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Exploring $SU(N)$ adjoint correlators in $3d$

We use numerical bootstrap techniques to study correlation functions of scalars transforming in the adjoint representation of $SU(N)$ in three dimensions. We obtain upper bounds on operator dimensions for various representations and study their dependence on $N$. We discover new families of kinks, one of which could be related to bosonic QED${}_3$. We then specialize to the cases $N=3,4$, which have been conjectured to describe a phase transition respectively in the ferromagnetic complex projective model $CP^2$ and the antiferromagnetic complex projective model $ACP^{3}$. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to small regions overlapping with the lattice predictions.