Anomaly-induced Thermodynamics in Higher Dimensional AdS/CFT

Gim Seng NgMcGill University

Northeast Gravity Workshop, 23rd April 2016

Based on : 1311.2940, 1407.6364 and 1505.02816 (with T. Azeyanagi, R. Loganayagam and M. J. Rodriguez)

Motivations

BHs have entropy ! Entropy-matching …

Often extremal or/and susy… How about non-extremal/non-SUSY finite temperature entropy matching?

BTZ entropy is reproduced by the universal Cardy’s formula

Higher-dimensional generalizations of Cardy’s formula?

Higher-dimensional AdS/CFT “Cardy entropy-matching”?

Outline

Chiral Half of Cardy’s Formula in

Replacement Rule from

2n

2n+1 2n

chiral half of Cardy

Consider a 2d CFT on a circle of radius R

Let us put it at finite temperature and rotation/boost

At high temperature:

R

Time

T � 1/R

Gravitational Anomaly(``Chiral Half’’/anomalous-part)

Weyl Anomaly

SCardy � 2πR1� R2Ω2

�cR + cL24 (4πT)

�+2πR2Ω1� R2Ω2

�cR � cL24 (4πT)

�

chiral HALF of Cardy

SCFT2,anom � 2πR2Ω1� R2Ω2

�cR � cL24 (4πT)

�

Generalizations to higher-d CFT (on sphere)Replacement rule

To understand the replacement rule, we need to review the following two things:

T=0 anomalies

Anomalous hydrodynamics

[Surowka, Loganayagam, Jensen, Yarom,…]

ANOMALY INFLOW

Anomalies are captured by Chern-Simons terms

Append an extra auxiliary direction.

The (2n+1) theory is anomaly-free, but with Chern-Simons terms

Non-conservation of the (2n)-theory is captured by the ``inflow’’ of charges into the extra auxiliary direction:

�μJμ|QFT2n � j�j�

QFT2nj� � �

�Panom�F

Panom = dICS

ICS

Panom F R

[Callan and Harvey]

anomalous hydro

Hydro: effective long-wavelength description:{ uα,T,μ, . . .}�Aα, gαβ, . . .

�

Hydro derivative expansion:

Leading anomalous contribution: parity-odd vorticity :

Jα = quα + . . . +(Jα)anom + . . .

Tαβ = Euαuβ + p�gαβ + uαuβ

�+ . . . +(Tαβ)anom + . . .

Vα � �αβ...uβ (�u)n�1...

anomalous hydro

(Jα)anom = ��F [T,μ]

�μ Vα + . . .

(Tαβ)anom =

�F � μ

��F�μ

�

T� T

��F�T

�

μ

��uαVβ + Vαuβ

�+ . . .

is like an anomalous Gibbs free energyRecall that standard relations:F

Q = ��G�μ , E = G� μ

��G�μ

�

T� T

��G�T

�

μ, S = �

��G�T

�

μ

Jα = quα + . . . +(Jα)anom + . . .

Tαβ = Euαuβ + p�gαβ + uαuβ

�+ . . . +(Tαβ)anom + . . .

Sanom = ��

�F�T

�

μ

��CFTVμdxμ = �

��F�T

�

μ

�

i(2πR2Ωi)

anomalous hydro

(Jα)anom = ��F [T,μ]

�μ Vα + . . .

(Tαβ)anom =

�F � μ

��F�μ

�

T� T

��F�T

�

μ

��uαVβ + Vαuβ

�+ . . .

Question: What is ?F

Sanom = ��

�F�T

�

μ

��CFTVμdxμ = �

��F�T

�

μ

�

i(2πR2Ωi)

chiral HALF of Cardy

Example: 2d CFT Cardy’s formula

Sanom = ��F�T (2πR2Ω)

SCFT2,anom � 2πR2Ω�cR � cL24 (4πT)

�

F = � cR � cL2(2π)242(2πT)2 = cg [R2]

��trR2�2(2πT)2

anomaly polynomial for 2d grav. anomaly

Replacement rule ORchiral half of cardy’s formula

[Surowka, Loganayagam, Jensen, Yarom,…]

(Jα)anom = ��F [T,μ]

�μ Vα + . . .

(Tαβ)anom =

�F � μ

��F�μ

�

T� T

��F�T

�

μ

��uαVβ + Vαuβ

�+ . . .

Vα � εαβ...uβ(�u)n�1

Sanom = ��

�F�T

�

μ

��CFTVμdxμ = �

��F�T

�

μ

�

i(2πR2Ωi)

F = Panom[F,R]|F�μ, [R2k]�2(2πT)2k

REPLACEMENT RULE

2d: Panom = cAF2 + cg [R2], F = cAμ2 + cg2(2πT)2

4d: Panom = cAF3 + cMF [R2], F = cAμ3 + cMμ� 2(2πT)2

6d: Panom � cg [R4], F = cg2(2πT)4

(Jα)anom = ��F [T,μ]

�μ Vα + . . .

