Defect Conformal Manifolds

• Defect conformal manifolds are spaces parameterizing local deformations of defects in CFTs with inherited geometric and operator structures that govern symmetry breaking and marginal dynamics.
• Their analysis employs geometric reductions, spectral flows, and anomaly matching to derive canonical metrics and invariant structures that classify defect phenomena.
• They impact physical observables by encoding entanglement properties, transport responses, and fusion rules that quantify universal behavior in quantum field theories.

A defect conformal manifold is a parameter space of conformal defects or interfaces in a conformal field theory (CFT), equipped with a natural geometric and operator-theoretic structure inherited from the underlying field theory and its symmetries. Unlike the conventional conformal manifold—which parameterizes exactly marginal deformations of a full CFT—defect conformal manifolds are formed by deformations local to the defect, interface, or boundary, which may involve breaking global symmetries, non-invertible symmetry structures, spectral deformations, or anomaly-enforced moduli. Their structure is central to the classification of defect CFTs, the paper of universal transport and entanglement phenomena, and the formulation of higher-categorical field-theoretic frameworks.

1. Geometric and Algebraic Foundations

The structure of defect conformal manifolds can be traced to several geometric and algebraic origins:

• Möbius and Conformal Cartan Reductions: For submanifolds of conformal manifolds, a canonical approach is to reduce the ambient conformal Cartan geometry to a subbundle (the Möbius reduction) along the defect (1006.5700). The resulting “primitive data” include an induced conformal structure, a weightless normal bundle with a metric connection, and a trace-free second fundamental form. Bonnet-type theorems state that, given solutions to the Gauß–Codazzi–Ricci equations for this data, the immersion (up to Möbius transformations) is uniquely fixed, underpinning a geometric description of the defect conformal manifold.
• Spectral and Integrable Deformations: Möbius-flat submanifolds (immersed submanifolds with flat Möbius structure and normal bundle) admit spectral flows: a one-parameter family of deformations of the intrinsic geometry, preserving all conformal compatibility conditions (1006.5700). This spectral deformation maps the defect conformal data continuously within the manifold and connects to integrable systems.
• Canonical Metrics and Conformal Invariants: In the context of compact even-dimensional manifolds and Riemann surfaces, the selection of canonical metrics via minimization of the harmonic energy functional leads to conformal invariants that serve as coordinates on the moduli space of conformal (defect) structures (1105.5904). The critical value of the functional quantifies the “defect” of formality in the wedge algebra of harmonic forms, offering a quantitative probe of the geometric structure of defect conformal manifolds.
• Symmetry Breaking Coset Construction: When a defect breaks a global symmetry $G$ to a subgroup $H$, the resulting space of exactly marginal local defect deformations is the coset $G/H$, often carrying a homogeneous Kähler geometry with a Zamolodchikov metric determined by two-point functions of marginal operators on the defect (Drukker et al., 2022). The curvature tensor of this coset is fixed by the integrated four-point function of these operators, and explicit constructions appear in Wilson loop and surface operator examples in maximally supersymmetric gauge theories.
• Non-Invertible and Phantom Symmetries: In 1+1 dimensions, even in the absence of continuous bulk symmetries, defect conformal manifolds may arise from “phantom” (non-invertible, emergent in the folded theory) symmetries. Marginal, non-locally generated defect operators can parametrize nontrivial interface manifolds via continuous deformations that are protected only at the level of the folded (doubled) CFT (Antinucci et al., 14 May 2025).

