Boundaries and Interfaces with Localized Cubic Interactions in the $O(N)$ Model

We explore a new approach to boundaries and interfaces in the $O(N)$ model where we add certain localized cubic interactions. These operators are nearly marginal when the bulk dimension is $4-\epsilon$, and they explicitly break the $O(N)$ symmetry of the bulk theory down to $O(N-1)$. We show that the one-loop beta functions of the cubic couplings are affected by the quartic bulk interactions. For the interfaces, we find real fixed points up to the critical value $N_{\rm crit}\approx 7$, while for $N> 4$ there are IR stable fixed points with purely imaginary values of the cubic couplings. For the boundaries, there are real fixed points for all $N$, but we don't find any purely imaginary fixed points. We also consider the theories of $M$ pairs of symplectic fermions and one real scalar, which have quartic $OSp(1|2M)$ invariant interactions in the bulk. We then add the $Sp(2M)$ invariant localized cubic interactions. The beta functions for these theories are related to those in the $O(N)$ model via the replacement of $N$ by $1- 2M$. In the special case $M=1$, there are boundary or interface fixed points that preserve the $OSp(1|2)$ symmetry, as well as other fixed points that break it.