The square root of the vacuum I. Equivariance for the κ -symmetry superdistribution
aa r X i v : . [ h e p - t h ] M a r THE SQUARE ROOT OF THE VACUUMI. EQUIVARIANCE FOR THE κ -SYMMETRY SUPERDISTRIBUTION RAFA L R. SUSZEK
Abstract.
A complete and natural geometric and physical interpretation of the tangential gaugesupersymmetry, also known as κ -symmetry, of a large class of Green–Schwarz(-type) super- σ -modelsfor the super- p -brane in a homogeneous space of a (supersymmetry) Lie supergroup is establishedin the convenient setting of the topological Hughes–Polchinski formulation of the super- σ -model andillustrated on a number of physical examples. The supersymmetry is identified as an odd superdis-tribution in the tangent sheaf of the supertarget of the super- σ -model, generating – through its weakderived flag – the vacuum foliation of the supertarget. It is also demonstrated to canonically liftto the vacuum restriction of the extended Hughes–Polchinski p -gerbe associated with the superback-ground of the field theory, and that in the form of a canonical linearised equivariant structure thereon,canonically compatible with the residual global supersymmetry of the vacuum. Contents
1. Introduction 12. Supergeometry `a la Cartan 73. The two faces of the physical model 204. The vacuum of the super- σ -model 375. Supersymmetries of the super- σ -model and of its vacuum 555.1. The global-supersymmetry group 555.2. The κ -symmetry superdistribution – an odd resolution of the vacuum 586. The higher geometry behind the physics 687. Supersymmetry-invariant gerbification of linearised κ -symmetry 858. Conclusions & Outlook 96Appendix A. The Zhou super-0-brane in s ( AdS × S ) . 98References 1001. Introduction
Symmetry, a polyseme with a broad semantic field in the physical parlance stretching between –on one hand – the concept of ‘correspondence’ between independent entities that implies ‘balance’ intheir quantity , and – on the other hand – the concept of ‘ambiguity’ and ‘unphysical redundancy’of equivalent representations of a given entity that calls for ‘reduction’ , is widely recognised as oneof the most robust organising principles in the mathematical modelling of physical phenomena. Fromadmissible interaction vertices and superselection rules, through conserved charges and constraintsimposed upon quantum-mechanical correlators, all the way to the exclusion of theories exhibitinganomalies and the removal of pure-gauge degrees of freedom, it helps to organise the physical contentof the theory and its dynamics in terms of the representation theory of the symmetry structure, typicallygiven as a group or an algebra. In extreme cases, a symmetry can be employed towards a completeresolution of the theory, as exemplified by the BPZ approach of Ref. [BPZ84] to the derivation ofcorrelators in a conformal field theory in two dimensions (in which the conformal group becomes infinite-dimensional), especially in the presence of an extension of the Virasoro algebra of symmetries modelledon a Kaˇc–Moody algebra, cp Ref. [KZ84], and by the localisation of the theory on a Cauchy hypersurface(an equitemporal slice carrying complete data of a state) through identification of ‘propagation’ of astate with a gauge transformation in the three-dimensional Chern–Simons topological gauge field theoryon a cylinder over a (punctured) Riemann surface, cp Ref. [Fre95]. Therefore, it is of utmost importance In this denotation, the ‘symmetry’ is customarily qualified as ‘global’ or ‘rigid’. Here, the usual qualifiers are ‘local’ or ‘gauge’. o have a good understanding of the symmetries of a field theory under study, and of their amenabilityto a consistent quatisation.An important class of physical models in which symmetries have a readily identifiable geometricorigin and their transcription to the quantum r´egime is systematised by higher geometry and cohomol-ogy is formed by non-linear σ -models describing simple geometro-dynamics of compact p -dimensionaldistributions of topological charge and energy (such as, e.g. , charged and massive pointlike particles at p =
0, loops and paths at p =
1, membranes at p = etc .) in external fields permeating the ambientmanifold M in which the charge propagates and coupling to the charge currents thus induced, i.e. , a ( p + ) -form field and a gravitational field. Geometrisation of the classes in the de Rham cohomologyof M determined by the ( p + ) -form field, requisite for a rigorous definition of the Dirac–Feynmanamplitude in these theories, cp Refs. [Alv85, Gaw88], leads to the emergence of the so-called p -gerbes– geometric structures that generalise, in a well-defined sense, principal C × -bundles encountered in thelagrangean description of the charged pointlike particle, cp Refs. [Mur96, MS00, Ste00, Joh02, Gaj97].Being defined entirely in terms of smooth differential-geometric objects over the target space of the σ -model, the p -gerbes determine geometric (pre)quantisation of the ( p + ) -dimensional field theory, cp Ref. [Gaw88, Sus11a], and in this manner enable us to distinguish, as quantum-mechanically consistent,those symmetries that lift (or transgress ) to isomorphisms in the (weak) ( p + ) -category of p -gerbesover M (in the case of global symmetries) resp. give rise to full-fledged equivariant structures overthe nerve of the action groupoid engendered by the action of the symmetry group on the target space(in the case of gauged symmetries), cp Refs.[GSW11, GSW10, GSW13, Sus12, Sus13]. This universalhigher-geometric scheme with a neat cohomological underpinning, readily extendible to more generaldualities between different theories (represented by a class of worldvolume defects) such as, e.g. , T-duality, gives us extensive control over (symmetries of) any given σ -model of interest and a robustmethod of charting the moduli space of theories of this type. The σ -models, finding numerous physicalapplications from the theory of condensed matter, through the description of long-distance phenom-ena in two-dimensional statistical mechanics at a critical point and the modelling of the effective fieldtheory of certain collective excitations of discrete one-dimensional integrable systems (spin chains), tocritical bosonic string theory, are prototypes of the field theories that we intend to study in the presentpaper.Symmetry considerations acquire critical significance in the construction and study of field theorieswith the fibre of the covariant configuration bundle given by a single orbit of a symmetry group.Among these, we find the much studied Wess–Zumino–Witten (WZW) σ -model of Ref. [Wit84] ofcharged-loop mechanics on a compact Lie group and the formally rather intricate Green–Schwarz(-type)super- σ -model of, i.a. , Refs. [GS84a, GS84b, BST86, BST87, AETW87, BLN+97, MT98a, dWPPS98,Cla99, AF08, GSW09, FG12, DFGT09, Sus17] for the super- p -brane in a homogeneous space G / Hof a Lie supergroup G ( i.e. , a group object in the category of supermanifolds, with the structuresheaf valued in the category of supercommutative Z / Z -graded algebras) relative to a subgroup H ⊂∣ G ∣ of its (Graßmann-even) body ∣ G ∣ , the latter field theory being defined for mappings, from an(inner hom-)functorial mapping supermanifold [ Ω p , G / H ] ( cp Ref. [Fre99]), of the ( p + ) -dimensionalworldvolume Ω p into the supermanifold G / H equipped with a transitive action of the supersymmetryLie supergroup G and tensorial data (a metric and a super- ( p + ) -cocycle) descended from it. Whilethe status of field theories of the latter type as models of natural phenomena seems at best dubitableat this moment, they do find a spectacular direct application in a predictive description of physicalsystems, closely akin to observable ones, involving strongly interacting elementary coloured particles,outside the perturbative r´egime. A connection between the two universes: the perturbative superstringresp. super- p -brane theory and the non-perturbative (supersymmetric) Yang–Mills theory is establishedby means of the celebrated AdS/CFT correspondence whose in-depth elucidation, still missing todate, is one of the key motivations behind our interest in this particular class of supersymmetric fieldtheories. Setting aside the subtle technical issue of a functional (or, indeed, functorial) definitionof the super- σ -model , we immediately notice the fundamental difference between the two types oftheories mentioned above, affecting their higher-geometric description: While the presence of a globalsymmetry in the σ -model described by the compact target Lie group does not – in the light of TheCartan–Eilenberg Theorem – result in an actual choice of the cohomology in which to perform theaforementioned geometrisation of the 3-form field coupling to the loop current in the group manifold, thesame structural property of the super- σ -model on (a homogeneous space of) the inherently noncompact Lie supergroup confronts us with the non trivial choice between the standard de Rham cohomology This will be discussed at great length in the main text. nd its supersymmetric refinement. The dramatic discrepancy between the two is best exemplifiedby the super-Minkowski space with its trivial de Rham cohomology and nontrivial Cartan–Eilenbergcohomology, and the Green–Schwarz super- ( p + ) -cocycle defining a non -zero class in the latter. Basedon a topological interpretation of the discrepancy and a reinterpretation of the super- σ -model (as asuper- σ -model on a supermanifold with the topology of the Graßmann-odd fibres faithfully encodingthe discrepancy) implied by it, originally due to Rabin and Crane, cp Refs. [RC85, Rab87], and furtherinspired by the canonical analysis of the field-theoretic realisation of supersymmetry in the super- σ -model in terms of a Poisson (super)algebra of Noether charges with a wrapping-charge anomaly sourcedby monodromies of the embedding field along the Graßmann-odd fibres (assumed compact), carried outby the Author in Ref. [Sus18a], a geometrisation scheme for the Green–Schwarz super- ( p + ) -cocycleswas postulated in a series of papers [Sus17, Sus19, Sus18a, Sus18b] which essentially puts the standardconstruction of a p -gerbe from the Graßmann-even category, originally due to Murray and Stevenson, cp Refs. [Mur96, MS00], and subsequently formally generalised by Gajer in Ref. [Gaj97], in the category ofLie supergroups, with surjective submersions replaced by Lie-supergroup extensions. The fundamentalmechanism of the geometrisation scheme consists in (super)algebraisation of the Cartan–Eilenbergclass of the Green–Schwarz super- ( p + ) -cocycle on the supersymmetry Lie supergroup G via theclassical correspondence between the Cartan–Eilenberg cohomology CaE ● ( G ) of G and the Chevalley–Eilenberg cohomology H ● ( g , R ) of its tangent Lie (super)algebra g with values in the trivial module R , augmented with the equally classical correspondence between classes in H ( g , R ) and (super)centralLie-(super)algebra extensions of g . Thus, a sequence of Lie-superalgebra extensions, each correspondingto a partial (term-wise) cohomological trivialisation of the original Green–Schwarz super- ( p + ) -cocycleand assumed integrable to a Lie-supergroup extension, should ultimately lead to the emergence of aLie supergroup ̂ G, epimorphically mapped onto G, on which the pullback of the Green–Schwarz super- ( p + ) -cocycle trivialises in the Cartan–Eilenberg cohomology CaE ● (̂ G ) . The surjective submersion ̂ G Ð → G should then be taken as the basis of the standard construction of a p -gerbe, with the sequential-extension mechanism applied on each level of that construction. The resulting higher-geometric object– the Cartan–Eilenberg super- p -gerbe – still has to be descended to the physical target space G / H,for which the extensions indicated earlier ought to be equivariant with respect to (the adjoint actionof) the tangent Lie algebra h of the isotropy group H of the homogeneous space. The scheme wassuccessfully applied in the super-Minkowskian setting (in Ref. [Sus17]) and in the case of the super-2-cocycle for the Zhou super- σ -model of the super-0-brane in s ( AdS × S ) (in Ref. [Sus18b]), resulting ina collection of concrete manifestly supersymmetric super- p -gerbes with various desirable propreties suchas, in the super-Minkowskian case (for p ∈ { , } ), equivariance with respect to the Lie supergroup ofsupertranslations realised in the adjoint, cp Ref. [Sus19], in conformity with the intuition coming fromthe study of the WZW σ -model on the Lie group, cp Refs. [GSW10, GSW13, Sus12]. The constructionfor the Zhou super-2-cocycle, on the other hand, was demonstrated to asymptote, in the limit of ahomogeneous blow-up of the curved supertarget s ( AdS × S ) , dual to the standard ˙In¨on¨u–Wignercontraction on the underlying Lie superalgebra, to the super-0-gerbe on the super-Minkowskian space.Contractibility of the super- p -gerbe over a curved Lie supergroup with a super-Minkowskian blow-upto its known Green–Schwarz counterpart over that blow-up was subsequently incorporated into thegeometrisation scheme as a physically motivated organising principle. Just to re-emphasise, the crucialfeature of the geometrisation scheme thus obtained is the canonical invariance of the super- p -gerbeunder the action of the supersymmetry group of the super- σ -model, indispensable for the physicalinterpretation of the supersymmetry as a (pre)quantisable symmetry of the field theory.Besides establishing ‘correspondence’ between independent degrees of freedom of the embedding fieldof the super- σ -model of an arbitrary Graßmann parity, which is the task of the global supersymmetrymentioned above, supersymmetry is also bound to take on the other rˆole alluded to at the beginning ofthe present Introduction, i.e. , that of a mechanism of ‘reduction’ of the excessive (Goldstone) degreesof freedom of the (Graßmann-)odd type that arise in the process of localisation of the (Graßmann-even)body of the embedding field in the (classical) vacuum of the theory. This is an elementary consequenceof the universal feature of the supersymmetry algebra by which the anticommutator of (Graßmann-odd)supercharges yields, in particular, the (Graßmann-even) ‘momentum’. The ‘odd’ gauge supersymmetrythat does the job was discovered very early on by de Azc´arraga and Lukierski in Ref. [dAL83] (for thesuperparticle), and subsequently also by Siegel in Refs. [Sie83, Sie84] as a ‘hidden’ supersymmetry ofthe Green–Schwarz super- σ -model for the superparticle and the superstring with a super-Minkowskian That is predicted by the correspondence on the level of the super- σ -models, cp , e.g. , Ref. [MT98a]. upertarget. As it did not preserve separately the metric or the topological terms of the super- σ -model in the standard Nambu–Goto formulation, but only a suitably relatively normalised combinationthereof, its existence was later employed as a constraint in the construction of super- σ -models forhigher-dimensional p -branes and for curved supertargets, cp , e.g. , Refs. [MT98a, PR99, Zho99]. Fora long time, the ‘odd’ gauge supersymmetry, dubbed κ -symmetry in the string-theory literature, haskept its status of a useful odd ity, definable solely through its function, which is that of restorationof a supersymmetric ‘balance’ in the localised vacuum of the super- σ -model through identification of(typically) a half of the Graßmann-odd degrees of freedom as pure gauge, and otherwise exhibiting avariety of intertwined peculiarities: ● it appeared in the infinitesimal (un-integrated) form exclusively; ● the anticommutator of two κ -symmetry transformations would yield, depending on the struc-ture of the super- σ -model, either a bare worldvolume diffeomorphism, or one corrected by alocal transformation from the sector of the Lie algebra h of the isotropy group of the targethomogeneous space (the algebra h invariably acting as a model of an infinitesimal hiddengauge symmetry) not preserving the vacuum, or both additionally augmented with an extralocal transformation vanishing only in the vacuum of the theory; ● the ‘ κ -symmetry algebra’ spanned on κ -symmetries, worldvolume diffeomorphisms and – oc-casionally – correcting hidden gauge transformations as above would close, if at all, only uponimposition of field equations of the theory and further constraints.Its additional ‘peculiarity’: the ‘wrong sign’ of the κ -symmetry variation in comparison with the ‘rightsign’ of a global-supersymmetry transformation was soon interpreted as a sign of its geometric origin,to wit, linearisation of a right translation on the Lie supergroup G in which the homogeneous spaceG / H is embedded (locally) with the help of a section of the surjective submersion G Ð→ G / H, cp Ref. [McA00, GKW06a]. The latter ad hoc interpretation does not really seem to have enhanced ourunderstanding of the actual geometric nature of κ -symmetry or explained the above peculiarities, andso as of this writing, it is still being described in the literature as ‘a “hidden” fermionic symmetrywith no evident geometric interpretation’ but with a well-defined field-theoretic rˆole to play.In the present paper, we intend to unequivocally establish the status of the above ‘odd’ gaugesupersymmetry entirely in terms of the geometry of the supertarget of the super- σ -model, withoutchanging the latter’s deep nature of a theory with a functorial mapping supermanifold as the domain,and subsequently verify the existence of a higher-geometric realisation of the gauge symmetry in thefamiliar form of an equivariant structure on the p -gerbe associated with the super- σ -model. In the lightof the previous remarks on the behaviour, under a κ -symmetry transformation, of the two terms in theDirac–Feynman amplitude of the super- σ -model in the Nambu–Goto formulation, this task requires atranscription of the field theory into an equivalent picture in which the information on the supertargetmetric is carried by extra components of the embedding field and the metric term in the amplitude isreplaced by the standard pullback, along the extended embedding field, of a topological object. Thisseemingly wild scenario is actually realised in a large class of super- σ -models of the type discussed forwhich purely topological duals exist with the information on the metric structure and its coupling tothe super- p -brane worldvolume encoded in a choice of a ( p + ) -dimensional Graßmann-even subspace t ( ) vac in the complement t of the isotropy algebra h in g , termed the vacuum subspace and ad-stabilisedby a subalgebra h vac ⊂ h with the corresponding Lie subgroup H vac ⊂ H, a family of non-dynamicalGoldstone modes associated with the spontaneous breakdown h ↘ h vac of the hidden gauge symmetryand hence labelled by h / h vac , and a trivial h vac -equivariant Cartan–Eilenberg super- p -gerbe determineduniquely by the geometrised volume form on the vacuum subspace, cp Ref. [Sus19]. Such a reformulationof the super- σ -model was first considered by Hughes and Polchinski in Ref. [HP86] in the context ofpartial spontaneous breaking of global supersymmetry, later elaborated significantly by Gauntlett etal. in Ref. [GIT90], and more recently used by McArthur ( cp Ref. [McA00, McA10]) and West et al. ( cp Refs. [Wes00, GKW06b, GKW06a]) in the construction of (super-) σ -models for (super-) p -branes It ought to be remarked that the so-called ‘superembedding formalism’ has been developed by Sorokin, cp Ref. [STV89] and also Ref. [Sor00] for a comprehensive review, in which a ‘canonical’ Graßmann-odd extension of theGraßmann-even worldvolume permits to model κ -symmetry as an ‘odd’ superdiffeomorphism of the extended (super-)worldvolume, and – in this manner – de-geometrise κ -symmetry entirely, as seen from the supertarget perspective. Thepurely internal supersymmetry is now transmitted back to the supertarget only upon embedding the (super-)worldvolumein it in a consistent manner. This reformulation goes, in a sense, in the direction precisely opposite to the one that wewish to pursue, motivated by the conviction of the necessity to gerbify (pre)quatisable symmetries, and the adherence tothe original interpretation, due to Freed, of the super- σ -model as a theory defined for the mapping-supermanifold functor rather than a set of (super)mappings with a rigidly fixed superdomain. n the broader context of nonlinear realisations of (super)symmetries that had been investigated froma variety of angles in Refs. [Sch67, Wei68, CWZ69, CCWZ69, SS69a, SS69b, ISS71, VA72, VA73,IK78, LR79, UZ82, IK82, SW83, FMW83, BW84]. The general conditions to be satisfied by thetriple ( G , H , H vac ) for the duality to obtain were identified by the Author in Ref. [Sus19, Thms. 5.1& 5.2], and the dual super- σ -model was given the name of the Hughes–Polchinski formulation of theGreen–Schwarz super- σ -model. Its manifestly topological nature opens an avenue for geometrisationof the entire content of the field theory, that is its field equations, states and symmetries, and for are-interpretation – intuited from the study of well-known examples of topological gauge field theories(such as, e.g. , the three-dimensional Chern–Simons theory) – of a distinguished class of its geometrisedgauge supersymmetries in terms of the local (super)geometry of the critically embedded worldvolume,or the vacuum of the theory. The obvious expectation is that this class would contain the Hughes–Polchinski dual of the ‘odd’ κ -symmetry of the Nambu–Goto formulation, and that in a distinguishedrˆole. Furthermore, with the higher-geometric object for the complete dual Dirac–Feynman amplitude inhand, originally constructed in Ref. [Sus19, Sec. 6.2] and dubbed the extended Hughes–Polchinski (HP) p -gerbe, the possibility arises to straightforwardly verify the existence of a lift of these distinguishedgauge supersymmetries to an equivariant structure of sorts on the extended HP p -gerbe. A rigorousconcretisation of the general ideas articulated above has been the prime objective of the work reportedin the present paper.The paper is organised as follows. ● In Section 2, we review the supergeometry of a homogeneous space of a Lie supergroup in theconvenient language of super-Harish–Chandra pairs. In particular, we give a precise descriptionof a decomposition of the supersymmetry algebra g required by the dual formulations ofthe super- σ -model of interest and subsequently describe in great detail the construction ofa (complete) family of local trivialisations (2.15) and (2.16) of the principal superfibrationsG Ð→ G / K with the Graßmann-even structure group K ∈ { H , H vac } associated with thisdecomposition and employed in the explicit statement of the duality. We also indicate thoseelements of the (super-)Cartan calculus on G that descend to the homogeneous space, andso can be used in the construction of a field theory with G / K as the fibre of the covariantconfiguration bundle. ● In Section 3, we define the field theory of interest in its both formulations alluded to aboveand state the duality between them in what constitutes an extension, given in Thm. 3.4, of theformer result of Ref. [Sus19, Thms. 5.1 & 5.2] to a larger class of super- σ -models. Upon giv-ing the definition (3.11) of a supermanifold Σ HP for the topological Hughes–Polchinski (HP)formulation (determined by a collection sB ( HP ) p,λ p of supergeometric data termed the HP su-perbackground) on which all our subsequent supergeometric and field-theoretic analysis takesplace, and a convenient description of its tangent sheaf T Σ HP in Prop. 3.6, we then pro-vide a differential-geometric interpretation of the duality in terms of a sub-superdistributionCorr HP / NG ( sB ( HP ) p,λ p ) ⊂ T Σ HP introduced in Def. 3.10. The Section concludes with a detailedexposition, in Examples 3.11-3.16, of the relevant supergeometric structure in the existingsuper- σ -models of the type considered. ● In Section 4, the Euler–Lagrange equations of the Green–Schwarz super- σ -model for the super- p -brane in the HP formulation are derived under some natural assumptions, later shown to besatisfied in all the examples reviewed previously. Among the conditions, we find the physicallymotivated requirement of existence of the ‘reduction’ mechanism in the vacuum by means ofan ‘odd’ gauge symmetry, mentioned above. The description of the classical vacuum of thefield theory thus obtained, and summarised in Prop. 4.2, is subsequently reformulated in termsof a superdistribution Vac HP / NG ( sB ( HP ) p,λ p ) ⊂ Corr HP / NG ( sB ( HP ) p,λ p ) , introduced as the vacuumsuperdistribution in Def. 4.3, to which the tangents of the critical embeddings are restricted bythe field equations of the theory. Various symmetry and equivariance properties of this explicit geometrisation of the field equations of the super- σ -model are considered, and the conditionsof integrability of the vacuum superdistribution into what we are right to call, in Def. 4.6, thevacuum foliation of the supertarget are established, in a purely superalgebraic language, inProp. 4.7. The abstract considerations are, once again, illustrated on the physical Examples4.10-4.18. In Section 5, a complete geometrisation of the supersymmetries – both global and local – ofthe super- σ -model in the HP formulation is worked out. A universal phenomenon of gauge-symmetry enhancement is discovered for the field configurations, distinguished by the HP/NGduality, with tangents in Corr HP / NG ( sB ( HP ) p,λ p ) , leading to the definition of an enhanced gauge-symmetry superdistribution G S( sB ( HP ) p,λ p ) ⊂ Corr HP / NG ( sB ( HP ) p,λ p ) that realises the gauge sym-metries and its odd-generated sub-superdistribution aligned with the vacuum superdistribu-tion – the κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) ⊂ Vac HP / NG ( sB ( HP ) p,λ p ) . The latter is the sought-after geometrisation of κ -symmetry . Its explicit description formulated inDef. 5.6 (on the basis of a field-theoretic calculation), in conjunction with the definition of a dis-tinguished global-supersymmetry subspace S HPG ⊂ Γ (T Σ HP ) induced by the global symmetriesand its restriction S HP , vacG ⊂ Γ ( Vac ( sB ( HP ) p,λ p )) to the vacuum foliation described in Prop. 5.5under the name of the residual global-supersymmetry subspace and modelled on a Lie super-algebra s vac , are the concrete substance of the said geometrisation. The various componentsthereof are subsequently put in physically meaningful relations with one another and with theformerly estalished elements of geometrisation of the super- σ -model, whereby conditions forthe global-supersymmetry invariance of the vacuum superdistribution and of its κ -symmetrysub-superdistribution are determined in Props. 5.4 and 5.8, respectively. The ‘odd’ gauge su-persymmetry of the super- σ -model is definitively demystified in Prop. 5.9 which identifies the κ -symmetry superdistribution as a universal odd(-generated) component of the superdistri-bution of enhanced gauge symmetries of the super- σ -model which in the physically favouredcircumstances of integrability (and then automatically global supersymmetry) of the vacuumsuperdistribution generates the latter – through the so-called weak derived flag with the limit κ −∞ ( sB ( HP ) p,λ p ) ≡ Vac HP / NG ( sB ( HP ) p,λ p ) modelled on a Lie su-superalgebra gs loc ( sB ( HP ) p,λ p ) ⊂ g ofvacuum-generating gauge supersymmetries, or the κ -symmetry superalgebra – and so en-velops the vacuum foliation of the supertarget, which wins it its name: the square root ofthe vacuum featuring in the title of the present paper. The various findings of the Section ofphysical relevance are summarised in the fundamental Thm. 5.10. The Section closes with phys-ical Examples 5.11-5.19. The identification stated above rests upon a number of assumptionsconcerning the superbackground, listed explicitely in this section and verified in the examples.An important scenario in which these assumptions are not satisfied – the only such case amongthe concrete physical systems explored – in scrutinised in Example 5.20. ● In Section 6, we recall the essential aspects of the gerbe-theoretic approach to σ -models andrecapitulate the geometrisation scheme of Refs. [Sus17, Sus19, Sus18a, Sus18b] for the super- σ -models under study, abstracting, along the way, several concrete definitions from the generalconsiderations presented in the original papers. In the exposition, special emphasis is laidon the higher-geometric realisation of σ -model symmetries in both instantiations: families of p -gerbe (1-)isomorphisms labelled by the global-symmetry group for global symmetries andequivariant structures on the p -gerbe over nerves of the action groupoid of the symmetry groupon the base of the p -gerbe for gauge symmetries. ● In Section 7, we take the first step on the path towards a full-fledged gerbification of thevacuum-generating κ -symmetry, the latter being understood as the involutive superdistribution κ −∞ ( sB ( HP ) p,λ p ) modelled on the Lie superalgebra gs loc ( sB ( HP ) p,λ p ) , consistent with the residualglobal supersymmetry of the vacuum realised by the residual global-supersymmetry subspace S HP , vacG . Thus, upon restricting the extended HP p -gerbe of the super- σ -model to the vacuumfoliation Σ HPvac of the HP section Σ HP , we equip the latter with a linearised κ -symmetry-equivariant structure, linearised-invariant with respect to the residual global supersymmetry ofthe vacuum. The definition of the latter object is derived through a cohomologically consistentlinearisation (in the vicinity of the group unit of the local-symmetry group) of a standardequivariant structure on a p -gerbe in the Graßmann-eve setting, and an analogous linearisation(in the vicinity of the group unit of the global-symmetry group) of a group-invariant structureon that equivariant structure. Our considerations lead to the statement of existence of a canonical gs loc ( sB ( HP ) p,λ p ) -equivariant structure on the vacuum restriction of the extended HP p -gerbe in Prop. 7.7, further proven canonically s vac -invariant in Prop. 7.11. The results ofimmediate physical relevance are summarised in the fundamental Thm. 7.13. ● In Section 8, we give a concise summary of our findings and indicate directions of future researchopened and motivated by them. In the Appendix, we provide an overview of the differential-geometric and superalgebraic struc-ture behind the super- σ -model for the Zhou super-0-brane in s ( AdS × S ) that violates someof the previous assumptions made with regard to the super- σ -model data, and yet realises inan interestingly extended form the geometrisation scenario of delineated in Sections 4, 5 and7. The Appendix is to be regarded as an appetiser for a future investigation of a wider classof super- σ -models in the topological HP formulation in the spirit of the work reported herein.2. Supergeometry `a la Cartan
In the present paper, we aim to study distinguished lagrangean field theories that model simplegeometric dynamics of charged extended objects of superstring theory, termed super- p -branes. Thefield theories of interest exhibit supersymmetry in a nonlinear realisation, and so it seems only naturalto begin our discussion with a word on the supermanifolds of relevance to the construction of thecorresponding configuration bundles in the adapted Cartan framework. For that purpose, we introducea Lie supergroup G that will play the rˆole of the supersymmetry group of the physical model. ALie supergroup can be understood abstractly (and naturally) as a group object in the category sMan of supermanifolds (cp Ref. [Sus17, Sec. 3]) with the body given by a Lie group ∣ G ∣ , and so, in particular,it comes equipped with a multiplication (supermanifold) morphism µ ∶ G × G Ð→ G with the sheafcomponent µ ∗ ∶ O G Ð→ O G × G ≅ O G ̂⊗ O G (the tensor product is a suitable completion of the standard(super)tensor product of sheaves) and a unit morphism e G ≡ ̂ e ∶ R ∣ Ð→ G defining the groupunit e ∈ ∣ G ∣ , one of the ∣ G ∣ -worth Hom sMan ( R ∣ , G ) ≡ {̂ g ∶ R ∣ Ð→ G } g ∈∣ G ∣ of topological points.There is an obvious notion of a Lie-supergroup morphism, and so there arises the category of Liesupergroups which we shall denote as sLieGrp .Among vector fields on the supermanifold G, i.e. , among global sections of the sheaf of superderiva-tions sDer O G ≡ T G of the structure sheaf, we have the left-invariant sections satisfying the condition ( id O G ⊗ L ) ○ µ ∗ = µ ∗ ○ L , and their right-invariant counterparts for which ( R ⊗ id O G ) ○ µ ∗ = µ ∗ ○ R .
The R -linear span of the former shall be denoted as Γ ( T G ) G . The Lie superbracket of vector fieldscloses on these sections and thus gives rise to the tangent Lie superalgebra sLie G of G,sLie G = ( Γ ( T G ) G , [ ⋅ , ⋅ }) , an object from the category sLieAlg of Lie superalgebras. We have a canonical isomorphism ofsupervector spaces L ⋅ ∶ T e G Ð→ Γ (T G ) G ∶ X z→ ( id O G ⊗ X ) ○ µ ∗ . Similarly, every right-invariant vector field can be obtained from a unique vector tangent to G at thetopological unit as R X = ( X ⊗ id O G ) ○ µ ∗ , X ∈ g . Following Kostant (cp Ref. [Kos77]), we shall, equivalently, think of a Lie supergroup as an objectdescribed in
Definition 2.1. A super-Harish–Chandra pair is a pair (∣ G ∣ , g ) composed of- a Lie group ∣ G ∣ with the tangent Lie algebra ∣ g ∣ ;- a Lie superalgebra g , to be termed the supersymmetry algebra , with the Graßmann-homogeneous components: g ( ) (even) and g ( ) (odd), g = g ( ) ⊕ g ( ) such that(sHCp1) g ( ) ≡ ∣ g ∣ ;(sHCp2) there exists a realisation ρ ⋅ ∶ ∣ G ∣ Ð→ Aut sLieAlg g ∶ g z→ ρ g of ∣ G ∣ on g that extends its adjoint realisation on the tangent Lie algebra, ∀ g ∈∣ G ∣ ∶ ρ g ↾ g ( ) ≡ T e Ad g . sHCp morphism between super-Harish–Chandra pairs (∣ G A ∣ , g A ) , A ∈ { , } , with the respectiveunits e A and realisations ρ A ⋅ ∶ ∣ G A ∣ Ð→ Aut sLieAlg g A , is a pair ( Φ , φ ) ∶ (∣ G ∣ , g ) Ð→ (∣ G ∣ , g ) composed of- a Lie-group homomorphism Φ ∶ ∣ G ∣ Ð→ ∣ G ∣ ;- a Lie-superalgebra homomorphism φ ∶ g Ð→ g such that(sHCpm1) φ ↾ g ( ) ≡ T e Φ;(sHCpm2) ∀ g ∈∣ G ∣ ∶ φ ○ ρ g = ρ ( g ) ○ φ .Super-Harish–Chandra pairs together with the associated sHCp morphisms form the category ofsuper-Harish–Chandra pairs , to be denoted as sHCp . ◇ We have the fundamental
Theorem 2.2 (Kostant ’77) . There exists an equivalence of categories K ∶ sHCp ≅ ÐÐ→ sLieGrp . Remark 2.3.
