This paper studies the physics hidden at sharp corners, or cusps, on one-dimensional defects in scale‑invariant quantum systems. When a straight line defect bends, the corner hosts its own quantum excitations. Those excitations are measured by numbers called cusp anomalous dimensions. The authors derive general bounds that relate those corner dimensions to a simpler quantity: the universal defect Casimir energy, which is the energy between a defect and its mirror image in flat space.

To get these results the authors use two basic ideas. First, they use reflection positivity (also called Osterwalder–Schrader positivity), a standard consistency condition for unitary quantum field theories. Second, they develop and apply what they call the cusp operator expansion (COE). The COE is a way to cut and glue defect geometries and to express states on a sphere pierced by defects in terms of operators that create cusps. Combining these ideas gives constraints on which cusp dimensions are allowed.

One clear result is a no‑crossing statement for cusps that join a defect to its conjugate, or mirror. The paper proves that the lightest singlet cusp — a cusp carrying no conserved charges besides its scaling dimension — cannot be overtaken by any charged (non‑singlet) cusp as the corner angle changes. In other words, singlet cusps are provably the lowest dimension states in that family.

Their main technical achievement is a universal lower bound on the dimension of a right‑angle cusp. The bound ties that cusp dimension to the defect Casimir energy and is valid in any scale‑invariant theory and in any spacetime dimension covered by their assumptions. The bound is “optimal” in the sense that it cannot be improved given the same assumptions. The authors construct an explicit analytic “magic” functional, inspired by techniques from recent bootstrap work, and show the bound is saturated by two‑dimensional boundary conformal field theories.