The Carrollian Superplane and Supersymmetry

THE CARR OLLIAN SUPERPLANE AND SUPERSYMMETR Y ANDREW JAMES BR UCE Institute of Mathematics, Polish A c ademy of Scienc es ul. ´ Sniade ckich 8, 00-656 Warszawa, Poland Abstract This note pro vides an in trinsic construction of the Carrollian sup erplane Π S ≃ R 2 | 4 as a sup ermanifold generalisation of the Carrollian plane. Mo ving aw a y from the c → 0 limit of relativistic spinors, we define Carroll spinors as sections of a degenerate Clifford mo dule. W e sho w that the Carrollian sup erplane is a principal R 1 | 2 -bundle. Once clo ck forms and a complementary basic one-form are sp ecified, there is a pair of o dd vector fields that generate nov el N = 2 Carrollian sup ersymmetry transformations, not all of whic h come from an In¨ on ¨ u–Wigner contraction of a Poincar ´ e sup eralgebra. Keyw ords: Sup ermanifolds; Carrollian Geometry; Clifford Algebras: Sup ersymmetry MSC 2020: Primary: 58A50 Se c ondary: 15A66; 53Z05 I dar e say you never even sp oke to Time! Lewis Carroll, Alice’s Adv en tures in W onderland, (1865) 1. Introduction The notion of a Carrollian manifold, largely due to Duv al et al. [10, 11, 12], is a manifold equipp ed with a degenerate metric whose k ernel is spanned by a nowhere v anishing complete v ector field. The earlier foundational works of L ´ evy-Leblond [18], Sen Gupta [20], and Henneaux [14] must also b e highlighted. Null hypersurfaces, such as punctured future or past light-cones in Minko wski spacetime, and the even t horizon of a Sc hw arzschild black hole, are examples of Carrollian manifold s. A large part of the renew ed interest in suc h manifolds is their role in flat space holography: the b oundary theory liv es on future null infinity I + ∼ = R × S d − 2 , which comes with a Carrollian structure, for example [3] and references therein. Carrollian physics has attracted a lot of attention from v arious p ersp ectives, including h ydro dynamics and condensed matter ph ysics–for a review, the reader may consult Bagc hi et al. [2]. Carrollian manifolds are often considered as the non-relativistic limit c → 0 of Lorentzian manifolds; lo osely , the Carrollian limit remov es the temp oral comp onents of a metric, leaving only a degenerate metric with a rank-1 kernel. Ho w ever, viewing Carrollian manifolds as just ‘brok en’ Lorentzian manifolds can obscure the in trinsic geometry . F or example, Carrollian manifolds naturally ha v e a principal R -bundle structure, as highlighted by Ciam b elli et al. [8]. In this note, we construct the Carroll sup erplane Π S ≃ R 2 | 4 , as a sup ermanifold generalisation of the Carroll plane. In particular, the Grassmann o dd co ordinates transform as Carroll spinors, whic h w e carefully define using the degenerate Clifford algebra of the Carroll plane. W e will refer to this degenerate Clifford algebra as a Carroll–Clifford algebra. The approach taken is to construct the geometry in trinsically rather than as a limiting pro cedure of a standard sup erspace. W e show that the Carroll sup erplane is a principal R 1 | 2 -bundle (where the group pro duct is the ab elian pro duct giv en by shifts) equipp ed with a degenerate inv ariant metric whose kernel is spanned by the vertical v ector fields. Via selecting an Ehresmann connection, we construct the clo ck forms, whic h are a triple of inv ariant one-forms asso ciated with the one even and t w o o dd times. Moreo v er, once a basic one-form has b een chosen, a geometric (equiv ariant) form of sup ersymmetry can b e constructed; this is reminiscen t of the geometric formulation of sup ergra vity . In particular, the ‘extra data’ of a connection and a one-form are required and considered to b e part of the intrinsic geometry . Up on the assumption that the clo c k forms E-mail address : andrewjamesbruce@googlemail.com . Date : March 24, 2026. 