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

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