A note on defect stability in $d=4-\varepsilon$

We explore the space of scalar line, surface and interface defect field theories in $d=4-\varepsilon$ by examining their stability properties under generic deformations. Examples are known of multiple stable line defect Conformal Field Theories (dCFTs) existing simultaneously, unlike the case of normal multiscalar field theories where a theorem by Michel guarantees that the stable fixed point is the unique global minimum of a so-called $A$-function. We prove that a suitable modification of Michel's theorem survives for line defect theories, with fixed points locally rather than globally minimizing an $A$-function along a specified surface in coupling space and provide a novel classification of the fixed points in the hypertetrahedral line defect model. For surface defects Michel's theorem survives almost untouched, and we explore bulk models for which the symmetry preserving defect is the unique stable point. In the case of interface theories, we prove that for any critical bulk model there can exist no fixed points stable under generic deformations for $N\geq 6$.