In this paper we study bulk viscosity in a thermal QCD model with large number of colors at two extreme limits: the very weak and the very strong 't Hooft couplings. The weak coupling scenario is based on kinetic theory, and one may go to the very strong coupling dynamics via an intermediate coupling regime. Although the former has a clear description in terms of kinetic theory, the intermediate coupling regime, which uses lattice results, suffers from usual technical challenges that render an explicit determination of bulk viscosity somewhat difficult. On the other hand, the very strong 't Hooft coupling dynamics may be studied using string theories at both weak and strong string couplings using gravity duals in type IIB as well as M-theory respectively. In type IIB we provide the precise fluctuation modes of the metric in the gravity dual responsible for bulk viscosity, compute the speed of sound in the medium and analyze the ratio of the bulk to shear viscosities. In M-theory, where we uplift the type IIA mirror dual of the UV complete type IIB model, we study and compare both the bulk viscosity and the sound speed by analyzing the quasi-normal modes in the system at strong IIA string coupling. By deriving the spectral function, we show the consistency of our results both for the actual values of the parameters involved as well for the bound on the ratio of bulk to shear viscosities. 65 5.1 The mirror type IIA model and its M-theory uplift 66 5.2 Quasi-normal modes, attenuation constant and the sound speed 70 5.3 The case with a vanishing bare resolution parameter 76 5.4 Shear viscosity, entropy and the bulk viscosity bound 80 6. Type IIA spectral function and the viscosity bound at strong coupling with non-zero flavors 85 6.1 Background gauge fluxes and perturbations on the flavor branes 86 6.2 Equation of motion for gauge field fluctuations 88 6.3 On-shell action and the strong coupling spectral function 95 6.4 The strong string coupling limit and pure classical supergravity 103 -1 -7. Conclusions and discussions 107 A. A Gauge-Invariant Combination of Scalar Modes of Metric Perturbations 112 A.1 The equation of motion for the fluctuation mode H tt 113 A.2 The equation of motion for the combined mode H s 115 A.3 The equations of motion for the remaining fluctuation modes 116 B. A derivation of the on-shell action and the Green's function 118C. Effective number of three-brane charges with background threeforms and the horizon radius 1201 Violation of this bound is seen in the presence of higher derivative terms, discussed first in [23]. In the absence of these terms, the KSS [22] bound continues to hold at strong 't Hooft coupling.2 The non-conformal string theory studied in [30] is different from what we consider here. In [30] it's the N = 2 * supersymmetric gauge theory obtained by a mass deformation of N = 4 Super Yang Mills theory. See also [31] and [32] for an even earlier study on bulk viscosity from first principles.

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In this paper we provide a quantum field theoretical study on the shear and bulk relaxation times. First, we find Kubo formulas for the shear and the bulk relaxation times, respectively. They are found by examining response functions of the stress-energy tensor. We use general properties of correlation functions and the gravitational Ward identity to parametrize analytical structures of the Green functions describing both sound and diffusion mode. We find that the hydrodynamic limits of the real parts of the respective energy-momentum tensor correlation functions provide us with the method of computing both the shear and bulk viscosity relaxation times. Next, we calculate the shear viscosity relaxation time using the diagrammatic approach in the Keldysh basis for the massless λφ 4 theory. We derive a respective integral equation which enables us to compute ητ π and then we extract the shear relaxation time. The relaxation time is shown to be inversely related to the thermal width as it should be.

The N ¼ 4 super Yang-Mills plasma is studied in the regime of weak coupling. Collective excitations and collisional processes are discussed. Since the Keldysh-Schwinger approach is used, the collective excitations in both equilibrium and nonequilibrium plasma are under consideration. The dispersion equations of gluon, fermion, and scalar fields are written down and the self-energies, which enter the equations, are computed in the hard loop approximation. The self-energies are discussed in the context of effective action which is also given. The gluon modes and fermion ones appear to be the same as those in the QCD plasma of gluons and massless quarks. The scalar modes are as of a free relativistic massive particle. The binary collisional processes, which occur at the lowest nontrivial order of the coupling constant, are reviewed and then the transport properties of the plasma are discussed. The N ¼ 4 super Yang-Mills plasma is finally concluded to be very similar to the QCD plasma of gluons and light quarks. The differences mostly reflect different numbers of degrees of freedom in the two systems.