't Hooft expansion of $SO(N)$ and $Sp(N)$ $\mathcal{N}=4$ SYM revisited

We study the 't Hooft expansion of $d=4$ $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory with the gauge group $SO(N)$ or $Sp(N)$. We consider the $1\mathrm{/}{N}_5$ expansion with fixed $g_s{N}_5$, where $g_s$ denotes the string coupling of bulk type IIB string theory on $Ad{S}_5\times {\mathbb{R}\mathbb{P}}^5$ and $N_5$ refers to the RR 5-form flux through ${\mathbb{R}\mathbb{P}}^5$. $N_5$ differs from $N$ due to a shift coming from the RR charge of O3-plane. As an example, we consider the $1\mathrm{/}{N}_5$ expansion of the free energy of $\mathcal{N}=4$ SYM on $S^4$ and the $1\mathrm{/}2$ BPS circular Wilson loops in the fundamental representation of $SO(N)$ or $Sp(N)$. We find that the $1\mathrm{/}{N}_5$ expansion is more “closed string like” than the ordinary $1\mathrm{/}N$ expansion.