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M. Sakaguchi 1
Penrose Limit and AdS Superalgebras
阪口 真(KEK) 2002/03/19 KEK理論研究会2002
based on hep-th/0202190
with
M.Hatsuda(KEK) and K.Kamimura(Toho Univ.)
1. Introduction
2. pp-wave background
3. Penrose limit of AdS5 � S5 background
4. Penrose limit of Super-AdS5 � S5 algebra
5. brane actions
6. Conclusion
M. Sakaguchi 2
1 Introduction
Maximally supersymmetric 11-dimensional backgrounds.
� at Minkowski (and their toroidal compacti�cations)
� AdS4 � S7
� AdS7 � S4
� pp-wave background Kowalski-Glikman '84
Figueroa-O'Farrill and Papadopoulos '0105
Maximally supersymmetric IIB backgrounds.
� at Minkowski (and their toroidal compacti�cations)
� AdS5 � S5
� pp-wave background Blau, Figueroa-O'Farrill, Hull and Papadopoulos '0110
M. Sakaguchi 3
� Superstrings on plane-wave RR-background Metsaev '0112
exactly solvable Metsaev and Tseytlin '0202
� pp-wave background as a Penrose limit [Penrose '76]
of supergravity background [G�uven '00]
of AdS�S background Blau, Figueroa-O'Farrill, Hull and Papadopoulos '0201
Blau, Figueroa-O'Farrill and Papadopoulos '0202
Cveti�c, L�u and Pope '0203
� AdS/CFT (scaling corresponding to the Penrose limit)
massive string modes and gauge invariant operators
Berenstein, Maldasena and Nastase '0202
Itzhaki, Klevanov and Mukhi '0202
Gomis and Ooguri '0202
Pando Zayas and Sonnenschein '0202
Orbifolded Kim, Pankiewicz, Rey and Theisen '0203
Takayanagi and Terashima '0203
Floratos and Kehagias '0203
Compacti�ed Michelson '0203
M. Sakaguchi 4
super-pp-wave algebra from super AdS�S algebra
Hatsuda, Kamimura and MS '0202
Maximally supersymmetric backgrounds
AdS � S
. &
pp-wave �! at
+
Spacetime Superalgebras
super-AdS � S algebra
. &
super-pp-wave algebra �! super-Poincar�e algebra
M. Sakaguchi 5
2 pp-wave background
Super-pp-wave background (IIB)
ds2 = 2dx�dx+ � 4�28X
m=1(xm)2(dx�)2 +
8Xm=1
(dxm)2;
F5 = �dx�(dx1dx2dx3dx4 + dx5dx6dx7dx8)
particle mechanics
L =
12e[2 _x+ _x� � 4�2(xm)2( _x�)2 + ( _xm)2]
) H =1
2[2p�
+ (pm)2 + 2�2(xm)2]
harmonic oscillator in transvers directions
M. Sakaguchi 6
isometry algebra
pp-wave algebra G0 = go (so(4)� so(4))
g : [e�
; em] = e�m
; [e�; e�m
] = �4�2em; [e�m
; en] = �4�2�mne+;
[Mmn; ep] = 2�npem; [Mmn; e�
p] = 2�npe�
m;
so(4)� so(4) : [Mij;Mkl] = 4�jkMil; [Mi0j0;Mk0l0] = 4�j0k0Mi0l0:
where i = 1; 2; 3; 4, i0 = 5; 6; 7; 8 and m = (i; i0)
� metric on M = G=K (group G of g and subgroup K of k = fe�i g )
� = ex+e+ex�e�exmem ) ��1d� = ��e� + !me�m (� = +;�;m)
) ds2 = �������;
� dimension of the algebra 1 + 1 + 8 + 8 + 6 + 6 = 30
cf) for AdS5 � S5, the dimension is 5 + 5 + 10 + 10 = 30
M. Sakaguchi 7
� Maximally supersymmetric (M = +;�; 1; :::; 8)
DM� = rM� +
