A Cartan tale of the orbifold superstring

A geometrisation scheme internal to the category of Lie supergroups is discussed for the supersymmetric de Rham cocycles on the super-Minkowski group 𝕋 which determine the standard super-p-brane dynamics with that target, and interpreted within Cartan’s approach to the modelling of orbispaces of group actions by homotopy quotients. The ensuing higher geometric objects are shown to carry a canonical equivariant structure for the action of a discrete subgroup of 𝕋, which results in their descent to the corresponding orbifolds of 𝕋 and in the emergence of a novel class of superfield theories with defects.


Introduction
Among the many instantiations of the symmetry principle in physics, the modelling of dynamics with internal degrees of freedom (iDOFs) represented by orbispaces of group actions figures as a particularly subtle conceptually yet robust one: Indeed, while it grants us access to potentially singular configurational geometries and necessitates, in the case of continous symmetries, the incorporation of extra (gauge-field) iDOFs with their own intricate dynamics, it does, all this complexity notwithstanding, sit at the core of, i.a., the Standard Model of fundamental interactions, and so plays the rôle of an organising principle in our unified picture of elementary processes in a vast energy range.Being rooted in Cartan's idea of the homotopy quotient [1], the modelling naturally leads to the emergence of meshes of codimension-1 spacetime defects in the field theory with a gauge(d) symmetry, at which there occur field discontinuities determined by the action of the symmetry group.This entails further structural subtleties in the dynamics in the presence of non-tensorial couplings, such as, e.g., the topological Wess-Zumino (WZ) couplings in the nonlinear σ-model, employed successfully in the description of a wide range of dynamical systems -from the Affleck-Haldane critical field theory of collective excitations of quantum spin chains all the way to classical (super)string theory.Here, Cartan's mixing construction [2] needs to be lifted to the higher-geometric (HG) and -categorial objects (a.k.a.p-gerbes and their morphisms) which represent the background (p + 2)-form fields over the space of iDOFs, and so it undergoes a categorification in the form of an equivariant structure on these objects, whose components decorate the aforementioned gauge-symmetry defects [3,4].
In the present note, we embed the above general discussion in the setting of the Green-Schwarz (GS) superfield theory [5] of extended distributions of supercharge (a.k.a.super-p-branes) in the super-Minkowski target Lie supergroup T ≡ R d,1 D d,1 (with D d,1 superchargescp.later), the relevance of these superfield theories hinging firmly upon their asymptotic relation to the super-σ-models with curved supertargets with bodies AdS m × S n , which underly the physically useful yet mathematically still largely elusive AdS/CFT correspondence.As the point of departure, we take the geometrisation scheme developed in [6] for the non-trivial classes in the Cartan-Eilenberg (CaE) cohomology CaE • (T) ≡ H • (T) T of that Lie supergroup which determine the relevant WZ couplings.The scheme, to be regarded as a generalisation of the intrinsically FDA construction of the so-called extended superspaces due to de Azcárraga et al. [7], yields distinguished Murray-Stevenson-type p-gerbe objects [8,9] in the category sLieGrp of Lie supergroups, dubbed CaE superp-gerbes in [6].Upon reviewing the logic of their construction, we identify the resultant HG objects as p-gerbes over the super-minkowskian base endowed with a canonical equivariant structure for the discrete Kostelecký-Rabin supersymmetry group Γ KR ⊂ T of [10].As such, they are to be viewed -in the spirit of an HG extension of Cartan's construction worked out in [3,4] -as models of p-gerbes over the Rabin-Crane super-orbifold T Γ KR of their base.The super-orbifold has compact Graßmann-odd fibres [11] and so seems to be singular as a supergeometry [12] -this is a situation in which Cartan's idea becomes particularly relevant and useful, even indispensable.The above identification is a nontrivial extension of Rabin's original proposal [13] to view CaE • (T) as the differential cohomology equivariant with respect to Γ KR , and therefore [2] -as a model of the de Rham cohomology of T Γ KR .Finally, the derivation of the canonical Γ KR -equivariant structure on the super-p-gerbes of the GS super-σ-model provides us with a novel construction of a superfield theory with Γ KR -jump defects, which is to be understood, along the lines of [14,4], as an effective model of super-p-brane dynamics on T Γ KR .Superstring theory is thus shown to naturally inhabit such singular supergeometries, which are yet to be explored.

