Carroll Gauge Theory

• Carroll Gauge Theory is the systematic gauging of the ultra-relativistic Carroll algebra that yields non-Lorentzian gravity models with distinct electric and magnetic sectors.
• The framework constructs a Cartan geometry with tailored vielbeine and connections, enforcing unique curvature constraints and topological invariants.
• Applications span flat space holography to novel matter couplings, offering practical insights into Carrollian particle dynamics and field theory duals.

Carroll Gauge Theory refers to a suite of geometric, algebraic, and dynamical structures emerging via the gauging of the Carroll algebra—a contraction of the Poincaré algebra in the limit where the speed of light vanishes, resulting in ultra-relativistic symmetries. This framework underpins a distinct class of non-Lorentzian gravity theories, the kinematics and dynamics of Carrollian particles and fields, and provides an organizing principle for flat space holography and field theory duals at null infinity.

1. Algebraic Origin: Carroll Algebra and Its Gauging

The Carroll algebra is obtained from the Poincaré algebra by an “ultra-relativistic” contraction: $P_0 = \omega H$, $M_{a0} = \omega G_a$, $\omega \to \infty$. The generators consist of $H$ (time translations), $P_a$ (spatial translations), $G_a$ (Carroll boosts), and $J_{ab}$ (spatial rotations), with key commutators such as $[J, P] = -2\delta P$ and $[G, P] = -\delta H$ (Bergshoeff et al., 2014). Unlike the Galilei (Bargmann) case, the Carroll algebra does not admit a central extension, implying that its boost and rotation connections cannot always be solved in terms of the vielbeine.

To gauge the Carroll algebra, one introduces a one-form connection valued in the algebra: $$A_\mu = \tau_\mu H + e_\mu^a P_a + \omega_\mu^a G_a + \omega_\mu^{ab} J_{ab}$$ Here, $\tau_\mu$ is the temporal vielbein, $e_\mu^a$ are spatial vielbeins, $\omega_\mu^a$ and $\omega_\mu^{ab}$ are the boost and rotation connections. Their local transformation rules together with suitable curvature constraints are chosen such that local translations correspond to spacetime diffeomorphisms (Bergshoeff et al., 2014, Hartong, 2015).

2. Carrollian Geometries and Cartan Gauge Theory

Gauging procedures define a Cartan geometry modeled on the Klein pair $(\mathfrak{g}, \mathfrak{h})$, where $\mathfrak{g}$ is a Carroll-type symmetry algebra and $\mathfrak{h}$ a stabilizer. The Cartan connection $A$ uniquely encodes both the soldering forms and connection fields: $$A = \frac{1}{2} \omega^{ab} J_{ab} + \psi^a G_a + \theta^a P_a + \xi H$$ Its curvature $F$ decomposes analogously. All gauge-invariant four-forms constructed from $A$ and $F$ define Lagrangians for Carrollian gravity; these include Carrollian analogues of Hilbert–Palatini, Holst, cosmological constant terms, as well as topological invariants such as analogues of Gauss–Bonnet and Pontryagin terms (Figueroa-O'Farrill et al., 2022). Notably, the construction yields two extra, intrinsically Carrollian, topological invariants without Lorentzian counterparts.

This procedure applied to the flat, de Sitter–Carroll, and anti-de Sitter–Carroll algebras yields lagrangians of uniform structure. In the “magnetic” Carrollian sector, the torsion constraint forces the Carrollian extrinsic curvature $K_{ij}$ to vanish. In contrast, the “electric” sector cannot be constructed by this first-order approach: the required lagrangians always eliminate dynamical time evolution of the spatial metric, enforcing the $K_{ij}=0$ constraint off-shell (Figueroa-O'Farrill et al., 2022, Campoleoni et al., 2022).

3. Dynamical Sectors: Electric and Magnetic Carroll Gravity

Carroll gauge theory naturally splits into “electric” and “magnetic” sectors, distinguished by their dynamical treatment of time and space derivatives (Henneaux et al., 2021, Campoleoni et al., 2022):

• In the magnetic sector (first-order formalism, Hamiltonian $c\to 0$ limit), only spatial curvatures survive in the constraint algebra. The momentum (boost) connection remains undetermined, and the extrinsic curvature vanishes: $K_{ij}=0$. The action reads

$$Z_\text{mag} = \int dt\,d^Dx\, [\pi^{ij} \dot{g}_{ij} - N \mathcal{J}_\text{mag} - N^i \mathcal{J}_i]$$

with $$\mathcal{J}_\text{mag} = - (16\pi G_\text{mag})^{-1} \sqrt{g}(R - 2\Lambda)$$ (Campoleoni et al., 2022).

• In the electric sector (second-order), the loss of boost connection dynamics allows extrinsic curvature $K_{ij}$ to fluctuate, leading to nontrivial time evolution. However, as shown in the gauging approach, the electric Carrollian theory cannot be realized in the first-order Cartan formalism due to intrinsic algebraic obstructions (Figueroa-O'Farrill et al., 2022).