2. Constraints from Conformal Symmetry and Ward Identities

The conformal symmetry imposes explicit constraints on the form, allowed deformations, and correlation functions of defects:

• Embedding Space and Cross-Ratio Formalism: Defect-expansion techniques and embedding space methods reveal that spherical (or planar in the flat limit) defects are naturally characterized by their O($m$)×SO($d-m+1,1$) invariance, with all correlation functions expressible via a finite set of conformal cross-ratios (Gadde, 2016). The number of independent cross-ratios for a pair of defects is given by $\min(m, k, d-m+2, d-k+2)$, enabling a systematic operator expansion and classification.
• Defect Operator Expansion and Conformal Blocks: Extended defects admit a “defect expansion,” analogous to the OPE, where the defect can be expressed as a sum over bulk local operators weighted by differential operators determined by conformal invariance (Gadde, 2016). The conformal blocks associated to defect correlators satisfy Casimir differential equations whose structure governs the analytic forms of all correlation functions involving defects. For three-point functions involving two bulk and one defect insertion, the resulting blocks are explicitly given in terms of Appell functions $F_4$ of two variables (Buric et al., 2020), reflecting the higher complexity of defect operator moduli.
• Image Method in DCFTs: A generalization of the 2d doubling trick allows higher-dimensional DCFT correlators to be systematically related to those of a conventional (defectless) CFT, by mapping each bulk insertion to a mirror-symmetrized pair (Nishioka et al., 2022). This reduces the complexity of defect correlator computations to those involving standard CFT techniques, and can directly realize the moduli of defect positions and parameters.

3. Deformations, Marginality, and Classification

The existence and classification of defect conformal manifolds are governed by the space of exactly marginal defect (and, in some cases, bulk) operators:

• Marginality and β-functions: For a deformation to be exactly marginal (and thus to generate a genuine conformal manifold), all β-functions associated with the deformation parameters must vanish (Karch et al., 2018). The cancellation of logarithmic divergences imposes sum-rules on the defect-bulk OPE coefficients and the defect three-point coefficients, generalizing the standard bulk conformal manifold analysis to include boundary and defect-localized terms.
• Anomaly-Enforced Moduli: In the presence of a bulk 't Hooft anomaly, continuous defect moduli may be enforced: defects are required to break the bulk symmetry, and the resulting conformal manifold is fixed (up to geometry) by the anomaly structure (Copetti, 21 Jul 2025). The modulated effective action, constructed to restore gauge invariance, includes universal terms determined by the anomaly coefficient. These directly set the defect conformal data (metric, connection), and their correlation functions are computable in terms of the anomaly.
• No-Go Theorems and Higher-Dimensional Obstructions: In higher dimensions ($d>2$), the existence of non-invertible, exactly marginal defect operators is heavily constrained by unitarity and topological arguments. Aside from certain free field examples (where the folded theory admits extra U(1) symmetries), general interacting theories do not support phantom conformal manifolds in the absence of bona fide continuous symmetries (Antinucci et al., 14 May 2025).

4. Entanglement, Universal Terms, and Physical Signatures

Defect conformal manifolds leave precise signatures in entanglement entropy, mutual information, and transport observables:

• Universal Terms in Entanglement Entropy: For codimension-two defects, entanglement (Renyi) entropies acquire universal logarithmic terms, whose coefficients are determined by the scaling dimensions of associated twist operators inserted at the tips of the causal diamond of the entangling region (Long, 2016). The conjectured universality of these terms across the defect conformal manifold implies strong constraints on how entanglement measures can vary under marginal deformations of the defect, reflecting robust geometric features.
• Zero-Mode and Topological Entanglement Corrections: In lattice realizations of defect CFTs (e.g., free-fermion chains with a conformal defect), zero-mode-induced degeneracy leads to analytic, O(1) corrections in the entanglement and Renyi entropies, which are nontrivially dependent on subsystem parity and invariant under scaling (Capizzi et al., 2023). These corrections directly encode universal data about the defect, distinct from the moduli of the bulk theory.
• Anomaly-Induced Boundary and Hall Responses: Bulk 't Hooft anomalies, modulated along the defect conformal manifold, fix transport signatures on the defect, such as quantized boundary charge pumping in (1+1)d and non-dissipative boundary Hall currents in higher dimensions (Copetti, 21 Jul 2025). The anomaly appears as a functional constraint on the modulated effective action and leads to measurable differences in physical quantities as one traces nontrivial loops in the moduli space of defect couplings.

