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]
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
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
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Duality of the U(1) anomaly :
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
Explanation of the Model
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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②
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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
? 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
STU model, which can get to 5d RN-AdS with CS term
hep-th/9810277 hep-th/9910219
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 …
Calculation
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
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
Einstein eq.
Maxwell eq.
Vector type : and
EOM :
New variable :
We try to diagonalize this EOM.
How to calculate
combination of variables
1st 2nd 3rd 0th
Diagonalized EOM !!
Hydrodynamic analysis
Pick up long-wave, which leads to seeing fluid behavior Diagonalization of EOM
(Decoupled EOM)
Actual forms of solution
Results
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]
Diffusion eq.
Diffusion constant
• Effect of CS term in dispersion relation
Denominator-part gives dispersion relation.
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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②
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 :
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.
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.
Full Analysis of Green function
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
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…)