Super-Carrollian Manifolds in Supersymmetric Geometry

• Super-Carrollian manifolds are non-Lorentzian supersymmetric structures characterized by a degenerate even metric and a non-singular odd vector field that defines the null direction.
• Researchers examine these structures via intrinsic definitions and flat superspace contractions, unveiling treatments of compatible affine connections, inevitable torsion, and symmetry constraints.
• These frameworks offer fresh insights into supersymmetry, Carrollian geometry on null hypersurfaces, and potential extensions through graded Lie algebroid approaches.

Searching arXiv for recent and foundational papers on super-Carrollian geometry and closely related Carrollian structures. Search query: "super-Carrollian manifold Carrollian superspace super-Carroll arXiv" super-Carrollian manifolds are non-Lorentzian supergeometric structures in which metric degeneracy is governed by supersymmetry. In the most explicit intrinsic formulation presently available, a super-Carrollian manifold is a quadruple $(M,g,Q,P)$ where $M$ is a supermanifold of dimension $n|1$, $g$ is a degenerate even metric, $Q$ is a non-singular odd vector field, $P$ is the even vector field determined by $[Q,Q]=2P$, and the kernel of $g$ is exactly $\mathrm{Span}\{Q\}$ (Bruce, 19 Aug 2025). A complementary flat-space formulation constructs $4D,\mathcal N=1$ super-Carrollian superspace by contracting Minkowski superspace so that the odd generators close only on time translation, providing an explicit superspace model of the same ultra-relativistic idea (Koutrolikos et al., 2023).

1. Intrinsic definition and conceptual core

The intrinsic supermanifold definition replaces the classical Carrollian kernel generator, which is even, by a non-singular odd vector field. Thus the radical of the degenerate metric is a rank-$M$0 distribution generated by $M$1, and the supersymmetry relation

$M$2

shows that the odd null direction is non-integrable: its square is an even translation rather than zero. By Shander’s local classification, non-singular odd vector fields are locally either homological,

$M$3

or supersymmetric,

$M$4

and super-Carrollian geometry chooses the second case. The result is a geometry in which the distinguished null direction is itself a supersymmetry generator rather than merely a background vector field (Bruce, 19 Aug 2025).

A related but distinct flat construction starts from the $M$5 super-Poincaré algebra and performs a Carroll contraction for which

$M$6

Here the odd bracket image is the one-dimensional time-translation direction only. This yields a flat super-Carrollian superspace model rather than an intrinsic curved supermanifold definition, but the structural message is similar: the odd sector determines the distinguished Carrollian time direction (Koutrolikos et al., 2023).

The available constructions therefore agree on one central point: super-Carrollian geometry is not merely Carrollian geometry with extra odd coordinates. The odd distribution is constitutive of the null structure itself. In the intrinsic approach the kernel of the metric is generated by an odd field, while in the contracted superspace approach the odd brackets collapse onto the Carrollian time flow.

2. Local normal forms and flat superspace models

On an intrinsic super-Carrollian manifold, adapted Shander coordinates $M$7 can be chosen with

$M$8

and the associated supersymmetric covariant derivative

$M$9

In these coordinates, the most general super-Carrollian metric takes the form

$n|1$0

while the reduced metric is

$n|1$1

Except possibly in reduced dimension $n|1$2, this reduced metric is pseudo-Riemannian. The paper also gives explicit examples on $n|1$3, $n|1$4, and a flat homogeneous model on $n|1$5 whose reduced metric is Lorentzian and flat for $n|1$6 (Bruce, 19 Aug 2025).

The contracted superspace model uses coordinates

$n|1$7

with Carroll boosts acting by

$n|1$8

and supercharges realized as

$n|1$9

The corresponding covariant derivatives are

$g$0

with

$g$1

This is the clearest flat superspace realization of a super-Carrollian structure obtained directly by contraction (Koutrolikos et al., 2023).

A superconformal variant appears in Carrollian superspace with coordinates $g$2 and odd generators

$g$3

satisfying

$g$4

This construction supplies a flat super-Carrollian conformal superspace and its finite and infinite symmetry algebras, but not yet an intrinsic curved differential geometry (Bagchi et al., 2022).

3. Connections, torsion, and compatible geometry

The intrinsic theory distinguishes three notions: supersymmetry compatibility,

$g$5

metric compatibility in the graded sense, and full compatibility, meaning both simultaneously. Compatible affine connections always exist. Starting from an arbitrary affine connection $g$6, one chooses a dual one-form $g$7 with $g$8 and defines

$g$9

to make $Q$0 parallel. A further even $Q$1-tensor can then be added to restore metric compatibility, and degeneracy makes the resulting linear system underdetermined because kernel-valued components do not enter the metric pairing. This yields existence, but not uniqueness, of compatible affine connections (Bruce, 19 Aug 2025).

