Null Strings Gauged and Reloaded, I: Null Strings Have Carroll-Weyl Gauge Symmetry

Abstract: Null strings, strings with Carrollian worldsheets, are traditionally described by the Isberg-Lindström-Sundborg-Theodoridis (ILST) action, which is obtained via a tensionless limit of standard tensile strings. In a recent work, we observed that the ILST action enjoys an overlooked partial-gauge symmetry whose existence calls into question the consistency of {standard} null strings analysis found in the literature. In this paper, we show that the Carrollian geometry provides us with two Weyl scaling options, in contrast to a single Weyl scaling available for the ordinary tensile string worldsheet. Defining the null string theory by the action that realizes the two Carroll-Weyl scalings as well as the 2D diffeomorphisms as local (gauge) symmetries, we construct the new null string action. We show that the ILST action is obtained after fixing one of the two Carroll-Weyl scalings of the action that we construct, and that the residual part of this symmetry is precisely the overlooked partial-gauge symmetry. We hence clarify the Carroll-geometric origin of the overlooked symmetry and pave the way for a consistent quantization of null strings.

• The paper establishes dual Carroll-Weyl gauge symmetry as the core insight, emphasizing the necessity of two distinct scalings in null string theory.
• It introduces a new gauged null string action that incorporates both volume-modulating and volume-preserving Carroll-Weyl scalings while preserving diffeomorphism invariance.
• The findings highlight significant implications for consistent quantization and extensions to null p-branes, urging a revision of prior tensile string analyses.

Carroll-Weyl Gauge Symmetry in Null String Theory

Overview

This work rigorously establishes the Carroll-Weyl gauge symmetry structure in null string theories, emphasizing how their worldsheet geometry—Carrollian rather than Lorentzian—enables a richer gauge symmetry. It scrutinizes the Isberg-Lindström-Sundborg-Theodoridis (ILST) action, derives a generalized action implementing dual Carroll-Weyl scalings, and clarifies the geometric origin and significance of previously overlooked codimension-1 gauge invariance. The analysis underpins the need for consistent treatment of null strings, especially in high-energy regimes and in the presence of strong gravitational backgrounds, and outlines implications for quantization and null p-brane extensions.

Carrollian Worldsheet Geometry and Symmetry Structure

The transition from tensile to tensionless strings leads to a fundamentally different worldsheet geometry. Null strings reside on Carrollian manifolds characterized by a degenerate metric $h_{ab}$, a kernel vector $v^a$, and a clock form $n_a$. Importantly, unlike Lorentzian worldsheets, Carrollian geometry admits two distinct Weyl scalings:

• The volume-modulating ($\phi$) scaling: $n_a, \ell_a \to e^\phi n_a, e^\phi \ell_a$, $v^a \to e^{-\phi} v^a$
• The volume-preserving ($\chi$) scaling: $n_a \to e^\chi n_a$, $\ell_a \to e^{-\chi} \ell_a$, $v^a \to e^{-\chi} v^a$

These two scaling symmetries are absent in ordinary string theory, where only a single Weyl scaling exists.

Construction of Gauged Carroll-Weyl Null String Action

The ILST action, derived from the tensionless limit of standard strings, formally respects diffeomorphism and Weyl symmetry but misses the full gauge structure inherent to Carrollian geometry. The paper constructs a new null string action by gauging both Carroll-Weyl scalings and 2D diffeomorphisms, introducing a Carroll-Weyl gauge field $v^a$0:

$v^a$1

where $v^a$2, $v^a$3, and the fields transform appropriately under both Carroll-Weyl scalings. The action is invariant with respect to:

• Diffeomorphisms
• Volume-modulating Carroll-Weyl scaling ($v^a$4)
• Volume-preserving Carroll-Weyl scaling ($v^a$5), with $v^a$6

Gauge fixing one Carroll-Weyl symmetry (setting $v^a$7) reduces this action to the ILST form, illuminating the origin of the residual codimension-1 gauge symmetry observed earlier.

Equations of Motion and Constraints

Varying the gauged action produces equations characteristic of gauge systems:

• Covariant conservation for $v^a$8: $v^a$9
• Constraints from variation with respect to $n_a$0 and $n_a$1:
  • $n_a$2 for each $n_a$3; $n_a$4
  • $n_a$5

Gauge degrees of freedom associated with $n_a$6 do not propagate classically; their residual symmetry in the gauge-fixed ILST action corresponds to transformations $n_a$7 with $n_a$8.

Implications and Future Directions

The identification of dual Carroll-Weyl gauge symmetry has immediate and broad implications:

• Consistency of Null String Quantization: The proper symmetry structure established here is foundational for consistent quantization. Prior analyses, ignoring the codimension-1 symmetry, may require revision.
• Symmetry Enhancement at Carrollian Point: The volume-preserving Carroll-Weyl scaling only appears at $n_a$9; it cannot be deduced from Lorentzian theories. This symmetry enhancement restricts the validity of simply taking limits of Lorentzian observables, necessitating careful treatment in the Carrollian regime.
• Extensions to Null p-Branes: The methodology and Carroll-Weyl symmetry generalize to null p-branes—relevant for higher-dimensional tensionless objects—suggesting similar symmetry revision is required for brane literature (Dutta et al., 2024).
• Quantization and Physical Hilbert Space: The symmetry analysis re-frames strategies for quantizing null strings and branes, which is crucial in settings such as near-black-hole geometries, AdS backgrounds, and regimes near the Hagedorn temperature.

Future developments will include rigorous Hamiltonian analysis (as planned in companion work), exploration of the quantization process under the new symmetry structure, and re-evaluation of null brane dynamics.

Conclusion

This study clarifies the gauge symmetry content of null string theory, exposing the necessity of dual Carroll-Weyl scaling in Carrollian geometries. The proposed gauged null string action remedies shortcomings in previous formulations and paves the way for consistent classical and quantum analysis. These insights not only reconcile the foundational structure of null string theory but also initiate a re-examination of null brane and tensionless string literature, fundamentally impacting the theoretical landscape of high-energy string dynamics and gravitational physics (2605.25817).