Carroll Spinors: Geometry & Applications

• Carroll spinors are spinor fields defined within Carrollian geometry characterized by a degenerate metric and ultralocal dynamics.
• The modified Clifford algebra yields distinct electric and magnetic sectors, fundamentally altering symmetry structures and spinor representations.
• Applications span ultra-relativistic limits, flat space holography, tensionless string theory, and condensed matter systems with flat band dynamics.

Carroll spinors refer to spinor fields defined within the Carrollian framework, in which the Lorentzian spacetime structure (characterized by the Minkowski metric) degenerates to a Carroll metric with vanishing temporal signature. This degeneration fundamentally alters the Clifford algebra, the transformation properties of spinors, and the symmetry structure under which they are classified. Carroll spinors have become central in studies of ultra-relativistic limits, flat space holography, tensionless string theory, near-horizon dynamics, and condensed matter systems exhibiting ultralocal band structures.

1. Carroll Geometry and Symmetry

Carroll spacetime is defined by a degenerate metric $h_{\mu\nu}$ and a Carroll vector $v^{\nu}$ in its kernel, such that $h_{\mu\nu}\,v^{\nu}=0$. In the $c\to 0$ limit of Minkowski spacetime, the metric reduces to purely spatial form:

$$\lim_{c\to 0}(-c^2 t^2 + \delta_{ij}x^ix^j) = h = \delta_{ij}x^ix^j.$$

The Carroll group acts as the isometry group of this degenerate geometry. Conformal extensions are defined by vector fields $\xi$ satisfying:

$$\mathcal{L}_\xi h_{\mu\nu} = -2\lambda\, h_{\mu\nu},\qquad \mathcal{L}_\xi v^{\mu} = \lambda\, v^{\mu},$$

which connect to the BMS symmetry algebra. This “ultralocal” geometry leads to collapse of the lightcone structure, a haLLMark of Carrollian physics.

2. Clifford Algebra Structure and Carroll Spinors

The Clifford algebra in the Carroll framework is governed by the degenerate metric and yields nontrivial representations. In $1+1$ dimensions:

$$\{\gamma_a,\gamma_b\} = 2\,h_{ab}\,\mathbb{1}, \qquad h_{ab} = \mathrm{diag}(0,1),$$

and for raised indices,

$$\{\gamma^a,\gamma^b\} = 2\,V^{ab}\,\mathbb{1}, \qquad V^{ab} = \mathrm{diag}(-1,0).$$

This structure gives rise to nilpotent gamma matrices and unique inhomogeneous representations, for instance,

$$\gamma_0 = \begin{pmatrix}0 & a \ 0 & 0\end{pmatrix},\quad \gamma_1 = \mathrm{diag}(1,-1),\quad \gamma^0 = i\,\gamma_1,\quad \gamma^1 = -i\,\gamma_0^T.$$

Such modifications alter the representation theory: Carroll spinors cannot admit a standard chiral decomposition or usual mass terms, reflecting the underlying geometry's degeneracy.

3. Electric and Magnetic Carroll Spinors

There exist two principal Carrollian spinor sectors:

• Magnetic Carroll spinors: These retain spatial derivatives and arise from inserting upper-index gamma matrices into the Dirac-like action,

$$S = \int d^2x\, \bar{\Psi}(i\,\gamma^a \partial_a - m)\Psi.$$

In components ($\Psi = (\psi_0,\psi_1)^T$),

$$S_\mathrm{mag} = i\int d^2x\, [\psi_0^* \partial_t\psi_1 + \psi_1^*\partial_t\psi_0 - \psi_0^*\partial_x\psi_0 - m(\psi_0^*\psi_1 - \psi_1^*\psi_0)].$$

One field acts as a Lagrange multiplier, enforcing $\partial_t \psi_0 = 0$.

• Electric Carroll spinors: These involve time derivatives alone, using lower-index gamma matrices,

$$S_\mathrm{el} = \int d^2x\, \bar{\Psi}\,\gamma_0\,(-\partial_0 - i m)\Psi,$$

with the action reducing to

$$S_\mathrm{el} = \int d^2x\, (i\,\psi_1^*\partial_t\psi_1 - m\psi_1^*\psi_1).$$

The existence of two limits reflects a dichotomy akin to electric and magnetic Galilean limits in non-relativistic dynamics and arises from distinct rescalings on spinor components under $c\to 0$.

