On $M$-theory dual of large-$N$ thermal QCD-like theories up to $\mathcal{O}(R^4)$ and $G$-structure classification of underlying non-supersymmetric geometries

Construction of a top-down holographic dual of thermal QCD-like theories (equivalence class of theories which are UV-conformal, IR-confining and have fundamental quarks) {\it at intermediate 't Hooft coupling} and the $G$-structure (torsion classes) classification of the underlying geometries (in the Infra Red (IR)/non-conformal sector in particular) of the {\it non-supersymmetric} string/${\cal M}$-theory duals, have been missing in the literature. We take the first important steps in this direction by studying the ${\cal M}$ theory dual of large-$N$ thermal QCD-like theories at intermediate gauge and 't Hooft couplings and obtaining the ${\cal O}(l_p^6)$ corrections arising from the ${\cal O}(R^4)$ terms to the"MQGP"background (${\cal M}$-theory dual of large-$N$ thermal QCD-like theories at intermediate gauge/string coupling, but large 't Hooft coupling) of \cite{MQGP}. The main Physics lesson learnt is that there is a competition between non-conformal IR enhancement and Planckian and large-$N$ suppression and going to orders beyond the ${\cal O}(l_p^6)$ is necessitated if the IR enhancement wins out. The main lesson learnt in Math is in the context of the differential geometry ($G$-structure classification) of the internal manifolds relevant to the string/${\cal M}$-theory duals of large-$N$ thermal QCD-like theories, wherein we obtain for the first time inclusive of the ${\cal O}(R^4)$ corrections in the Infra-Red (IR), the $SU(3)$-structure torsion classes of the type IIA mirror of \cite{metrics} (making contact en route with Siegel theta functions related to appropriate hyperelliptic curves, as well as the Kiepert's algorithm of solving quintics), and the $G_2/SU(4)/Spin(7)$-structure torsion classes of the seven- and eight-folds associated with its ${\cal M}$ theory uplift.