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We explore a new approach to boundaries and interfaces in the $O(N)$ modelwhere we add certain localized cubic interactions. These operators are nearlymarginal 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-loopbeta functions of the cubic couplings are affected by the quartic bulkinteractions. For the interfaces, we find real fixed points up to the criticalvalue $N_{\rm crit}\approx 7$, while for $N> 4$ there are IR stable fixedpoints 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 imaginaryfixed points. We also consider the theories of $M$ pairs of symplectic fermionsand one real scalar, which have quartic $OSp(1|2M)$ invariant interactions inthe bulk. We then add the $Sp(2M)$ invariant localized cubic interactions. Thebeta functions for these theories are related to those in the $O(N)$ model viathe replacement of $N$ by $1- 2M$. In the special case $M=1$, there areboundary or interface fixed points that preserve the $OSp(1|2)$ symmetry, aswell as other fixed points that break it.

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