Massdeformed ABJM theory and LLM geometries: exact holography
Received: December
Massdeformed ABJM theory and LLM geometries: exact holography
Dongmin Jang 0 1 3
Yoonbai Kim 0 1 3
OKab Kwon 0 1 3
D.D. Tolla 0 1 2 3
with SO 0 1
=Zk 0 1
Open Access 0 1
c The Authors. 0 1
0 Suwon 440746 , South Korea
1 Institute of Basic Science, Sungkyunkwan University
2 University College, Sungkyunkwan University
3 Department of Physics, BK21 Physics Research Division
We test the gauge/gravity duality between the N = 6 massdeformed ABJM theory with Uk(N ) U k(N ) gauge symmetry and the 11dimensional supergravity on LLM geometries SO(4)=Zk isometry, in the large N limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of massdeformed ABJM theory and from the implementation of KaluzaKlein holography to the LLM geometries. We focus on the chiral primary operator with convanishing as expected by the supersymmetry. We extend our results to the case of k 6= 1 for Coulomb branch of the N = 4 super YangMills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators.
exact; holography; AdSCFT Correspondence; Brane Dynamics in Gauge Theories; MTheory

= 1. We show that hO( =1)
Contents
1 Introduction
2 Vacua of the mABJM theory and the LLM geometries
Supersymmetric vacua of the mABJM theory
2.2 LLM geometries and their droplet picture
3 Vevs of CPOs in mABJM theory 3.1 CPOs in ABJM theory 3.2 vevs of CPO in mABJM theory
4 KK reduction and gauge invariant modes
11dimensional gravity equations of motion
Fluctuations on AdS4
Expansion in S7 spherical harmonics
Gauge invariant uctuations
KK reduction
The equations for spin zero elds
The equations for spin one elds
The equations for spin two elds
5 Exact KK holography for LLM geometries Asymptotic expansion of the LLM geometries Asymptotic expansions for the physical modes in 4 dimensions 5.3 Comparison with eld theory results
6 Conclusion
A Spherical harmonics on S7
A.1 Scalar spherical harmonics
A.2 Vector spherical harmonics
A.4 CI1=2 and C( =1)
B Asymptotic expansions
C Proof of (5.48)
A.3 Scalar spherical harmonics on S7 with SO(4)
SO(4) symmetry
Introduction
ity in (d+1)dimensional spacetime and a quantum
eld theory (QFT) on the ddimensional
string/M theory on AdSd+1
X , with a compact internal manifold X , and conformal eld
X . However,
rank of the gauge group.
AdSd+1
operator on the eld theory side there is a corresponding gravity
eld among these KK
gravity theory [13{15]. For a CPO of conformal dimension
, the vev which is obtained
which is asymptotically AdS5
S5 [13, 14].
N = 6 supersymmetry and Uk(N )
U k(N ) gauge symmetry, where k is the ChernSimons
due to the existence of these discrete vacua.
SO(4)=Zk
obtained [22, 23]. See also [24{29] for related works.
with conformal dimension
motion on AdS4
S7 background in 11dimensional supergravity. In order to obtain the
When we are interested only in the CPO with conformal dimension
= 1, the linearized
equations are su cient. However, for CPOs with
2, one has to consider nonlinear
ground. Here we focus on the
S5
backobtain an exact holographic relation which is given by
for some speci c types of the LLM solutions.
1For the linearized equations of motion on AdS4
S7 background in de Donder gauge, see [30, 31]. It is
in the de Donder gauge is not straightforward.
the vevs of the operators in the case
renormalization to read the vevs of CPO with
the LLM solutions and compare the
eld theory and the gravity results. In section 6,
We also include
of equation (5.48).
Vacua of the mABJM theory and the LLM geometries
U(N ) k gauge group is a superconformal CS
M2branes on the C4=Zk orbifold
xed point [19]. One interesting feature of the ABJM
which break the global SU(4) symmetry to SU(2)
U(1). Solving the classical
section, we brie y review the correspondence.
