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Siam Physics Congress 2016
Dimensional Reduction
of Eleven Dimensional Supergravity
on SU(2) Group Manifold
and N=4 Gauged Supergravity
Patharadanai Nuchino
Dr. Parinya
Karndumri
Department of Physics, Faculty of Science, Chulalongkorn University
June 8, 2016
Room Anek, Baansuan-Khunta and Golf Resort Hotel, Ubon Ratchathani, Thailand
Introduction of Supergravity theory
Supergravity
=
General relativity
(gravity)
Supergravity multiplet
+
Supersymmetry
symmetry between
boson and fermion
graviton
gravitino
(Spin=2)
(Spin=3/2)
The simplest case in 4D, N=1
Supercharge
Supergravity can be extend
Higher dimensional theory
with extra dimensions
N>1 theory
with more fields
D=11 is maximum
N=8 is maximum for 4D
http://jac_leon.perso.neuf.fr/gravitation/images/space-time.gif
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Gauged
Supergravity
Ex.
=
N>1
Supergravity
contents
Vector (gauge) fields
+
Gauge
symmetry
4D N=4 SO(4) gauged
Supergravity multiplet
a graviton (Spin=2)
4 gravitinos (Spin=3/2)
6 gauge fields (Spin=1)
SO(4) gauge symmetry
4 spinor fields (Spin=1/2)
2 scalar fields (Spin=0)
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Gauged Supergravity in AdS/CFT
AdS/CFT correspondence
[1, 2, 3]
the new duality relating
Derivation are important
Supergravities
Λ<1
in D dimensional
anti-de Sitter (AdS) background
quantum
http://quantum-bits.org/wp-content/uploads/2015/09/ads-cft.png
Superconformal field theories
on the AdS boundary D-1 dimensions
higher dimensional
ungauged
supergravity
Dimensional
reduction
lower dimensional
gauged supergravity
admit AdS background
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What is Dimensional Reduction ?
Consideration
a gravitational theory
a gravitational theory
in D spacetime dimensions
with additional
gives rise
scalar fields
where n spatial dimensions are
&
compactified to be
vector fields
a very small compact space
in D+n spacetime dimensions
corresponds
symmetry on compact space
with
gauge symmetry
http://www.pbs.org/wgbh/nova/assets/img/full-size/imagining-otherdimensions-merl.jpg
http://www.vcharkarn.com/uploads/17/17496.jpg
http://www.diffusion.ens.fr/vip/images/10.11.jpg
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4019/280x242/strings.gif
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11D or IIB (10D)
supergravity
reduction
on n-dimensional
sphere (Sn)
lower dimensional
gauged supergravity
[4]
admit AdS vacuum solutions
some notable cases
S4 reduction
11 dimensional
supergravity
IIB (10 dimensional)
supergravity
S7 reduction
S5 reduction
gauged supergravity
in 7D
[5, 6]
gauged supergravity
in 4D
[7]
gauged supergravity
in 5D
[8]
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Objective
Study
[9]
11 dimensional
S7 reduction
supergravity
the unique maximaldimensional supergravity
(the most fundamental)
N=4 SO(4)
gauged supergravity
in 4 dimensions
the half-maximal 4D
supergravity
[10]
AdS/CFT correspondence
Conformal Field Theory
in 3 dimensional
spacetime
the quantum field theory on 2 dimensional surface
now used for
study the superconductivity of 2D superconductor [11,12]
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Dimensional Reduction Procedure
SU(2) SU(2)
the compact space
S7 ~ R × S3 × S3
from the fact that S3 is a group manifold of Lie group SU(2)
The reduction ansatze
(expressions of higher dimensional fields in terms of lower dimensional fields)
11D
Supergravity
Equations of motion
yield
N = 4 SO(4)
gauged supergravity
in 4 dimensions
Equations of motion
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Conclusions
• Consistent dimensional reduction
of the eleven dimensional supergravity
giving rise to
the N=4 gauged supergravity in 4 dimensions
was obtained
• The two SU(2)s form SO(4) ~ SU(2) x SU(2)
gauge group of the lower dimensional
theory
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Application
-> Lift lower dimensional solutions up to the 11 dimensional theory
11D
N = 4 SO(4)
gauged supergravity
in 4 dimensions
solutions
satisfy
Equations of motion
Ex.
Supergravity
Lift up
Reduction ansatze
satisfy
Equations of motion
Consistent
Lift up
[13]
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solutions
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References
[1] Maldacena, J. 1998. “The large N limit of superconformal
field theories and supergravity”. Advances in Theoretical and
Mathematical Physics. 2: 231-252.
[8] Baguet, A., Hohm, O., and Samtleben, H. 2015. “Consistency
type IIB reductions to maximal 5D supergravity”. Physical
Review D. 92: 065004.
[2] Gubser, S. S., Klebanov, I. R., and Polyakov, A. M. 1998.
“Gauge theory correlators from noncritical string theory”.
Physics Letters B. 428: 105-114.
[9] Cremmer, E., Julia, B., and Scherk, J. 1978. “Supergravity
Theory in Eleven-Dimensions”. Physics Letters B. 76: 409-412
[3] Witten, E. 1998. “Anti-de Sitter space and holography”.
Advances in Theoretical and Mathematical Physics. 2: 253291.
[10] Cvetic, M., Lu, H. and Pope, C. N. 2000. “Four-dimensional
N=4 SO(4) Gauged Supergravity from D=11”. Nuclear Physics B.
574: 761-781.
[11] Gubster, S. S., Pufu, S. S., and Rocha, F. D. 2010. “Quantum
[4] Cvetic, M., et al. 1999. “Embedding AdS Black Holes in Ten critical superconductors in string theory and M-theory”.
and Eleven Dimensions”. Nuclear Physics B. 558: 96-126.
Physical Letter B. 683: 201-204.
[5] Nastase, H., Vaman, D., and Nieuwenhuizen, P. V. 1999.
[12] Gauntlett, J., Sonner, J., and Wiseman, T. 2010. ”Quantum
“Consistent nonlinear KK reduction of 11d supergravity on
Criticality and Holographic Superconductors in M-theory”
AdS7×S4 and self-duality in odd dimensions”. Physics Letters B. Journal of High Energy Physics. 2010: 60.
469: 96-102.
[13] Lu, H. and Zhang, X. 2014. “Exact Collapse Solutions in D=4,
[6] Nastase, H., Vaman, D., and Nieuwenhuizen, P. V. 2000.
N=4 Gauged Supergravity and Their Generalizations” Journal of
4
“Consistent of the AdS7×S reduction and the origin of selfHigh Energy Physics. 2014: 099.
duality in odd dimensions”. Nuclear Physics B. 581: 179-239.
[7] De Witt, B. and Nicolai, H. 1987. “The consistency of the S7
truncation in D=11 supergravity”. Nuclear Physics B. 281: 211240.
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Thank you
Every questions and comments
are welcome
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