How to get an interacting conformal line defect for free theories

1. Problem Statement

The paper addresses a fundamental question in the classification of Defect Conformal Field Theories (DCFTs): Can interacting conformal line defects exist within a bulk free quantum field theory?

Previous work (specifically Ref. [3]) established a "triviality condition" for defects in free bulk theories. The argument relied on two main observations:

1. The bulk equation of motion (EOM) restricts the spectrum of operators in the Defect Operator Expansion (DOE).
2. Regularity conditions on three-point functions (specifically bulk-defect-defect) impose a "double twist condition" on scaling dimensions.

Under the assumption of inversion symmetry (or time-reversal symmetry), these constraints force the defect operators appearing in the DOE of the bulk field to be Generalized Free Fields (GFF). This implies that while the defect might exist, the interaction is trivial in the sense that correlation functions factorize via Wick's theorem, and no non-trivial anomalous dimensions arise for the relevant operators.

The authors challenge the assumption of inversion symmetry, arguing that if this symmetry is broken, the "triviality proof" fails, potentially allowing for non-trivial interacting line defects even in free bulk theories.

2. Methodology

The authors employ a combination of conformal bootstrap techniques, exact field redefinitions, and perturbative quantum field theory calculations.

• Conformal Analysis of Cross-Ratios:

  • They analyze the kinematic constraints on bulk-defect-defect three-point functions $\langle \phi(x_1) \hat{O}_2(t_2) \hat{O}_3(t_3) \rangle$.
  • Standard conformal invariance allows a cross-ratio $\zeta$. However, for line defects ($p=1$), they identify a special cross-ratio $\nu$ that is invariant under the conformal group $SO(1,2) \times SO(d-1)$ but changes sign under inversion ($t \to -t$).
  • They demonstrate that the "double twist condition" (which forces GFF behavior) arises only when requiring invariance under inversion. By relaxing this requirement, the correlation functions can be smooth functions of $\nu$ without singularities that force the spectrum to be trivial.

• Construction of a Toy Model:

  • To prove existence, they construct a specific model in $d=4$ dimensions:
    • Bulk: A free massless scalar field $\phi$.
    • Defect: A 1D line containing a complex fermion $\psi$.
    • Interaction: A localized Yukawa coupling $g \bar{\psi}\psi \phi$ and a linear source term $h\phi$ on the defect.
  • Exact Solution: They utilize a non-local field redefinition $\psi(t) = \exp\left(ig \int^t \phi(\tau) d\tau\right) \psi'(t)$ to map the interacting system to a decoupled system of free fields. This allows for exact computation of correlation functions.

• Perturbative Verification:

  • They perform explicit one-loop and two-loop calculations of the beta function and wavefunction renormalization to verify that the coupling $g$ is exactly marginal in $d=4$, confirming the conformal nature of the defect.

3. Key Contributions and Results

A. Breaking the Triviality Proof

The authors show that the proof of triviality in Ref. [3] relies critically on the assumption that correlation functions are invariant under inversion.

• They introduce the cross-ratio $\nu = \frac{(t_2 - t_1)(t_1 - t_3) + y_1^2}{(t_2 - t_3)|y_1|}$.
• Unlike the standard cross-ratio $\zeta$, $\nu$ is not invariant under time reversal.
• In the absence of inversion symmetry, the bulk-defect-defect three-point function does not require the "double twist condition" to avoid singularities. Consequently, the defect operator spectrum is not restricted to Generalized Free Fields.

B. The Toy Model Results

The constructed Yukawa model provides a concrete counter-example to the triviality conjecture:

1. Non-Trivial Anomalous Dimensions: The fermion operators $\psi$ and $\bar{\psi}$ acquire an anomalous scaling dimension:
   $$\Delta_\psi = \Delta_{\bar{\psi}} = \frac{g^2}{8\pi^2}$$
   This confirms the theory is interacting, as free fields would have dimension 0 (in 1D).
2. Protected Operators: The composite operator $\bar{\psi}\psi$ has dimension 0 (protected), acting as a conserved charge, but it is distinct from the identity operator. This violates another assumption of the triviality proof (that the only dimension-zero operator is the identity).
3. Exact Correlators: Using the field redefinition, they derive exact expressions for three-point functions, such as:
   $$\langle \phi(x_1) \psi(t_2) \bar{\psi}(t_3) \rangle \propto \frac{\tanh^{-1}(1/\nu)}{|t_2 - t_3|^{\Delta_\psi} |y_1|^{\dots}}$$
   These expressions are smooth in $\nu$ but explicitly break time-reversal symmetry (due to the $\tanh^{-1}$ and step functions), matching the theoretical prediction that inversion symmetry is not required.
4. Marginality: The beta function for the coupling $g$ vanishes exactly in $d=4$ ($\beta_g = 0$). This is confirmed by:
   • The exact solution showing no renormalization of the bulk scalar.
   • Explicit two-loop calculations showing $Z_\psi = Z_g$ (Ward identity), leading to a vanishing beta function.

C. Stress Tensor and Displacement Operator

The authors compute the displacement operator $D_i$ (which measures the response of the defect to shape deformations). They find that the displacement operator has a non-zero two-point function only if the source term $h \neq 0$. The $g$-function (a measure of the defect's degrees of freedom) is calculated to be zero (or constant), indicating the defect does not change the central charge of the bulk theory in a way that depends on $g$.

4. Significance and Implications

• Existence of Interacting Free-Bulk DCFTs: The paper definitively proves that interacting conformal line defects can exist in free bulk theories, provided inversion symmetry is broken. This expands the landscape of known DCFTs beyond the previously assumed "Generalized Free Field" sector.
• Role of Symmetry: It highlights that inversion symmetry is a stronger constraint than conformal symmetry itself. Relaxing discrete symmetries allows for richer operator spectra and non-trivial interactions.
• Maxwell Theory Extension: The authors discuss the extension to free Maxwell theory in $d=4$. While the scalar model generalizes via a Wilson line dressing, the gauge invariance of the Maxwell field complicates the construction of a simple interacting defect. They suggest that while a trivial GFF sector likely exists, non-trivial interacting defects in Maxwell theory remain an open and promising area for future research.
• AdS/CFT Connection: The "Note Added" mentions that the lack of parity symmetry discussed here relaxes "locality constraints" in AdS$_2$, suggesting implications for holographic duals of free theories.

Conclusion

Bartlett-Tisdall, Ge, and Herzog successfully demonstrate that the "triviality" of defects in free theories is not a universal law but a consequence of assuming inversion symmetry. By constructing an exactly solvable Yukawa model with a line defect, they show that non-trivial anomalous dimensions and interacting correlation functions can exist in a free bulk environment. This work fundamentally alters the understanding of defect conformal field theories and opens the door to exploring interacting defects in other free theories, such as Maxwell and fermionic bulk theories.