Line Defects, Conformal Quantum Mechanics, and Model Handoffs

The same $SL(2,\mathbb R)$ covariance can describe three physically different situations: an autonomous Euclidean operator algebra in one dimension, a quantum-mechanical system with conformal time evolution, or the local operator algebra living on a line defect inside a higher-dimensional CFT. Their correlators may share blocks and crossing equations, but their states, observables, symmetry labels, and consistency conditions are not interchangeable.

Required background. One-Dimensional Conformal Symmetry and SL(2,R) supplies the representation data. Primaries, Correlators, and Ordering Sectors fixes the ordered correlators. Helpful background. Support and Codimension introduces extended operators; Conformal Boundaries and Defects develops the ambient defect geometry.

Three uses of the same algebra

An intrinsic one-dimensional CFT data set consists of local operators on the line or circle, an OPE with a stated convergence domain, a vacuum functional, crossing-compatible ordered correlators, and—when unitarity is claimed—a reflection-positive inner product. It can be studied abstractly without specifying a Lagrangian.

Conformal quantum mechanics instead begins with a Hilbert space, a Hamiltonian, and operators depending on time. The generators $H$, $D$, and $K$ close $\mathfrak{sl}(2,\mathbb R)$, but the physical Hamiltonian need not be the compact generator used to organize a lowest-weight representation. Time ordering, self-adjoint domains, the vacuum, and the spectrum of $H$ are dynamical information. They do not follow from a Euclidean block decomposition; the representation and domain distinctions are developed in Andrzejewski 2016, §§2–4.

A conformal line defect in a $d$-dimensional ambient CFT preserves

$$SO(2,1)\times SO(d-1),$$

for a straight line, up to covers and discrete symmetries. Defect primaries carry a one-dimensional scaling dimension $\widehat\Delta$ and a representation of the transverse rotation group $SO(d-1)$. Their correlators along the line use $SL(2,\mathbb R)$ blocks, but their OPE coefficients are data of the pair “ambient CFT plus defect,” not of the ambient CFT alone.

The ambient information carried by a defect

The line supports several related expansions:

• a defect OPE among operators $\widehat{\mathcal O}$ inserted on the line;
• a bulk-to-defect OPE that resolves a bulk operator approaching the line;
• bulk correlators in the presence of the defect, with both bulk- and defect-channel decompositions.

Only the first resembles an autonomous 1D four-point problem. Even there, the allowed transverse-spin sectors and selection rules remember the ambient embedding.

Broken translations normal to the line define the displacement operator $D^i$. In a standard flat-defect normalization its Ward identity is

$$\partial_\mu T^{\mu i}(x) =\delta^{(d-1)}(x_\perp)D^i(x_\parallel),$$

where $i$ is transverse. Conformal covariance fixes $\widehat\Delta_D=2$ for a line and places $D^i$ in the vector representation of $SO(d-1)$. The overall sign and delta-function normalization depend on the convention for translating the defect. Detailed tensor structures, bulk-to-defect blocks, and displacement sum rules belong to Boundaries, Defects, and Interfaces.

Classification by observable content

Question	Intrinsic 1D CFT data	Conformal quantum mechanics	Line-defect CFT data
Coordinate	Euclidean line or circle	Physical time, with a chosen Hamiltonian	Coordinate along an embedded line
State input	Vacuum functional or radial state construction	Hilbert space, domains, and time evolution	Ambient vacuum with the defect inserted
Extra symmetry labels	Internal and discrete symmetries	Model-dependent conserved charges	Transverse $SO(d-1)$ representations
Characteristic OPE	Local 1D OPE	Operator products may be time-ordered and domain-sensitive	Defect OPE plus bulk-to-defect OPE
Stress-tensor information	Not guaranteed by global $SL(2,\mathbb R)$ alone	Hamiltonian is explicit; a local stress tensor need not exist	Ambient $T_{\mu\nu}$ and displacement Ward identity
Positivity test	Euclidean reflection positivity	Positive Hilbert norm plus self-adjoint evolution	Ambient reflection positivity compatible with the line
Data dependence	Intrinsic spectrum and OPE coefficients	Hamiltonian and chosen state	Ambient theory, defect type, and transverse sector

The most economical classification is the weakest one supported by the observables. A four-point function satisfying $SL(2,\mathbb R)$ crossing establishes a consistent one-dimensional correlator problem. It does not by itself identify a quantum-mechanical Hamiltonian or an ambient line defect.