Supersymmetric vacua of the mABJM theory
as follows
Y A = (Za; W ya);
Lbos =
Vbos = 0, is written as
ZaZyZb
ZbZyZa =
WaZbWb
WbZbWa = 0;
W yaWbW yb
W ybWbW ya =
ZbWbZa
ZaWbZb = 0;
Z0a =
W0ya =
(n+1) matrices,
GRVV matrices
(n) = BB
(n) = BB
where n = 0; 1;
1. The vacuum solutions are given by
remains to be supersymmetric [22].
LLM geometries and their droplet picture
The LLM solution with SO(2; 1)
SO(4) isometry in 11dimensional supergravity
and W ya are N
following two constraints,
The solution contains Nn rectangular matrices of the type Ma
(n) and N n0 rectangular
matri
N matrices, the occupation numbers, Nn and N n0, should satisfy the
N =
Nn + N n0
X Nn =
as the dual gravity theory [23].
The LLM geometry with Zk orbifold is given by
ds2 =
Gtt( dt2 + dw12 + dw22) + Gxx(dx~2 + dy~2) + G
ds2S3=Zk + G~~ds2S~3=Zk ;
are given by
where ds2S3=Zk and ds2S~3=Zk
are metrics of the two S3's with Zk orbifold and the warp factors
Gtt =
q 12 + Z 12=3
Z2 12=3
Gxx = @
Z2 12=3
Z 12=3
given by
f (x~; y~) = p1
0 =
i=1 2p(x~ x~i)2 + y~2
i=1 2p(x~ x~i)2 + y~2
?2 dZ with y~x~ = 1.
The corresponding 4form
eld strength is given by
F4 =
^ dt ^ dw1 ^ dw2 + 0
1 V d(y~2e2G) + h2e3G ?2 d(y~2e 2G) ^ d 3
1 V d(y~2e 2G)
h2e 3G ?2 d(y~2e2G) ^ d ~ 3;
system.2 The Zk quotient acts as
where d 3 =
(sin =8)d ^ d ^ d , d ~ 3 =
[23, 36]. The 4form
relations [37{39]
h2 =
14 d 2 + sin2 d 2 + (d + cos d )2 .
sin d + sin cos d ; 2 =
+ sin sin d ; 3 = d
+ cos d
with ranges of the angles, 0
e2G = 211 + Z
a value
12 denoted by
representation. See the
gure 1. Since the function Z(x~; 0) is
12 if x~ < x~1 and it is 12
x~F = x~1 + X(x~2i+1
starting at the
than k these parameters should satisfy the condition 0
k, which is the same
fNn; N n0g, the onetoone correspondence is given by
fln; ln0g () fNn; N n0g:
gure 1 for the parametrization of droplet picture and Young diagram.
Vevs of CPOs in mABJM theory
to AdS5
SO(4)=Zk
The Young diagram corresponding to the droplet picture (a).
theory in the large N limit were tested in a systematic way [13, 14].
theories are away from the UV
xed point and they preserve the full supersymmetry with
renormalization procedure.
In this section, we construct the CPO with
vacua of mABJM theory for general k in the large N limit.
CPOs in ABJM theory
The gauge invariant CPOs of conformal dimension
in the ABJM theory are given by
( ) = CA1; ;An
where A; B
= 1;
( )B1; ;Bn are symmetric in lower as well as upper indices
It is well known fact that the coe cients CA1; ;An
( )B1; ;Bn are identi ed with the similar
coe cients CI
cal harmonics;
In particular, the coe cients of the CPO with
the operator is given by
, we obtain
im = O
where h
theory, respectively, and
i0 denote the vevs of operators in the mABJM theory and the ABJM
( ) is an operator containing at least one Y^ A or Y^ yA. The N1
( =1) =
vevs of CPO in mABJM theory
Here we calculate the vevs of CPO with
expanded as
Y A = Y0A + Y^ A;
( )(Y A)im = O
For CPO with
( =1)
im =
N n0)n(n + 1):
nite k in the
large N limit.
