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It is well established that the $O(N)$ Wilson-Fisher (WF) CFT sits at a kinkof the numerical bounds from bootstrapping four point function of $O(N)$vector. Moving away from the WF kinks, there indeed exists another family ofkinks (dubbed non-WF kinks) on the curve of $O(N)$ numerical bounds. Differentfrom the $O(N)$ WF kinks that exist for arbitary $N$ in $2<d<4$ dimensions, thenon-WF kinks exist in arbitrary dimensions but only for a large enough$N>N_c(d)$ in a given dimension $d$. In this paper we have achieved a thoroughunderstanding for few special cases of these non-WF kinks. The first case isthe $O(4)$ bootstrap in 2d, where the non-WF kink turns out to be the $SU(2)_1$Wess-Zumino-Witten (WZW) model, and all the $SU(2)_{k>2}$ WZW models saturatethe numerical bound on the left side of the kink. We further carry outdimensional continuation of the 2d $SU(2)_1$ kink towards the 3d $SO(5)$deconfined phase transition. We find the kink disappears at around $d=2.7$dimensions indicating the $SO(5)$ deconfined phase transition is weakly firstorder. The second interesting observation is, the $O(2)$ bootstrap bound doesnot show any kink in 2d ($N_c=2$), but is surprisingly saturated by the 2d freeboson CFT (also called Luttinger liquid) all the way on the numerical curve.The last case is the $N=\infty$ limit, where the non-WF kink sits at$(\Delta_\phi, \Delta_T)=(d-1, 2d)$ in $d$ dimensions. We manage to write downits analytical four point function in arbitrary dimensions, which equals to thesubtraction of correlation functions of a free fermion theory and generalizedfree theory. An important feature of this solution is the existence of a fulltower of conserved higher spin current. We speculate that a new family of CFTswill emerge at non-WF kinks for finite $N$, in a similar fashion as $O(N)$ WFCFTs originating from free boson at $N=\infty$.

Non-Wilson-Fisher kinks of $O(N)$ numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs