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

Abstract

The $\mathcal{M}$-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'' ${\mathcal{B}}_{\mathrm{KK}}(r)$ whose squared norm factorises exactly into external and internal factors. ${\mathcal{B}}_{\mathrm{KK}}(r)$ 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. ${\mathcal{B}}_{\mathrm{KK}}$ 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.

v2:1+51 pages, LaTeX; 1 figure; Mathematics subject classification (2020) corrected