1 2 THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y are closed, we obtain the exp ected form of Carrollian sup ersymmetry (also referred to as C- sup ersymmetry [17]), sc hematically { Q, Q } = Ψ( x ) ∂ t . Note, how ev er, the ‘structure constants’ are no w, in general, functions of p osition–thus w e hav e a Lie–Rinehart pair rather than a Lie algebra. There is a lot of freedom in the basic one-form, and it may b e chosen constant, whic h is consisten t with an In¨ on ¨ u–Wigner contraction of a P oincar ´ e sup eralgebra. The x -dep endence of the basic one-form/structure functions is at o dds with Poincar ´ e symmetry , and thus not all Carrollian sup ersymmetries hav e a relativistic paren t. In other words, the class of Carrollian sup ersymmetries is larger than just those that arise from the c → 0 limit. W e remark that a similar, but distinct approach to Carrollian spinors w as given by Bagchi et al. [1], Banerjee et al. [4] & Grumiller et al. [13]. In particular, the only metric-like structure on the Carroll plane is the degenerate metric, and so the degenerate Clifford algebra mo dule is generated b y gamma matrices with lo w er indices only . The limit approach to fermions w as carefully discussed by Bergsho eff [6]. Carrollian conformal sup eralgebras were studied by Zheng & Chen [21], and they provided nov el algebras that are not constructed as the c → 0 limit of a Poincar ´ e sup eralgebra. Thus, the results of this note sit comfortably with Zheng & Chen’s constructions, although they work in d = 3 , 4. Concha & Rav era [9] classify kinematical sup eralgebras via semigroup expansions (S-expansions) in their study of non-Loren tzian su- p ergra vity theories. Sup ersymmetric Carrollian theories, using c → 0 limits, were studied by Kasik ci et al. [16] and Koutrolikos & Na jafizadeh [17]. While the field of Carrollian sup ersymmetry is rapidly ev olving, it is fair to say the sub ject is relatively p o orly understo o d. Much like the Lorentzian case, sup ersymmetric Carrollian theories are exp ected to ha ve b etter renormalisation prop erties. F or instance, the b oson- fermion cancellation mechanism ma y lead to well-defined asymptotic c harges whic h otherwise are div ergen t for purely b osonic theories. Within the flat space holograph y programme, Carrollian sup ersymmetry ma y place constraints on the dual theories, restricting the admissible classes of theories on the b oundary . Our use of supermanifolds: W e understand supermanifolds as locally superringed spaces, see for example Carmeli [7], and we will assume a working knowledge of sup ergeometry . Ho w ever, w e will mostly work using co ordinates and not employ deep results from the general theory of sup ermanifolds. Principal bundles in the category of sup ermanifolds are cov ered by Bartocci et al. [5, Chapter VI I]. W e will denote the Garssmann parit y of an ob ject by ‘tilde’, i.e., e O ∈ Z 2 = { 0 , 1 } . 