Given the ubiquity of Kostant’s supergroups in field-theoretic models of systems withsupersymmetry, and – in particular – with view to our subsequent considerations, it will be appositeto explicit at least some of the structures that arise in the constructive proof of the above theorem. Inso doing, we adopt the conventions of Ref. [CCF11].Thus, in one direction, the equivalence K − assigns to a Lie supergroup G the super-Harish–Chandra pair (∣ G ∣ , sLie G ) composed of its body Lie group ∣ G ∣ and its tangent Lie superalgebra together with the mappingsAd ⋅ ∶ ∣ G ∣ Ð→ Aut sLieAlg ( sLie G ) given bysAd g ∶ sLie G ↺ ∶ L z→ c ∗ g − ○ L ○ c ∗ g , g ∈ ∣ G ∣ , where c ∗ g = ( ev g ⊗ id O G ⊗ ev g − ) ○ ( id O G ⊗ µ ∗ ) ○ µ ∗ is expressed in terms of the evaluation ev g of sections of O G at the topological point g , assigning toa section f the unique real number ev g ( f ) ≡ f ( g ) ∈ R such that the corrected section f − f ( g ) is notinvertible in any neighbourhood of g , cp Ref. [CCF11, Lem. 4.2.2].Going in the opposite direction is significantly more involved. Let (∣ G ∣ , g ) be a super-Harish–Chandra pair with the realisation ρ ⋅ ∶ ∣ G ∣ Ð→ Aut sLieAlg ( g ) , and denote by U ( g ) (resp. by U ( g ( ) ) )the universal enveloping algebra of the Lie superalgebra g (resp. of the Graßmann-even compo-nent g ( ) of g ), as defined in Ref. [Sch79, Chap. I § sAlg assoc in what follows.Furthermore, the Z / Z -graded vector space U ( g ) is equipped with a natural Hopf-superalgebra struc-ture, with the product m U ( g ) ≡ ⋅ U ( g ) ∶ U ( g ) ⊗ U ( g ) Ð→ U ( g ) descended from the (tensor) producton the tensor algebra of g , and the coproduct ∆ U ( g ) ∶ U ( g ) Ð→ U ( g ) ⊗ U ( g ) and the antipode S U ( g ) ∶ U ( g ) ↺ determined by the respective restrictions to the generators X ∈ g of U ( g ) (we areidentifying g with its image in U ( g ) under the canonical injection g ↪ U ( g ) ) and the algebra unit,∆ U ( g ) ( ) ∶ = ⊗ , ∆ U ( g ) ( X ) ∶ = X ⊗ + ⊗ X , S U ( g ) ( ) = , S U ( g ) ( X ) = − X , the former being a superalgebra homomorphism, and the latter – an even super-antimultiplicativemapping, i.e. , one satisfying, for any homogeneous u, v ∈ U ( g ) , the identity S U ( g ) ○ m U ( g ) ( u, v ) = (− ) ∣ u ∣⋅∣ v ∣ m U ( g ) ( S U ( g ) ( v ) , S U ( g ) ( u )) , in which ∣ u ∣ , ∣ v ∣ ∈ { , } are the Graßmann parities. To begin with, we identify two distinguished objectsin the object class of the category U ( g ( ) )− Mod of (left) U ( g ( ) ) -modules, to wit: the envelopingalgebra U ( g ) of g with the action engendered by m U ( g ) (and using the obvious embedding U ( g ( ) ) ↪ U ( g ) ), and, for any open set ∣ U ∣ ⊂ ∣ G ∣ , the space C ∞ (∣ U ∣ , R ) of smooth functions on ∣ U ∣ with the action nduced by the natural action of the left-invariant vector fields associated with elements of g ( ) . Next,we form the coinduced U ( g ) -module Hom U ( g ( ) )− Mod ( U ( g ) , C ∞ (∣ U ∣ , R )) = ∶ O G (∣ U ∣) of Ref. [Kos82, Sec. 1] with the (left) U ( g ) -actionU λ ⋅ ∶ U ( g ) × O G (∣ U ∣) Ð→ O G (∣ U ∣) which, for a homogeneous u ∈ U ( g ) , readsU λ u ∶ O G (∣ U ∣) ↺ ∶ χ z→ (− ) ∣ u ∣ χ ○ m U ( g ) (⋅ , u ) , (2.1)and use the coproduct on U ( g ) to induce on it the structure of a supercommutative superalgebra,m O G (∣U∣) ≡ ⋅ O G ∶ O G (∣ U ∣) ⊗ O G (∣ U ∣) Ð→ O G (∣ U ∣) , m O G (∣U∣) ( f ⊗ f ) ≡ f ⋅ O G f = m C ∞ (∣U∣ , R ) ○ ( f ⊗ f ) ○ ∆ U ( g ) , where m C ∞ (∣U∣ , R ) ≡ ⋅ C ∞ ∶ C ∞ (∣ U ∣ , R ) ⊗ C ∞ (∣ U ∣ , R ) Ð→ C ∞ (∣ U ∣ , R ) , m C ∞ (∣U∣ , R ) ( f ⊗ f ) ≡ f ⋅ C ∞ f ∶ ∣ U ∣ ∋ g z→ f ( g ) ⋅ f ( g ) ∈ R is the standard pointwise product of (smooth) functions. We then define the supermanifoldG ∶ = (∣ G ∣ , O G ) with the structure sheaf O G ≡ Hom U ( g ( ) )− Mod ( U ( g ) , C ∞ (⋅ , R )) ∶ T (∣ G ∣) Ð→ sAlg scomm ∶ ∣ U ∣ z→ O G (∣ U ∣) (here, T (∣ G ∣) is the category of open sets in ∣ G ∣ with inclusions as morphisms, and sAlg scomm isthe category of supercommutative superalgebras). The canonical (parity-factorised) structure of thelatter is determined by the isomorphismHom U ( g ( ) )− Mod ( U ( g ) , C ∞ (∣ U ∣ , R )) ≅ ÐÐ→ C ∞ (∣ U ∣ , R ) ⊗ ⋀ ● g ( ) ∗ ∶ f z→ f ○ γ (2.2)of Ref. [Kos82, Lem. 2], written in terms of the supercoalgebra homomorphism γ ∶ ⋀ ● g ( ) Ð→ U ( g ) (2.3)that R -linearly extends the assignment (here, X i ∈ g ( ) ⊂ U ( g ) , i ∈ , k ) γ ( X ∧ X ∧ ⋯ ∧ X k ) ∶= k ! ∑ σ ∈ S k sgn ( σ ) X σ ( ) ⋅ U ( g ) X σ ( ) ⋅ U ( g ) ⋯ ⋅ U ( g ) X σ ( k ) . Finally, we promote G to the rank of a Lie supergroup by defining the structure (supermanifold)morphisms µ ∶ G × G Ð→ G , Inv ∶ G ↺ and ε ∶ R ∣ Ð→ G with the obvious (Lie-group) bodycomponents and with the sheaf components µ ∗ ∶ O G Ð→ O G ̂⊗ O G , Inv ∗ ∶ O G ↺ , ε ∗ ∶ O G Ð→ R that evaluate on O G ∋ f as ( µ ∗ ( f )( u ⊗ v ))( g, h ) = ( f ○ m U ( g ) ( U ρ h − ( u ) , v ))( g ⋅ h ) , ( Inv ∗ ( f )( u ))( g ) = ( f ○ U ρ g ○ S U ( g ) ( u ))( g − ) ,ε ∗ ( f ) = ( f ( ))( e ) for u, v ∈ U ( g ) , g, h ∈ ∣ G ∣ , for e the group unit of ∣ G ∣ , and forU ρ ⋅ ∶ ∣ G ∣ Ð→ Aut sAlg assoc ( U ( g )) the unique extension of ρ ⋅ to the universal enveloping algebra of g . These were called produced representations in Ref. [Sch79]. emark 2.4. For the better part of our discussion, we shall need a geometric perspective on theformal concepts and constructions that we stumble upon in our considerations, on which we shall baseour intuitions. Such a perspective is provided by a collection of fundamental category-theoretic resultsthat introduce a familiar structure into the supergeometric universe.The first of these results is
Theorem 2.5 (Yoneda’s Lemma for sMan ) . There exists a fully faithful covariant functor, termedthe
Yoneda embedding , from the category sMan into the category of presheaves Presh ( sMan ) on the latter category, Yon ⋅ ∶ sMan ↪ Presh ( sMan ) , with the object component Yon M ∶ = Hom sMan (⋅ , M ) , M ∈ Ob sMan and the morphism component Yon φ ∶ = Hom sMan (⋅ , φ ) ≡ φ ○ , φ ∈ Hom sMan ( M , M ) . The Lemma enables us to replace the study of a supermanifold M with the study of the Ob sMan -indexed family of sets { Yon M ( S )} S ∈ Ob sMan of the so-called S -points of M . Note that the topologicalpoints in M are among the S -points for any supermanifold S . Indeed, point x ∈ ∣ M ∣ can be identifiedwith a unique morphism ̂ x ∶ R ∣ Ð→ M from the terminal supermanifold R ∣ into M , and theexistence of ̂ x implies the existence of the canonical morphism S ⇢ R ∣ ̂ x ÐÐ→ M . In the case of aLie supergroup M ≡ G ∈ Ob sLieGrp , every set Yon G ( S ) of S -points of G is actually a group , andmorphisms between Lie supergroups are mapped to group homomorphisms.The next important result is Theorem 2.6.
There exists a fully faithful contravariant functor from the category sMan of super-manifolds into the category sAlg scomm , A ⋅ ∶ sMan ↪ sAlg scomm , with the object component A M ∶ = O M (∣ M ∣) , M ∈ Ob sMan and the morphism component A (∣ φ ∣ ,φ ∗ ) ∶ = φ ∗∣ M ∣ ∶ O M (∣ M ∣) Ð→ O M (∣ M ∣) , (∣ φ ∣ , φ ∗ ) ∈ Hom sMan ( M , M ) . The theorem reduces the analysis on supermanifolds to the analysis of the corresponding superalgebrasof global sections of their structure sheaves and superalgebra homomorphisms between them. Further-more, it yields (in conjunction with the Hadamard Lemma) a natural description of the set Yon R p ∣ q ( S ) of S -points in the model supermanifold R p ∣ q , and so – when combined with the previous result – pavesthe way to a ‘functional’ formulation of the (local) differential calculus on supermanifolds.The way goes through yet another fundamental Theorem 2.7 (The Local Chart Theorem) . There is a bijection between – on the one hand – superman-ifold morphisms φ ∈ Hom sMan ( U , U ) between superdomains U A ≡ (∣ U A ∣ , C ∞ (⋅ , R ) ⊗ ⋀ ● R q A ) , ∣ U A ∣ ⊂ R p A , A ∈ { , } with the respective canonical coordinates { θ α A A , x a A A } ( α A ,a A )∈ ,q A × ,p A of Graßmann par-ities (∣ θ α A A ∣ , ∣ x a A A ∣) = ( , ) and – on the other hand – collections of global sections from O U (∣ U ∣) : q Graßmann-odd ones {̃ θ α } α ∈ ,q and p Graßmann-even ones {̃ x a } a ∈ ,p , satisfying the condition ∀ m ∈∣ U ∣ ∶ (̃ x ( m ) , ̃ x ( m ) , . . . , ̃ x p ( m )) ∈ ∣ U ∣ . It leads to a straightforward extension of the standard local description of a manifold in terms of localcoordinates and that of mappings between manifolds expressed in terms of such coordinates on thedomain and codomain of the mapping, the extension in question consisting in the incorporation ofGraßmann-odd coordinates into the analysis on par with the standard Graßmann-even ones. Thus, amorphism φ ≡ (∣ φ ∣ , φ ∗ ) ∶ M Ð→ M between supermanifolds M A , A ∈ { , } of the respective superdimensions ( p A ∣ q A ) , with a restriction φ ↾ U ∶ U Ð→ U , U A ≡ (∣ U A ∣ , O M A ↾ ∣ U A ∣ ) , A ∈ { , } This is the category of contravariant functors from sMan to Set , with natural transformations as morphisms. o domains U A of the respective local coordinate charts κ A ∶ U A ≅ ÐÐ→ (∣ W A ∣ , C ∞ (⋅ , R ) ⊗ ⋀ ● R × q A ≡ O W A ) ≡ W A , ∣ W A ∣ ⊂ R × p A , with the corresponding canonical coordinates { θ α A A , x a A A } ( α A ,a A )∈ ,q A × ,p A , acquires a coordinate pre-sentation in the form of a collection of p + q mappings { φ ∗ ( θ α ) , φ ∗ ( x a )} ( α ,a )∈ ,q × ,p of the respective parities ∣ φ ∗ ( θ α )∣ = ∣ φ ∗ ( x a )∣ =
0, determined by the composite morphism φ ∶ = κ ○ φ ○ κ − . As sections of O W , the φ ∗ ( θ α ) and φ ∗ ( x a ) admit coordinate presentations φ ∗ ( θ α ) = q ∑ k = θ α θ α ⋯ θ α k φ ( ) α α ...α k ( x a ) , φ ∗ ( x a ) = q ∑ l = θ α θ α ⋯ θ α l φ ( ) α α ...α l ( x b ) , with φ ( ) α α ...α l = = φ ( ) α α ...α l + for l ∈ N . This imitates a ‘mapping’ ( θ , x ) z→ ( θ ( θ , x ) , x ( θ , x )) ≡ ( φ ∗ ( θ ) , φ ∗ ( x )) of ‘points’ in the M A . The scheme induces a (local) mapping between S -pointsHom sMan ( S , U ) ∋ ψ Ð→ φ ○ ψ ∈ Hom sMan ( S , U ) , with a coordinate descriptionHom sMan ( S , W ) ∋ ψ ≡ κ ○ ψ Ð→ φ ○ ψ ∈ Hom sMan ( S , W ) fixed by the formulæ ( φ ○ ψ ) ∗ ( θ α ) = q ∑ k = ψ ∗ ( θ α ) ψ ∗ ( θ α ) ⋯ ψ ∗ ( θ α k ) φ ( ) α α ...α k ( ψ ∗ ( x a )) , ( φ ○ ψ )( x a ) = q ∑ l = ψ ∗ ( θ α ) ψ ∗ ( θ α ) ⋯ ψ ∗ ( θ α l ) φ ( ) α α ...α l ( ψ ∗ ( x b )) that correspond to a ‘mapping’ ( ψ ∗ ( θ ) , ψ ∗ ( x )) z→ (( φ ○ ψ ) ∗ ( θ ) , ( φ ○ ψ ) ∗ ( x )) . These considerations define the point of departure for a ‘standard’ differential calculus augmented withthe sign conventions of Ref. [Sus17, App. A].The supermanifolds appearing in the field-theoretic setting of immediate interest come each with adistinguished atlas of local coordinate charts modelled on (a subspace of) the tangent Lie superalgebra g = g ( ) ⊕ g ( ) of the supersymmetry supergroup, and so the above reasoning reduces our task to thedifferential calculus of (typically quite explicit) ‘functions’ of dim g ( ) Graßmann-even and dim g ( ) Graßmann-odd coordinates, with the relevant vector fields and differential forms locally spanned onthe corresponding coordinate derivations resp. differentials. Details follow below.Having introduced the supersymmetry group, we may, next, consider a Lie subgroup H ⊂ ∣ G ∣ , tobe interpreted as the (hidden) gauge group of the physical theory and termed the isotropy group ,with the tangent Lie algebra h ⊂ g ( ) , and its Lie subgroup H vac ⊂ H (with the tangent Lie algebra h vac ⊂ h ), to be termed the vacuum isotropy group , the latter modelling the gauge symmetriespreserved by the classical solution of the field theory, to be called the vacuum in what follows. Withthese ingredients in hand, we may finally indicate the supermanifolds of immediate interest to us –these are the homogeneous spaces M H ∶ = G / H and M H vac ∶ = G / H vac . As in the purely Graßmann-even setting, they can be regarded as bases of the respective principal(super)fibrations K / / G π G / K (cid:15) (cid:15) M K , K ∈ { H , H vac } (2.4) ith the structure group K (with the tangent Lie algebra k ∈ { h , h vac } ), cp Refs. [Kos77, Kos82] (butsee also, e.g. , Ref. [CCF11] for a modern perspective). This paves the way to the (redundant) realisationof the homogeneous spaces as collections of local (super)sections σ K i ∶ U K i Ð→ G , π G / K ○ σ K i = id U K i , i ∈ I K of the surjective submersion π G / K associated with a covering { U K i } i ∈ I K ≡ U K of the base M K by opensuperdomains. The latter are supermanifolds of the form U K i = (∣ U K i ∣ , O M K ↾ ∣ U K i ∣ ) with {∣ U K i ∣} i ∈ I K ≡ ∣ U K ∣ a trivialising (open) cover of the body principal K-bundleK / / ∣ G ∣ π ∣ G ∣/ K ≡ ∣ π G / K ∣ (cid:15) (cid:15) ∣ M K ∣ ≡ ∣ G ∣/ K , K ∈ { H , H vac } , (2.5)and with O M K ∶ T (∣ M K ∣) Ð→ Alg scomm ∶ ∣ U ∣ z→ { f ∈ O G ( π − ∣ G ∣/ K (∣ U ∣)) ∣ ∀ J ∈ k ∶ L J ( f ) = ∧ ∀ k ∈ K ∶ r ∗ k ( f ) = f } ≡ O M K (∣ U ∣) the structure sheaf of the homogeneous space M K , introduced in Ref. [FLV07, Sec. III.A] ( cp alsoRef. [CCF11, Sec. 9.3]) and determined in terms of the left-invariant vector fields L J on G associatedwith vectors of T e G ≡ g ⊃ k ∋ J as per L J ≡ ( id O G ⊗ J ) ○ µ ∗ and of the sheaf component of the naturalright action r g ≡ µ ○ ( id G × ̂ g ) ∶ G × R ∣ ≅ G Ð→ G , g ∈ ∣ G ∣ of ∣ G ∣ ⊃ K on G that descends to the usual right regular action of the body group on itself, ∣ r ⋅ ∣ ≡ ∣ r ∣ ⋅ ∶ ∣ G ∣ × ∣ G ∣ Ð→ ∣ G ∣ ∶ ( g, f ) z→ g ⋅ f ≡ ∣ µ ∣( g, f ) ≡ ∣ r f ∣( g ) ≡ ∣ r ∣ f ( g ) , and admits an explicit realisation in Kostant’s model (locally) given by r ∗ g ∶ O G (∣ U ∣) ↺ ∶ f z→ ∣ r ∣ ∗ g ○ f ○ U ρ g − , The first indication of the existence of such sections appeared already in Ref. [Kos77, Prop. 3.9.2]. Amore tractable construction, which we are about to recapitulate and elaborate below, was given inRef. [FLV07, Sec. III.A]. It is along these sections, in the S -point picture advocated before, that wepull back suitable K-basic covariant tensor fields from the supergroup supermanifold G to M K andultimately employ them, in a manner reviewed at length in Ref. [Sus19, Sec. 5], in the construction of alagrangean model of the dynamics of interest. While the models that we have in mind do not dependon the choice of the sections, establishing a correspondence between them that will be instrumental inour considerations calls for a judicious choice thereof in which that correspondence is particularly easyto write out. Towards this end, we take a closer look at and make – with hindsight – certain furtherassumptions with regard to the decomposition of the supersymmetry algebra g in the presence of itsdistinguished Lie subalgebras h and h vac ⊂ h . Thus, we write g = t ⊕ h , where the supervector space t = t ( ) ⊕ t ( ) , a direct-sum complement of the isotropy algebra h within g , is assumed to be an h -module, [ h , t ] ⊂ t , which qualifies the decomposition as reductive . The isotropy algebra decomposes further as h = d ⊕ h vac (2.6)into the vacuum isotropy algebra h vac and its direct-sum complement d . The former is determinedas the isotropy subalgebra within the Lie algebra h of the Graßmann-even component of a (physically)distinguished vacuum subspace t vac = t ( ) vac ⊕ t ( ) vac ⊂ t n the h -module t , [ h vac , t ( ) vac ] ⊂ t ( ) vac . We write the corresponding direct-sum complements within t as t = e ⊕ t vac , e = e ( ) ⊕ e ( ) . The decompositions t ( ) = e ( ) ⊕ t ( ) vac , t ( ) = e ( ) ⊕ t ( ) vac correspond to a pair of projectors P ( A ) = P ( A ) ○ P ( A ) ∈ End t ( A ) , A ∈ { , } such that t ( ) vac ≡ Im P ( ) , t ( ) vac ≡ Im P ( ) . In what follows, we assume both projections to be nontrivial, i.e. , t ( ) vac ⊊ t ( ) , t ( ) vac ⊊ t ( ) . Assuming d to be an h vac -module, [ h vac , d ] ⊂ d , as well as independent preservation of e ( ) by h vac , [ h vac , e ( ) ] ⊂ e ( ) , we obtain another reductive decomposition of the supersymmetry algebra, to wit, g = f ⊕ h vac , f = t ⊕ d . With view to subsequent field-theoretic applications, and in particular to the analysis of a correspon-dence between various formulations of the field theory of interest, we augment the above and impose
The Even Effective-Mixing Constraints:
We assume the vacuum isotropy algebra h vac to pre-serve the decomposition of t ( ) into the Graßmann-even vacuum subspace t ( ) vac and its direct-sumcomplement e ( ) , [ h vac , t ( ) vac ] ! ⊂ t ( ) vac , [ h vac , e ( ) ] ! ⊂ e ( ) , (2.7)and its direct-sum complement d in the isotropy algebra h to ad-rotate the two subspaces into oneanother, [ d , t ( ) vac ] ! ⊂ e ( ) , [ d , e ( ) ] ! ⊂ t ( ) vac . (2.8) Remark 2.8.
As a consistency condition, implied by the Jacobi identity for triples from d × d × t ( ) vac and d × d × e ( ) , we derive from the above the additional constraints: [ d , d ] ⊂ h vac . (2.9)For the sake of later bookkeeping, we set ( D, δ, δ, d, p, q ) ∶ = ( dim g − , dim t − , dim f − , dim t ( ) − , dim t ( ) vac − , dim t ( ) vac ) and denote the respective homogeneous basis vectors (generators) of the various subalgebras and sub-spaces as g = ⊕ DA = ⟨ t A ⟩ , h = ⊕ D − δS = ⟨ J S ⟩ , , h vac = ⊕ D − δS = ⟨ J S ⟩ , d = ⊕ D − δ ̂ S = D − δ + ⟨ J ̂ S ⟩ , f = ⊕ δµ = ⟨ t µ ⟩ , t = ⊕ δA = ⟨ t A ⟩ , t vac = ⊕ p + qA = ⟨ t A ⟩ , t ( ) = ⊕ da = ⟨ P a ⟩ , t ( ) = ⊕ δ − dα = ⟨ Q α ⟩ , (2.10) t ( ) vac = ⊕ pa = ⟨ P a ⟩ , t ( ) vac = ⊕ qα = ⟨ Q α ⟩ , e ( ) = ⊕ d ̂ a = p + ⟨ P ̂ a ⟩ , e ( ) = ⊕ δ − d ̂ α = q + ⟨ Q ̂ α ⟩ . e also introduce the structure constants [ t A , t B } = f CAB t C of g in the above basis, with the obvious symmetries: f CBA = ( − ) ∣ A ∣⋅∣ B ∣+ f CAB , written in terms of the Graßmann parities ∣ A ∣ ≡ ∣ t A ∣ and ∣ B ∣ ≡ ∣ t B ∣ of the homogeneous generators t A .Finally, we write, for any g ∈ ∣ G ∣ , ρ g ( Q α ) = S ( g ) βα Q β . We may, now, use the above algebra to give a very natural and convenient definition of local sectionsof the two principal bundles (2.4). Thus, first of all, we pick up an open neighbourhood ∣ U e ∣ ⊂ ∣ G ∣ ofthe group unit e ∈ ∣ G ∣ sufficiently small to support local coordinates { χ A } A ∈ ,D ≡ { χ a ≡ x a } a ∈ ,d ∪ { χ α ≡ θ α } α ∈ ,δ − d ∪ { χ ̂ S ≡ φ ̂ S } ̂ S ∈ D − δ + ,D − δ ∪ { χ T ≡ ψ T } T ∈ ,D − δ in which the involutive (super)distribution D h vac generated by the left-invariant vector fields L J T ≡ L T , T ∈ , D − δ is spanned by the coordinate derivations { ∂∂ψ T } T ∈ ,D − δ and the Graßmann-even coor-dinates { x a , φ ̂ S } ( a, ̂ S ) ∈ ,d × D − δ + ,D − δ chart ∣ U e ∣ faithfully, that is each (vector) value of the coordinatesfrom a parameter domain ∣ W e ∣ ⊂ R × d + × R × δ − δ corresponds to a different H vac -coset. The existence ofsuch a neighbourhood follows directly from the supergeometric variant of The Local Frobenius Theo-rem [FLV07, Thm. A.9] ( cp also Ref. [CCF11, Thms. 6.1.12]). Consider the sub-supermanifold V e of Gdefined as the common zero locus of the coordinates ψ T , that is the supermanifold with the body givenby the corresponding submanifold ∣ V e ∣ ⊂ ∣ U e ∣ of the body Lie group ∣ G ∣ and the structure sheaf inducedfrom O G ↾ ∣ U e ∣ by setting all the ψ T to zero. The sub-supermanifold is modelled on the superdomain W e ∶ = (∣ W e ∣ , C ∞ ( ⋅ , R ) ⊗ ⋀ ● R × δ − d ≡ O W e ) , (2.11)that is there exists a superdiffeomorphism ̟ e ∶ W e ≅ ÐÐ→ V e . Its body component ∣ ̟ e ∣ ∶ ∣ W e ∣ ≅ ÐÐ→ ∣ V e ∣ may be chosen in the form of a smooth section of the bodyprojection ∣ π G / H vac ∣ , ∣ π G / H vac ∣ ○ ∣ ̟ e ∣ = id ∣ W e ∣ . It is then particularly straightforward to prove, as was done in Ref. [FLV07, Sec. III.A], the existenceof a superdiffeomorphism ξ e ≡ ( id ∣ W e ∣ , ξ ∗ e ) ∶ U H vac ≡ (∣ W e ∣ , O G / H vac ↾ ∣ W e ∣ ) ≅ ÐÐ→ W e , which identifies W e as a model of the neighbourhood U H vac of the unital coset H vac ≡ e ⋅ H vac inG / H vac (a right H vac -invariant section of O G over ∣ W e ∣ is determined uniquely by its evaluation atthe section ∣ V e ∣ ). Central to the proof is the superdiffeomorphic nature of the supermanifold morphism γ e ≡ µ ○ ( V e × H vac ) ∶ V e × H vac ≅ ÐÐ→ (∣ V e ∣ ⋅ H vac , O G ↾ ∣ V e ∣⋅ H vac ) ≡ π − / H vac ( U H vac ) ⊂ Gwritten in terms of the canonical superembeddings X ∶ X ↪ G , X ∈ { H vac , V e } . The latter defines (the inverse of) a local trivialisation of the principal H vac -bundle (2.4) over U H vac as per τ − ≡ γ e ○ ( ̟ e ○ ξ e × id H vac ) ∶ U H vac × H vac ≅ ÐÐ→ π − / H vac ( U H vac ) ⊂ G , and so also a local section of the principal H vac -bundle, σ H vac ≡ τ − ○ ( id U Hvac0 × ̂ e ) ∶ U H vac × R ∣ ≅ U H vac Ð→ G , cp Ref. [FLV07, Lem. 3.2]. It is to be noted that the entire non-canonical information on the latteris encoded in the superparametrisation ̟ e . Below, we present one such superparametrisation thatis particulary suited to subsequent physical considerations. It was introduced operationally in the unctor-of-point picture in Refs. [Sus17, Sus19] and termed the exponential superparametrisation ibidem . It was subsequently employed in concrete calculations in Refs. [Sus18a, Sus18b].The point of departure of the explicit construction of a local section of the principal H vac -bundle(2.4) over U H vac is a local section of the body fibration (2.5) of the standard form ∣ σ H vac ∣ ∶ ∣ W e ∣ Ð→ ∣ G ∣ ∶ ( x a , φ ̂ S ) z→ e x a P a ⋅ e φ ̂ S J ̂ S , with the exponential map defined, as usual, in terms of the (unital-time) flowse X ( g ) ≡ Φ ∣ L ∣ X ( t = g ) of the point g ∈ ∣ G ∣ along the integral lines of the left-invariant vector fields ∣ L ∣ X ∈ Γ ( T ∣ G ∣) associatedwith the respective elements X ∈ g ( ) , so that ∣ σ H vac ∣( x a , ψ ̂ S ) ≡ Φ ∣ L ∣ φ ̂ SJ ̂ S (
1; Φ ∣ L ∣ xaPa ( e )) . We may, next, extend the above body section to a full-blown mapping between the correspondingsupermanifolds upon – by a mild abuse of the notation – replacing the points ( x a , φ ̂ S ) ∈ ∣ W e ∣ withcoordinate functions on ∣ W e ∣ denoted by the same symbols,e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S ≡ (∣ σ H vac ∣( ⋅ ) ≡ e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S , ∣ σ H vac ∣ ∗ ) ∶ (∣ W e ∣ , C ∞ ( ⋅ , R )) Ð→ (∣ G ∣ , C ∞ ( ⋅ , R )) , where – as the notation suggests – for any f ∈ C ∞ (∣ U ∣ ; R ) on ∣ U ∣ ⊂ ∣ σ vac0 ∣(∣ W e ∣) , ∣ σ H vac ∣ ∗ f ≡ f ○ ∣ σ H vac ∣ . In order to make direct contact with Kostant’s construction, it suffices to invoke isomorphism (2.2) for g ( ) ≡
0, whereby we obtain the identity (∣ G ∣ , C ∞ ( ⋅ , R )) ≅ (∣ G ∣ , Hom U ( g ( ) )− Mod ( U ( g ( ) ) , C ∞ ( ⋅ , R ))) . At this stage, it is fairly straightforward to write out a super-extension of the body morphism e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S . Indeed, as long as we work with the local models (2.2) and (2.11), all we need is an isomor-phism ⋀ ● g ( ) ∗ ≅ ⋀ ● R × δ − d for g ( ) ≡ ⊕ δ − dα = ⟨ Q α ⟩ that yields a superalgebra homomorphism whentensored with the body morphism ∣ σ H vac ∣ ∗ . This we choose, in keeping with the original considerationsin Refs. [Sus17, Sus19, Sus18a, Sus18b] and the physics literature , in the distinguished forme θ α ⊗ Q α ∶ ⋀ ● g ( ) ∗ ≅ ÐÐ→ ⋀ ● R × δ − d ∶ λ + δ − d ∑ k = λ β β ...β k q β ∧ q β ∧ ⋯ ∧ q β k z→ λ + δ − d ∑ k = (− ) k ( k + ) λ β β ...β k θ β θ β ⋯ θ β k , expressed in terms of the dual basis { q α } α ∈ ,δ − d of g ( ) ∗ , q α ( Q β ) = δ α β , (2.12)and with the sums taken over sequences of spinor indices ordered as 1 ≤ β < β < . . . < β l ≤ δ − d .In order to explain the somewhat non-obvious signs appearing in the above formula and justify thesuggestive notation e θ α ⊗ Q α , we need to pull back the above isomorphism to Kostant’s structure sheaf.Below, we do that for the composite mapping ∣ σ H vac ∣ ∗ ⊗ e θ α ⊗ Q α . To this end, we employ (2.2), withthe explicit definition (2.3) of γ , alongside the evaluation map ev g ∶ O G (∣ U ∣) Ð→ R ∶ f z→ f ( )( g ) . After some trivial manipulations, we obtain a mappinge θ α ⊗ Q α ⋅ e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S ≡ ̟ e = (∣ σ H vac ∣ , ̟ ∗ e ) ∶ W e Ð→ Gwith the sheaf component ̟ ∗ e ∶ = ( id O W e ⊗ ev e ) ○ ( δ − d ∑ k = k ! ( θ α ⊗ Q α ) ○ ( θ α ⊗ Q α ) ○ ⋯ ○ ( θ α k ⊗ Q α k )) ○ r ∗ e xa (⋅)⊗ Pa ○ r ∗ e ψ ̂ S (⋅)⊗ J ̂ S . Note its structural relation with the normal-coordinate description of the Lie supergroup (with a connected body)due to Berezin and Kaˇc given in Ref. [BK70]. The status of the exponential superparametrisation is very rarely discussed rigorously in the physics (mainlysuperstring-theoretic) literature, which is where the specific sections considered herein are employed in the construc-tion of the relevant action functionals, and then it is usually related to Berezin’s concept of an exponential mapping. ere, it is understood that the above evaluates on an arbitrary f ∈ O G (∣ U ∣) , ∣ U ∣ ⊂ ∣ σ H vac ∣(∣ W e ∣) , withthe decomposition (in which the sum is taken over sequences of spinor indices ordered as 1 ≤ β < β < . . . < β l ≤ δ − d for which the corresponding l -forms q β ∧ q β ∧ ⋯ ∧ q β l are linearly independent) f ○ γ = δ − d ∑ l = f β β ...β l ⊗ q β ∧ q β ∧ ⋯ ∧ q β l as ̟ ∗ e ( f ) = δ − d ∑ k = (− ) k ( k − ) k ! θ α θ α ⋯ θ α k ( ⋅ ) U λ Q α ⋅ U ( g ) Q α ⋅ U ( g ) ⋯⋅ U ( g ) Q αk ( f ○ U ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S )( )( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) = δ − d ∑ k = (− ) k ( k + ) k ! θ α θ α ⋯ θ α k ( ⋅ ) f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ⋯ ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α k ))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) = δ − d ∑ k = ( − ) k ( k + ) θ β < θ β < ⋯ < θ β k ( ⋅ ) S ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) γ β S ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) γ β ⋯ S ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) γ k β k f ○ γ ( Q γ ∧ Q γ ∧ ⋯ ∧ Q γ k )( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) = δ − d ∑ k = ( − ) k ( k + ) θ β θ β ⋯ θ β k ( ⋅ ) S ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) γ β S ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) γ β ⋯ S ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) γ k β k f γ γ ...γ k ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) , which, incidentally, explains the signs introduced earlier. Given the last formula, it is easy to convinceoneself that ̟ ∗ e is (or, more precisely, gives rise to) a superalgebra homomorphism. Indeed, we have,for any f , f ∈ O G (∣ U ∣) , ̟ ∗ e ( f ⋅ O G f ) ≡ δ − d ∑ k = (− ) k ( k + ) k ! θ α θ α ⋯ θ α k ( ⋅ ) m C ∞ (∣ U ∣ , R ) (( f ⊗ f )( k ∏ l = ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l ) ⊗ + ⊗ ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l ))))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) , which – in consequence of the anti commutativity of the Graßmann-odd sections θ α and due to theimplication f ( u ) ≠ Ô⇒ ∣ f ∣ = ∣ u ∣ – can be rewritten in the form ̟ ∗ e ( f ⋅ O G f ) = δ − d ∑ k = (− ) k ( k + ) k ! θ α θ α ⋯ θ α k ( ⋅ ) m C ∞ (∣ U ∣ , R ) (( f ⊗ f )( k ∑ l = ( kl ) ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ⋯⋯ ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l ) ⊗ ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ⋯ ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α k )))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) = δ − d ∑ k = k ∑ l = (− ) k ( k + ) + l ∣ f ∣ l ! ( k − l ) ! θ α θ α ⋯ θ α l θ α l + θ α l + ⋯ θ α k ( ⋅ ) ( f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ⋯⋯ ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l )) ⋅ C ∞ f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ⋯ ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α k )))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) ≡ δ − d ∑ k = k ∑ l = (− ) k ( k + ) + l ( k − l ) l ! ( k − l ) ! θ α θ α ⋯ θ α l ( ⋅ ) f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ⋯⋯ ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l ))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) θ α l + θ α l + ⋯ θ α k f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ⋯ ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α k ))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) . However, k ( k + ) + l ( k − l ) = l ( l + ) + ( k − l )( k − l + ) + l ( k − l ) , and so upon invoking the trivial implication k > δ − d Ô⇒ ∏ ki = θ α i ≡
0, we finally obtain ̟ ∗ e ( f ⋅ O G f ) = δ − d ∑ k = k ∑ l = (− ) l ( l + ) l ! θ α θ α ⋯ θ α l ( ⋅ ) f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α ) ⋅ U ( g ) ⋯⋯ ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l ))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) ⋅ (− ) ( k − l )( k − l + ) ( k − l ) ! θ α l + θ α l + ⋯ θ α k f ( ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α l + ) ⋅ U ( g ) ⋯ ⋅ U ( g ) ρ e − xa (⋅) Pa ⋅ e φ ̂ S (⋅) J ̂ S ( Q α k ))( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) ≡ ̟ ∗ e ( f ) ⋅ ̟ ∗ e ( f ) , as claimed.In fact, a moment’s thought reveals that the Graßmann-odd component e θ α ⊗ Q α of ̟ e is a naturalsuper-completion of the standard exponential parametrisation of a neighbourhood of the Lie-group unit,taking into account the polynomial nature of the Graßmannian component ⋀ ● g ( ) ∗ of the structuresheaf, and that the notation e θ α ⊗ Q α ⋅ e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S used above acquires the same status as its classicalcounterpart e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S in the S -point picture to which we now pass. In particular, we have theexpected Proposition 2.9.
In the above notation, the supermanifold morphism e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S ⋅ e θ α ⊗ Q α ≡ ̃ ̟ e = (∣ σ H vac ∣ , ̃ ̟ ∗ e ) ∶ W e Ð→ G with the sheaf component ̃ ̟ ∗ e ∶ = ( id O W e ⊗ ev e ) ○ r ∗ e xa (⋅)⊗ Pa ○ r ∗ e ψ ̂ S (⋅)⊗ J ̂ S ○ ( δ − d ∑ k = k ! ( θ α ⊗ Q α ) ○ ( θ α ⊗ Q α ) ○ ⋯ ○ ( θ α k ⊗ Q α k )) satisfies the identity e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S ⋅ e θ α ⊗ Q α = e θ α ⊗ ρ e xa ⊗ Pa ⋅ e φ ̂ S ⊗ J ̂ S ( Q α ) ⋅ e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S . Proof.
