Exact holography and entanglement entropy from one-point functions

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1 Exact holography and entanglement entropy from one-point functions O-Kab Kwon (Sungkyunkwan University) In collaboration with Dongmin Jang, Yoonbai Kim, Driba Tolla arxiv: , IBS-KIAS Joint workshop on particle physics, and cosmology Feb. 5 (Sun) Feb. 10 (Fri), 2017, High1 Resort

2 Outline Entanglement entropy Gauge/gravity duality for finite N Holographic entanglement entropy with PDE effect Gauge/gravity duality and the entanglement entropy Summary

3 Entanglement Entropy (EE) and Holographic EE Quantum entanglement is a physical phenomenon that occurs when pairs of particles interact. Then the quantum state of each particle cannot be described independently. è entanglement entropy (EE) è CMT, quantum gravity, information theory, etc Density matrix of a pure state : i tot = ih Ex) two spin system =(a "i A + b #i A ) (c "i B + d #i B ) 6= 0, 2,, 3 2, è entangled state!

4 In field theory B A H tot = H A H B Integrating out degrees of freedom in region B A Tr B tot Associated von Neumann entropy: S A = Tr A A ln A è Replica method

5 Replica trick: Path integral method CFT and deformed CFT in 2-dimensions Higher dimensional (d 3) case: è CFT (3d) using localization method è free theory using Replica trick : numerical work Only possible for limiting cases and difficult in QFT!! è Holographic EE ( 06 Ryu-Takayanagi s proposal)

6 Holographic EE S A = Area( A) 4G N ds 2 AdS = R 2 AdS [Ryu-Takayanagi 06] dt 2 + dx i dx i + du 2 u 2 A is the codimension 2 minimal surface Leading contribution reproduces area law terms Recover the EE in 2d CFT Strong subadditivity Many checks and results for higher dim.

7 Motivation RT conjecture is based on the gauge/gravity duality Duality properties of field theory and gravity theory: Quantum field theory duality Quantum gravity Strongly coupled gauge theory Large N Weakly curved gravity (classical gravity) Very useful but difficult to check the duality! For some BPS objects which have no quantum corrections, it was possible to check the duality using supersymmetry and conformal symmetry in the large N limit.

8 Motivation Therefore, in the calculation of the HEE using the RT conjecture, taking the large N limit is necessary. Can we apply the RT conjecture for finite N? No example up to now! How can we show that the RT conjecture is also applicable to the case of finite N? The first step is to show the exact gauge/gravity for finite N.

9 Gauge/gravity duality for finite N ABJM (CFT) + mass deformation = mabjm (non-cft) [Hosomichi et al 08] [Gomis et al 08] LLM geometry with [Lin-Lunin-Maldacena, 2004] Discrete Higgs vacua denoted by numerical values è Quantitative comparison for finite N orbifold [Gomis et al 08] [Cheon-Kim-Kim 2011]

10 Gauge/gravity duality for finite N To show the dual relation of two theories, correspondence between vacua of QFT and BPS solutions in gravity is not enough. mabjm theory: 3-dim. gauge theory LLM geometry: BPS solution in 11-dim. SUGRA KK-reduction 4-dim. Gravity theory

11 Gauge/gravity duality for finite N To show the dual relation of two theories, correspondence between vacua of QFT and BPS solutions in gravity is not enough. mabjm theory: 3-dim. gauge theory LLM geometry: BPS solution in 11-dim. SUGRA duality KK-reduction 4-dim. Gravity theory

12 Gauge/gravity duality for finite N Diagonalized equations in 4-dim. gravity AdS/CFT correspondence ( : radius of ) scalar field ˇ2 è Dual CPO with conformal dimension 1

13 Gauge/gravity duality for finite N Gauge/gravity mapping (GKP-W relation) LLM geometry Vev of CPO with in field theory side Fix the normalization factor (k=1):

14 Gauge/gravity duality for finite N The relations are satisfied for all possible supersymmetric vacuum solutions è infinite examples Number of vacuum for large N Exact dual relations è no 1/N corrections only depends on the shape of the Young-diagram è does not depend on the area (N)

15 Gauge/gravity duality for finite N Extension to k>1: [Jang, Kim, OK, Tolla 2016] One evidence of the gauge/gravity duality for finite N and k between the mass-deformed ABJM and LLM First example of exact duality for finite N! How can apply exact duality to HEE?

16 Holographic entanglement entropy of the mabjm w 2 (2+1)d boundary w 1 HEE proposal u Induced metric u 1 r LLM geometries In 11-dimensions

17 LLM geometry Half-BPS solutions with SO(2,1)XSO(4)XSO(4) isometry in 11-dimensional supergravity [04, Lin-Lunin-Maldacena] x (, ) y To calculate the HEE in the LLM geometry we have to solve PDE equations

18 This solution is completely determined by two functions: droplet Young diagram new parametrization

19 Calculation of HEE (small mass expansion for the Disk) HEE proposal [Ryu-Takayanagi 06] Induced metric Mapping for the Disk case:

20 Minimal area: Equation of motion:

21 Small mass expansion for general droplet

22 Exact solution of the PDE for the general droplet case up to 2 th order Solution satisfying the boundary condition u( = 0) = 0 and u( = 1) = 0

23 HEE for the LLM geometry Entanglement entropy Renormalized Entanglement entropy [C.Kim, K.Kim, OK 2016]

24 HEE for the LLM geometry Entanglement entropy Renormalized Entanglement entropy Positive for all LLM solutions è Monotonically decreasing behavior of c-function

25 Gauge/gravity duality and EE Entanglement entropy of the LLM geometry in 11d SUGRA (RT formula) Exact holography for the vev of CPO

26 Gauge/gravity duality and EE Entanglement entropy of the vacua in mabjm theory(rt formula) Exact holography for the vev of CPO Can we express the EE in terms of the one-point functions of the gauge invariant operators?

27 Gauge/gravity duality and EE Entanglement entropy of the vacua in mabjm theory(rt formula)???

28 Gauge/gravity duality and EE Leading contribution to the Einstein s equation G µ T µ Gauge invariant quantities KK reduction Read solutions of 4d EOM from LLM geometries???

29 Gauge/gravity duality and EE Entanglement entropy of the vacua in mabjm theory(rt formula) S = S( ho i i)uptoµ 2 0 order (for finite N) [Takayanagi et al 2013] EOM for EOM for up to linear order [D. Jang, Y. Kim, OK, D.Tolla, work in progress]

30 Discussion Exact holography for finite N and k between the mass-deformed ABJM theory and the LLM geometry in 11-dim. supergravity Can the RT formula be applied for finite N? S $ g µ Clarify the relation using the exact holography up to linear order è extend it to quadratic order (for finite N) Extension form the small mass (UV limit) expansion to the large mass (IR limit) expansion?

31 Thank you