2. The Carroll Plane and Carroll Superplane 2.1. The Canonical Carrollian Geometry of the Plane. The underlying smo oth manifold structure is M ∼ = R 2 , which we equip with standard global co ordinates ( t, x ). The canonical (w eak) Carrollian structure on M is giv en by g = δ x ⊗ δ x and κ = ∂ t . The reader can directly observ e that ker( g ) = Span { κ } . The admissible changes of co ordinates we consider are the extende d Carr ol l tr ansformations t ′ = t − α f ( x ) , x ′ = x − β , (2.1) where α and β are constants with units of time and length, resp ectiv ely . Here f ∈ C ∞ ( R ) is an arbitrary smo oth function. The sp ecific transformations t ′ = t − γ f ( x ) are referred to as sup ertr anslations , and show that the isometry group here is infinite dimensional; the symmetries w e consider are BMS-lik e. Note that T aylor expanding the smo oth function, w e obtain temp oral shifts and Carrollian b o ost as the degree zero and one terms. That is, the group defined by (2.1) includes the standard Carrollian group Ca rr 2 , see L ´ evy-Leblond [18] A direct calculation giv es (2.2) ∂ t ′ = ∂ t , ∂ x ′ = ∂ x +  α ∂ x f ( x )  ∂ t , THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y 3 meaning that κ = ∂ t is in v ariant under the extended Carroll transformations. In short, the Carrollian structure ( g , κ ) is preserved under (2.1). Observ e that we hav e a principal R -bundle prj : M → Σ ∼ = R , where the right action in co ordinates is ( t, x ) ◁ r = ( t + r , x ). The vertical transformation part of the extended Carroll transformations (2.1) w e thus interpret as gauge transformations. Moreo v er, the degenerate metric is righ t-in v arian t, or in more physical language, the metric is stationary . 2.2. The Carroll–Clifford Algebra. F or an ov erview of the theory of Clifford algebras, the reader ma y consult Lounesto [19]. W e will fix a field F := R or C . W e define the Carroll–Clifford algebra C C l ( F ) := C l 1 , 0 , 1 ( F ) as the asso ciative F -algebra with generators  1 , e t , e x  , sub ject to the relations (2.3) { e t , e t } = 0 , { e t , e x } = 0 , { e x , e x } = 2 1 . Th us, a general element of the Carroll–Clifford algebra is of the form A = a 1 + b e t + c e x + d e t e x , with a, b, c and d ∈ F . Note that we ha ve what is referred to in the literature as a degenerate Clifford algebra. A k ey observ ation here is that we hav e a canonical algebra isomorphism ϕ : C C l ( F ) ∼ − → Λ 1 ( F ) ⊗ F C l 1 , 0 , 0 ( F ) , giv en b y the asso ciative algebra map (2.4) ϕ ( 1 ) := 1 ⊗ F 1 , ϕ ( e t ) := θ ⊗ F 1 , ϕ ( e x ) := 1 ⊗ F e x . Neglecting the explicit reference to the isomorphism and tensor pro duct, we will write C C l ( F ) ∋ A = a + b θ + c e x + d θ e x . The Carroll–Clifford algebra comes with a natural Z 2 × Z 2 -grading defined on the generators as (2.5) deg( 1 ) = (0 , 0) , deg( θ ) = (1 , 0) , deg( e x ) = (0 , 1) . Remark. The assignment of the grading is unique; the degrees of θ and e x ma y b e exchanged. It is also imp ortan t to note C C l ( F ) is not a Z 2 × Z 2 -graded comm utativ e algebra. The Carr ol lian b o ost gener ator is defined as (2.6) S tx := 1 4 [ θ , e x ] = 1 2 θ e x (= − S xt ) . Note that due to the nilp otency of θ , ( S tx ) 2 = 0. Directly , we observe that [ S tx , θ ] = 1 2  θ e x θ − θ 2 e x  = − θ 2 e x = 0 , [ S tx , e x ] = 1 2  θ e 2 x − e x θ e x  = θ e 2 x = θ . Comparing the ab ov e with (2.2), w e see that we hav e a cov arian t vector representation of su- p ertranslations, i.e., this “mimics” ho w partial deriv ativ es transform. 