Contact Structure-induced Symplectic Nearly Half-Flat $SU(3)$ Structure in Finite-$N$ Thermal ${\cal M}$-Theory

RESUMO

The ${\cal M}$-theory dual of thermal QCD-like theories constructed in arXiv:1306.4339 , 2004.07259 involves, as shown in arXiv:2211.13186 , a closed $G_2$-structure seven-fold $\mathcal{M}_7$ supporting Contact 3-Structures (C3S) that induce a ``transverse'' $SU(3)$-structure. We prove that this induced structure belongs to the symplectic nearly half-flat (S(nearly)HF) class: $d\omega_\Phi=0$ exactly, while $d\Omega_+$ is suppressed as $\mathcal{O}(e^{-N^{1/3}/\mathcal{O}(1)})$ in the infrared "MQGP limit", leaving $W_\Phi^{SU(3)}=W_2^-$ as the sole non-vanishing Chiossi--Salamon torsion class and completing the $G$-structure classification of the MQGP dual initiated in arXiv:1507.02692 , 2004.07259 , 2211.13186 . As a physical application, Kaluza--Klein reduction of the eleven-dimensional Hodge dual of the differential of the $G_2$-structure positive three-form on $\mathcal{M}_7$, organised through the $G_2$/S(nearly)HF torsion data, yields a radial ``magnetic field'' $\mathcal{B}_{\rm KK}(r)$ whose squared norm factorises exactly into external and internal factors. $\mathcal{B}_{\rm KK}(r)$ satisfies a Bessel-type ODE with a regular singular point at the horizon; normalisability and infrared regularity select $J_2$, with harmonicity of the aforementioned internal factor recovered from the large-$N$ suppression. $\mathcal{B}_{\rm KK}$ is related to the magnetic field of the type~IIA flavour $D6$-branes via a non-linear power law, furnishing a top-down holographic mechanism for generating weak magnetic fields in finite-temperature string theory.