slides

Holographic Entanglement Entropy
For 𝑊𝐴𝑑𝑆3
Wei Song
YMSC, Tsinghua University
with Qiang Wen and Jianfei Xu
Strings 2016, Tsinghua University
1
Bekenstein
Bardeen-Carter-Hawking
Strominger-Vafa
Maldacena
Introduction
𝑆𝐵𝐻
𝐴
=
4𝐺
AdS/CFT
𝑆𝑇𝐻
Holographic
Dualities
Strings 2016, Tsinghua University
2
Guica-Hartman-WS-Strominger
Bredberg-Hartman-WS-Strominger
Castro-Maloney-Strominger
Introduction
𝑆𝐵𝐻
𝐴
=
4𝐺
AdS/CFT
𝑆𝑇𝐻
Holographic
Dualities
ASG
Kerr/CFT
Holography for astrophysical black
holes, e.g., GRS 1905+105
Strings 2016, Tsinghua University
3
Introduction
Anninos-Li-Padi-WS-Strominger
Compere-Guica-Rodriguez
𝐴
=
4𝐺
AdS/CFT
𝑆𝐵𝐻
𝑆𝑇𝐻
Holographic
Dualities
SL(2,R)xU(1)
Kerr/CFT
ASG
WAdS/CFT
Detournay-Compere
Hofman-Strominger
Deournay-Hartman-Hofman
Compere-WS-Strominger
WAdS/WCFT
CFT:
Virasoro-Virasoro
WCFT:
Virasoro-Kac-Moody
Strings 2016, Tsinghua University
4
Ryu-Takayanagi
Casini-Huerta-Myers
Lewkowycz-Maldacena
Motivation
𝑆𝐵𝐻
𝐴
=
4𝐺
AdS/CFT
𝑆𝑇𝐻
Holographic
Dualities
𝑆𝑅𝑇
𝐴𝑚𝑖𝑛
=
4𝐺
Kerr/CFT
WAdS/CFT
𝑆𝐸𝐸
WAdS/WCFT
Strings 2016, Tsinghua University
5
Motivation
𝑆𝐵𝐻
WS-Wen-Xu
𝐴
=
4𝐺
AdS/CFT
𝑆𝑇𝐻
Holographic
Dualities
?
Kerr/CFT
WAdS/CFT
𝑆𝐸𝐸
WAdS/WCFT
Strings 2016, Tsinghua University
6
Outline
• Review of WAdS3
• EE in WAdS/CFT
• EE in WAdS/WCFT
Strings 2016, Tsinghua University
7
WAdS3 holography
2
𝑑𝑠𝑊𝐴𝑑𝑆
3
=
SL(2,R) 𝐿 × U(1) 𝑅
2
𝑑𝑠ƸAdS
3
+
SL(2,R) 𝐿 x SL(2,R) 𝑅
𝜆2 𝐴2
𝐴: 𝑆𝐿 2, 𝑅 𝐿
Invariant 1-form
SL(2,R) 𝑅 → 𝑈(1) 𝑅
WAdS3 /CFT: Vir 𝐿 -Vir 𝑅
Evidence:
𝑆𝐵𝐻 = 𝑆𝐶𝑎𝑟𝑑𝑦
EE: WS−Wen−Xu ? = 3𝑐 log(𝑎𝜖)
Compere-Guica-Rodriguez
Anninos−Li−Padi−WS−Strominger
Calabrieze−Cardy
WAdS3 /WCFT: Vir 𝐿 - Kac-Moody 𝐿
Evidence:
𝑆𝐵𝐻 = 𝑆𝐷𝐻𝐻
EE: WS−Wen−Xu
? = 𝑆𝐶𝐻𝐼
Castro−Hofman−Iqbal
AdS3 /WCFT
Compere-WS-Strominger
Detournay-Compere
Deournay-Hartman-Hofman
Strings 2016, Tsinghua University
Entanglement entropy in WAdS/CFT
• Assumptions: WAdS/CFT
• Method: Lewkowycz-Maldacena adapted
• Results: The bulk calculation agrees with the CFT expectation
Strings 2016, Tsinghua University
9
The generalized gravitational entropy
Lewkowycz-Maldacena
AdS/CFT, Einstein
• Replica trick extended to the bulk
generalized gravitational entropy
• Selecting a special curve
Extremal!
• Calculate the generalized gravitational entropy
presymplectic structure
= 𝑆𝑅𝑇
Strings 2016, Tsinghua University
10
Main assumption: holographic duality exist
Role of consistent asymptotic boundary conditions
• set up a dictionary
bulk metric
AdS, Einstein
boundary metric
• Replica trick extended to the bulk
• Selecting a special curve
boundary terms only cancels out when the two
metrics satisfy the same asymptotic b.c.
At n=1, the near cone expansion has to be
compatible with the asymptotic expansion
• Calculate the generalized gravitational entropy
Ambiguity in the presymplectic structure
Strings 2016, Tsinghua University
11
The generalized gravitational entropy in
WAdS/CFT
• Consistent boundary conditions
WS-Wen-Xu
Compere-Guica-Rodriguiz
boundary metric
• Selecting a special curve
NOT geodesic in WAdS,
But geodesic in AdS
Compatibility
𝐿෨
=
= 𝑆𝑞,𝑚 ≠ 𝑆𝑅𝑇
4𝐺
Strings 2016, Tsinghua University
12
Main Modifications to Lewkowycz-Maldacena for WAdS/CFT
• the expansion near the special surface has to be compatible with
the asymptotic expansion;
• periodic conditions are imposed to coordinates in the phase space
with diagonalized symplectic structure, not to all fields appearing in
the action;
• evaluating the entanglement functional using the boundary term
method amounts to evaluating the presymplectic structure at the
special surface, where some additional exact form may contribute.
Strings 2016, Tsinghua University
13
Entanglement entropy in WAdS/WCFT
• Assumptions: WAdS/WCFT, AdS/WCFT
• Method: Casini-Huerta-Myers approach adapted
• Results: The bulk calculation agrees with the WCFT calculation given
by Castro-Hofman-Iqbal
Strings 2016, Tsinghua University
14
CFT
z→ 𝑓 𝑧 , 𝑧ҧ → 𝑔(𝑧)ҧ
WCFT
𝑥→𝑓 𝑥 ,
𝑡 → 𝑡 + 𝑔(𝑥)
Detournay-Compere
Hofman-Strominger
Deournay-Hartman-Hofman
By carefully analyzing the moduli properties and keeping track of the
anomalis, a Cardy-like formula was derived
moduli transformation
Strings 2016, Tsinghua University
15
EE on WCFT
Casini-Huerta-Myers
Castro-Hofman-Iqbal
• EE on an interval
warped conformal mapping
• EE on
= Thermal entropy on
Strings 2016, Tsinghua University
16
WS-Wen-Xu
A bulk calculation
• WAdS with coordinates (U,V, ρ)
• Bulk coordinate transformation preserves two explicit U(1) isometries
SL(2,R)x U(1) quotient
• Black string with coordinates (u,v, r) and thermal entopy:
Strings 2016, Tsinghua University
17
Strings 2016, Aug.1-Aug.5, Tsinghua University
18
Summary of our results
• Assuming WAdS/CFT, we take the LM approach, and derive a holographic
calculation for the entanglement entropy. The bulk result agrees with the CFT
results.
• Assuming WAdS/WCFT, we take the CHM approach, and derive a
holographic calculation for entanglement entropy. The bulk result agrees with
the WCFT results.
Strings 2016, Tsinghua University
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Thank you!
Strings 2016, Tsinghua University
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