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For a single free scalar field in $d \geq 2$ dimensions, almost all theunitary conformal defects must be `trivial' in the sense that they cannot holdinteresting dynamics. The only possible exceptions are monodromy defects in $d\geq 4$ and co-dimension three defects in $d \geq 5$. As an intermediate resultwe show that the $n$-point correlation functions of a conformal theory with ageneralized free spectrum must be those of the generalized free theory.

Line and surface defects for the free scalar field