LLM geometries in M-theory
and probe branes inside them
Jun-Bao Wu
IHEP, CAS
Nov. 24, 2010, KITPC
Based on
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B. Chen, E. O Colgain, JW, H. Yavartanoo,
JHEP04(2010)078, 1001.0906.
E. O Colgain, JW, H. Yavartanoo,
JHEP08(2010)114, 1005.4527.
E. O Colgain, JW, H. Yavartanoo,
1010.5982.
Outline
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Vanishing of a particular flux in 11d LLM
geometries
Probe branes in Maldacena-Nunez
background
Conclusions and discussions
11d LLM geometry
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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.
Holographic dual of N=2
theories
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Gaiotto (2009) constructed a huge class of
4d N=2 gauge theories by wrapping N M5
branes on a (punctured) Riemann surface.
Gaiotto and Maldacena (2009) suggested
the dual geometries fall into double-Wickrotated LLM solutions (S5 becomes AdS5,
and M4 becomes Euclidean).
Dual geometries
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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.
Fluxes
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From [Gaiotto, Maldacena]
No such a flux
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We show that there are no solutions
with such a flux.
Aside remark:
LLM noticed that if there is such a flux,
the geometry is singular. So in certain
sense, this singularity is ruled out by
the sixteen supercharges (and the
isometry).
11d supergravity
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The bosonic sector of the 11d SUGRA
includes the metric g and a 3-form
potential C with field strength F(4)=dC.
The action for this sector is:
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Killing spinor equation:
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Ansatz
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LLM looked for half-BPS solutions with
isometry SO(6)*SO(3), so they began
with the following ansatz
Decomposition
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The decomposition of the gamma
matrices:
We decompose the 11d Killing spinor
using Killing spinors on S5 and S2:
Reduction of KSE
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The 11d Killing spinor equations now
reduce to:
The bispinors
(scalars and vectors)
Algebraic relations among
scalars
Algebraic relations among
vectors
Vanishing of I
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For general case, we have
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If we assume I is nonzero,
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By solving the above algebraic
equations, we get
Gaiotto’s N=2 dualities
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Gaiotto studied a huge class of N=2 theory
obtained 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.
Simplest example
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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.)
S-duality (I)
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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.
S-duality (II)
More complicated quiver
TN theory
The case without punctures
Maldacena-Nenuz background
A bit more on the geometry
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S4 part of the six-dimetional internal
space:
Non-local operators/probe
branes
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There are non-local operators (objects)
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]
Killing spinors
M5 branes
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We focus on M5-brane in this MN
background.
There are self-dual 3-form h field in the
worldvolume of M5-brane.
The equations of motion are quite
complicated, so we do not give the
details.
BPS condition
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The supersymmetries preserved by the
M5 brane are determined by the
following condition
Half-BPS AdS3 probe
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The brane is along AdS3 (inside AdS5) Σ2
and directions with θ=π/2 :
Field theory dual
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Half of the supersymmetries are broken
by this brane, while SU(2)*U(1) Rsymmetry is preserved.
The brane should be dual to some twodimensional operators in the field
theory side. Maybe it is dual to half-BPS
surface operator.
Back reaction
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It is interesting to study the ¼-BPS
solution 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.
Two known solutions
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We began with the bispinors and using the
tool 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 …
Summary
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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.
Future directions
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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.
THE END
Thank you very much!