KK reduction and gauge invariant modes
on AdS4
11dimensional gravity equations of motion
M(an), while W0ya contains N n0 of the matrix M(an), we have
Tr(Z0aZay0
W0yaWa0) =
N n0)Tr(Ma M(an))
(n)
N n0)n(n + 1);
2 gpqR =
2 gpqFrstuFrstu + 4FpstuFqstu ;
@p(eFpqrs) +
3The ABJM theory is dual to Mtheory on AdS4
We will extend our results to general k case eventually.
S =
16 G11
~p1p2p3q1 q4r1 r4 Cp1p2p3 Fq1 q4 Fr1 r4 ;
G11 =
where we used the index notation p; q; r;
= 0;
; 10, ~0123 10 =
1 is the LeviCivita
symbol. The 11dimensional Newton's gravitational constant is
equations of motion for the metric and the 4form
eld strength:
radius of S7.
= pjgAdS4 j ~
is the LeviCivita tensor for the AdS4 space, and L is the
where e
write the AdS4
g. Using the index notation ( ; ; ;
= 0;
S7 solution of the equations of motion in (4.3) as follows
ds2 =
dt2 + dw12 + dw22 + d 2 + L2ds2S7 ;
; and it is zero otherwise:
Fluctuations on AdS4
We consider a solution which is asymptotically AdS4
S7 so that we can write it as
gpq = gpq + hpq;
Cpqr = Cpqr + cpqr
Fpqrs = Fpqrs + fpqrs;
where hpq; cpqr; fpqrs represent deviations from the AdS4
S7 geometry and they become
following equations of motion for hpq and fpqrs up to linear order,
r r
r rphqr +r rqhpr
Rrshrs +rrrshrs
FrstuF rstuhpq 4gpqhrsF rtuvF stuv
+rar h a +rar h a (r r +rara)h
+rarbhab +(r ra +rar )h a (r r +rara)(h +hbb)
= 0;
r r ha + r rah
b b
+ r r hab + r rah b
= 0;
whereas the AdS4
S7 values of those objects are denoted by normal font symbols.
= 0;
combination which is de ned as
T(ab) =
(Tab + Tba)
where gab is a metric on S7. The notation [abc
indices, a; b; c;
, for instance,
] means antisymmetrization among
T[ab] =
r f abc + rdf dabc +
Expansion in S7 spherical harmonics
h (x; y) = hI1 (x)Y I1(y);
haa(x; y) = I1(x)Y I1(y);
(x; y) = s~I1 (x)Y I1(y);
= 0;
) = 0;
r f ab + rcf c ab = 0;
(x; y) = 4r[ sI1 ](x)Y I1 (y);
a(x; y) = 3r[ vI7 ](x)YaI7 (y)
f abc(x; y) = r tI35 (x)Y[aI3b5c](y)
3tI21 (x)r[aYbIc2]1 (y);
fabcd(x; y) = 4tI35 (x)r[aYbIc3d5](y);
c3 ! d (2). However, the 4form
eld strength is invariant under this transformation.
binations which are de ned as
t~I21 (x)
t~I35 (x):
s~I1 (x)
v~I7 (x) + r[ v~I7] (x);
and projecting onto the scalar harmonics Y I1 , we obtain
L2 hI1 + r r hI1 + r r hI1
6 I1 +
= 0;
r r . We have used the AdS4
in (4.20) gives the following equation for scalar elds
S7 solutions in (4.4) and the results of
8 I1 +
= 0:
raY I1 and gabYbI7 , respectively
5 I1 +
6 I1 +
1 I7 +
= 0;
= 0:
elements: gabY I1 , r
(aY b)I7 and Y (ab)I27 ,
r r hI1 = 0;
2r sI1 =
2r vI7 =
6 I1 +
5 I1 sI1 + hI1 +
5 I1 ;
2r v^I7 = 0;
following set of equations
r sI1 = r s^I1
= 0;
I35 = (I35+3)2
and we have used the relation
ra1 YaI23a53a4 =
to obtain the two equations in (4.34).