Configurational & dynamical aspects of Cartan's mixing construction
The subject of our interest is a lagrangean field theory with the space of iDOFs given by the space of orbits M Γ of an action λ ∶ Γ × M → M ∶ (g, x) ↦ λ g (x) ≡ g ⊳ x of a group Γ on a smooth manifold M .If λ is not free and proper, there may be no smooth structure on and hence no direct access to M Γ.Under such circumstances, we invoke Cartan's mixing construction [1,2] and model the field theory on classes of field configurations in M .Thus, the point of departure is a field theory with M as the space of iDOFs, i.e., a field bundle π F ∶ F → Σ with typical fibre M over a metric spacetime (Σ, g) together with a functional A DF ≡ exp(i S) ∶ Γ(F ) → U(1), termed the Dirac-Feynman amplitude (DFA), whose critical points are the classical field configurations.The DFA is expressed in terms of an action functional S, itself determined by a lagrangean density as S[φ] = ∫ Σ L(φ, Tφ).The identification of A DF as the fundamental object is in keeping with Dirac's quantum-mechanical interpretation of S, and it paves the way towards a rigorous description of charge dynamics over topologically non-trivial spacetimes [15], as in the Aharonov-Bohm experiment.We assume A DF to be λinvariant and subsequently 'descend' the field theory from M to M Γ.To this end, we consider Cartan's mixing diagram [2] EΓ in which π EΓ ∶ EΓ → BΓ is the universal principal Γ-bundle over the classifying space BΓ of Γ, with a contractible total space EΓ on which Γ acts, and in which EΓ × λ M ≡ (EΓ × M ) Γ is the quotient manifold for the diagonal action of Γ.The latter manifold fibres over BΓ with typical fibre M , and admits a fibrewise action of the universal adjoint bundle AdEΓ ≡ EΓ× Ad Γ, locally modelled on λ.Whenever M Γ is smooth, with the orbit projection ̟ ∼ smooth and Γ-principal, it also fibres over M Γ as π ∶ EΓ × λ M → M Γ ∶ [(p, x)] ↦ Γ ⊳ x, which implies the homotopy equivalence EΓ × λ M ∼ h M Γ.The equivalence underlies Cartan's identification of the universal associated bundle EΓ × λ M as a smooth model of the potentially singular quotient geometry M Γ, carrying information on the latter's homotopyinvariant structures -hence, e.g., the definition of the orbispace de Rham cohomology In the field-theoretic setting of interest, we work with avatars of Cartan's universal construction in the form of pullbacks of (1) along maps Φ ∶ Σ → BΓ, i.e., with gauge bundles P Φ ≡ Φ * EΓ and the associated bundles P Φ M ≡ Φ * (EΓ × λ M ) whose global sections Γ(P Φ M ) acquire the status of (matter) fields of the theory with Γ gauged.These are acted upon by the gauge group Γ(AdP Φ ), capturing vertical auto-equivalences Aut vert BunΓ(Σ) (P Φ ) ≅ Γ(AdP Φ ) of the field-theoretic pullback.The relevance of these structures to the attainment of the original goal is readily appreciated when M Γ is smooth: In this case, sections φ ≡ (id In a generic situation, the picture is subtler but no less meaningful: Taking into account the classic bijection Ψ λ ∶ Γ(P Φ M ) ≅ Hom Γ (P Φ , M ) between matter fields and Γ-equivariant maps P Φ → M , and using sections σ i ∶ O i → P Φ , i ∈ I of P Φ over a trivialising cover {O i } i∈I , we represent φ locally by smooth maps . Thus, configurations of the orbispace field theory are effectively realised as equivalence classes, with respect to the action of Σ-dependent (local) profiles in Γ, of piecewise smooth profiles in M over domains O i of (arbitrary) trivialisations of P Φ M , with discontinuities at arbitrarily located (within the O ij ) 'domain walls' controlled by profiles g ij in Γ (which 'smoothen out' upon projection to M Γ).In this manner, Cartan's construction gives rise to a field theory with topological gauge-symmetry defects, i.e., codimension-1 loci of field discontinuity which can be continuously deformed within the spacetime of the theory without affecting the value of its DFA, cp.[14,4] for details.It deserves to be emphasised that it is the existence of nontrivial gauge bundles P Φ which entails the emergence of the so-called twisted sector (with properly discontinuous field configurations) in the orbispace field theory.