2.3. Carroll–Clifford Mo dules. W e will consider the v ector space F 4 as Z 2 -graded and, more refined, Z 2 × Z 2 -graded; F 4 ∋ v =  v 0 v 1  =     v 00 v 11 v 01 v 10     . A Carr ol l–Cliffor d mo dule is defined as a Z 2 × Z 2 -grading preserving action (2.7) ρ : C C l ( F ) − → End( F 4 ) 4 THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y Resp ecting the Z 2 × Z 2 -grading places strict constraints on the form of the action. In particular, emplo ying the standard P auli spin matrices ρ ( 1 ) =  1 0 0 1  , ρ ( θ ) =  0 ± σ + ± σ + 0  , ρ ( e x ) =  0 σ 3 σ 3 0 .  (2.8) A quic k calculation giv es (2.9) ρ ( S tx ) = 1 2  ∓ σ + 0 0 ∓ σ +  . W e then observe that ρ ( S tx ) v =     ∓ v 11 0 ∓ v 10 0     . T o build the Carroll spinor bundle, note that all the c hoices of ρ ( θ ) are similar; this essentially follo ws from the matrices all b eing rank 2 and nilp otent. Different choices here will only result in differen t c hoices of fibre co ordinates on the Carroll spinor bundle that differ by a sign. The Carr ol l spinor bund le on the Carrollian plane, we define as the vector bundle π : S → M ∼ = R 2 , with typical fibre S p ∼ − → R 4 ( p ∈ R 2 ) and adapted co ordinates ( t, x ; v 00 , v 11 , v 01 , v 10 ) together with admissible co ordinate transformation of the form t ′ = t − α f ( x ) x ′ = x − β , (2.10a) v ′ 00 = v 00 + 1 2 α∂ x f ( x ) v 11 , v ′ 11 = v 11 , (2.10b) v ′ 01 = v 01 + 1 2 α∂ x f ( x ) v 10 , v ′ 10 = v 10 . (2.10c) The transformation rule for the fibre co ordinates ma y b e written as v ′ = v + α ∂ x f ( x ) ρ ( S tx ) v , where the exact signs in the transformation rules are defined by the choice of ρ ( S tx ). Note that as the matrix ρ ( S tx ) is nilp otent, we do not need to exp onen tiate the infinitesimal transformation to obtain global ones, as is required in the Lorentzian case. Th us, the transformation rules (2.10b) and (2.10c) are shear transformations rather than rotations. While this realisation is already present in the c → 0 approac h to Carrollian spinors, this fact is not stressed enough. Moreo v er, lo oking at the dimensions, [ α ] = [ T ], we see that (2.11) [ v 00 ] [ v 11 ] = [ v 01 ] [ v 10 ] = [ T ] [ L ] , that is, the ratios of the units must b e ‘slo wness’, i.e, in v erse sp eed. As we do not hav e a fundamen tal sp eed, we cannot construct a meaningful dimensionless ratio. It is imp ortant to note that the admissible coordinate transformations do not preserve the Z 2 × Z 2 -grading, but rather only the Z 2 -grading defined b y total degree. Thus, the Carroll spinor bundle is a Z 2 -graded v ector bundle. As a graded vector bundle, we hav e the decomp osition S ≃ S 0 ⊕ S 1 . Similarly , w e define the complexified Carroll spinor bundle S C b y defining the typical fibres ( S C ) p ∼ − → R 4 ⊗ R C . THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y 5 2.4. The Carrollian sup erplane. Using the Batc helor–Gaw edzki theorem, we define the Carr ol lian sup erplane as Π S . Th us we hav e a sup ermanifold equipp ed with co ordinates ( t, x, ζ i , η j ), with i, j ∈ { 1 , 2 } , together with the admissible co ordinate transformations t ′ = t − α f ( x ) , x ′ = x − β , (2.12) ζ i ′ = ζ i + 1 2 α∂ x f ( x ) η i , η j ′ = η j . W e assign units to the Grassmann o dd co ordinates [ ζ i ] = [ T ] and [ η j ] = [ L ]. Observe that w e hav e a principal R 1 | 2 -bundle structure, where the group structure on R 1 | 2 , given using the functor of p oints, is ( r 1 , ϵ i 1 ) · ( r 2 , ϵ i 2 ) := ( r 1 + r 2 , ϵ i 1 + ϵ i 2 ). That is, w e consider the strictly ab elian group structure of translations. The principal action can b e