4. Discrete Symmetries: Charge Conjugation, Parity, and Chirality

Carroll spinors admit modified discrete symmetry operators:

• Charge conjugation: Defined as $$\mathcal{C}^{-1} \gamma^a \mathcal{C} = -(\gamma^a)^T$$, for a suitable choice (e.g., $\mathcal{C} = -\Lambda$).
• Parity: Spinors transform as $$\Psi(t, x) \mapsto \Psi^P = \mathcal{P}\,\Psi(t,-x)$$, with $\mathcal{P} = i\gamma^0$.
• Chirality: There is no operator $\gamma^*$ that squares to unity and anticommutes with all Clifford algebra generators, hence no true chiral decomposition is available. Bilinears such as $\bar{\Psi}\Psi$ become pseudoscalars under parity, and $\bar{\Psi}\gamma^\mu\Psi$ pseudovectors.

5. Carroll–Ising Model and Carroll Conformal Field Theory

As an application, a massless magnetic Carroll fermion realizes the Carroll–Ising model at criticality:

$$S_\mathrm{CI} = \int d^2x\, (\psi_0 \partial_t\psi_1 + \psi_1 \partial_t\psi_0 - \psi_0 \partial_x\psi_0),$$

with equations of motion $\partial_t\psi_0 = 0$, $\partial_t\psi_1 = \partial_x\psi_0$. The stress tensor solves Carroll Ward identities, and its mode expansion exhibits a BMS$_3$ symmetry algebra, a key result for flat space holography.

6. Carroll ELKO Spinors

The construction of Carroll ELKO spinors (eigenspinoren des Ladungskonjugationsoperators), inspired by Lorentzian ELKO introduced by Ahluwalia and Grumiller, proceeds via eigenspinor equations in the Carroll framework using a charge conjugation operator $C = \mathcal{C} \Lambda^T K$ ($K$: complex conjugation). In $1+1$ dimensions, the eigenstates are Majorana. In four dimensions, a “lower” Carroll–Clifford algebra is explicitly realized, splitting the four-component Carroll spinor as $\Psi = (\phi_0,\phi_1)^T$, and ELKO candidates are constructed as

$$C\,\lambda^\pm = \pm \lambda^\pm,\qquad C\,\rho^\pm = \pm \rho^\pm.$$

Notably, Carroll ELKO spinors are invariant only under a subgroup of the full Carroll group—a contraction of the Lorentzian SIM(2) invariance, admitting only restricted Carroll boosts and rotations.

7. Quantum and Supersymmetric Extensions

The quantum Carroll equation, derived as a group-theoretic quantization of the Carroll algebra,

$-i\hbar\,\partial_s \Psi(x,s) = m\Psi(x,s),$

admits ultralocal solutions $\Psi(x,s) = e^{ims/\hbar}\phi(x)$, where only the phase evolves under Carrollian “time,” reflecting ultralocality. Magnetic Carroll fermion representations are reducible yet indecomposable under Carroll group transformations, forming Jordan block–like structures. These features extend into supersymmetric Carrollian field theories and, via brane formalism, to superstrings in the tensionless (Carrollian) regime.

8. Applications and Outlook

Carroll spinors materialize in condensed matter (e.g., flat-band systems like magic angle bilayer graphene), gravitational physics (near-horizon soft hair physics, flat holography), and tensionless string theory. Their ultralocality (flat energy bands in momentum space) and constrained dynamics (e.g., fields becoming Lagrange multipliers) are essential to the theoretical description of ultra-relativistic regimes and emergent phenomena in flat space limits.

9. Summary and Future Directions

Carroll spinors are constructed by systematic Clifford-algebra degenerations, with explicit representations and field actions that differ fundamentally from Lorentzian analogues. Both electric and magnetic sectors exist, the latter showing indecomposable representation structure, and many standard discrete symmetry features (e.g., chirality) are lost or modified. The novel Carroll ELKO construction demonstrates that pseudo-Majorana eigenspinors of charge conjugation operators have invariance only under restricted Carroll group subalgebras. The identification and paper of Carroll spinors has deep implications for ultrarelativistic physics, flat space holography, degeneracy phenomena in condensed matter, and new symmetry principles in high-energy and gravitational theory (Grumiller et al., 23 Sep 2025). Future investigations are expected to address central extension effects in contracted symmetry algebras, potential applications in quantized Carroll field theories, and links to supersymmetric Carrollian models and corresponding geometric structures.