Null Line Defects in Lorentzian Field Theory

• Null line defects are one-dimensional loci along lightlike directions in spacetime that preserve enhanced conformal symmetry and yield localized shockwave effects.
• They are constructed via an ultraboost limit of timelike defects or through direct solutions with massless sources, producing delta-function supported field strengths.
• Their maximal symmetry constrains correlators and observables in Lorentzian field theories and links ultrarelativistic dynamics with nonrelativistic scaling behavior.

Null line defects are one-dimensional loci or submanifolds along null directions in Lorentzian or pseudo-Riemannian manifolds (most prominently in Lorentzian conformal field theory, quantum field theory, and continuum mechanics) where the properties, interactions, or symmetries of a physical system are altered in a highly localized fashion. In contrast to spacelike or timelike line defects, null defects align with the propagation of light and display enhanced symmetry, producing unique consequences for fields, correlation functions, and associated conservation laws. Although terminology such as "null line defect" appears in several domains, its technical realization, its effect on local and nonlocal observables, and its mathematical treatment depend intricately on the underlying physical and algebraic structure of the host system.

1. Geometric Definition and Symmetry of Null Line Defects

A null line defect is a codimension-two object localized along a null (lightlike) direction in spacetime. In a Lorentzian manifold with coordinates $(x^+, x^-, x^\perp)$ (where $x^+ = t + x$), a null line defect may be supported on $x^+ = 0, x^\perp = 0$, extending along the $x^-$ axis. Geometric analysis shows that such null lines preserve a much larger subgroup of the Lorentzian conformal group than either timelike or spacelike lines. Specifically, for a null line on $x^+ = x^i = 0$, the preserved subalgebra contains:

• translations along the defect,
• dilatations,
• "null rotations" mixing transverse and defect directions,
• a subgroup of special conformal transformations,
• an $SO(d - 2)$ rotation symmetry in the transverse space.

This maximal symmetry emerges from the fact that the light cone is conformally invariant, in contrast to massive trajectories. As a result, field-theoretic and algebraic observables defined on or near such a defect are severely constrained by conformal Ward identities, often leading to trivial or "shockwave"-supported solutions (Erramilli et al., 4 Sep 2025).

2. Null Line Defect Construction: Ultraboost and Distributional Limits

Two rigorous approaches to constructing null line defects in gauge theory and field theory are distinguished:

1. Ultraboost Limit of Timelike Defects: Starting from a massive (timelike) Wilson line, which physically corresponds to the worldline of a heavy charged particle, one takes the limit where the boost velocity $u \to 1$. The classical gauge potential for a particle moving at velocity $u$ reads

$A_u^\mu(x) = -\frac{g F^2}{4\pi} \frac{v^\mu}{R_u}$

with $R_u^2 = (u x^0 + x^1)^2 + (1 - u^2)|x^\perp|^2$. Differentiating yields the associated field strength. In the $u \to 1$ limit, the result is a pure shockwave:

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

indicating all nontrivial field strength is localized on the null hyperplane $x^+ = 0$ (Erramilli et al., 4 Sep 2025).

1. Direct Construction for Massless Sources: For a source $J_1^- = g \delta(x^+) \delta^2(x^\perp)$, the Lorenz-gauge solution is

$A_1^-(x) = -\frac{g}{4\pi x^+} \Theta_*(x^+)$

where $\Theta_*$ is supported for $x^+ > 0$ except on the defect. The field strength coincides with the shockwave found by ultraboost, and, crucially, the solution's scale-dependent contact terms (e.g., logarithmic $\mu$-dependent pieces) are pure gauge when acting in the restricted test function space $\mathcal{S}_{\mathrm{null}}(\mathbb{R}^4)$ intrinsic to null defects. This resolves issues of scale ambiguities and ensures gauge invariance (Erramilli et al., 4 Sep 2025).

The physical consequence is that whether obtained by limiting procedure or direct construction, any nontrivial field strength or observable is a distribution supported solely on the null plane.

