Mixed Carroll–Galilei Symmetry

• Mixed Carroll–Galilei symmetry is defined as the interplay between the Carroll (ultra-relativistic) and Galilei (non-relativistic) spacetime groups, with distinct geometric and algebraic structures.
• The duality is realized via the Bargmann framework, where exchanging Carroll time and Newtonian time bridges the algebraic and field-theoretical descriptions.
• Practical applications include non-Lorentzian field theories, gravity models, holography, and quantum deformations, underpinning exotic kinematical behavior in modern physics.

Mixed Carroll–Galilei symmetry refers to the deep structural interplay, duality, and occasional unification between two distinct non-Einsteinian spacetime symmetry groups—the Carroll group (arising in the ultra-relativistic limit, $c\to0$) and the Galilei group (arising in the non-relativistic limit, $c\to\infty$)—together with their associated geometric, algebraic, and field-theoretical constructs. This symmetry concept has emerged as a central organizing principle in recent advances in non-Lorentzian physics, string theory, flat and de Sitter holography, quantum group deformations, and the paper of nonrelativistic/ultrarelativistic field theories.

1. Carroll and Galilei Groups: Definitions and Geometric Framework

The Carroll group is defined via a contraction of the Poincaré group in the limit $c\to 0$ (or, equivalently, by a scaling $C\to0$ with $s = C x^0$ as the "Carroll time"), resulting in a degenerate geometric structure where the propagation of free particles in space is "frozen"—that is, the light cone collapses onto the time axis. In contrast, the Galilei (Newton–Cartan) group arises in the $c\to\infty$ limit after appropriate scaling, leading to a geometry of absolute time and Galilei-invariant spatial intervals.

The intrinsic geometric definition of the Carroll manifold uses a $(d+1)$-dimensional manifold equipped with a degenerate metric tensor $g$ of rank $d$, whose kernel is generated by a global vector field $\xi$ (Carroll time direction), and an automorphism group preserving both $g$ and $\xi$. The Carroll and Galilei groups act as automorphism groups of these respective structures, with the Carroll group in flat space given as $$\text{Carr}(d+1) \cong E(d)\ltimes \mathbb{R}^{d+1}$$, where $E(d)$ denotes the $d$-dimensional Euclidean group.

2. Duality Structure and Unified Framework

A deep duality exists between Carrollian and Galilean kinematics, most transparently realized through the Bargmann framework. Here, both Newton–Cartan and Carrollian geometries are seen as distinct "slicings" or projections of a $(d+2)$-dimensional (Bargmann) manifold endowed with a Lorentzian metric and a preferred null direction. Explicitly, the exchange $t \leftrightarrow s$ (where $t$ is Newtonian time and $s$ is Carroll time) interchanges the Galilean and Carrollian descriptions.

Boost transformations further illustrate the duality:

• Galilei: $x' = x + b t$, $t' = t$
• Carroll: $x' = x$, $s' = s - b \cdot x$

In the Bargmann setting, both are realized via different choices of quotienting or slicing the null structure, and both allow for central and non-central extensions governed by the structure of the underlying metric Lie algebra.

3. Mixed Carroll–Galilei Symmetry: Algebraic and Mathematical Structures

Recent work demonstrates that both the Carrollian and Galilean Lie algebras can be derived from a parent Bargmannian Lie algebra, defined as a one-dimensional double extension of a metric Lie algebra $(\mathfrak{g}_0, \langle-,-\rangle_0)$ by a skew-symmetric derivation $D_0$: $$\mathfrak{g} = \mathfrak{g}_0 \oplus \mathbb{R} D \oplus \mathbb{R} Z$$ with

$$[X,Y] = [X,Y]_0 + \langle D_0 X,Y\rangle_0 Z,\quad [D,X]=D_0 X,\quad [Z,-]=0$$

This structure canonically produces both Carrollian algebras as central extensions and Galilean algebras as quotients by the central element $Z$ (Figueroa-O'Farrill, 2022).

Generalizations include Leibnizian Lie algebras, which mediate non-canonical correspondences between Carroll and Galilei algebras—a flexibility essential in various physical applications, e.g., extended symmetries in field theories and gravity sectors.

4. Dynamical Realizations and Field Theories

Carrollian Electromagnetism and Field Theories

Carrollian electromagnetism, as constructed via the Carroll limit of Maxwell's equations, yields distinctive equations where time derivatives are suppressed or "frozen," e.g.,

$$\nabla\times \mathbf{E} + \partial_s \mathbf{B} = 0,\quad \nabla\cdot \mathbf{B} = 0,\quad \nabla\times \mathbf{B} = 0, \quad \nabla\cdot \mathbf{E} = 0$$

and these remain invariant under Carroll boosts $s\to s-b\cdot x$ (Duval et al., 2014).

Analogous spinor and supersymmetric Carroll and Galilei field theories can be systematically constructed by contraction or partition–constrain strategies, leading to sectors dominated by either time or space derivatives. For example, in the Carroll gravity limit, matter actions become dependent solely on time derivatives, while the Galilei gravity limit survives only through spatial derivatives (Bergshoeff et al., 2017, Koutrolikos et al., 2023).

Supersymmetric and Brane Theories

Supersymmetric extensions naturally embed the Carroll–Galilei duality via carefully chosen contraction schemes of the super-Poincaré algebra and via construction of new multiplets and tensor calculus methods, including real scalar multiplets in extended (e.g. $N=2$) Carroll supersymmetry (Zorba et al., 23 Sep 2024). At the brane level, the formal map between Galilei and Carroll algebras is achieved by exchanging longitudinal and transverse indices at the level of generators, providing explicit duality between, e.g., particles and strings in 3D and 4D gravity (Bergshoeff et al., 2020).