The same paper proves that torsion is unavoidable. If $Q$2, then

$Q$3

Even if one only assumes metric compatibility, one still obtains

$Q$4

for some function $Q$5, which cannot vanish because $Q$6 is even and $Q$7 is odd. Thus there is no super-Carrollian analogue of a torsion-free Levi-Civita connection. Another consequence is that $Q$8 cannot be Killing: $Q$9 since

$P$0

The supersymmetric non-integrability of the null distribution is therefore directly visible in both torsion and symmetry (Bruce, 19 Aug 2025).

Bosonic Carrollian geometry supplies useful comparison points. Special Carrollian manifolds are tuples $P$1 with

$P$2

while potential Carroll structures replace the parallel splitting form by a potential condition,

$P$3

In both cases the torsion is constrained to be minimal, and once the relevant 1-form is chosen the compatible connection is uniquely determined (Blitz et al., 27 Jan 2026). This suggests that a curved super-Carrollian geometry may eventually require not only a degenerate supermetric and a null generator, but also a super-analogue of a splitting or potential 1-form.

A second structurally important bosonic result is the Lie-algebroid reformulation of singular Carrollian geometry. There one places the degenerate metric and null line on a Lie algebroid $P$4, with null line bundle $P$5, and defines the Carroll distribution by the anchor image

$P$6

The singularity is then encoded in the anchor, not in the kernel of the metric. Compatible Carrollian connections always exist in this framework as well (Bruce, 4 Oct 2025). This suggests an algebroid route for future super-Carrollian generalizations with singular null data.

4. Superfields, multiplets, and field-theoretic realizations

The flat contracted superspace of $P$7 super-Carroll theory admits a chirality-like constraint. A C-chiral superfield is defined by

$P$8

with component expansion

$P$9

This constraint is preserved by Carroll boosts,

$[Q,Q]=2P$0

so it defines an irreducible flat super-Carroll multiplet. The associated superspace Lagrangian

$[Q,Q]=2P$1

produces the electric Carroll Wess–Zumino model

$[Q,Q]=2P$2

In Majorana notation,

$[Q,Q]=2P$3

and the off-shell algebra closes as

$[Q,Q]=2P$4

(Koutrolikos et al., 2023).

The same framework also supports a magnetic Carroll construction, but only after enlarging the superfield system with compensators. The obstruction is that G-chirality is not preserved by Carroll boosts: $[Q,Q]=2P$5 As a result, magnetic Carroll supersymmetry is realized by a compensator system rather than by a naive chiral superfield alone. The literature therefore distinguishes two non-Lorentzian supersymmetric sectors: electric Carroll naturally fits C-supersymmetry, while magnetic Carroll requires a more elaborate G-supersymmetric construction (Koutrolikos et al., 2023).

At the superconformal level, the finite $[Q,Q]=2P$6 Carrollian superconformal algebra in $[Q,Q]=2P$7 contains bosonic generators

$[Q,Q]=2P$8

and fermionic generators

$[Q,Q]=2P$9

with key anticommutators

$g$0

An infinite-dimensional lift replaces time translations, boosts, and temporal special conformal transformations by supertranslation-like operators $g$1, and the fermionic generators by $g$2 satisfying

$g$3

This provides the flat super-Carrollian conformal and super-BMS algebraic backbone, though not a curved supergeometry (Bagchi et al., 2022).

5. Relation to ordinary Carrollian geometry and null hypersurfaces

Bosonic Carrollian geometry is now understood as the intrinsic geometry of null hypersurfaces, and this viewpoint supplies much of the structural background for super-Carrollian questions. A Carrollian structure is a triple

$g$4

with $g$5 a symmetric degenerate 2-tensor of corank $g$6 and $g$7 a nowhere-vanishing vector field spanning its kernel. To perform tensor calculus, one introduces a 1-form $g$8 with

$g$9

forming a ruled Carrollian structure $\mathrm{Span}\{Q\}$0, together with the projector

$\mathrm{Span}\{Q\}$1

The preferred torsionless intrinsic connection is not metric-compatible in the ordinary sense; rather, it satisfies

$\mathrm{Span}\{Q\}$2

This is the modern intrinsic bosonic connection calculus that any curved super-Carrollian geometry would have to supersymmetrize (Ciambelli et al., 24 Oct 2025).

