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We construct unitary, stable, and interacting conformal boundary conditionsfor a free massless scalar in four dimensions by coupling it to edge modesliving on a boundary. The boundary theories we consider are bosonic andfermionic QED$_3$ with $N_f$ flavors and a Chern-Simons term at level $k$, inthe large-$N_f$ limit with fixed $k/N_f$. We find that interacting boundaryconditions only exists when $k\neq 0$. To obtain this result we compute the$\beta$ functions of the classically marginal couplings at the firstnon-vanishing order in the large-$N_f$ expansion, and to all orders in $k/N_f$and in the couplings. To check vacuum stability we also compute the large-$N_f$effective potential. We compare our results with the the known conformalbootstrap bounds.

Conformal boundary conditions for a 4d scalar field