3. Conformal and Correlator Constraints

Null line defects in conformal field theory (CFT) inherit a set of conformal isometries that leaves the null line invariant, but this maximal null symmetry "nearly trivializes" the structure of allowed correlators:

• Correlation functions involving operators restricted (or coupled) to the null defect are supported at coincident points or may yield vanishing contributions except on specific singular loci.
• Ward identities enforce strict dependence on the light-cone coordinate and prohibit generic dependence on other directions.
• In some settings (e.g., the null Wilson line in gauge theory), physical consistency demands interpreting correlators as tempered distributions with support restricted to test function subspaces adapted to the defect (Erramilli et al., 4 Sep 2025).

For scalar pinning defects, the ultraboosted (null) limit leads to solutions with support only on the light-ray, with associated boundary conditions or jump conditions arising as specialized limits of their timelike analogues (Erramilli et al., 4 Sep 2025).

4. Null Wilson Lines and Shockwave Solutions

A canonical example of a null line defect is the null Wilson line, interpreted as the trajectory of a massless charged particle in gauge theory. The associated observable is the path-ordered exponential along the light-ray: $$W_\mathrm{null} = \mathcal{P} \exp \left[ i \int dx^- A^-(x^+, x^-, x^\perp) \right]$$ The electromagnetic potential for this defect can be obtained either as an ultrarelativistic limit (boost $u \to 1$) of the massive Liénard–Wiechert potential, or by directly solving Maxwell's equations with a massless source. The physical field strength is always a delta-function-supported shockwave: $$F_{+i} = \frac{g F^2}{2\pi} \frac{x_i}{|x^\perp|^2} \delta(x^+)$$ All additional gauge-dependent or scale-dependent ($\mu$) terms are eliminated upon passing to the appropriate test function space $\mathcal{S}_{\mathrm{null}}$, ensuring that the null Wilson line respects the maximal conformal symmetries of the light-ray and forms a legitimate conformal defect operator (Erramilli et al., 4 Sep 2025).

5. Classification, Nonrelativistic Connections, and Shockwave Genericity

The enhanced symmetry of null defects yields strong similarities with nonrelativistic systems. The subgroup of conformal isometries that fix a null line coincides structurally with (projective) Galilean or Schrödinger groups in one fewer spatial dimension (Erramilli et al., 4 Sep 2025). This implies a close connection between the dynamical constraints on null line defects and nonrelativistic scaling/structure, especially in the context of ultraboosted limits.

The paper demonstrates the genericity of shockwave-type solutions for any operator or correlator localized on the null line. Physically, this means that any excitation or source placed along a null trajectory imprints its influence entirely on the light cone, with the rest of the spacetime unaffected except at points reachable by null propagation.

6. Distributions, Test Function Spaces, and Consistency Conditions

A major technical aspect in handling null defects is the distributional nature of the solutions and the necessity to use restricted test function spaces:

• The correct mathematical framework for null defect correlators requires defining them as distributions on $\mathcal{S}_{\mathrm{null}}(\mathbb{R}^4)$, restricting the allowed test functions to those supported or smooth near the null locus.
• This formalism resolves the well-known ambiguities in singular limits, such as ultraboosting the gauge potential (removal of $\log(\mu^2 |x^\perp|^2)$ terms as physically irrelevant).
• Compatibility between Gauss’ law, conformal symmetry, and ultrarelativistic limits is realized at the level of these restricted spaces and the associated "sewing" conditions at the defect (Erramilli et al., 4 Sep 2025).

7. Broader Context and Outstanding Questions

Null line defects play a crucial role in the paper of lightlike (ultrarelativistic) limits of observables, infrared dynamics in gauge theory (e.g., cusp anomalous dimensions, soft theorems), and have implications for nonperturbative constraints such as the uniqueness of shockwave solutions. The constraint to distributions with support on the null locus ensures that the set of observables is nontrivial only in scattering- or infrared-relevant regimes.

The paper of null defects also clarifies longstanding problems (such as the ultraboost of gauge potentials in classical electromagnetism) and connects conformal and nonrelativistic limits, suggesting that further exploration of null polygonal defects and their algebraic properties may yield deeper insights into conformal dynamics, quantum field theory, and potential applications in integrable systems (Erramilli et al., 4 Sep 2025).

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In summary, null line defects are characterized by their support along lightlike directions, their preservation of maximal conformal symmetry, and their propagation of sharply localized (shockwave) effects. Their mathematical realization mandates a careful distributional approach, with all physical observables restricted to the null locus and compatible test function spaces, providing a unifying perspective for a broad class of ultrarelativistic, conformally invariant phenomena.