Carrollian Bosonic String: A Tensionless Model

• Carrollian bosonic string is a tensionless model defined on Carrollian manifolds with ultralocal worldsheet dynamics and degenerate metric structures.
• It features extended BMS3 gauge symmetries that uniquely govern its finite-dimensional quantum spectrum and constraint algebra.
• BRST quantization reveals a cohomology encoding spacetime deformations and background fluxes, linking quantum states to modifications of Carrollian geometry.

The Carrollian bosonic string is a string-theoretic model defined on a two-dimensional worldsheet and a target spacetime, both equipped with Carrollian structures. In such geometries, the metric is ultra-local—formally, the speed of light is set to zero—yielding a degenerate causal cone and novel symmetry properties. Carrollian string theories arise in tensionless or ultra-relativistic limits and, as shown by recent research, possess residual gauge symmetries isomorphic to the three-dimensional extended BMS algebra. Quantization via BRST techniques leads to sharply distinctive spectra and representations compared to the usual bosonic string, with all physical states associated to zero energy and a finite-dimensional cohomology interpreted as spacetime deformations and background fluxes.

1. Classical Formulation and Worldsheet Structure

The classical Carrollian bosonic string is described by a sigma model whose dynamical variables are embedding maps $X^\mu(\sigma)$ from a Carrollian worldsheet to a Carrollian target spacetime. The worldsheet is equipped with a degenerate metric structure—concretely, a clock one-form and a Carroll vector field—which defines non-propagating time evolution. The action is typically constructed in first-order (phase-space) form and, after proper gauge fixing (Carrollian analogue of conformal gauge), takes the schematic form: $$S = \int d^2\sigma \, \mathcal{E} \left[ \frac{1}{2}\tau^2 \left( v^\alpha_\alpha X^i P_i + e^\alpha e^\beta_\alpha (\partial_\beta X^i)(\partial_\beta X^i) \right) + \frac{1}{2} v^\alpha v^\beta_\alpha (\partial_\beta X^0)(\partial_\beta X^0) \right]$$ where $\tau$ is the string tension, $v^\alpha$ and $e^\alpha$ are Carrollian frame fields, and the action includes a Lagrange multiplier enforcing trivial time evolution for transverse fields $X^i$ (Figueroa-O'Farrill et al., 4 Sep 2025).

The ultralocal (tensionless) feature implies that the theory’s causal structure lacks light-cone propagation; the worldsheet equations of motion constrain spatial evolution while temporal derivatives are suppressed or absent.

2. Carrollian Gauge Symmetries and Residual Algebra

Imposing Carrollian conformal gauge exposes an extended set of residual gauge symmetries. After gauge fixing, the algebra of residual diffeomorphisms is isomorphic to the extended BMS$_3$ algebra. Its generators (typically denoted $L_n$, $M_n$) satisfy: $$[L_n, L_m] = (n - m) L_{n + m},\qquad [L_n, M_m] = (n - m) M_{n + m},\qquad [M_n, M_m] = 0$$ Central extensions enter upon quantization; the operator product expansions for energy–momentum fields $T(z)$ and $M(z)$ are

$$T(z)T(w) \sim \frac{\frac{1}{2}c_L}{(z-w)^4} + \cdots,\qquad T(z)M(w) \sim \frac{\frac{1}{2}c_M}{(z-w)^4} + \cdots$$

with criticality fixing $c_M = 0$ and $c_L = 52$, yielding a 26-dimensional target space (Figueroa-O'Farrill et al., 4 Sep 2025).

This algebra replaces the double Virasoro symmetry of standard string theory and governs the constraint structure and quantization of the Carrollian bosonic string.

3. BRST Quantization and Cohomological Structure

Quantization is performed using BRST methods tailored to the BMS$_3$ symmetry structure. Two sets of ghost fields ($b, c$ and $B, C$) of conformal weights (2, –1) encode the gauge symmetry. The BRST current takes the canonical form: $$J_{\mathrm{BRST}}(z) = c(z) T(z) + C(z) M(z) + \text{(ghost terms)}$$ and its zero mode satisfies $Q_{\mathrm{BRST}}^2 = 0$ precisely at critical central charge values. The full BRST complex, constructed from oscillator creation modes acting on momentum eigenstates, is graded by ghost number and depends sensitively on the ghost zero modes ($c_0, C_0$), yielding a four-fold degeneracy ("pictures") in cohomology (Figueroa-O'Farrill et al., 4 Sep 2025).