written, using the functor of p oin ts, as (ignoring the base co ordinates) (2.13) ( t, ζ i ) ◁ ( r , ϵ i ) := ( t + r , ζ i + ϵ i ) , Where w e assign units [ r ] = [ ϵ i ] = [ T ]. The reader can quic kly v erify that the coordinate c hanges are compatible with the action. Moreo v er, as the co ordinate changes are essentially shifts, the co cycle conditions are clearly satisfied. W e then observ e that the (canonical) right inv ariant degenerate metric on R 2 | 4 is of the form (2.14) g := δ x ⊗ δ x ± 2 δ η 1 ⊗ δ η 2 , where we ha ve employ ed the Z 2 -graded tensor pro duct. W e then observe that the kernel is of rank 1 | 2 and explicitly giv en by k er( g ) = Span  ∂ t , ∂ ζ 1 , ∂ ζ 2  . W e then observe that the structure underlying the Carroll sup erplane is the Carrollian plane. 2.5. Connections and Clo ck F orms. Directly , w e observe that under the changes of co ordinates (2.12), the partial deriv atives transform as ∂ t ′ = ∂ t , ∂ x ′ = ∂ x + α∂ x f ( x ) ∂ t − 1 2 α∂ 2 x f ( x ) η i ∂ ζ i , (2.15a) ∂ ζ i ′ = ∂ ζ i , ∂ η j ′ = ∂ η j − 1 2 α∂ x f ( x ) ∂ ζ j . (2.15b) Th us, as exp ected, we require an Ehresmann connection, which w e refer to as the Carr ol lian c onne ction or Carr ol lian c ovariant derivative that realised the decomp osition V ect (Π S ) ∼ − → V ect V (Π S ) ⊕ V ect H (Π S ) , in to v ertical and horizontal vector fields. W e thus define (2.16) ∇ x = ∂ x + A t x ∂ t + A i x ∂ ζ i , ∇ η j = ∂ η j + A k j ∂ ζ k , where the connection co efficients A = ( A t x , A i x , A k j ), in general dep end on all the co ordinates. Under c hanges of co ordinates, one can quickly deduce (2.17) A ′ t x = A t x − α∂ x f , A ′ i x = A i x + 1 2 α∂ 2 x f η i , A ′ k j = A k j + 1 2 α∂ x f δ k j . The in v ariant basis of one-forms, once a connection has b een selected, gives the notion of the clo ck forms (2.18) τ 0 := d t − d x A t x , τ i := d ζ i − d x A i x − d η j A i j , As a R 1 | 2 -v alued one-form, we define the clo ck form as τ := diag ( τ 0 , τ 1 , τ 2 ). The F r ob enius–Carr ol l curvatur e of the connection is defined as R : Vect H (Π S ) × V ect H (Π S ) → V ect V (Π S ) (2.19) ( X , Y ) 7− → prj V  [ X , Y ]  , where prj V : Vect (Π S ) → Vect V (Π S ) is the canonical pro jection. The curv ature measures the non-closure of the horizon tal distribution under the Lie brack et. A Carrollian connection is said 6 THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y to b e flat if the F rob enius–Carroll curv ature (2.19) v anishes, i.e., R ( X , Y ) = 0 for all horizontal v ector fields. As standard for connections, observe that τ 0 ( X ) = 0 and τ i ( X ) = 0, if and only if X ∈ V ect H (Π S ); the reader can quic kly chec k this using lo cal co ordinates. As, d ω ( X , Y ) = X  ω ( Y )  − ( − 1) e X e Y Y  ω ( X )  − ω ([ X , Y ]) , for all ω ∈ Ω 1 (Π S ) and X , Y ∈ V ect H (Π S ), w e observ e that (2.20) d τ 0 ( X , Y ) = − τ 0 ([ X , Y ]) , d τ i ( X , Y ) = − τ i ([ X , Y ]) . Th us, d τ = 0 is equiv alent to the F rob enius–Carroll curv ature v anishing. The clo ck form is said to b e stationary if the Carrollian connection is a principal connection, i.e., we require Vect H (Π S ) is equiv arian t. In lo cal co ordinates, this means that the components A are indep endent of ( t, ζ i ). More inv ariantly , defining κ := ( ∂ t , ∂ ζ 1 , ∂ ζ 2 ) T , the stationary condition is L κ τ = 0. Giv en a clo ck form, there is an underlying clo ck form on the Carrollian plane. T o see this, observ e that in any admissible co ordinate