sI1 = 0;
r vI7 = 0;
tI21 = 0;
r vI21 = 0;
r vI7 = 0;
t~I35 = 0;
h^I1 = hI1
Since the only nonvanishing component of the 4form
eld strength is F
~hI1 = r
~tI27 = 0;
~vI7 = 2 (Iv7);
~vI7 = r
~sI1 = 2 (Is1);
~sI1 = r
~ I1 = 2 I1 (Is1):
^I1 =
v^I7 = vI7
where s~I1 = sI1
nontrivial equations in second part of (4.36) are
Then, we obtain the following results
a = I1 F
r[ ~sI1 ] =
= 0;
Gauge invariant
uctuations are
. Up to linear order, these gaugedependent
degrees of freedom transform as,
~hpq = (rp q + rq p);
~fpqrs =
parameter p(x; y) in terms of the spherical harmonics on S7 as
(x; y) = I1 (x)Y I1 (y);
gaugedependent coe cients of the spherical harmonics
linear order.
KK reduction
motion for physical modes.
The equations for spin zero
of motions for two gauge invariant scalar elds
14 ^I1 = 0;
5 I1 ^I1 = 0;
where the gauge invariant scalar elds are
^I1 = (18hI1
^I1 = ( I1
where we have introduced
(I1 + 6)(I1 + 12)
I1 = 0;
I1 = 0;
I1 =
I1 =
(I1 + 7) 18(I1
1) ^I1 + 7 ^I1
14(I1 + 3)
18(I1 + 7) ^I1 + 7 ^I1
14(I1 + 3)
three more elds which are already diagonal and gauge invariant
I27(I27 + 6)
tI27 = 0;
(I35 + 3)(I35
3) tI35 = 0;
where tI27
introduced in [13, 14]
SI = sI + JIJnJm tJn tJm + LIJnJm r tJn
s s
where s
I represent any of the 11dimensional
elds, and SI is the corresponding
a pseudotensor), we note that the rst three elds
last two elds T+I35 ; T I35 are pseudoscalar elds.
The equations for spin one
same. Therefore, the
elds I1 ; I1 ; tI27 ; tI+35 ; tI35 are the correct 4dimensional spinzero
elds at the linear order and are denoted as
I1 ; I1 ; T I27 ; T+I35 ; T I35 , respectively. Based
I1 ; I1 ; T I27 are scalar elds while the
u^I7 + 12
3 u^I7 = 0;
v^I7 = 0;
I7(I7+6) 1 , and we have introduced the following gauge invariant
combinawith v^I7 = v~I7 + r[ v~I7] :
I72 + 12I7 + 23
vI7 = 0;
uI7 = 0;
vI7 =
4(I7 + 3)
uI7 =
2(I7 + 7)v^I7 + u^I7
4(I7 + 3)
tI21 = 0;
where I7 =
vector mode
where tI21 is equivalent to tI21 in (4.19).
third one is a pseudovector eld.
