Do null defects dream of conformal symmetry? (2509.04578v1)

Abstract: We initiate the study of null line defects in Lorentzian conformal field theories in various dimensions. We show that null lines geometrically preserve a larger set of conformal isometries than their timelike and spacelike counterparts, explain a connection to non-relativistic systems, and constrain correlation functions using conformal Ward identities. We argue that having conformal symmetry, and especially maximal conformal symmetry, is extremely constraining -- nearly trivializing systems. We consider the (3+1)d scalar pinning field and null Wilson line examples in depth, compare their results to ultraboosted limits of timelike and spacelike systems, and argue that shockwave-type solutions are generic. A number of physical consistency conditions compel us to consider defect correlators as distributions on a restricted subspace of Schwartz test functions. Consequently, we provide a resolution to the longstanding problem of ultraboosted limits of gauge potentials in classical electromagnetism. We briefly analyze semi-infinite sources for the scalar in ($4-\epsilon$)-dimensions, consider solutions on the Lorentzian cylinder, and introduce the ''perfect null polygon'' which emerges for compatibility between Gauss' law and ultraboosted limits.

• The paper establishes that null defects in Lorentzian CFTs preserve a maximal conformal symmetry that nearly trivializes defect observables.
• Using conformal Ward identities, the authors demonstrate that allowed correlation functions reduce to shockwave-type distributions supported on the null plane.
• The study resolves ultraboost limit challenges through a distributional approach, linking null defects to non-relativistic and holographic frameworks.

Null Line Defects and Conformal Symmetry in Lorentzian CFTs

Introduction and Motivation

This work initiates a systematic paper of null line defects in Lorentzian Conformal Field Theories (CFTs) across various spacetime dimensions. Null defects, defined as line-like loci moving at the speed of light, are fundamentally Lorentzian objects with no Euclidean analog. Their paper is motivated by both physical and mathematical considerations: they model ultra-relativistic impurities, massless charged particles, and provide a bridge to non-relativistic systems via lightfront quantization and null reduction. The paper demonstrates that null lines preserve a surprisingly large subalgebra of the conformal group, leading to strong kinematic constraints on defect observables and nearly trivializing the associated defect CFTs.

Kinematics and Symmetry Algebra of Null Defects

A central result is the identification of the maximal conformal symmetry algebra preserved by a geometric null line in Minkowski space. Unlike timelike or spacelike lines, which preserve a product of 1d conformal and transverse rotation algebras, a null line preserves the so-called "null defect algebra":

$$\mathfrak{n}_d = \left( \mathfrak{sl}(2,\mathbb{R}) \times \mathbb{R} \times \mathfrak{so}(d-2) \right) \ltimes \mathfrak{h}_{d-2}$$

where $\mathfrak{h}_{d-2}$ is a $(d-2)$-dimensional Heisenberg algebra. This algebra is much larger than the symmetry preserved by non-null lines, and its structure is closely related to the Schrödinger algebra of non-relativistic CFTs in one lower dimension. The null defect splits spacetime into two regions separated by a shockwave plane, and the physics is often confined to or discontinuous across this plane.

Figure 1: Plots of $h(\tau)$ and $\phi_{\ell,0}^c(\tau)$ for an adiabatic defect with a specific choice of parameters, illustrating the causal response of a conformally coupled scalar on the Lorentzian cylinder.

Constraints from Conformal Ward Identities

The paper systematically analyzes the constraints imposed by the null defect algebra on correlation functions using conformal Ward identities. For infinite null line defects, the maximal symmetry is so restrictive that all non-trivial one-point functions must vanish, and two-point functions are characterized by discontinuities across the shockwave plane. The only allowed non-trivial solutions are shockwave-type distributions, supported on the null plane $x^+ = 0$. This is in stark contrast to timelike or spacelike defects, where non-trivial profiles are generically allowed.

The analysis is extended to pinning field deformations, null Wilson lines, and higher-spin operators. It is shown that only the null Wilson line can preserve the full $\mathfrak{n}_d$ symmetry in the UV, while scalar and higher-spin deformations necessarily break some of the maximal symmetry. The null Wilson line in pure Maxwell theory yields a field strength one-point function:

$$F_{+i} = \frac{g}{2\pi} \frac{x_i}{|x_\perp|^2} \delta(x^+)$$

which is compatible with ultrarelativistic limits and Ward identities, but does not preserve the full maximal symmetry, instead leading to a "perfect null polygon" configuration on the Lorentzian cylinder.

