Gauge/gravity duality in
Einstein-dilaton theory
Chanyong Park (CQUeST)
@ Workshop on String theory and cosmology (Pusan, 2012.06.14)
Ref.
S. Kulkarni, B. –H. Lee, CP, and R. Roychowdhury
arXiv:1205.3883.
Motivation
The AdS/CFT correspondence is the fascinating and important subject in
studying the strongly interacting QFT, where the dual QFT of the AdS space
becomes conformal.
However, the real physics usually appears as a non-conformal theory except
several critical points (like UV and IR fixed points and the phase transition
point).
Can we generalize the AdS/CFT correspondence to the non-conformal case?
There exists a generalization of this duality, the so-called gauge/gravity
duality. Under this generalized concept,
- what is the dual gravity theory to the non-conformal QFT?
- which parameter of the gravity can describe the non-conformality of QFT?
To answer these questions is the main goal of this work.
• Outline
1. Einstein-dilaton theory and its dual QFT
2. Brief Review of the linear response theory
3. Transport coefficients of the dual QFT
4. Conclusion
1. Einstein-dilaton theory and its dual QFT
Consider the Einstein-dilaton theory
with a Liouville-type dilaton potential
where
and
are the cosmological constant and an arbitrary constant.
From now on, we take into account the negative cosmological constant
and set
for simplicity.
Einstein equation and equation of motion for dilaton
To solve these equations, we take the following ansatz
with
Non-black brane solution
Gubser bound
the above solution is well-defined only for
AdS space limit
for
, the above solution reduces to the 4-dimensional AdS space.
Symmetry
- Poincare symmetry of the boundary coordinates
- the scale symmetry
These isometry of the AdS space appears as the conformal symmetry of the
dual QFT.
This is called the AdS/CFT correspondence.
The generalization of the AdS/CFT correspondence
There exists the generalized concept, the so-called gauge/gravity duality,
which has been widely used in the holographic study on the strongly
interacting QFT,
ex) Sakai-Sugimoto model, Lifshitz and Schrodinger geometries, …
Assuming the gauge/gravity duality even in the Einstein-dilaton theory,
we can find that the dual QFT of it is the relativistic non-conformal QFT,
because the Poincare symmetry of the boundary space still remains but the scale
symmetry is broken.
Now, we investigate the black brane solution of the Einstein-dilaton
theory, whose dual theory is represented as the relativistic nonconformal QFT at finite temperature.
The black brane solution of the Einstein-dilaton theory
with
where
is the black brane mass and a constant
is introduced for later
convenience
and
is a regularized volume in
appropriate infrared cutoff
plane with an
Thermodynamics
- Hawking temperature
- Bekenstein-Hawking entropy
Usually, the black hole (or black brane) provides a well-defined analogous
thermodynamic system, so the black hole should satisfy the first thermodynamic
law
Notice that following the gauge/gravity duality, the thermodynamics of the black
hole can be identified with that of the dual QFT.
- the thermal energy of the dual QFT
- the free energy of the dual QFT
- using the definition of pressure
the equation of state parameter
of the dual QFT (or black brane) becomes
As a result, we can see that the dual theory to the Einstein-dilaton gravity is
generally a non-coformal QFT with the above equation of state parameter.
Especially, for
the dual QFT reduces to conformal theory, whose
energy-momentum tensor is traceless.
Thermodynamics instability of the dual QFT
Due to the Gubser bound
the range of the equation of state parameter is given by
In this parameter range, the specific heat is given by
For
For
(the crossover value), the specific heat is singular.
the dual QFT is thermodynamically unstable due to the
negative specific heat.
Only for
, the dual QFT is thermodynamically stable.
2. Brief Review of the linear response theory
Goal :
study on the linear response of the dual non-conformal QFT
Transport coefficients
- which are typical parameters in an effective low energy description (such
as hydrodynamics or Langevin equations)
- once they are specified, they completely determine the macroscopic
behavior of the medium.
Ex) DC conductivity, Shear viscosity, ...
Here,
we concentrate on the DC
constant.
conductivity and charge diffusion
• Field theory Setup
Consider a quantum field theory containing an operator
an external classical source
with
.
At the level of linear response theory
- the one-point function of
is linear in
- when expressed in Fourier modes, the proportionality constant is simply the
thermal retarded correlator
of
The low frequency limit of this correlator is of physical importance, as it
defines a transport coefficients
which implies that if we apply a time varying source
, the response of
the system in the low energy limit is given by
For the DC conductivity,
the gauge/gravity duality implies that a bulk vector fluctuation
a source of the current
where
behaves as
in the dual QFT.
is the AC conductivity because it depend on the frequency and
related to the retarded Green function
Note that
is the AC conductivity because it depend on the frequency,
and related to the retarded Green function
In the zero momentum
, the zero frequency limit of it reduces to the
DC conductivity
Goals :
Using the gauge/gravity duality, investigate the charge diffusive mode of
the strongly interacting dual QFT.
3. Transport coefficients of the dual QFT
Turn on the U(1) vector fluctuations on the previous black brane background
After taking the
gauge and the following Fourier mode expansions
with
the equations of motion can be divided into two parts: the longitudinal and
transverse one
- Longitudinal modes
- Transverse mode
In the hydrodynamic limit (
equations perturbatively
), we can solve these
Boundary conditions
- Incoming BC at the horizon, which breaks the unitarity of the dual QFT.
- Dirichlet BC at the asymptotic boundary, which fixes the source of dual QFT.
- The retarded Green function of longitudinal modes
in the low frequency and low momentum limit (hydrodynamic limit),
the retarded Green functions become
The longitudinal Green function has a
charge diffusive pole governed by the
following dispersion relation
with
the charge diffusion constant
- The charge diffusion constant implies that the quasi normal mode (charge
current of the dual QFT) eventually diffuses away back into the thermal
equilibrium with a half-life time
- The retarded Green function of transverse mode
- There is no pole.
- The DC conductivity is given by
4. Discussion
In this work,
-We showed that the Einstein-dilaton theory is dual to the relativistic nonconformal QFT with the equation of state parameter
- We find several transport coefficients depending on the nonconformality.
Future directions,
- other transport coefficients
(shear viscosity, momentum diffusion constant)
- physical properties of the dual QFT beyond the hydrodynamic limit
Thank you !