Anomalies, Chern-Simons Terms, and
Black Hole/Entanglement Entropy!
Tatsuo Azeyanagi (ENS) !
!
Based on works with !
R. Loganayagam (IAS&ICTS), G.S. Ng (McGill), M.J. Rodriguez (AEI&Utah)!
Workshop “Gravity, Quantum Fields and Entanglement” @ Lorentz Center, Leiden, Netherlands, January 4th, 2016 !
Motivation!
CFT2n with Gravitational/Mixed Anomaly !
at Finite Temperature!
AdS/CFT!
Gravity with Gravitational/Mixed Chern-Simons Term
on AdS2n+1 BH Background!
20th Anniversary of Strominger-Vafa!
Strominger-Vafa and Beyond!
“Traditional” Black Hole Microstate Counting!
[Strominger-Vafa], [Strominger], …, [Cardy] (cf. [Brown-Henneaux(1986)]) !
BTZ BH, Some BPS BHs Entropy ⇔ Cardy Formula for CFT2 !
Beyond “Traditional” BH Microstate Counting!
・Warped AdS3 BH Entropy ⇔ Cardy-like Formula for Warped CFT2!
[Li-Song-Strominger],…, [Detournay-Hartman-Hofman] … !
・AdS4 BPS BH Entropy　⇔　SUSY Localization for BPS Index !
[Cacciatori-Klemm]…, [Benini-Hristov-Zaffaroni]!
・Chern-Simons Contribution to BH Entropy ⇔ “Replacement Rule” !
[Solodukhin], [Tachikawa], [TA-Loganayagam-Ng], [Chapman-Neiman-Oz]… !
[Loganayagam-Surkowski], [Jensen-Loganayagam-Yarom]…!
10th Anniversary of Ryu-Takayanagi!
Ryu-Takayanagi and Beyond!
Holographic Entanglement Entropy for Einstein Gravity!
[Ryu-Takayanagi], [Lewkowycz-Maldacena]!
EE for Region A ⇔ Co-dim 2 Minimal Surface Area!
Area of !
minimal surface !
A!
Higher-Derivative Terms and Holographic EE !
・Higher-Curvature Gravity!
[de Boer-Kulaxizi-Parnachev], [Hung-Myers-Smolkin] …, !
[Bhattacharyya-Kaviraj-Sinha], [Fursaev-Pastrushev-Solodukhin] [Dong], [Camps], … !
・ Gravitational/Mixed Chern-Simons Terms!
[Castro-Detournay-Iqbal-Perlmutter], [Guo-Miao], [TA-Loganayagam-Ng], … !
Outline!
1)  Brief Review on !
Anomaly-Induced Transports and Replacement Rule!
2) Gravity Dual with Chern-Simons Terms!
3) Holographic EE for Chern-Simons Terms!
Anomaly-Induced Transports
and Replacement Rule!
Anomalies in QFT!
(Quantum) Anomalies in QFT2n!
Today’s focus: Global Anomalies!
More precisely, Gravitational Anomaly, Mixed U(1)-Gravitational Anomaly!
Anomalies at Zero Temperature!
Beautiful structure associated with anomalies: !
・Adler-Bardeen Theorem → Anomalies are one-loop exact!
・ Systematic study based on Anomaly Inflow Mechanism (next slide)!
Anomaly Inflow Mechanism!
[Callan-Harvey]!
Chern-Simons (CS) Term!
Bulk current!
from CS term!
one-loop exact!
Anomalies are characterized by Anomaly Polynomials!
Examples!
Anomaly!
Chern-Simons Term!
Anomaly Polynomial!
2d gravitational!
4d mixed !
6d gravitational!
: U(1) field-strength 2-form!
・・・
・・・
・・・
: U(1) potential 1-form !
: connection 1-form!
: curvature 2-form!
Anomalies at Finite Temperature!
Big development recently!!
Anomaly-Induced Transport!
[Son-Surowka, Bhattacharyya et.al.!
Erdmenger et.al., Torabian-Yee, …]!
In hydrodynamic limit, anomalies generate new type of transports!
(example) U(1) Current or Entropy Current!
Without anomalies!
Extra term in the presence of anomalies!
Replacement Rule!
Anomaly-induced transport coefficients are!
completely determined from anomaly polynomial!
[Jensen-Loganayagam-Yarom] !
Stress Tensor!
U(1) Current!
Entropy Current!
where!
(example)!
Short Summary!
・Existence of Anomaly-Induced Transports!
・Replacement Rule for Anomaly-Induced Transport Coefficients!
! ! ! ! ! !Question:!
Replacement Rule from Gravity Dual?!
Some 5d analysis !
[Chapman-Neiman-Oz], [Karzeev-Yee], !
[Landsteiner-Megias-Melgar-Pena-Benitez], … !
Gravity Dual with Chern-Simons Terms!
Setup!
CFT Side!
Theory: 2n-dim CFT with grav./mixed anomaly (at finite temperature)
Configuration: U(1) charged rotating (conformal) fluid
Gravity Side!
Theory: (2n+1)-d Einstein-Maxwell-Chern-Simons theory!
with negative cosmological const.
Choice of CS terms → Same as anomaly inflow!
Configuration: U(1) charged rotating black hole on AdS2n+1
Equations of Motion!
Einstein eq. and Maxwell eq.!
・Maxwell part of stress-energy tensor!
・CS contribution to stress tensor and U(1) current!
