llm geometries in m-theory and probe branes inside them
DESCRIPTION
LLM geometries in M-theory and probe branes inside them. Jun-Bao Wu IHEP, CAS Nov. 24, 2010, KITPC. Based on. B. Chen, E. O Colgain, JW, H. Yavartanoo, JHEP 04 (2010)078, 1001.0906. E. O Colgain, JW, H. Yavartanoo, JHEP 08 (2010)114, 1005.4527. E. O Colgain, JW, H. Yavartanoo, - PowerPoint PPT PresentationTRANSCRIPT
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LLM geometries in M-theory and probe branes inside them
Jun-Bao WuIHEP, CAS
Nov. 24, 2010, KITPC
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Based on B. Chen, E. O Colgain, JW, H. Yavartanoo, JH
EP04(2010)078, 1001.0906.
E. O Colgain, JW, H. Yavartanoo, JHEP08(2010)114, 1005.4527.
E. O Colgain, JW, H. Yavartanoo, 1010.5982.
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Outline
Vanishing of a particular flux in 11d LLM geometries
Probe branes in Maldacena-Nunez background
Conclusions and discussions
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11d LLM geometry
Lin, Lunin and Maldacena (2004) found a large class of half-BPS solutions with isometry SO(6)*SO(3)*R of 11d SUGRA.
The geometry is warped product of S5, S2 and M4.
This geometry plays an important role in AdS/CFT correspondence.
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Holographic dual of N=2 theories
Gaiotto (2009) constructed a huge class of 4d N=2 gauge theories by wrapping N M5 branes on a (punctured)(punctured) Riemann surface.
Gaiotto and Maldacena (2009) suggested the dual geometries fall into double-Wick-rotated LLM solutions (S5 becomes AdS5, and M4 becomes Euclidean).
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Dual geometries
For cases without punctures, the dual geometries are solutions obtained by Maldacena and Nunez (2000), which are special cases of double-Wick-rotated LLM solutions.
For case with punctures, the full dual geometries haven’t been obtained.
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Fluxes
From [Gaiotto, Maldacena]
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No such a flux We show that there are no solutions w
ith such a flux. Aside remark: LLM noticed that if there is such a flux,
the geometry is singularsingular. So in certain sense, this singularity is ruled out by the sixteen supercharges (and the isometry).
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11d supergravity The bosonic sector of the 11d SUGRA i
ncludes the metric g and a 3-form potential C with field strength F(4)=dC.
The action for this sector is:
Killing spinor equation:
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Ansatz LLM looked for half-BPS solutions wit
h isometry SO(6)*SO(3), so they began with the following ansatz
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Decomposition
The decomposition of the gamma matrices:
We decompose the 11d Killing spinor using Killing spinors on S5 and S2:
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Reduction of KSE
The 11d Killing spinor equations now reduce to:
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The bispinors (scalars and vectors)
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Algebraic relations among scalars
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Algebraic relations among vectors
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Vanishing of I
For general case, we have
If we assume I is nonzero,
By solving the above algebraic equations, we get
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Gaiotto’s N=2 dualities Gaiotto studied a huge class of N=2 theory o
btained from wrapping M5 branes on (punctured) Riemann surface.
Only a small fraction of these theories have known descriptions in terms of UV Lagrangian.
Gaiotto found generalization of various known S-dualities.
Non-perturbative results can be obtained from M-theory.
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Simplest example
SU(2) theory with 4 flavors is corresponding to a sphere with 4 punctures. (In the right figure, SO(4)*SO(4) subgroup of flavor group SO(8) is picked out.)
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S-duality (I)
S-duality SL(2, Z) group acts on
SL(2, Z) acts through triality on SO(8) flavor group, and exchanges quarks, monopoles and dyons.
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S-duality (II)
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More complicated quiver
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TN theory
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The case without punctures
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Maldacena-Nenuz background
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A bit more on the geometry
S4 part of the six-dimetional internal space:
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Non-local operators/probe branes There are non-local operators (object
s) with various dimensions in these N=2 field theories: Wilson-’t Hooft loops, surface operators, domain walls …
In certain conditions they should be dual to probe M2 or M5 branes.
The M2 branes dual to loop operators: [Drukker, Morrison, Okuda]
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Killing spinors
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M5 branes We focus on M5-brane in this MN back
ground. There are self-dual 3-form h field in th
e worldvolume of M5-brane. The equations of motion are quite co
mplicated, so we do not give the details.
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BPS condition
The supersymmetries preserved by the M5 brane are determined by the following condition
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Half-BPS AdS3 probe
The brane is along AdS3 (inside AdS5) Σ2 and directions with θ=π/2 :
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Field theory dual
Half of the supersymmetries are broken by this brane, while SU(2)*U(1) R-symmetry is preserved.
The brane should be dual to some two-dimensional operators in the field theory side. Maybe it is dual to half-BPS surface operator.
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Back reaction It is interesting to study the ¼-BPS sol
ution of 11d SUGRA describing the back reaction of this BPS M5 brane.
It should be warped product of AdS3, S2 and a six-dimensional internal space including Σ2.
We tried to search such solution following the ideas of LLM.
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Two known solutions We began with the bispinors and using the t
ool of G-structures. We re-obtained two known solutions: 1. SU(3)-structure: AdS3*S2*CY3 [Maldacena,
Strominger, Witten] 2. SU(2)-structure: the one studied by [Gauntlett, etal][Kim3] The wanted solution is not in either class. We are still searching for it …
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Summary
We showed that there are no certain flux in LLM geometries (closed the previous loophole).
We studied the probe branes in a special LLM background.
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Future directions
Continue to study the gravity dual for the case with punctures. Related works:
[Donos, Simon] [Reid-Edwards et al]
Further studies on the correspondence between non-local operators and probe branes.
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THE END
Thank you very much!