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Holographic effect of Chern-Simons term in 5d RN-AdSbackgroundto magnetic conductivity and photo emission rate Shingo TakeuchiJune 4th ’10Nat’l Taiwan Univ.and Nat’l Tsing Hua Univ.

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Holographic effect of Chern-Simons term in 5d RN-AdS background to magnetic conductivity and photo emission rate

Shingo Takeuchi

June 4th ’10

Nat’l Taiwan Univ. and Nat’l Tsing Hua Univ.

JHEP 1004:071,2010 arXiv:0910.3722 [hep-th]

NPB820,2009 arXiv:0901.0610 [hep-th]

APCTP, Korea → KITPC-CAS, Beijing China, this December

arXiv:100X.XXXX [hep-th]

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D=4 N=2 SYM with finite density at finite temperature 5d RN-AdS, which is Schwarzschild AdS BH with U(1) charge.

dual

5d RN-AdS

STU model

CS term appears in the case coming from STU model There are two models, both can get to 5d RN-AdS

Introduction and Motivation

SAdS BH + D7

U(1) anomaly in the dual field theory next slide

Its analysis

We will study 5d RN-AdS with Chern-Simons term.

Holographic anomaly effect in dual field theory

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Ref.) D.T.Son and P.Surowka arXiv:0906.5044 [Hep-th]

Relativistic quantum hydrodynamics

U(1) anomaly under existence of U(1) current

5d RN-AdS Field theory side :

Gravity side :

5d Einstein-Maxwell-AdS theory with CS term

Duality of the U(1) anomaly :

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2. Numerical analysis in full frequency and momentum

• The dual gravity model

• Hydrodynamics of RN-AdS with CS term

• Effect of CS term in the dual field theory

3. Summary and future work

Abstract

1. Hydrodynamics

Holographic U(1) anomaly effect of CS term in 5d RN-AdS

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Explanation of the Model

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SAdS BH x S5+D7

(Schwarzschild AdS blackhole with U(1) charge) Our background :

5d Schwarzschild AdS BH

How do we introduce finite density ?

dual

One way is to put probe D7-branes on 5d SAdS BH x S5.

5d RN-AdS

Another way is STU model

with reduction of S5-part, referred later.

Gravity model and dual field theory

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5d RN-AdS and action

Action (5d Einstein-Maxwell-AdS theory with CS term) :

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When taking variation, derivative of variation appears, giving non-zero contribution under b.d. condition . This means that we cannot get EOM. Then what do we do with it ?

GH term is added, which has a rule to cancel such a term.

This is regularization.

Counter term and GH term

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? D7

D3

Is this possible ?

The way in which 5d Chern-Simon term appears (if it is possible) :

Answer is no.

Is there CS term in the 5d SAdS BH x S5 +D7 ?

STU model

Therefore, there is no Chern-Simons term in 5d SAdS BH x S5 with D7.

Then, how do we take in Chern-Simons term ?

• We put D7-branes on 5d SAdS BH x S5

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STU model, which can get to 5d RN-AdS with CS term

hep-th/9810277 hep-th/9910219

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RN-AdS

STU model

However in the case coming from STU model, we can have CS term.

We study how effect from CS term in hydrodynamics on RN-AdS.

The both two can get to RN-AdS :

• STU model

• 5d Schwarzschild AdS BH x S5 + D7

Summary for our model and what we will do

5d SAdS BH x S5 + D7

My comment : Two kinds of U(1) charge in the absence or existence of CS term

This becomes the difference in meaning of dual U(1) charge, U(1) R-charge or U(1) baryon charge …

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Calculation

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tensor Classical perturbation

3 types of perturbation Fourier expansion

vector type : tensor type : scalar type :

scalar

Symmetry for rotation of x-y plane ( Decoupling into 3 types )

Setup for calculation

Gauge fixing

c.f.) Discussion by the helicity base

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vector type :

tensor type : scalar type :

Only if vector mode is considered, effect of CS term enters.

Maxwell eq. :

Argument of gauge field :

Gauge fields in the 3 type of perturbations :

The perturbation mode where the effect of CS-term enters

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Einstein eq.

Maxwell eq.

Vector type : and

EOM :

New variable :

We try to diagonalize this EOM.

How to calculate

combination of variables

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1st 2nd 3rd 0th

Diagonalized EOM !!

Hydrodynamic analysis

Pick up long-wave, which leads to seeing fluid behavior Diagonalization of EOM

(Decoupled EOM)

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Actual forms of solution

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Results

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Classical sol.

Effect of CS term in dual Quark-Gluon-Plasma (Results)

Green function GKP-W relation

Real-time AdS/CFT

[Son, Starinets], [Herzog, Son]

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Diffusion eq.

Diffusion constant

• Effect of CS term in dispersion relation

Denominator-part gives dispersion relation.

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Definition :

Chiral Magnetic Conductivity arXiv:0808.3382 [hep-ph] The Chiral Magnetic Effect arXiv:0907.0494 [hep-lat] Numerical evidence of chiral magnetic

effect in lattice gauge thoery arXiv:0907.5007 [hep-ph] Chiral Magnetic Conductivity

vector type : tensor type :

scalar type :

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Relation among CS term, U(1) anomaly and the conductivity

U(1) anomaly

Magnetic field aligns magnetic moment (spin) of quarks along the direction of magnetic field.

CS term

dual

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This is consistent with ours. However we have further obtained magnetic conductivity.

Summary for study of the effect of CS term in RN-AdS

It has been turned out that it appears in vector mode.

(1) Dispersion relation at 3rd order (2) Magnetic conductivity

Particularly, it appears in following two points as

We have studied the effect of CS term in hydrodynamics of RN-AdS.

Bindusar Sahoo, Ho-Ung Yee arXiv:0910.5915[hep-th]

published at almost the same time Dispersion relation

Effect of CS term to instability in blackhole-type spacetime

Future work :

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Tachyonic mode appears caused by CS term (A novel instability) S.Nakamura, H.Ooguri and C.-S. Park

arXiv:0911.0679 [Hep-th]

• Perturbations on 5d RN-AdS only with classical electronic field

• Analysis of BF bound

(threshold for appearance of imaginary mass of the perturbation)

Now, I am working with Feng-Li and Nakamura-san on the instability in a blackhole-type space-time.

Ongoing work

Perturbation is only U(1) gauge field

Perturbation are both gauge field and gravity

BF bound is kept, which means no tachyonic mode and 5d RN-AdS can remain stable.

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Holographic anomaly effect to photo emission rate in Chern Simons–term in 5d RN-AdS

(Ongoing work)

Sang-Jin Sin, Kwanghuan Jo and S.T.

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Full Analysis of Green function

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Results

S.C.Huot, P.Kovtun, A.Starinets, G.D.Moorse Yaffe A.Parnachev, D.A.Sahakyan

J.Mas, Jonathan Shock, J.Tarrio, D Zoakos Our current study

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Other works in progress concerning hQCD

Exact zero temperature in 5d RN-AdS

Quark number susceptibility with back-reaction coming from quark mass and number of flavor D-breans

[ Sang-Jin Sin and S.T. ]

[ Youngman kim, Kyung-il Kim and S.T. ] [arXiv:1008(?).XXXX]

[arXiv:1XXX.XXXX]

EOM in three type of perturbation :

Then the singular in EOM becomes more hard. ( horizon locates at u=1 ) How do we get rid of or manage to this singular ?

• Matching two solutions at horizon and at boundary.

• Rewrite EOM to Schrodinger type eq. without time-dependence.

Difficult !!

(gr-qc/9603021) (arXiv:0910.0645, arXiv:0910.0645…)

參考文獻

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