Whenever Γ is smooth, the above configurational construction is not the end of the story: Describing the dynamics of the iDOFs from M Γ calls for the incorporation of a Γ(AdEΓ)-equivariant connection on EΓ × λ M canonically induced from a principal one on EΓ.As the symmetry group of our immediate interest is discrete, which we assume with regard to Γ henceforth, we do not elaborate this point, referring the Reader to the literature, e.g., [3,4].
The construction of the orbispace field theory uses differential-geometric structures over M , which enter the definition of the mother DFA.In an attempt to 'descend' the latter to M Γ, we may encounter two distinct situations: (i) the dynamics is determined by λ-invariant tensors on M , e.g., the standard 'kinetic' term given by a metric tensor G ∈ Γ(T * M ⊗ M,R T * M ) on the space of iDOFs with Isom(M, G ) ⊃ Γ; (ii) the dynamics is determined by non-tensorial couplings sourced by (higher-)geometric objects over M which couple to the 'states' of the field theory, and the DFA is invariant under automorphisms of these objects, including lifts of the λ g .In the former case, the descent is straightforward: Any Γ-invariant tensor is automatically Γ-basic for Γ discrete, and so the DFA for [Σ, M ], as it stands, describes the iDOFs modulo λ.In particular, in the previously described defect picture of Cartan's model, there is no need to augment the patchwise definition of the twisted sector over the O i with any extra data supported on the defect graph -the discrete jumps at the defect lines do not introduce any inconsistencies, and the lines themselves can be deformed freely within the intersections O ij of the trivialisation frames.In the latter scenario, on the other hand, the situation may become substantially more involved, which we demonstrate on a specific example, with view to subsequent considerations.
We focus on the lagrangean field theory -the so-called 2d σ-model -of smooth mappings [Σ, M ] from a closed 1 two-dimensional manifold Σ ∈ ∂ −1 ∅ into a metric manifold (M, g), which describes minimal embeddings of the worldsheet Σ in the target space M distorted by Lorentz-type forces induced by a torsion field , and such a choice may be enforced upon us, e.g., by the requirement of conformality of the quantised theory, the topological WZ term in the DFA, which models the coupling of H, does not admit a simple tensorial definition -instead, the amplitude takes the form [15] given in terms of the surface holonomy , inherited from the standard pair groupoid Pair(YM ) through restriction and used to define pullback operators , and we have additional structure: a principal C × -bundle L with a principal connection form A L ∈ Ω 1 (L) of curvature 1 The assumption of closedness of Σ, adopted for simplicity, may readily be dropped in a systematic manner [14].