The equations for spin two
following linear order equations for gauge invariant tensor elds
^I1 = 0;
^I1 = 0;
where we have introduced the following gauge invariant tensor elds
and we have de ned
We note that ^I1
in (4.44).
of traceless tensor modes
are the gauge invariant scalar elds de ned
where ^(I1 ) = ^I1
^I1 and ^(I1 ) = ^I1
^I1 . Then we introduce the transverse
( ) = 0,
and diagonalized linear equation,
30(I1 +2)(I1 +4) r( r )
( ) = 0;
where MI21 = I1(I1+6) 8
L2
T I21 , and one tower of spintwo mode HI1 .
order, HI1
) are the correct 4dimensional spin two KK modes.
ity yields, three towers of scalar modes
I1 ; I1 ; T I27 , two towers of pseudoscalar modes
Exact KK holography for LLM geometries
= 1, are
Asymptotic expansion of the LLM geometries
of the Legendre polynomials as follows
Z( ; ) =
V ( ; ) =
+ X [(n + 1) Pn+1 ( )
2 nPn ( ) + (n
1) Pn 1 ( )] Cn
have de ned [39]
= L4r~3 ,
Cn =
where A = kN
the parameters Cn's satisfy an identity:
where G
= L166G4xx .
ds2 =
dt2 + dw12 + dw22 + G
+ G ds2S3 + G~~ds2S~3 ; (5.4)
C12 = 2:
direction is well de ned. In a geometry which is asymptotically AdSd+1
the FG coordinate system is given by
X , the metric in
ds2 =
transformations
= (z; );
= (z; );
which should satisfy two conditions
Then we obtain
= 0:
ds2 =
dt2 + dw12 + dw22
+ g2 (z; ) d 2 + g3 (z; ) ds2S3 + g4 (z; ) ds2S~3 ;
g1 (z; ) = Gtt ( (z; ) ; (z; )) ;
g2 (z; ) = Gxx ( (z; ) ; (z; ))
g3 (z; ) = G
( (z; ) ; (z; )) ;
g4 (z; ) = G~~ ( (z; ); (z; )) :
tions (5.7) in the asymptotic region, we use the following ansatze,
(z; ) =
8C2C14 + 12C3C13 + 3 C22
4C2C3C1 + C23
4C32 + 3C2C4
60C2C14
100C3C13 + 9 11C22 + 5C4 C12
12C2C3C1
27C23 + 56C32
45C2C4
60C2C14
100C3C13 + 9 11C22 + 5C4 C12
12C2C3C1
27C23 + 56C32
45C2C4 (1
where the ai and bi are determined from (5.7),
a1 =
a2 =
b1 =
b2 =
for I1 = 0; 2; 4;
hi0j =
hi2j =
hi4j =
h0 =
h2 =
h4 =
0 17C16 51C2C14 28C3C13 +72C22C12 +42C2C3C1 45C23 7C32 +O
L2 0 2C13 3C2C1 +C3 +O
0 28C16 84C2C14 +28C3C13 +9 7C22 15C4 C12 +228C2C3C1
135C23 128C32 +135C2C4 +O
gijhiIj1 = 4Lz22 ijhiIj1,
51C2C14
28C3C13 + 72C22C12 + 42C2C3C1
3C2C1 + C3 + O( 03);
84C2C14 + 28C3C13 + 9 7C22
15C4 C12 + 228C2C3C1
128C32 + 135C2C4 + O( 04):
The values of the scalars sI1 are obtained from
pendix B,
0 =
4 =
by expanding,
14C16 42C2C14 +4C3C13 +39C22C12 6C2C3C1 10C23 +C32
124C16 +372C2C14 124C3C13 +9 15C4 31C22 C12 84C2C3C1
+104C32 +135 C23 C2C4
r r h(ab) = sI1rarbr(arb)Y I1 = 6sI1 I1
a b
Then using appendix B we obtain
s0 =
s2 =
s4 =
u~0 =
u~2 =
u~4 =
21C2C14 + 12C22C12 + 6 2C12
3C2 C3C1 + 5C23 + 3C32
3C2C1 + C3 + O( 03);
12C2C14 + 4C3C13 + 9 C22 + 15C4 C12
276C2C3C1
+ 136C32 + 135 C23
In order to read the values of the scalars I1, we take the trace of hab,
haa = I1Y I1 = gab gab
gab = gabgab
14C16 42C2C14 +4C3C13 +39C22C12 6C2C3C1 10C23 +C32 +O( 04); (5.22)
2C13 3C2C1 +C3 +O( 03);
124C16 +372C2C14 124C3C13 +9 15C4 31C22 C12 84C2C3C1
+104C32 +135 C23 C2C4
To determine the graviton mode, in addition to h
we also need the values of the
tensor elds u~
de ned in (4.58). We can rewrite the de nition in (4.58)as
u~I1 I1Y I1 =
a =
and noting that, for the LLM geometry, the only non zero f
following results
I1 Y I1 . Using this into (5.30)
a is ftw1w2 , we obtain the
u~iIj1 I1 Y I1 = 3!
