Degression of Any Symmetric Spin in anti-de Sitter Space

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10 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Degression of Any Symmetric Spin in anti-de Sitter Space
A.Artsukevich
Lebedev Physical Institute
Joint Institute for Nuclear Research,Dubna,July 19,2011
based on A.A.and M.A.Vasiliev (to appear soon)
Introduction
A conventional way to consider a higher-dimensional theory in lower dimension
is by the Kaluza-Klein mechanism going from R
d+k
to R
d
(S
1
)
k
as well as its
further generalizations to less trivial compact manifolds.It may be however
interesting analyze similar procedure starting from AdS space.
The procedure is divided in the steps
I
Foliation of AdS geometry
I
Splitting of the world indices
I
Splitting of the ber indices
I
Choose the convenient variables
I
Fourier expansion
Foliation of AdS geometry
The description of AdS
d+1
geometry
b
d
b


AB
+
b


A
C
^
b


CB
= 0;
b


AB
= 
b


BA
;A;B;:::= 0;:::;d +1:
The covariant denition of the frame eld and Lorentz connection
V
A
V
A
= 1;
b
H
A
=
b
DV
A
;
b


AB
L
=
b


AB
+V
A
b
H
B
V
B
b
H
A
:
The covariant splitting of ber indices
U
A
U
A
= 1;V
A
U
A
= 0:
The description of AdS
d
geometry
d!
AB
+!
A
C
^!
CB
= 0:
DU
B
= 0;h
A
= DV
A
;!
AB
L
=!
AB
+V
A
h
B
V
B
h
A
:
The splitting of the world indices
b
d = d +d'@
'
;
b


AB
=

AB
+d'
AB
;


AB
=!
AB
+(1 cos
1
('))

V
A
h
B
V
B
h
A

+tan(')

U
A
h
B
U
B
h
A

;

AB
= cos
1
(')

U
A
V
B
U
B
V
A

;'2 [0;2):
Symmetric Bosonic Massless Field in AdS
d+1
c
W
A(s1);B(s1)
1-form carrying the representation of o(d;2)
b
R
A(s1);B(s1)
=
b
D
c
W
A(s1);B(s1)
;

c
W
A(s1);B(s1)
=
b
D
A(s1);B(s1)
:
On-Mass-Shell theorem
b
R
A(s1);B(s1)
= s
2
b
H
C
^
b
H
D
B
A(s1)C;B(s1)D
;

A(2)
B
A(s);B(s)
= 0;V
B
B
A(s);B(s)
= 0:
Weyl module
b
D
L
B
A(i +s);B(s)
= (i +s +1)
b
H
C

B
A(i +s)C;B(s)
+
s
i +2
B
A(i +s)B;B(s1)C

+
(i +1)(d +i +2s 3)
d +2i +2s 1
h
b
H
A
B
A(i +s1);B(s)

i +s 1
d +2i +2s 3

A(2)
L
b
H
C
B
A(i +s2)C;B(s)

s
d +i +2s 3

AB
L
b
H
C
B
A(i +s1);B(s1)C
+
s(i +s 1)
(d +2i +2s 3)(d +i +2s 3)

A(2)
L
b
H
C
B
A(i +s2)B;B(s1)C

;i = 0;1;:::
Cell Operator Algebra
Consider the generating function T
(k;l )
= T
A(k);B(l )
y
A
1
:::y
A
k
p
B
1
:::p
B
l
,
k;l = 0;1;:::,where T
A(k);B(l )
irrep of o(d;2) and
X
A
b

+
1 A
T
(k;l )
2 T
(k+1;l )
;X
A
b


1 A
T
(k;l )
2 T
(k1;l )
;
X
A
b

+
2 A
T
(k;l )
2 T
(k;l +1)
;X
A
b


2 A
T
(k;l )
2 T
(k;l 1)
:
b


1 A
=
@
@y
A
+
1
^x +2
p
B
@
@y
B
@
@p
A
;
b


2 A
=
@
@p
A
;
^x = ^y ^p,^y = y
A @
@y
A
,^p = p
A @
@p
A
.The explicit form of
b

+
1 A
and
b

+
2 A
is more
complicated.
The formulation of symmetric massless eld in AdS
d+1
b
R
(s1;s1)
=
b
D
c
W
(s1;s1)
;
c
W
(s1;s1)
=
b
D
(s1;s1)
;
b
R
(s1;s1)
=
b
H
B
^
b
H
A
b