system, the comp onent of the connection A i x is a collection of ev en (lo cal) functions, thus it can b e expanded as A i x ( t, x, ζ , η ) = A i 0 ,x ( t, x ) + T erms Inv olving Odd Co ordinates , th us A i 0 ,x ( t, x ) is a collection of lo cal functions on R 2 that transforms in the appropriate wa y . That is, we hav e a connection on the Carrollian plane. Moreo ver, A i x is a collection of o dd lo cal functions, and so v anishes on the Carrollian plane. If the Carrollian connection on Π S , then the reduced Carrollian connection on R 2 is also principal. 2.6. Geometric Sup ersymmetry. Let us fix a principal Carrollian connection A and a basic (with resp ect to the principal bundle structure) Ψ = d x Ψ x ( x, η ) + d η i Ψ i ( x, η ) . By conv en tion, Ψ is taken as Grassmann o dd, meaning that Ψ x ( x, η ) is even and Ψ i ( x, η ) is Grassmann o dd. Note that as a basic one-form, Ψ is equiv arian t, and the comp onents transform as scalars. Moreov er, w e will take the basic form to ha ve units [ T ], so the comp onents ha v e units [ T ][ L ] − 1 . W e then define the o dd vector fields, which we refer to as the sup er gener ators (2.21) Q i := ∇ η i + Ψ i ∂ t , whic h is globally defined and is inv ariant under co ordinate transformations. Directly w e calculate (2.22) { Q i , Q j } =  ∂ η i Ψ j + ∂ η j Ψ i  ∂ t +  ∂ η i A k j + ∂ η j A k i  ∂ ζ k ∈ V ect V (Π S ) . Note that Q 2 ∼ ∂ t + ∂ ζ , is not quite of generally exp ected form for a version of Carrollian sup ersymmetry . Sp ecifically , in more standard approac hes, the Carrollian sup ercharge squares to temp oral translations, see for example [17]. Observe that if the Carrollian connection is flat, then w e obtain (2.23) { Q i , Q j } =  ∂ η i Ψ j + ∂ η j Ψ i  ∂ t = 2 Ψ ( ij ) ( x ) ∂ t , where Ψ i ( x, η ) = η k Ψ ki ( x ) due to Ψ i b eing Grassmann o dd, and we define the symmetrisation as standard, i.e., 2 Ψ ( ij ) = Ψ ij + Ψ j i . Note that one may c ho ose each Ψ ij and so each Ψ ( ij ) to b e constan ts. In particular, one may choose Ψ ( ij ) = δ ij , so that the (2.23) is of the classical form. Ev en with non-constant comp onen ts Ψ i , observ e that [ Q i , { Q j , Q k } ] = 0, which is imp ortan t when defining the asso ciated Lie–Rinehart structure. The asso ciated sup ersymmetry transformations expressed via co ordinates are thus t 7→ t + ξ i Ψ i ( x, η ) , x 7→ x , (2.24) ζ i 7→ ζ i + ξ k A i k ( x, η ) , η j 7→ η j + ξ j , THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y 7 where ξ i are Grassmann o dd parameters with units [ η ] = [ L ]. Note that these sup ertrans- formations resp ect the principal bundle structure. Sp ecifically , they are compatible with the R 1 | 2 -principal action. Restricting to Carrollian b o osts for simplicity , i.e., f = x , we define the vector fields (2.25) H = ∂ t , P = ∇ x , B = x∂ t − 1 2 η i ∂ ζ i , Q j = ∇ η j + Ψ j ∂ t , X k = ∂ ζ k , and observ e that the non-zero Lie brack ets are [ P , B ] = H , [ P , Q i ] = P (Ψ i ) H , (2.26) [ B , Q j ] = 1 2 X j , { Q k , Q l } =  Q k (Ψ l ) + Q l (Ψ k )  H . Recall that a Lie–Rinehart sup erpair is a pair ( A , V ) where A is an asso ciative sup ercommutativ e algebra, and V is a Lie sup eralgebra such that V is an A -mo dule, and V acts as deriv ations on A . In particular, [ u, ψ v ] = a ( u ) ψ v + ( − 1) e ψ e u ψ [ u, v ], where a : V → Der( A ) is referred to as the anc hor map. F or details, the reader ma y consult [15] and references therein. The Carr ol l–Lie–Rinehart sup erp