Interacting Conformal Line Defects in CFT

• Interacting conformal line defects are codimension-2 loci in CFTs characterized by nontrivial operator spectra and enhanced correlation functions.
• They are constructed through lattice models, supersymmetric Wilson loops, and localized interactions, providing a platform for studying symmetry breaking and RG flows.
• Bootstrap techniques, integrable systems, and defect fusion analyses yield universal constraints and novel fixed points in quantum field theory.

Interacting conformal line defects are codimension-2 loci in conformal field theories (CFTs) where local interactions, operator spectra, and correlation functions exhibit dynamics intrinsically richer than those of both the “ambient” CFT and of trivial or generalized free defects. Their paper tests and extends general principles of quantum field theory, conformal invariance, operator product expansion (OPE), representation theory, RG flows, and even higher-categorical and integrable structures. The theory and phenomenology of interacting line defects has advanced through synergistic developments in lattice models, algebraic QFT, analytic bootstrap, numerical bounds, and the systematic paper of perturbative and nonperturbative fixed points.

1. Construction and Physical Realization

A conformal line defect in a d-dimensional CFT is a one-dimensional locus L (e.g., the path traced by a Polyakov or Wilson loop, or the intersection of frustrated bonds in a lattice model) that preserves an SO(2,1) × SO(d–1) subgroup of the full conformal group SO(d+1,1). Interacting line defects arise in a variety of contexts:

• Lattice Constructions and Dualities: Twist (monodromy) defects in the 3d Ising model are defined as the boundary of a surface along which nearest-neighbor couplings are frustrated (their sign flipped). In dual terms, these are Wilson lines of the Z₂ gauge theory (Billó et al., 2013).
• Defects as Boundaries/Interfaces or via Localized Interactions: Interacting line defects can be engineered by coupling free or interacting bulk fields to lower-dimensional systems—e.g., by adding relevant deformations localized along the line, or by coupling bulk fields to minimal models or SYK-like systems at the defect (Bashmakov et al., 2 Oct 2024, Ge et al., 27 Mar 2024).
• Supersymmetric and Wilson Loop Defects: In N=2 or N=4 SYM, supersymmetric Wilson lines act as conformal line defects. Modifying their couplings or framing can generate a web of RG flows and new interacting fixed points (Castiglioni et al., 21 Feb 2025).
• Transdimensional and Intersecting Defects: Continuously interpolating defect dimension p = 2+δ and studying intersection-induced RG flows on “edges” or “corners” generalizes the landscape of possible interacting fixed points (Sabbata et al., 26 Nov 2024, Shachar, 21 Nov 2024).

These constructions allow for robust realization and classification of interacting line defects in both free and interacting bulk QFTs.

2. Symmetry, Operator Content, and OPE Structure

In the presence of a line defect, the ambient conformal group SO(d+1,1) is broken, typically to SO(2,1) × SO(d–1). The defect supports its own spectrum of local operators, which are organized by their scaling dimensions Δ and transverse “spin” (under SO(d–1) rotation). This spectrum is highly nontrivial even when the bulk theory is free or nearly free:

• On a lattice, local defect operators are constructed as gauge-invariant combinations of neighboring spins respecting enhanced symmetry groups such as D₈ (in the 3d Ising twist defect) (Billó et al., 2013).
• Primary defect excitations include Z₂-odd spin-½ operators and the ubiquitous displacement operator D (of protected dimension Δ_D = 2 and spin 1), the latter universally reflecting the breaking of bulk translation invariance.
• The OPE of bulk fields near the defect organizes into

$$\text{bulk field} \sim \sum\limits_{j} |x^\perp|^{\Delta_{\hat{\mathcal{O}}_j} - \Delta_{\mathcal{O}}} \, \left[\text{phase/spin-dependent structure} \times \hat{\mathcal{O}}_j\right],$$

where $\hat{\mathcal{O}}_j$ are defect primaries (Billó et al., 2013, Billò et al., 2016).

The nontriviality of defect operator spectra is often revealed by numerical evaluation of scaling dimensions—frequently integer-like or only mildly renormalized compared to free theory—reflecting a near-free behavior in some settings (Billó et al., 2013), but with strongly interacting OPE data in others (Bashmakov et al., 2 Oct 2024, Ge et al., 27 Mar 2024).

3. Correlation Functions and Conformal Invariance

Correlation functions in systems with a line defect are subject to stringent constraints from residual symmetry:

• Two-point functions of defect operators take universal power-law forms, $$\langle\hat{\mathcal{O}}(0)\hat{\mathcal{O}}(x)\rangle \sim |x|^{-2\Delta}$$, allowing direct extraction of low-lying anomalous dimensions (Billó et al., 2013, Barrat et al., 2023).
• Bulk–defect mixed correlators: Conformal symmetry fixes the analytic structure,

$$\langle\mathcal{O}(x) \hat{\mathcal{O}}(0)\rangle = C^{\mathcal{O}}_{\hat{\mathcal{O}}} |x^\mu|^{-2\Delta_{\hat{\mathcal{O}}}} |x^i|^{\Delta_{\hat{\mathcal{O}}}-\Delta_{\mathcal{O}}},$$

with precise phase and amplitude relations determined by the lattice or continuum symmetry prescriptions (Billó et al., 2013, Billò et al., 2016).

• Four-point functions of defect operators are organized in terms of conformal blocks, which in defect setups correspond to eigenfunctions of Calogero–Sutherland integrable Hamiltonians. This unlocks a suite of analytic tools for extracting OPE data (Isachenkov et al., 2018).

For higher-point and mixed correlators, analytically tractable regimes exist—especially at large N or for small deformations of a straight defect—enabling calculation of expectation values and defect “entropies” to all orders in perturbation theory or as a systematic expansion about the straight line (Gabai et al., 2023, Gabai et al., 12 Jan 2025).

