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We study the critical properties of scalar field theories in $d+1$ dimensionswith $O(N)$ invariant interactions localized on a $d$-dimensional boundary. Bya combination of large $N$ and epsilon expansions, we provide evidence for theexistence of non-trivial $O(N)$ BCFTs in $1<d<4$. Due to having free fields inthe bulk, these models possess bulk higher-spin currents which are conserved upto terms localized on the boundary. We suggest that this should lead to a setof protected spinning operators on the boundary, and give evidence that theiranomalous dimensions vanish. We also discuss the closely related long-range$O(N)$ models in $d$ dimensions, and in particular study a weakly coupleddescription of the $d=1$ long range $O(N)$ model near the upper critical valueof the long range parameter, which is given in terms of a non-local non-linearsigma model. By combining the known perturbative descriptions, we provide someestimates of critical exponents in $d=1$.

$O(N)$ Models with Boundary Interactions and their Long Range Generalizations