RENORMALIZED ENTANGLEMENT ENTROPY
FLOW IN MASS-DEFORMED ABJM THEORY II
(MORE DETAILS)
Aspects of Holography, July 17, 2014, Postech, Korea
In collaboration with O. Kwon, C. Park. H. Shin
arXiv:1404.1044 and arXiv: 1407.xxxx
Kyung Kim Kim
(Gwangju Institute of Science and Technology)
CONTENTS
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LLM geometry for the mass-deformed ABJM theory
Discrete vacua for the mass-deformed ABJM theory
Holographic entanglement entropy for the Strip case.
Cutting the minimal surface and the mass
deformation effect.
Holographic entanglement entropy for the Disk case.
Validity in terms of the curvature and the PDE effect.
Future direction
LLM GEOMETRY FOR THE MASSDEFORMED ABJM THEORY
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The dual geometries are given by solutions of the 11
dimensional super gravity with appropriate ansatz using
the bubbling geometry technique.
Ignoring the Z_k modding, the M2 brane world-volume
theory has SO(8) R symmetry.
By the mass deformation, this R-symmetry is broken to
SO(4)XSO(4).
So the metric has two S3 spheres and the 4-form field
strength also contains the S3 volume forms .
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The 11 dimensional killing spinor has a structure roughly
- ( S3 Killing spinor) X ( B : spinor in the other space)
Using this B, one can construct spinor bilinears
~ B (Gamma Matrices) B
This construction gives (pseudo) scalars, (pseudo) vector, 2
forms, 3 forms, …, and so on, which are part of the metric
and the field strength.
Finally, one can obtain following solution.
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As Dr. Kwon introduced in his talk, Considering Z_k
modding and SO(2,1) isometry to the Ansatz give the
solution we will consider as follows.
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Thus the z(x,y) and V(x,y) determine the geometry and the
field strength.
By Susy and the equation of motion, z(x,y) and V(x,y) are
related to each other. They are not independent.
The equation for V(x,y) is nothing but the Laplace equation
in the cylindrical coordinates.
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So V(x,y) is given by the scalar potential produced by
charges sitting on y=0 line.
To make the asymptotic geometry AdS4, the sum of the
charges should be 1.
DISCRETE VACUA FOR THE MASSDEFORMED ABJM THEORY
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From the Kwon’s talk
There are infinite number of discrete vacua encoded by
VEV of the scalar field.
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GRVV matrix
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Vacuum solutions
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For super symmetric vacua
( Kims and Cheon)
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This gives a constraint for x_i s.
Matching with gravity solution
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N_n = l_n
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HOLOGRAPHIC ENTANGLEMENT
FOR THE STRIP CASE.
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To consider the minimal surface,
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The induced metric is given by
ENTROPY
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We are considering target space map for the strip as follows.
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To avoid Partial differential equation, we take into account
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This is valid, when the minimal surface is far from the
droplets and the charge distribution is close to symmetric
configuration.
Since our interest is behavior near UV fixed point and
symmetric charge distribution so far.
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The minimal surface
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Induced metric and the minimal surface action
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Convenient coordinate choice
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where R is AdS radius.
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The minimal surface action
To cover more general case, we use the Legendre
polynomials .
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We are considering the UV behavior
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We may approximate f as follows.
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For k=1, the droplet configuration can be described by
Young diagram with fixed area.
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Using this approximation, the minimal surface is given by
And we have a conserved hamiltonian with an appropriate
boundary condition.
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One can find u’ and l in terms of u and u_0
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It is easy to integrate the minimal surface as follows.
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Substituting l, the minimal surface, the holographic
entanglement for the strip case.
The renormalized entanglement entropy is
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For symmetric droplet case.
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So the free energy( c function )
CUTTING THE MINIMAL SURFACE AND THE
MASS DEFORMATION EFFECT.
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Ryu and Takayanagi provided a useful way to introduce
the mass deformation in a bottom-up approach.
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Cutting = Mass deformation
Comparing the bottom-up approach with the top-down
approach.
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The shift for the mass deformation is
This provides the correlation length in the bottom-up
approach.
For the symmetric case
HOLOGRAPHIC ENTANGLEMENT
FOR THE DISK CASE.
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The target space map is given by
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The minimal surface Lagrangian is
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In the small mass approximation
ENTROPY
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For the conformal case without mass deformation, the
minimal surface is very simple as follows.
In order to consider the leading order mass deformation
effect, we defined following deformed surface.
This additional part is governed by following equation of
motion.
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To obtain the minimal surface, we have to solve the
equation.
Fortunately, we can solve it.
The solution is as follows.
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Because of the AdS metric factor, the minimal surface has
a counter-intuitive shape as follows.
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For general droplets, the holographic entanglement
entropy is given by
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For k= 1 and the symmetric droplet case,
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The entanglement entropy is
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The c-function is
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For general case
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For the symmetric case, the c-function or the partition
function shows the monotonic decreasing behavior.
For the general case, it is not guaranteed. We need to
investigate the validity of our computation and the
modification of the coefficient.
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VALIDITY IN TERMS OF THE CURVATURE
AND THE PDE EFFECT.
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In order to consider validity of our calculation, we have to
think about two points.
One is the PDE effect and the other one is the
supergravity approximation.
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The PDE effect
We have ignored the distorted effect from the asymmetric
configuration of the droplets.
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The exact consideration should include solving the PDE
directly with r( w_2, \alpha)
It is a possible subject, so we leave it as a next topic.
Anyway our result is valid for the case which is close to the
symmetric configuration.
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The next one we should be careful is whether the
supergravity approximation is valid or not.
In this paper, it was pointed out that the application of the
AdS/CFT for this problem is valid for only weakly curved
geometries.
In fact, there are two curvature scales from UV and IR.
UV curvature scale is given by AdS radius R and it goes to
zero, when we take large N limit.
The IR curvature scale, however, is given by the droplet
configuration. So it is not guaranteed that such a geometry
is weakly curved in the large N limit.
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Otherwise, we have to consider higher derivative
corrections and such a case is beyond the Supergravity
approximation.
To have clear understanding, let us consider one droplet
configurations.
For k=1, this is described by a rectangular young tabuleux.
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The IR curvature scale is given by scalar curvature near
y=0.
The scalar curvature is given by
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For the rectangular Young diagram
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When w =
The scalar curvature is
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Therefore
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the IR geometry is not weakly curved
under Large N limit.
We cannot trust supergravity result for this droplet
configuraiton.
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In this droplet configuration, even the entanglement
entropy has also strange behavior.
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For this case
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The REE is
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In the case
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The REE are divergent for
or
FUTURE DIRECTION(OR ONGOING)
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More exact description for the UV behavior
- Consideration of the small PDE effect.
Approximated IR behavior for the symmetric droplet.
Full RG flow of REE from numerical calculation( ongoing )
-Solving PDE numerically
Consideration of higher derivative corrections.
Be our future collaborators.
Thank you for your attention !