On Intersecting Conformal Defects

We study the physics of 2 and 3 mutually intersecting conformal defects forming wedges and corners in general dimension. For 2 defects we derive the beta function of the edge interactions for infinite and semi-infinite wedges and study them in the tricritical model in $d=3-ε$ as an example. We discuss the dependency of the edge anomalous dimension on the intersection angle, connecting to an old issue known in the literature. Additionally, we study trihedral corners formed by 3 planes and compute the corner anomalous dimension, which can be considered as a higher-dimensional analog of the cusp anomalous dimension. We also study 3-line corners related to the three-body potential of point-like impurities.

💡 Research Summary

The paper investigates the renormalization group (RG) behavior of intersecting conformal defects in general dimensions, focusing on configurations where two or three planar defects meet to form wedges, edges, and trihedral corners. The authors start by constructing a semi‑infinite planar defect D with an edge δD and introduce bulk scalar primary operators that deform both the defect and its edge. Using operator product expansion (OPE) techniques and conformal perturbation theory, they derive the beta function for the edge coupling hα. A key observation is that when two bulk operators on the defect approach the edge, a boundary contribution appears in the OPE, leading to additional logarithmic divergences that modify the RG flow of hα. The resulting beta function contains the usual linear term proportional to the deviation εα from marginality, quadratic terms involving defect‑defect couplings gi gj, and mixed terms gi hβ, reflecting the interplay between bulk, defect, and edge degrees of freedom.

Next, the authors consider two intersecting planar defects D1 and D2 that meet at a codimension‑two intersection I with relative angle α. By parametrizing the geometry so that the intersection lies in the first p‑1 coordinates and allowing one defect to have extra transverse directions, they compute the OPE of operators on D1 and D2 when they collide at I. The calculation yields a new angular factor W(α) that multiplies the logarithmic divergence. Consequently, the beta function for the intersection coupling hα acquires an explicit dependence on sin α and (π − α), resolving a long‑standing puzzle in the literature about angle‑dependent anomalous dimensions. The authors verify that in the limit α→π the wedge reduces to a single infinite plane and the edge disappears, as expected.

The paper then treats the wedge formed by two semi‑infinite planes (a combination of the previous setups). By performing the OPE in this geometry and integrating over the semi‑infinite coordinates, they obtain a unified beta function that includes the term π − α sin α gi gj, confirming the smooth interpolation between the separate‑plane and wedge cases.

Having established the RG structure for two‑defect systems, the authors move to three mutually intersecting defects, defining a trihedral corner where three 2‑dimensional planes meet at a point. They show that at cubic order in the weak couplings gi the free energy acquires a logarithmic divergence proportional to g(1) g(2) g(3) ⟨O1 O2 O3⟩. This defines a “corner anomalous dimension” Γ_corner, which they interpret as a higher‑dimensional analogue of the cusp anomalous dimension familiar from Wilson line physics. The explicit result is

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