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I study the two-dimensional defects of the $d$ dimensional critical $O(N)$model and the defect RG flows between them. By combining the$\epsilon$-expansion around $d = 4$ and $d = 6$ as well as large $N$techniques, I find new conformal defects and examine their behavior acrossdimensions and at various $N$. I discuss how some of these fixed points relateto the known ordinary, special and extraordinary transitions in the 3d theory,as well as examine the presence of new symmetry breaking fixed pointspreserving an $O(p) \times O(N-p)$ subgroup of $O(N)$ for $N \le N_c$ (with theestimate $N_c = 6$). I characterise these fixed points by obtaining theirconformal anomaly coefficients, their 1-point functions and comment on thecalculation of their string potential. These results establish surfaceoperators as a viable approach to the characterisation of interface criticalphenomena in the 3d critical $O(N)$ model.

Surface defects in the $O(N)$ model