Straightforward. (cid:3)
Accordingly, we shall – by a mild abuse of the notation in which the superdiffeomorphism ξ e isdropped and W e is taken as an element of the trivialising cover of the principal H vac -bundle (2.4) nearH vac – write σ H vac ( θ, x, φ ) ≡ e θ α ⊗ Q α ⋅ e x a ⊗ P a ⋅ e φ ̂ S ⊗ J ̂ S (2.13)and interpret this formula as an explicit coordinate description of a trivialising section of the principalH vac -bundle (2.4) over U vac0 .Next, we fix a collection { g i } i ∈ I Hvac , including g ≡ e ( i.e. , I H vac ∋ g i ∈ ∣ G ∣ with the property ⋃ i ∈ I Hvac [∣ l ∣] g i (∣ U H vac ∣) = ∣ M H vac ∣ , written in terms of the natural left action [∣ l ∣] ⋅ of the body group ∣ G ∣ on the body homogeneous space ∣ M H vac ∣ induced from the left regular action ∣ l ∣ ⋅ ∶ ∣ G ∣ × ∣ G ∣ Ð→ ∣ G ∣ ∶ ( f, g ) z→ f ⋅ g ≡ ∣ µ ∣( f, g ) ≡ ∣ l ∣ f ( g ) It suffices to consider Taylor’s expansion of the f β β ...β k ( e x a (⋅) P a ⋅ e φ ̂ S (⋅) J ̂ S ) around ( x, φ ) = ( , ) and invokeEq. (2.1). f that group on itself as [∣ l ∣] ⋅ ∶ ∣ G ∣ × ∣ M H vac ∣ Ð→ ∣ M H vac ∣ ∶ ( f, g H vac ) z→ ∣ l ∣ f ( g ) H vac . The supermanifolds U H vac i ∶ = (∣ U H vac i ∣ ≡ [∣ l ∣] g i (∣ U H vac ∣) , O M Hvac ↾ ∣ U Hvac i ∣ ) , i ∈ I H vac compose the desired trivialising cover for the fibration G Ð→ G / H vac . Indeed, we may now propagatethe previously constructed local section σ vac0 ≡ σ H vac ∶ U H vac Ð→ G , with the fundamental property π G / H vac ○ σ vac0 = id U Hvac0 , to all the remaining superdomains of the cover by means of the superdiffeomorphisms l g i ∶ = µ ○ (̂ g i × id G ) ∶ R ∣ × G ≅ G Ð→ G , i ∈ I H vac , lifting the formerly introduced body action ∣ l ∣ ⋅ of ∣ G ∣ from ∣ G ∣ to G, together with its quotientcounterpart [ l ] g i ∶ = [ ℓ ] H vac ⋅ ○ (̂ g i × id M Hvac ) ∶ R ∣ × M H vac ≅ M H vac Ð→ M H vac , the latter being defined in terms of the unique action of G on M H vac , [ ℓ ] H vac ⋅ ∶ G × M H vac Ð→ M H vac , that closes, for K = H vac , the commutative diagram (in sMan )G × G ℓ ⋅ ≡ µ / / id G × π G / K (cid:15) (cid:15) G π G / K (cid:15) (cid:15) G × M K [ ℓ ] K ⋅ / / M K . (2.14)Its existence was stated in Ref. [Kos77, Prop. 3.10.1] ( cp also Ref. [FLV07, Prop. 3.4 & 3.5] and Ref. [CCF11,Prop. 9.3.5]). With these in hand, we define σ H vac i ≡ σ vac i ∶ = l g i ○ σ vac0 ○ ([ l ] H vac g i ) − ∶ U H vac i Ð→ Gand readily check π G / H vac ○ σ vac i ≡ π G / H vac ○ l g i ○ σ vac0 ○ ([ l ] H vac g i ) − = [ l ] H vac g i ○ π G / H vac ○ σ vac0 ○ ([ l ] H vac g i ) − = id U Hvac i . In the S -point picture, and in the local coordinates ( χ µi ) ≡ ( θ αi , x ai , φ ̂ Si ) introduced earlier (assigned an extra index i ∈ I H vac to formally distinguish the various local incarna-tions of the same coordinate), we may, therefore, write σ vac i ( χ i ) = g i ⋅ e θ αi ⊗ Q α ⋅ e x a ⊗ P a ⋅ e φ ̂ Si ⊗ J ̂ S , i ∈ I H vac . (2.15)It is crucial to note that upon freezing the last subset of coordinates at φ ̂ Si = , ̂ S ∈ D − δ + , D − δ ,we induce local sections of the other principal bundle σ H i ≡ σ ✟✟❍❍ vac i ∶ = σ vac i ↾ φ i = ∶ π G / H ( σ vac i (U H vac i )) ≡ U H i Ð→ G , with the S -point presentation σ ✟✟❍❍ vac i ( θ i , x i ) ∶ = σ vac i ( θ i , x i , ) ≡ g i ⋅ e θ αi ⊗ Q α ⋅ e x a ⊗ P a , i ∈ I H vac (2.16)in terms of the local coordinates ( χ Ai ) ≡ ( θ αi , x ai ) This particular induction mechanism shall be encountered in the field-theoretic analysis to follow. emark 2.10. Our findings can be rephrased as follows: The canonical surjective submersion (2.4)defines a family of principal K-bundles indexed by Ob sMan , each coming with a preferred trivial-ising cover and the associated local trivialisations described in terms of the ( ∣ G ∣ -shifted) exponentialsuperparametrisations discussed above.We are now ready to study, in the convenient geometric S -point picture, the differential calculuson the homogeneous spaces M K , K ∈ { H , H vac } . Various physically relevant elements thereof can bedescended from the super-variant of the standard Cartan calculus on the mother supersymmetry groupG along the local sections σ H vac i (resp. σ H i ), whenever the pair ( G , K ) is reductive, as assumed above.In order to facilitate the discussion of the descent, we denote g = l ⊕ k , ( l , k ) ∈ {( t , h ) , ( f , h vac )} , l = ⊕ dim l − ζ = ⟨ T ζ ⟩ , k = ⊕ dim k Z = ⟨ J Z ⟩ and decompose the g -valued left-invariant Maurer–Cartan 1-form on G as θ L = θ ζ L ⊗ T ζ + θ Z L ⊗ J Z . whereupon we identify the k -valued 1-formΘ K ∶ = θ Z L ⊗ J Z as a principal connection 1-form on the total space G of the principal K-bundle (2.4), and the remainingcomponents as ρ ⋅ -tensors with respect to the defining fibrewise right action of the structure group Kon G, r K ⋅ ∶ G × K Ð→ G ∶ ( g, k ) z→ r K k ( g ) ≡ g ⋅ k , (2.17)that is we have r K ∗⋅ θ ζ L ( g, k ) = ( ρ k − ) ζ ζ ′ θ ζ ′ L ( g ) for ρ k ( T ζ ′ ) = ∶ ( ρ k ) ζ ζ ′ T ζ . These observations give us a definition of a horizontal lift of sections of the tangent sheaf (or vectorfields) over U K , Hor σ K i ( ⋅ ) ∶ T ⋅ U K i Ð→ T σ K i ( ⋅ ) G ∶ V ( ⋅ ) z→ ( id T G − ̂ Θ K )( σ K i ( ⋅ )) ○ T ⋅ σ K i ( V ) , (2.18)expressed in terms of the identity endomorphism id T G ≡ θ A L ⊗ L A , L A ≡ L t A and of the vertical-projectorfield ̂ Θ K ∶ = θ Z L ⊗ L Z , alongside a natural class of K-basic contravariant n -tensor fields on G that descend to the homogeneousspace M K , to wit, those given by ( R -)linear combinations ω ( n ) = ω ζ ζ ...ζ n θ ζ L ⊗ θ ζ L ⊗ ⋯ ⊗ θ ζ n L , ζ , ζ , . . . , ζ n ∈ , dim l − , of the K-horizontal left-invariant (component) 1-forms θ ζ L , ζ ∈ , dim l −
1, dual to the left-invariantvector fields L ζ ≡ L T ζ generated by vectors from l , L T ζ ⌟ θ ζ ′ L = δ ζ ′ ζ , with (constant) K-invariant tensors as coefficients, i.e. , with – for any k ∈ K – ω ζ ζ ...ζ n ( ρ k ) ζ ζ ′ ( ρ k ) ζ ζ ′ ⋯ ( ρ k ) ζ n ζ ′ n = ω ζ ′ ζ ′ ...ζ ′ n . Locally, the descent is effected by the sections σ H vac i , i ∈ I H vac , providing us – in the S -point picture –with the standard local Vielbeine, σ K ∗ i θ A L ( ξ i ) = ∶ d ξ ζi K E Aζ ( ξ i ) , A ∈ , D , (2.19)on U K i coordinatised by ξ i = χ i (for K = H vac ) resp. ξ i = χ i (for K = H). In what follows, we denotethe Vielbeine H vac E Aµ relevant to one of the formulations of the field theory of interest as H vac E Aµ ≡ E Aµ (2.20) o unclutter the notation. The ensuing n -tensor fields on the homogeneous space M K do not dependon the choice of the local section along which we pull them back to it, and hence glue smoothly overnon-empty intersections U K ij to globally smooth tensor fields on the homogeneous space. As such, theybecome natural building blocks of a field theory with the typical fibre of the configuration bundle givenby G / K.We close this differential-(super)geometric intermezzo by commenting on the structure of the tangentsheaf
T M K of M K . Among its global sections, we find the distinguished vector fields K [ ℓ ] K ⋅ X ∶ = ( X ⊗ id O M K ) ○ ([ ℓ ] K ⋅ ) ∗ , X ∈ T e G . This is a special instance of the general situation in which an action λ ⋅ ≡ ( ∣ λ ∣ ⋅ , λ ∗⋅ ) ∶ G × M Ð→ M of a Lie supergroup G on a supermanifold M gives rise to the fundamental vector fields K λ ⋅ X ∶ = ( X ⊗ id O M ) ○ λ ∗⋅ , X ∈ T e G , (2.21)with the property Proposition 2.11. [CCF11, Thm. 8.2.3]
In the hitherto notation, the mapping K λ ⋅ ⋅ ∶ g Ð→ Γ ( T M ) ∶ L X z→ K λ ⋅ X is an antimorphism of Lie superalgebras, satisfying the identity λ ∗⋅ ○ K λ ⋅ X = ( R X ⊗ id O M ) ○ λ ∗⋅ . An obvious example of a supermanifold with an action of a Lie supergroup is G itself – we have theleft regular action ℓ ⋅ ≡ µ ∶ G × G Ð→ Gand its right regular counterpart ℘ ⋅ ≡ µ ○ τ ∶ G × G Ð→ G , where τ ∶ G × G ↺ is the standard transposition. In these cases, we have the intuitive results K ℓ ⋅ X ≡ R X , K ℘ ⋅ X ≡ L X . (2.22)The important peculiarity of the homogeneous space M K is that the fundamental vector fields K [ ℓ ] K ⋅ X , X ∈ T e G actually span the tangent sheaf.3.
The two faces of the physical model
Our hitherto considerations provide us with all the conceptual and computational tools requisitefor the formulation and canonical analysis of two classes of supersymmetric field theories of immediateinterest. Both are lagrangean (field) theories of S -points ξ ∈ [ Ω p , M K ](S) of the mapping super-manifold [ Ω p , M K ] defined, for Ω p an arbitrary p -dimensional closed ( ∂ Ω p = ∅ ) oriented manifold(the worldvolume ) and M K as introduced previously (and termed the supertarget in this context),as the inner-Hom functor [ Ω p , M K ] ≡ Hom sMan ( ⋅ × Ω p , M K ) ∶ sMan Ð→ Set , to be evaluated on odd hyperplanes S ≡ R ∣ N of an arbitrary superdimension ( ∣ N ) , N ∈ N × , cp Ref. [Fre99]. In other words, we may think of the theories in question as countable families (indexed by N × ) of theories of generalised superembeddings of the (Graßmann-)odd-extended worldvolume R ∣ N × Ω p in the supertarget M K , and we shall write ξ ∈ [ Ω p , M K ] with this understanding. Their definitioncalls for a pair of H-basic (and so also H vac -basic) contravariant tensors on G, to wit, a degeneratesymmetric rank-2 tensor g = g ab θ a L ⊗ θ b L , g ba = g ab , with – for any k ∈ G – g ab ( ρ k ) ac ( ρ k ) bd = g cd , nd a de Rham-exact Graßmann-even (super-) ( p + ) -form χ ( p + ) = ( p + ) ! χ A A ...A p + θ A L ∧ θ A L ∧ ⋯ ∧ θ A p + L ∈ Ω p + ( G ) , termed the Green–Schwarz super- ( p + ) -cocycle , with – for any k ∈ G – χ A A ...A p + ( ρ k ) A B ( ρ k ) A B ⋯ ( ρ k ) A p + B p + = χ B B ...B p + and with an H-basic global primitive β ( p + ) = β A A ...A p + θ A L ∧ θ A L ∧ ⋯ ∧ θ A p + L ∈ Ω p + ( G ) , d β ( p + ) = χ ( p + ) such that – for any k ∈ G – β A A ...A p + ( ρ k ) A B ( ρ k ) A B ⋯ ( ρ k ) A p + B p + = β B B ...B p + . These define uniquely the corresponding tensors: g , H ( p + ) and B ( p + ) on M H satisfying the identities π ∗ G / H g = g , π ∗ G / H H ( p + ) = χ ( p + ) , π ∗ G / H B ( p + ) = β ( p + ) . (3.1)By the end of the day, then, we arrive at what has been and shall be referred to as the Green–Schwarzsuper- p -brane superbackground , sB ( GS ) p = (M H , g , χ ( p + ) ) . (3.2)Besides the above, the definition of the theories that we have in mind employs a canonical object asso-ciated with the pair ( G , t ( ) vac ) given by the (rescaled) volume form on the Graßmann-even component t ( ) vac of the vacuum subspace t vac ⊂ t ⊂ g , β ( p + ) ( HP ) = ( p + ) ! ǫ a a ...a p θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L , (3.3)written in terms of the standard totally antisymmetric symbol ǫ a a ...a p = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ sign ( . . . pa a . . . a p ) if { a , a , . . . , a p } = , p Hughes–Polchinski super- ( p + ) -form . If, as we shall assume henceforth, therestriction of the adjoint action of H vac on g ( ) to the Graßmann-even component t ( ) vac of the vacuumsubspace is unimodular , i.e. , we have ∀ h ∈ H vac ∶ det ( T e Ad h ↾ t ( ) vac ) = , (3.4)the HP super- ( p + ) -form descends to the homogeneous space M H vac , that is there exists a uniquesuper- ( p + ) -form B ( p + ) ( HP ) on M H vac such that π ∗ G / H vac B ( p + ) ( HP ) = β ( p + ) ( HP ) . The latter, in conjunction with the descendant H vac ( p + ) such that π ∗ G / H vac H ( p + ) vac = χ ( p + ) and H ( p + ) vac = d B ( p + ) vac11 Actually, we only need χ ( p + ) to be de Rham-closed, and hence locally exact. In the best studied examples of physicalrelevance, though, the Green–Schwarz super- ( p + ) -cocycle is exact and admits a global primitive, non-supersymmetric ingeneral. We choose to incorporate this empirical fact in the very definition of the superbackground, so that the usual deRham-cohomological issues do not obscure the Cartan–Eilenberg-cohomological ones that arise as we try to understandthe supersymmetry of the super- σ -model from the higher-geometric perspective. or B ( p + ) vac such that π ∗ G / H vac B ( p + ) vac = β ( p + ) , determines the Hughes–Polchinski super- p -brane background sB ( HP ) p,λ p = (M H , χ ( p + ) + λ p d β ( p + ) ( HP ) ≡ ̂ χ ( p + ) ) . (3.5)Under circumstances to be made precise in what follows, the field theories come in (essentially)dual pairs consisting of a theory with K = H and another one of the latter type for the correspondingK = H vac . Accordingly, we define pairs of super- σ -models related by a duality and, in the subsequentsections, exploit that duality towards an elucidation of the (super)symmetry content of the theories.We begin with Definition 3.1.
The
Green–Schwarz super- σ -model in the Nambu–Goto formulation forthe super- p -brane in the Green–Schwarz superbackground sB ( NG ) p as above, with – in particular – g = t ⊕ h reductive, is the lagrangean theory of mappings ξ ∈ [ Ω p , M H ] determined by the principle ofleast action applied, in the S -point picture, to the Dirac–Feynman amplitude A ( NG ) ,p DF [ ξ ] ∶ = e i S ( NG ) GS , p [ ξ ] determined by the action functional S ( NG ) GS ,p [ ξ ] ∶ = ∫ Ω p √ det ( p ) ( ξ ∗ g ) + ∫ Ω p ξ ∗ B ( p + ) . Equivalently, upon picking up an arbitrary open (superdomain) cover U H ≡ {U H i } i ∈ I H of M H thattrivialises the principal H-bundle (2.4) (for K = H) together with the corresponding local sections σ H i ∶ U H i Ð→ G of π G / H , and an arbitrary tessellation △ Ω p of Ω p , with the k -cell sets T k , k ∈ , p + U H along ξ , as captured by the existence of a map ı ⋅ ∶ △ Ω p Ð→ I H satisfying thecondition ∀ τ ∈ T p + ∶ ∣ ξ ∣( τ ) ⊂ ∣ U H ı τ ∣ , the action functional of the theory can be expressed in terms of tensors g and β ( p + ) as ( ξ τ ≡ ξ ↾ τ ) S ( NG ) GS ,p [ ξ ] ≡ ∑ τ ∈ T p + ( ∫ τ √ det ( p ) (( σ H ı τ ○ ξ τ ) ∗ g ) + ∫ τ ( σ H ı τ ○ ξ τ ) ∗ β ( p + ) ) . ◇ The previous definition is accompanied by
Definition 3.2.
The
Green–Schwarz super- σ -model in the Hughes–Polchinski formulationfor the super- p -brane in the Hughes–Polchinski superbackground sB ( HP ) p as above at λ p ∈ R × , with– in particular – g = f ⊕ h vac reductive and for a unimodular adjoint action of H vac on t ( ) vac , is thelagrangean theory of mappings ̂ ξ ∈ [ Ω p , M H vac ] determined by the principle of least action applied, inthe S -point picture, to the Dirac–Feynman amplitude A ( HP ) ,p,λ p DF [̂ ξ ] ∶ = e i S ( HP ) ,λp GS ,p [̂ ξ ] determined by the action functional S ( HP ) ,λ p GS ,p [̂ ξ ] ∶ = ∫ Ω p ̂ ξ ∗ ( λ p B ( p + ) ( HP ) + B ( p + ) vac ) . Equivalently, upon picking up an arbitrary open (superdomain) cover U H vac ≡ { U H vac i } i ∈ I Hvac of M H vac that trivialises the principal H vac -bundle (2.4) (for K = H vac ) together with the corresponding localsections σ H vac i ∶ U H vac i Ð→ G of π G / H vac , and an arbitrary tessellation △ Ω p of Ω p , with the k -cell sets T k , k ∈ , p +
1, subordinate to U H vac along ̂ ξ , as captured by the existence of a map ı ⋅ ∶ △ Ω p Ð→ I H vac satisfying the condition ∀ τ ∈ T p + ∶ ∣̂ ξ ∣( τ ) ⊂ ∣ U H vac ı τ ∣ , he action functional of the theory can be expressed in terms of tensors β ( p + ) ( HP ) and β ( p + ) as ( ̂ ξ τ ≡ ̂ ξ ↾ τ ) S ( HP ) GS ,p [̂ ξ ] ≡ ∑ τ ∈ T p + ∫ τ ( σ H vac ı τ ○ ̂ ξ τ ) ∗ ( λ p β ( p + ) ( HP ) + β ( p + ) ) . ◇ The Dimensional Constraint:
We may further constrain the admissible choices of superbackgroundsby restoring linear dimensions of the various coordinates and – as a consequence – also of the corre-sponding basis super-1-forms entering the definition of the GS super- ( p + ) -cocycle. Thus, upon setting [ x a ] ≡ [ θ a L ] = = [ θ α L ] ≡ [ θ α ] , we readily conclude that the only super- σ -models (of the type considered) with both terms in theaction functional of dimensionality m p + (or, equivalently, with a dimension less relative normalisationcoefficient) are those whose GS super- ( p + ) -cocycles are wedge products of 2 s spinorial components θ α L and p + − s vectorial components θ a L of the Maurer–Cartan super-1-form θ L (recall that thesuper- ( p + ) -cocycles are h -horizontal by assumption), with s constrained by the equality s + p + − s ! = p + , whence the requirement s ! = , (3.6)which we impose henceforth (resp. verify in the most studied examples). Consequently, we always write χ ( p + ) = p ! χ αβa a ...a p Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L for some H-invariant tensors χ αβa a ...a p ≡ χ ( αβ )[ a a ...a p ] , α, β ∈ , δ − d, a , a , . . . , a p ∈ , d .It ought to be noted that among the known consistent superbackgrounds there are also those thatexplicitly violate the above constraint, to wit, the Zhou super-0-brane of Ref. [Zho99] in s ( AdS × S ) ,the Metsaev–Tseytlin D3-brane of Ref. [MT98b] in s ( AdS × S ) , the M2-brane of Ref. [dWPPS98] ins ( AdS × S ) or in s ( AdS × S ) , and the M5-brane of Ref. [Cla99] in the same supertargets. In the firstof these ‘counterexamples’, the GS super-2-cocycle is a linear combination of a bi-spinorial term and abi-vectorial term, conspiring to render the combination de Rham-closed. The construction of a super-symmetric topological (Wess–Zumino) term of the super- σ -model for the D3-brane, on the other hand,calls for the incorporation of the curvature of a principal C × -bundle over the D3-brane worldvolume,the latter super-2-form admitting only an implicit (homotopy-formula) presentation in terms of thecomponents of the LI Maurer–Cartan super-1-form on SU ( , ∣ ) . Finally, in the eleven-dimensionalsetting, the super- σ -model uses the M-theory 4-form with a non-trivial geometrisation reflected by asubtle Dirac’s charge-quantisation condition imposed upon it, cp Refs. [Wit97b, Wit97a]. We hope toreturn to these more involved structures in the future. In order to give the Reader a foretaste of what wemay stumble upon along the way, we scrutinise the much tractable four-dimensional ‘counterexample’from the above list in the Appendix.The two field-theoretic constructs introduced above were first set in correspondence by
Theorem 3.3 (Ref. [Sus19, Thms. 5.1 & 5.2]) . If the superbackgrounds sB ( NG ) p and sB ( HP ) p,λ p are asdescribed above (that is – in particular – both decompositions g = f ⊕ k are reductive and the adjointaction of H vac on t ( ) vac is unimodular), there exists a T e Ad H -invariant scalar product g on t ( ) suchthat t ( ) vac ⊥ g e ( ) , and the Maximal Mixing Constraint ⟨ P ̂ a ∣ ∃ ( b, ̂ S ) ∈ ,p × D − δ + ,D − δ ∶ f b ̂ S ̂ a ≠ ⟩ = e ( ) (3.7) The de Rham class of the 4-form becomes integral only after a twist by one quarter of the first Pontryagin class ofthe supertarget. In what folows, we refer to this situation as the T e Ad H -invariance of the vacuum splitting . s obeyed, then there exists a unique value of the parameter λ p for which the GS super- σ -model in theNG formulation of Def. 3.1 with g = g is (classically) equivalent to the GS super- σ -model in the HPformulation of Def. 3.2 partially reduced through imposition of the Inverse Higgs Constraints ( σ H vac ı τ ○ ̂ ξ ) ∗ θ ̂ a L ! = , ̂ a ∈ p + , d . More specifically, the DF amplitude A ( HP ) , p ,λ p DF written in the gauge σ H vac i ≡ σ vac i , i ∈ I H vac of Eq. (2.15) then reduces to the DF amplitude A ( NG ) , pDF written in the gauge σ H i ≡ σ ✟✟❍❍ vac i , i ∈ I H vac of Eq. (2.16) . Taking into account the manifest violation of the Maximal Mixing Constraint in many physicallyinteresting and important situations, we readily generalise the result of Ref. [Sus19] in the form of
Theorem 3.4.
Let the superbackgrounds sB ( NG ) p and sB ( HP ) p,λ p be as described above (that is – inparticular – assume both decompositions g = f ⊕ k to be reductive and the adjoint action of H vac on t ( ) vac to be unimodular), write d − t ( ) vac ∶ = ⟨ P ̂ a ∣ ∃ ( b, ̂ S ) ∈ ,p × D − δ + ,D − δ ∶ f b ̂ S ̂ a ≠ ⟩ ≡ d − L ⊕ ̂ a = p + ⟨ P ̂ a ⟩ for some ≤ L ≤ d − p (with L = d − p corresponding to the degenerate d − t ( ) vac = ) and subsequentlydecompose e ( ) ≡ d − t ( ) vac ⊕ l ( ) , assuming that both d − t ( ) ≡ t ( ) vac ⊕ d − t ( ) vac and l ( ) are T e Ad H -invariant (and so also ad h -invariant), and – finally – let g vac be a T e Ad H -invariantscalar product on d − t ( ) such that t ( ) vac ⊥ g vac d − t ( ) vac . Then, there exists a unique value of the parameter λ p for which the GS super- σ -model in the HPformulation of Def. 3.2 restricted to field configurations obeying the Body-Localisation Constraints ( σ H vac ı τ ○ ̂ ξ ) ∗ θ ̂ a L ! = , ̂ a ∈ d − L + , d (3.8) and further reduced (partially) through imposition of the Inverse Higgs Constraints ( σ H vac ı τ ○ ̂ ξ ) ∗ θ ̂ a L ! = , ̂ a ∈ p + , d − L (3.9) is (classically) equivalent to the GS super- σ -model in the NG formulation of Def. 3.1 with g such that g ↾ d − t ( ) ≡ g vac , restricted to field configurations subject to the (same) Body-Localisation Constraints ( σ H ı τ ○ ξ ) ∗ θ ̂ a L ! = , ̂ a ∈ d − L + , d , where it is to be understood that the DF amplitude A ( HP ) , p ,λ p DF written in the gauge σ H vac i ≡ σ vac i , i ∈ I H vac of Eq. (2.15) reproduces the DF amplitude A ( NG ) , pDF written in the gauge σ H i ≡ σ ✟✟❍❍ vac i , i ∈ I H vac ofEq. (2.16) under the correspondence.Proof. Obvious. (cid:3)
Remark 3.5.
The generalisation formulated above is intended to cover a situation in which the bodyof the supertarget is a cartesian product ∣ G / H ∣ = (∣ G ∣/ H ) × (∣ G ∣/ H ) of two reductive homogeneous spaces of the Lie groups ∣ G A ∣ , A ∈ { , } (with the respective Lie algebras ∣ g A ∣ ) relative to the respective cartesian factors H A of the isotropy groupH = H × H , and the vacuum of the super- σ -model is constrained to be embedded entirely in one of the factors, say ∣ G ∣/ H , in such a manner that in the obvious notation (referring directly to the previously introducedone) ∣ g A ∣ = t ( ) A ⊕ h A , ith t ( ) ⊕ t ( ) ≡ t ( ) , h ⊕ h ≡ h , we have h vac = h ⊕ h , h ⊂ h ⊕ d ≡ h and t ( ) vac ⊂ t ( ) vac ⊕ e ( ) ≡ t ( ) , e ( ) ≡ e ( ) ⊕ t ( ) , with d − t ( ) vac ⊂ e ( ) ⊊ e ( ) . Under such circumstances, switching on and subsequently integrating out Goldstone modes along d will not induce the terms of the metric integrand in the NG super- σ -model along t ( ) , and so the latterhave to be frozen out by hand in the metric term of the NG super- σ -model through imposition of theBody-Localisation Constraints. This certainly is an invasive manipulation on the GS field theory underconsideration. However, we should bear in mind that the correspondence is employed solely towardsgeometrisation of the description of the vacuum of that field theory ensuing from the variational analysisof its DF amplitude, and that of its local supersymmetry. When considered from this vantage point, themanipulation acquires the interpretation of a mere (partial) localisation of (the body of) the vacuum.The correspondence established in the last theorem admits a reformulation that opens up an avenuefor a geometrisation of field-theoretic statements made in the purely topological setting of the HPformulation that we shall find particularly convenient and robust in our subsequent considerations. Itbegins with the definition of the family of sub-supermanifolds V i ∶ = l g i ( V e ) ≡ σ vac i ( U H vac i ) ⊂ G , i ∈ I H vac , (3.10)with V ≡ V e , that faithfully present the supertarget M H vac patchwise. Their disjoint unionΣ HP ∶ = ⊔ i ∈ I Hvac V i , (3.11)which we shall call the Hughes–Polchinski section in what follows, is the arena on which all super-geometric phenomena of interest to us take place, and so it is certainly worth a closer inspection. Wehave
Proposition 3.6.
In the hitherto notation, the tangent sheaf T Σ HP ≡ ⊔ i ∈ I Hvac
T V i , T V i = T σ vac i ( T M
Hvac ↾ U Hvac i ) of the Hughes–Polchinski section Σ HP is spanned, over its component V i , on vector fields T µ i = L µ ↾ V i + T Sµ i L S , µ ∈ , δ , (3.12) with sections T Sµ i ∈ O G ( V i ) uniquely fixed by the tangency condition T µ i ! ∈ T V i in the form T Sµ i ( σ vac i ( χ i )) = ( E ( χ i ) − ) νµ E Sν ( χ i ) , where E Aµ is the Vielbein field introduced in Eqs. (2.19) and (2.20) and E ( χ i ) − is the inverse of thequadratic matrix E ( χ i ) ≡ ( E νµ ( χ i )) ν ∈ ,δµ ∈ ,δ . roof. We adapt the reasoning given in Ref. [Sus18a, Sec. 2]. Present T µ i as the pushforward of avector field tangent to the base, T µ i ( σ vac i ( χ i )) ≡ T χ i σ vac i ( T µ i ( χ i )) , that we write in a local coordinate system as T µ i ( χ i ) = ∆ νµ i ( χ i ) ⃗ ∂∂χ νi , and ∆ νµ i E Aν ( χ i ) ≡ T µ i ⌟ σ vac ∗ i θ A L ( χ i ) = T µ i ⌟ θ A L ( σ vac i ( χ i )) ≡ δ Aµ + T Sµ i ( σ vac i ( χ i )) δ AS , valid for any A ∈ , D . Setting A ≡ λ ∈ , δ , we obtain∆ νµ i E λν ( χ i ) = δ λµ . (3.13)At this stage, the only thing that has to be proven is the invertibility of E . To this end, we compute,by a variation on the above, T χ i σ vac i ( ⃗ ∂∂χ µi ) = E νµ ( χ i ) L ν ( σ vac i ( χ i )) + E Sµ ( χ i ) L S ( σ vac i ( χ i )) . Now, the horizontal lift of the coordinate vector fields determined by the principal H vac -connection asin Eq. (2.18), Hor σ vac i ( χ i ) ( ⃗ ∂∂χ µi ) = E νµ ( χ i ) L ν ( σ vac i ( χ i )) , yields a basis of the horizontal subspace H σ vac i ( χ i ) G in the horizontal subsheaf H G of the tangentsheaf T G, of dimension dim H σ vac i ( χ i ) G = dim f , whence the anticipated property of the reduced Vielbein E . (cid:3) It is natural, from our field-theoretic point of view, to distinguish those among global sections of thetangent sheaf of the HP section that descend to the homogeneous space M H vac . This we do in Definition 3.7.
Adopt the hitherto notation and let { h ij } i,j ∈ I Hvac be the transition mappings of theprincipal H vac -bundle (2.4) (with K = H vac ) for the family { σ vac i } i ∈ I Hvac of the local sections introducedpreviously, so that we have σ vac j ( χ j ) = σ vac i ( χ i ) ⋅ h ij ( χ i ) (at every S -point) in the intersection of U H vac i and U H vac j . Consider an arbitrary vector field W ∈ Γ ( T Σ HP ) with restrictions W ↾ V i = W µ T µ i expressed in terms of sections W µ ∈ O G ( V i ) . We call W an H vac -descendable vector field on Σ HP if the identity ( ρ h ij ( χ i ) − ) νµ W µ ( χ i ) = W ν ( χ j ) holds true over the intersection of U H vac i and U H vac j for any i, j ∈ I H vac . ◇ A related concept of prime relevance to our later analysis is introduced in the next
Definition 3.8.
In the hitherto notation, let D ⊂ T Σ HP be an arbitrary superdistribution (in the sense of Ref. [CCF11, Def. 6.1.1]). We call D an H vac -descendable superdistribution over Σ HP if the relation T χ i r H vac h ij ( χ i ) ( D σ vac i ( χ i ) ) ⊆ D σ vac j ( χ j ) , written in terms of the defining action r H vac ⋅ of Eq. (2.17), obtains over the intersection of U H vac i and U H vac j for any i, j ∈ I H vac , or – equivalently – if the following implications hold true W µ T µ i ( σ vac i ( χ i )) ∈ D χ i Ô⇒ ( ρ h ij ( χ i ) − ) νµ W µ T ν j ( σ vac j ( χ j )) ∈ D χ j over the intersections. Remark 3.9.
The notion of an H vac -descendable superdistribution is closely related to the familiarnotion of an h vac -invariant superdistribution , i.e. , of a distribution D with the property [ h vac ↾ Σ HP , D] ⊆ D , with h vac ↾ Σ HP standing for the restriction of the vertical distribution of the principal H vac -bundle (2.4)(with K = H vac ) to the HP section.We may, at last, return to our discussion of the super- σ -model and rephrase the thesis of Thm. 3.4upon giving one last Definition 3.10.
Adopt the hitherto notation. The correspondence superdistribution of sB ( HP ) p,λ p is the superdistribution within the tangent sheaf T Σ HP of the Hughes–Polchinski section of Prop. 3.6defined as Corr HP / NG ( sB ( HP ) p,λ p ) ∶ = Ker ( P ge ( ) ○ θ L ↾ T Σ HP ) ⊂ T Σ HP (3.14)in terms of the projector P ge ( ) ∶ g ↺ onto e ( ) , with the kernel Ker P ge ( ) = t ( ) ⊕ t ( ) vac ⊕ h whose proper subspace t ( ) ⊕ t ( ) vac ⊕ d ≡ corr HP / NG ( sB ( HP ) p,λ p ) models Corr HP / NG ( sB ( HP ) p,λ p ) locally. ◇ The correspondence between the two formulations of the GS super- σ -model stated in Thm. 3.4 pertainsto those mappings ̂ ξ ∈ [ Ω p , M H vac ] for which the images of the tangents of the superpositions σ vac i ○ ̂ ξ are contained in the correspondence superdistribution – we shall refer to the entirety of such mappings(in the context of the opening paragraph of the present section) as the HP/NG correspondencesector . In order for the correspondence to be physically meaningful, it is necessary that the localisationconstraints: (3.8) and (3.9) which we impose locally over Ω p can be continued across the trivialisingpatches U H vac i and do not depend on the choice of the local gauge ( i.e. , on the choice of the σ vac i ).Thus, altogether, we demand that the correspondence superdistribution descend to M H vac . Inspectionof the structure of the modelling supervector space corr HP / NG ( sB ( HP ) p,λ p ) readily shows that our demandshould be formulated as The Descendability Constraint:
We require the splitting (2.6) of the isotropy algebra to be ad h vac -invariant in the sense expressed by the relation [ h vac , d ] ⊂ d . (3.15)The last few definitions, related by the correspondence theorem (and subject to the DimensionalConstraint), demarcate the environment in which all our subsequent physical considerations are placed,and the rest of the present section provides the relevant supergeometric substrate. In order to put someflesh on the abstract logical skeleton laid thus, let us, prior to launching a canonical (super)symmetryanalysis of the field theories of interest and identifying the higher geometry behind them, take a lookat a bunch of physically relevant examples that illustrate the abstract ideas and constructions. Example 3.11. The Green–Schwarz super-0-brane in sMink ( , ∣ ) . ● The mother super-Harish–Chandra pair:sISO ( , ∣ ) ≡ ( ISO ( , ) ≡ R , ⋊ SO ( , ) , siso ( , ∣ )) , onsisting of the Poincar´e group ISO ( , ) of the Minkowski space R , and the super-Poincar´ealgebra siso ( , ∣ ) = ⊕ α = ⟨ Q α ⟩ ⊕ ⊕ a = ⟨ P a ⟩ ⊕ ⊕ a,b = ⟨ J ab = − J ba ⟩ with the structure equations { Q α , Q β } = ( C Γ a ) αβ P a , [ P a , P b ] = , [ J ab , J cd ] = η ad J bc − η ac J bd + η bc J ad − η bd J ac , [ J ab , P c ] = η bc P a − η ac P b , [ P a , Q α ] = , [ J ab , Q α ] = ( Γ ab ) βα Q β , expressed in terms of the generators { Γ a } a ∈ , of the Clifford algebra Cliff ( R , ) of the Minkowski space R , with the metric η = diag ( − , + , + , + , + , + , + , + , + , + ) (used to lower and raise vector indices throughout), their commu-tators Γ ab = [ Γ a , Γ b ] , the chirality operator Γ = Γ Γ ⋯ Γ and a charge-conjugation matrix C (used to lower and raise spinor indices throughout) in aMajorana-spinor representation in which C T = − C and Γ a ≡ C Γ a = ( C Γ a ) T , so that – in particular – C Γ a C − = − Γ a T and Γ ≡ C Γ = − Γ T11 C ; ● The T e Ad SO ( , ) -invariance of the vacuum splitting – obvious; ● The homogeneous spaces: the NG onesMink ( , ∣ ) ≡ sISO ( , ∣ )/ SO ( , ) ≡ R ( , ∣ ) , H ≡ SO ( , ) , h ≡ so ( , ) = ⊕ a,b = ⟨ J ab = − J ba ⟩ , with the body ∣ sMink ( , ∣ )∣ = R , ≡ Mink ( , ) , which happens to be a Lie supergroup, with global coordinates { θ α , x a } ( α,a ) ∈ , × , in which µ ∗ ∶ ( θ α , x a ) z→ ( θ α ⊗ + ⊗ θ α , x a ⊗ + ⊗ x a − θ α ⊗ Γ aαβ θ β ) , Inv ∗ ∶ ( θ α , x a ) z→ ( − θ α , − x a ) , or, equivalently (in the S -point picture), with the group operations ( θ α , x a ) ⋅ ( θ β , x b ) = ( θ α + θ α , x a + x a − θ α Γ aαβ θ β ) , ( θ α , x a ) − = ( − θ α , − x a ) , and the HP one sISO ( , ∣ )/ SO ( ) , H vac ≡ SO ( ) , h vac ≡ ⊕ ̂ a, ̂ b = ⟨ J ̂ a ̂ b = − J ̂ b ̂ a ⟩ , = ⊕ ̂ a = ⟨ J ̂ a ⟩ , t ( ) vac = ⟨ P ⟩ , d − t ( ) vac = ⊕ ̂ a = ⟨ P ̂ a ⟩ ≡ e ( ) ; ● The exponential superparametrisation(s) ( ̂ b ∈ , σ vac0 ( θ α , x a , φ ̂ b ) = e θ α ⊗ Q α ⋅ e x a ⊗ P a ⋅ e φ ̂ b ⊗ J ̂ b ; ● The superbackgrounds: the NG one sB ( NG ) = ( sMink ( , ∣ ) , η ab θ a L ⊗ θ b L , Σ L ∧ Γ Σ L ≡ χ ( ) GS ) ,θ L = Σ α L ⊗ Q α + θ a L ⊗ P a + θ ab L ⊗ J a < b with a non supersymmetric global curving β ( ) GS ( θ, x, φ ) = θ Γ d θ (and no supersymmetric one), and the HP one sB ( HP ) ,λ = ( sISO ( , ∣ )/ SO ( ) , χ ( ) GS + λ ( Σ L ∧ Γ Σ L − δ ̂ a ̂ b θ ̂ a L ∧ θ ̂ b L ) ≡ ̂ χ ( ) GS ) , ⟨ P ⟩ ⊥ η ⊕ ̂ a = ⟨ P ̂ a ⟩ ; ● The Body-Localisation Constraints: none.