The null-hypersurface picture is particularly sharp in gravitational-wave embeddings. Dodgson waves admit a canonical lightlike foliation by pseudo-invariant torsionfree Carrollian manifolds, and conversely any pseudo-invariant torsionfree Carrollian manifold can be embedded inside a Dodgson wave (Morand, 2018). In a complementary approach, shear-free null hypersurfaces in four-dimensional Einstein spacetimes are analyzed by Cartan and Newman–Penrose methods, yielding explicit coframes, structure group, and an induced Carrollian structure with a unique pair of Ehresmann and affine connections (Blitz et al., 2024). These are bosonic results, but they establish a recurring pattern: degenerate metric data alone do not determine the geometry; one also needs a preferred null generator, a horizontal splitting, and connection data adapted to that splitting.

A bundle-theoretic reformulation develops the same theme upstairs on a principal $\mathrm{Span}\{Q\}$3-bundle $\mathrm{Span}\{Q\}$4 with degenerate metric $\mathrm{Span}\{Q\}$5 satisfying

$\mathrm{Span}\{Q\}$6

where the vertical bundle is generated by the Euler vector field $\mathrm{Span}\{Q\}$7. Given a principal connection $\mathrm{Span}\{Q\}$8, the nondegenerate metric

$\mathrm{Span}\{Q\}$9

produces a canonical torsionless affine connection on the total space, generally not compatible with the original degenerate metric (Bruce, 27 May 2025). A related construction uses

$4D,\mathcal N=1$0

to build a Lorentzian metric on the total space and thereby define Hodge-theoretic operators that are obstructed on the degenerate base (Bruce, 29 Jul 2025). A plausible implication is that super-Carrollian Hodge theory, if developed, may likewise require an enlarged bundle or superbundle with nondegenerate pairing data.

6. Variants, limitations, and open directions

The present literature contains distinct super-Carrollian constructions rather than a single unified curved theory. The intrinsic supermanifold model $4D,\mathcal N=1$1 is an $4D,\mathcal N=1$2 geometry with odd radical and inevitable torsion. The contracted superspace model is a flat $4D,\mathcal N=1$3 construction in which odd brackets close on time translation only. The superconformal model furnishes flat Carrollian superspace and its finite and infinite symmetry algebras, but no intrinsic curved super-Carrollian manifold. These approaches are clearly related, but they are not yet parts of one formalism (Bruce, 19 Aug 2025); (Koutrolikos et al., 2023); (Bagchi et al., 2022).

Their limitations are explicit. The intrinsic theory is restricted to one odd direction,

$4D,\mathcal N=1$4

because the kernel is generated by a single odd field. The contracted superspace theory is explicitly flat and does not introduce curved supervielbeins, super-Carroll clock forms, or supergravity. The superconformal construction does not define a Berezinian measure, curved torsion constraints, or an intrinsic super-Carrollian conformal class. Thus the central unresolved problem is the curved theory: a differential-geometric synthesis of odd radical, graded connection, torsion constraints, and null supersurface geometry.

Several bosonic frameworks indicate possible routes forward. The Lie-algebroid formulation separates regular null data from singular anchor images, suggesting a graded algebroid extension for singular super-Carrollian geometries (Bruce, 4 Oct 2025). Potential and special Carrollian structures show how compatible connections may be encoded by additional splitting or potential forms (Blitz et al., 27 Jan 2026). The almost-commutative Lie-Rinehart framework explicitly includes supergeometry as the $4D,\mathcal N=1$5 special case and defines Carrollian $4D,\mathcal N=1$6-Lie-Rinehart pairs with degenerate metric and free cyclic kernel module, providing a rigorous algebraic template for even super-Carrollian structures (Bruce, 22 Oct 2025). This suggests an algebraic route to graded Carrollian geometry, though not yet to the odd-kernel geometry of the intrinsic super-Carrollian manifold.

A final limitation concerns null infinity. Supersymmetric dynamics can already be organized on bosonic Carrollian null infinity: loop amplitudes of supersymmetric gauge theory and gravity have been written in Carrollian position-space variables on null infinity with bosonic coordinates $4D,\mathcal N=1$7, degenerate metric $4D,\mathcal N=1$8, and kernel vector $4D,\mathcal N=1$9. But this remains a bosonic Carrollian base carrying supersymmetric amplitudes, not a genuine super-Carrollian null-infinity superspace (Nenmeli et al., 9 Apr 2026).

The current state of the subject is therefore sharply defined. Super-Carrollian geometry already has an intrinsic local model in which an odd supersymmetry generator spans the radical of an even degenerate metric, and any compatible connection is necessarily torsionful (Bruce, 19 Aug 2025). It also has explicit flat superspace and superconformal realizations whose odd brackets close on the Carrollian time direction and support superfields, multiplets, and super-BMS-type algebras (Koutrolikos et al., 2023); (Bagchi et al., 2022). What remains open is the full curved theory unifying these ingredients into a single notion of super-Carrollian manifold with graded connection calculus, intrinsic splitting data, and null hypersurface interpretation.