Endable Conformal Line Defects in CFT
- Endable conformal line defects are one-dimensional observables in CFT that can terminate at endpoints, blending bulk and boundary field dynamics.
- Key methodologies such as the bootstrap, ε-expansion with Padé resummation, and fuzzy sphere regularization combine to yield precise scaling dimensions and operator product coefficients.
- These insights reveal that while asymmetric defects sustain stable conformal boundaries, symmetric (Z₂) defects tend toward instability, matching lattice and field-theoretic predictions.
Endable conformal line defects are one-dimensional extended operators in conformal field theory (CFT) that can terminate at distinguished endpoints, serving as nonlocal observables which interpolate between conventional bulk operator insertions and extended topological or topological–like defects. Their paper synthesizes conformal symmetry with boundary and defect field theory, entangling algebraic, analytic, geometric, and numerical approaches. Endable line defects incorporate both bulk and boundary (endpoint) degrees of freedom and encode crucial information about dualities, universality, and RG flows in the bulk CFT as well as about coupled boundary/defect CFTs. Recent advances demonstrate the unified power of bootstrap, effective field theory, Monte Carlo, and semiclassical limits for their direct characterization.
1. Structural Definition and Universal Properties
An endable conformal line defect is a one-dimensional defect (typically with or without endpoints) which is compatible with the preserved subgroup SO(2,1) × SO(d−1) of the conformal group in d dimensions, and that may have associated endpoint operators of prescribed scaling dimensions. Unlike infinite or closed defect lines, endable “open” line defects can terminate on primary “endpoint” operators, with all local correlation functions transforming covariantly under conformal symmetry.
A central property is the intrinsic crossing symmetry that arises when correlators involve both defect-local and endpoint operators, mixing bulk, defect, and boundary channels. For the 3D Ising CFT with a pinning-field line defect, the endpoint corresponds either to the minimal energy operator at the boundary of the defect or, in Z₂-symmetric realizations, to a domain wall creation operator.
A distinguishing feature of endable conformal line defects is the existence of a complete defect OPE (expansion of bulk operators near the defect in terms of defect-local operators) and a boundary OPE (expansion at the endpoint), with positivity and unitary constraints requiring consistency between the bulk, defect, and endpoint spectra and OPE data (Lanzetta et al., 20 Aug 2025).
2. Bootstrap Approaches Incorporating Endpoints
Breakthroughs in numerical and analytic bootstrap methodology now allow concrete nonperturbative bounds on the data of CFTs containing endable line defects. By extending the conformal four-point bootstrap to include configurations with endpoints, it is possible to exploit novel crossing symmetry relations that mix bulk, defect, and endpoint CFT data, while preserving positivity necessary for semidefinite programming techniques.
The bootstrap equations relate, in particular, four-point functions of endpoint operators (or mixed bulk–endpoint–defect correlators) to the full set of conformal data. The platform is sufficiently general to accommodate the pinning field line defect of the 3d Ising model. It enables extraction of both the scaling dimension Δ₀0+ of the minimal endpoint operator and Δ₀+– of the domain wall operator, which respectively control the lowest boundary and domain wall excitations at defect endpoints. These scaling dimensions are subject to rigorous numerical bounds, and these results are robust under variations in the assumed spectral gaps and OPE coefficient signs (Lanzetta et al., 20 Aug 2025).
3. Analytical Results, Epsilon Expansion, and Pade Resummation
Perturbative (ε-expansion) results for minimal endpoint and domain wall scaling dimensions provide an essential analytic complement to bootstrap bounds. Specifically, the O(ε) estimates for endpoint and domain wall operator dimensions, once resummed using [1,1] Padé approximants and fixed to match known values in d=2, closely approach the rigorous 3d bootstrap results:
| Parameter | ε-expansion (Padé) | Bootstrap Bound |
|---|---|---|
| Δ₀0+/Δ₀+– | ~0.128 | ~0.132(7) |
| Δ₀+– | ~0.85 | ~0.818(28) |
This matching substantiates the reliability of both perturbative and nonperturbative methods and helps interpolate results across dimensions. A salient observation is that while Δ₀0+ displays non-monotonicity as d flows from 4 to 2, Δ₀+– increases monotonically and the resummed ε-expansion remains consistent with nearly rigorous bounds in 3d. This tight agreement implies that the endpoint bootstrap, complemented by perturbative expansion, provides a precise quantitative framework for the operator content of endable line defects in interacting CFTs (Lanzetta et al., 20 Aug 2025).
4. Physical Spectra, Defect Instabilities, and Z₂ Symmetry
Examining the spectrum of endpoint and domain wall operators reveals key physical consequences about defect phases and instabilities. In the 3D Ising CFT, the Z₂-symmetric defect consisting of a direct sum of two conjugate pinning field defects exhibits signature features in the scaling dimensions:
- The ratio Δ₀0+/Δ₀+– is small, indicating that the endpoint operator is much lighter than the domain wall excitation.
- Bootstrap and resummation evidence show that this Z₂-symmetric defect is unstable to domain wall proliferation—i.e., the direct sum is not an extremal case in the bootstrap allowed region, but rather a composite with lower-lying excitations, matching expectations from lattice and field-theoretic arguments.
These observations provide decisive evidence against stable long range order for such symmetric defects, emphasizing that only certain "asymmetric" (explicitly symmetry-breaking) pinning field defects host stable endable conformal boundary conditions (Lanzetta et al., 20 Aug 2025).
5. Connections to Other Techniques and Models
The endpoint bootstrap results integrate and validate independent approaches, including fuzzy sphere regularization studies of the magnetic defect in the 3D Ising model (Hu et al., 2023). The fuzzy sphere method yields explicit spectra for defect and endpoint operators, and the operator dimensions extracted via this approach agree with bootstrap and resummed ε-expansion results. The convergence of these techniques grounds the universality of endable line defect spectra.
Further, similar endable line defects in models with continuous global symmetries and explicit defect deformations have been constrained via numerical bootstrap (including the interplay of displacement, tilt, and endpoint operators) and systematic perturbative expansions, affirming that the fusion of analytic and numerical tools is essential for comprehensive analysis (Gimenez-Grau et al., 2022).
6. Defect Endpoint Operators and Universal Constraints
Integral and sum-rule constraints on defect correlators, derived from ambient conformal symmetry and functional bootstrap methods, provide additional nontrivial checks and predictions on scaling dimensions and OPE coefficients (Gabai et al., 12 Jan 2025). Inclusion of endpoints requires explicit accounting of endpoint operators in crossing and variation equations, ensuring the complete bootstrap set for defect CFT data. This web of constraints, together with numerical and perturbative results, ensures the universality and consistency of endable line defect spectra, even as the full parameter space of boundary and domain wall conditions is explored.
7. Outlook and Implications
The endpoint bootstrap for endable conformal line defects establishes a rigorous, model-independent platform for the classification and computation of defect spectra, endpoint operator content, and domain wall dynamics. The agreement of bootstrap, resummed analytic, and fuzzy sphere results in nontrivial CFTs such as the 3D Ising model provides compelling evidence of their reliability and relevance. The extension of these methods to more general CFTs, more intricate defect networks, and higher dimensions promises powerful new probes of universality, duality, and boundary critical phenomena. These results further underscore the central role of endable conformal line defects in the ongoing synthesis of analytic, numerical, and algebraic approaches to modern conformal field theory.