u~Iz1z I1 Y I1 =
@ @zFtw1w2 + 3 @ Ftw1w2
@zFtw1w2 + 3 z Ftw1w2
0 = 0, we can not read u~0 .
3C1C2 + C3 + O( 03)
12C2C14 + 4C3C13 + 9 C22 + 15C4 C12
276C2C3C1
u~i2j =
u~i4j =
u~z2z =
u~z4z = p10
discussed in subsection 4.5.
totic expansions for I1 and
3C1C2 + C3:
+ 135C23 + 136C32
135C2C4 + O( 04)
3C1C2 + C3 + O( 03);
12C2C14 + 4C3C13 + 9 C22 + 15C4 C12
276C2C3C1
+ 135C23 + 136C32
135C2C4 + O( 04):
because for the LLM geometry h a is zero.
only nonvanishing
0 = O( 02);
0 = O( 02);
2 = O( 03);
2 =
4 = O( 02);
4 = O( 02);
on the gravity side is related to the conformal dimension
of the dual gauge invariant
operator by
m2RA2dSd+1 =
m2L2 = 2 (2
spin zero elds are as follows:
For the scalar eld
I1 we have
6) = 2 (2
fI1 = 2; 4; 6;
(I1 + 12)(I1 + 6) = 2 (2
fI1 = 0; 2; 4;
(I27 + 6)I27 = 2 (2
fI27 = 2; 4; 6
For the pseudoscalar eld T I35 we have
For the pseudoscalar eld T+I35 we have
(I35 + 3)(I35
3) = 2 (2
(I35 + 9)(I35 + 3) = 2 (2
For the scalar eld
I1 we have
For the scalar eld T I27 we have
= 1; 2;
I1 and T I27 are also not the candidates. The only scalar
I1 , with I1 =
subsection 4.5.2.
I1 + 12
I27 + 6
I35 + 3
I35 + 9
( ) = O( 03);
( ) = O( 04):
Comparison with eld theory results
conformal dimension
is determined by the coe cient
in the asymptotic
expansion of a dual gauge invariant scalar elds
on the gravity side, i.e.
im = p N ( )
11dimensional supergravity, and
is de ned as
dual relation for the M2brane theory [19, 46, 47].
trivial asymptotic expansions at linear order in
0. At quadratic order or higher, more of
We have obtained the asymptotic expansion of the scalar eld
2 in (5.36) while the vev
In order to x the normalization factor N, we use the identity
hO(1)im = p N (1) =
ln0)i =
N 3=2
4 2 n=0
N n0)i =
72N 0 X1 hn(n + 1)(ln
Now recalling that
with conformal dimension
= 1 is given by
1442 . Therefore, the vev of the CPO
im =
large N limit [47].
dual relation is then given by
im =
3 =
= 1
CPO with
0 expansion of the
because the theory is supersymmetric.
CPO with
the identity (5.3) while C3 is determined by the vevs of CPO with
= 1. Therefore, the
vevs of CPO with
and k (
these issues for future study [32].
with those of mABJM theory on S3.
Acknowledgments
and NRF2014R1A1A2059761 (O.K.).
