Universal criticality of thermodynamic geometry for boundary conformal field theories in gauge/gravity duality

According to more recent AdS/CFT interpretation cite{Karch:2015rpa}, in which varying cosmological constant $Lambda$ in the bulk corresponds to varying the curvature radius governing the space on which the field theory resides, we study the criticality of thermodynamic curvatures for thermal boundary conformal field theories (CFT) that are dual to $d$-dimensional charged anti-de Sitter (AdS) black holes, embedded in $D$-dimensional M-theory/superstring inspired models having $AdS_{d}times mathbb{S}^{d+k}$ spacetime with $D=2d+k$. Analogous with criticality features acquired for charged AdS black holes in the bulk cite{HosseiniMansoori:2020jrx}, the normalized intrinsic curvature $R_N$ and extrinsic curvature $K_N$ of the boundary CFT has critical exponents 2 and 1, respectively. In this respect, the universal amplitude of $R_Nt^2$ is $frac{1}{2}$ and $K_Nt$ is $-frac{1}{2}$ when $trightarrow0^-$, whereas $R_Nt^2approx frac{1}{8}$ and $K_Ntapproxfrac{1}{4}$ in the limit $trightarrow0^+$ in which $t=T/T_c-1$ is the temperature parameter with the critical temperature, $T_{c}$. Interestingly, these critical amplitudes are independent of the number of thermal CFT dimensions and are remarkably similar to one given for higher dimensional charged AdS black holes in the bulk.