The holonomy can now be understood as the image ı([x * G]) ≡ Hol G (x(Σ)) of the pullback element [x * G] in the group W 3 (Σ; curv = 0) of isoclasses of flat 1-gerbes over Σ (with the binary operation ⊗ on representatives) under the canonical isomorphism ı ∶ W 3 (Σ; 0) ≅ H 2 (Σ, U(1)) ≡ U(1).For the cohomologically trivial H = db with b ∈ Ω 2 (M ), we obtain the trivial gerbe ), and the expected formula It is to be emphasised that the fully fledged bicategorial structure is requisite for a consistent modelling -in terms of decorated surface holonomies -of defects with selfintersections in the spacetime Σ of the theory whose 'bulk' dynamics is determined by the 1-gerbe: In this case, the target space becomes stratified as M ⊔Q ⊔⊔ n≥3 T n , reflecting the decomposition of Σ, effected by the embedded defect graph G , into (closed) 2d domains Σ i , 1d domain walls (or defect lines) ℓ i,j = Σ i ∩ Σ j ⊂ G and their junctions v i1,i2,...,in of valence n ∈ N ≥3 .Each connected bulk (sub)stratum M i ⊂ M comes with its own metric and a 1-gerbe, to be pulled back to Σ i ; each defect-line (sub)stratum Q i,j ⊂ Q comes with a 1-cell of Grb ∇ (Q i,j ), to be pulled back to ℓ i,j ; and, finally, each junction (sub)stratum T i1,i2,...,in ⊂ T n comes with a 2-cell of Grb ∇ (T i1,i2,...,in ), to be pulled back to v i1,i2,...,in , cp. [14] for details.The fundamental benefit of working with 1-gerbes is that they canonically determine prequantisation of the 2d field theory through cohomological transgression , a canonical assignment to (the isoclass of) G of (the isoclass of) a line bundle over the (single-loop) configuration space LM ≡ [S 1 , M ] with connection of curvature ∫ S 1 ev * H, as discovered in the seminal work [15].The latter fact justifies adherence, subsequent to this discovery, to the principle of 'gerbification' (i.e., construction of a bicategorial lift) applied in the analysis of all properties and structures of the classical 2d field theory which are anticipated to survive in the quantum régime, and explains the tremendous success of the thus based HG approach to its study, pioneered and developed by Gawȩdzki.
Among the bicategorial lifts, we find that of global symmetries of the σ-model: Given Γ as above, the lift consists of a family {Φ g ∶ G ≅ λ * g G} g∈Γ of invertible 1-cells of Grb ∇ (M ), which transgress to automorphisms of L G .The data {Φ g } g∈Γ are the point of departure for the descent M ↘ M Γ in the HG setting, as governed by the equivalence Grb ∇ (M Γ) ≅ Grb ∇ (M ) Γ−equiv -a specialisation of the more general one from [3] to the case of Γ discrete -between the bicategory of 1-gerbes over the quotient manifold M Γ (whenever it exists) and that of 1-gerbes over M equipped with a Γ-equivariant structure, i.e., simplicial 1-gerbes over the nerve N • (Γ⋉ λ M ) ≡ Γ ×• × M of the action groupoid Γ⋉ λ M , represented by diagrams , and we have additional structure: a 1-cell Υ in Grb ∇ (Γ × M ) which establishes equivalence between λ * G and pr * 2 G, and a 2-cell υ in Grb ∇ (Γ ×2 × M ) which renders the bicategorial lift (G, Υ) of λ homomorphic and associative.In the light of the above bicategorial correspondence, existence of the Γ-equivariant structure ensures descent of the pullback 1-gerbe pr * 2 G from over EΓ × M to EΓ × λ M , and so also after the field-theoretic pullback Φ, whereby there arises a 1-gerbe G over the field bundle . The rôle of (G, Υ, υ) becomes more intuitively clear in the local picture with gauge-symmetry defects -indeed, the triple is precisely what is needed to consistently define the aforementioned decorated surface holonomy in the presence of defect lines ℓ i,j ∋ y at which the bulk embedding field x i ∶ O i → M jumps as x i (y) = λ gij (y) (x j (y)) by the (locally) constant transition mappings g ij ∶ O ij → Γ: With the dynamics in the domains Σ i determined by the x * i G, we endow the defect lines ℓ i,j with the respective 1-cells (g ij , x j ) * Υ, and the elementary (valence-3) junctions v i,j,k with the 2-cells (g ij , g jk , x k ) * υ.The associator condition ∆ (3) υ = id, satisfied by υ over Γ ×3 × M , ensures that 2-cells required for junctions of higher valence can be consistently induced from the elementary ones through (vertical) 2-cell composition dictated by a limiting procedure in which the junction of valence ≥ 4 is recovered by contracting all internal edges of its arbitrary binary-tree resolution, at no cost in the value of the DFA (owing to the topological nature of the defects), cp.[14] for details and [17] for a general treatment of simplicial σ-models.In summary, we see that the data (G, Υ, υ) effectively determine an orbispace σ-model with target M Γ.