5. Fusion, Higher Categories, and Effective Field Theory

The interplay and composition of defect conformal manifolds are formalized via fusion, higher-categorical structures, and effective field theory:

• Algebraic Fusion: The operation of fusing two defects—implemented via convolution (or categorical composition) of functors between conformal nets—yields a new defect, and under finite-index conditions, the closedness and associativity allow the formation of a 3-category of nets, defects, sectors, and intertwiners (Bartels et al., 2013). This categorical framework provides organizing principles for the moduli spaces of defect conformal manifolds in both physics and mathematics.
• Effective Field Theory for Fusion: The fusion of two nearby conformal defects is described by an effective field theory for the interpolating (“fused”) defect, with a derivative expansion in their separation (Kravchuk et al., 7 Jun 2024). Weyl invariance imposes strong constraints on the effective action: the leading term scales as $\ell^{-p}$ (where $\ell$ is the fusion distance, $p$ the defect dimension), with higher-derivative corrections. Anomaly-matching between the UV and IR defects is enforced via explicit terms in the action, leading to robust identification of scheme-independent invariants (such as certain anomaly coefficients) on the defect conformal manifold. When fusion breaks (part of) the transverse rotation symmetry, new, tilt-dependent anomaly terms enter, further enriching the structure of allowed defect moduli.
• Topological and RG Flow Defects: In string-theoretic and 2d CFT constructions, topological defects—constructed, for example, via Landau–Ginzburg matrix factorizations or via the unfolding of D-brane boundary states—categorify the phase space of defect moduli (Cabrera, 2017). Fusion rules for these algebraic interfaces parameterize the space of RG flows between UV and IR fixed points, providing a moduli-theoretic realization of the defect conformal manifold.

6. Compactness, Metric, and Moduli Space Geometry

The geometric structure of defect conformal manifolds is often rich, with compactness, Kähler or homogeneous coset structure, and nontrivial topology:

• Compact Conformal Manifolds: In examples with supersymmetric (especially 4d N=2 to N=1) defects, the moduli space of conformally equivalent defects can realize compact Kähler geometries, such as complex projective spaces, realized via quotient constructions (symplectic reduction or D-term constraints) (Buican et al., 2014). Compactness is certified by the completeness and bounded diameter of the Zamolodchikov metric on the manifold.
• Metric Geometry: The Zamolodchikov metric on the defect conformal manifold is directly fixed via the two-point function of exactly marginal defect operators. For symmetry-breaking cosets, the metric can be brought to a conformally flat form, with explicit curvature determined by integrated four-point functions (Drukker et al., 2022). Anomaly analysis may also enforce quantized holonomy or discrete topological classes in the moduli.
• Anomalous Moduli Space Geometry: The presence of bulk or defect anomalies may force the moduli space to have nontrivial topology (e.g., noncontractible loops), with physical consequences for charge pumping and transport under adiabatic deformations (Copetti, 21 Jul 2025). Effective actions, anomalies, and consistency conditions collectively ensure that the conformal manifold carries both the global information of the quantum field theory and the local conformal data of the defect.

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In summary, defect conformal manifolds are formed by the space of exactly marginal local deformations associated with defects, interfaces, or boundaries in conformal field theories, often constrained or organized by symmetry breaking, anomaly matching, non-invertible or phantom symmetries, and fusion or spectral deformation data. They are detected and parameterized by canonical metrics and invariants, their geometry is tightly constrained by conformal symmetry and anomalies, and their physical signatures are imprinted in observables ranging from transport (Hall response, charge pumping) to entanglement structure and RG-flow moduli. The interplay of differential geometry, homological algebra, operator formalism, and effective field theory provides a unified toolkit for the construction, classification, and physical interrogation of these manifolds across dimensions and contexts.