Spherical harmonics on S7
The spherical harmonics on S7 are de ned as follows
r2Y I1 =
I1 Y I1 =
I7 YaI7 =
I21 Y[aI2b1] =
I35 Y[aI3b5c] =
2)Y(Ia2b7);
3)Y[aI3b5c];
raY I7 = raY(Ia2b7) = raY[aI2b1] = raY[aI3b5c] = 0;
a
gabY(Ia2b7) = gabY[aI2b1] = gabY[aI3b5c] = 0:
antiSymmetric 2
antiSymmetric 3
Now consider
1 Z
Scalar spherical harmonics
The scalar spherical harmonics on S7 are the restriction of
Y I1 =
; 8) are the Cartesian coordinates of IR8 and
the coe cients CI1
i1 iI1 are totally symmetric and traceless. In order to evaluate integrals
1 Z
xi2m =
2m 1(m + 3)!
all possible pairing ;
Y I1 Y J1 =
!7LI1+J1 S7 i1 iI1 CjJ11 jJ1 xi1
CI1
only when I1 = J1
Y I1 Y J1 =
!7L2I1 S7 i1 iI1 CjJ11; jJ1 xi1
CI1
2I1 1(I1 + 3)! CiI11; iI1 CjJ11 jJ1
all possible pairing :
total number of such terms is I1!. Therefore we get
Y I1 Y J1 =
2I1 1(I1 + 3)! hCI1 CJ1 i;
where hCI1 CJ1 i = CI1
i1 iI1 . Actually, we normalize the scalar harmonics such that
by parts.
Vector spherical harmonics
Consider a vector eld in IR8
p = 1;
VagabVb =
Vpe^pagabe^bqVq:
pq = pq
VagabVb =
npnq)Vq:
Now we can make the following replacement
Recalling that xpY I7 = 0, the second piece is zero. In general we can drop the second
p
1 Z
gabYaI7 YbJ7 =
npnq)VpI;7i1 iI7 VqJ;j71 jI7 xi1
xjI7 : (A.14)
with V I7 traceless and totally symmetric in the i1;
p
; iI7 indices. It also satisfy xpY I7 =
p
Y I7 = e^pY I7 =
a a p
a = 1;
e^pa =
harmonics we use the following procedure. Lets consider
in (A.8) we can write
Therefore we have
1 Z
gabYaI7 YbJ7 =
!7L2I7 S7 VpI;7i1 iI7 VpJ;j71 jI7 xi1
V I7 V J7 :
spherical harmonics.
5We will use the index notation where p; q;
are the IR8 indices and a; b;
are the S7 indices.
Scalar spherical harmonics on S7 with SO(4)
which satisfy the following harmonic equation
I1(I1 + 6)
Y I1 =
I1(I1 + 6)
Y I1 = 0;
ds2S7 = L
2 d 2 + cos2 ds2S3 + sin2 ds2S~3 :
of S7 is related to the
= cos . Then we have
coordinate of LLM as
ds2S7 = L
Imposing the SO(4)
SO(4) symmetry, the spherical harmonics depend only on the
coordinate, which implies
), the second solution is a polynomial
and the rst few terms are
which gives
I1(I1 + 6)
=)
I1(I1 + 6)
Y I1 = 0
Y I1 ( ) = 0:
2 =
Y 2 =
= 2 2 Y 2;
Y I1 ( ) = N I1 2F1
Y I1 ( ) = N I1 2F1
For I1 = 4i, (i = 0; 1; 2;
), the rst solution is a polynomial and the rst few terms are
Y 0 = 1;
Y 4 =
For later convenience lets invert these relation and write the following
In subsection 3.1, we have stated the fact that the coe cients CI1
scalar spherical harmonics in (A.2) are related to the coe cients CB1
with conformal dimension
the solution in (A.22) is obtained by imposing the SO(4)
SO(4) symmetry, we rewrite
the scalar harmonics in (A.2) in a form that manifests this symmetry
i1 iI1 , which de nes the
( )A1 An of the CPOs
with the R8 coordinates restricted to S7 are written as follows
Y 2 =
Ci2j xixj +
x2 = L
x4 = L
x6 = L
; x8 = L
: (A.25)
x1 = L
x3 =
x5 = L
x7 =
ment gives
gives C121 = 2p1 2 .