The super-Minkowskian super-p-gerbes: construction & supersymmetry
We now come to the main point in our story, at which we embed the previous constructions in the Z 2Z-graded geometric category, and add supersymmetry as the fundamental symmetry of the ensuing superfield theory.More specifically, let Σ ∈ ∂ −1 ∅ be a (p + 1)-dimensional smooth manifold and fix a supermanifold M = ( M , O M ) of superdimension (m n), with a body (manifold) M and a structure sheaf O M , locally modelled on (R ×m , C ∞ (⋅) ⊗ B n ) ≡ R m n for the rank-n Graßmann algebra B n ≡ ⋀ • R ×n .Assume further the existence of a Lie supergroup G = ( G , O G ) (a group object in the category sMan of supermanifolds) acting on M as λ ∶ G × M → M. The supermanifold M should come with an even rank-2 symmetric tensor g ∈ Γ(T * M ⊗ T * M) (0) and an even integral de Rham (p + 2)cocycle H p , both λ-invariant.Given the Graßmann-even nature of Σ, probing the soul of the supertarget M (i.e., the nilpotent component of O M ) calls for an inner-Hom functorial structure of the space of fields, which we take, after [18], in the form of the generalised mapping supermanifold [Σ, M] ≡ Hom sMan (Σ × ⋅, M), to be evaluated on a family {R 0 N } N ∈N × of superpoints, nested as R 0 N1 ↪ R 0 N2 for N 1 < N 2 .Thus, we may think of the superfield theory as (the direct limit of) a family of 'ordinary' field theories.With these, we define a dynamics which generalises (2).
An obvious choice of the target, motivated by the postulate of irreducibility, is a homogeneous space M ≡ G K of the supersymmetry group G relative to a closed subgroup K ⊂ G of its body, which -for (sLie G, Lie K) , where the Γ a ≡ CΓ a , assumed symmetric, are products of the generators Γ a of the Clifford algebra = 0 in S d,1 , which restricts the spectrum of admissible pairs (d, p) [7], we establish the crucial property: . The property attests to the existence of an intricate topology lurking beneath the plain supergeometry T, which we elucidate below, following the extensive study [6].
There is a canonical choice of the (quasi-)metric on T, given by the LI lift g = η ab e a ⊗ e b of η, and the requirement of restoration of supersymmetry in the classical vacuum of the superfield theory for (g, H p ) through κ-symmetry projection fixes the remaining choice of the (p + 2)-cocycle H p for the WZ term [19]: In all admissible cases p < 10, we find H p = q p χ p for some q p ≠ 0, and so while we might write the resultant GS super-σ-model in a quasi-supersymmetric form (2) with a trivial WZ term given by the pullback of a de Rham primitive of H p , the constitutive nature of T-invariance forces us to face the problem of 'geometrising' [H p ], conceptually identical with the one encountered in the un-graded setting for the ordinary differential cohomology.Technically, the solution to it proposed in [6] generalises the construction of the so-called extended superspaces in [7] and, as such, hinges on the classic bijection between the 2nd cohomology group H 2 (sLie G, a) of the tangent Lie superalgebra sLie G of a Lie supergroup G with values in its trivial (super)commutative module a and the set of equivalence classes of (super)central extensions of sLie G through a, the latter being captured by short exact sequences 0 → a → sLie G → sLie G → 0 of Lie superalgebras.For a of (super)dimension 1, the former is the 2nd Chevalley-Eilenberg cohomology group CE 2 (sLie G) of sLie G, which is canonically isomorphic with CaE 2 (G), and so we arrive at an algorithm of sequential geometrisation [6]: The procedure delineated above maya priori -fail at some elementary (H 2 -)stage due to nonintegrability of a Lie-superalgebra extension.As it happens, in the study, reported in [6], of the GS (p + 2)-cocycles H p for p ∈ {0, 1, 2} determined by the requirement of vacuum supersymmetry one does not encounter such obstructions.Thus, in all these physically distinguished cases, we obtain a p-gerbe object G p in sLieGrp, described by a Murray diagram with a Lie supergroup at each node, all arrows representing Lie-supergroup epimorphisms, and all superdifferential forms LI.Such higher geometric objects were dubbed CaE super-p-gerbes