Carroll fermions, expansions and the lightcone

Abstract: We investigate fermions on Carrollian manifolds. We complement previous intrinsic analysis by deriving Carrollian fermion actions from a relativistic Dirac theory via a systematic expansion in the speed of light ($c$). We then study relativistic fermions in light-cone coordinates and their connection to Carrollian fermions in one lower dimension. This follows from the recent observation that the Poincaré algebra, written in lightcone coordinates contains (two) co-dimension one Carroll sub-algebras. Our results establish a clear bridge between intrinsic Carrollian constructions, small $c$-expansion and light-cone dynamics. In the process, we understand why Carrollian fermions in $D$-dimensions have features that relate them to relativistic fermions in both $D$ and $(D+1)$ dimensions.

• The paper establishes a systematic derivation of Carrollian fermions from the relativistic Dirac theory via controlled c-expansion, capturing both leading electric and subleading magnetic sectors.
• It reveals novel algebraic structures, including degenerate Clifford algebras and representation doubling in odd dimensions, which are crucial for flat space holography.
• The analysis connects lightcone reduction with Carrollian dynamics, providing insights into ultra-local propagators and enhanced symmetry structures.

Carroll Fermions, Systematic Expansions, and Lightcone Dynamics

Introduction and Motivation

This work (2604.14301) establishes a systematic bridge between the theory of fermions on Carrollian manifolds, their emergence via controlled limits of the relativistic Dirac theory, and a detailed lightcone analysis. Carrollian symmetries—arising in the $c\to 0$ (ultra-relativistic) limit where the Lorentzian lightcone collapses—have found critical applications in asymptotically flat holography, null surface physics, and effective field theories on black hole horizons. Despite the surge of interest in Carrollian structures and their role in flat spacetime holography, a detailed, systematic construction of fermions (as opposed to scalars or gauge fields) in these geometries has been lacking. This paper addresses that gap through methodological analysis at both the algebraic and dynamical/holographic levels.

Carrollian Symmetries and Clifford Algebra Structure

The Carroll algebra emerges from a contraction of the Poincaré algebra as $c\to 0$. The underlying geometry is encoded by a degenerate metric, with the Carrollian manifold specified by $h_{\mu\nu}$ (a rank $D{-}1$ degenerate 'spatial' metric) and a null vector $\theta^\mu$ spanning the kernel. The degenerate nature dictates a split between temporal and spatial structures, and this non-invertibility percolates to the Clifford algebra:

• The relativistic Clifford algebra, $\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}$, degenerates to

$\{\gamma^\mu,\gamma^\nu\} = 2 h^{\mu\nu}$

for the lower sector and

$\{\gamma_\mu,\gamma_\nu\} = 2\Theta_{\mu\nu}$

for the upper sector, where $h$ and $\Theta$ are orthogonal projectors.

This results in two classes of inequivalent Carrollian fermions: "lower gamma" (built on $c\to 0$0) and "upper gamma" (on $c\to 0$1). Their corresponding Clifford algebras enforce nontrivial matrix nilpotency and essential modifications to spinor representations. Notably, in odd spacetime dimensions, the structure of the Carroll algebra requires a representation that is higher-dimensional than in the relativistic case (e.g., 3D Carrollian fermions require a 4D representation rather than 2D).

Systematic Derivation via $c\to 0$2-Expansion

The authors perform a controlled expansion of the relativistic Dirac action in powers of the speed of light, tracking both leading and subleading terms—a strategy analogous to the post-Newtonian expansion in gravity but here targeting the Carrollian 'ultra-local' regime.

Key points and results include:

• Necessity of Odd Powers: Unlike bosonic cases, fermionic $c\to 0$3-expansions must include odd powers due to the structure of the Clifford algebra. This subtlety is tied to the expansion of gamma matrices enforced by the degenerate metric.
• Representation Doubling in Odd Dimensions: The degenerate Clifford algebra for Carrollian fermions in odd dimensions does not admit an irreducible representation of the same dimension as its relativistic counterpart, leading to a dimension-doubled minimal representation and new algebraic features.
• Electric and Magnetic Sectors: The leading term in the expansion—manifestly Carroll invariant—corresponds to the "electric" Carroll fermion ("lower gamma"/inhomogeneous representation), while the sub-leading "magnetic" theory corresponds to the "upper gamma" sector. Remarkably, both leading and next-to-leading order terms in the expansion remain Carroll invariant, in contrast to the bosonic case where only the leading term enjoys invariance.
• Gamma-less Carroll Fermions: The work also introduces and analyzes a Carrollian fermion action without any gamma matrices, showing that it behaves as a "Grassmann-valued scalar" in the purely Carrollian regime.

