R ENORMALIZED E NTANGLEMENT E NTROPY F LOW IN M ASS - DEFORMED ABJM T HEORY II (M ORE DETAILS ) Aspects of Holography, July 17, 2014, Postech, Korea In collaboration with O. Kwon, C. Park. H. Shin arXiv: and arXiv: 1407.xxxx Kyung Kim Kim (Gwangju Institute of Science and Technology)

CONTENTS 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 MASS - DEFORMED ABJM THEORY 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.

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.

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.

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.

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.

D ISCRETE VACUA FOR THE MASS - DEFORMED ABJM THEORY From the Kwon’s talk There are infinite number of discrete vacua encoded by VEV of the scalar field. GRVV matrix

Vacuum solutions

For super symmetric vacua ( Kims and Cheon) This gives a constraint for x_i s. Matching with gravity solution N_n = l_n

H OLOGRAPHIC ENTANGLEMENT ENTROPY FOR THE S TRIP CASE. To consider the minimal surface, The induced metric is given by

We are considering target space map for the strip as follows. To avoid Partial differential equation, we take into account 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.

The minimal surface

Induced metric and the minimal surface action Convenient coordinate choice where R is AdS radius.

The minimal surface action To cover more general case, we use the Legendre polynomials.

We are considering the UV behavior We may approximate f as follows.

For k=1, the droplet configuration can be described by Young diagram with fixed area.

Using this approximation, the minimal surface is given by And we have a conserved hamiltonian with an appropriate boundary condition.

One can find u’ and l in terms of u and u_0 It is easy to integrate the minimal surface as follows.

Substituting l, the minimal surface, the holographic entanglement for the strip case. The renormalized entanglement entropy is

For symmetric droplet case. So the free energy( c function )

C UTTING THE MINIMAL SURFACE AND THE MASS DEFORMATION EFFECT. Ryu and Takayanagi provided a useful way to introduce the mass deformation in a bottom-up approach.

Cutting = Mass deformation Comparing the bottom-up approach with the top-down approach.

The shift for the mass deformation is This provides the correlation length in the bottom-up approach. For the symmetric case

H OLOGRAPHIC ENTANGLEMENT ENTROPY FOR THE D ISK CASE. The target space map is given by The minimal surface Lagrangian is In the small mass approximation

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.

To obtain the minimal surface, we have to solve the equation. Fortunately, we can solve it. The solution is as follows.

Because of the AdS metric factor, the minimal surface has a counter-intuitive shape as follows.

For general droplets, the holographic entanglement entropy is given by For k= 1 and the symmetric droplet case, The entanglement entropy is

The c-function is For general case

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. ?

V ALIDITY IN TERMS OF THE CURVATURE AND THE PDE EFFECT. 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.

The PDE effect We have ignored the distorted effect from the asymmetric configuration of the droplets.

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.

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.

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.

The IR curvature scale is given by scalar curvature near y=0. The scalar curvature is given by

For the rectangular Young diagram When w = The scalar curvature is Therefore the IR geometry is not weakly curved under Large N limit. We cannot trust supergravity result for this droplet configuraiton.

In this droplet configuration, even the entanglement entropy has also strange behavior. For this case The REE is

In the case The REE are divergent for or

F UTURE DIRECTION ( OR O NGOING ) 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 !