in [6].We close this section with an explicit example of the above geometrisation mechanism -that of the CaE super-1-gerbe G 1 -which paves the way to a topological interpretation of the advocated geometrisation scheme and a novel superfield theory.Thus, assume the Fierz identity for p = 1 and consider the CaE 3-cocycle H 1 ≡ σ α ∧ F α , written in terms of the odd 2-cocycles F α = η ab Γ a αβ σ β ∧ e b .Their CE counterparts ω α give a supercentral extension (Yt ≡ t⊕⊕ YT ω α + pr * 2 θ α (dual to the Z α ), written in terms of the MC 1-forms θ α ≡ dξ α on R 0 D d,1 (with global coordinates ξ α ).On YT, we find the trivialisation α − ξ α ) such that 0 ≠ [F (1) ] ∈ CaE 2 (Y [2] T), and so its CE counterpart ω defines a rank-1 central extension (l ≡ Y [2]  [2] T × C × → Y [2] T. The binary operation Lm on L follows from left-invariance of A L = pr * 1 A + pr * 2 θ C × (the dual of Z), with A(θ, x, ξ (1) , ξ (2) α ) and the standard MC 1-form θ C × on C × .The LI 1-form provides a trivialisation π * L F (1) = dA L , and we check ∆ (2) A = 0, whence also the groupoid structure µ L = 1, readily verified to be a Lie-supergroup isomorphism.Altogether, we obtain a CaE super-1-gerbe (YT, π YT , B 1 , L, π L , A L , µ L ) as in (3).
The construction of G 1 for the superstring, and analogous constructions of the super-0-gerbe for the superparticle, and of the super-2-gerbe for the M -theory supermembrane form the basis of an in-depth HG study of κ-symmetry [19] and of a systematic reconstruction of maximally supersymmetric super-minkowskian defects [17] along the lines of [14], i.e., using a supersymmetric multiplicative structure on the super-p-gerbe.We here, instead, pursue the conceptual question: Can we ascribe to the above geometrisation the meaning of a resolution of a nontrivial topology, and thereby establish a deeper correspondence with Murray's construction?An HG object associated with an integral class in the de Rham cohomology resolves the homology dual of that class.This simple observation puts flesh on the bones of the previous question, and immediately suggests a negative answer -indeed, R d,1 D d,1 has no nontrivial topology.And yet. . .A more elementary question in the same vein was asked by Rabin in [13], subsequent to the work [10] on lattice supersymmetric field theory in which discrete subgroups of T had come up naturally, to wit: Is there a discrete subgroup Γ ⊂ T with the property Ω • (T) Γ ≡ Ω • (T) T ?Clearly, an affirmative answer to this question would imply, by Cartan's logic, that we can think of CaE • (T) as a model of H • (T Γ).In the remainder of this note, we review Rabin's original answer, and examine our scheme of geometrisation of CaE • (T) from this newly acquired angle.
Let us consider a nested family of sets Yon T (R 0 L1 ) ⊂ Yon T (R 0 L2 ), L 1 < L 2 of superpoints in T, in the image of the Yoneda embedding Yon T (⋅) ≡ Hom sMan (⋅, T) of T in the category of presheaves on sMan (a.k.a.generalised supermanifolds).It deserves to be emphasised that these are precisely the sets probed by the GS superσ-model in Freed's approach.Technically, the restriction to Yon T (R 0 L ) is effected through an explicit realisation of O T in a fixed rank-L Graßmann algebra B L with generators β i , i ∈ 1, L, so that θ α ∈ B L (1) and x a ∈ B L (0) and we obtain the model L (1) of Yon T (R 0 L ).We may then define the Kostelecký-Rabin group at level L as the subset Z by the basis of B L , only to find the desired identity Ω [13], cp.[6] for a proof.The family {Γ KR (L) } L∈N inherits a nesting from {T L } L∈N , and so we may pass, after [13], to the direct limit Γ KR ≡ lim → Γ KR (L) .At finite level, Ω • (T L ) T L models the exterior algebra of T L Γ KR (L) .An explicit construction of such an orbifold was given in [11], and demonstrates its geometric intricacy.In the direct limit, there arises the Rabin-Crane super-orbifold T Γ KR ≡ lim → T L Γ KR (L) , and we arrive at the anticipated interpretation of CaE • (T) as a model of H • (T Γ KR ) [13].At this point, it becomes natural to expect that the geometrisations G p of the GS classes in CaE • (T) from Sec. 3 are particular models of p-gerbes over T Γ KR .