A disciplined handoff test

Before importing data from a model, ask:

1. What is inserted? Local Euclidean operators, Heisenberg operators, or defect-local operators?
2. What prepares the state? A reflection-positive vacuum, a ground state of $H$, a thermal density matrix, or an ambient path integral with a defect?
3. Which $SL(2,\mathbb R)$ generator evolves the system? Translation, dilation, or a compact combination?
4. What extra labels survive? Fermion parity, internal charges, transverse spin, supersymmetry, or topological sectors?
5. Which completeness relation justifies the OPE? A discrete sum, a continuous spectral integral, or a mixed resolution?
6. Where does positivity come from? Specify the reflection or Hilbert-space adjoint that turns coefficients into squares.

If an answer depends on a microscopic model, the derivation and normalization remain with that model’s subject volume. This chapter accepts only the resulting one-dimensional conformal data after those questions are answered.

Examples with different interpretations

The de Alfaro–Fubini–Furlan inverse-square model realizes $\mathfrak{sl}(2,\mathbb R)$ on a quantum-mechanical Hilbert space. Choosing a compact combination of $H$ and $K$ yields a discrete basis, but it does not make the physical translation Hamiltonian identical to circle dilatation de Alfaro, Fubini, and Furlan 1976, §§2–4.

A Wilson or impurity line in a higher-dimensional CFT supports defect primaries and a displacement operator. The one-dimensional block decomposition is then one projection of a larger data set. The general ambient and defect tensor structures are developed by Billò et al. 2016, §§2–5.

Generalized-free 1D data provide a third case: they solve the intrinsic crossing equations exactly, while supplying neither a particular Hamiltonian nor an ambient embedding. “Exactly solvable” therefore does not determine physical interpretation.

Common pitfalls

Identifying $D$ with the physical Hamiltonian without checking the frame. Dilatations generate logarithmic scale evolution. A quantum-mechanical model may use $P$ as $H$ and a different compact combination for its discrete basis.

Dropping transverse representations from line data. Two defect primaries with the same $\widehat\Delta$ but different $SO(d-1)$ representations belong to different sectors and obey different selection rules.

Inferring an ambient defect from a 1D crossing solution. Defect crossing is necessary for a defect operator algebra but does not reconstruct the ambient stress tensor, bulk-to-defect OPE, or displacement normalization.

Exercises

A one-dimensional spectrum contains a dimension-two vector under $SO(4)$. Is that enough to identify a displacement operator for a line in five dimensions?

Solution

No. The dimension and transverse representation match the kinematics of a displacement operator, but identification also requires an ambient stress tensor and the broken-translation Ward identity with the candidate on its right-hand side. Without those data it is only a dimension-two $SO(4)$ vector.

Why can a discrete spectrum of a compact $SL(2,\mathbb R)$ generator coexist with a continuous spectrum of the physical Hamiltonian?

Solution

The compact generator and the parabolic translation generator are different self-adjoint operators in the same representation. Discreteness is a spectral property of the chosen generator, not of the Lie algebra as a whole.

References

• Andrzejewski, K. “Quantum Conformal Mechanics.” Annals of Physics 367 (2016): 227–250. arXiv. DOI.
• Billò, M., Gonçalves, V., Lauria, E., and Meineri, M. “Defects in Conformal Field Theory.” Journal of High Energy Physics 2016, 091 (2016). arXiv. DOI.
• de Alfaro, V., Fubini, S., and Furlan, G. “Conformal Invariance in Quantum Mechanics.” Il Nuovo Cimento A 34 (1976): 569–612. DOI.