EoM is covariant while CS Lagrangian is not
Gravity Dual of Anomalous Fluid (1)!
How to construct rotating charged AdS BH dual to rotating charged fluid? !
Fluid/Gravity: AdS/CFT in Hydrodynamic Limit !
[Bhattacharya-Hubeny-Minwalla-Rangamani]!
Recipe!
(1)! Static AdS BH!
(2)!
(3)! Derivative expansion !
(in Eddington-Finkelstein)!
BH!
boost!
to solve EoM!
BH!
BH!
metric!
(NOT solution)!
Boundary stress-energy tensor & U(1) current = Those for fluid!
where
= fluid velocity !
Gravity Dual of Anomalous Fluid (2)!
Detail of Steps!
[TA-Loganayagam-Ng-Rodriguez]!
(1)  Start with EoM for Einstein-Maxwell theory and !
charged AdS BH solution !
(2) Carry out fluid/gravity expansion (up to 2nd order)!
(3)Take into account CS term and solve EoM under the ansatz!
(Leading order terms proportional to pseudo-vector):!
projection matrix!
electric potential!
“BH Entropy is Noether Charge”!
Before talking about CS terms…!
BH Entropy for Covariant Lagrangian!
[Wald, Lee-Wald, Iyer-Wald]!
・Covariant Lagrangian!
・Killing vector of BH!
Differential Noether charge!
(1st law)!
Wald’s BH entropy formula!!
・ Differential Noether charge is covariant!
Noether Procedure!
How to construct differential Noether charge?!
Definition of differential Noether charge!
where pre-symplectic current
Variation of !
Lagrangian!
EoM!
is defined by!
Boundary
term!
Both diff. Noether charge and pre-symplectic current
have ambiguity to add total derivative terms!
Wald Formalism and Extension!
Key Point of Wald Formalism!
[Lee-Wald, Iyer-Wald]!
Definition of Wald’s pre-symplectic current!
Covariant
→ !Covariant
→ !Covariant
!
Generalization to CS Term!
・ Straightforward generalization of Wald formalism![Tachikawa] (see also [Solodukhin])!
!→ BH entropy formula for CS term proposed. “Tachikawa formula”!
・In 5d and higher, one needs to take appropriate coordinate & gauge !
to obtain Tachikawa formula … !
[Bonora et. al.]!
(Non-)Covariance!
Origin of Non-Covariance!
[TA-Loganayagam-Ng-Rodriguez]!
Non-Covariant…!
Non-Covariant
→ !Non-Covariant
Manifestly Covariant Formalism!
→ ! Non-Covariant
!
[TA-Loganayagam-Ng-Rodriguez]!
CS contribution to EoM 〜 Derivatives of anomaly polynomial!
Covariant!!
Start with EoM and rewrite RHS in the total derivative form!
Covariant
→ !Covariant
→ !Covariant
!
Chern-Simons BH Entropy!
Evaluation of our differential Noether charge at bifurcation horizon!
BH Entropy Formula for CS Term!
For general anomaly polynomial!
We Covariantly Derived Tachikawa Entropy Formula! !
Replacement Rule from Gravity!
Fluid/Gravity Solution!
[TA-Loganayagam-Ng]!
Non-anomalous part!
Anomalous part!
Non-anomalous part!
Anomalous part!
Differential Noether Charge!
Linear Terms in　!
Two potential !
contributions!
Reproduces Replacement Rule!!
Holographic Entanglement
Entropy for CS Terms!
Holographic Entanglement Entropy(1)!
Entanglement Entropy and Replica Method
A!
n-replicated geometry!
Ryu-Takayanagi Formula (for Einstein Gravity)!
[Ryu-Takayanagi]!
Area of !
minimal surface !
A!
Holographic Entanglement Entropy (2)!
Proof of Ryu-Takayanagi Formula!
[Lewkowycz-Maldacena],[Dong]!
see also [Fursaev], [Fursaev-Pastrushev-Solodukhin]!
AdS/CFT!
contribution from singularity is not included!
regularized cone!
→ Generalization to general higher-curvature Lagrangian!
[Dong], [Camps],…!
Holographic EE for CS Term (1)!
Holographic EE for CS Term!
Direct (n-1)-expansion is easy in 3d … but much harder in higher dim.!
[Castro-Detournay-Iqbal-Perlmutter]!
(example) 7d gravitational CS term!
CS term!
Anomaly polynomial!
many terms…!
Strategy for General CS Term!
simple!!
[TA-Loganayagam-Ng], [Guo-Miao]!
Uplift one dimension　→　“Anomaly polynomial is Lagrangian”!
We can also use some result by Dong and Camps!!
Holographic EE for CS Term (2)!
Holographic EE Formula for　! [TA-Loganayagam-Ng]!
BH entropy part!
Lorentz-Frame Dependence!
[Castro-Detournay-Iqbal-Perlmutter], !
[TA-Loganayagam-Ng]!
Anomaly-induced EE changes under Lorentz boost!
Lorentz boost orthogonal to
= gauge trans.!
CFT computation [Castro-Detournay-Iqbal-Perlmutter], [Iqbal-Wall], [Nishioka-Yarom], [Hughes-Leigh-Parrikar-Ramamurthy]!
Summary!
・ Manifestly covariant diff. Noether charge for CS terms!
! → Tachikawa BH entropy formula proved covariantly !
・ Holographic derivation of replacement rule!
・ Holographic EE for CS terms !
→　Anomaly polynomial is useful! !
Thank You !!