RENORMALIZED ENTANGLEMENT ENTROPY OF MASS-DEFORMED ABJM THEORY YongPyong-High1 2015, Joint Winter Conference on Particle Physics, String and Cosmology In collaboration with Chanju Kim, O-Kab Kwon Based on arXiv15xx,xxxx, Phys.Rev. D90 (2014) 4, 046006 , Phys.Rev. D90 (2014) 12, 126003

with O. Kwon, Chanyong Park and hyenjoon Shin. Kyung Kiu Kim (Gwangju Institute of Science and Technology) CONTENTS Entanglement Entropy Holographic Entanglement Entropy(Ryu-Takay anagi formula) 1+1 d case(Mass-deformation for 1+1 d by c utting)

C theorem in 1+1d F theorem in 2+1d Renormalized entanglement entropy for F LLM geometry and mass deformed ABJM REE near UV Summary ENTANGLEMENT ENTROPY Quantity measuring spatial entanglement of

a state The state(Usually ground state) is given by CAB A X B where A and B are basis wave fu nctions of Hilbert spaces Ha and Hb. Ha and Hb are apart from each other spatially. Definition Simple example : two points and two spins Maximum entanglement entropy

Minimum entanglement entropy with a trivia l Hamiltonian HOLOGRAPHIC ENTANGLEMENT ENTR OPY(RYU-TAKAYANAGI FORMULA)

Area of minimal surface in AdS = Entangleme nt entropy of CFT A: entangling surface Agreements with CFTs in 2d 3d 4d .

From the field theories(2 dimensions and hig her dimensions Minimal surfaces have same structure of the se UV divergences. 1+1 D CASE

From field theory 94C. Holzhey, F. Larsen and F. Wilczek, Minimal surface(geodesic) in AdS3

Mass deformation Computation through HEE = log ( tan s/2 )

C THEOREM IN 1+1 D Decreasing c function under RG flow Zamolodchikov 86, Cardy 88, Komargodski-S chwimmer 11 Is there a 3 dimensional version? F-theorem in 2+1 d

F= - log Z : free energy decreses ? F theorem (Jafferis-Klebanov-Pufu-Safdi 11, Myers-Sinh a 10 ) F_UV > F_IR For CFT_3 [Casini-Huerta-Myers 11 ] RENORMALIZED ENTANGLEMENT ENT

ROPY FOR F Liu-Mezei 12 We will follow this prescription. The goal of this work is to show that F can be a c function by controlling mass deformation.

UV -> IR : The mass becomes bigger CFT -> massive theory We will take the ABJM theory as a probe theor y. LLM GEOMETRY FOR THE MASS-DEFO RMED ABJM THEORY

The dual geometries are given by solutions of the 11 d imensional super gravity with an appropriate ansatz u sing 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 t o 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 rou ghly - ( 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) ve ctor, 2 forms, 3 forms, , and so on, which are part of the metric and the field strength. Finally, one can obtain following solution. Considering Z_k modding and SO(2,1) isometry to th e Ansatz give the solution we will consider as follows.

Thus the z(x,y) and V(x,y) determine the geometry an d 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 equ ation 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 th e charges should be 1. DISCRETE VACUA FOR THE MASS-DEF ORMED ABJM THEORY There are infinite number of discrete vacua encoded b

y VEV of the scalar field. GRVV matrix Vacuum solutions

For super symmetric vacua ( S. Kim H. Kim and Cheon) This gives a constraint for x_i s. Matching with gravity solution

N_n = l_n HOLOGRAPHIC ENTANGLEMENT ENT ROPY FOR THE STRIP CASE.

To consider the minimal surface, The induced metric is given by We are considering target space map for the strip as follow s.

To avoid Partial differential equation, we take into account This is valid, when the minimal surface is far from the dropl ets and the charge distribution is close to symmetric config uration.

Since our interest is behavior near UV fixed point and sym metric charge distribution so far. The minimal surface

For symmetric droplet case. So the free energy( c function ) HOLOGRAPHIC ENTANGLEMENT ENT ROPY FOR THE DISK CASE.

The target space map is given by The minimal surface Lagrangian is In the small mass approximation

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

DISK case The c-function is For general case

Because of the AdS metric factor, the minimal surface has a counter-intuitive shape as follows. For general droplets, the holographic entanglement en tropy 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 mo dification of the coefficient.

? F theorem is valid near UV fixed point for sy mmetric configuration . Its not true for general configuraiton. REE is a good choice for a c function. But PD E effect?

The PDE effect We have ignored the distorted effect from the asymm etric configuration of the droplets. FOR PDE EFFECT F theorem is valid near UV fixed point for all t he vacuum configuraitons .

REE is a good choice for a c function. But PD E effect? Now OK! Up to

Symmetric case SUMMARY We calculated HEE for mass deformed ABJM t heory with small mass approximation. The UV behavior are consistent with F theore m. ( decreasing c function F ) for all the vac ua. Stationarity

SUMMARY IR behavior and finite mass computation are on-going. Thank You for Listening!