Bulk Viscosity, Speed of Sound and Contact Structure at Intermediate Coupling

In the context of strongly coupled thermal QCD-like theories, the bulk viscosity($\zeta$)-to-shear viscosity($\eta$) ratio using the type IIA-theory dual of thermal QCD-like theories was shown to vary like $\frac{1}{3} - c_s^2$ (arXiv:1807.04713) ($c_s$ being the speed of sound), and the same ($\frac{\zeta}{\eta}$) at weak coupling using kinetic theory and finite temperature field theory, was shown and is known to vary like $\left(\frac{1}{3} - c_s^2\right)^2$ (arXiv:1807.04713). The novelties of the results of this paper are that we not only show for the first time from ${\cal M}$ theory that at intermediate coupling, with the inclusion of the ${\cal O}(R^4)$ corrections, the result obtained for $\frac{\zeta}{\eta}$ interpolates between the strong and weak coupling results in a way consistent with lattice computations of $SU(3)$ Gluodynamics (arXiv:0710.3717 [hep-lat]) within statistical errors (and within the temperature range permissible by our ${\cal M}$-theory uplift), but also observe that this behavior is related to the existence of Contact 3-Structures that exist only at intermediate coupling effected by the intermediate-$N$ "MQGP" limit of (arXiv:2211.13186[hep-th]). We also obtain an explicit dependence of $\frac{\zeta}{\eta}$ on (fractional powers of) the temperature-dependent/running gauge coupling (and its temperature derivative), and verify that the weak-coupling result dominates at large temperatures. We further conjecture that the aforementioned fractional-power-dependence of $\frac{\zeta}{\eta}$ on the gauge coupling is related to the lack of ``$N$-connectedness'' in the parameter space of Contact 3-Structures (as shown in arXiv:2211.13186[hep-th]).