Lightcone Analysis and Connection to Intrinsic Carrollian Structures

A significant portion of the paper is devoted to illuminating the relationship between lightcone formulations of the Poincaré algebra and Carrollian subalgebras:

• Carroll Subalgebras from Lightcone Decomposition: When the Poincaré algebra is expressed in lightcone coordinates, it encapsulates two co-dimension one Carrollian subalgebras (associated with each null direction). This enforces a physical and geometric isomorphism: Carrollian fermions in $c\to 0$4 dimensions are closely related to relativistic fermions in $c\to 0$5 dimensions via lightcone reduction.
• Clifford Structure in Null Coordinates: The relativistic gamma matrices, expressed in the lightcone basis, decompose into degenerate sub-Clifford algebras for each null direction—precisely matching the lower/upper Carrollian Clifford algebra structures.
• Spinor Decomposition and Dynamical Content: In lightcone quantization, only half the spinor components remain dynamical; this echoes the structure of Carrollian fermions derived via $c\to 0$6-expansion, providing a physical rationale for the unique features observed in odd-dimensional Carrollian theories.
• Null Contraction of Dirac Theory: Explicit contraction along a null direction ($c\to 0$7, $c\to 0$8) yields decoupled Weyl spinors whose equations are ultra-local in the Carrollian sense. The propagators exhibit strict localization in transverse directions, and component analysis clarifies the chiral nature of surviving degrees of freedom.

Carrollian Field Theories: Symmetries and Quantum Aspects

The constructed Carrollian fermionic theories admit rich symmetry structures:

• Full finite and infinite-dimensional Carroll and BMS/BMS-like symmetry enhancements (including conformal and supertranslation/superrotation extensions).
• Subtleties in discrete symmetries (parity, time-reversal, charge conjugation) are addressed, including the matching of Majorana conditions and mapping between two Carrollian theories related by null direction exchange.
• Carrollian quantum field theory remains subtle due to degenerate metrics and 'ultra-locality', but the lightcone formulation offers a direct handle for the construction of propagators and, as suggested, the formulation of perturbation theory.
• The work directly constructs the Carrollian fermion propagator, showing its singular "ultra-local" structure.

Practical and Theoretical Implications

This analysis underlines several crucial statements:

• Unified Perspective: The formalism validates the viewpoint that Carrollian fermions are not just an abstract algebraic oddity but inherit concrete physical structure as lightcone-reduced relativistic fermions, and are systematically derivable via analytic expansions.
• Flatspace Holography and Null Geometries: The connection to BMS symmetries, conformal Carrollian structures, and the existence of intrinsic Carrollian fermions all play a central role in current approaches to flat space holography. This work supplies a systematic method for incorporating spinor fields in this context, directly relevant for extensions involving supersymmetry, black hole horizon physics, and non-Lorentzian hydrodynamics.
• Representation Doubling and Null Dynamics: The minimal dimension doubling in odd Carrollian dimensions has clear implications for attempts to formulate SUSY or higher-spin Carrollian field theories and suggests care in constructing physical states in those regimes.
• Quantization and Interactions: The lightcone picture provides a template for quantizing Carrollian theories and suggests that established lightfront methods may be adaptable, countering skepticism about the viability of Carrollian QFT. The authors also highlight future directions with regards to gauge couplings, quantum corrections, and holographic correlators.

Conclusion

This paper provides a definitive, systematic treatment of Carrollian fermions, reconciling intrinsic, algebraic, and lightcone approaches. By setting up a controlled expansion from relativistic theories and demonstrating the equivalence with lightcone null reductions, it supplies a detailed blueprint for the consistent formulation of Carrollian spinor field theories and elucidates their representation-theoretic and dynamical properties. These results are foundational for further studies of Carrollian field theories, especially in the context of flat space holography, quantum aspects of non-Lorentzian systems, and the interplay between null geometry and field-theoretic dynamics.