L
1 B
b

L
2 A
B
(s;s)
;
b
D
L
B
(s+i;s)
=
b
H
A
b

L
1 A
B
(i +s+1;s)
+
(^x +1)(^z 3)
^z +^x 1
b
H
A
b

+L
1 A
B
(i +s1;s)
;
i = 0;1;:::,^z = d +^y +^p.
Splitting space indices
We single out the components of forms along the (d+1)th direction
b
R
(s1;s1)
= R
(s1;s1)
d'^T
(s1;s1)
;
c
W
(s1;s1)
= W
(s1;s1)
+d'C
(s1;s1)
:
R
(s1;s1)
= DW
(s1;s1)
;T
(s1;s1)
= DC
(s1;s1)

b
D
'
W
(s1;s1)
:
The relation of the AdS
d+1
covariant derivative to that of AdS
d
b
DX
A
= (D+d'
b
D
'
)M
A
B
e
X
B
= M
A
B
(D +d'@
'
)
e
X
B
;
R
A(s1);B(s1)
= M
A
1
A
0
1
:::M
B
s1
B
0
s1
e
R
A
0
(s1);B
0
(s1)
and the same formulas for T,W,C,B.
On mass shell theorem
e
R
(s1;s1)
= D
f
W
(s1;s1)
= cos
2
(')h
A
^h
B
b


1 A
b


2 B
e
B
(s;s)
;
e
T
(s1;s1)
= D
e
C
(s1;s1)
@
'
f
W
(s1;s1)
= cos
2
(')h
A
h
b


1 A
b


2
(
e
U) 
b


1
(
e
U)
b


2 A
i
e
B
(s;s)
;
where
e
U
A
= U
B
M
BA
= cos
1
(')U
A
+tan(')V
A
.
Splitting ber indices
The branching rule for o(d;2)#o(d 1;2) in the case of two-row rectangular
Young tableaux
s-1
s-1
)
s1
L
t=0
s-1
t
The vector U
A
allows to make the branching in the covariant way
e
R
(s1;s1)
=
s
X
t=1
(
b

+
2
(U))
st
er
(s1;t1)
;
f
W
(s1;s1)
=
s
X
t=1
(
b

+
2
(U))
st
ew
(s1;t1)
;
e
T
(s1;s1)
=
s
X
t=1
(
b

+
2
(U))
st
e
t
(s1;t1)
;
e
C
(s1;s1)
=
s
X
t=1
(
b

+
2
(U))
st
ec
(s1;t1)
:
The decomposition of Weyl tensor
The decomposition of Weyl tensor in accordance with the branching rule
e
B
(s;s)
=
s
X
t=0
(
b

+
2
(U))
st
e
b
(s;t)
:
e
B
(s;s)
carries o(d;1) irrep.)
b


2
e
V
e
B
(s;s)
= 0:It gives for
e
b
(s;t)


1 V
e
b
(s;t)
= 0;
(s t)(s t +1)
+
2
(V)
e
b
(s;t1)
sin(')(s t)(^z ^x +s t 3)
e
b
(s;t)
+
(^z ^x +s t 3)(^z ^x +s t 2)
(^z ^x 1)


2
(V)
e
b
(s;t+1)
= 0;t = 0;:::;s 1
e
b
(s;t)
are not irreps of o(d 1;1).We introduce b
(s;t)
with


1 V
b
(s;t)
= 

2 V
b
(s;t)
= 0 and
e
b
(s;t)
=
t
X
k=0
(
+
2
(V))
tk

t
k
(^x;^z)b
(s;k)
;
where
t
k
(^x;^z) = N
t
k
(^x;^z)C
^z^x3
2
tk
(sin(')).
The choose of convenient variables
The equations in AdS
d
take the form of mixture for dierent spins
er
(s1;t1)
= h
A
^h
B
(
t
X
k=0
cos
2
(')


+
2
(V)

tk
(:::)
t
k
(^x;^z +2)

