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.

Second order transport coefficients of nonconformal relativistic fluids in various dimensions from Dp-brane

“…which is easy to verify. The D4-brane tensor perturbation (3.26) is the same as in the compactified case [13,51] which can be shown by first using (3.27) and then the arctangent addition formula. So one can draw the conclusion that compactifying one direction of D4brane will not effect the coefficients of viscous tensors, i.e., η, τ π and λ 1,2 will not change compared with the compactified D4-brane case.…”

“…Following the standard prescription of fluid/gravity correspondence [7], we promote the boost parameters u µ and r H to be x µ dependent and expand them to the first order in ∂ µ like in [51], then we get…”

“…Here we use different conventions for setting the perturbations like k and w µ from that of [13,51]. After expanded to first order, (3.3) becomes 4) in which all the first order perturbations only depend on r. This is because the expansion actually happens at a specific point on the boundary (which we set by default the original point).…”

“…From the above we can see that the vector perturbation diverges for p = 5 and can not be normalized for p = 6. Since we use different convention for setting the vector perturbation in (3.3), that's why the above for p = 4 is not the same as in [51]. The advantage of this convention is that it can circumvent the appearance of divergent part at the second order, compared with [7,13,51].…”

“…References of this kind are [13,16,37,[39][40][41][42][43][44][45][46][47][48][49][50][51], in which [39][40][41][42]47] are about the relativistic fluid that dual to supergravity backgrounds. And [13,16,37,[44][45][46][47][48][49][50][51] are about NS5-brane [44], Dp-brane [16,37,[46][47][48][49][50] and compactified D4-brane [13,45,51]. To be more specific, [37, 44-46, 48, 50, 51] study the first order transport coefficients while [13,16,47,49] investigate the second order ones, in which [16,…”