From (A.22) we notice that Y 2 depends only on the
coordinate of S7. Therefore, the
C121 = C222 = C323 = C424;
C525 = C626 = C727 = C828;
Ci2j = 0; for i 6= j:
C525 while the orthonormality condition
spherical harmonics in terms of C4 coordinates as
Y 2 =
where the C4 coordinates are given by
Comparing (A.24) and (A.27) we obtain
y1 = x1 + ix2;
y2 = x3 + ix4;
y3 = x5 + ix6;
y4 = x7 + ix8:
C(1)1 =
C(1)3 =
C121 + C222 =
C525 + C626 =
C(1)2 =
C(1)4 =
C323 + C424 =
C727 + C828 =
Hence the CPO of conformal dimension
= 1 is given by (3.3).
Asymptotic expansions
as well as for the various components of the 4form
eld strength Fpqrs in (2.11) using the
but we keep only up to
20 for our purpose. Applying the ansatze (5.10) to the de ning
the warp factors in FG coordinate system,
( 0z)2 28C16 84C2C14+28C3C13+9 7C22 15C4 C12+228C2C3C1
9C23+20C32 27C2C4
135C23 128C32+135C2C4 2+
12C16+36C2C14+4C3C13 3 13C22+3C4 C12+12C2C3C1
+7C23 8C32+9C2C4 +
(2C13 3C2C1+C3)(1+ )
(2C13 3C2C1+C3)(1 )
3C23+8C32 9C2C4
63C23 8C32+15C2C4
135C23 104C32+135C2C4 2+
same manner
Ftw1w2z (z; ) =
2 2C13 3C2C1 +C3
Ftw1w2 (z; ) =
2C13 3C2C1 +C3
16C16 48C2C14 +28C3C13 +27 C22 C4 C12 +12C2C3C1
15C23 20C32 +27C2C4
4C16 12C2C14 +4C3C13 +9 C22 +15C4 C12 276C2C3C1
+135C23 +136C32 135C2C4
4C16 12C2C14 +4C3C13 +9 C22 +15C4 C12 276C2C3C1
+135C23 +136C32 135C2C4
z (z; ) =
(1+ )2 sin
(z; ) =
(1+ ) sin
F~~ ~z (z; ) =
F~~ ~ (z; ) =
2C13 3C2C1 +C3 (1+6 )+
2C13 3C2C1 +C3 (4+9 )+
2C13 3C2C1 +C3 (1 6 )+
2C13 3C2C1 +C3 (4 9 )+
Proof of (5.48)
C1 = p
we rewrite C1 as
C1 = p
m2i) =
x~1 + X(x~2i+1
x~2i) :
3 =
region. See gure 1. Then we can write 3 as
N 3=2 3 =
zero, we can rewrite the above equation as
N 3=2 3 = X ( 1)i+1
N 3=2
Using the relation
we obtain
n (n + 1) =
N 3=2
3 = 4
2j jmij 1 2NB+1 jmij 13
i=1 n=0
i=2j+1 n=0
the second double summation, we write
N 3=2
3 = 4
2j mi 1 2NB+1 mi 13
i=1 n=0
i=2j+1 n=0
Next lets expand the summations over i, which gives
2m2j+1 1 m2j+2 1 m2j+3 1
5 n (n + 1) : (C.9)
the second square bracket unpaired, to obtain
N 3=2
3 =
n= m2
n= m4
n=m2j+2
5 n (n + 1)
n= m2j
n=m2NB
5 n (n + 1) :
we obtain
N 3=2
3 =
n(n + 1)(ln
which is what we have in (5.48).
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