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Abstract

We study the spectrum of local operators living on a defect in a generic
conformal field theory, and their coupling to the local bulk operators. We
establish the existence of universal accumulation points in the spectrum at
large $s$, $s$ being the charge of the operators under rotations in the space
transverse to the defect. Our tools include a formula that inverts the bulk to
defect OPE, analogous to the Caron-Huot formula for the four-point function.
Analyticity of the formula in $s$ implies that the scaling dimensions of the
defect operators are aligned in Regge trajectories $\widehat{\Delta}(s)$. These
results require the correlator of two local operators and the defect to be
bounded in a certain region, a condition that we do not prove in general. We
check our conclusions against examples in perturbation theory and holography,
and we make specific predictions concerning the spectrum of defect operators on
Wilson lines. We also give an interpretation of the large $s$ spectrum in the
spirit of the work of Alday and Maldacena.