Example 3.12. The Green–Schwarz super- p -brane with p ∈ , in sMink ( d, ∣ D d, ) . ● The mother super-Harish–Chandra pair:sISO ( d, ∣ D d, ) ≡ ( ISO ( d, ) ≡ R d, ⋊ SO ( d, ) , siso ( d, ∣ D d, )) , consisting of the Poincar´e group ISO ( d, ) of the Minkowski space R d, and the super-Poincar´ealgebra siso ( d, ∣ D d, ) = D d, ⊕ α = ⟨ Q α ⟩ ⊕ d ⊕ a = ⟨ P a ⟩ ⊕ d ⊕ a,b = ⟨ J ab = − J ba ⟩ with the structure equations { Q α , Q β } = ( C Γ a ) αβ P a , [ P a , P b ] = , [ J ab , J cd ] = η ad J bc − η ac J bd + η bc J ad − η bd J ac , [ J ab , P c ] = η bc P a − η ac P b , [ P a , Q α ] = , [ J ab , Q α ] = ( Γ ab ) βα Q β , expressed in terms of the generators { Γ a } a ∈ ,d of the Clifford algebra Cliff ( R d, ) of theMinkowski space R d, with the metric η = diag ( − , + , + , . . . , + ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ d times ) , their commutatorsΓ ab = [ Γ a , Γ b ] and a charge-conjugation matrix C in a Majorana-spinor representation of dimension D d, inwhich C Γ a C − = − Γ a T , C T = − ǫ d C , ǫ d = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ − √ ( ( d + ) π ) if d ∈ { , , , , } − √ ( ( d + ) π ) if d ∈ { , , } , with ( d, p ) chosen in such a manner that ( C Γ a a ...a p ) T = C Γ a a ...a p , that is with ǫ d ! = ( − ) ( p − )⋅( p − ) , We adopt the conventions of Refs. [Wes99, CdAIPB00]. nd in which the following Fierz identity obtains: η ab Γ a ( αβ Γ ba a ...a p − γδ ) = ● The T e Ad SO ( d, ) -invariance of the vacuum splitting – obvious; ● The homogeneous spaces: the NG onesMink ( d, ∣ D d, ) ≡ sISO ( d, ∣ D d, )/ SO ( d, ) ≡ R ( d, ∣ D d, ) , H ≡ SO ( d, ) , h ≡ so ( d, ) = ⊕ da,b = ⟨ J ab = − J ba ⟩ , with the body ∣ sMink ( d, ∣ D d, )∣ = R d, ≡ Mink ( d, ) , which is, again, a Lie supergroup with the structure as in the previous example, and the HPone sISO ( d, ∣ D d, )/( SO ( p, ) × SO ( d − p )) , H vac ≡ SO ( p, ) × SO ( d − p ) , h vac ≡ ⊕ pa,b = ⟨ J ab ⟩ ⊕ ⊕ d ̂ a, ̂ b = p + ⟨ J ̂ a ̂ b ⟩ , d = ⊕ pa = ⊕ d ̂ b = p + ⟨ J a ̂ b ⟩ , t ( ) vac = ⊕ pa = ⟨ P a ⟩ , d − t ( ) vac = ⊕ d ̂ a = p + ⟨ P ̂ a ⟩ ≡ e ( ) ; ● The exponential superparametrisation(s) ( ( b, ̂ c ) ∈ { , } × , d ): σ vac0 ( θ α , x a , φ b ̂ c ) = e θ α ⊗ Q α ⋅ e x a ⊗ P a ⋅ e φ b ̂ c ⊗ J b ̂ c ; ● The superbackgrounds: the NG one sB ( NG ) p = ( sMink ( d, ∣ D d, ) , η ab θ a L ⊗ θ b L , Σ L ∧ Γ a a ...a p Σ L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ≡ χ ( p + ) GS ) with a non supersymmetric global curving β ( p + ) GS ( θ, x, φ ) = p + ∑ pk = θ Γ a a ...a k a k + a k + ...a p d θ ∧ d x a a ...a k ∧ e a k + a k + ...a p ( θ, x ) , d x a a ...a k ≡ d x a ∧ d x a ∧ ⋯ ∧ d x a k ,e a k + a k + ...a p ≡ e a k + ∧ e a k + ∧ ⋯ ∧ e a p , e a ( θ, x ) = d x a + θ Γ a d θ (and no supersymmetric one), and the HP one sB ( HP ) p,λ p = ( sISO ( d, ∣ D d, )/( SO ( p, ) × SO ( d − p )) ,χ ( p + ) GS + λ p p ! ǫ a a ...a p ( Σ L ∧ Γ a Σ L − δ ̂ c ̂ d θ a ̂ c L ∧ θ ̂ d L ) ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ≡ ̂ χ ( p + ) GS ) , p ⊕ a = ⟨ P a ⟩ ⊥ η d ⊕ ̂ a = p + ⟨ P ̂ a ⟩ ; ● The Body-Localisation Constraints: none.
Remark 3.13.
Implicit in the structure of the Lie superalgebra siso ( d, ∣ D d, ) is the assumption ( C Γ a ) T = C Γ a which further constrains ǫ d to be equal to 1. This assumption can be maintained, in conjunction withthe constraints already imposed, only for p ∈ N + p ∈ N +
2. The remaining possibilities for whichwe might contemplate more general symmetry conditions for the Γ a and a supersymmetry algebrawith the anticommutator of the supercharges spanned exclusively on topological charges turn out tobe ruled out by a simple algebraic argument given in Ref. [Wes99, Sec. 1.5]. The argument not onlyrestricts the admissible values of p as p ≡ p ≡ d − p ≡ d − p ≡ Example 3.14. The Zhou super-1-brane in s ( AdS × S ) . The mother super-Harish–Chandra pair:SU ( , ∣ ) ≡ ( SO ( , ) × SO ( ) , su ( , ∣ ) ) , consisting of the product Lie group SO ( , ) × SO ( ) and the Lie superalgebra su ( , ∣ ) = ⊕ ( α ′ ,α ′′ ,I ) ∈ { , } × ⟨ Q α ′ α ′′ I ⟩ ⊕ ⊕ a ′ ∈ { , } ⟨ P a ′ ⟩ ⊕ ⊕ a ′′ ∈ { , } ⟨ P a ′′ ⟩ ⊕ ⟨ J = − J ⟩ ⊕ ⟨ J = − J ⟩ with the structure relations { Q α ′ α ′′ I , Q β ′ β ′′ J } = (( C γ a ⊗ ) α ′ α ′′ Iβ ′ β ′′ J P a − i ( C ⊗ σ ) α ′ α ′′ Iβ ′ β ′′ J J − i ( C γ ⊗ σ ) α ′ α ′′ Iβ ′ β ′′ J J ) , [ P , P ] = J , [ P , P ] = − J , [ P a ′ , P a ′′ ] = , [ J , J ] = , (3.16) [ J , P a ′ ] = η a ′ P − η a ′ P , [ J , P a ′′ ] = δ a ′′ P − δ a ′′ P , [ J , P a ′ ] = = [ J , P a ′′ ] , [ P a , Q α ′ α ′′ I ] = i (̃ γ ′ γ a ⊗ σ ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J , [ J ab , Q α ′ α ′′ I ] = ( γ ab ⊗ ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J , expressed in terms of the Pauli matrix σ = ( − ii ) , the generators { γ a ′ ≡ Γ a ′ ⊗ , γ b ′′ ≡ Γ Γ ⊗ Γ b ′′ } ( a ′ ,b ′′ ) ∈ { , } × { , } of the Clifford algebra Cliff ( R , ) of the Minkowski space R , with the metric η = diag ( − , + , + , + ) and their commutators γ ab = [ γ a , γ b ] , a, b ∈ , { Γ a ′ } a ′ ∈ { , } of the Clifford algebra Cliff ( R , ) (in the 2-dimensional spinor representation) and the gener-ators { Γ a ′′ } a ′′ ∈ { , } of the Clifford algebra Cliff ( R , ) (also in the 2-dimensional spinor representation), and interms of (the tensor components of) the chirality operator γ ≡ γ γ γ γ = ̃ γ ′ ̃ γ ′′ , ̃ γ ′ = Γ Γ ⊗ , ̃ γ ′′ = ⊗ Γ Γ of Cliff ( R , ) , as well as of the charge conjugation matrix C = C ′ ⊗ C ′′ = − C T given as the product of the charge conjugation matrices C ′ = − C ′ T of Cliff ( R , ) and C ′′ = C ′′ T of Cliff ( R , ) , and such that C ′ Γ a ′ = ( C ′ Γ a ′ ) T , C ′′ Γ d ′′ = ( C ′′ Γ d ′′ ) T , C ′ Γ b ′ c ′ = ( C ′ Γ b ′ c ′ ) T , whereas C ′′ Γ a ′′ b ′′ = −( C ′′ Γ a ′′ b ′′ ) T , o that – in particular – C γ a C − = − γ a T ; ● The T e Ad SO ( , ) × SO ( ) -invariance of the vacuum splitting – obvious; ● The homogeneous spaces: the NG ones ( AdS × S ) = SU ( , ∣ ) /( SO ( , ) × SO ( )) , H = SO ( , ) × SO ( ) , h ≡ so ( , ) ⊕ so ( ) = ⟨ J ⟩ ⊕ ⟨ J ⟩ , with the body ∣ s ( AdS × S )∣ = SO ( , )/ SO ( , ) × SO ( )/ SO ( ) ≡ AdS × S , and the HP one(s): the super-1-brane entirely in/over AdS SU ( , ∣ ) /( SO ( , ) × SO ( )) , H vac = SO ( , ) × SO ( ) , h vac = ⟨ J ⟩ ⊕ ⟨ J ⟩ , d = , t ( ) vac = ⟨ P , P ⟩ , d − t ( ) vac = ; the super-1-brane astride SU ( , ∣ ) , H vac = , h vac = , d = ⟨ J , J ⟩ , t ( ) vac = ⟨ P , P ⟩ , d − t ( ) vac = ⟨ P , P ⟩ ≡ e ( ) ; ● The exponential superparametrisation(s): the super-1-brane entirely in/over AdS σ vac0 ( θ α ′ α ′′ I , x a ) = e θ α ′ α ′′ I ⊗ Q α ′ α ′′ I ⋅ e x a ⊗ P a ; the super-1-brane astride σ vac0 ( θ α ′ α ′′ I , x a , φ , φ ) = e θ α ′ α ′′ I ⊗ Q α ′ α ′′ I ⋅ e x a ⊗ P a ⋅ e φ ⊗ J + φ ⊗ J ; ● The superbackgrounds: the NG one sB ( NG ) = ( s ( AdS × S ) , η ab θ a L ⊗ θ b L , Σ L ∧ ( Cγ a ⊗ σ ) Σ L ∧ θ a L ≡ χ ( ) Zh ) ,θ L = Σ α ′ α ′′ I L ⊗ Q α ′ α ′′ I + θ a L ⊗ P a + θ ⊗ J + θ ⊗ J with a supersymmetric global predecessor of the curving β ( ) Zh = Σ L ∧ ( C ̃ γ ′ ⊗ σ ) Σ L on SU ( , ∣ ) that descends to s ( AdS × S ) , and the HP one(s): the super-1-brane entirely in/over AdS (with ǫ = − sB ( HP ) ,λ = ( SU ( , ∣ ) /( SO ( , ) × SO ( )) , χ ( ) Zh − λ ǫ a ′ b ′ Σ L ∧ ( C γ a ′ ⊗ ) Σ L ∧ θ b ′ L ≡ ̂ χ ( ) Zh ( ) ) , ⟨ P , P ⟩ ⊥ η ⟨ P , P ⟩ ; the super-1-brane astride (with ǫ = − sB ( HP ) ,λ = ( SU ( , ∣ ) , χ ( ) Zh − λ ( ǫ a ′ b ′′ Σ L ∧ ( C γ a ′ ⊗ ) Σ L ∧ θ b ′ L − θ ∧ θ ∧ θ − θ ∧ θ ∧ θ ) ≡ ̂ χ ( ) Zh ( ) ) , ⟨ P , P ⟩ ⊥ η ⟨ P , P ⟩ ; ● The Body-Localisation Constraints: .14.1. the super-1-brane entirely in/over AdS : θ a ′′ L ≈ , a ′′ ∈ { , } ; the super-1-brane astride: none. Example 3.15. The Park–Rey super-1-brane in s ( AdS × S ) . ● The mother super-Harish–Chandra pair: ( SU ( , ∣ ) × SU ( , ∣ )) ≡ ( SO ( , ) × SO ( ) , ( su ( , ∣ ) ⊕ su ( , ∣ )) ) consisting of the product Lie group SO ( , ) × SO ( ) and the Lie superalgebra ( su ( , ∣ ) ⊕ su ( , ∣ )) = ⊕ ( α ′ ,α ′′ ,α ′′′ ,I ) ∈ { , } × ⟨ Q α ′ α ′′ α ′′′ I ⟩ ⊕ ⊕ a ′ ∈ { , , } ⟨ P a ′ ⟩ ⊕ ⊕ a ′′ ∈ { , , } ⟨ P a ′′ ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ = − J b ′ a ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ ∈ { , , } ⟨ J a ′′ b ′′ = − J b ′′ a ′′ ⟩ with the structure relations { Q α ′ α ′′ α ′′′ I , Q β ′ β ′′ β ′′′ J } = ( C γ a ′ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J P a ′ − ( C γ a ′′ γ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J P a ′′ − i ( C γ a ′ b ′ γ ⊗ σ ) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J J a ′ b ′ + i ( C γ a ′′ b ′′ γ ⊗ σ ) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J J a ′′ b ′′ , [ P a , P b ] = ε ab J ab , ε ab = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ + a, b ∈ { , , } − a, b ∈ { , , } , [ J ab , J cd ] = η ad J bc − η ac J bd + η bc J ad − η bd J ac , [ J ab , P c ] = η bc P a − η ac P b , (3.17) [ P a ′ , Q α ′ α ′′ α ′′′ I ] = i ( γ a ′ γ ⋅ ( ⊗ σ ) ⊗ σ ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I Q β ′ β ′′ β ′′′ J , [ P a ′′ , Q α ′ α ′′ α ′′′ I ] = − i ( γ a ′′ ⋅ ( ⊗ σ ) ⊗ σ ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I Q β ′ β ′′ β ′′′ J , [ J ab , Q α ′ α ′′ α ′′′ I ] = ( γ ab ⊗ ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I Q β ′ β ′′ β ′′′ J , expressed in terms of the Pauli matrices σ = ( ) , σ = ( − ii ) , σ = ( − ) , the generators { γ a ′ ≡ Γ a ′ ⊗ ⊗ σ , γ b ′′ ≡ ⊗ Γ b ′′ ⊗ σ } ( a ′ ,b ′′ ) ∈ { , , } × { , , } of the Clifford algebra Cliff ( R , ) of the Minkowski space R , with the metric η = diag ( − , + , + , + , + , + ) and their commutators γ ab = [ γ a , γ b ] , a, b ∈ , { Γ a ′ } a ′ ∈ { , , } of the Clifford algebra Cliff ( R , ) (in the 2-dimensional spinor representation) and the gener-ators { Γ a ′′ } a ′′ ∈ { , , } of the Clifford algebra Cliff ( R , ) (also in the 2-dimensional spinor representation), and interms of the chirality operator γ ≡ − ⊗ σ f Cliff ( R , ) , as well as of the charge conjugation matrix C = C ′ ⊗ C ′′ ⊗ σ = C T given as the product of the charge conjugation matrices C ′ = − C ′ T of Cliff ( R , ) and C ′′ = − C ′′ T of Cliff ( R , ) , such that C ′ Γ a ′ = ( C ′ Γ a ′ ) T , C ′′ Γ a ′′ = ( C ′′ Γ a ′′ ) T ,C ′ Γ b ′ c ′ = ( C ′ Γ b ′ c ′ ) T , C ′′ Γ d ′′ e ′′ = ( C ′′ Γ d ′′ e ′′ ) T , so that – in particular – C γ a C − = − γ a T ; ● The T e Ad SO ( , ) × SO ( ) -invariance of the vacuum splitting – obvious; ● The homogeneous spaces: the NG ones ( AdS × S ) = ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , H = SO ( , ) × SO ( ) , h ≡ so ( , ) ⊕ so ( ) = ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ ∈ { , , } ⟨ J a ′′ b ′′ ⟩ , with the body ∣ s ( AdS × S )∣ = SO ( , )/ SO ( , ) × SO ( )/ SO ( ) ≡ AdS × S , and the HP one(s): the super-1-brane entirely in/over AdS ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , H vac = SO ( , ) × SO ( ) , h vac = ⟨ J ⟩ ⊕ ⊕ a ′′ ,b ′′ ∈ { , , } ⟨ J a ′′ b ′′ ⟩ , d = ⟨ J , J ⟩ , t ( ) vac = ⟨ P , P ⟩ , d − t ( ) vac = ⟨ P ⟩ ⊊ e ( ) ; the super-1-brane astride ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) , H vac = SO ( ) × SO ( ) , h vac = ⟨ J , J ⟩ , d = ⟨ J , J , J , J ⟩ , t ( ) vac = ⟨ P , P ⟩ , d − t ( ) vac = ⟨ P , P , P , P ⟩ ≡ e ( ) ; ● The exponential superparametrisation(s): the super-1-brane entirely in/over AdS σ vac0 ( θ α ′ α ′′ α ′′′ I , x a , φ , φ ) = e θ α ′ α ′′ α ′′′ I ⊗ Q α ′ α ′′ α ′′′ I ⋅ e x a ⊗ P a ⋅ e φ ⊗ J + φ ⊗ J ; the super-1-brane astride σ vac0 ( θ α ′ α ′′ α ′′′ I , x a , φ , φ , φ , φ ) = e θ α ′ α ′′ α ′′′ I ⊗ Q α ′ α ′′ α ′′′ I ⋅ e x a ⊗ P a ⋅ e φ ⊗ J + φ ⊗ J + φ ⊗ J + φ ⊗ J ; ● The superbackgrounds: the NG one sB ( NG ) = ( s ( AdS × S ) , η ab θ a L ⊗ θ b L , Σ L ∧ ( Cγ a ′ ⋅ ( ⊗ σ ) ⊗ σ ) Σ L ∧ θ a ′ L − Σ L ∧ ( Cγ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ ) Σ L ∧ θ a ′′ L ≡ χ ( ) PR ) ,θ L = Σ α ′ α ′′ α ′′′ I L ⊗ Q α ′ α ′′ α ′′′ I + θ a L ⊗ P a + θ a ′ b ′ L ⊗ J a ′ < b ′ + θ a ′′ b ′′ L ⊗ J a ′′ < b ′′ ith a supersymmetric global predecessor of the curving β ( ) PR = − i Σ L ∧ ( C γ ⊗ σ ) Σ L on ( SU ( , ∣ ) × SU ( , ∣ )) that descends to s ( AdS × S ) , and the HP one(s): the super-1-brane entirely in/over AdS (with ǫ = − sB ( HP ) ,λ = (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) ,χ ( ) PR − λ ǫ a ′ b ′ ( Σ L ∧ ( C γ a ′ ⋅ ( ⊗ σ ) ⊗ ) Σ L ∧ θ b ′ L − θ a ′ L ∧ θ ∧ θ b ′ ) ≡ ̂ χ ( ) PR ( ) ) , ⟨ P , P ⟩ ⊥ η ⟨ P ⟩ ; the super-1-brane astride sB ( HP ) ,λ = (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) χ ( ) PR + λ ( Σ L ∧ ( C γ ⋅ ( ⊗ σ ) ⊗ ) Σ L ∧ θ + Σ L ∧ ( C γ γ ⋅ ( ⊗ σ ) ⊗ ) Σ L ∧ θ − δ a ′ b ′ θ a ′ L ∧ θ ∧ θ b ′ L − δ a ′′ b ′′ θ ∧ θ a ′′ L ∧ θ b ′′ L ) ≡ ̂ χ ( ) PR ( ) ) , ⟨ P , P ⟩ ⊥ η ⟨ P , P , P , P ⟩ ; ● The Body-Localisation Constraints: the super-1-brane entirely in/over AdS : θ a ′′ L ≈ , a ′′ ∈ { , , } ; the super-1-brane astride: none. Example 3.16. The Metsaev–Tseytlin super-1-brane in s ( AdS × S ) . ● The mother super-Harish–Chandra pair:SU ( , ∣ ) ≡ ( SO ( , ) × SO ( ) , su ( , ∣ )) , consisting of the product Lie group SO ( , ) × SO ( ) and the Lie superalgebra su ( , ∣ ) = ⊕ ( α ′ ,α ′′ ,I ) ∈ , × , × { , } ⟨ Q α ′ α ′′ I ⟩ ⊕ ⊕ a ′ = ⟨ P a ′ ⟩ ⊕ ⊕ a ′′ = ⟨ P a ′′ ⟩ ⊕ ⊕ a ′ ,b ′ = ⟨ J a ′ b ′ = − J b ′ a ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ = − J b ′′ a ′′ ⟩ with the structure relations ( a, b ∈ , { Q α ′ α ′′ I , Q β ′ β ′′ J } = i (( C γ a ′ γ ) α ′ α ′′ Iβ ′ β ′′ J P a ′ − ( C γ a ′′ ) α ′ α ′′ Iβ ′ β ′′ J P a ′′ ) + ε ab ( C γ ab ) α ′ α ′′ Iβ ′ β ′′ J J ab , [ P a , P b ] = ε ab J ab , ε ab = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ + a, b ∈ , − a, b ∈ ,
90 otherwise , [ J ab , J cd ] = η ad J bc − η ac J bd + η bc J ad − η bd J ac , [ P a , J bc ] = η ab P c − η ac P b , (3.18) [ P a ′ , Q α ′ α ′′ I ] = − i ( γ a ′ γ ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J , [ P a ′′ , Q α ′ α ′′ I ] = i ( γ a ′′ ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J , [ J ab , Q α ′ α ′′ I ] = ( γ ab ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J . expressed in terms of the Pauli matrices as above, the generators { γ a ′ ≡ Γ a ′ ⊗ ⊗ σ , γ b ′′ ≡ ⊗ Γ b ′′ ⊗ σ } ( a ′ ,b ′′ ) ∈ , × , f the Clifford algebra Cliff ( R , ) of the Minkowski space R , with the metric η = diag ( − , + , + , + , + , + , + , + , + , + ) and their commutators γ ab = [ γ a , γ b ] , a, b ∈ , { Γ a ′ } a ′ ∈ , of the Clifford algebra Cliff ( R , ) (in the 4-dimensional spinor representation, in which theyare traceless) and the generators { Γ a ′′ } a ′′ ∈ , of the Clifford algebra Cliff ( R , ) (also in the 4-dimensional spinor representation, in whichthey are traceless), and in terms of the chirality operator γ ≡ − ⊗ σ of Cliff ( R , ) , as well as of the charge conjugation matrix C = C ′ ⊗ C ′′ ⊗ i σ = − C T of Cliff ( R , ) given as the product of the charge conjugation matrices C ′ = − C ′ T of Cliff ( R , ) and C ′′ = − C ′′ T of Cliff ( R , ) (as well as the Pauli matrix σ ), such that C ′ Γ a ′ = −( C ′ Γ a ′ ) T , C ′′ Γ a ′′ = −( C ′′ Γ a ′′ ) T , and C ′ Γ a ′ b ′ = ( C ′ Γ b ′ a ′ ) T , C ′′ Γ a ′′ b ′′ = ( C ′′ Γ a ′′ b ′′ ) T , so that – in particular – C γ a C − = − γ a T ; ● The T e Ad SO ( , ) × SO ( ) -invariance of the vacuum splitting – obvious; ● The homogeneous spaces: the NG ones ( AdS × S ) = SU ( , ∣ )/( SO ( , ) × SO ( )) , H = SO ( , ) × SO ( ) , h ≡ so ( , ) ⊕ so ( ) = ⊕ a ′ ,b ′ = ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ , with the body ∣ s ( AdS × S )∣ = SO ( , )/ SO ( , ) × SO ( )/ SO ( ) ≡ AdS × S , and the HP one(s) the super-1-brane entirely in/over AdS :SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) , H vac = SO ( , ) × SO ( ) × SO ( ) , h vac = ⟨ J ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ = − J b ′ a ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ = − J b ′′ a ′′ ⟩ , d = ⟨ J , J , J , J , J , J ⟩ , t ( ) vac = ⟨ P , P ⟩ , d − t ( ) vac = ⟨ P , P , P ⟩ ⊊ e ( ) ; the super-1-brane astride:SU ( , ∣ )/( SO ( ) × SO ( )) , H vac = SO ( ) × SO ( ) , h vac = ⊕ a ′ ,b ′ = ⟨ J a ′ b ′ = − J b ′ a ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ = − J b ′′ a ′′ ⟩ , d = ⊕ a ′ = ⟨ J a ′ ⟩ ⊕ ⊕ b ′′ = ⟨ J b ′′ ⟩ , t ( ) vac = ⟨ P , P ⟩ , d − t ( ) vac = ⊕ a ′ = ⟨ P a ′ ⟩ ⊕ ⊕ b ′′ = ⟨ P b ′′ ⟩ ≡ e ( ) ; The exponential superparametrisation(s): the super-1-brane entirely in/over AdS ( (̂ b ′ , ̂ c ′ ) ∈ { , , } ): σ vac0 ( θ α ′ α ′′ I , x a , φ ̂ b ′ , φ ̂ c ′ ) = e θ α ′ α ′′ I ⊗ Q α ′ α ′′ I ⋅ e x a ⊗ P a ⋅ e φ ̂ b ′ ⊗ J ̂ b ′ + φ ̂ c ′ ⊗ J ̂ c ′ ; the super-1-brane astride ( (̂ b ′ , ̂ c ′′ ) ∈ , × , σ vac0 ( θ α ′ α ′′ I , x a , φ b ′ , φ c ′′ ) = e θ α ′ α ′′ I ⊗ Q α ′ α ′′ I ⋅ e x a ⊗ P a ⋅ e φ ̂ b ′ ⊗ J ̂ b ′ + φ ̂ c ′′ ⊗ J ̂ c ′′ ; ● The superbackgrounds: the NG one sB ( NG ) = ( s ( AdS × S ) , η ab θ a L ⊗ θ b L , i Σ L ∧ C γ a ′ Σ L ∧ θ a ′ L − i Σ L ∧ C γ a ′′ γ Σ L ∧ θ a ′′ L ≡ χ ( ) MT ) ,θ L = Σ α ′ α ′′ I L ⊗ Q α ′ α ′′ I + θ a L ⊗ P a + θ a ′ b ′ L ⊗ J a ′ < b ′ + θ a ′′ b ′′ L ⊗ J a ′′ < b ′′ with a supersymmetric global predecessor of the curving β ( ) MT = − Σ L ∧ C γ Σ L on SU ( , ∣ ) that descends to s ( AdS × S ) , and the HP one(s): the super-1-brane entirely in/over AdS (with ǫ = − sB ( HP ) ,λ = ( SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) ,χ ( ) MT − λ ǫ a ′ b ′ ( i Σ L ∧ C γ a ′ γ Σ L ∧ θ b ′ L − δ ̂ c ′ ̂ d ′ θ a ′ L ∧ θ ̂ c ′ L ∧ θ b ′ ̂ d ′ L ) ≡ ̂ χ ( ) MT ( ) ) , ⟨ P , P ⟩ ⊥ η ⊕ ̂ a = ⟨ P ̂ a ⟩ ; the super-1-brane astride: sB ( HP ) ,λ = ( SU ( , ∣ )/( SO ( ) × SO ( )) ,χ ( ) MT + λ ( i Σ L ∧ C γ γ Σ L ∧ θ + i Σ L ∧ C γ Σ L ∧ θ − δ ̂ a ′ ̂ b ′ θ ̂ a ′ L ∧ θ ∧ θ ̂ b ′ L − δ ̂ a ′′ ̂ b ′′ θ ∧ θ ̂ a ′′ L ∧ θ ̂ b ′′ L ) ≡ ̂ χ ( ) MT ( ) ) , ⟨ P , P ⟩ ⊥ η ⊕ ̂ a ′ = ⟨ P ̂ a ′ ⟩ ⊕ ⊕ ̂ b ′′ = ⟨ P ̂ b ′′ ⟩ ; ● The Body-Localisation Constraints: the super-1-brane entirely in/over AdS : θ a ′′ L ≈ , a ′′ ∈ , the super-1-brane astride: none. Remark 3.17.
In all the examples considered above, h vac is spanned on the generators J ab of h withboth indices taken from the same subset (vacuum-vacuum resp. transverse-transverse), i.e. , those of thetype J ab resp. J ̂ a ̂ b , whereas d is spanned on the generators with mixed indices (vacuum-transverse), i.e. , those of the type J a ̂ b . Accordingly, the Descendability Constraint is universally satisfied in conse-quence of the relations [ J ab , J c ̂ d ] = η bc J a ̂ d − η ac J b ̂ d , [ J ̂ a ̂ b , J c ̂ d ] = δ ̂ b ̂ d J c ̂ a − δ ̂ a ̂ d J c ̂ b . We are now ready to discuss, in all generality, the supersymmetry of the super- σ -models introducedabove. 4. The vacuum of the super- σ -model Having defined the two classes of supersymmetric field theories of interest to us and stated a cor-respondence between them, we may, next, derive the dynamics of the super- p -brane in either one ofthe dual pair of models and study its (super)symmetries. As we ultimately intend to elucidate the(higher-)geometric nature of these symmetries, with emphasis on the local (or gauge) ones that onlypreserve a suitably relatively normalised combination of the two terms in the action functional, wechoose to work in the Hughes–Polchinski formulation of Def. 3.2 in which both terms are of the sametopological nature. The advantage of working in this formulation, to become apparent presently, is anintrinsically geometric form of the ensuing Euler–Lagrange equations and a neat geometric interpreta-tion of the infinitesimal symmetries. Their derivation, as well as the symmetry analysis of the model, ecome tractable upon imposing a few additional constraints on the supersymmetry Lie superalgebra.These we discover one by one in the explicit calculations that follow.We begin by writing out the logarithmic variation of the DF amplitude of the GS super- σ -model inthe HP formulation engendered by an arbitrary section δ ̂ ξ ∈ [ Ω p , T M H vac ] (in the S -point picture).As the amplitude is a pure differential character, cp Ref. [Sus19, Sec. 2] and Sec. 6, we obtain − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∫ Ω p ̂ ξ ∗ ( δ ̂ ξ ⌟ ̂ H ( p + ) vac ) , where ̂ H ( p + ) vac ≡ H ( p + ) vac + λ p d B ( p + ) ( HP ) . In order to cast the result in a form amenable to further analysis, we use the fact that the tensorialcomponents of the HP superbackground are pulled back from the mother supergroup G and pushforward the variation field δ ̂ ξ to the sub-supermanifolds V i of Eq. (3.10) over which we contract thepushforwards with the H vac -basic super- ( p + ) -form ̂ χ ( p + ) ≡ π ∗ G / H vac ̂ H ( p + ) vac . The pushforward decomposes in the basis of the tangent sheaf T Σ HP described in Prop. 3.6 as T χ i σ vac i δ ̂ ξ ( χ i ) = ∶ δθ αi T α i ( σ vac i ( χ i )) + δx ai T a i ( σ vac i ( χ i )) + δφ ̂ Si T ̂ S i ( σ vac i ( χ i )) ≡ ( δ ̂ ξ ζi L ζ + ∆ Ti L T )( σ vac i ( χ i )) , with the (H vac -)vertical correction ∆ Ti L T ensuring that the pushforward is actually tangent to V i ⊂ G, cp Prop. 3.6. The latter is linear in each of the independent coordinate variations δθ αi , δx ai and δφ ̂ Si andin the kernel of the super- ( p + ) -form with which we contract the variation below, and so its presencedoes not affect linear independence of the coordinate variations. Thus, we find the local formula − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ ( δ ̂ ξ ζı τ L ζ ⌟ ̂ χ ( p + ) ) . Taking into account The Dimensional Constraint and the structure of the HP super- ( p + ) -form(alongside the super-Maurer–Cartan equations), that altogether imply ̂ χ ( p + ) ≡ p ! ( χ αβa a ...a p Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L + ( − ) ∣ A ∣ ⋅ ∣ B ∣ + λ p f a AB ǫ a a a ...a p θ A L ∧ θ B L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) , we establish the identity − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ [ p ! δθ αı τ ( χ αβa a ...a p Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L + λ p f a αβ ǫ a a a ...a p Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) + ( p − ) ! δx aı τ ( χ αβaa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L + ( − ) ∣ A ∣ ⋅ ∣ B ∣ λ p f bAB ǫ aba a ...a p − θ A L ∧ θ B L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − λ p p f baB ǫ ba a ...a p θ B L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) + ( p − ) ! δx ̂ aı τ ( χ αβ ̂ aa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − λ p p f b ̂ aB ǫ ba a ...a p θ B L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) − λ p p ! δφ ̂ Sı τ f b ̂ SB ǫ ba a ...a p θ B L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ] . We shall now systematically derive from it the Euler–Lagrange equations of the super- σ -model in hand.We begin with those implied by the requirement of the vanishing of the above variation for δθ αi = = δx ai and δφ ̂ Si ≠
0. The relevant variation now reduces to − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = − λ p p ! ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ δφ ̂ Sı τ f b ̂ S ̂ a ǫ ba a ...a p θ ̂ a L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L38 nd infers, in consequence of the identification of the HP super- ( p + ) -form with the volume elementon the vacuum ( i.e. , on the embedded super- p -brane) that implies β ( p + ) ( HP ) ≠
0, the first subset of theEuler–Lagrange equations: θ ̂ a L ≈ , ̂ a ∈ p + , d − L . (4.1)In these, we readily recognise the Inverse Higgs Constraints (IHC) of Thm. 3.3, to be augmented withthe Body-Localisation Constraints (BLC) θ ̂ a L ≈ , ̂ a ∈ d − L + , d . (4.2)Just to reiterate and – in so doing – explain the notation that we shall be using throughout, we extractfrom our previous discussion the following Notational Convention:
The inscription θ µ L ≈ , written for some µ ∈ , δ , means that the vacuum embedding ̂ ξ ∈ [ Ω p , M H vac ] of the GS super- σ -model in the HP formulation is such that the image of the tangent of σ vac i ○ ̂ ξ, i ∈ I H vac lies in thesuperdistribution Ker θ µ L ↾ T Σ HP within the tangent sheaf T Σ HP of the HP section.The first set of field equations, taken together with the BLC, has a fundamentally different statusfrom that of the remaining field equations to be derived below – indeed, it enforces correspondencewith the original NG formulation. Their imposition reduces the variation to the form − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ [ p ! δθ αı τ ( χ αβa a ...a p Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L + λ p f a αβ ǫ a a a ...a p Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) + ( p − ) ! δx aı τ ( χ αβaa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L + ( − ) ∣ A ∣ ⋅ ∣ B ∣ λ p f bAB ǫ aba a ...a p − θ A L ∧ θ B L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − λ p p f bac ǫ ba a ...a p θ c L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L (4.3) − λ p p f baS ǫ ba a ...a p θ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) + ( p − ) ! δx ̂ aı τ ( χ αβ ̂ aa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − λ p p f b ̂ ac ǫ ba a ...a p θ c L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L − λ p p f b ̂ a ̂ S ǫ ba a ...a p θ ̂ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L )] from which we readily extract another set of Euler–Lagrange equations by setting δx ai = = δx ̂ ai and δθ αi ≠ θ a L , to wit, ( χ αβa a ...a p + λ p f a αβ ǫ a a a ...a p ) Σ β L ≈ , α ∈ , δ − d , a , a , . . . , a p ∈ , p . (4.4)The latter result begs elucidation as it does not, on the face of it, have an obvious geometric meaning.Therefore, we interrupt our derivation to discuss the anticipated geometric form of the field equationsin the formulation chosen, and the inevitable breakdown of translational symmetries effected by theirsolution, whereby we discover a path to its straightforward interpretation.It is natural to expect, based – in general – on the geometric nature of the field theories under con-sideration and – quite concretely – on the hitherto results ( cp also the remaining set of field equations,Eq. (4.11)), that the Euler–Lagrange equations determine the body of the embedded worldvolume ofthe super- p -brane (up to the choice of the initial condition) covariantly through identification of itstangent within the tangent sheaf of the HP section as the intersection of kernels of the componentsof the Maurer–Cartan super-1-form along the direct-sum completion of the vacuum subspace t ( ) vac . Bysupersymmetry, this scheme should have its Graßmann-odd counterpart. Indeed, the freezing of theGraßmann-even degrees of freedom transverse to the vacuum ( i.e. , in particular, those along e ( ) )results in a spontaneous breakdown of supersymmetry which is transmitted, via the anticommutator { t ( ) , t ( ) } ⊂ t ( ) ⊕ h , o the Graßmann-odd sector, and so the field equations in the odd sector ought to effect a ‘localisation’of the embedded worldvolume in what we presciently distinguished as the vacuum supspace t ( ) vac ≡ Im P ( ) on p. 12. Accordingly, we put forward The κ -Symmetry Constraints: We assume the Graßmann-odd component of the Euler–Lagrangeequations of the super- σ -model to define the corresponding sector of the vacuum of the field theoryby (co)identifying its normal in the tangent sheaf of the HP section, with the algebraic model t ( ) vac . Tothis end, we demand that the identity χ αβa a ...a p + λ p f a αβ ǫ a a a ...a p ! = ( δ − d − P ( ) ) γα ∆ βγa a ...a p ≡ ( δ − d − P ( ) ) γβ ∆ αγa a ...a p (4.5)be satisfied for some non-singular tensor ∆ βγa a ...a p , i.e. such that for any p -tuple ( a a . . . a p ) ∈ , p the endomorphism ∆ a a ...a p = ∆ αβa a ...a p δ βγ q α ⊗ Q γ ∈ End ( t ( ) ) (written in the notation of Eq. (2.12)) is invertible, and for a unique (up to a sign) value λ p ∈ R × .The reduction of the odd degrees of freedom thus effected must be strictly ( i.e. , supersymmetrically)correlated with the similar reduction in the even sector encoded in Eqs. (4.1)-(4.2), and so we furtherrequire { t ( ) vac , t ( ) vac } ! ⊂ t ( ) vac ⊕ h . (4.6)Clearly, whenever a projector can be read off from Eq. (4.4), the latter field equations can be rewrittenin the compact form ( δ − d − P ( ) ) αβ Σ β L ≈ , α ∈ , δ − d . (4.7)At this stage, in order to be able to proceed with our analysis, we make yet another simplifyingassumption (intuited from inspection of the best known examples), which we term, by structuralanalogy with the standard differential calculus on metric manifolds, The No-Curvature and No-Torsion Constraints:
We assume the vacuum embedding to be (lo-cally) flat and torsion-free up to a gauge transformation, in the sense expressed by the followingidentities [ t ( ) vac , t ( ) vac ] ! ⊂ h vac ! ⊃ [ e ( ) , e ( ) ] . (4.8) Remark 4.1.
