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We start by reviewing the recently obtained Lorentzian inversion formula for four-point functions in conformal field theories (CFTs) and what it generically implies for their spectrum.
We then turn to defect CFTs, and show the existence of universal accumulation points in the defect spectrum at large "transverse" spin, i.e. at large charge of the operators under rotations in the space transverse to the defect, by means of an inversion formula for the bulk to defect OPE.