How to get an interacting conformal line defect for free theories

TL;DR

This work shows that interacting conformal line defects can exist in free bulk theories if inversion symmetry is broken, by leveraging an inversion-odd cross ratio $\nu$ that governs bulk-defect-defect three-point functions. A free scalar bulk with a Yukawa line defect illustrates how a nonlocal field redefinition can render the Yukawa coupling exactly marginal in $d=4$, while the defect fermions acquire anomalous dimensions and a defect operator spectrum that avoids the constraints of previous no-go theorems. The analysis demonstrates a persistent generalized free-field sector for defect operators, accompanied by nontrivial three-point functions and a nonzero displacement operator when appropriate, and shows the $G$-function is independent of the Yukawa coupling at $d=4$. The results motivate exploring interacting line defects in free Maxwell theories and other bulk-free setups, with potential extensions to multiple flavors, varied dimensions, and finite-temperature contexts.

Abstract

We argue that interacting conformal line defects in free quantum field theories can exist, provided that inversion symmetry is broken. Important for our demonstration is the existence of a special cross ratio for bulk-defect-defect three point functions that is invariant under the conformal group but picks up a sign under inversion. We examine the particular case of a free scalar field in detail, and provide a toy model example where this bulk field interacts via a Yukawa term with fermions on the line. We expect nontrivial line defects may also exist for free Maxwell theory in four dimensions and free bulk fermions.

Figures (6)

• Figure 1: Contour plots for a) the Euclidean cross ratio $\nu_E$ when $(\tau_2, \tau_3) = (1,-1)$ and b) the Minkowski cross ratio $\nu$ when $(t_2, t_3) = (1,-1)$. Darker colors are more negative. Lighter colors are more positive.
• Figure 2: Cross ratio $\zeta$ in the conformal frame as a function of transverse distance $r$.
• Figure 3: One loop self energy and counterterm.
• Figure 4: One-loop vertex and counterterm.
• Figure 5: Two loop self energy and counter terms.