As a consistency condition, implied by the Jacobi identity for triples from t ( ) vac × e ( ) × d ,we derive from the the above the additional constraint: [ t ( ) vac , e ( ) ] ! ⊂ d . (4.9)These, in conjunction with the formerly derived field equations, lead to a much simplified expressionfor the variation: − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ [ λ p p ! δx aı τ ( p f bSc ǫ aba a ...a p − θ S L ∧ θ c L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − f baS ǫ ba a ...a p θ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L ) + ( p − ) ! δx ̂ aı τ ( χ αβ ̂ aa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − λ p p f b ̂ a ̂ S ǫ ba a ...a p θ ̂ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L )] , which reduces further upon taking into account the assumed unimodularity of the adjoint action ofH vac on t ( ) vac (as expressed in Eq. (3.4)), − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ [ ( p − ) ! δx ̂ aı τ ( χ αβ ̂ aa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L15
One must keep in mind that the very definition of the super- σ -model embeddings presupposes a Graßmann-oddextension of the Graßmann-even worldvolume, as recalled in the opening paragraphs of Sec. 3. λ p p f b ̂ a ̂ S ǫ ba a ...a p θ ̂ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L )] . Note that the variation along t ( ) vac has vanished identically, and so we see that translations along t ( ) vac preserve the DF amplitudes after partial reduction (namely, after imposition of the IHC, of the BLCand of the spinorial field equations (4.4)).At this point, there remains one last condition to be imposed to conclude our derivation. As theonly source of the spinorial indices carried by the tensors χ αβ ̂ aa a ...a p − and by the structure constants f ̂ aαβ are the generators of the relevant Clifford algebra (and the charge conjugation matrix), we arerather naturally led to impose on the GS super- ( p + ) -cocycle The Γ -Constraints: We assume the following identities χ αγ ̂ aa a ...a p − P ( ) γβ = χ γβ ̂ aa a ...a p − ( δ − d − P ( ) ) γα (4.10)to hold true.When used together with the Euler–Lagrange equations (4.7), this yields − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = λ p p ! ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ δx ̂ aı τ f b ̂ S ̂ a ǫ ba a ...a p θ ̂ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L , and so, upon recalling the interpretation of the volume form in the vacuum, we arrive at the last setof field equations: θ ̂ S L ≈ , ̂ S ∈ D − δ + , D − δ . (4.11)We summarise our findings in Proposition 4.2.
Adopt the assumptions and the notation of Def. 3.2. If the Hughes–Polchinskisuperbackground sB ( HP ) p,λ p satisfies the Even Effective-Mixing Constraints (2.7) - (2.9) , the DimensionalConstraint (3.6) , the Descendability Constraint (3.15) , the No-Curvature and No-Torsion Constraints (4.8) and (4.9) , and the Γ -Constraints (4.10) , then the Euler–Lagrange equations of the correspondingGreen–Schwarz super- σ -model in the Hughes–Polchinski formulation for the super- p -brane in sB ( HP ) p,λ p restricted to field configurations subject to the Body-Localisation Constraints (4.1) read ( θ ̂ a L , ( χ αβa a ...a p + λ p f a αβ ǫ a a a ...a p ) Σ β L , θ ̂ S L ) ≈ , (̂ a, α, a k , ̂ S ) ∈ p + , d × , δ − d × , p × D − δ + , D − δ . If also the κ -Symmetry Constraints (4.5) and (4.6) are satisfied, the equations can be cast in the form P ge ⊕ d ○ θ L ≡ ( θ ̂ a L , ( δ − d − P ( ) ) αβ Σ β L , θ ̂ S L ) ≈ , (̂ a, α, ̂ S ) ∈ p + , d × , δ − d × D − δ + , D − δ , (4.12) where P ge ⊕ d ∶ g ↺ is the projector onto e ⊕ d with the kernel Ker P ge ⊕ d = t vac ⊕ h vac . Proof.
Given above. (cid:3)
We are thus led to
Definition 4.3.
Adopt the hitherto notation and assume that the Hughes–Polchinski superback-ground sB ( HP ) p,λ p satisfies all the Constraints listed in Prop. 4.2. The Hughes–Polchinski vacuumsuperdistribution of sB ( HP ) p,λ p is the sub-superdistribution of the correspondence superdistributionCorr HP / NG ( sB ( HP ) p,λ p ) of Def. 3.10 given byVac ( sB ( HP ) p,λ p ) ∶ = Ker ( P ge ⊕ d ○ θ L ↾ T Σ HP ) ⊂ Corr HP / NG ( sB ( HP ) p,λ p ) ⊂ T Σ HP (4.13)and modelled on t vac . ◇ emark 4.4. In order to neatly describe the HP vacuum superdistribution and related objects, weshall work with an eigenbasis of the projector P ( ) . Elements of a basis of Im P ( ) ≡ t ( ) vac shall bedenoted as Q α ≡ Λ βα Q β , α ∈ , q , and those of a basis of Ker P ( ) ≡ t ( ) vac as ̂ Q ̂ α ≡ ̂ Λ β ̂ α Q β , ̂ α ∈ q + , δ − d , where the Λ βα and the ̂ Λ β ̂ α are suitable numerical coefficiens.There are three physically motivated regularity criteria that we are compelled to invoke with regardto Vac ( sB ( HP ) p,λ p ) at this stage. We discuss at length the first two of them, postponing the last one toSec. 5 in which we shall have gathered the requisite formal and conceptual tools to make its analysisstructural. The first of these is H vac -descendability which is indispensable if we wish to have access tovacua stretching across several trivialising patches U vac i of an a priori nontrivial principal H vac -bundle(2.4), and even over a single patch it appears necessary to preserve the status of H vac as the model ofa gauge symmetry of the vacuum . We have Proposition 4.5.
Adopt the notation of Def. 4.3. If the Hughes–Polchinski vacuum superdistribution
Vac ( sB ( HP ) p,λ p ) is H vac -descendable in the sense of Def. 3.8, then [ h vac , t ( ) vac ] ⊂ t ( ) vac . Proof.
Obvious. (cid:3)
The second natural criterion is involutivity , which for an arbitrary superdistribution D ⊂ T Σ HP isexpressed by the relation [D , D} ⊂ D . In the light of the supergeometric variant of The Frobenius Theorem ( cp Ref. [CCF11, Thms. 6.1.12 &6.2.1]), this property ensures existence of a foliation of the HP section by the integral sub-supermanifoldsof D . From the point of view of the underlying field theory, this is to be understood as a foliation byvacua corresponding to different initial conditions. We put it in a separate Definition 4.6.
Adopt the hitherto notation. The foliation of the Hughes–Polchinski section Σ HP by the integral leaves of the Hughes–Polchinski vacuum superdistribution Vac ( sB ( HP ) p,λ p ) , whenever itexists, shall be termed the Hughes–Polchinski vacuum foliation of Σ HP and denoted as Σ HPvac (the disjoint union of its leaves), with the embedding ι vac ∶ Σ HPvac ↪ Σ HP . (4.14) ◇ We have
Proposition 4.7.
Adopt the notation of Def. 4.3. The Hughes–Polchinski vacuum superdistribution
Vac ( sB ( HP ) p,λ p ) is involutive and hence determines the Hughes–Polchinski vacuum foliation of Def. 4.6iff the following relations – to be termed the Vacuum-Superalgebra Constraints henceforth – aresimultaneously satisfied: { t ( ) vac , t ( ) vac } ⊂ t ( ) vac ⊕ h vac , [ t ( ) vac , t ( ) vac ] ⊂ t ( ) vac , [ h vac , t ( ) vac ] ⊂ t ( ) vac . When put in conjunction with the Constraints listed in Prop. 4.2 and assumed herein, they endow the vacuum supervector space of sB ( HP ) p,λ p defined as vac ( sB ( HP ) p,λ p ) ≡ t vac ⊕ h vac ⊂ g with the structure of a Lie sub-superalgebra, to be referred to as the vacuum superalgebra of sB ( HP ) p,λ p ,whenever it exists. roof. The vacuum superdistribution is a sub-superdistribution of the manifestly involutive superdis-tribution T Σ HP locally generated (over O G (V i ) ) by the vector fields T µ i of Prop. 3.6. Each of thelatter is identified uniquely by its horizontal component L µ ↾ V i with µ ∈ , δ , and so it suffices to checkwhich generators of f appear in the supercommutators [ t vac , t vac } (corresponding to the supercom-mutators [ L A , L B } of the horizontal components) and the commutators [ h vac , t vac ] (corresponding tothe commutators [ L S , L A ] of a horizontal component with a vertical correction). This yields the firststatement of the proposition. The second part is obvious. (cid:3) Remark 4.8.
The Vacuum-Superalgebra Constraints shall be encountered again when we come toinvestigate a peculiar odd local supersymmetry of the super- σ -model in the HP formulation that arisesin the correspondence sector of its configuration space. Meanwhile, we indicate a simple way of realisingthe above (and previous) constraints, suggested by the analysis of known examples – it consists inimposing the constraints P ( ) γα f aγβ = f aαγ P ( ) γβ , P ( ) γα f ̂ aγβ = f ̂ aαγ ( δ − d − P ( ) ) γβ , (4.15) P ( ) γα f Sγβ = f Sαγ P ( ) γβ , P ( ) γα f ̂ Sγβ = f ̂ Sαγ ( δ − d − P ( ) ) γβ , (4.16) P ( ) γα f βSγ = f γSα P ( ) βγ , P ( ) γα f β ̂ Sγ = f γ ̂ Sα ( δ − d − P ( ) ) βγ , (4.17) P ( ) γα f βaγ = f γaα P ( ) βγ . (4.18)In the remainder of the present paper, we regard the regular case in which the HP vacuum foliationis both H vac -descendable and involutive as the physically most appealing and natural one, hoping toreturn to the less obvious irregular case in a future study. In order to be able to quantify departuresfrom regularity observed in some of the superbackgrounds listed in Sec. 3 that we analyse one by one atthe end of this section, we need an adaptation, to the present supergeometric context, of the differential-geometric concepts that we introduce below after Ref. [Tan70]. This extra conceptual investment willpay back in the analysis of local supersymmetries of the super- σ -model in the next section. Definition 4.9.
Let M be a smooth manifold and D ⊂ T M a distribution over it. The weakderived flag of D is the filtration D ● ∶ D ≡ D − ⊂ D − ⊂ . . . ⊂ D − j ⊂ . . . of T M with components defined recursively as D − j ∶ = D − j + + [D , D − j + ] , j > . Given x ∈ M , denote by D − j ( x ) the R -linear span of all iterated Lie brackets, of length not greaterthan j , of the (local) generators of D evaluated at x . A distribution D is called regular if ∀ j ∈ N × ∀ x,y ∈M ∶ dim R D − j ( x ) = dim R D − j ( y ) . The bounded function µ ∶ M Ð→ N × ∶ x z→ min { j ∈ N × ∣ D − j − ( x ) = D − j ( x ) } shall be termed the height of D ● . A distribution D is called bracket-generating if ∀ x ∈M ∶ D − µ ( x ) ( x ) = T x M . For any regular bracket-generating distribution D , and any x ∈ M , write q − ( x ) ∶ = D − ( x ) , q − j ( x ) ∶ = D − j ( x )/ D − j + ( x ) , j > m ( x ) ∶ = µ ( x ) ⊕ j = q − j ( x ) , endowing that latter vector space with the Lie bracket [ ⋅ , ⋅ ] m ( x ) ∶ m ( x ) × m ( x ) Ð→ m ( x ) with restrictions [ ⋅ , ⋅ ] m ( x ) ↾ q − j ( x ) × q − j ( x ) ∶ q − j ( x ) × q − j ( x ) Ð→ q − j − j ( x ) ( X ( − j ) ( x ) + D − j + ( x ) , X ( − j ) ( x ) + D − j + ( x )) z→ [ X ( − j ) , X ( − j ) ]( x ) + D − j − j + ( x ) , written in terms of (local) sections X ( − j A ) A ∈ Γ loc (D − j A ) , A ∈ { , } . The graded nilpotent Lie algebra ( m ( x ) , [ ⋅ , ⋅ ] m ( x ) ) , generated by its subspace q − ( x ) (and hence termed fundamental ), is called the symbol of thedistribution D at the point x . It is customary to choose at a given x ∈ M a local basis B x of T x M adapted to the weak derived flag of D evaluated at that point, i.e. , to form the correspondingfiltration of local bases B − x ⊂ B − x ⊂ . . . ⊂ B x , D − j ( x ) = ⟨ B − jx ⟩ R , and subsequently present q − j ( x ) as q − j ( x ) ≡ ⟨ B − jx ∖ B − j + x ⟩ R . We adopt this convention in what follows.Whenever there exists a fundamental graded nilpotent Lie algebra (of minimal degree − µ ∈ Z < ) ( m ≡ µ ⊕ j = q − j , [ ⋅ , ⋅ ] m ) with the property ∀ x ∈M ∶ m ( x ) ≅ m , written in the category of graded nilpotent Lie algebras, we call D a distribution of constantsymbol m . ◇ Thus, we shall be interested in – among other things – the limit
Vac −∞ ( sB ( HP ) p,λ p ) ≡ Vac − µ ( sB ( HP ) p,λ p ) ⊂ T Σ HP of the weak derived (super)flag Vac ● ( sB ( HP ) p,λ p ) of the regular superdistribution Vac ( sB ( HP ) p,λ p ) and itsrelation to the mother tangent sheaf T Σ HP in the familiar examples. Invariably, the regular vacuumsuperdistribution itself and the various components of its weak derived flag are sewn from the respectiverestrictions to the superdomains V i , differing solely in the index i ∈ I H vac carried by the local generators.Therefore, whenever disclosing the anatomy of the breakdown of integrability, we give the local structure(in the form of an O G (V i ) -linear span) exclusively. Example 4.10. The vacuum superdistribution for the Green–Schwarz super-0-brane in sISO ( , ∣ )/ SO ( ) . The identity χ αβ + λ f αβ = αβ + λ Γ αβ ≡ αγ ( λ − Γ Γ ) γβ , yields – for λ ∈ {− , } – the possible projectors P ( ) ± = ± Γ Γ with the propertiesΓ P ( ) ± = ( − P ( ) ± ) Γ , Γ ̂ a P ( ) ± = P ( ) ± Γ ̂ a , C P ( ) ± C − = ( − P ( ) ± ) T , and so upon choosing, for the sake of concreteness, P ( ) ≡ P ( ) + = + Γ Γ , we readily verify ● the κ -Symmetry Constraints, { P ( ) γα Q γ , P ( ) δβ Q δ } = P ( ) γβ Γ αγ P ∈ ⟨ P ⟩ ⊕ ⊕ a,b = ⟨ J ab ⟩ ; the Even Effective-Mixing Constraints, [ J ̂ a ̂ b , P ] = ∈ ⟨ P ⟩ ∋ δ ̂ a ̂ b P = [ J ̂ a , P ̂ b ] , [ J ̂ a ̂ b , P ̂ c ] = δ ̂ b ̂ c P ̂ a − δ ̂ a ̂ c P ̂ b ∈ ⊕ ̂ d = ⟨ P ̂ d ⟩ ∋ P ̂ a = [ J ̂ a , P ] ; ● the No-Curvature and No-Torsion Constraints – trivial; ● the Γ-Constraints – trivial; ● the Vacuum-Superalgebra Constraints, with { P ( ) γα Q γ , P ( ) δβ Q δ } ∈ ⟨ P ⟩ ⊕ ⊕ ̂ a, ̂ b = ⟨ J ̂ a ̂ b ⟩ and [ J ̂ a ̂ b , P ( ) βα Q β ] = Γ β ̂ a ̂ b α P ( ) γβ Q γ ∈ Im P ( ) , [ P , P ( ) βα Q β ] = ∈ Im P ( ) ; ● SO ( ) -descendability.Consequently, the HP vacuum superdistribution with restrictionsVac ( sISO ( , ∣ )/ SO ( ) , ̂ χ ( ) GS ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i ⟩ is an SO ( ) -descendable integrable superdistribution associated with the Lie superalgebra vac ( sISO ( , ∣ )/ SO ( ) , ̂ χ ( ) GS ) = ⊕ α = ⟨ Q α ⟩ ⊕ ⟨ P ⟩ ⊕ ⊕ ̂ a, ̂ b = ⟨ J ̂ a ̂ b ⟩ . Example 4.11. The vacuum superdistribution for the Green–Schwarz super- ( k + ) -branein sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) for k ∈ { , , } . The identity χ αβa a ...a k + + λ k + f bαβ ǫ ba a ...a k + = ( k + ) ! ǫ ba a ...a k + Γ bαγ ⎛⎝ λ k + ( k + ) ! Dd, − Γ Γ ⋯ Γ k + ⎞⎠ γβ , yields – for λ k + ∈ { − ( k + ) ! ≡ λ − k + , ( k + ) ! ≡ λ + k + } – the possible projectors P ( ) λ ± k + = Dd, ± Γ Γ ⋯ Γ k + with propertiesΓ a P ( ) λ ± k + = ( D d, − P ( ) λ ± k + ) Γ a , Γ ̂ a P ( ) λ ± k + = P ( ) λ ± k + Γ ̂ a , C P ( ) λ ± k + C − = ( D d, − P ( ) λ ± k + ) T , and so upon choosing P ( ) ≡ P ( ) λ + k + = Dd, + Γ Γ ⋯ Γ k + , we check ● the κ -Symmetry Constraints, { P ( ) γα Q γ , P ( ) δβ Q δ } = P ( ) γβ Γ aαγ P a ∈ k + ⊕ a = ⟨ P a ⟩ ⊕ d ⊕ a,b = ⟨ J ab ⟩ ; ● the Even Effective-Mixing Constraints, [ J ab , P c ] = η bc P a − η ac P b ∈ ⊕ k + d = ⟨ P d ⟩ ∋ = [ J ̂ a ̂ b , P c ] , [ J a ̂ b , P ̂ c ] = δ ̂ b ̂ c P a ∈ ⊕ k + d = ⟨ P d ⟩ , [ J ab , P ̂ c ] = ∈ ⊕ d ̂ d = k + ⟨ P ̂ d ⟩ ∋ δ ̂ b ̂ c P ̂ a − δ ̂ a ̂ c P ̂ b = [ J ̂ a ̂ b , P ̂ c ] , [ J a ̂ b , P c ] = − η ac P ̂ b ∈ ⊕ d ̂ d = k + ⟨ P ̂ d ⟩ ; ● the No-Curvature and No-Torsion Constraints – trivial; ● the Γ-Constraints, ( C Γ ̂ a Γ a Γ a ⋯ Γ a k ) αγ P ( ) γβ = ( C P ( ) Γ ̂ a Γ a Γ a ⋯ Γ a k ) αβ = (( D d, − P ( ) ) T C Γ ̂ a Γ a Γ a ⋯ Γ a k ) αβ ≡ ( C Γ ̂ a Γ a Γ a ⋯ Γ a k ) γβ ( D d, − P ( ) ) γα ; the Vacuum-Superalgebra Constraints, with { P ( ) γα Q γ , P ( ) δβ Q δ } ∈ k + ⊕ a = ⟨ P a ⟩ ⊕ k + ⊕ a,b = ⟨ J ab ⟩ ⊕ d ⊕ ̂ a, ̂ b = k + ⟨ J ̂ a ̂ b ⟩ , and [ J ab , P ( ) βα Q β ] = Γ βab α P ( ) γβ Q γ ∈ Im P ( ) , [ J ̂ a ̂ b , P ( ) βα Q β ] = Γ β ̂ a ̂ b α P ( ) γβ Q γ ∈ Im P ( ) and [ P a , P ( ) βα Q β ] = ∈ Im P ( ) ; ● ( SO ( k + , ) × SO ( d − k − )) -descendability.Consequently, the HP vacuum superdistribution with restrictionsVac ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i = Dd, ⊕ α = ⟨ T α i ⟩ ⊕ k + ⊕ a = ⟨ T a i ⟩ is an ( SO ( k + , ) × SO ( d − k − )) -descendable integrable superdistribution associated with the Liesuperalgebra vac ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) = Dd, ⊕ α = ⟨ Q α ⟩ ⊕ k + ⊕ a = ⟨ P a ⟩ ⊕ k + ⊕ a,b = ⟨ J ab ⟩ ⊕ d ⊕ ̂ a, ̂ b = k + ⟨ J ̂ a ̂ b ⟩ . Example 4.12. The vacuum superdistribution for the Green–Schwarz super- ( k + ) -branein sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) for k ∈ { , } . The identity χ αβa a ...a k + + λ k + f bαβ ǫ ba a ...a k + = ( k + ) ! ǫ ba a ...a k + Γ bαγ ⎛⎝ λ k + ( k + ) ! Dd, − Γ Γ ⋯ Γ k + ⎞⎠ γβ , yields – for λ k + ∈ { − ( k + ) ! ≡ λ − k + , ( k + ) ! ≡ λ + k + } – the possible projectors P ( ) λ ± k + = Dd, ± Γ Γ ⋯ Γ k + with propertiesΓ a P ( ) λ ± k + = P ( ) λ ± k + Γ a , Γ ̂ a P ( ) λ ± k + = ( D d, − P ( ) λ ± k + ) Γ ̂ a , C P ( ) λ ± k + C − = P ( ) T λ ± k + , and so upon choosing P ( ) ≡ P ( ) λ + k + = Dd, + Γ Γ ⋯ Γ k + , we check ● the κ -Symmetry Constraints, { P ( ) γα Q γ , P ( ) δβ Q δ } = P ( ) γβ Γ aαγ P a ∈ k + ⊕ a = ⟨ P a ⟩ ⊕ d ⊕ a,b = ⟨ J ab ⟩ ; ● the Even Effective-Mixing Constraints, [ J ab , P c ] = η bc P a − η ac P b ∈ ⊕ k + d = ⟨ P d ⟩ ∋ = [ J ̂ a ̂ b , P c ] , [ J a ̂ b , P ̂ c ] = δ ̂ b ̂ c P a ∈ ⊕ k + d = ⟨ P d ⟩ , [ J ab , P ̂ c ] = ∈ ⊕ d ̂ d = k + ⟨ P ̂ d ⟩ ∋ δ ̂ b ̂ c P ̂ a − δ ̂ a ̂ c P ̂ b = [ J ̂ a ̂ b , P ̂ c ] , [ J a ̂ b , P c ] = − η ac P ̂ b ∈ ⊕ d ̂ d = k + ⟨ P ̂ d ⟩ ; ● the No-Curvature and No-Torsion Constraints – trivial; ● the Γ-Constraints, ( C Γ ̂ a Γ a Γ a ⋯ Γ a k + ) αγ P ( ) γβ = ( C ( D d, − P ( ) ) Γ ̂ a Γ a Γ a ⋯ Γ a k + ) αβ = (( D d, − P ( ) ) T C Γ ̂ a Γ a Γ a ⋯ Γ a k + ) αβ ≡ ( C Γ ̂ a Γ a Γ a ⋯ Γ a k + ) γβ ( D d, − P ( ) ) γα ; the Vacuum-Superalgebra Constraints, with { P ( ) γα Q γ , P ( ) δβ Q δ } ∈ k + ⊕ a = ⟨ P a ⟩ ⊕ k + ⊕ a,b = ⟨ J ab ⟩ ⊕ d ⊕ ̂ a, ̂ b = k + ⟨ J ̂ a ̂ b ⟩ , and [ J ab , P ( ) βα Q β ] = Γ βab α P ( ) γβ Q γ ∈ Im P ( ) , [ J ̂ a ̂ b , P ( ) βα Q β ] = Γ β ̂ a ̂ b α P ( ) γβ Q γ ∈ Im P ( ) and [ P a , P ( ) βα Q β ] = ∈ Im P ( ) ; ● ( SO ( k + , ) × SO ( d − k − )) -descendability.Consequently, the HP vacuum superdistribution with restrictionsVac ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i = Dd, ⊕ α = ⟨ T α i ⟩ ⊕ k + ⊕ a = ⟨ T a i ⟩ is an ( SO ( k + , ) × SO ( d − k − )) -descendable integrable superdistribution associated with the Liesuperalgebra vac ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) = Dd, ⊕ α = ⟨ Q α ⟩ ⊕ k + ⊕ a = ⟨ P a ⟩ ⊕ k + ⊕ a,b = ⟨ J ab ⟩ ⊕ d ⊕ ̂ a, ̂ b = k + ⟨ J ̂ a ̂ b ⟩ . Example 4.13. The vacuum superdistribution for the Zhou super-1-brane in SU ( , ∣ ) /( SO ( , ) × SO ( )) . The identity χ α ′ α ′′ Iβ ′ β ′′ Ja ′ + λ f b ′ α ′ α ′′ Iβ ′ β ′′ J ǫ b ′ a ′ = ( C γ b ′ ⊗ ) α ′ α ′′ Iγ ′ γ ′′ K ǫ b ′ a ′ ( λ + γ γ ⊗ σ ) γ ′ γ ′′ Kβ ′ β ′′ J , yields – for λ ∈ { − , } – the possible projectors P ( ) ∓ = ± γ γ ⊗ σ with properties ( γ a ⊗ ) P ( ) ± = ( − P ( ) ± ) ( γ a ⊗ ) , ( γ ̂ a ⊗ ) P ( ) ± = P ( ) ± ( γ ̂ a ⊗ ) , ( C ⊗ ) P ( ) ± ( C ⊗ ) − = ( − P ( ) ± ) T , and so upon choosing P ( ) ≡ P ( ) − = + γ γ ⊗ σ , we check ● the κ -Symmetry Constraints, { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δδ ′ Lβ ′ β ′′ J Q δδ ′ L } = P ( ) γ ′ γ ′′ Kβ ′ β ′′ J (( C γ a ⊗ ) α ′ α ′′ Iγ ′ γ ′′ K P a − i ( C ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K J − i ( C γ ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K J ) ∈ ⟨ P , P ⟩ ⊕ ⟨ J , J ⟩ ; ● the Even Effective-Mixing Constraints, [ J , P a ] = δ a P − η a P ∈ ⟨ P , P ⟩ ∈ = [ J , P a ] , [ J , P ̂ a ] = ∈ ⟨ P , P ⟩ ∈ δ ̂ a P − δ ̂ a P = [ J , P ̂ a ] ; ● the No-Curvature and No-Torsion Constraints, with [ P , P ] = J ∈ ⟨ J , J ⟩ ∋ − J = [ P , P ] ; the Γ-Constraints, ( C γ ̂ a ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K P ( ) γ ′ γ ′′ Kβ ′ β ′′ J = (( C ⊗ ) P ( ) γ ̂ a ⊗ σ ) α ′ α ′′ Iβ ′ β ′′ J = (( − P ( ) ) T ( C γ ̂ a ⊗ σ )) α ′ α ′′ Iβ ′ β ′′ J ≡ ( C γ ̂ a ⊗ σ ) γ ′ γ ′′ Kβ ′ β ′′ J ( − P ( ) ) γ ′ γ ′′ Kα ′ α ′′ I ; ● the Vacuum-Superalgebra Constraints, with { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δ ′ δ ′′ Lβ ′ β ′′ J Q δ ′ δ ′′ L } ∈ ⟨ P , P ⟩ ⊕ ⟨ J , J ⟩ and [ J , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ γ ⊗ ) β ′ β ′′ Jα ′ α ′′ I P ( ) γ ′ γ ′′ Kβ ′ β ′′ J Q γ ′ γ ′′ K ∈ Im P ( ) , [ J , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ γ ⊗ ) β ′ β ′′ Jα ′ α ′′ I P ( ) γ ′ γ ′′ Kβ ′ β ′′ J Q γ ′ γ ′′ K ∈ Im P ( ) , and [ P a , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = i ( γ γ γ a ⊗ σ ) γ ′ γ ′′ Kα ′ α ′′ I P ( ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ∈ Im P ( ) ; ● ( SO ( , ) × SO ( )) -descendability.Consequently, the HP vacuum superdistribution with restrictionsVac ( SU ( , ∣ ) /( SO ( , ) × SO ( )) , ̂ χ ( ) Zh ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is an ( SO ( , ) × SO ( )) -descendable integrable superdistribution associated with the Lie superalgebra vac ( SU ( , ∣ ) /( SO ( , ) × SO ( )) , ̂ χ ( ) Zh ( ) ) = ⊕ α = ⟨ Q α ⟩ ⊕ ⟨ P , P ⟩ ⊕ ⟨ J , J ⟩ . Example 4.14. The vacuum superdistribution for the Zhou super-1-brane in SU ( , ∣ ) . The identity χ α ′ α ′′ Iβ ′ β ′′ Ja + λ f bα ′ α ′′ Iβ ′ β ′′ J ǫ ba = ( C γ a ⊗ σ ) α ′ α ′′ Iβ ′ β ′′ J + λ ( C Γ b ⊗ ) α ′ α ′′ Iβ ′ β ′′ J ǫ ba ≡ ( C γ b ⊗ ) α ′ α ′′ Iγ ′ γ ′′ K ǫ ba ( λ + γ γ ⊗ σ ) γ ′ γ ′′ Kβ ′ β ′′ J , yields – for λ ∈ { − , } – the possible projectors P ( ) ∓ = ± γ γ ⊗ σ with properties ( γ a ⊗ ) P ( ) ± = ( − P ( ) ± ) ( γ a ⊗ ) , ( γ ̂ a ⊗ ) P ( ) ± = P ( ) ± ( γ ̂ a ⊗ ) , ( C ⊗ ) P ( ) ± ( C ⊗ ) − = ( − P ( ) ± ) T , and so upon choosing P ( ) ≡ P ( ) − = + γ γ ⊗ σ , we check ● the κ -Symmetry Constraints, { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δδ ′ Lβ ′ β ′′ J Q δδ ′ L } = P ( ) γ ′ γ ′′ Kβ ′ β ′′ J (( C γ a ⊗ ) α ′ α ′′ Iγ ′ γ ′′ K P a − i ( C ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K J − i ( C γ ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K J ) ∈ ⟨ P , P ⟩ ⊕ ⟨ J , J ⟩ ; the Even Effective-Mixing Constraints [ J , P ] = P ∈ ⟨ P , P ⟩ ∋ = [ J , P ] , [ J , P ] = ∈ ⟨ P , P ⟩ ∋ − P = [ J , P ] , [ J , P ] = P ∈ ⟨ P , P ⟩ ∋ = [ J , P ] , [ J , P ] = ∈ ⟨ P , P ⟩ ∋ P = [ J , P ] ; ● the No-Curvature and No-Torsion Constraints, with [ P , P ] = ∈ ∋ [ P , P ] ; ● the Γ-Constraints, ( C γ ̂ a ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K P ( ) γ ′ γ ′′ Kβ ′ β ′′ J = (( C ⊗ ) P ( ) γ ̂ a ⊗ σ ) α ′ α ′′ Iβ ′ β ′′ J = (( − P ( ) ) T ( C γ ̂ a ⊗ σ )) α ′ α ′′ Iβ ′ β ′′ J ≡ ( C γ ̂ a ⊗ σ ) γ ′ γ ′′ Kβ ′ β ′′ J ( − P ( ) ) γ ′ γ ′′ Kα ′ α ′′ I . The Vacuum-Superalgebra Constraints, on the other hand, are not satisfied in view of the above, { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δ ′ δ ′′ Lβ ′ β ′′ J Q δ ′ δ ′′ L } ∉ ⟨ P , P ⟩ , with [ P a , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = i ( γ γ γ a ⊗ σ ) γ ′ γ ′′ Kα ′ α ′′ I ( − P ( ) ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ∈ Im ( − P ( ) ) , and so the HP vacuum superdistribution with restrictionsVac ( SU ( , ∣ ) , ̂ χ ( ) Zh ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is nonintegrable. Its weak derived flag with (local) components of the symbol q − ( i ) ≡ Vac ( SU ( , ∣ ) , ̂ χ ( ) Zh ( ) ) ↾ V i , q − ( i ) = ⊕ ̂ α = ⟨ T ̂ α i ⟩ ⊕ ⟨ T i , T i ⟩ , q − ( i ) = ⟨ T i , T i ⟩ is bracket-generating for the tangent sheaf of the HP section,Vac −∞ ( SU ( , ∣ ) , ̂ χ ( ) Zh ( ) ) = T Σ HP . Example 4.15. The vacuum superdistribution for the Park–Rey super-1-brane in ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) . The identity χ α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ Ja ′ + λ f b ′ α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J ǫ b ′ a ′ = ( C γ b ′ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K ǫ b ′ a ′ ( λ − γ γ ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J , yields – for λ ∈ { − , } – the possible projectors P ( ) ± = ± γ γ ⊗ σ with properties ( γ a ⊗ ) P ( ) ± = ( − P ( ) ± ) ( γ a ⊗ ) , ( γ ̂ a ⊗ ) P ( ) ± = P ( ) ± ( γ ̂ a ⊗ ) , ( C ⊗ ) P ( ) ± ( C ⊗ ) − = ( − P ( ) ± ) T , and so upon choosing P ( ) ≡ P ( ) + = + γ γ ⊗ σ , we check ● the κ -Symmetry Constraints, { P ( ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I Q γ ′ γ ′′ γ ′′′ K , P ( ) δδ ′ δ ′′′ Lβ ′ β ′′ β ′′′ J Q δδ ′ δ ′′′ L } = P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J ( ( C γ a ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P a + i ( C γ γ ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K J i ( C γ a ′′ b ′′ γ ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K J a ′′ b ′′ ) ∈ ⟨ P , P ⟩ ⊕ ⊕ a ′ ,b ′ = ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ ; ● the Even Effective-Mixing Constraints, [ J , P a ] = δ a P − η a P ∈ ⟨ P , P ⟩ ∋ = [ J a ′′ b ′′ , P c ] , [ J a , P ̂ b ] = δ ̂ b P a ∈ ⟨ P , P ⟩ ∋ δ ̂ b ̂ c P a = [ J a ̂ b , P ̂ c ] , [ J , P ̂ a ] = ∈ ⊕ ̂ d = ⟨ P ̂ d ⟩ ∋ δ b ′′ ̂ c P a ′′ − δ a ′′ ̂ c P b ′′ = [ J a ′′ b ′′ , P ̂ c ] , [ J a , P b ] = − δ ab P ∈ ⊕ ̂ d = ⟨ P ̂ d ⟩ ∋ − δ ac P ̂ b = [ J a ̂ b , P c ] ; ● the No-Curvature and No-Torsion Constraints, with [ P , P ] = J ∈ ⟨ J ⟩ ⊕ ⊕ b ′′ ,c ′′ = ⟨ J b ′′ c ′′ ⟩ ∋ = [ P , P a ′′ ] , [ P a ′′ , P b ′′ ] = − J a ′′ b ′′ ∈ ⟨ J ⟩ ⊕ ⊕ c ′′ ,d ′′ = ⟨ J c ′′ d ′′ ⟩ ; ● the Γ-Constraints, ( C γ ⋅ ( ⊗ σ ) ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J = (( C ⊗ ) P ( ) ( γ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J = (( − P ( ) ) T ( C γ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J ≡ ( C γ ⋅ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J ( − P ( ) ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I , − ( C γ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J = − (( C ⊗ ) P ( ) ( γ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J = − (( − P ( ) ) T ( C γ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J ≡ − ( C γ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J ( − P ( ) ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I ; ● the Vacuum-Superalgebra Constraints, with { P ( ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I Q γ ′ γ ′′ γ ′′′ K , P ( ) δδ ′ δ ′′′ Lβ ′ β ′′ β ′′′ J Q δδ ′ δ ′′′ L } ∈ ⟨ P , P ⟩ ⊕ ⟨ J ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ and [ J , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I Q β ′ β ′′ β ′′′ J ] = ( γ γ ⊗ ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J Q γ ′ γ ′′ γ ′′′ K ∈ Im P ( ) , [ J a ′′ b ′′ , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I Q β ′ β ′′ β ′′′ J ] = ( γ a ′′ γ b ′′ ⊗ ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J Q γ ′ γ ′′ γ ′′′ K ∈ Im P ( ) , and [ P a , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ α ′′′ I Q β ′ β ′′ β ′′′ J ] = i ( γ a γ ⋅ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I P ( ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J ∈ Im P ( ) ; ● ( SO ( , ) × SO ( )) -descendability.Consequently, the HP vacuum superdistribution with restrictionsVac (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is an ( SO ( , ) × SO ( )) -descendable integrable superdistribution associated with the Lie superalgebra vac (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , ̂ χ ( ) PR ( ) ) = ⊕ α = ⟨ Q α ⟩ ⊕ ⟨ P , P ⟩ ⊕ ⟨ J ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ . xample 4.16. The vacuum superdistribution for the Park–Rey super-1-brane in ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) . The identities χ α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J − λ f α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J = ( C γ γ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K ( λ + γ γ γ ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J ,χ α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J + λ f α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J = ( C γ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K ( λ + γ γ γ ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J , yield – for λ ∈ { − , } – the possible projectors P ( ) ∓ = ± γ γ γ ⊗ σ with properties ( γ a ⊗ ) P ( ) ± = P ( ) ± ( γ a ⊗ ) , ( γ ̂ a ⊗ ) P ( ) ± = ( − P ( ) ± ) ( γ ̂ a ⊗ ) , ( C ⊗ ) P ( ) ± ( C ⊗ ) − = P ( ) T ± , and so upon choosing P ( ) ≡ P ( ) + = + γ γ γ ⊗ σ , we check ● the κ -Symmetry Constraints, { P ( ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I Q γ ′ γ ′′ γ ′′′ K , P ( ) δ ′ δ ′′ δ ′′′ Lβ ′ β ′′ β ′′′ J Q δ ′ δ ′′ δ ′′′ L } = P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J (( C γ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P − ( C γ γ ⋅ ( ⊗ σ ) ⊗ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P − i ( C γ a ′ γ ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K J a ′ + i ( C γ a ′′ γ ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K J a ′′ ) ∈ ⟨ P , P ⟩ ⊕ ⊕ a ′ ,b ′ = ⟨ J a ′ ,b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ ; ● the Even Effective-Mixing Constraints, [ J , P a ] = ∈ ⟨ P , P ⟩ ∈ = [ J , P a ] , [ J a ′ , P ̂ b ] = δ a ′ ̂ b P ∈ ⟨ P , P ⟩ ∋ δ a ′′ ̂ b P = [ J a ′′ , P ̂ b ] , [ J , P ̂ a ] = δ ̂ a P − δ ̂ a P ∈ ⟨ P , P , P , P ⟩ ∋ δ ̂ a P − δ ̂ a P = [ J , P ̂ a ] , [ J a ′ , P b ] = − η b P a ′ ∈ ⟨ P , P , P , P ⟩ ∋ − δ b P a ′′ = [ J a ′′ , P b ] ; ● the No-Curvature and No-Torsion Constraints, with [ P , P ] = ∈ ⟨ J , J ⟩ ∋ J = [ P , P ] , [ P , P ] = [ P , P ] = [ P , P ] = [ P , P ] = ∈ ⟨ J , J ⟩ ∋ − J = [ P , P ] ; ● the Γ-Constraints, ( C γ ̂ a ′ ⋅ ( ⊗ σ ) ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J = (( C ⊗ ) ( − P ( ) ) ( γ ̂ a ′ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J = (( − P ( ) ) T ( C γ ̂ a ′ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J ≡ ( C γ ̂ a ′ ⋅ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J ( − P ( ) ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I , ̂ a ′ ∈ { , } , − ( C γ ̂ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ ) α ′ α ′′ α ′′′ Iγ ′ γ ′′ γ ′′′ K P ( ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J − (( C ⊗ ) ( − P ( ) ) ( γ ̂ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J = − (( − P ( ) ) T ( C γ ̂ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ )) α ′ α ′′ α ′′′ Iβ ′ β ′′ β ′′′ J ≡ − ( C γ ̂ a ′′ γ ⋅ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kβ ′ β ′′ β ′′′ J ( − P ( ) ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I , ̂ a ′′ ∈ { , } ; ● ( SO ( ) × SO ( )) -descendability.The Vacuum-Superalgebra Constraints, on the other hand, are not satisfied in view of the above, { P ( ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I Q γ ′ γ ′′ γ ′′′ K , P ( ) δ ′ δ ′′ δ ′′′ Lβ ′ β ′′ β ′′′ J Q δ ′ δ ′′ δ ′′′ L } ∉ ⟨ P , P ⟩ ⊕ ⟨ J , J ⟩ , with [ J , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ γ ⊗ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I P ( ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J ∈ Im P ( ) , [ J , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ γ ⊗ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I P ( ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J ∈ Im P ( ) but also [ P , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = i ( γ γ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I ( − P ( ) ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J ∈ Im ( − P ( ) ) , [ P , P ( ) β ′ β ′′ β ′′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = − i ( γ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I ( − P ( ) ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J ∈ Im ( − P ( ) ) , and so the HP vacuum superdistribution with restrictionsVac (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is ( SO ( ) × SO ( )) -descendable but not integrable. Its weak derived flag with (local) components ofthe symbol q − ( i ) ≡ Vac (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i , q − ( i ) = ⊕ ̂ α = ⟨ T ̂ α i ⟩ ⊕ ⟨ T i , T i , T i , T i ⟩ , q − ( i ) = ⟨ T i , T i , T i , T i ⟩ is bracket-generating for the tangent sheaf of the HP section,Vac −∞ (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) , ̂ χ ( ) PR ( ) ) = T Σ HP . Example 4.17. The vacuum superdistribution for the Metsaev–Tseytlin super-1-brane in SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) . The identity χ α ′ α ′′ Iβ ′ β ′′ Ja ′ + λ f b ′ α ′ α ′′ Iβ ′ β ′′ J ǫ b ′ a ′ = i ( C γ b ′ γ ) α ′ α ′′ Iγ ′ γ ′′ K ǫ b ′ a ′ ( λ − γ γ γ ) γ ′ γ ′′ Kβ ′ β ′′ J , yields – for λ ∈ { − , } – the possible projectors P ( ) ± = ± γ γ γ with properties γ a P ( ) ± = P ( ) ± γ a , γ ̂ a P ( ) ± = ( − P ( ) ± ) γ ̂ a , C P ( ) ± C − = P ( ) T ± , and so upon choosing P ( ) ≡ P ( ) + = + γ γ γ , we check ● the κ -Symmetry Constraints, { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δδ ′ Lβ ′ β ′′ J Q δδ ′ L } = P ( ) γ ′ γ ′′ Kβ ′ β ′′ J ( i ( C γ a γ ) α ′ α ′′ Iγ ′ γ ′′ K P a + ( C γ ) α ′ α ′′ Iγ ′ γ ′′ K J ) ∈ ⟨ P , P ⟩ ⊕ ⊕ a ′ ,b ′ = ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ ; the Even Effective-Mixing Constraints, [ J , P a ] = δ a P − η a P ∈ ⟨ P , P ⟩ ∋ = [ J ̂ a ̂ b , P c ] , [ J a ̂ b , P ̂ c ] = δ ̂ b ̂ c P a ∈ ⟨ P , P ⟩ , [ J , P ̂ a ] = ∈ ⊕ ̂ d = ⟨ P ̂ d ⟩ ∋ δ ̂ b ̂ c P ̂ a − δ ̂ a ̂ c P ̂ b = [ J ̂ a ̂ b , P ̂ c ] , [ J a ̂ b , P c ] = − η ac P ̂ b ∈ ⊕ ̂ d = ⟨ P ̂ d ⟩ ; ● the No-Curvature and No-Torsion Constraints, with [ P , P ] = J ∈ ⟨ J ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ ⟩ ⊕ ⊕ b ′′ ,c ′′ = ⟨ J b ′′ c ′′ ⟩ ∋ = [ P , P a ′′ ] = [ P , P a ′′ ] = [ P , P a ′′ ] , [ P , P ] = J ∈ ⟨ J ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ ∋ J = [ P , P ] , [ P , P ] = J ∈ ⟨ J ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ ⟩ ⊕ ⊕ c ′′ ,d ′′ = ⟨ J c ′′ d ′′ ⟩ ∋ − J a ′′ b ′′ = [ P a ′′ , P b ′′ ] ; ● the Γ-Constraints, ( C γ ̂ a ′ ) α ′ α ′′ Iγ ′ γ ′′ K P ( ) γ ′ γ ′′ Kβ ′ β ′′ J = ( C ( − P ( ) ) γ ̂ a ′ ) α ′ α ′′ Iβ ′ β ′′ J = (( − P ( ) ) T ( C γ ̂ a ′ )) α ′ α ′′ Iβ ′ β ′′ J ≡ ( C γ ̂ a ′ ) γ ′ γ ′′ Kβ ′ β ′′ J ( − P ( ) ) γ ′ γ ′′ Kα ′ α ′′ I , ̂ a ′ ∈ { , , } , − ( C γ a ′′ γ ) α ′ α ′′ Iγ ′ γ ′′ K P ( ) γ ′ γ ′′ Kβ ′ β ′′ J = − ( C ( − P ( ) ) γ a ′′ γ ) α ′ α ′′ Iβ ′ β ′′ J = − (( − P ( ) ) T ( C γ a ′′ γ )) α ′ α ′′ Iβ ′ β ′′ J ≡ − ( C γ a ′′ γ ) γ ′ γ ′′ Kβ ′ β ′′ J ( − P ( ) ) γ ′ γ ′′ Kα ′ α ′′ I ; ● the Vacuum-Superalgebra Constraints, with { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δδ ′ Lβ ′ β ′′ J Q δδ ′ L } ∈ ⟨ P , P ⟩ ⊕ ⟨ J ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ and [ J , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ ) β ′ β ′′ Jα ′ α ′′ I P ( ) γ ′ γ ′′ Kβ ′ β ′′ J Q γ ′ γ ′′ K ∈ Im P ( ) , [ J ̂ a ̂ b , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ ̂ a ̂ b ) β ′ β ′′ Jα ′ α ′′ I P ( ) γ ′ γ ′′ Kβ ′ β ′′ J Q γ ′ γ ′′ K ∈ Im P ( ) , and [ P a , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = − i ( γ a γ ) γ ′ γ ′′ Kα ′ α ′′ I P ( ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ∈ Im P ( ) ; ● ( SO ( , ) × SO ( ) × SO ( )) -descendability.Consequently, the HP vacuum superdistribution with restrictionsVac ( SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is an ( SO ( , ) × SO ( ) × SO ( )) -descendable integrable superdistribution associated with the Lie su-peralgebra vac ( SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) = ⊕ α = ⟨ Q α ⟩ ⊕ ⟨ P , P ⟩ ⊕ ⟨ J ⟩ ⊕ ⊕ a ′ ,b ′ ∈ { , , } ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ . Example 4.18. The vacuum superdistribution for the Metsaev–Tseytlin super-1-brane in SU ( , ∣ )/( SO ( ) × SO ( )) . The identities χ α ′ α ′′ Iβ ′ β ′′ J − λ f α ′ α ′′ Iβ ′ β ′′ J = i ( C γ ) α ′ α ′′ Iγ ′ γ ′′ K ( λ + γ γ ) γ ′ γ ′′ Kβ ′ β ′′ J ,χ α ′ α ′′ Iβ ′ β ′′ J + λ f α ′ α ′′ Iβ ′ β ′′ J = i ( C γ γ ) α ′ α ′′ Iγ ′ γ ′′ K ( λ + γ γ ) γ ′ γ ′′ Kβ ′ β ′′ J , yield – for λ ∈ { − , } – the possible projectors P ( ) ∓ = ± γ γ ith properties γ a P ( ) ± = ( − P ( ) ± ) γ a , γ ̂ a P ( ) ± = P ( ) ± γ ̂ a , C P ( ) ± C − = ( − P ( ) ± ) T , and so upon choosing P ( ) ≡ P ( ) − = + γ γ , we check ● the κ -Symmetry Constraints, { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δ ′ δ ′′ Lβ ′ β ′′ J Q δ ′ δ ′′ L } = P ( ) γ ′ γ ′′ Kβ ′ β ′′ J ( i ( C γ γ ) α ′ α ′′ Iγ ′ γ ′′ K P − i ( C γ ) α ′ α ′′ Iγ ′ γ ′′ K P + ( C γ a ′ ) α ′ α ′′ Iγ ′ γ ′′ K J a ′ − ( C γ a ′′ ) α ′ α ′′ Iγ ′ γ ′′ K J a ′′ ) ∈ ⟨ P , P ⟩ ⊕ ⊕ a ′ ,b ′ = ⟨ J a ′ ,b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ ; ● the Even Effective-Mixing Constraints, [ J ̂ a ̂ b , P c ] = ∈ ⟨ P , P ⟩ , [ J a ̂ b , P ̂ c ] = δ ̂ b ̂ c P a ∈ ⟨ P , P ⟩ , [ J ̂ a ̂ b , P ̂ c ] = δ ̂ b ̂ c P ̂ a − δ ̂ a ̂ c P ̂ b ∈ ⊕ a ′ = ⟨ P a ′ ⟩ ⊕ ⊕ b ′′ = ⟨ P b ′′ ⟩ , [ J a ̂ b , P c ] = − η ac P ̂ b ∈ ⊕ a ′ = ⟨ P a ′ ⟩ ⊕ ⊕ b ′′ = ⟨ P b ′′ ⟩ ; ● the No-Curvature and No-Torsion Constraints, with [ P , P ] = ∈ ⊕ c ′ ,d ′ = ⟨ J c ′ d ′ ⟩ ⊕ ⊕ c ′′ ,d ′′ = ⟨ J c ′′ d ′′ ⟩ ∋ δ a ′ ̂ a δ b ′ ̂ b J a ′ b ′ − δ a ′′ ̂ a δ b ′′ ̂ b J a ′′ b ′′ = [ P ̂ a , P ̂ b ] ; ● the Γ-Constraints, (written for ̂ a ′ ∈ { , , , } and ̂ a ′′ ∈ { , , , } ) ( C γ ̂ a ′ ) α ′ α ′′ Iγ ′ γ ′′ K P ( ) γ ′ γ ′′ Kβ ′ β ′′ J = ( C P ( ) γ ̂ a ′ ) α ′ α ′′ Iβ ′ β ′′ J = (( − P ( ) ) T ( C γ ̂ a ′ )) α ′ α ′′ Iβ ′ β ′′ J ≡ ( C γ ̂ a ′ ) γ ′ γ ′′ Kβ ′ β ′′ J ( − P ( ) ) γ ′ γ ′′ Kα ′ α ′′ I , − ( C γ ̂ a ′′ γ ) α ′ α ′′ Iγ ′ γ ′′ K P ( ) γ ′ γ ′′ Kβ ′ β ′′ J = − ( C P ( ) γ ̂ a ′′ γ ) α ′ α ′′ Iβ ′ β ′′ J = − (( − P ( ) ) T ( C γ ̂ a ′′ γ )) α ′ α ′′ Iβ ′ β ′′ J ≡ − ( C γ ̂ a ′′ γ ) γ ′ γ ′′ Kβ ′ β ′′ J ( − P ( ) ) γ ′ γ ′′ Kα ′ α ′′ I ; ● ( SO ( ) × SO ( )) -descendability.The Vacuum-Superalgebra Constraints, on the other hand, are not satisfied in view of the above, { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δ ′ δ ′′ Lβ ′ β ′′ J Q δ ′ δ ′′ L } ∉ ⟨ P , P ⟩ ⊕ ⊕ a ′ ,b ′ = ⟨ J a ′ b ′ ⟩ ⊕ ⊕ a ′′ ,b ′′ = ⟨ J a ′′ b ′′ ⟩ , with [ J ̂ a ̂ b , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ ̂ a γ ̂ b ) γ ′ γ ′′ Kα ′ α ′′ I P ( ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ∈ Im P ( ) but also [ P , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = − i ( γ γ ) γ ′ γ ′′ Kα ′ α ′′ I ( − P ( ) ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ∈ Im ( − P ( ) ) , [ P , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = i ( γ ) γ ′ γ ′′ Kα ′ α ′′ I ( − P ( ) ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ∈ Im ( − P ( ) ) , and so the HP vacuum superdistribution with restrictionsVac ( SU ( , ∣ )/( SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is nonintegrable. Its weak derived flag with (local) components of the symbol q − ( i ) ≡ Vac ( SU ( , ∣ )/( SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i , q − ( i ) = ⊕ ̂ α = ⟨ T ̂ α i ⟩ ⊕ ⊕ a ′ = ⟨ T a ′ i ⟩ ⊕ ⊕ b ′′ = ⟨ T b ′′ i ⟩ , q − ( i ) = ⊕ a ′ = ⟨ T a ′ i ⟩ ⊕ ⊕ b ′′ = ⟨ T b ′′ i ⟩ s bracket-generating for the tangent sheaf of the HP section,Vac −∞ ( SU ( , ∣ )/( SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) = T Σ HP . Supersymmetries of the super- σ -model and of its vacuum In the present section, we investigate in full detail (super)symmetries of the GS super- σ -model,with view to their geometrisation and subsequent lifting to the higher-geometric structures underlyingthe topological term in the DF amplitude that we recall in Sec. 6. Particular emphasis shall be laidon the peculiar local (or gauged) supersymmetry that appears – through a universal mechanism ofenhancement of gauge symmetry – in the HP/NG correspondence sector and restores balance betweenbosonic and fermionic degrees of freedom of the vacuum of the super- σ -model, as first observed by deAzc´arraga and Lukierski in Ref. [dAL83], later discussed by Siegel in Refs. [Sie83, Sie84], and recentlyelaborated by McArthur in Ref. [McA00] and West et al. in Refs. [GKW06a] from the more geometricperspective . The symmetry shall be demonstrated to geometrise in the HP formulation, but that ina rather non-trivial manner, to wit, as a distinguished (Graßmann-)odd-generated superdistributioncontained properly in the HP vacuum superdistribution Vac ( sB ( HP ) p,λ p ) over the HP section. In general,the superdistribution is neither H vac -descendable nor integrable but in the physically preferred cir-cumstances in which Vac ( sB ( HP ) p,λ p ) defines a vacuum foliation of the HP section that descends to thesupertarget M H vac , the symmetry superdistribution is seen to bracket-generate Vac ( sB ( HP ) p,λ p ) and –in consequence – envelop the vacuum, all that in a manner manifestly compatible with the (linearised)global supersymmetry.5.1. The global-supersymmetry group.
The very formulation of the super- σ -model in terms oftensor products of the components of the left-invariant Maurer–Cartan super-1-form along f contractedwith H vac -invariant tensors encodes the global supersymmetry of the field theory modelled by the Liesupergroup G acting by left translations [ ℓ ] ⋅ ≡ [ ℓ ] H vac ⋅ as in Eq. (2.14). Indeed, there exist local liftsof [ ℓ ] ⋅ to Σ HP determined, up to the action of the local gauge group [U H vac i , H vac ] , by the condition g ⋅ σ vac i ( χ i ) ⋅ h ij ( χ i ; g ) = σ vac j ( χ j ( g [ ⊳ ] χ i )) (5.1)written, in the S -point picture, for arbitrary g ∈ G (an S -point) and i ∈ I H vac , and for j ∈ I H vac suchthat π G / H vac ( g ⋅ σ vac i ( χ i )) ∈ U H vac j , and with [ ⊳ ] representing the induced action (in that picture), sothat for any k ∈ I H vac with the same property we find h ik ( χ i ; g ) = h ij ( χ i ; g ) ⋅ h jk ( χ j ( g [ ⊳ ] χ i )) , where the h jk are the gluing mappings of the principal H vac -bundle (2.4) (with K = H vac ). Theintegrands in the action functional of the super- σ -model being (right-)H vac -basic (so that they descendto the homogeneous space) and – by assumption – quasi-supersymmetric , that is left-invariant up toa total external derivative in such a manner that the action functional for Ω p closed is left-G-invariant,the induced left action of G on the supertarget M H vac preserves the DF amplitude. We shall discussthe deeper higher-geometric meaning of this global supersymmetry of the DF amplitudes in Sec. 6.Meanwhile, note that the non-linear realisation of the global-supersymmetry group on the HP sectionΣ HP admits an ‘infinitesimal’ presentation in terms of the corresponding ‘fundamental’ vector fields,specified in Proposition 5.1.
In the hitherto notation, let the K A , A ∈ , D be the fundamental vector fields ofthe induced G -action [ ℓ ] ⋅ of G on M H vac . The local tangent lift K A i ( σ vac i ( χ i )) ≡ T χ i σ vac i (K A ( χ i )) (5.2) of the fundamental vector field K A to V i , i ∈ I H vac takes the form K A i = R A ↾ V i + Ξ SA i L S , A ∈ , D , (5.3) with the sections Ξ SA i ∈ O G (V i ) given by the formulæ Ξ SA i ( σ vac i ( χ i )) = H νA ( σ vac i ( χ i )) ( E ( χ i ) − ) µν E Sµ ( χ i ) − H SA ( σ vac i ( χ i )) Cp Ref. [Sus19] for a fairly complete list of references. That is the limit of the weak derived flag of the superdistribution coincides with the vacuum superdistribution. n terms of the change-of-basis sections H BA ∈ O G (V i ) defined as R A = ∶ H BA L B . We shall call the R -linear span of the vector fields K A ∈ Γ (T Σ HP ) , K A ↾ V i = K A i the global-supersymmetry subspace of T Σ HP and denote it as S HPG = ⟨ K A ∣ A ∈ , D ⟩ ⊂ Γ ( T Σ HP ) . Proof.
The general structure of the reference formula (5.3) follows from Eqs. (2.22) and (5.1). Theproof develops along similar lines as that of Prop. 3.6. Thus, we write K A ( χ i ) = ∆ µA i ( χ i ) ⃗ ∂∂χ µi and proceed with a calculation similar to the one leading to Eq. (3.13), whereby we arrive at the identity∆ µA i E Bµ ( χ i ) ≡ K A ⌟ σ vac ∗ i θ B L ( χ i ) = K A i ⌟ θ B L ( σ vac i ( χ i )) ≡ H BA ( σ vac i ( χ i )) + Ξ SA i ( σ vac i ( χ i )) δ BS , valid for any B ∈ , D . Setting B ≡ ν ∈ , δ , we obtain∆ µA i E νµ ( χ i ) = H νA ( σ vac i ( χ i )) . (cid:3) It is to be emphasised that while the maps ℓ ⋅ and [ ℓ ] ⋅ are left actions in the standard sense, the locallifts of the latter to the HP section do not compose, in general, a bona fide action of the supersymmetrygroup due to the inherent ambiguity in the definition of the target index j in Eq. (5.1). What does survive the physically motivated restriction to Σ HP is a linearised realisation of the supersymmetrygroup on Σ HP derived above. It is this linearised realisation that we shall work with when discussingthe global supersymmetry of the various physically distinguished superdistributions over Σ HP . For that,however, we first need to formalise meaningfully the notion of a linearised global (super)symmetry inthe case of a (super)distribution, which we do in Definition 5.2.
Let M be a supermanifold, and let S ⊂ Γ (T M) be a Lie superalgebra. A superdis-tribution D ⊂ T M shall be called S -symmetric if the following identities are satisfied: ∀ V∈ S ∶ [V , D} ⊂ D , so that the flows of the vector fields spanning S preserve D . In particular, let G be a Lie supergroup(with the tangent Lie superalgebra g ) that acts on M , inducing the fundamental vector fields K λX , X ∈ g of Eq. (2.21) that compose the R -linear subspace S G ∶ = { K λX ∣ X ∈ g } ⊂ Γ ( T M ) closed under the supercommutator and modelled on g . An S G -symmetric superdistribution D ⊂ T M shall be termed globally linearised- G -symmetric , or g -invariant . ◇ Taking into accout the correspondence between the ‘fundamental’ vector fields K A on Σ HP and theproper fundamental vector fields K A on M H vac , we adapt the above standard definition to the presentsituation as follows. Definition 5.3.
Adopt the notation of Props. 3.6 and 5.1 and let S HPG ⊂ Γ ( T Σ HP ) be the global-supersymmetry subspace of Prop. 5.1. A superdistribution D ⊂ T Σ HP shall be called globallylinearised-supersymmetric if D is S HPG -symmetric in the sense of Def. 5.2. ◇ The above definition enables us to formulate and study the third and last from the list of regularitycriteria, first mentioned on p. 42, that can and should be applied to the HP vacuum superdistribution.This extra criterion is preservation of the vacuum superdistribution under global supersymmetry andexistence of a residual global supersymmetry in the vacuum ( i.e. , on any of the integral supermanifoldsof that superdistribution). Below, we return to the discussion begun in the previous section.The much reassuring general answer to the first of the two questions resulting from the global-supersymmetry criterion is given in roposition 5.4. The HP vacuum superdistribution
Vac ( sB ( HP ) p,λ p ) of Def. 4.3 is globally linearised-supersymmetric in the sense of Def. 5.3 iff [ h vac , t ( ) vac ] ⊂ t ( ) vac . In particular, an H vac -descendable HP vacuum superdistribution is globally linearised-supersymmetric.Proof. All vector fields from S HPG lie in T Σ HP , as do sections of Vac ( sB ( HP ) p,λ p ) . Hence, taking intoaccount Prop. 3.6 and the standard (trivial) supercommutation relations of the right-invariant vectorfields R B with the left invariant ones L C , we conclude that the supercommutator of a ‘fundamental’vector field K A i of Eq. (5.3) with a section τ Ai T A i , written in terms of arbitrary τ Ai ∈ O G (V i ) , isdetermined uniquely by the horizontal components L ν ↾ V i that can be obtained by supercommutingvertical components of K A i with the horizontal components L A ↾ V i of τ Ai T A i . Thus, we arrive at therequirement [ h vac , t vac ] ⊂ t vac (5.4)that boils down to the one from the claim of the proposition by the definition of h vac . The second partof the claim follows directly from Prop. 4.5. (cid:3) In the light of the last proposition, the vacuum foliation, whenever it exists, is preserved as a whole bythe flow of global supersymmetry, that is, vacua are carried into one another. It remains to distinguishthose of the flows that preserve a particular vacuum. We do that it in
Proposition 5.5.
Adopt the hitherto notation, and in particular – that of Def. 4.3 and Prop. 5.1. Atangent lift K X ≡ X A K A of the fundamental vector field X A K A , engendered by the induced action [ ℓ ] ⋅ of G on M H vac , alongthe family { σ vac i } i ∈ I Hvac is tangent to the vacuum foliation of the HP section Σ HP (the latter beingassumed to exist) iff X ≡ X A t A is a constant (over Σ HP ) solution to the set of δ − p − q − linearequations P ge ⊕ d ( X H ( σ vac i ( χ i ))) = , written in terms of the endomorphism H ( σ vac i ( χ i )) = τ A ⊗ H BA ( σ vac i ( χ i )) t B ∶ g ↺ in which { τ A } A ∈ ,D is the basis of g ∗ dual to { t A } A ∈ ,D , τ A ( t B ) = δ AB . We shall call the linear span of these vector fields the residual global-supersymmetry subspace of
Vac ( sB ( HP ) p,λ p ) and denote it as S HP , vacG ⊂ Γ ( Vac ( sB ( HP ) p,λ p )) . (5.5) The subspace is closed under the supercommutator, and so modelled on a Lie superalgebra, to be termedthe residual global-supersymmetry subalgebra of g and denoted as s vac ⊂ g , (5.6) Proof.
