α( ) = eb
− − − −
Inparti
ularweget Ea = −1 ω(+)a | |ω(,)a | | and −|r| =
. Using the relation 2e − − and
hanging variables as
exp |e|b2 , x± = 2be±r, θ(+|)|=(cid:0) e√τ2r and θ(cid:1)(−) = e√θ2r we get the (cid:28)nal κ
| | | | | |
gaug(cid:0)e (cid:28)xe(cid:1)d expression for the dreibein and the gravitini
i
E± = |e|r dx±± √2s(±)θ(±)∗n −→d −←−d θ(±)n (3.8)
(cid:18) (cid:16) (cid:17) (cid:19)
1 dr
E2 =
e r (3.9)
| |
0
χ(+) = e√2r
n | | (cid:18) dθn(+) (cid:19) (3.10)
χ( ) = e√2r dθn(−)
n− | | 0 (3.11)
(cid:18) (cid:19)
8
κ
With theseexpressionsinhand wetrytointegratethe gauge(cid:28)xed version
of the
omplex 3 (cid:28)eld strength
G=ǫ VαdAβ =G VaVbVc+2iΨ Γ ΨVa
αβ + abc c a
Vα b b b b
( ± are
onstant). This is approa
hbbisbpossible beκ
ausebthe operation of pull-
ba
k on the surfa
e whi
h (cid:28)xes the gauge of the symmetry
ommutes with
the external derivatives.
We noti
e immediately that there is a part whi
h is
losed but not exa
t as
S
3
it is easy to see from the fa
t that one of the addends is the volume form .
4 5
If we use proje
tive
oordinates for the 3-sphere , the (cid:28)nal answer is
G=d ǫ VαAβ =
αβ +
2 e 1
= 3eρǫijkEiEjEk+d 2e3ω(cid:0)(+)+ω(−)−(cid:1) − e2ei(ϕ(+)+ϕ(−))χ(N+)χM(−)η(s)Nη(−s)M =
(cid:18) | | | | (cid:19)
e e (3.12)
16e dyi dyj dyk
= ρ ǫ ∧ ∧ +
3e3 ijk (1 y2)3
| | −
i i
+d −2e|e|r2 dx++ √2θ(+)∗n −→d −←−d θ(+)n ∧ dx−+ √2θ(−)∗n −→d −←−d θ(−)n
(cid:26) (cid:20)+2i e r dθ(+) nd(cid:16)θ( )η (cid:17) ηm (cid:21) + (cid:20)dθ(+)dθ( ) mηn η(cid:16) (cid:17) (cid:21)
e2 ∗ m− (s)cn (−s)c n − ∗ (s) (−s)m
| | h
−idθn(+)dθm(−e)ηn(s)eη(m−s)c − idθ(+)∗ndθ(−)e∗mηe(s)cnη(−s)m (3.13)
io
e e e e
4 The a
tion.
We have now all the ne
essary ingredients to be able to write down the a
tion
on this ba
kground using the geφne=ral foπrm given in ([18℄) or the spe
i(cid:28)
a
tion
derived in app. E (after setting −2):
1
S = d2ξ√ ggαβ
− 2 ×
ZΣ2
4 Our
onveyn2ti=onηsijfyoirytjh=eS−3δ
ijoyseiytjmanifoldwiththeKillingindu
ed metri
,i.e. negative
de(cid:28)nite,are ( )
2 dyi
Ei =
|e|1−y2
4y[idyj]
̟ij =
1−y2
5 A
tually the sign of the se
ond aedid(ϕen(+d)+oϕf(e−q)). (3.12) is not (cid:28)xed be
ause there is a
possiblesignambiguityinthevalueof whi
h
anbe(cid:28)xedasdis
ussedinapp.
D.
9