The path towards verification of the latter expectation leads through categorification of the action λ ≡ m T whose cotangent lift enters the definition of CaE • (T).
As before, we focus on the case p = 1, and adopt the S-point picture for the sake of simplicity.The categorification now assumes the form of a family Taking into account the definition of pullback as a universal object, we may next exploit the existence of the natural lifts Ym (t,0) and Lm (t,0,0,1) of the λ t to YT and L, respectively, in conjunction with the left-invariance of B 1 and A L , to choose (YT, B 1 ) as the surjective submersion of the pullback 1-gerbe and (L, A L ) as its principal C × -bundle, and thus to canonically identify λ * t G 1 with G 1 [6].This yields the 1-cells Φ t ≡ id G1 , determined fully by the bundle L and the trivial groupoid structure µ L ≡ 1 of G 1 [16].Such very special form of the categorification of supersymmetry furnished by the CaE super-1-gerbe (and the other G p alike), which ultimately rests upon the internalisation of Murray's definition of a 1-gerbe (resp.that of a p-gerbe) in sLieGrp, is the first distinctive feature of the geometrisation scheme advocated.
The descent of the G p to the Rabin-Crane super-orbifold has an important superfield-theoretic implication: The triple E ΓKR can be used to consistently define, upon pullback to Σ with Γ KR -jump defects, a super-σ-model with iDOFs modelled on T Γ KR , even in the absence of a smooth supermanifold structure on the latter, cp. the detailed study [14].In this manner, the geometrisation scheme delineated in this note promises to open a new direction in both: the study of non-Rothstein supergeometries and the modelling of (super)charged dynamics thereon.
reductiveallows us to employ the K-basic component of the left-invariant (LI) Cartan calculus on G in the construction of the DFA for G K, as realised by local sections of G → G K. This is the logic underlying, i.a., the definition of the important class of super-σ-models with targets s(AdS m × S n ), central to the formulation of the AdS/CFT correspondence.In what follows, we focus on the simplest homogeneous geometry -the super-Minkowski space sISO(d, 1 D d,1 ) Spin(d, 1) ≡ T, with its global generators of O T : the odd θ α , α ∈ 1, D d,1 and the even x a , a ∈ 0, d, and the Lie-supergroup structure.Its binary operation m T has the sheaf component m * T , η) with the skew charge-conjugation matrix C, all in a Majorana-spinor representation S d,1 of dim S d,1 = D d,1 .The corresponding Lie superalgebra t ≡ sLie T = ⊕ D d,1 α=1 ⟨Q α ⟩ ⊕ ⊕ d a=0 ⟨P a ⟩ has the wellknown structure equations: {Q α , Q β } t = Γ a αβ P a and [Q α , P a ] t = 0 = [P a , P b ] t .The relevant algebra of LI differential forms Ω • (T) T = ⟨σ α , e a ⟩ has generators σ α (θ, x) = dθ α and e a (θ, x) = dx a + 1 2 θ α Γ a αβ dθ β ≡ η −1 ab e b (θ, x).Upon assuming the Γ a1a2...ap ≡ CΓ a1a2...ap symmetric and imposing the Fierz identities η ab Γ a (αβ Γ ba1a2...ap−1 γδ)
known as the Green superalgebra.It integrates to a Lie-supergroup extension π YT ≡ pr 1 ∶ YT ≡ T×R 0 D d,1 → T with a binary operation Ym fixed by the requirement of left-invariance of the 1-forms ζ α = π *

4 .
Categorification of SUSY & descent to the Rabin-Crane super-orbifold y effected by the transition mappings g ij ∶ O ij → Γ of P Φ associated with the local gauges σ i .The local matter fields f i undergo local gauge transformations