Distributional Nature of Defect Correlators

A key technical insight is that defect correlators in the presence of null defects must be understood as distributions on a restricted subspace of Schwartz test functions, denoted $\mathcal{S}_{\rm null}(\mathbb{R}^d)$. This restriction is necessary to avoid divergences and arbitrary scales, and to ensure compatibility with the expected symmetries. For example, the shockwave solution for a free scalar in $(3+1)d$,

$$\phi_s(x^+, x^-, x^\perp) = \frac{h}{4\pi} \delta(x^+) \log \mu^2 x_\perp^2$$

is only meaningful as a distribution on $\mathcal{S}_{\rm null}$, and the scale $\mu$ becomes physically irrelevant. This approach resolves longstanding issues in the ultraboosted limits of gauge potentials in classical electromagnetism.

Ultralimit and Shockwave Solutions

The paper carefully analyzes the ultraboosted limits of timelike and spacelike defects, showing that the null defect solutions are recovered as distributional limits. Shockwave-type solutions are generic, and the appearance of a lightlike Kubo formula is demonstrated for perturbative expansions. The paper of semi-infinite defects and defects on the Lorentzian cylinder reveals the necessity of adiabatic switching to avoid non-dissipating ripples and to maintain physical consistency.

The Perfect Null Polygon and Global Considerations

On the Lorentzian cylinder, the ultraboosted limit of two timelike Wilson lines leads to a "perfect null polygon" configuration, which preserves a larger set of symmetries than two infinitely long null rays. This configuration is necessary for compatibility with Gauss' law on compact spatial slices and provides a geometric realization of maximal symmetry in the defect sector.

(1+1)d Case and Representation Theory

In $(1+1)d$, the null defect algebra reduces to $$\mathfrak{sl}(2,\mathbb{R})_- \times \{\bar{J}_0, \bar{J}_1\}$$, and the paper classifies the projective unitary irreducible representations of the right-moving part. The branching of bulk lowest-weight representations to the defect algebra is analyzed, providing a putative description of defect local operators and the defect Hilbert space. The Ward identities for bulk local operators are solved, confirming the vanishing of one-point functions except for contact terms, and the structure of two-point functions is elucidated.

Implications and Future Directions

The results have several important implications:

• Nearly Trivialization of Defect CFTs: Maximal conformal symmetry for null defects almost entirely trivializes the defect sector, with only shockwave-type solutions allowed.
• Resolution of Ultralimit Pathologies: The distributional approach resolves issues in ultraboosted limits of gauge potentials and provides a consistent framework for null defect correlators.
• Connections to Non-Relativistic and Holographic Systems: The relationship to Schrödinger symmetry and non-relativistic CFTs suggests that techniques and results may be transferable, potentially resolving longstanding issues in lightfront quantization and null reduction.
• Collider Physics and SCET: The preservation of chiral dilatations in null defects aligns with fixed points in Soft Collinear Effective Theory, indicating relevance for collider observables and factorization.
• Global Geometric Structures: The perfect null polygon on the Lorentzian cylinder provides a geometric realization of maximal symmetry and is necessary for global consistency.

Theoretical developments may include a deeper understanding of defect OPEs, Hilbert space structure, and the lifting of non-relativistic CFT results to relativistic settings. Practically, the framework may inform the analysis of ultra-relativistic probes, shockwave backgrounds, and the structure of observables in high-energy and condensed matter systems.

Conclusion

This work establishes the foundational kinematics and symmetry constraints for null line defects in Lorentzian CFTs, revealing that maximal conformal symmetry is highly constraining and generically trivializes defect observables. The distributional approach to defect correlators resolves longstanding pathologies in ultraboosted limits and provides a consistent framework for analyzing null defects. The connections to non-relativistic systems, holography, and collider physics suggest rich avenues for future research, both in the formal structure of defect CFTs and in their physical applications.