Obvious. (cid:3)
We shall, next, examine another superdistribution within T Σ HP distinguished by field-theoretic con-siderations. .2. The κ -symmetry superdistribution – an odd resolution of the vacuum. Our next ob-jective is the identification and investigation of a regular ( i.e. , non-coincidental) enhancement of theresidual hidden gauge-symmetry, modelled on the Lie algebra h vac , of the GS super- σ -model in theHP formulation that occurs upon restriction of field configurations ̂ ξ ∈ [ Ω p , M H vac ] to the HP/NGcorrespondence sector and takes the form of a superdistribution GS( sB ( HP ) p,λ p ) ⊂ Corr HP ( sB ( HP ) p,λ p ) with the defining property: arbitrary variations δ ̂ ξ ∈ [ Ω p , GS( sB ( HP ) p,λ p )] leave the DF amplitude un-changed (to the linear order) for field configurations from the correspondence sector . We shall call itthe enhanced gauge-symmetry superdistribution of sB ( HP ) p,λ p in what follows and demand that itdescend to M H vac and be globally linearised-supersymmetric. The obvious reason to perform partialreduction of the field theory under study through imposition of the IHC and the BLC and require H vac -descendability of the ensuing gauge symmetry is that we are ultimately interested in the physical gaugesymmetry of the dual NG super- σ -model. Once identified, we shall subsequently intersect GS( sB ( HP ) p,λ p ) with the vacuum superdistribution Vac ( sB ( HP ) p,λ p ) , obtaining an object which – generically – will turnout to be spanned on those of the generators T µ i of T Σ HP listed in Prop. 3.6 that carry labels µ of t ( ) , and in any event, it will contain a distinguished sub-superdistribution κ ( sB ( HP ) p,λ p ) ⊂ Vac ( sB ( HP ) p,λ p ) of this form. Its tentative physical interpretation as an infinitesimal odd local symmetry of the vac-uum of an effectively topological field theory prompts three intertwined geometric and field-theoreticquestions, and it is only an affirmative answer to all three of them that grants κ ( sB ( HP ) p,λ p ) its status ofa gauge supersymmetry of the field theory under study. First of all, we must enquire – once again, butthis time with regard to a proper substructure – if κ ( sB ( HP ) p,λ p ) descends to the physical supertarget M H vac . If this is the case, we are bound to ask if it does so in a manner compatible with the globalsupersymmetry present, or – in other words – if it is globally linearised-supersymmetric. By the argu-ment from the proof of Prop. 5.4, we know that a positive answer to the first question actually implies apositive answer to the second one. The last of the three questions regards the limit of its weak derived(super)flag, κ −∞ ( sB ( HP ) p,λ p ) ⊆ T Σ HP . In fact, it makes sense to speak of κ ( sB ( HP ) p,λ p ) as a proper symmetry of the vacuum iff the limitstays within Vac ( sB ( HP ) p,λ p ) , in which case it foliates the vacuum by gauge orbits. Under the physicallypreferred circumstances in which all three questions have been answered in the positive, we are dealingwith a sub-superdistribution κ ( sB ( HP ) p,λ p ) ⊆ Vac ( sB ( HP ) p,λ p ) whose weak derived flag is closed under the supercommutator and hence modelled on a Lie sub-superalgebra gs vac ( sB ( HP ) p,λ p ) ⊆ vac ( sB ( HP ) p,λ p ) of the vacuum superalgebra of Prop. 4.7. At this point, it seems fit to pause briefly in order to articu-late an intuition drawn from experience with standard topological gauge field theories to which theGS super- σ -model in the HP formulation appears to bear structural affinity. The crucial observationis that in a topological field theory, in the absence of local degrees of freedom, propagation of con-figurations localised at Cauchy slices of the theory’s spacetime is realised, or – indeed – replaced by(a class of) gauge transformations, cp , e.g. , the extensively studied three-dimensional Chern–Simonstopological gauge field theory (of Ref. [Fre95]) on a cylinder over a (punctured) Riemann surface for anexplicit instantiation of this phenomenon. This leads us to anticipate that the (H vac -descendable andglobally linearised-supersymmetric) superdistribution κ ( sB ( HP ) p,λ p ) should fill up the vacuum superdis-tribution, i.e. , that it should be bracket-generating for Vac ( sB ( HP ) p,λ p ) , so that the vector fields spanningthe limit of its weak derived flag envelop the embedded worldvolume, winning it its name – the squareroot of the vacuum – that features in the title of the present paper. This may seem like a lot to The Reader is advised to consult Atiyah’s foundational paper [Ati89] for an axiomatic distillate obtained from thevarious ad hoc definitions of a topological field theory employed in the physics literature, with due emphasis on (and arigorus rendering of) its constitutive properties that give a meaning to the name. xpect (in particular, this expectation presupposes some sort of ‘completeness’ of the Lie superalge-bra gs vac ( sB ( HP ) p,λ p ) as a model of residual gauge transformations preserving the vacuum) but explicitcomputations carried out for the superbackgrounds from those of Examples 3.11-3.16 that possess asupersymmetric vacuum foliation actually confirm our intuition. With this reassuring note in mind,we now pass to the derivation of GS( sB ( HP ) p,λ p ) and κ ( sB ( HP ) p,λ p ) .The point of departure of our analysis is formula (4.3) for an arbitrary variation of the DF amplitudein which on top of the restriction to the HP correspondence section, we also impose (on the supersym-metry algebra) the Even Effective-Mixing Constraints (2.7)-(2.9), the Descendability Constraint (3.15),the No-Curvature and No-Torsion Constraints (4.8) and (4.9), as well as the κ -Symmetry Constraints(4.5) and (4.6), to the effect: − i δ δ ̂ ξ log A ( HP ) ,p,λ p DF [̂ ξ ] = ∑ τ ∈ T p + ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ [ p ! δθ αı τ ( δ − d − P ( ) ) γα ∆ βγa a ...a p Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L + ( p − ) ! δx aı τ (( δ − d − P ( ) ) γβ ∆ αγaa a ...a p Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L ) + ( p − ) ! δx ̂ aı τ ( χ αβ ̂ aa a ...a p − Σ α L ∧ Σ β L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p − L − λ p p f b ̂ a ̂ S ǫ ba a ...a p θ ̂ S L ∧ θ a L ∧ θ a L ∧ ⋯ ∧ θ a p L )] . The sole subtlety in the above expression lurks in its second line in which for any ( p − ) -tuple of (even)vacuum indices ( a , a , . . . , a p ) ∈ , p we have a pair ( a , a ) ∈ , p of complementary indices, with,say, ǫ a a a ...a p =
1. In the light of the assumptions made with regard to the tensors ∆ αβb b ...b p , wemay write, for a ( p + ) -tuple as above, δx aı τ ( δ − d − P ( ) ) γβ ∆ αγaa a ...a p = ( δx a ı τ ∆ αγa a a ...a p + δx a ı τ ∆ αγa a a ...a p ) ( δ − d − P ( ) ) γβ = ∆ αγa a a ...a p ( δx a ı τ δ γε + δx a ı τ ( ∆ − a a a ...a p ) γδ ∆ δεa a a ...a p ) ( δ − d − P ( ) ) εβ ( no summation over the repeated indices a and a ), and so the possibility arises of having a ‘chiral’gauge symmetry with δx a ı τ = − tr t ( ) ( ∆ − a a a ...ap ∆ a a a ...ap ( δ − d − P ( ) )) δ − d − q δx a ı τ (recall that we assumed P ( ) ≠ δ − d ), generated by vector fields T a a = T a − tr t ( ) ( ∆ − a a a ...ap ∆ a a a ...ap ( δ − d − P ( ) )) δ − d − q T a . Above, we identify the matrices ∆ αβb b ...b p with the corresponding endomorphisms of t ( ) . Inspectionof the examples leads us to think of such symmetries as exceptional and treat them separately, whichjustifies imposition of The Even Achirality Constraints:
We assume, in the hitherto notation, the following identitiesΠ ( a ,a ∣ a ,a ,...,a p ) ≡ tr t ( ) ( ∆ − a a a ...a p ∆ a a a ...a p ( δ − d − P ( ) )) ! = ( p + ) -tuple ( a , a , . . . , a p ) ∈ , p .With the Constraints in force, we readily infer from the above that, in a generic situation, the enhancedgauge-symmetry superdistribution is sewn from its local restrictions of the form GS ( sB ( HP ) p,λ p ) ↾ V i = q ⊕ α = ⟨ T α i ⟩ ⊕ D − δ ⊕ ̂ S = D − δ + ⟨ T ̂ S i ⟩ (5.8)confirmed – without a single exception – through scrutiny of Examples 3.11-3.16. Thus, we recover the full hidden gauge symmetry of the dual formulation (the vertical component being realised trivially)augmented with translations along the Graßmann-odd generators of the vacuum superdistribution. Ouridentification of the invariances as infinitesimal gauge symmetries stems from the following observation:Their very derivation implies that they can be regarded as ‘infinitesimal’ right translations of fieldconfigurations from the HP/NG correspondence sector in the directions of the supervector subspace gs ( sB ( HP ) p,λ p ) ≡ t ( ) vac ⊕ d ⊂ g . (5.9) hile left translations do not depend on the choice of the local sections of the principal H vac -bundle(2.4) (with K = H vac ) and, consequently, descend to the homogeneous space along the canonical pro-jection π M Hvac , the right ones do depend on the precise form of the σ vac i , and so the inherent locality ofthese transformations is an irremovable consequence of the hidden gauge freedom of the super- σ -model( cp also Ref. [McA00, Sec. 3]). In view of the peculiar relation between the lagrangean density andthe presymplectic form of the GS super- σ -model in the topological HP formulation, this identifica-tion is in keeping with the standard definition of gauge symmetries as generators of the kernel of thepresymplectic form of the field theory, cp Ref. [Gaw72].Upon taking into account the structure of the vacuum superdistribution, we are led to
Definition 5.6.
Adopt the hitherto notation, and in particular that of Prop. 3.6. The superdistribution κ ( sB ( HP ) p,λ p ) ⊂ Vac ( sB ( HP ) p,λ p ) , κ ( sB ( HP ) p,λ p ) ↾ V i = q ⊕ α = ⟨ T α i ⟩ , (5.10)spanned on generators of Graßmann-odd local symmetries of the HP/NG correspondence sector ofthe (generic) Green–Schwarz super- σ -model in the Hughes–Polchinski formulation of Def. 3.2, shall becalled the κ -symmetry superdistribution of sB ( HP ) p,λ p . Whenever the limit of its weak derived flagis contained in the vacuum superdistribution, that is κ −∞ ( sB ( HP ) p,λ p ) ⊆ Vac ( sB ( HP ) p,λ p ) , we call the ensuing Lie superalgebra gs vac ( sB ( HP ) p,λ p ) ⊆ vac ( sB ( HP ) p,λ p ) the vacuum gauge-symmetry superalgebra , or – for historical reasons – the κ -symmetry super-algebra . ◇ Remark 5.7.
Parenthetically, let us note that vacua with q ≡ dim t ( ) vac out of the D d, ≡ dim t ( ) supercharges (or ‘supersymmetries’) of the theory left unbroken are usually referred to as qD d, -BPSstates in the physics parlance. Hence, in the regular ( i.e. , H vac -descendable and integrable) case, wemight speak of the BPS fraction of the vacuum of sB ( HP ) p,λ p ,BPS ( sB ( HP ) p,λ p ) ∶ = qD d, ≡ D d, tr t ( ) P ( ) . Our derivation of the κ -symmetry superdistribution paves the way to further analysis, subordinated tothe primary goal of answering the three basic structural questions formulated above. Upon invokingour former considerations and results, we readily arrive at Proposition 5.8.
In the hitherto notation and under the assumption that the Even Effective-MixingConstraints (2.7) - (2.9) , the Descendability Constraint (3.15) , the No-Curvature and No-Torsion Con-straints (4.8) and (4.9) , the κ -Symmetry Constraints (4.5) and (4.6) , as well as the Even AchiralityConstraints (5.7) are satisfied, the κ -symmetry superdistribution of the HP superbackground sB ( HP ) p,λ p given in Eq. (5.10) is globally linearised-supersymmetric in the sense of Def. 5.3 iff [ h vac , t ( ) vac ] ⊂ t ( ) vac . Thus, κ ( sB ( HP ) p,λ p ) is globally linearised-supersymmetric iff Vac ( B ( HP ) p,λ p ) is. In particular, an H vac -descendable κ -symmetry superdistribution is globally linearised-supersymmetric.Proof. Follows directly from Prop. 5.4 and its proof. (cid:3)
We also have
Proposition 5.9.
In the hitherto notation and under the assumption that the Even Effective-MixingConstraints (2.7) - (2.9) , the Descendability Constraint (3.15) , the No-Curvature and No-Torsion Con-straints (4.8) and (4.9) , the κ -Symmetry Constraints (4.5) and (4.6) , as well as the Even AchiralityConstraints (5.7) are satisfied, the limit κ −∞ ( sB ( HP ) p,λ p ) of the weak derived flag κ ● ( sB ( HP ) p,λ p ) of the -symmetry superdistribution κ ( sB ( HP ) p,λ p ) of the HP superbackground sB ( HP ) p,λ p , given in Eq. (5.10) , iscontained in the HP vacuum superdistribution Vac ( sB ( HP ) p,λ p ) of Def. 4.3, κ −∞ ( sB ( HP ) p,λ p ) ⊆ Vac ( sB ( HP ) p,λ p ) , if the Vacuum-Superalgebra Constraints of Prop. 4.7 are satisfied. If, in addition, the endomorphisms f a ≡ f aαβ δ βγ q α ⊗ Q γ ∈ End ( t ( ) ) , a ∈ , p (expressed in the notation of Eq. (2.12) ) satisfy the Odd Achirality Constraints ∀ a ∈ ,p ∃ f − a ∈ End ( t ( ) ) ∶ f − a ○ f a = id t ( ) , ∀ a,b ∈ ,p ∃ λ a ∈ R × ∶ Π ab ≡ tr t ( ) ( f − a ○ P ( ) T ○ f b ○ P ( ) ) ! = λ a δ ab ,κ ( sB ( HP ) p,λ p ) is bracket-generating for Vac ( sB ( HP ) p,λ p ) , κ −∞ ( sB ( HP ) p,λ p ) = Vac ( sB ( HP ) p,λ p ) , and the κ -symmetry superalgebra of sB ( HP ) p,λ p coincides with the vacuum superalgebra of sB ( HP ) p,λ p , asdefined ibidem, gs vac ( sB ( HP ) p,λ p ) = vac ( sB ( HP ) p,λ p ) . Proof.
As for the first part, it is fully analogous to that of Prop. 5.4. (cid:3)
In view of the physical significance of the various results scattered in the propositions written out andcited heretofore, we summarise our findings in
Theorem 5.10.
Let G = (∣ G ∣ , O G ) be a Lie supergroup with the tangent Lie superalgebra g = g ( ) ⊕ g ( ) ,and let H and H vac ⊂ H be two Lie subgroups of the body ∣ G ∣ of G that correspond to two reductivedecompositions of g , g = f ⊕ k , ( f , k ) ∈ {( t , h ) , ( t ⊕ d , h vac )} , in which h = d ⊕ h vac and h vac are the tangent Lie algebras of H and H vac , respectively, satisfying therelations [ h vac , h vac ] ⊂ h vac , [ h vac , d ] ⊂ d , [ d , d ] ⊂ h vac , with the direct-sum complement t of h in g further decomposing as t = t vac ⊕ e into supervector sub-spaces t vac = t ( ) vac ⊕ t ( ) vac , e = e ( ) ⊕ e ( ) with the properties [ h vac , t ( ) vac ] ⊂ t ( ) vac , [ h vac , e ( ) ] ⊂ e ( ) , [ d , t ( ) vac ] ⊂ e ( ) , [ d , e ( ) ] ⊂ t ( ) vac , and such that the relations [ t ( ) vac , t ( ) vac ] ⊂ h vac ⊃ [ e ( ) , e ( ) ] , [ t ( ) vac , e ( ) ] ⊂ d , { t ( ) vac , t ( ) vac } ⊂ t ( ) vac ⊕ h hold true. Consider the principal (super)bundles K / / G π G / K (cid:15) (cid:15) G / K , K ∈ { H , H vac } , coming with the respective families { σ vac i ∶ U H vac i Ð→ G } i ∈ I Hvac and { σ ✟✟❍❍ vac i ∶ U H i Ð→ G } i ∈ I Hvac of sec-tions associated with open covers {U K i } i ∈ I K of the respective bases, defined on p. 18. The Green–Schwarzsuper- σ -model in the Hughes–Polchinski formulation of Def. 3.2 for the superbackground sB ( HP ) p,λ p of q. (3.5) satisfying the κ -Symmetry Constraints (4.5) and (4.6) as well as the Γ -Constraints (4.10) –defined for objects of the mapping supermanifold [ Ω p , Σ HP ] of the ( p + ) -dimensional worldvolume Ω p into the Hughes–Polchinski section Σ HP = ⊔ i ∈ I Hvac σ vac i (U H vac i ) ⊂ G and equivalent to the Green–Schwarz super- σ -model in the Nambu–Goto formulation of Def. 3.1 for thesuperbackground sB ( GS ) p of Eq. (3.2) in the HP/NG correspondence sector composed of mappings withtangents restricted to the correspondence superdistribution Corr HP / NG ( sB ( HP ) p,λ p ) = Ker ( P ge ( ) ○ θ L ↾ T Σ HP ) ⊂ T Σ HP of Def. 3.10 (upon postcomposition with tangents of the σ vac i ) – determines the vacuum superdistribution Vac ( sB ( HP ) p,λ p ) = Ker ( P ge ⊕ d ○ θ L ↾ T Σ HP ) ⊂ Corr HP / NG ( sB ( HP ) p,λ p ) of Def. 4.3 that is globally linearised-supersymmetric iff [ h vac , t ( ) vac ] ⊂ t ( ) vac , which is also a necessary condition for it to descend to M H vac . The vacuum superdistribution is in-volutive, and hence defines the vacuum foliation of the Hughes–Polchinski section of Prop. 4.7 iff theadditional conditions { t ( ) vac , t ( ) vac } ⊂ t ( ) vac ⊕ h vac , [ t ( ) vac , t ( ) vac ] ⊂ t ( ) vac , [ h vac , t ( ) vac ] ⊂ t ( ) vac are satisfied, making the vacuum supervector space vac ( sB ( HP ) p,λ p ) = t vac ⊕ h vac ⊂ g into the vacuum (Lie) superalgebra.Furthermore, the Green–Schwarz super- σ -model in the Hughes–Polchinski formulation distinguishesthe enhanced gauge-symmetry superdistribution GS( sB ( HP ) p,λ p ) ⊂ Corr HP ( sB ( HP ) p,λ p ) of local symmetries of its HP/NG correspondence sector. The associated κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) = Ker (( id g − P gt ( ) vac ) ○ θ L ↾ T Σ HP ) of Def. 5.6 is globally linearised-supersymmetric iff [ h vac , t ( ) vac ] ⊂ t ( ) vac , which is also a necessary condition for it to descend to M H vac . The limit κ −∞ ( sB ( HP ) p,λ p ) of its weakderived flag coincides with the limit Vac −∞ ( sB ( HP ) p,λ p ) of the weak derived flag of the vacuum superdis-tribution if the conditions ∀ a ∈ ,p ∃ f − a ∈ End ( t ( ) ) ∶ f − a ○ f a = id t ( ) , ∀ a,b ∈ ,p ∃ λ a ∈ R × ∶ tr t ( ) ( f − a ○ P ( ) T ○ f b ○ P ( ) ) = λ a δ ab , are satisfied, and so – whenever this happens – it lies in that superdistribution iff the latter is involutive,in which case the κ -symmetry superdistribution is bracket-generating for Vac ( sB ( HP ) p,λ p ) , that is κ −∞ ( sB ( HP ) p,λ p ) = Vac ( sB ( HP ) p,λ p ) , (5.11) and the κ -symmetry superalgebra gs vac ( sB ( HP ) p,λ p ) of Def. 5.6 coincides with the vacuum superalgebra, gs vac ( sB ( HP ) p,λ p ) = vac ( sB ( HP ) p,λ p ) . Altogether, we conclude that physical considerations favour – as leading to a meaningful ‘localisation’of the vacuum within the supertarget which is compatible with the global supersymmetry present –superbackgrounds with involutive (and H vac -descendable) vacuum superdistributions, and these arebracket-generated by their κ -symmetry sub-superdistributions that envelop the embedded vacua, inconformity with our TFT intuition. The vacua are expected, furthermore, to exhibit global (linearised)supersymmetry, realised by the residual global-supersymmetry subspace S HP , vacG ⊂ Vac ( sB ( HP ) p,λ p ) . Given he simple and highly constrained (super)algebraic model of the κ -symmetry superdistribution and ofits weak derived flag, it is completely straightforward to identify the sources of a potential obstructionagainst both: integrability of the vacuum superdistribution and identity (5.11). The first of theseanomalies is the projection a int ∶ = P ge ( ) ([ t ( ) vac , t ( ) vac ]) that quantifies the violation of the second of the Vacuum-Superalgebra Constraints of Prop. 4.7. Notethat it also encodes the potential violation of the first of these Constraints as a non-zero projection P gd ({ t ( ) vac , t ( ) vac }) ≠ d ofthe mother gauge-symmetry algebra h yields P ge ( ) ([{ t ( ) vac , t ( ) vac } , t ( ) vac ]) ≠ a int = , we are immediately led – via the super-Jacobi identity restricted to the triple t ( ) vac × t ( ) vac × t ( ) vac – to thecontradiction P ge ( ) ([{ t ( ) vac , t ( ) vac } , t ( ) vac ]) ⊂ P ge ( ) ({[ t ( ) vac , t ( ) vac ] , t ( ) vac }) ⊂ P ge ( ) ({ t ( ) vac , t ( ) vac }) ⊂ P ge ( ) ( t ( ) vac ⊕ h ) = . The last (and independent) anomaly is the projection a susy ∶ = P ge ( ) ([ h vac , t ( ) vac ]) that captures the potential obstruction against global (linearised) supersymmetry. Prior to establishingthe first step towards a consistent higher-geometric lift of our symmetry analysis in the next section, wepresent below a complete list of the superalgebraic models and anomalies of the κ -symmetry superdis-tributions for the superbackgrounds from Examples 3.11-3.16, writing them out in the convention a int = ( X a [ P a , P ( ) βα Q β ]) , a susy = ( Φ S [ J S , P ( ) βα Q β ]) , with X a and Φ S arbitrary (real) parameters. Note that all regular cases have the BPS fraction . Example 5.11. The square root of the Green–Schwarz super-0-brane in sISO ( , ∣ )/ SO ( ) . The tensor f ≡ Γ satisfies the Odd Achirality ConstraintsΠ = tr t ( ) ( Γ Γ ( + Γ Γ )) = tr t ( ) ( + Γ Γ ) = . The Even Achirality Constraints are satisfied trivially. The κ -symmetry superdistribution with restric-tions κ ( sISO ( , ∣ )/ SO ( ) , ̂ χ ( ) GS ) ↾ V i = ⊕ α = ⟨ T α i ⟩ is an SO ( ) -descendable superdistribution with the limit of its weak derived flag with restrictions κ −∞ ( sISO ( , ∣ )/ SO ( ) , ̂ χ ( ) GS ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i ⟩ ≡ Vac ( sISO ( , ∣ )/ SO ( ) , ̂ χ ( ) GS ) ↾ V i . Example 5.12. The square root of the Green–Schwarz super- ( k + ) -brane in sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) for k ∈ { , } . The tensors∆ a a ...a k + ≡ ( k + ) ! ǫ aa a ...a k + Γ a , f a ≡ Γ a satisfy the Even Achirality Constraints,Π ( a ,a ∣ a ,a ,...,a k + ) = − tr t ( ) ( Γ a Γ a ( D d, − Γ Γ ⋯ Γ k + )) = η a a tr t ( ) ( Γ a Γ a ⋯ Γ a k + ) = ( − ) k − η a a tr t ( ) ( Γ a Γ a ⋯ Γ a k + Γ a ) ≡ − Π ( a ,a ∣ a ,a ,...,a k + ) = , I.a. , by the super-Jacobi identities of the mother Lie superalgebra g . nd the Odd Achirality Constraints,Π ab = tr t ( ) ( Γ a Γ b ( D d, + Γ Γ ⋯ Γ k + )) = δ ba tr t ( ) ( D d, + Γ Γ ⋯ Γ k + ) = D d, δ ab . The κ -symmetry superdistribution with restrictions κ ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i = Dd, ⊕ α = ⟨ T α i ⟩ is an ( SO ( k + , ) × SO ( d − k − )) -descendable superdistribution with the limit of its weak derivedflag with restrictions κ −∞ ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i = Dd, ⊕ α = ⟨ T α i ⟩ ⊕ k + ⊕ a = ⟨ T a i ⟩ ≡ Vac ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i . Example 5.13. The square root of the Green–Schwarz super- ( k + ) -brane in sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) for k ∈ { , } . The tensors∆ a a ...a k + ≡ ( k + ) ! ǫ aa a ...a k + Γ a , f a ≡ Γ a satisfy the Even Achirality Constraints,Π ( a ,a ∣ a ,a ,...,a k + ) = − tr t ( ) ( Γ a Γ a ( D d, − Γ Γ ⋯ Γ k + )) = η a a tr t ( ) ( Γ a Γ a ⋯ Γ a k + ) = η a a tr t ( ) ( Γ a k + T Γ a k T ⋯ Γ a T ) = ( − ) k + η a a tr t ( ) ( C Γ a k + Γ a k ⋯ Γ a C − ) = ( − ) k + + k ( k + ) η a a tr t ( ) ( Γ a Γ a ⋯ Γ a k + ) ≡ − Π ( a ,a ∣ a ,a ,...,a k + ) = , and the Odd Achirality Constraints,Π ab = tr t ( ) ( Γ a Γ b ( D d, + Γ Γ ⋯ Γ k + )) = δ ba tr t ( ) ( D d, + Γ Γ ⋯ Γ k + ) = D d, δ ab . The κ -symmetry superdistribution with restrictions κ ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i = Dd, ⊕ α = ⟨ T α i ⟩ is an ( SO ( k + , ) × SO ( d − k − )) -descendable superdistribution with the limit of its weak derivedflag with restrictions κ −∞ ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i = Dd, ⊕ α = ⟨ T α i ⟩ ⊕ k + ⊕ a = ⟨ T a i ⟩ ≡ Vac ( sISO ( d, ∣ D d, )/( SO ( k + , ) × SO ( d − k − )) , ̂ χ ( p + ) GS ) ↾ V i . Example 5.14. The square root of the Zhou super-1-brane in SU ( , ∣ ) /( SO ( , ) × SO ( )) . The tensors ∆ ≡ C γ ⊗ , ∆ ≡ − C γ ⊗ , f a ≡ C γ a ⊗ satisfy the Even Achirality Constraints,Π ( ∣ − ) = − tr t ( ) (( γ γ ⊗ ) ⋅ ( − γ γ ⊗ σ )) = − ( ( γ γ ) + σ ) = , and the Odd Achirality Constraints,Π ab = tr t ( ) (( γ a γ b ⊗ ) ( + γ γ ⊗ σ )) = δ ba tr t ( ) ( + γ γ ⊗ σ ) = δ ab . The κ -symmetry superdistribution with restrictions κ ( SU ( , ∣ ) /( SO ( , ) × SO ( )) , ̂ χ ( ) Zh ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ is an ( SO ( , ) × SO ( )) -descendable superdistribution with the limit of its weak derived flag withrestrictions κ −∞ ( SU ( , ∣ ) /( SO ( , ) × SO ( )) , ̂ χ ( ) Zh ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ Vac ( SU ( , ∣ ) /( SO ( , ) × SO ( )) , ̂ χ ( ) Zh ( ) ) ↾ V i . Example 5.15. No square root for the Zhou super-1-brane in SU ( , ∣ ) . The tensors∆ ≡ C γ ⊗ , ∆ ≡ − C γ ⊗ , f a ≡ C γ a ⊗ satisfy the Even Achirality Constraints,Π ( ∣ − ) = − tr t ( ) (( γ γ ⊗ ) ⋅ ( − γ γ ⊗ σ )) = − ( ( γ γ ) + σ ) = , and the Odd Achirality Constraints,Π ab = tr t ( ) (( γ a γ b ⊗ ) ( + γ γ ⊗ σ )) = δ ba tr t ( ) ( + γ γ ⊗ σ ) = δ ab . The κ -symmetry superdistribution with restrictions κ ( SU ( , ∣ ) , ̂ χ ( ) Zh ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ does not bracket-generate the HP vacuum superdistribution due to the anomaly a int = ( i X a ( γ γ γ a ⊗ σ ) γ ′ γ ′′ Kα ′ α ′′ I ( − P ( ) ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ) , a susy = . Instead, the limit of its weak derived flag envelops the HP section, κ −∞ ( SU ( , ∣ ) , ̂ χ ( ) Zh ( ) ) = T Σ HP . Example 5.16. The square root of the Park–Rey super-1-brane in ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) . The tensors∆ ≡ C γ ⋅ ( ⊗ σ ) ⊗ , ∆ ≡ − C γ ⋅ ( ⊗ σ ) ⊗ , f a ≡ C γ a ⋅ ( ⊗ σ ) ⊗ satisfy the Even Achirality Constraints,Π ( ∣ − ) = − tr t ( ) ((( ⊗ σ ) ⋅ γ γ ⋅ ( ⊗ σ ) ⊗ ) ⋅ ( − γ γ ⊗ σ )) = − tr t ( ) (( γ γ ⊗ ) ⋅ ( − γ γ ⊗ σ )) = − ( ( γ γ ) + σ ) = , and the Odd Achirality Constraints,Π ab = tr t ( ) ((( ⊗ σ ) ⋅ γ a γ b ⋅ ( ⊗ σ ) ⊗ ) ⋅ ( + γ γ ⊗ σ )) = δ ba tr t ( ) ( + γ γ ⊗ σ ) = δ ab . The κ -symmetry superdistribution with restrictions κ (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is an ( SO ( , ) × SO ( )) -descendable superdistribution with the limit of its weak derived flag withrestrictions κ −∞ (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ ≡ Vac (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i . Example 5.17. No square root for the Park–Rey super-1-brane in ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) . The tensors∆ ≡ C γ γ ⋅ ( ⊗ σ ) ⊗ , ∆ ≡ C γ ⋅ ( ⊗ σ ) ⊗ ,f ≡ C γ ⋅ ( ⊗ σ ) ⊗ , f ≡ − C γ γ ⋅ ( ⊗ σ ) ⊗ satisfy the Even Achirality Constraints,Π ( ∣ − ) = tr t ( ) ((( ⊗ σ ) ⋅ γ γ γ ⋅ ( ⊗ σ ) ⊗ ) ⋅ ( − γ γ γ ⊗ σ )) = tr t ( ) (( γ γ γ ⊗ ) ⋅ ( − γ γ γ ⊗ σ )) = ( i tr ( Γ ) tr ( Γ ) + σ ) = i tr ( Γ ) tr ( Γ ) = − i tr ( C ′ Γ C ′− ) tr ( Γ ) = − i tr ( Γ ) tr ( Γ ) = , nd the Odd Achirality Constraints,Π = − tr t ( ) ((( ⊗ σ ) ⋅ γ γ γ ⋅ ( ⊗ σ ) ⊗ ) ⋅ ( + γ γ γ ⊗ σ )) = − tr t ( ) (( γ γ γ ⊗ ) ⋅ ( + γ γ γ ⊗ σ )) = − ( i tr ( Γ ) tr ( Γ ) + σ ) = , Π aa ≡ . The κ -symmetry superdistribution with restrictions κ (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) , ̂ χ ( ) PR ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is ( SO ( ) × SO ( )) -descendable but does not bracket-generate the HP vacuum superdistribution dueto the anomaly a int = ( i X ( γ γ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I ( − P ( ) ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J , − i X ( γ ( ⊗ σ ) ⊗ σ ) γ ′ γ ′′ γ ′′′ Kα ′ α ′′ α ′′′ I ( − P ( ) ) β ′ β ′′ β ′′′ Jγ ′ γ ′′ γ ′′′ K Q β ′ β ′′ β ′′′ J ) , a susy = . Instead, the limit of its weak derived flag envelops the HP section, κ −∞ (( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( ) × SO ( )) , ̂ χ ( ) PR ( ) ) = T Σ HP . Example 5.18. The square root of the Metsaev–Tseytlin super-1-brane in SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) . The tensors∆ ≡ i C γ γ , ∆ ≡ − i C γ γ , f a ≡ i C γ a γ satisfy the Even Achirality Constraints,Π ( ∣ − ) = − tr t ( ) ( γ γ γ γ ⋅ ( − γ γ γ )) = − tr t ( ) ( γ γ ⋅ ( − γ γ γ )) = − tr t ( ) γ = σ = , and the Odd Achirality Constraints,Π ab = tr t ( ) ( γ γ a γ b γ ⋅ ( + γ γ γ )) = δ ba tr t ( ) ( + γ γ γ ) = δ ab . The κ -symmetry superdistribution with restrictions κ ( SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ is an ( SO ( , ) × SO ( ) × SO ( )) -descendable superdistribution with the limit of its weak derived flagwith restrictions κ −∞ ( SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i , T i ⟩ ⊕ ⟨ T i , T i ⟩ ≡ Vac ( SU ( , ∣ )/( SO ( , ) × SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i . Example 5.19. No square root for the Metsaev–Tseytlin super-1-brane in SU ( , ∣ )/( SO ( ) × SO ( )) . The tensors∆ ≡ i C γ , ∆ ≡ i C γ γ , f ≡ i C γ γ , f ≡ − i C γ satisfy the Even Achirality Constraints,Π ( ∣ − ) = tr t ( ) ( γ γ γ ⋅ ( − γ γ )) = ( i tr ( Γ ) tr ( Γ ) − σ ) = , and the Odd Achirality Constraints,Π = − tr t ( ) ( γ γ γ ⋅ ( + γ γ )) = − tr t ( ) ( i tr ( Γ ) tr ( Γ ) + σ ) = , Π aa = . he κ -symmetry superdistribution with restrictions κ ( SU ( , ∣ )/( SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) ↾ V i = ⊕ α = ⟨ T α i ⟩ is ( SO ( ) × SO ( )) -descendable but does not bracket-generate the HP vacuum superdistribution dueto the anomaly a int = ( − i X ( γ γ ) γ ′ γ ′′ Kα ′ α ′′ I ( − P ( ) ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J , i X ( γ ) γ ′ γ ′′ Kα ′ α ′′ I ( − P ( ) ) β ′ β ′′ Jγ ′ γ ′′ K Q β ′ β ′′ J ) , a susy = . Instead, the limit of its weak derived flag envelops the HP section, κ −∞ ( SU ( , ∣ )/( SO ( ) × SO ( )) , ̂ χ ( ) MT ( ) ) = T Σ HP . There is one super- σ -model with an integrable vacuum from the previous list of Examples 4.10-4.18that we left out above as it fails to satisfy both Achirality Constraints. In view of its relevance, as afundamental super- σ -model of the superstring, we discuss it separately below. Example 5.20. The square root of the Green–Schwarz super-1-brane in sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) . The tensors∆ ≡ − , ∆ ≡ , f a ≡ Γ a satisfy the identities Π ( , ∣ − ) = − tr t ( ) ( Γ Γ ( D d, − Γ Γ )) = − D d, and Π ab = tr t ( ) ( Γ a Γ b ( D d, + Γ Γ )) = D d, ( δ ba − η bb ǫ ab ) , and so manifestly violate the Even and Odd Achirality Constraints. In consequence, the correspondencesector of the super- σ -model exhibits an additional Graßmann-even symmetry generated by the vectorfields T + T , whence the enhanced gauge-symmetry superdistribution in the exceptional form GS ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) = Dd, ⊕ α = ⟨ T α ⟩ ⊕ ⟨ T + T ⟩ ⊕ ⊕ ( a, ̂ b ) ∈ { , } × ,d ⟨ T a ̂ b ⟩ whereas the limit of the weak derived flag of the ( SO ( , ) × SO ( d − )) -descendable κ -symmetrysuperdistribution κ ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) ≡ Dd, ⊕ α = ⟨ T α ⟩ is given by κ −∞ ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) = Dd, ⊕ α = ⟨ T α ⟩ ⊕ ⟨ T − T ⟩ ⊊ Vac ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) . The two vector fields: T ± ∶ = T ± T are complementary, and it is natural to think of them as target-superspace counterparts of the chiralworldsheet diffeomorphisms (in, say, the static gauge). Thus, κ ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) alone is seen to generate one chiral half of the integrable vacuum of the superstring. Asnoted already in Ref. [Sus19, Remark 6.2] (in a different description adopted ibidem ), the appearanceof the chiral field T − in the weak derived flag of the κ -symmetry superdistribution is to be understoodin this context as a variant of the chiral Sugawara extension. learly, upon extending the purely Graßmann-odd κ -symmetry superdistribution by the span of T + ,that is by defining the extended κ -symmetry superdistribution κ ext ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) ∶ = Dd, ⊕ α = ⟨ T α ⟩ ⊕ ⟨ T + ⟩ , we obtain the anticipated result κ −∞ ext ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) ≡ Vac ( sISO ( d, ∣ D d, )/( SO ( , ) × SO ( d − )) , ̂ χ ( ) GS ) . The higher geometry behind the physics
The (super)background of a (super-) σ -model is but the lowest rung in a hierarchy of geometricstructures over the (super)target co-defining the field theory. As we are about to recapitulate, thehigher structures, associated with the (super)background’s ( p + ) -form component, encode essentialinformation on a distinguished quantisation scheme for the (super-) σ -model, and so any discussion ofa constitutive property of the latter is necessarily incomplete, and potentially even wrong, until welift it to those higher structures. This is particularly true of classical symmetries whose consistenttransposition to the quantum r´egime is a key component of the quantisation scheme. Accordingly, inthe last two sections of the present paper, we take the first step on the path towards a full-fledgedrealisation of the enhanced gauge symmetry of the GS super- σ -model in its HP formulation on objectsthat geometrise the super- ( p + ) -cocycles of the HP superbackground through a construction whichfeatures a beautiful interplay between higher cohomology and differential geometry. Below, we merelyrecapitulate those aspects thereof that are of immediate relevance to the intended symmetry analysis.The higher cohomology behind the topological term in the Dirac–Feynman amplitude of the purely bosonic (or Graßmann-even) σ -model was originally identified by Alvarez in Ref. [Alv85] and, morestructurally, by Gaw¸edzki in Ref. [Gaw88], and later geometrised by Murray and Stevenson in Refs. [Mur96,MS00] in a manner amenable to various subsequent generalisations and extensions. The point of de-parture is the interpretation of the said term in the Dirac–Feynman amplitude, A NG ,p DF , top ∶ = e i ( S ( NG ) σ,p − S ( NG ) σ,p, metr ) ∶ [ Ω p , M] Ð→ U ( ) ,S ( NG ) σ,p, metr [ x ] ∶ = ∫ Ω p √ det ( p ) ( x ∗ γ ) , x ∈ [ Ω p , M] , in a background B p = (M , γ, χ ( p + ) ) composed of a target space given by a standard metric ( C ∞ -)manifold (M , γ ) and of a de Rham ( p + ) -cocycle χ ( p + ) ∈ Z p + (M) with periodsPer ( χ ( p + ) ) ⊂ π Z . The term admits a local presentation A NG ,p DF , top [ x ] = e i ∫ Ω p x ∗ β ( p + ) for x ∈ [ Ω p , U] ⊂ [ Ω p , M] , where U ⊂ M is an open subset with the property χ ( p + ) ↾ U = d β ( p + ) for some β ( p + ) ∈ Ω p + (M) . It yields the so-called ( p + ) -volume holonomy A NG ,p DF , top ≡ Hol G ( p ) , over x ( Ω p ) ∈ Z p + (M) , of an abelian p -gerbe G ( p ) , the latter being a geometrisation of (the coho-mology class of) χ ( p + ) in the spirit of Ref. [Gaj97], inspired by the pioneering papers [Mur96, MS00]. The idea was reviewed at great length in Ref. [Sus17], where, moreover, a long bibliographical list was drawn. he holonomy is an example of a Cheeger–Simons differential character of degree p + π Z ( cp Ref. [CJ85]),Hol G ( p ) [ x ] = ∶ h G ( p ) ( x ( Ω p )) , h G ( p ) ∈ Hom
AbGrp ( Z p + (M) , U ( )) with the property ∀ c ( p + ) ∈ C p + ( M ) ∶ h G ( p ) ( ∂ c ( p + ) ) = ε ( p ) ( c ( p + ) ) , expressed in terms of the ( p + ) -cochain ε ( p ) ∈ Hom
AbGrp ( C p + (M) , U ( )) given by ε ( p ) ≡ e i ∫ ⋅ curv ( G ( p ) ) ∶ C p + (M) Ð→ U ( ) ∶ c ( p + ) z→ e i ∫ c ( p + ) curv ( G ( p ) ) , for curv (G ( p ) ) ≡ χ ( p + ) the curvature of G ( p ) . A little more abstractly, the holonomy is the imageHol G ( p ) [ x ] ≡ ι p ([ x ∗ G ( p ) ]) of the isoclass of the flat gerbe x ∗ G ( p ) over Ω p under the canonical isomorphism ι p ∶ W p + ( Ω p ; 0 ) ≅ ÐÐ→ U ( ) between the group W p + ( Ω p ; 0 ) of isoclasses of flat abelian p -gerbes over Ω p and U ( ) . It may begiven an entirely explicit form, though, in terms of a trivialisation of curv (G ( p ) ) over some opencover {O i } i ∈ I ≡ O M of M ( e.g. , a good one, i.e. , one with all non-empty multiple intersections O i ∩ O i ∩ ⋯ ∩ O i N ≡ O i i ...i N , N ∈ N × contractible, which always exists on a C -manifold by TheWeil–de Rham Theorem) that consists of sheaf-cohomological data of G ( p ) . The relevant cohomologyis the real Deligne–Beilinson hypercohomology, i.e. , the direct limit, over refinements of (good) opencovers, of the cohomologies of the total complexes of the bicomplexes formed by an extension of thebounded Deligne complex D( p + ) ● : Z M ≡ D( p + ) − R M ≡ D( p + ) Ω (M) ≡ D( p + ) Ω (M) ≡ D( p + ) ⋯ Ω p + (M) ≡ D( p + ) p + π id Z M ≡ d (− ) d ≡ d ( ) d ≡ d ( ) d ≡ d ( ) d ≡ d ( p ) , of sheaves of local integer constants, locally smooth maps (containing the former as the image of thesheaf counterpart of the injection Z ∋ n z→ πn ∈ R ) and k -forms (for k ∈ , p +
1) on M in thedirection of the ˇCech cohomology associated with a (fixed) good open cover O M , cp. Ref. [Joh02]. Thelatter cohomologies being defined for groups of k -cochainsˇCD k (O M , D( p + ) ● ) = ⊕ ( m,n ) ∈ N × ({ − } ∪ N ) m + n = k ˇ C m (O M , D( p + ) n ) , with ˇ C m (O M , D( p + ) n ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ Z ⊔ i ,i ,...,im + ∈ I O i i ...im + for n = − C ∞ ( ⊔ i ,i ,...,i m ∈ I O i i ...i m , R ) for n = n ( ⊔ i ,i ,...,i m ∈ I O i i ...i m ) for 0 < n ≤ p +
10 for n > p + , and the Deligne coboundary operators D ( k ) ∶ ˇCD k (O M , D( p + ) ● ) Ð→ ˇCD k + (O M , D( p + ) ● ) , The notion of a pullback in the (higher) category of p -gerbes was also recalled in Ref. [Sus17]. D ( k ) ↾ ˇ C m ( O M , D ( p + ) n ) = d ( n ) + ( − ) n + ˇ δ ( m ) , m + n = k , given in terms of the ˇCech coboundary operatorsˇ δ ( m ) ∶ ˇ C m (O M , D( p + ) n ) Ð→ ˇ C m + (O M , D( p + ) n ) ∶ ( ̟ ( n ) i i ...i m ) z→ ( m + ∑ l = ( − ) l ̟ ( n ) i i ...i m ̂ il ↾ O i i ...im + ≡ ( ˇ δ ( m ) ̟ ( n ) ) i i ...i m + ) . We have the useful
Proposition 6.1.