(Tαβ)anom =

�F � μ

��F�μ

�

T� T

��F�T

�

μ

��uαVβ + Vαuβ

�+ . . .

Sanom = ��

�F�T

�

μ

��CFTVμdxμ = �

��F�T

�

μ

�

i(2πR2Ωi)

Outline

Chiral Half of Cardy’s Formula in

Replacement Rule from- Testing AdS/CFT in the presence of anomalies - What bulk geometric structure captures the replacement rule?

2n

2n+1 2n

A holographer’s recipe

1. Find your favorite AdS bulk theories

2. Find the relevant AdS BH solutions

3. Calculate responses, charges, thermodynamics…

4. Match/predict CFT results

Issue 1: Need gauge-gravitational Chern-Simons terms

Issue 2: Need AdS-Kerr-Newman+CS solutions (with all rotations/charges turned on)

Issue 3: Holographic renormalization/charges / entropy for Chern-Simons terms

step 1: favorite Ads gravity setup

Toy model: D=2n+1 Einstein-Maxwell+negative c.c. +CS terms Ics

Equations of motion:

Rab � 12 (R� 2Λ) gab = 8πGN

�(TM)ab + (TH)ab

�

�bFab = g2YM (JH)a

(TM)ab

JH = � �

��Panom

�F

�

(TH)ab = �cΣ(ab)c Σ(ab)c = �2 �

��Panom�Rab

�

CS/``Hall’’ contributions:

Maxwell contribution:

Panom = dICS

step 2: fluid/gravity BH solution (no anomalies)

Gravity dual of charged rotating fluid (for now, without anomalies)

ds2 = �2uμdxμdr + r2��f(r,m, q)uμuν + Pμν

�dxμdxν +

A = Φ(r, q)uμdxμ + Pμν = gμν + uμuν

Φ(r, q) =q

r2n�2 f(r,m, q) = 1� mr2n +

12κq

q2r2(2n�1)

Non-trivial bulk radial dependence:

Horizon values:Φ(rH) = μ, ΦT(rH) = 2πT ΦT(r) � r2

2dfdr

step 2: fluid/gravity BH solution (with anomalies)

Gravity dual of anomalous charged rotating fluid

ds2 = �2uμdxμdr + r2��f(r,m, q)uμuν + Pμν

�dxμdxν +

A = Φ(r, q)uμdxμ +

Leading contributions from the CS-terms :

+gV(r,m, q)(uμVν + uνVμ)dxμdxν + . . .

+aV(r,m, q)Vμdxμ + . . .

gV, aV

Rab � 12 (R� 2Λ) gab = 8πGN

�(TM)ab + (TH)ab

�

�bFab = g2YM (JH)a

Bulk replacement rule:

TH � �2r

��G�ΦT

�JH � �r

��G�Φ

�

G � Panom�F � Φ, [R2k] � 2Φ2kT

�ΦT(r) � r2

2dfdr

step 3&4: currents and replacement rule

�Tαβ

�anom =

�Tαβ

�

anom=

�G� Φ �G

�Φ � ΦT�G�ΦT

�

r=rH

�uαVβ + uβVα

�

G(r = rH) = F [F � μ, [R2k] � 2(2πT)2k]

Bulk replacement rule metric corrections

boundary stress-tensor/currents

(Jα)anom � gμα (Frμ)anom | = ��

�G�Φ

�

r=rHVα

Bulk replacement rule-> boundary replacement rule !!

step 3&4: entropy

Tachikawa Formula (see also Solodukhin and Bonora-Cvitan-Prester-Pallua-Smolic) (note: NOT Wald’s formula applied to Ics)

SCS =

� ��

k=1(8πk)ΓNR2k�2

N�Panom� [R2k]

Panom = dIcs, ΓN,RN =

Applying this to the fluid/gravity metric we found gives

Sanom = ��

�F�T

�

μ

��CFTVμdxμ = �

��F�T

�

μ

�

i(2πR2Ωi)

Computations are opaque, long and tedious … (what bulk structures imply the replacement for entropy?)

summary, current and future work

Higher-dimensional chiral-half of Cardy’s formula (Replacement rule)

Construct (anomalous) fluid/gravity solutions in the bulk for the Einstein-Maxwell-CS theory

Found ``bulk replacement rule’’ which implies the boundary replacement rule

Current/Future: Time-dependence (non-stationary) and/or higher-order?More realistic AdS/CFT setup … add matter and etc (should not alter the conclusions)

Anomaly-induced entanglement entropy [Azeyanagi-Loganayagam-Ng, 1507.02298]