Table: Key Features of Carroll Gravity Sectors

Sector	Dynamical Variables	Characteristic Equation	Geometric Constraint
Magnetic	$g_{ij}$, $\pi^{ij}$	$K_{ij} = 0$	Arbitrary boost conn.
Electric	$g_{ij}$	$K_{ij}$ fluctuates; second order	Boost conn. eliminated

4. Matter Couplings and Conformal Program

Matter couplings in Carroll gauge theory can be systematically constructed via a conformal approach using compensators and local isotropic dilatations (Bergshoeff et al., 23 Dec 2024). The relevant conformal Carroll algebra includes transformations for scale (dilatation $D$) and special conformal ($K$) symmetries, in addition to the base Carroll symmetry.

• Electric sector: Coupling a massless Carroll scalar via a covariantized time derivative and subsequent gauge-fixing yields the electric Carroll gravity action, with the Lagrangian proportional to the square of the Carrollian torsion component.
• Magnetic sector: The Lagrangian involves both a massless scalar and an independent Lagrange multiplier, the latter being absorbed by the boost spin-connection after full conformal gauge-fixing. The magnetic Carroll gravity Lagrangian then organizes itself as a sum of Carrollian rotation and boost curvatures.

A haLLMark is the presence of nontrivial intrinsic torsion—certain components of torsion cannot be eliminated, and these influence transformation rules and dependent connection fields. In contrast to the Galilei case, where such compensator-based coupling does not hold in general, the Carroll framework admits a unique absorption feature in the magnetic case.

5. Particle and Field Dynamics

The coupling of Carroll gauge fields to matter leads to nontrivial dynamics for both particles and fields:

• Carroll Particles: In flat background, single massive Carroll particles remain “ultralocal” (do not move). Upon coupling to nontrivial Carroll gauge backgrounds, the action is covariantized:

$$S = \int d\tau\ [p_\mu \dot{x}^\mu - \frac12 e^{-1}\tau(t, x)\tau(t, x)g^{\mu\nu}p_\mu p_\nu - M]$$

The dynamics now involve geodesic motion in the background Carroll geometry: $\dot{x}^\mu = e\tau^\mu p$, etc. (Bergshoeff et al., 2014).

• Carroll Gauge Fields: Carroll gauge theory generalizes to p-form fields and Yang–Mills systems. In these cases, different “Carrollian sectors” (electric, magnetic, and mixed) arise by scaling field components differently in the contraction limit (1901.10147), and actions can be formulated covariantly by introducing compensator fields.

In 2+1 dimensions, Carrollian gauge groups admit nontrivial central extensions, leading to additional Casimir invariants and facilitating more exotic physical sectors, such as anyonic or noncommutative structures (Marsot, 2021).

6. Holography, Quantum Aspects, and Topological Terms

Carroll gauge theory has direct relevance to flat spacetime holography:

• The conformal Carroll group realized at null infinity is isomorphic to the BMS group, connecting Carroll-invariant field theories to boundary duals of gravitational theories in asymptotically flat space (1901.10147, Campoleoni et al., 2022).
• In 3d gravity, relaxation of the Bondi gauge reveals that the boundary metric can be promoted to a Lorentz or Carroll frame (dyad). The dynamics of the promoted Cartan frame naturally encodes quantum anomalies in Carroll (boost) symmetry, which are cohomologically equivalent to the usual Weyl anomaly. These anomalies remain in the flat limit and predict quantum anomalies in conformal Carrollian field theories (Campoleoni et al., 2022).
• Carroll gauge theory supports topological terms (Carrollian Gauss–Bonnet, Pontryagin, Nieh–Yan, and two uniquely Carrollian invariants) with possible implications for boundary dynamics and novel phases.

7. Applications and Future Directions

Carroll gauge theory forms a robust platform for multiple research directions:

• Flat Space Holography: The infinite-dimensional conformal Carroll group and its enhancement of boundary symmetries suggest Carrollian field theories as natural holographic duals for gravitational dynamics at null infinity.
• Ultra-relativistic Limits and Null Surfaces: The geometry induced on null hypersurfaces of Lorentzian spacetimes is automatically Carrollian. This connects Carroll gauge theory to the asymptotic structure of black holes, BMS symmetry, and soft theorems.
• Integrable QFT and Constraints: Canonical analyses reveal that “magnetic” Carrollian field theories are free of second-class constraints, simplifying their quantization and analysis compared to their Lorentzian parents (Majumdar, 3 Jul 2025).
• Emergent Phases and Condensed Matter Analogs: The freezing of degrees of freedom and the identification of Carroll symmetry in systems such as shallow water hydrodynamics and fracton phases illustrate the broader applicability of Carrollian geometry concepts.

Unresolved directions include the extension to full (super)gravity, supersymmetric Carroll gauge theories, detailed matching to holographic data, and classification of all possible matter couplings.

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In summary, Carroll Gauge Theory is the rigorous construction and analysis of ultra-relativistic gravitational and gauge-theoretic models by gauging the Carroll algebra. It enables the systematic paper of ultra-relativistic limits, novel gravitational dynamics on null geometry, and provides a concrete algebraic and geometric underpinning for both boundary field theories in flat holography and new classes of non-Lorentzian QFTs. Key features include the role of degenerate metrics, the partition into magnetic/electric dynamical sectors, and rich geometric/topological structure inaccessible in Lorentzian frameworks.