Carroll supergravities

Technical Summary: Carroll Supergravities

Problem Statement
The paper addresses the construction of Carrollian limits for $N=1$ supergravity in four spacetime dimensions ($D=4$). While the electric and magnetic Carrollian limits of Einstein gravity have been previously established, the corresponding limits for supergravity theories—specifically those involving the gravitino field and local supersymmetry—had not been explicitly derived. The authors aim to construct these limits to understand how supersymmetry behaves in the Carrollian regime, where the speed of light effectively vanishes, and to determine the resulting constraint algebras and covariant actions.

Methodology
The authors employ the Inönü-Wigner contraction of the Poincaré superalgebra, specifically rescaling the boost generators, energy operator, and supercharges to obtain the Carroll superalgebra. A key methodological choice is to perform this contraction within the Hamiltonian formulation of supergravity rather than the Lagrangian formulation.

The procedure involves the following steps:

1. Hamiltonian Derivation: The authors first derive the Hamiltonian form of $N=1, D=4$ supergravity. They utilize the "time gauge" ($h^{(0)}_m = 0$) to decompose the action into space and time components. This involves redefining variables (momenta and spinor fields) to achieve a canonical symplectic structure where kinetic terms take the standard $p\dot{q}$ form.
2. Contraction Limits: Two distinct limits are performed by rescaling the lapse function $N$ and the fermionic Lagrange multiplier $\chi$ (related to the time-component of the gravitino) while taking the coupling constant $\kappa^2$ to specific limits:
   • Electric Limit: $\kappa^2 \to \infty$ with $N_E = 2\kappa^2 N$ and $\chi_E = 2\kappa^2 \chi$ held fixed.
   • Magnetic Limit: $\kappa^2 \to 0$ with $N_M = (2\kappa^2)^{-1} N$ held fixed.
3. Constraint Analysis: The authors analyze the resulting constraint algebras for both limits, focusing on the commutation relations between the Hamiltonian constraint ($H$), the diffeomorphism constraint ($H_k$), and the supersymmetry generator ($S$).
4. Covariant Reconstruction: For the electric limit, the authors reconstruct a manifestly covariant first-order action and subsequently a second-order action, utilizing Carrollian tetrads and gamma matrices defined in the appendices.

Key Contributions and Results

• Explicit Derivation of Limits: The paper provides the first explicit derivation of the electric and magnetic Carrollian limits for $N=1$ supergravity.
  • In the Electric Limit, the resulting theory is "ultralocal." Spatial gradients in the Hamiltonian and supersymmetry constraints disappear, leaving only algebraic terms and time derivatives. The quartic fermion terms remain, while the kinetic term for the metric is modified.
  • In the Magnetic Limit, spatial gradients are retained in the constraints, but the quartic fermion terms vanish. The extrinsic curvature is forced to vanish on-shell, similar to the magnetic limit of pure gravity.
• Constraint Algebra: A central result is the simplification of the constraint algebra in the Carrollian limit. In both limits, the algebra of the supersymmetry generators $S$ and the Hamiltonian $H$ becomes:
  $$\{S_A(x), S_B(y)\} = -i\delta_{AB} H \delta(x-y)$$
  $$\{H(x), H(y)\} = 0$$
  This indicates that the supersymmetry generators act as the "square roots" of the Hamiltonian, analogous to the standard Poincaré supergravity, but without the spatial diffeomorphism term ($H_k$) appearing on the right-hand side of the supersymmetry anticommutator.
• Carroll Spinors: The authors confirm that in the Carrollian limit, the spinor charges do not transform under Carroll boosts. The spinors are "boost-inert," transforming only under the spatial rotation subgroup $so(3)$. This property is naturally inherited from the contraction of the Poincaré superalgebra.
• Covariant Formulation: The electric limit is successfully rewritten in a manifestly covariant form. The action is expressed using Carrollian tetrads and specific "Carroll gamma matrices" ($\Gamma^\mu, \rho^\mu$). The spinor part of the action involves Lie derivatives along the normal vector and the Carrollian extrinsic curvature, formulated without reference to the time gauge.
• Energy Positivity: The paper notes that the rigid supersymmetry algebra in the limit implies the positivity of energy ($P^0 \geq 0$). However, in the electric limit, the surface integrals for energy and supercharge vanish, rendering the positivity statement trivial ($0 \geq 0$). In the magnetic limit, where spatial gradients persist, the energy is given by non-vanishing surface integrals, allowing for a non-trivial positivity argument.

Significance and Claims
The authors claim that their approach via the Hamiltonian formulation is "direct" and "powerful" for deriving Carrollian limits, as it naturally separates space and time generators, facilitating the contraction process. They emphasize that the gauge invariance and the first-class nature of the constraints are preserved automatically in the limit, requiring no additional checks.

The paper positions these results as a foundational step for exploring supersymmetric Carroll theories. Specifically, the authors suggest that the derived commutation relations could serve as a tool to investigate "Carroll swiftons" (Carroll tachyons with positive energy) and may impact the quantization of Carroll field theories. The methodology is presented as generalizable to extended supergravity models (e.g., $D=10$), though the paper focuses strictly on the $N=1, D=4$ case.

The authors remain modest regarding the magnetic limit, acknowledging that writing it in a manifestly covariant off-shell form is "more intricate" due to the on-shell vanishing of the extrinsic curvature and potential divergences in the supersymmetry transformations. They suggest that a gauging of the super-Carroll algebra might be required to fully resolve these issues in future work.