Gauge/gravity duality in Einstein-dilaton theory Chanyong Park Workshop on String theory and cosmology (Pusan, ) Ref. S. Kulkarni, B. –H. Lee, CP, and R. Roychowdhury arXiv:

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 non- conformal 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 plane with an appropriate infrared cutoff

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 (the crossover value), the specific heat is singular. For the dual QFT is thermodynamically unstable due to the negative specific heat. Only for, the dual QFT is thermodynamically stable.

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. 2. Brief Review of the linear response theory Ex) DC conductivity, Shear viscosity,... Here, we concentrate on the DC conductivity and charge diffusion constant. Goal : study on the linear response of the dual non-conformal QFT

Field theory Setup Consider a quantum field theory containing an operator with an external classical source. 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 behaves as a source of the current in the dual QFT. where 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 the equations of motion can be divided into two parts: the longitudinal and transverse one with

- Longitudinal modes - Transverse mode In the hydrodynamic limit ( ), we can solve these equations perturbatively

Boundary conditions - The retarded Green function of longitudinal modes - 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. 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 - We find several transport coefficients depending on the nonconformality. In this work, -We showed that the Einstein-dilaton theory is dual to the relativistic non- conformal QFT with the equation of state parameter Future directions, - other transport coefficients (shear viscosity, momentum diffusion constant) - physical properties of the dual QFT beyond the hydrodynamic limit

Thank you !