Fluid/gravity correspondence: A nonconformal realization in compactified D4 branes

Abstract

We develop the framework of boundary derivative expansion (BDE) formalism of fluid/gravity correspondence in compactified D4-brane system, which is a nonconformal background used in top-down holographic QCD models. Such models contain the D4-D6 model and the Sakai-Sugimoto (SS) model, with the background of the compactified black D4 branes under the near horizon limit. By using the dimensional reduction technique, we derive a 5D Einstein gravity minimally coupled with 3 scalar fields from the 10D D4-brane background. Following the BDE formalism of fluid/gravity correspondence in the conformal background, we directly derive all the first order transport coefficients for nonconformal gluonic matter. The results of the ratio of the bulk to shear viscosity and the sound speed agree with those obtained from the Green-Kubo method. This agreement guarantees the validity of the BDE formalism of fluid/gravity duality in the nonconformal D-brane background, which can be used to calculate the second order transport coefficients in nonconformal background.

Fluid/gravity correspondence: second order transport coefficients in compactified D4-branes

“…Re-expressed (2.16) in Eddington-Finkelstein coordinates dv = dt + dr r 3/2 f (r) with the coordinates are boosted as dv → −u µ dx µ , dx i → P i µ dx µ , one will 4 Q4 is defined through the normalization condition for F4: 2κ 2 µ4Nc = S 4 F4, where 2κ 2 = (2π) 7 l 8 s and µ4 = ((2π) 4 l 5 s ) −1 is the D4-brane charge. 5 This has been explained in [57]. One can calculate the Ricci scalar and the square of the Rieman tensor which separately gives R = − 5(14r 3 +r 3 H ) 6r 11/3 and RMNP QR M NP Q = 25(62r 6 +2r 3 r 3 H +125r 6 H ) 108r 22/3 , from which we can easily see that at r → 0, both of these two approach to zero.…”

“…If the reader wants to learn more about it, we recommend her/him to Ref. [57], where we develop a nonconformal version of the fluid/gravity correspondence by using the compactified, near-extremal black D4-brane.…”

“…The boundary stress tensor of the 5D reduced gravity theory can be derived out from Eq. (2.5) [57] as…”

“…In this work, we would like to offer a non-asymptotically AdS 5 background to holographically study the second order nonconformal relativistic fluid. With this purpose and based on our previous work [57] where we generalize the BDE formalism of fluid/gravity correspondence [25] in compactified D4-brane at first order, we are going to move to the second order in the same background and calculate the transport coefficients. Through Ref.…”

“…Through Ref. [57] and this work, we want to offer a nonconformal counterpart to Bhattacharyya et al's AdS 5 construction for the BDE formalism [25] and improve our knowledge about the second order transport properties for nonconformal relativistic fluid.…”