i192e�FML1���L4�L1���L4� = 0 8 32 �
32 Majorana supercharges Q =�
Q1
Q2�
Super-pp-wave algebra (IIB) G0 with
[e�; Q] = �Q(I + J) i�2; [ei; Q] = ��Q�i�+I i�2;
[ei0; Q] = �Q�i0�+J i�2; [e�m; Q] = 2�2Q�m�+ 11;
[Mmn; Q] =1
2Q�mn 11
fQ;Qg = �C�+ 11e+ � C�� 11e� � C�m 11em
�1
2�C�iI i�2e�
i +
12�C�i
0J i�2e�
i0
��C�ij�+I i�2Mij + �C�i0j0
�+J i�2Mi0j0
where I = �1234, J = �5678
M. Sakaguchi 8
3 Penrose Limit of AdS5 � S5 background
metric
AdS5 � S5 metric
ds2 = R2"
�d�2 + sin2 � (dr2
1 + r2+ r2d32) + d 2
1 + sin2 1d42
#
where d32 = d3(�1; �2; �3)2 and d42 = d4( 2; 3; 4; 5)2
let
u = 1 + �; v = 1 � �; r = sinh�
thends2 = R2
�dudv + sin2u� v
2
(d�2 + sinh2�d32) + sin2u + v
2
d42�
M. Sakaguchi 9
Penrose limit (along the null geodesic parametrized by u)
scale
ds2 = 2ds2; v = 2v; u = u; � = �;
i = i; (i = 2; 3; 4; 5); �i = �i; (i = 1; 2; 3):
then take ! 0
d�2 + sinh2�d32 ! (d�2 + �d32) � ds2(E 4); fy1; :::; y4g
d42 ! (d 22
+ d 32
+ d 42
+ d 52
) � ds2(E 4); fy5; :::; y8g
ds2 = R22
4dudv + sin2u
2
8Xm=1
(dym)23
5
let x� = Ru2; x+ = Rh
v � 14sin u(ym)2i
; xm = Rym sin u2
then
ds2 = 2dx�dx+ � 4�2(xm)2(dx�)2 + (dxm)2:
M. Sakaguchi 10
5-form ux F5
RR- ux F5 =
12R
�dvol(AdS5) + dvol(S5)�
ds2 � ������� ) F5 =
12R
��0�1�2�3�4 + �5�6�7�8�9�
F5 �
14F5 =
14(R5 sin4 � sinh3 � cos2 �1 cos�2d�d�d�3d�4d�5
+R5 sin4 1 cos3 2 cos2 3 cos 4d 1d 2d 3d 4d 5)
! �dx�(dx1dx2dx3dx4 + dx5dx6dx7dx8)
Super-pp-wave background (IIB) has been derived from AdS5�S5 back-
ground.
M. Sakaguchi 11
4 Penrose Limit of Super-AdS5 � S5 algebra
Super-AdS5 � S5 algebra [Metsaev-Tseytlin]
[Pa; Pb] = �2Mab; [Pa0; Pb0] = ��2Ma0b0;
[Pa;Mbc] = �abPc � �acPb; [Pa0;Mb0c0] = �a0b0Pc0 � �a0c0Pb0;
[Mab;Mcd] = �bcMad + 3-terms; [Ma0b0;Mc0d0] = �b0c0Ma0d0 + 3-terms;
[QI ; Pa] = �i
2QJ(i�2)JI a; [QI ; Pa0] = ��1
2QJ(i�2)JI a0;
[QI ;Mab] = �1
2QI ab; [QI ;Ma0b0] = �1
2QI a0b0;
fQ��0I ; Q��0Jg = �2i(C a)�� C 0�0�0(11)IJPa + 2C�� (C 0 a0
)�0�0(11)IJPa0
+ �(C ab)�� C 0�0�0(i�2)IJMab + �C�� (C 0 a0b0
)�0�0(i�2)IJMa0b0
M. Sakaguchi 12
4.1 bosonic part
Let P� = 1p2(P9 � P0); P �i = Mi0; P �i0= Mi09;
Scale Pm ! 1Pm; P �m ! 1P �m; P+ ! 1
2P+;
[Pi; P+] =�22
p2P �i ; [Pi0; P+] = ��22
p2P �i0 ;
[P �i ; P+] = �2p
2Pi; [P �i0 ; P+] =2p
2Pi0;
[Pm; P�] = � �2p2P �m; [P �m; P�] =1p
2Pm;
[Pi; Pj] = �22Mij; [Pi0; Pj0] = ��22Mi0j0;
[P �i ; P�
j ] = 2Mij; [P �i0 ; P�
j0] = �2Mi0j0;
[P �i ; Pj] = �1p
2�ij(P+�2P�); [P �i0 ; Pj0] = � 1p
2�i0j0(P++2P�);
oso(4)� so(4)
M. Sakaguchi 13
Two limits
� �! 0 as at limit
� ! 0 as Penrose limit
the pp-wave algebra under
Pm ! 12p
2em; P �m ! 1
4p
2�2e�m; P�! 1
2p
2e�
M. Sakaguchi 14
4.2 fermionic part
De�ning 9+1 dimensional gamma matrices
�a = a 1 �1
�a0
= 1 a0 �2
C = C C 0 i�2
chiral projection
h� =1
2(1� �11); �11 = �0 � � � �9 = 11 11 �3; Q = Qh+
