On the ubiquity of (almost) contact structures in hot MQCD at intermediate coupling

In the context of fluxed Formula: see text-structure Formula: see text-theory duals of intermediate-Formula: see text (the number of color Formula: see text-branes in the parent type IIB dual of M. Mia, K. Dasgupta, C. Gale and S. Jeon, Five easy pieces: The dynamics of quarks in strongly coupled plasmas, Nucl. Phys. B 839 (2010) 187, arXiv:hep-th/0902.1540) thermal Quantum Chromodynamics (QCD)-like theories, inclusive of quartic-in-curvature corrections, we show the explicit existence of (Almost) Contact 3-Structures (A)C3S. A novel result in this context is that in the Infra-Red, within the parameter space of AC3S arising from Formula: see text structures the subspaces of C3S and AC3S are not mutually “Formula: see text-path connected” because the former exists only for intermediate Formula: see text while the latter exists only for very large Formula: see text. Using a proposition of X. de la Ossa, M. Larfors and M. Magill, Almost contact structures on manifolds with a [Formula: see text structure, Adv. Theor. Math. Phys. 26(1) (2022) 143–216, arXiv:2101.12605 hep-th], we also discuss the reduction of the Formula: see text structure to “transverse” Formula: see text structure induced by (A)C3S in the IR. Apart from providing a novel insight into the differential geometry of the relevant Formula: see text-theory non-supersymmetric manifolds, our results lead us to conjecture that the intermediate-Formula: see text limit which supports C3S is farther away from the non-supersupersymmetric AdS “swampland” than theFormula: see text-limit which supports AC3S. We also point out that a rich variety of high-temperature phenomena such as post Page-time “Islands” black hole entanglement entropy, the “Lyapunov exponent” and “butterfly velocity” pertaining to an “unusually” chaotic behavior discussed earlier by one of the co-authors AM and his former group members in G. Yadav and A. Misra, Entanglement entropy and Page curve from the M-theory dual of thermal QCD above Tc at intermediate coupling, Phys. Rev. D 107(10) (2023) 106015, arXiv:2207.04048 [hep-th; G. Yadav, S. S. Kushwah and A. Misra, Pole-skipping and chaos in hot Formula: see text, J. High Energy Phys. 05 (2024) 015, doi:10.1007/JHEP05(2024)015, arXiv:2311.09306 hep-th], respectively, use precisely the aforementioned intermediate-Formula: see text limit supporting C3S. Further, the conjectural correspondence: (non-analytic branch-point singularities gauge-coupling dependence of the Formula: see text(curvatureFormula: see text-corrected complexified “bulk viscosity” in terms of speed of sound S. S. Kushwah and A. Misra, Bulk viscosity, speed of sound, and contact structure at intermediate coupling, Phys. Rev. D 110(12) (2024) 126010, doi: 10.1103/PhysRevD.110.126010, arXiv: 2403.10541 [hep-th] and strong magnetic field photoproduction spectral density S. S. Kushwah and A. Misra, Photoproduction, paramagnetic anisotropic plasma, IR log-gravitational-DBI renormalization and [Formula: see text-structure induced (Almost) Contact 3-Structures in hot strongly magnetic MQCD at intermediate coupling, preprint (2025), arXiv: 2503.07732 hep-th], analytic gauge-coupling dependence of the Formula: see text(curvature 4 ) non-renormalized complexified pressure/energy of a paramagnetic plasma Formula: see textC3S in the aforementioned intermediate-Formula: see text MQGP limit, AC3S in Formula: see text MQGP limitFormula: see text mimicking the aforementioned lack of Formula: see text-path connectedness, is reiterated.