Conformal boundary conditions for a 4d scalar field

We construct unitary, stable, and interacting conformal boundary conditions for a free massless scalar in four dimensions by coupling it to edge modes living on a boundary. The boundary theories we consider are bosonic and fermionic QED _3 3 with N_f Nf flavors and a Chern-Simons term at level k k , in the large- N_f Nf limit with fixed k/N_f k/Nf . We find that interacting boundary conditions only exist when k≠ 0 k≠0 . To obtain this result we compute the β β functions of the classically marginal couplings at the first non-vanishing order in the large- N_f Nf expansion, and to all orders in k/N_f k/Nf and in the couplings. To check vacuum stability we also compute the large- N_f Nf effective potential. We compare our results with the the known conformal bootstrap bounds.