Flipped Carrollian Contraction

• Flipped Carrollian contraction is an ultrarelativistic limit (c → 0) that reverses the roles of time and space, yielding Carrollian and BMS-type symmetry structures.
• It unifies approaches in flat space holography, Carrollian field theories, and string quantization by reorganizing dynamical spectra and symmetry algebras.
• The method employs anti-normal ordering and dual scaling of generators to achieve altered BRST quantization, modified scattering amplitudes, and ultra-local dynamics.

The flipped Carrollian contraction is a family of ultrarelativistic (speed of light $c \rightarrow 0$) limits that produce Carrollian symmetry structures, often by interchanging the roles of time and space sectors, or by reversing the time direction in one of multiple sectors. This concept appears across the paper of algebra contractions, holography, gravity, field theory, integrable systems, and quantum strings. Its key features are the emergence of Carrollian (or BMS-like) symmetries, ultra-local dynamical structures, finite or drastically reorganized spectra, and close ties to flat space holography.

1. Carrollian Geometry and Algebra: Flipped vs Standard Contractions

In Carrollian geometry, the metric is degenerate, separating a spatial metric from a distinguished null (Carrollian time) direction. The flipped Carrollian contraction arises when the contraction parameter (typically $c$) is sent to zero with an assignment of scaling weights that exchanges or reverses the roles of time and space in the sectoral decomposition of the symmetry generators.

For example, in contractions of the Maxwell algebra in $D+1$ dimensions (Barducci et al., 2019), partitioning indices into “Minkowskian” and “Euclidean” sectors allows two inequivalent limits:

• Galilean contraction: spatial momenta and boosts are rescaled, leaving time untouched.
• Carrollian (flipped) contraction: time-sector momenta and boosts are rescaled as $\tilde{P}_\alpha = \frac{1}{\omega} P_\alpha$, $$\tilde{B}_{\alpha a} = \frac{1}{\omega} B_{\alpha a}$$, with tensor charges scaled using exponents $t, r, s$ in a fashion that exchanges $t\leftrightarrow s$ compared to the Galilean case.

This structural “flipping” establishes that in the Carroll limit, what was associated with spatial directions (in the Galilean case) is now associated with time, and vice versa. The resulting contracted algebras under such scaling are non-isomorphic at the quantum level unless a further equivalence is established by order-preserving procedures, as is articulated in the context of $W_N$-algebras (Fredenhagen et al., 17 Sep 2025).

2. Holographic Description and Boundary Fluid/Gravity Duality

The flipped Carrollian contraction directly underpins the smooth $k \to 0$ (zero cosmological constant) limit in the AdS fluid/gravity correspondence (Ciambelli et al., 2018):

• The AdS boundary, initially $2+1$-dimensional, contracts into a $2$-dimensional spatial surface $S$ equipped with Carrollian time, metric $a_{ij}$, and frame $b_i$, the triplet $(t, a_{ij}, b_i)$ specifying the Carrollian structure.
• The degenerate boundary fluid velocity disappears, leaving energy density, heat currents $Q_i$, and viscous stress tensors $\Sigma_{ij}$, which transform under Carrollian diffeomorphisms and relate via duality to geometric data (Cotton tensor descendants).
• The holographic dictionary and reconstruction of Ricci-flat spacetimes are implemented by expanding in derivatives, yielding metrics determined entirely by Carrollian fluid data, and explicitly dual to families of spacetimes, such as Kerr–Taub–NUT and Robinson–Trautman solutions.

This mechanism exposes the Carrollian structure not as an exotic limit but as essential for flat space holography, where the boundary field theory is Carrollian, and asymptotic symmetries algebra is extended BMS.

3. Representation Theory and BRST Quantization in String Models

Quantum Carrollian bosonic and supersymmetric strings involve worldsheet and target spaces of Carrollian type (Figueroa-O'Farrill et al., 4 Sep 2025, Chen et al., 19 Jan 2025):

• The flipped contraction at the level of worldsheet CFT can mean working with either the oscillator vacuum (standard normal ordering) or a flipped vacuum (highest-weight prescription), each with different consequences for spectrum truncation and state–operator correspondence.
• The gauge-fixed Carrollian string has residual diffeomorphisms generating BMS$_3$, and the BRST cohomology, constructed from ghost systems and the associated BRST operator,

$$Q = \oint \frac{dz}{2\pi i} \big(c(z) T(z) + C(z) M(z) + \cdots \big)$$

is nilpotent when the central charge $c_L = 52$, fixing the critical dimension to $D+1 = 26$.

• The spectrum is finite-dimensional, supported only at zero energy ($p_0 = 0$), with cohomology classes corresponding to unitary irreducible representations of the Carroll group (SO(25) or induced from SO(24)). Poincaré duality $H^n(p) \simeq H^{5-n}(p)$ is realized at the level of BRST cohomology.

Geometric deformations represented by non-trivial vertex operator classes are mapped to Carrollian structure deformations

$$\kappa^\mu = \delta^\mu_0 + \kappa^{(1)\,\mu}, \quad h_{\mu\nu} = \delta_{ij} \delta^i_\mu \delta^j_\nu + h^{(1)}_{\mu\nu}, \quad B_{\mu\nu} = B^{(1)}_{\mu\nu}$$

with moduli space interpretations.