4. Renormalization Group Flows and Fusion

Interacting line defects are natural loci of RG flows:

• Defect RG flows: Defects perturbed by relevant local operators undergo irreversible flows to new fixed points. This is characterized by a defect version of the g-theorem: there exists a canonical “defect entropy” or g-function whose monotonic decrease is governed by a gradient formula involving the two-point function of the defect stress tensor (Cuomo et al., 2021, Castiglioni et al., 21 Feb 2025).
• Fusion of Defects: When two defects approach each other and are “fused,” the path integral is invariant at the level of bare couplings, but the renormalization of local defect operators registers new UV divergences tied to the fusion process. The RG fixed point after fusion restores conformal invariance and typically leaves only a subset of original defect couplings nonzero, reflecting the “washing out” of explicit scales introduced by separation (Rousu, 2023, Bartels et al., 2013).
• Categorical and higher algebraic structures: The fusion of defects, their sectors (bimodules), and representation theory are encoded in the framework of conformal nets and von Neumann algebraic categories, allowing rigorous classification and equivalence statements for interacting defects (Bartels et al., 2013).

These RG and fusion phenomena underlie the universality and structural richness of the space of interacting conformal defects.

5. Bootstrap, Integrability, and Universal Constraints

The analytic and numerical bootstrap has achieved significant traction for interacting line defects:

• Numerical bootstrap: Crossing equations for correlators of displacement, tilt, and other canonical defect operators, often within the context of O(N) global symmetry, carve out allowed regions for scaling dimensions and OPE coefficients. Features such as "kinks” or “cusps” in these bounds strongly suggest the existence of physically interesting interacting defects, and comparison with ε-expansion and Monte Carlo results confirms quantitative agreement (Gimenez-Grau et al., 2022, Lanzetta et al., 20 Aug 2025).
• Casimir and block technology: For defect correlators, the conformal block decomposition reduces to analytic differential equations that, in nontrivial cases, map onto integrable systems (Calogero–Sutherland models), admitting explicit solutions and inversion formulas (Isachenkov et al., 2018).
• Universal integral constraints: The invariance of the defect partition function under conformal transformations (realized nonlinearly on a wiggled defect) yields exact integral constraints on correlation functions. These constraints supply sum rules for OPE data and, in some cases, fix previously undetermined structure constants at subleading order in ε or at strong coupling (Gabai et al., 12 Jan 2025).

Furthermore, the functional and analytic bootstrap for deformed or curved line defects demonstrates that, under assumptions of conformal symmetry and large-N factorization, the defect expectation value is often uniquely fixed as a functional of its shape (Gabai et al., 2023).

6. Applications, Extensions, and Emerging Directions

Interacting conformal line defects have broad implications and spur new research directions:

• Holography and AdS/CFT: In large N CS–matter theories and supersymmetric gauge theories, line defects correspond to open string worldsheets in AdS, making their conformal data essential for precision studies of holographic duality (Gabai et al., 2023).
• Thermal and finite-temperature effects: Temporal defects (e.g., Polyakov loops) at finite temperature induce a new class of OPE data—thermal defect one-point functions—governed by sum rules derived from the KMS condition. The defect’s influence on free energy and entropy density is captured by the stress-tensor one-point function expanded in the defect OPE (Barrat et al., 19 Jul 2024).
• Intersections and higher-codimension generalizations: When multiple defects intersect (for example, at edges or trihedral corners), novel RG flows, angular dependencies in anomalous dimensions, and higher-dimensional analogs of the cusp anomalous dimension arise (Shachar, 21 Nov 2024).
• Transdimensional defects: The paradigm of continuously variable defect dimension (p = 2+δ) provides a unified view that interpolates between line, surface, and interface defects. Operator dimensions and RG structure can be computed at both perturbative and large-N levels (Sabbata et al., 26 Nov 2024).
• Constraints from symmetry breaking: Defects that break global or discrete symmetries (such as inversion) permit non–trivial line defect fixed points even in free bulk theories. For example, in the absence of inversion symmetry, a special cross ratio (denoted ν) enables the realization of interacting defect sectors in free scalar theory through facilitated bulk–defect–defect three-point function structure (Bartlett-Tisdall et al., 3 Oct 2025).

These findings highlight that interacting line defects are both natural laboratories for deep aspects of conformal and quantum field theory and central ingredients in the characterization of universality classes of critical phenomena.

7. Summary Table: Key Properties and Approaches

Aspect	Description/Example	Key Reference
Defect Construction	Twist defect, frustrated bonds, Wilson loop, coupled lower-D CFT	(Billó et al., 2013, Bashmakov et al., 2 Oct 2024)
Operator Content	Defect primaries (spin, displacement), near-free or interacting	(Billó et al., 2013, Gimenez-Grau et al., 2022)
Correlator Geometry	Two- and four-point, mixed bulk–defect, Casimir/CS Hamiltonian	(Billò et al., 2016, Isachenkov et al., 2018)
RG and Fusion	Entropy (g-function), monotonic RG, defect fusion, sector fusion	(Cuomo et al., 2021, Rousu, 2023, Bartels et al., 2013)
Bootstrap/Integrability	Numerical bounds, singular features, Calogero–Sutherland approach	(Gimenez-Grau et al., 2022, Isachenkov et al., 2018)
Defect Data and Sum Rules	Constraint equations, finite temperature, integral sum rules	(Gabai et al., 12 Jan 2025, Barrat et al., 19 Jul 2024)

Interacting conformal line defects thus constitute a robust, versatile, and mathematically rich sector of modern conformal field theory, connecting rigorous algebraic, analytic, numerical, and physical methods that shape our understanding of universality, criticality, and duality in quantum field theory.