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$(a)$ A leading diagram in double line notation contributing to the one-point function of a single trace operator of length $L=6$ in the large $N$ limit of $\mathcal{N}=4$ SYM on $\mathbb{RP}^4$ without the gauging of charge conjugation. The pre-factor ensures the UV limit of the two-point functions is normalized to one. $(b)$ A leading diagram in the large $N$ limit contributing to the one-point function of the same single trace operator in $\mathcal{N}=4$ SYM on $\mathbb{RP}^4$ with the gauging of charge conjugation. In this example, the double line notation graph originates a surface with three faces (illustrated in distinct colors) as opposed to the four faces of the figure (a), hence producing a different large $N$ scaling.

This plot exhibits the profile of the scalars $\chi$ and $\alpha$ and the warp factor $X$ as function of the radial coordinate $r$ for $\mathcal{J}=1/6$. Importantly, the region for $r\leq r^{\ast}$ depicted in red is not physical: the scalar $\alpha$ develops a singularity precisely at $r=r^{\ast}$ and $\chi$ becomes complex for $r\leq r^{\ast}$.

This plot exhibits the profile of the five-dimensional dilaton $\varphi$ as function of the radial coordinate $r$ for $\mathcal{J}=1/6$. At $r= r^{\ast}$, the five-dimensional dilaton becomes singular $\varphi \rightarrow -\infty$. The region depicted in red is not physical.