Carroll supergravities

Abstract: The electric and magnetic carrollian limits of $N=1$ supergravity in $D=4$ spacetime dimensions are explicitly derived. The approach is general and applies also to extended supergravity models

• The paper derives the Carrollian limits of D=4, N=1 supergravity using Hamiltonian contraction techniques to obtain electric and magnetic regimes with distinct constraint algebras.
• The electric limit suppresses spatial derivatives while the magnetic limit retains them, illustrating the structural differences in gauge dynamics.
• Supersymmetry acts as the square root of the Hamiltonian constraint, ensuring energy positivity and paving the way for extensions to higher supergravity models.

Carroll Supergravity Limits in $D=4$, $N=1$: Hamiltonian Analysis and Implications

Introduction and Motivation

The analysis of non-Lorentzian limits of relativistic field theories, notably in the context of the Carroll group, has acquired significant relevance for applications in holography, asymptotic symmetries, and model building within ultra-relativistic regimes. This paper provides a comprehensive derivation of the electric and magnetic Carrollian limits of $N=1$, $D=4$ supergravity via an explicit Hamiltonian contraction procedure (2607.08329). The methodology facilitates a systematic treatment of the contraction applied not only to the underlying Poincaré algebra but to its supersymmetric extension, preserving local gauge symmetries and offering direct access to the structure of constraints and their algebra.

Hamiltonian Formulation of $N=1$ Supergravity

The canonical structure of $D=4$ supergravity is presented as a prelude for performing the Carrollian contractions. The supergravity action, composed of the Einstein-Hilbert, Rarita-Schwinger, and quartic fermion sectors, is recast into first-order Hamiltonian form. The kinetic term involves both metric variables (triad, spatial metric) and the Rarita-Schwinger field, with canonical momenta for the spatial metric receiving nontrivial contributions from fermion bilinears.

Key features of the Hamiltonian structure are:

• Canonical conjugacy between triad components $h^{(a)}_m$ and $\pi_{(a)}^m$.
• Fermionic self-conjugacy (graded Poisson bracket structure) after suitable field redefinitions.
• First-class constraint system: the Hamiltonian, diffeomorphism, supersymmetry, and local rotation generators each yield gauge transformations encoding the underlying spacetime, local supersymmetry, and $SO(3)$ invariance.

This structural clarity is pivotal, as the contraction modifies temporal and spatial components differently, and the Hamiltonian language allows for a manifestly regular contraction process.

Carrollian Contractions: Electric and Magnetic Limits

Electric Carroll Limit

The electric limit is effected by scaling the lapse function $N$ and conjugate spinor variable $N=1$0 appropriately, taking the limit $N=1$1. In this regime:

• The Hamiltonian and supersymmetry constraints become ultralocal, i.e., spatial derivatives drop out except in the diffeomorphism constraint.
• The quadratic and quartic fermion terms remain, but kinetic and spatial derivative terms are absent from $N=1$2 and $N=1$3.
• The constraint algebra drastically simplifies:

$N=1$4

and $N=1$5.

• The supersymmetry constraint is the square root of the Hamiltonian constraint, mirroring the Poincaré superalgebra modulo the spatial momentum.

Magnetic Carroll Limit

The magnetic limit, in which $N=1$6 is rescaled by $N=1$7 and $N=1$8, preserves spatial derivatives in the constraints:

• $N=1$9 reduces to the spatial Ricci scalar term, while $N=1$0 retains derivative structure in the gravitino.
• Quartic terms in fermions are absent; the resulting dynamics is further restricted by the vanishing extrinsic curvature ($N=1$1 on-shell).
• The constraint algebra matches that of the electric limit, but with distinct geometric and dynamical content.

In both limits, supersymmetry requirements guarantee the first class nature of the constraints, and the Hamiltonian approach ensures the gauge invariance is preserved throughout the contraction.

Covariant Reformulation and Carroll Spinor Structure

For the electric limit, the action is recast in manifestly covariant form using the Carrollian geometry developed in earlier work [Henneaux:1979vn]. The construction employs Carroll tetrads and Carroll gamma matrices, yielding a Carroll-covariant supergravity action in both the bosonic and fermionic sectors.

Carroll spinors—interpreted as inert under Carroll boosts—admit a transformation law under $N=1$2 only, and the covariant fermionic action is formulated accordingly. The covariant derivatives, Lie derivatives, and constraint structure are elucidated in this Carrollian framework.

In the magnetic case, achieving a fully covariant and off-shell formulation poses greater challenges due to the persistence of spatial derivatives and constraints arising from the vanishing extrinsic curvature; the on-shell structure is tractable, but a full off-shell covariant action requires further development, likely via gauging the super-Carroll algebra.

Constraint Algebra and Energy Positivity

A notable structural outcome is that, in both electric and magnetic limits, the supersymmetry generators continue to square to the Hamiltonian constraint. This directly parallels the $N=1$3 algebra of global supersymmetry, and thus the Carrollian limits inherit positivity properties for the energy, crucial for the viability of these limits as physical theories. In the magnetic case, energy positivity is manifest at the level of surface integrals on shell, while in the electric limit the absence of nontrivial spatial dynamics renders the statement vacuous.

Implications and Prospects

The explicit construction of Carrollian supergravities at the Hamiltonian level enables controlled studies of ultra-relativistic limits in supersymmetric gravitational systems. Potential avenues include:

• Exploration of new symmetry-enhanced phases in ultra-relativistic or tensionless limits of string and brane theories, informed by Carrollian structure.
• Investigation of quantization in Carrollian field theories, especially with supersymmetry to regulate or explain novel features in such models [deBoer et al, JHEP 09 (2023), 148].
• Application to holography in flat and non-Lorentzian geometries, leveraging the clear constraint and symmetry structure.
• Construction of Carrollian extensions of higher-dimensional or extended ($N=1$4) supergravity models, as suggested by the generality of the contraction procedure.

Conclusion

This work demonstrates that the electric and magnetic Carrollian limits of $N=1$5, $N=1$6 supergravity can be systematically derived via Hamiltonian contraction. Both limits yield distinct, gauge-invariant Carroll supergravity theories, each preserving a supersymmetric constraint algebra where supersymmetry acts as the square root of the Hamiltonian constraint. The analysis opens a pathway to a broad class of Carrollian supersymmetric field theories and provides tools for systematic investigation of their symmetries, dynamics, and quantization properties. The extension to higher $N=1$7 and $N=1$8, as well as the investigation of holographic and quantum features, constitutes a promising direction for further research.

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Reference: "Carroll supergravities" (2607.08329)