air is defined as follows. The asso ciativ e sup ercomm utativ e algebra is A := C ∞ (Π S ), and the Lie algebra is V := Span { H , P , B , Q i , X j } , together with their Lie brack ets (2.26). The anchor map a : V → Vect (Π S ) is given b y a ( Z ) ψ := Z ( ψ ), for all Z ∈ V and ψ ∈ C ∞ (Π S ). T o define the sup ercov ariant deriv ativ es in the case of closed clo cks, we require Ψ ij = Ψ ( ij ) and w e write Q i = ∇ η i + η j Ψ ( j i ) ( x ) ∂ t . The sup er c ovariant derivatives are defined as (2.27) D i := ∇ η i − η j Ψ ( j i ) ( x ) ∂ t . The reader can quic kly verify that { D i , D j } = − 2 Ψ ( ij ) ( x ) ∂ t , { D i , Q j } = (Ψ ( ij ) − Ψ ( j i ) ) ∂ t = 0 . 3. Concluding Remarks In this note, w e hav e, via the in trinsic construction of Carrollian spinors, built a supermanifold generalisation of the Carrollian plane (Π S , g , κ ). Once (equiv ariant) clo c ks hav e been chosen τ , alongside a complemen tary basic one-form Ψ, then a geometric sup ersymmetry can be constructed. The full structure should then be considered to b e (Π S , g , κ , τ , Ψ). The form of the sup ersymmetry , sp ecifically { Q i , Q j } = 2 Ψ ( ij ) ( x ) ∂ t , is nov el and sho ws that not all Carrollian sup ersymmetries can b e constructed via a c → 0 pro cedure. In particular, the x -dep endence demonstrates that these sup ersymmetries and not, in general, an In¨ on ¨ u–Wigner con traction of a Poincar ´ e sup eralgebra. It is possible to consider a “smaller” notion of the Carrollian superplane by considering Π S 0 ≃ Π S 1 . In this case, we hav e a principal R 1 | 1 -bundle, and the constructions in this note sp ecialise to that setting directly . How ev er, w e do not ha v e a degenerate metric whose k ernel is spanned b y the vertical vector fields. As the classical notion of a Carrollian manifold requires a degenerate metric, w e accept that we require four Grassmann o dd co ordinates. While the fo cus of this note has b een mathematics, sp ecifically degenerate Clifford algebras, principal bundles and sup ermanifolds, it is exp ected that the constructions presented here will b e of some use in building in trinsic Carrollian sup ersymmetric field theories. F or instance, considering sup erstatic (scalar) fields, so sup erfields Φ that satisfy X i (Φ) = 0. In co ordinates, w e ha ve Φ = ϕ ( t, x ) + η i ψ i ( t, x ) + 1 2 η i η j F j i ( t, x ) , whic h hav e their standard interpretation as a pair of even fields and a pair of o dd fields: note F 12 = − F 21 are the only non-v anishing comp onents of F j i . The comp onent-wise Carrollian sup ertransformations are defined as δ ξ Φ := ξ i Q i Φ, and explicitly are δ ξ ϕ = ξ i ψ i , δ ξ ψ j = ξ k  Ψ ( kj ) ∂ t ϕ − F kj  , δ ξ F ij = − ξ k  Ψ ( ki ) ∂ t ψ j − Ψ ( kj ) ∂ t ψ i  . 8 THE CARROLLIAN SUPERPLANE AND SUPERSYMMETR Y More generally , one can mimic standard superspace metho ds to construct in v arian t actions. The co ordinate Berezin v olume is in v ariant under the co ordinate changes (2.12) and the Carrollian sup ersymmetry transformations (2.24). W e can then define actions S [Φ] = Z L (Φ , D Φ) , whic h are automatically inv arian t under Carrollian sup ersymmetry . W ritten out in comp onents, the Euler–Lagrange equations will b e of ‘electric-t yp e’, i.e., contain only partial deriv atives with resp ect to time. Th us, although quite exotic Lagrangian densities can b e constructed, they cannot p ossess on-shell propagating degrees of freedom. 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