1 A


2 B

t2
X
k=0


+
2
(V)

tk2
(:::)
t2
k+2
(^x;^z)
+
2 A


1 B
)
b
(s;k)
;
e
t
(s1;t)
= cos
1
(')
t
X
k=0
(
+
2
(V))
tk
(:::)
t
k
(^x;^z)h
A


1 A
b
(s;k)
:
We require that in new variables the equations take the form
r
(s1;k1)
=
cos
2
(')
^z ^x +2s 2k 2
h
A
^h
B


1 A


2 B
b
(s;k)
;
t
(s1;k)
= 
cos
1
(')(^x +2)(^z ^x 1)
(^x +1)(^z ^x 2)(^z ^x +2s 2k 4)
h
A


1 A
b
(s;k)
:
There is the reversible transformation (r;t),(er;
e
t).
Fourier expansion
w
(s1;k1)
=
1
X
n=0
w
(s1;k1)
n
P
(k)
n
(');c
(s1;k1)
=
1
X
n=0
c
(s1;k1)
n
cos(')P
(k1)
n
(');
r
(s1;k1)
=
1
X
n=0
r
(s1;k1)
n
P
(k)
n
(');t
(s1;k1)
=
1
X
n=0
t
(s1;k1)
n
cos(')P
(k1)
n
(');
b
(s;k)
=
1
X
n=0
b
(s;k)
n
cos
2
(')P
(k)
n
('):
The conditions for P
(k)
n
(') follow form the curvatures
r
(s1;k1)
=:::+(:::) cos(') [@
'
+(d +2k 6) tan(')] h
A

+
2 A
^w
(s1;k2)
;
t
(s1;k)
=:::+(:::)@
'
w
(s1;k)
:
The equations
@
2
'
P
(k)
n
+(d +2k 5) tan(')@
'
P
(k)
n
+(
(k)
n
)
2
P
(k)
n
= 0;k = 0;1;:::;s:
The condition of normalizability of P
(k)
n
gives the spectrum
(
(k)
n
)
2
= (n +1)(n +d +2k 4):
On mass shell theorem and Weyl module
On mass shell theorem
r
(s1;k1)
n
=
1
d +2s 4
h
A
^h
B


1 A


2 B
b
(s;k)
n
;
t
(s1;k)
n
= 
(s k +1)(d +2k 1)
(s k)(d +2k 2)(d +2s 4)
h
A


1 A
b
(s;k)
n
:
Weyl module we deduce from the above equations
D
L
b
(s+i;k)
n
= h
A


1 A
b
(s+i +1;k)
n
+
i
n
(^x;^z)h
A

+L
1 A
b
(s+i 1;k)
n
+
i
n
(^x;^z)h
A


2 A
b
(s+i;k+1)
n1
+
i
n
(^x;^z)h
A

+L
2 A
b
(s+i;k1)
n+1
;
for k = 0;1;:::;s and i = 0;1;:::
The curvature
r
(s1;k1)
n
= Dw
(s1;k1)
n
+
(s k +1)(d +2k 3)
(k)
n
(d +2k 6)(d +s +k 2)
h
A

+
2 A
^w
(s1;k2)
n+1
;
t
(s1;k)
n
= Dc
(s1;k)
n
+
(s k +1)(d +2k 1)
(k)
n
(d +2k 2)(d +s +k 3)
h
A

+
2 A
c
(s1;k1)
n+1
+
(d +s +k 2)
(k+1)
n1
(s k)(d +2k 2)
w
(s1;k)
n1

d +2k 1
d +2k 2

+
2
(V)w
(s1;k1)
n

(s k +1)(d +2k 3)(d +2k 1)
(k)
n
(d +2k 6)(d +2k 4)(d +2k 2)(d +s +k 3)
(
+
2
(V))
2
w
(s1;k2)
n+1
:
They are invariant under gauge transformations
w
(s1;k1)
n
= D
(s1;k1)
n
+
(s k +1)(d +2k 3)
(k)
n
(d +2k 6)(d +s +k 3)
h
A

+
2 A

(s1;k2)
n+1
;
c
(s1;k)
n
= 
(d +s +k 2)
(k+1)
n1
(s k)(d +2k 2)

(s1;k)
n1
+
d +2k 1
d +2k 2

+
2
(V)
(s1;k1)
n
+
(s k +1)(d +2k 3)(d +2k 1)
(k)
n
(d +2k 6)(d +2k 4)(d +2k 2)(d +s +k 3)
(
+
2
(V))
2

(s1;k2)
n+1
: