Universality at large transverse spin in defect CFT

TL;DR

This work establishes universal large-spin structure in defect CFTs by deriving a Lorentzian inversion formula that reconstructs defect data from bulk two-point functions. The defect spectrum exhibits Regge trajectories ${\\widehat{\\Delta}}(s)$ with leading growth ${\\widehat{\\Delta}}(s)=s+\\Delta_{\\phi}+2m$, and the lightcone bootstrap shows accumulation points dictated by the bulk identity; the inversion formula resums this data and yields finite-$s$ corrections from individual bulk blocks. Applications to Wilson lines, replica defects, holographic line defects, and the Ising twist defect demonstrate consistency with perturbative and holographic results and yield concrete predictions for the spectrum and OPE coefficients. The AdS interpretation links large-$s$ defect states to semiclassical rotating particles, offering a unifying picture and suggesting further extensions to spinning external operators and bulk OPE inversion. The framework provides a robust analytic handle on defect data and has potential implications for a wide class of conformal defects and their holographic duals.

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 $\widehatΔ(s)$. 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.

Figures (6)

• Figure 1: The configuration of the insertions in $\braket{\phi(x_1)\phi(x_2)}$. We show a two-dimensional plane, transverse to the defect, where the two operators lie. The defect is space-like, it intersects the plane at the origin, and the two operators are placed at $(1,1)$ and $(z,\bar{z})$ respectively.
• Figure 2: The positive real axis on the $w$ complex plane, at fixed $r<1$, maps to the black solid line in the $(z,\bar{z})$ plane. The bulk OPE singularities correspond to the intersection of the line with the past and future lightcones of the operator $\phi(1,1)$.
• Figure 3: Deformation of the $|w|=1$ contour of \ref{['euclideaninv']} used to define $b_0({\widehat{\Delta}},s)$ and $b_\infty({\widehat{\Delta}},s)$ in \ref{['lorinveven']}. For the case of odd codimension it was necessary to add zero in the form of two contours $C_+$ and $C_-$\ref{['addzero']}.
• Figure 4: A three dimensional version of the configuration discussed in the text. In red, the line defect marks two lines along $\tau$ at $\theta=\frac{\pi}{2}$ and $\psi=0,\ \pi$. The particle spins fast along the $\varphi$ direction.
• Figure 5: (a) Depiction of the geometry in the coordinates of eqs. \ref{['eq:metric']} and \ref{['eq:adsfactormetric']}, at fixed $(\tau,\varphi)$. The metric has a conical singularity at $\xi=0$ (dashed line), and is asymptotically $AdS_4$ at large $\xi$. (b) Depiction of the geometry in global coordinates at fixed $(t,\varphi)$, see eqs. \ref{['defect_metric_sph']} and \ref{['metricglobal']}. The conical singularity now extends along the diameter of the sphere (dashed line).