BRST cohomology is computed using semi-infinite methods in momentum space, and the relevant homological structures are mapped explicitly in tables of ghost gradings and symmetry representations.

4. Physical Spectrum: Finiteness and Representation Theory

In contrast with the standard bosonic string (which exhibits an infinite tower of massive string states), the spectrum of the Carrollian bosonic string is finite-dimensional. All nontrivial physical states occur at zero "energy" ($p_0 = 0$), so the quantum string does not support dynamical propagation in the Carrollian time direction.

For zero transverse momentum ($p = 0$), the BRST cohomology organizes into representations of SO(25) (the transverse rotation group). For nonzero transverse momentum ($p = (0, p)$), the residual rotation symmetry is SO(24), and BRST representatives are interpreted as square-integrable sections over the 24-dimensional sphere in momentum space. These cohomology classes correspond to unitary irreducible representations of the 26-dimensional Carroll group, following a Mackey–Wigner representation-theoretic construction (Figueroa-O'Farrill et al., 4 Sep 2025).

Notably, the cohomology exhibits Poincaré duality: for ghost number $n$, the relative cohomology $H^n$ is isomorphic to $H^{5-n}$, which persists in the absolute cohomology via long exact sequences.

5. Geometric Interpretation: Spacetime Deformations and the Kalb–Ramond Field

A salient physical interpretation arises for BRST cohomology classes of ghost number 2 (and related pictures), which correspond to first-order deformations of the spacetime Carrollian structure. Explicitly, the classes can be mapped to:

• Deformations of the Carroll vector field ($\kappa^\mu$), although one component remains unexplained,
• Deformations of the "ruler" part (degenerate metric) $h_{ij}$,
• Deformations represented by the Kalb–Ramond two-form field $B_{\mu\nu}$.

These deformations describe fluctuations of the background geometry and fluxes to which the Carrollian string couples. Most of the physical states in the cohomology can thus be interpreted as generating infinitesimal changes in the underlying Carrollian spacetime or its background fields (Figueroa-O'Farrill et al., 4 Sep 2025).

Such geometric connection between quantum string states and target-space structure is generically absent or differently realized in Lorentzian string theories, underscoring the specific utility of the Carrollian framework for modeling non-relativistic gravity and field theories.

6. Implications, Distinctions, and Future Directions

The quantum Carrollian bosonic string formalism, with its critical dimension, ultralocal worldsheet, BMS$_3$ gauge algebra, finite-dimensional spectrum, and representation-theoretic richness, exhibits profound distinctions from standard bosonic string theory:

• The lack of infinite towers of oscillator excitations,
• Spectral truncation at zero energy,
• Organization of quantum states as representations of the Carroll group (not the Lorentz group),
• Natural realization of Poincaré duality and geometric cohomology interpretations.

These features suggest that the Carrollian bosonic string is uniquely suited to describing tensionless, ultra-relativistic regimes—such as near black hole horizons or flat holography—where standard relativity is degenerate. The finite-dimensional, cohomological nature of the spectrum further invites exploration of higher-spin generalizations, dynamical Carrollian gravity, and connections to non-Lorentzian quantum field theory, as well as the possible extension to supersymmetric models and interpolating limits between Carrollian and Galilean theories.

A plausible implication is that many of the amplitudes and physical quantities in Carrollian bosonic string theory could serve as sources for deformations of background fields in quantum gravity models with an ultra-local causal structure, and that the geometric coupling to the Kalb–Ramond field may tie the theory to more general non-relativistic string backgrounds. The representation-theoretic structure embedded in the BRST cohomology aligns Carrollian string physics with modern developments in Carrollian geometry, celestial holography, and algebraic classification of ultra-relativistic field theories.