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 $\Lambda$ the bulk corresponds curvature radius governing space on field theory resides, we study criticality of thermodynamic curvatures for thermal boundary conformal theories (CFT) that are dual $d$-dimensional charged anti-de Sitter (AdS) black holes, embedded $D$-dimensional M-theory/superstring inspired models having $AdS_{d}\times \mathbb{S}^{d+k}$ spacetime with $D=2d+k$. Analogous features acquired AdS holes \cite{HosseiniMansoori:2020jrx}, normalized intrinsic $R_N$ and extrinsic $K_N$ CFT has critical exponents 2 1, respectively. In this respect, universal amplitude $R_Nt^2$ is $\frac{1}{2}$ $K_Nt$ $-\frac{1}{2}$ when $t\rightarrow0^-$, whereas $R_Nt^2\approx \frac{1}{8}$ $K_Nt\approx\frac{1}{4}$ limit $t\rightarrow0^+$ $t=T/T_c-1$ temperature parameter temperature, $T_{c}$. Interestingly, these amplitudes independent number dimensions remarkably similar one given higher dimensional bulk.