4. Quantum and Free-Field Realizations of Flipped Carrollian Contractions

In the construction of Carrollian $W_N$ algebras (Fredenhagen et al., 17 Sep 2025), two contraction procedures are contrasted:

• Flipped Carrollian contraction: Reverse the time direction in one sector before taking the contraction, realized operationally by anti-normal ordering in one copy of the free-field realization (e.g., two Virasoro or $W_N$ algebras).
  • At the classical level, the resulting Carrollian and Galilean algebras are isomorphic.
  • At the quantum level, the flipped contraction produces a quantum Carrollian $W_N$-algebra isomorphic to the quantum Galilean one (central charges:

  $$\hat{c}_L = c - \bar{c}, \qquad \hat{c}_M = \lim_{\epsilon \to 0} \epsilon^2 (c + \bar{c})$$

), whereas the symmetric contraction with averaged ordering yields quantum Carrollian algebras whose structure constants match classical Carrollian values.

The free-field/Miura transformation approach allows for an explicit realization of the algebraic structures, construction of modules, and systematic comparison between Galilean and Carrollian contractions at the quantum level.

5. Symmetry, BMS Extensions, and Flat Holography

Flipped Carrollian contractions are intimately tied to extensions of Carrollian symmetry:

• At the level of global symmetry, the contraction can convert the Poincaré group into the Carroll group, with the further extension to conformal Carroll or infinite-dimensional BMS algebras (as in the residual worldsheet algebra in Carrollian strings).
• In 2$+$1 and 3$+$1-dimensional (A)dS/BMS algebras, Carrollian contractions can be systematically imposed via rescaling boost and translation generators. In 3$+$1-dimensions, only a quasi- or “flipped” contraction is possible, whereby the identification of time and space-like generator sectors is mixed compared to the lower-dimensional or Galilean case (Borowiec et al., 2023).

This extension of symmetry under the flipped contraction plays a defining role in:

• Holographic dualities for asymptotically flat spacetimes, connecting Carrollian field theories at null infinity (Carrollian CFTs) with the bulk gravitational S-matrix.
• Representation and computation of correlators with Carrollian symmetry, where, for example, the spectrum and correlation functions of string models are reorganized and truncated compared to their relativistic counterparts.

6. Applications, Implications, and the Ultra-Local Regime

The physical and mathematical consequences of the flipped Carrollian contraction include:

• Finite or drastically reorganized BRST spectra and novel dualities in string theory.
• Ultra-local quantum dynamics, as in Carrollian Dirac fermions coupled via a Yukawa-type interaction that yields a $\delta$-function–like potential in space and only time-dependent factors in dynamics (Ekiz et al., 8 Feb 2025).
• Reorganization of scattering amplitudes and loop corrections in Carrollian field theories, with explicit Feynman rules adapted to external states at null infinity and amplitudes that depend polynomially on the basic Carrollian two-point function (Liu et al., 6 Feb 2024).
• Generalized Carrollian field theories with arbitrary scaling dimensions, modifications to UV/IR behaviors, and well-defined renormalization group (Callan–Symanzik) flows in the boundary theory.
• Differential equations satisfied by Carrollian amplitudes, translated from celestial (Mellin) bases to Fourier (Carrollian) ones, yielding non-distributional, physically meaningful correlators in three-dimensional Carrollian CFT (Ruzziconi et al., 5 Jul 2024).
• Geometric and algebraic connections to higher-spin algebras and the extension of BMS symmetry, as in the realization that contractions of higher-spin AdS singleton algebras yield Carrollian (flat-space) higher-spin structures (Bekaert et al., 2022).

A plausible implication is that the flipped Carrollian contraction framework unifies approaches across flat space holography, Carrollian field theory, and ultra-local condensed matter systems with fractonic or ultra-local excitations due to analogous symmetry reduction mechanisms.

7. Summary Table: Contrasting Flipped and Standard Carrollian Contractions

Aspect	Standard Carrollian Contraction	Flipped Carrollian Contraction
Scaling of Generators	Time and boosts rescaled in same manner	Time direction reversed/anti-ordered in one sector
Algebraic Output (Quantum)	Carrollian algebras (possibly non-classical)	Sometimes isomorphic to Galilean algebra (quantum)
BRST Cohomology (Strings)	Possibly infinite (depending on vacuum)	Spectrum truncated; exhibits Poincaré duality
Symmetry Algebra	Carroll group, BMS, extended superalgebras	Carroll/BMS, with sectors interchanged or reversed
Physical Interpretation	Ultra-relativistic, null limit	Time/space roles exchanged, ultra-local phenomena

In conclusion, the flipped Carrollian contraction provides a systematic route for producing, analyzing, and exploiting the ultrarelativistic ($c \rightarrow 0$) limit in a wide range of theories, with characteristic features including the exchange or reversal of time and space, emergence of Carrollian/BMS-type symmetry structures, and the realization of ultra-local or dramatically reorganized dynamical spectra. The framework is central to recent advances in flat space holography, Carrollian strings and conformal field theories, field-theoretic models with ultra-local interactions, and the mathematical underpinnings of non-Lorentzian symmetry algebras.