Makoto Sakaguchi (Okayama Institute for Quantum Physics ...qft/2009/slides/Sakaguchi.pdfMakoto...
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Makoto Sakaguchi (Okayama Institute for Quantum Physics)
2009/7/9 @ 基研研究会「場の理論と弦理論」
based on
``Super Galilean conformal algebra in AdS/CFT,'' arXiv:0905.0188 [hep-th].
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Introduction
2
AdS/CFT correspondence
IIB string in AdS5 x S5
weakN=4 super Yang-Mills in 4-dim
strong
Applications AdS/CMPsuperconductor [Gubser '0801, Hartnoll-Herzog-Horowitz '0810, …]
quantum Hall effect [Keski-Vakkuri-Kraus '0805, Davis-Kraus-Shah '0809, …]
fermions at unitarity [Son '0804, Balasubramanian-McGreevy '0804,…]
Non-relativistic CFTSchrödinger symmetry
Galilean conformal symmetry
conformal
∪ NR limit
Poincare Galilei
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• symmetry of (free) Schrödinger equation [Hagen'72, Niederer'72]
• subalgebra of conformal algebra in one higher dimensions
NR limit
• relativistic FT Schrödinger-invariant FT (for example [Jackiw-Pi’90])
• z=2 case
Schrödinger symmetryGalilean symmetry { H, Pi, Jij, Gi }
dilatation D, Galilean special conformal K
central charge M
z: dynamical exponent
anisotropic scale transformation D
3
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Galilean conformal symmetryGalilean symmetry { H, Pi, Jij, Gi }
dilatation D, Galilean special conformal K
acceleration Ki
z=1 case
PLAN
• Inönü-Wigner contraction of conformal algebra (review)
• bulk realization of GCA
• supersymmetric GCA
4
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GCA from conformal algebra
5
confd+1
IW contraction
and
Galilean conformal algebra
cf: [Lukierski-Stichel-Zakrzewski '0511]
:
:
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GCA1+d
so(2,1) + so(d)
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confd+1 GCAd+1
contraction[Lukierski-Stichel-Zakrzewski '0511]
bulk realization of GCA
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AdSd+2
8
confd+1 GCAd+1
contraction[Lukierski-Stichel-Zakrzewski '0511]
bulk realization of GCA
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AdSd+2 ???
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confd+1 GCAd+1
contraction
contraction[Lukierski-Stichel-Zakrzewski '0511]
bulk realization of GCA
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AdSd+2 ???
10
confd+1 GCAd+1
contraction
contraction[Lukierski-Stichel-Zakrzewski '0511]
bulk realization of GCA
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AdSd+2 Newton-Hooke string
algebra
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confd+1 GCAd+1
contraction
contraction
[Gomis-Gomis-Kamimura '0507]
[Lukierski-Stichel-Zakrzewski '0511]
GCA is realized as NH string algebra in the bulk
bulk realization of GCA
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[Gomis-Gomis-Kamimura '0507]
AdS2 string worldsheet
close-up of AdS2 string worldsheet
NR string [GGK'0507, MS-Yoshida'0703,'0709,'0712]
GCA and Newton-Hooke string algebraAdSd+2
Newton-Hooke string algebraAdS2
so(d)
32 susy NH algebras [GGK ‘0507, MS-Yoshida ‘0605]
bo
un
da
ry
(0,d+1)
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NH string algebra
so(2,1) + so(d)
GCA1+d
so(2,1) + so(d)
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Close-up of AdS brane in AdSd+1
NH brane algebra semi-GCA
[Brugues-Gomis-Kamimura'0603]
semi-GCA is realized as NH brane algebra in the bulk
[Alishahiha-Davody-Vahedi’0903]
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supersymmetric GCA1+3 from psu(2,2|4)
psu(2,2|4) super GCA1+3
∪
AdS5 x S5
contraction
We are interested in
dynamical susy
susy on AdS2
ρ p=1 p=3 p=5 p=7
σ1 , σ3 (2,0) (2,4), (4,2)
iσ2 (3,1), (1,3) (5,3), (3,5)
1 (1,1) (1,5), (3,3), (5,1)
a pair of 16 component Majorana-Weyl spinors
1/2 BPS brane configurations in AdS5 x S5
[MS-Yoshida ‘0605]
[MS, de Azcarraga-Lukierski ‘0905]
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16
IW contraction
and
32 susy GCA1+3
GCA iso(5)
dynamical susy
1/2 susy subalgebra
so(2,1) so(3) so(5)
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16 susy GCA1+3
u(1) x su(2)GCA
[MS, de Azcarraga-Lukierski ‘0905]
su(2,2|2) psu(2,2|4)
IW contraction
32 susy GCA1+3
so(2,1) + so(3) + so(5) + 16 susy
16 susy GCA1+3 from su(2,|2)
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32 susy GCA from osp(8|4)
GCA
M2 (AdS2 x S1) in AdS4 x S7
16 susy GCA from osp(4|4)
GCA so(3) x iso(2)
8 susy GCA from osp(2|4)
GCA u(1)
super GCA1+2
close-up of S1 in S7
[MS ‘0905]
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32 susy semi-GCA from osp(8*|4)
iso(4)semi-GCA
M2 (AdS3) in AdS7 x S4
so(2,2) so(4) + so(4)
16 susy semi-GCA from osp(8*|2)
su(2)semi-GCA
super semi-GCA1+5[MS ‘0905]
AdS3
1/2 susy subalgebra
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Summary GCA is a boundary realization of NH string algebra
semi-GCA is a boundary realization of NH brane algebra
super GCA1+3 from SCA4, psu(2,2|4) and su(2,2|2)
string(AdS2) in AdS5
super GCA1+2 from SCA3, osp(8|4), osp(4|4) and osp(2|4)
M2(AdS2 x S1) in AdS4 x S7
super semi-GCA1+5 from SCA6, osp(8*|4) and osp(8*|2)
M2(AdS3) in AdS7
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super GCAs for other 1/2 BPS branes
NRCFT with GCA and NR string
Wilson line in SYM4
infinitely extended super GCA
and so on
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Thank you very much.