5. Quantum and Conformal Extensions

Quantum deformations (Hopf algebraic and noncommutative geometry approaches) display further mixed Carroll–Galilei phenomena. For example, $\kappa$-deformed Galilei and Carrollian spaces of worldlines can be directly constructed as contraction limits from the $\kappa$-Poincaré algebra; crucially, while the underlying $\kappa$-Minkowski spacetime algebra is shared, the noncommutative worldline algebras are dramatically different: the $\kappa$-Galilei worldlines show Euclidean Snyder-type noncommutativity, whereas the $\kappa$-Carroll worldlines are commutative (Ballesteros et al., 2022, Ballesteros et al., 2019).

Conformal extensions of Carroll and Galilei groups, labeled by an integer $k$, yield families of infinite-dimensional algebras such as BMS and NU groups. The duality is further encoded in the condition $\lambda + k\mu = 0$ relating scaling of the degenerate metric to that of its kernel/clock form (Duval et al., 2014). In 1+1 dimensions, conformal Carroll and conformal Galilei groups are isomorphic (after exchanging space and time), while in $d>1$ their structures differ and their dynamical exponents may vary (e.g., $z=2/k$).

6. Physical Applications and Implications

Fluid Dynamics and Holography

In 1D hydrodynamics, an explicit isomorphism of the Galilean and Carrollian algebras underlies a duality between Galilean and Carrollian fluids (Athanasiou et al., 8 Jul 2024). Here, dynamical variables (e.g., mass density, energy, heat currents) and the structure of the hydrodynamic equations themselves are mapped under a duality that exchanges equilibrium and non-equilibrium roles. The duality analysis reveals that Carrollian fluids can serve as boundary theories in flat or de Sitter holography, where the structure at null infinity is governed by Carrollian symmetry (Blair et al., 24 Jun 2025).

Gravity Theories and Chern-Simons Models

Both Carroll and Galilei gravity emerge as distinct but related limits of the Einstein–Hilbert action, leading to field equations that constrain geometry via undetermined spin connection components serving as Lagrange multipliers in a first- or second-order formalism (Bergshoeff et al., 2017). In three-dimensional gravity, the S-expansion method allows the derivation of Carroll, Galilei, and mixed Carroll–Galilei algebras from lower-dimensional Euclidean AdS or Poincaré algebras, facilitating the construction of Chern–Simons gravity models. These can include "exotic" terms and higher-order extensions with central charges, i.e., Post–Carroll–Newtonian algebras (Concha et al., 31 Dec 2024).

Noncommutative Geometry and Quantum Gravity

Snyder-type deformed phase spaces and quantum algebra contractions in both Lorentzian and non-Lorentzian settings reveal that mixed Carroll–Galilei symmetry is encoded at the Planck scale as a remnant mixing of space and time even in nonrelativistic limits (Ballesteros et al., 2019). Quantum deformations of (anti-)de Sitter, Galilei, and Carroll algebras are classified via contracted $r$-matrices; almost all coboundary deformations can be generated in this way (Trześniewski, 2023).

7. Summary Table: Central Features of Mixed Carroll–Galilei Symmetry

Aspect	Carroll Symmetry	Galilei Symmetry	Mixed Feature
Commuting Time/Space	Absolute space, 'frozen' motion	Absolute time, 'instant' evolution	Duality under $t \leftrightarrow s$; Bargmann connection
Boosts	$x' = x$, $s' = s - b\cdot x$	$x' = x + bt$, $t' = t$	In Bargmann, mapped by index exchange or slicing of null directions
Central Charges	Optional (central extensions possible)	Optional (Bargmann mass)	Both from a double extension of the same metric Lie algebra (Figueroa-O'Farrill, 2022)
Field Theory Dynamics	Only time-derivatives survive	Only spatial-derivatives survive	Transition and interpolation via contraction and partition–constrain methods
Quantum Extensions	Carrollian BMS, k-Carroll, conformal	Galilei BMS, k-Galilei, conformal	Dual under coordinate/translation role exchange, worldline/spacetime mapping
Gravity/CS Action	CS theory with Carrollian algebra	CS theory with Galilei algebra	Carroll–Galilei via expansion/contraction from 2D Euclidean AdS algebra

8. Outlook and Broader Significance

Mixed Carroll–Galilei symmetry is not a formal curiosity but a structural property with broad relevance:

• It underpins the geometry of ultra-relativistic/strong-coupling and nonrelativistic/weak-coupling regimes.
• It manifests in dual formulations of gravity, field theories, string/brane effective actions, and in the structure of quantum groups and noncommutative geometry.
• Via dualities and unified constructions, it helps explain the emergence and interplay of different kinematics in physical backgrounds, including null surfaces, holographic correspondences, and condensed matter systems.
• The unified Bargmann–Leibnizian framework and its deformations provide a systematic method for classifying and constructing exotic symmetry algebras enforcing non-Lorentzian kinematics.
• Extensions to conformal, supersymmetric, higher-spin, hydrodynamic, and quantum-deformed settings continue to elucidate the flexibility and naturalness of mixed Carroll–Galilei symmetry in physical law.

Thus, the concept is central in the modern understanding of spacetime symmetry beyond the Einsteinian paradigm (Duval et al., 2014, Figueroa-O'Farrill, 2022, Concha et al., 31 Dec 2024, Koutrolikos et al., 2023, Athanasiou et al., 8 Jul 2024, Blair et al., 24 Jun 2025).