fQ;Qg = �2iC a C 0 11Pa + 2C C 0 a0 11Pa0 + � � �
C�ah+ = C a C 0 �3h+ = C a C 0
C�a0h+ = iC C 0 a
0 11h+ = iC C 0 a0
= �2iC�� 11P� + � � �
= �2iC�+ 11P+ � 2iC�� 11P� � 2iC�m 11Pm + � � �
M. Sakaguchi 15
Light cone projection
P� =1
2����; �� =
1p2(�9 � �0); Q� = QP�
C��P� = CP��� = eP�C��; C��P� = 0
fQ�; Q�g = �2iC�� 11P� + � � �
fQ+; Q�g = �2iC�m 11Pm + � � �
M. Sakaguchi 16
Scale Q+ ! 1Q+; Q� ! Q�
[P+; Q�] = ��2
2p
2Q�(I � J) i�2; [P�; Q�] = +
�2p
2Q�(I + J) i�2;
[Pi; Q+] = +�2
2p
2Q��i��I i�2; [Pi0; Q+] = +�2
2p
2Q��i0��J i�2;
[Pi; Q�] = � �2p
2Q+�i�+I i�2; [Pi0; Q�] = +
�2p
2Q+�i0�+J i�2;
[P �i ; Q+] = � 2
2p
2Q��i�� 11; [P �i0; Q+] =
2
2p
2Q��i0�� 11;
[P �i ; Q�] = +
12p
2Q+�i�+ 11; [P �i0; Q�] =
12p
2Q+�i0�+ 11;
[Mij; Q�] =1
2Q��ij 11; [Mi0j0; Q�] =1
2Q��i0j0 11;
M. Sakaguchi 17
fQ+; Q+g = �2iC�+ 1P++�2 ip
2C�mn��
I i�2Mmn
fQ�
; Q�
g = �2iC�� 1P�
� �ip
2C�ij�+I i�2Mij
+ �ip
2C�i0j0�+J i�2Mi0j0
fQ+; Q�g = �2iC�m 1Pm � 2i�C�iI i�2P�
i + 2i�C�i0J i�2P�
i0
Two limits
� �! 0 as at limit
� ! 0 as Penrose limit
the super-pp-wave algebra under renaming
Pm ! 12p
2em; P �m ! 1
4p
2�2e�m; P�! 1
2p
2e�; Q!s
ip2Q
M. Sakaguchi 18
5 brane action
MC one-forms on super-AdS5 � S5
��1d� = LaPa + La0Pa0 +1
2LabMab +1
2La0b0Ma0b0 + L�Q�
Penrose limit scale
L+! 2L+; Lm
! Lm; Lm
�
� (Li0;Li09)! Lm
�
; L�
+ ! L�+
then take limit ! 0
WZ term for F/D-strings dLWZ = h(3) [Metsaev and Tseytlin '9805]
h(3) = �iLa( �L a�L) + La0( �L a0�L); � =�
�3 for F1
�1 for D1
= �iL+(L�C�+�L�)� iL�(L+C���L+)
�iLm(L+C�m�L�)� iLm(L�C�m�L+)
scale h(3) ! 2h(3)
pp and then all terms survive in the limit ! 0
F/D-string WZ term on the pp-wave RR-background dLWZ = h(3)
pp .
M. Sakaguchi 19
WZ term for D3-brane [Metsaev and Tseytlin '9806]
dLD3WZ = h(5) + h
(3)1 F ; F = dA�B2; h
(3)3 = dB2
h(5) =i
6LaLbLc( �L� abcL)�1
6La0Lb0Lc0( �L� a0b0c0L)
�12LaLbLa0( �L� ab a0L) +i
2LaLa0Lb0( �L� a a0b0L)
+1
30�a1���a5La1 � � �La5 + 1
30�a0
1���a05La01 � � �La05;
Scale h(5) ! 4h(5)
pp and then take the limit ! 0
h(5)
pp =i
6LmLnLp(LC�mnp i�2L) +i
2LmLnL+(LC�mn+ i�2L)
+i
2LmLnL�(LC�mn� i�2L) + iLmL+L�(LC�m+� i�2L)
+2p
2L�(L1L2L3L4 + L5L6L7L8)
D3-brane WZ term on the pp-wave RR-background dLD3WZ = h
(5)pp +h(3)
ppF .
M. Sakaguchi 20
6 Conclusion
�We clari�ed the relation of maximally IIB supersymmetric algebras as
IW contractions (coordinate independent)
super-AdS5 � S5
�!0
����! 10-dim super-Poincar�e
!0
����! super-pp-wave
� string and D3-brane on the super-pp-wave background
� super-AdS4� S7 and super-AdS7� S4 to the super-pp-wave algebra in
11-dim.
� completeness of maximally supersymmetric IIB backgrounds ?
classi�ed in [Kowalski-Glikman '84] for 11-dim
� AdS/CFT
� & c
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