Adopt the above notation and define – for a ˇCech–Deligne k -cochain (for k < p + ) A ( k ) ≡ ( α ( k ) i , α ( k − ) i ′ i ′ , . . . , α (− ) i ( k ) i ( k ) ...i ( k ) k + ) ∈ ˇCD k (O M , D( p + ) ● ) . and a smooth vector field V ∈ Γ (T M) – the ˇCech–Deligne ( k − ) -cochain V ⌟ A ( k ) ∶ = (V ⌟ α ( k ) i , V ⌟ α ( k − ) i ′ i ′ , . . . , V ⌟ α ( ) i ( k − ) i ( k − ) ...i ( k − ) k − , ) ∈ ˇCD k − (O M , D( p + ) ● ) . The k -cochain − L V A ( k ) ∶ = ( − L V α ( k ) i , − L V α ( k − ) i ′ i ′ , . . . , − L V α ( ) i ( k − ) i ( k − ) ...i ( k − ) k , ) ∈ ˇCD k (O M , D( p + ) ● ) . satisfies the identity − L V A ( k ) = D ( k − ) (V ⌟ A ( k ) ) + V ⌟ D ( k ) A ( k ) . Proof.
Straightforward. (cid:3)
In this setting, the local data of G ( p ) compose a ( p + ) -cocycle B ( p + ) ≡ ( β ( p + ) i , β ( p ) i ′ i ′ , . . . , β (− ) i ( p + ) i ( p + ) ...i ( p + ) p + ) ∈ Ker D ( p + ) ⊂ ˇCD p + (O M , D( p + ) ● ) . (1-)Isomorphic p -gerbes G ( p ) A , A ∈ { , } over a common base are described by (DB-)cohomologous ( p + ) -cocycles B ( p + ) A , A ∈ { , } (for a suitable choice of the cover), i.e. , there exists a ˇCech–Deligne p -cochain P ( p ) ( ) ≡ ( P ( p ) ( ) i , P ( p − ) ( ) i ′ i ′ , . . . , P (− ) ( ) i ( p ) i ( p ) ...i ( p ) p + ) ∈ ˇCD p (O M , D( p + ) ● ) (6.2)satisfying the identity B ( p + ) + D ( p ) P ( p ) ( ) = B ( p + ) . Similarly, (2-)isomorphic (1-)isomorphisms represented by p -cochains P ( p ) ( ) A , A ∈ { , } as above are(DB-)cohomologous (for a suitable choice of the cover), i.e. , there exists a ˇCech–Deligne ( p − ) -cochain P ( p − ) ( ) ≡ ( P ( p − ) ( ) i , P ( p − ) ( ) i ′ i ′ , . . . , P (− ) ( ) i ( p ) i ( p ) ...i ( p ) p ) ∈ ˇCD p − (O M , D( p + ) ● ) satisfying the identity P ( p ) ( ) + D ( p − ) P ( p − ) ( ) = P ( p ) ( ) , and this hierarchy continues all the way to ( p + ) -isomorphisms for which equivalence means properequality.The fundamental advantage of the gerbe-theoretic description of the σ -model, justifying the in-troduction of the somewhat heavy cohomological and geometric formalism, is the canonical way toprequantisation that it paves. The way leads through the transgression map τ p ∶ H p + (M , D( p + ) ● ) Ð→ H ( C p M , D( ) ● ) , rst noted and put to use for p = p -gerbe G ( p ) over M , represented by a class in H p + (M , D( p + ) ● ) , the isoclass of a principal C × -bundle C × / / L G ( p ) π L G( p ) (cid:15) (cid:15) C p M , (6.3)termed the transgression bundle , over the configuration space C p M ≡ [ C p , M ] of the σ -model attached to a Cauchy hypersurface C p ⊂ Ω p , with connection ∇ L G( p ) of curvaturecurv ( ∇ L G( p ) ) = ∫ C p ev ∗ p curv ( G ( p ) ) , written for ev p ∶ C p M × C p Ð→ M ∶ ( x, c ) z→ x ( c ) . The curvature of ∇ L G( p ) is to be compared with the (pre)symplectic form Ω σ = δϑ T ∗ C p M + π ∗ T ∗ C p M ∫ C p ev ∗ p curv ( G ( p ) ) of the σ -model over its space of states T ∗ C p M (written out in its simplest form) canonically projectingto the configuration space, π T ∗ C p M ∶ T ∗ C p M Ð→ C p M , the 2-form being expressed in terms of theso-called (kinetic-action) 1-form ϑ T ∗ C p M ∈ Ω ( T ∗ C p M ) with the familiar presentation ϑ T ∗ C p M [ x, p ] = ∫ C p Vol (C p ) ∧ p µ ( ⋅ ) δx µ ( ⋅ ) in the coordinates ( x µ , p ν ) on T ∗ C p M (here, p is the kinetic -momentum field over the Cauchyhypersurface). Suitably polarised sections of the line bundle associated with the principal C × -bundlegiven by the pullback, along the bundle projection π T ∗ C p M , of the transgression bundle to the classicalspace of states T ∗ C p M define the Hilbert space of the σ -model.The intrinsically quantum-mechanical nature of the geometrisation G ( p ) of the ( p + ) -cocycle χ ( p + ) ,derived from Dirac’s ingenious interpretation of the classical lagrangean density as a transport operatorbetween quantum states (later elaborated by Feynman), makes the higher geometry associated with G ( p ) a natural arena for a rigorous discussion of (pre)quantisable symmetries of the σ -model. Thelatter are customarily grouped into two categories: ● global symmetries – represented by those isometries of (M , γ ) , composing a group G σ acting smoothly as λ ⋅ ∶ G σ × M Ð→ M ∶ ( g, m ) z→ λ g ( m ) , that lift to p -gerbe (1-)isomorphismsΦ g ∶ λ ∗ g G ( p ) ≅ ÐÐ→ G ( p ) , g ∈ G σ , (6.4) with – for an open cover O M endowed with index maps ı g ∶ I ↺ such that λ g (O i ) ⊂ O ı g ( i ) , i.e. , defining an extension ̂ λ ∗ g ∶ ˇCD k (O M , D( p + ) ● ) ↺ The (pre)symplectic form for the lagrangean field theory of maps from [ Ω p , M ] defined by S ( NG ) σ,p can be derived inthe first-order formalism of Refs. [Gaw72, Kij73, Kij74, KS76, Szc76, KT79], cp also Ref. [Sau89] for a modern treatment. The principal C × -bundle π ∗ T ∗ C p M L G ( p ) (with the principal C × -connection 1-form corrected by the pullback ofthe action 1-form ϑ T ∗ C p M from its base) is to be understood as the frame bundle F L σ of the prequantum bundle L σ ≅ ( F L σ × C )/ C × . ( ̟ ( k ) i , ̟ ( k − ) i ′ i ′ , . . . , ̟ (− ) i ( k ) i ( k ) ...i ( k ) k + ) z→ ( λ ∗ g ̟ ( k ) ı g ( i ) , λ ∗ g ̟ ( k − ) ı g ( i ′ ) ı g ( i ′ ) , . . . , λ ∗ g ̟ (− ) ı g ( i ( k ) ) ı g ( i ( k ) ) ...ı g ( i ( k ) k + ) ) of the manifold map λ ⋅ – a sheaf-cohomological presentation furnished by a ˇCech–Deligne p -cochain P ( p ) ( ) g as in Eq. (6.2), satisfying the identity ̂ λ ∗ g B ( p + ) + D ( p ) P ( p ) ( ) g = B ( p + ) ; ● local symmetries induced from the global ones with a model G σ – represented by G σ -equivari-ant structures on G ( p ) that may be understood – in the cohomological picture – as completionsof the ( p + ) -cocycles B ( p + ) in the Deligne–Beilinson cohomology to those in its extension in thedirection of group cohomology (for a suitable choice of the open cover O M , cp Ref. [GSW13,App. I]) that are based on the sheaf-cohomological data of the p -gerbe 1-isomorphismΥ p ≡ Υ ( ) p ∶ λ ∗⋅ G ( p ) ≅ ÐÐ→ pr ∗ G ( p ) ⊗ I ( p ) ̺ − θ L ( p + ) (6.5) over the arrow manifold G σ × M of the action groupoid G σ ⋉ M (with the source and targetmaps given by pr and λ ⋅ , respectively, cp Def. 6.13), where I ( p ) ̺ − θ L ( p + ) is the trivial p -gerbe withthe global curving ̺ − θ L ( p + ) = p + ∑ k = ( − ) p − k k ! pr ∗ α A A ...A k ( p + − k ) ∧ pr ∗ ( θ A L ∧ θ A L ∧ ⋯ ∧ θ A k L ) ∈ Ω p + ( G σ × M ) written in terms of components θ A L , A ∈ , dim G σ of the left-invariant Maurer–Cartan 1-form θ L = θ A L ⊗ τ A ∈ Ω ( G σ ) ⊗ g σ corresponding to generators τ A of the Lie algebra g σ of G σ , as well as of the ( p − k ) -forms α A A ...A k + ( p − k ) = ( − ) k ( p − k − ) K λ ⋅ A ⌟ K λ ⋅ A ⌟ ⋯ ⌟ K λ ⋅ A k ⌟ κ A k + ( p ) , k ∈ , p that are all determined by the fundamental vector fields K λ ⋅ A ≡ K λ ⋅ τ A ∈ Γ ( T M) for λ ⋅ associatedwith the τ A in the usual manner and by the p -forms κ A ( p ) (assumed to exist) satisfying theconditions d κ A ( p ) = − K λ ⋅ A ⌟ curv (G ( p ) ) , A ∈ , dim G σ , cp Ref. [Sus19, Sec. 2] for details and examples.
Remark 6.2.
It is important to note that p -gerbes admitting a G σ -equivariant structure with a vanishing curving, ̺ − θ L ( p + ) ≡ , are precisely the ones that descend to the quotient manifold M/ G σ whenever the latter exists, that isevery gerbe over M with that property is (then) a pullback of a p -gerbe over M/ G σ and vice versa , cp Ref. [GSW10, Thm. 5.3] (and also Ref. [GSW13, Thms. 8.15 & 8.17]).The above identification of the higher-geometric realisations of σ -model symmetries was confirmed inRef. [Sus12], where it was demonstrated that the p -gerbe isomorphisms transgress to automorphisms ofthe prequantum bundle, and so (some of them) give rise to symmetries of the quantum theory, whereasequivariant structures provide target-space data for the gauge-symmetry defects of Ref. [Sus11b, Sus12,Sus13] that implement the gauging of the global symmetry G σ through a natural generalisation of theworldsheet-orbifold construction of Refs. [DHVW85, DHVW86] and of its more recent application inthe framework of the TFT quantisation of CFT in Ref. [FFRS09] ( cp also Ref. [JK06]). Equivariantstructures themselves were first introduced in the context of the gauging of global σ -model symmetriesin Refs. [GSW10, GSW13], where a mixture of geometric and categorial arguments enriched with field-theoretic considerations was invoked to prove that the structures do, indeed, ensure descent of the fieldtheory from the original target space M to the orbit space M/ G σ or, whenever the latter is not asmooth manifold, transform the original theory into a model of a field theory with M/ G σ as the targetspace through application of the Universal Gauge Principle worked out in those papers. ll the field-theoretic, higher-geometric and -cohomological ideas brought up above converge nat-urally and become particularly tightly entangled in the setting of super- σ -models on homogeneousspaces of Lie supergroups, considered in the present paper. Indeed, the couplings of the backgroundGreen–Schwarz (super-) ( p + ) -cocycles to the (super)charge currents sourced by the propagation ofsuper- p -branes ‘in’ the homogeneous spaces, defining the topological terms in the corresponding Dirac–Feynman amplitudes, call for a deep and precise understanding and a rigorous formal description justlike their Graßmann-even counterparts, and super symmetry – both global and local – is the organisingprinciple in the construction of the relevant field theories, constraining severly the admissible choicesof a consistent backround and the topological charge of the super- p -brane, or – more precisely – theratio of the gravitational and topological charge ( i.e. , the relative normalisation of the two terms in thesuper- σ -model action functional). This is the basic rationale behind the postulate and a systematicadvancement of the programme of a supersymmetry-equivariant geometrisation of the (super-) ( p + ) -form fields and their topological couplings to ‘embedded’ super- p -brane worldvolumes that was putforward in Ref. [Sus17] and subsequently elaborated from a variety of angles in Refs. [Sus19, Sus18a]and Ref. [Sus18b]. The programme combines the intuitions and techniques derived and borrowed fromits predecessor – the comprehensive study of the gerbe theory of the two-dimensional bosonic σ -modelwith the Wess–Zumino term in the action functional , laying emphasis on the reformulation andelucidation of global symmetries and of the Universal Gauge Principle in these field theories in thehigher-geometric (or gerbe-theoretic) language – with the Kleinian(-type) supergeometry of the saidhomogeneous spaces and the cohomology sensu largo of Lie supergroups. The fundamental idea under-lying the postulate is the physically motivated identification of the supersymmetry-invariant refinementof the standard de Rham cohomology as the appropriate cohomology in which to analyse and trivialisethe Green–Schwarz (super-) ( p + ) -cocycles, in conjunction with the interpretation of the discrepancybetween that refinement and the original de Rham cohomology in terms of the nontrivial topology(or homology) of the Graßmann-odd fibre of an orbifold of the supertarget space by the action of adiscrete subgroup of the supersymmetry group generated by ‘integral’ supertranslations , the orbifoldbeing understood as the implicit supertarget space of the field theory in question. The interpretation,originally advanced by Rabin in Ref. [Rab87] (on the basis of his earlier work with Crane on globalaspects of supergeometry, reported in Ref. [RC85]), and then rephrased and further corroborated bythe Author, in Ref. [Sus19], through the study of the wrapping(-charge) anomaly in the Poisson algebraof Noether charges of global supersymmetry in the canonical description of the super- σ -model in thepresence of non-vanishing monodromies of (Cauchy) states along the Graßmann-odd directions in thesupertarget space, led to the construction of higher-supergeometric objects, termed super- p -gerbes,that are in exactly the same structural relation to the Green–Schwarz (super-) ( p + ) -cocycles as theone between p -gerbes and de Rham ( p + ) -cocycles in the purely Graßmann-even setting, but withthe supersymmetry-invariant cohomology replacing the standard de Rham cohomology. Thus, by astraightforward generalisation of Ref. [Sus17, Def. 5.21], we arrive at the postulative Definition 6.3.
Let G be a Lie supergroup and M a supermanifold endowed with an action λ ∶ G × M Ð→ M of G, in the sense of Ref. [CCF11, Def. 7.2.7]. A G -invariant p -gerbe is an abelian (bundle) p -gerbe, in the sense of Ref. [Gaj97], with total spaces of all surjective submersions entering its definitionendowed with the respective (projective) lifts of λ , commuting with the defining C × -actions on the totalspaces of all the principal C × -bundles present, with respect to which the submersions are equivariant,all connections are invariant and all (connection-preserving) principal C × -bundle isomorphisms areequivariant. G -invariant k -isomorphisms between G -invariant p -gerbes for k ∈ , p + i.e. , it is a proper Lie supergrupthat is not a Lie group), the corresponding G-invariant p -gerbes are called super- p -gerbes , or su-persymmetric p -gerbes , and G-invariant k -isomorphisms between them are called super- p -gerbe k -isomorphisms , or supersymmetric p -gerbe k -isomorphisms . Of particular relevance are the σ -models with the background ( G , κ g ○ ( θ L ⊗ θ L ) , H ( ) = κ g ○ ([ θ L ∧ ,θ L ] ∧ θ L )) givenby a compact Lie group G with the Cartan–Killing metric ( κ g is the Killing form on the Lie algebra g of G) and thecanonical Cartan 3-form on it, i.e. , the so-called Wess–Zumino–Witten σ -models of Ref. [Wit84], cp also Ref. [Gaw99] foran in-depth exposition. For an interpretation of this somewhat imprecise definition of the subgroup – of the Kosteleck´y–Rabin type ( cp Ref. [KR84]) – in the S -point picture, we refer the Reader to Ref. [Sus17, Rem. 4.1]. Remark 6.4.
Using the structure of a G-invariant p -gerbe, we may readily construct p -gerbe 1-isomorphisms (6.4), lifting the action of the (super)symmetry group G to the geometric object G ( p ) from its base. This fact was illustrated (on 0- and 1-gerbes) in Ref. [Sus19, Secs. 4.1 & 4.2].The general definition, which can readily be derived from the hitherto developments of the supersym-metry-equivariant geometrisation programme , has the following important specialisations that playa prime rˆole in the analysis of concrete examples. Definition 6.5.
In the notation of Def. 6.3, a super-0-gerbe , or a supersymmetric 0-gerbe over M of curvature χ ( ) ∈ Z (M) , with G as the supersymmetry group, is a principal C × -bundle ( Y M , π Y M , A ( ) ) , described by the diagram C × / / Y M , A ( ) π Y M (cid:15) (cid:15) M , χ ( ) in which π Y M is a surjective submersion and A ( ) ∈ Ω ( Y M) is a principal C × -connection such that d A ( ) ∈ π ∗ Y M χ ( ) (the pullback having the usual realisation in the local-coordinate ( i.e. , S -point) picture), with thefollowing properties: ● the action λ admits a lift Y λ to the total space Y M determined by the commutative diagramG × Y M Y λ / / id G × π Y M (cid:15) (cid:15) Y M π Y M (cid:15) (cid:15) G × M λ / / M (6.6) that commutes with the defining action r Y M ∶ Y M × C × Ð→ Y M of the structure group C × on Y M , as expressed by the commutative diagramG × Y M × C × Y λ × id C × (cid:15) (cid:15) id G × r Y M / / G × Y M Y λ (cid:15) (cid:15) Y M × C × r Y M / / Y M ; ● the lift preserves the principal C × -connection, which reduces, in the local-coordinate ( i.e. , S -point) picture, to the standard property of invariance of the latter super-1-form under pullbackalong the mapping, induced by Yon λ (S) , that effects the action of a fixed g ∈ Yon G (S) onYon M (S) (both in a local-coordinate description), to be written – by a customary abuse ofnotation – as Y λ ∗ g A ( ) = A ( ) , g ∈ Yon G (S) . Cp , e.g. , the opening discussion on the notion of supersymmetry of a 0- and 1-gerbe over a supermanifold inRef. [Sus19, Sec. 4.1] and Ref. [Sus19, Sec. 4.2], respectively. n isomorphism between super-0-gerbes ( Y A M , π Y A M , A ( ) A ) , A ∈ { , } , endowed with therespective lifts Y A λ of λ , is a connection-preserving isomorphism Y M Φ / / π Y M (cid:15) (cid:15) Y M π Y M (cid:15) (cid:15) M id M M between the principal C × -bundles that intertwines the lifts, i.e. , satisfiesG × Y M Y λ / / id G × Φ (cid:15) (cid:15) Y M Φ (cid:15) (cid:15) G × Y M Y λ / / Y M . ◇ Definition 6.6.
In the notation of Def. 6.3, a super-1-gerbe , or a supersymmetric 1-gerbe ofcurvature χ ( ) ∈ Z (M) over M , with G as the supersymmetry group, is an abelian (bundle) 1-gerbe ( Y M , π Y M , B ( ) , L, π L , A ( ) L , µ L ) , described by the diagram µ L ∶ pr ∗ , L ⊗ pr ∗ , L ≅ ÐÐ→ pr ∗ , L (cid:15) (cid:15) C × / / L, A ( ) Lπ L (cid:15) (cid:15) Y [ ] M pr , - - pr , / / pr , Y [ ] M pr / / pr / / Y M , B ( ) π Y M (cid:15) (cid:15) M , χ ( ) in which π Y M is a surjective submersion, B ( ) ∈ Ω ( Y M) is the curving such that d B ( ) = π ∗ Y M χ ( ) , and L is a principal C × -bundle over the fibred square , described by the commutative diagram Y [ ] M pr " " ❊❊❊❊❊❊❊❊❊❊ pr | | ②②②②②②②②②② Y M π Y M " " ❊❊❊❊❊❊❊❊❊❊ Y M π Y M | | ②②②②②②②②②② M , and endowed with a principal C × -connection A ( ) L ∈ Ω ( L ) of curvature ( pr ∗ − pr ∗ ) B ( ) , d A ( ) L = π ∗ L ( pr ∗ − pr ∗ ) B ( ) , Cp , e.g. , Ref. [CCF11, Def. B.1.11]. nd a (connection-preserving) principal C × -bundle isomorphism µ L of (the tensor product of) thepullback principal C × -bundlespr ∗ i,j L ≡ Y [ ] M × pr i,j L pr / / pr (cid:15) (cid:15) L π L (cid:15) (cid:15) Y [ ] M pr i,j ≡ ( pr i , pr j ) / / Y [ ] M , ( i, j ) ∈ {( , ) , ( , ) , ( , )} over the fibred cube Y [ ] M , described by the commutative diagram Y [ ] M pr & & ▲▲▲▲▲▲▲▲▲▲▲▲ pr (cid:15) (cid:15) pr x x rrrrrrrrrrrr Y M π Y M & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ Y M π Y M (cid:15) (cid:15) Y M π Y M x x qqqqqqqqqqqqq M , that induces a groupoid structure on its fibres, being subject to the associativity constraintpr ∗ , , µ L ○ ( id pr ∗ , L ⊗ pr ∗ , , µ L ) = pr ∗ , , µ L ○ ( pr ∗ , , µ L ⊗ id pr ∗ , L ) over the fibred tetrahedron Y [ ] M (with its canonical projections pr i,j,k ≡ ( pr i , pr j , pr k ) ∶ Y [ ] M Ð→ Y [ ] M and pr i,j ≡ ( pr i , pr j ) ∶ Y [ ] M Ð→ Y [ ] M, i, j ∈ { , , , } ), described by the commutativediagram Y [ ] M pr * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ pr & & ▲▲▲▲▲▲▲▲▲▲▲▲ pr x x rrrrrrrrrrrr pr t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Y M π Y M * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Y M π Y M & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ Y M π Y M x x qqqqqqqqqqqqq Y M π Y M t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ M , with the following properties: ● the action λ admits a lift Y λ to the total space Y M determined by the commutativeDiag. (6.6), and so also to the fibred products Y [ n ] M , n ∈ { , , } , Y [ n ] λ ≡ ( Y λ ○ pr , , Y λ ○ pr , , . . . , Y λ ○ pr ,n ) ∶ G × Y [ n ] M Ð→ Y [ n ] M , and from Y [ ] M to the total space L ,G × L Lλ / / id G × π L (cid:15) (cid:15) L π L (cid:15) (cid:15) G × Y [ ] M Y [ ] λ / / Y [ ] M , the latter lift commuting with the defining action r L ∶ L × C × Ð→ L The tensor product L ⊗ L of principal C × -bundles L α , α ∈ { , } is defined, after Ref. [Bry93], as the (principal)bundle ( L × L )/ C × associated with L through the defining C × -action on L , cp also Ref. [Sus17, Rem. 5.5] andRef. [Keß19, Def. 6.2.1]. f the structure group C × on L , as expressed by the commutative diagramG × L × C × Lλ × id C × (cid:15) (cid:15) id G × r L / / G × L Lλ (cid:15) (cid:15) L × C × r L / / L ; ● the lifts Y [ ] λ and Lλ induce, in turn, the canonical lifts L i,j λ ≡ ( Y [ ] λ ○ ( pr × pr ) , Lλ ○ ( pr × pr )) ∶ G × pr ∗ i,j L ≡ G × ( Y [ ] M × pr i,j L ) Ð→ pr ∗ i,j L of λ to the respective pullback bundles, and so also a (diagonal) lift to the tensor-productbundle L , , λ ∶ G × ( pr ∗ , L ⊗ pr ∗ , L ) Ð→ pr ∗ , L ⊗ pr ∗ , L ; ● the lifts Y λ and Lλ preserve the curving B ( ) and the principal C × -connection A ( ) L , respectively,which – in the previously introduced notation – may be written as Y λ ∗ g B ( ) = B ( ) ∧ Lλ ∗ g A ( ) L = A ( ) L , g ∈ Yon G (S) ; ● the groupoid structure µ L is equivariant with respect to the lifted actions, µ L ○ L , , λ = L , λ ○ ( id G × µ L ) . An isomorphism between super-1-gerbes ( Y A M , π Y A M , B ( ) A , L A , π L A , A ( ) L A , µ L A ) , A ∈ { , } ,endowed with the respective lifts Y A λ and L A λ , is a gerbe 1-isomorphism Φ = ( YY , M , π YY , M , E, π E , A ( ) E , α E ) , composed of a surjective submersion π YY , M ∶ YY , M Ð→ Y , M over the fibred product Y , M ≡ Y M × M Y M pr ' ' PPPPPPPPPPPPPP pr w w ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Y M π Y M ( ( PPPPPPPPPPPPPPPP Y M π Y M v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ M , and a principal C × -bundle C × / / E, A ( ) Eπ E (cid:15) (cid:15) YY , M over its total space with a surjective submersion π E and a principal C × -connection A ( ) E ∈ Ω ( E ) of curvature pr ∗ B ( ) − pr ∗ B ( ) , d A ( ) E = π ∗ E ( pr ∗ B ( ) − pr ∗ B ( ) ) , together with a connection-preserving principal C × -bundle isomorphism α E ∶ ( π YY , M × π YY , M ) ∗ pr ∗ , L ⊗ pr ∗ E ≅ ÐÐ→ pr ∗ E ⊗ ( π YY , M × π YY , M ) ∗ pr ∗ , L ver the fibred square Y [ ] Y , M ≡ YY , M × M YY , M pr ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ pr u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ YY , M π Y M ○ pr ○ π YY , M ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ YY , M π Y M ○ pr ○ π YY , M u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ M satisfying the identity ( id pr ∗ E ⊗ π ∗ , , ○ pr ∗ , , µ L ) ○ ( pr ∗ , α E ⊗ id π ∗ , ○ pr ∗ , L ) ○ ( id π ∗ , ○ pr ∗ , L ⊗ pr ∗ , α E ) = pr ∗ , α E ○ ( π ∗ , , ○ pr ∗ , , µ L ⊗ id pr ∗ E ) over the fibred cube Y [ ] Y , M ≡ YY , M × M YY , M × M YY , M pr , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ pr (cid:15) (cid:15) pr r r ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ YY , M π Y M ○ pr ○ π YY , M , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ YY , M π Y M ○ pr ○ π YY , M (cid:15) (cid:15) YY , M π Y M ○ pr ○ π YY , M r r ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ M , equipped with the mappings π i,j = ( π YY , M × π YY , M ) ○ pr i,j , ( i, j ) ∈ {( , ) , ( , ) , ( , )} ,π , , = π YY , M × π YY , M × π YY , M , the gerbe 1-isomorphism Φ having the following properties: ● the action Y , λ ≡ ( Y λ ○ pr , , Y λ ○ pr , ) ∶ G × Y , M Ð→ Y , M , induced canonically by the Y A λ admits a lift YY , λ to YY , M , described by the commuta-tive diagram G × YY , M YY , λ / / id G × π YY , M (cid:15) (cid:15) YY , M π YY , M (cid:15) (cid:15) G × Y , M Y , λ / / Y , M , and the latter further lifts to the total space E ,G × E Eλ / / id G × π E (cid:15) (cid:15) E π E (cid:15) (cid:15) G × YY , M YY , λ / / YY , M , the resulting lift commuting with the defining action r E ∶ E × C × Ð→ E f the structure group C × on E , as expressed by the commutative diagramG × E × C × Eλ × id C × (cid:15) (cid:15) id G × r E / / G × E Eλ (cid:15) (cid:15) E × C × r E / / E ; ● the lifts YY , λ and Eλ , in conjunction with the formerly introduced ones, induce, in turn,the canonical lifts Y [ ] A,A + L A λ ≡ ( Y [ ] Y , λ ○ pr , , ( Y , λ ○ pr , , Y , λ ○ pr , ) ○ pr , , L A λ ○ pr , ) ∶ G × ( Y [ ] Y , M × π YY , M × π YY , M ( Y [ ] , M × pr A,A + L A )) Ð→ Y [ ] Y , M × π YY , M × π YY , M ( Y [ ] , M × pr A,A + L A ) , Y [ ] A Eλ ≡ ( Y [ ] Y , λ ○ pr , , Eλ ○ pr , ) ∶ G × ( Y [ ] Y , M × pr A E ) Ð→ Y [ ] Y , M × pr A E written for A ∈ { , } , for the fibred square Y [ ] , M ≡ Y , M × M Y , M pr ( ( PPPPPPPPPPPPPPP pr v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Y , M π Y M ○ pr ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ Y , M π Y M ○ pr v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ M and for the canonical lift Y [ ] Y , λ ≡ ( YY , λ ○ pr , , YY , λ ○ pr , ) ∶ G × Y [ ] Y , M Ð→ Y [ ] Y , M of YY , λ to Y [ ] Y , M , and so also canonical (diagonal) lifts Y [ ] , L Eλ ∶ G × ( π YY , M × π YY , M ) ∗ pr ∗ , L ⊗ pr ∗ E Ð→ ( π YY , M × π YY , M ) ∗ pr ∗ , L ⊗ pr ∗ E , Y [ ] , EL λ ∶ G × pr ∗ E ⊗ ( π YY , M × π YY , M ) ∗ pr ∗ , L Ð→ pr ∗ E ⊗ ( π YY , M × π YY , M ) ∗ pr ∗ , L to the tensor-product bundles are induced; ● the lift Eλ preserves the principal C × -connection A ( ) E , which – in the previously introducednotation – may be written as Eλ ∗ g A ( ) E = A ( ) E , g ∈ Yon G (S) ; ● the isomorphism α E is equivariant with respect to the lifted actions, α E ○ Y [ ] , L Eλ = Y [ ] , EL λ ○ ( id G × α E ) . ◇ The above constitute a most natural abstraction of the concrete results of Refs. [Sus17, Sus19, Sus18a,Sus18b], the latter being tailored to the peculiar supergeometric circumstances in which physicallyrelevant models are realised (by construction). These are, invariably, the circumstances in which theG-supermanifold of a supersymmetry Lie supergroup G of immediate interest is a homogeneous spaceG / H relative to an isotropy Lie subgroup H ⊂ ∣ G ∣ ≡ M H of its body, the mother supergroup fibringprincipally over the base M H in the manner discussed at length in Sec. 3. Furtermore, the correspond-ing super- ( p + ) -cocycles H ( p + ) (descended from those on G, cp Eq. (3.1)) define nontrivial classes inthe supersymmetry-invariant refinement of the de Rham cohomology of M H , at least as long as we in-sist – in keeping with the principles which underlie the construction of the associated super- σ -models,as postulated in the original papers and subsequently recalled and adapted to the higher-geometric ramework in Ref. [Sus18a] – that the curvings on homogeneous spaces with a super-Minkowskian limitof the (homogeneous) ˙In¨on¨u–Wigner blow-up should asymptote to their flat-superspacetime counter-parts, cp Ref. [Sus18a] for details of the argument. As a consequence, the task of geometrising H ( p + ) relative to the supersymmetrically-invariant refinement of the de Rham cohomology of M H , H ● dR ( M H , R ) G , boils down to associating with the nontrivial class [ χ ( p + ) ] CaE , H − basic of χ ( p + ) in the (right-)H-basic refinement of the Cartan–Eilenberg cohomology of G, that is the de Rham cohomologyCaE ● ( G ) H − basic ≡ H ● dR ( G , R ) GH − basic of (left-)G-invariant and H-basic superdifferential forms on G, a particular abelian super- p -gerbe overG that descends (as a super- p -gerbe) to the quotient G / H. By a super-counterpart of the argumentrecalled in Rem. 6.2, this requires that the super- p -gerbe over G be equipped with an H-equivariantstructure of a vanishing curving which should be taken supersymmetric ( i.e. , compatible with the globalsupersymmetry present, essentially along the lines of Ref. [Sus19, Sec. 4]). Being a resolution of χ ( p + ) in the (H-basic) Cartan–Eilenberg cohomology, the mother super- p -gerbe over G may acquire a Lie-superalgebraic description determined by the super-variant of – on the one hand – the classical equalitybetween the Cartan–Eilenberg cohomology CaE ● ( G ) and the Chevalley–Eilenberg cohomologyCE ● ( g ) ≡ H ● ( g ; R ) of the Lie superalgebra g with values in the trivial g -module R , as derived in Ref. [Sus17, App. C], and– on the other hand – an interpretation of classes in H ● ( g ; R ) in terms of superalgebraic extensionsof g , further assumed integrable to surjective submersions over G. The latter interpretation has beenconsidered in two as yet unrelated (in all generality) guises: ● the direct one-to-one correspondence between classes in CE p + ( g ) and the peculiar L ∞ -super-algebras introduced by Baez and Huerta in Refs. [BH11, Hue11] and termed slim Lie ( p + ) -superalgebras ( cp also Ref. [BC04] for a non- Z / Z -graded precursor of that correspondence)– some preliminary ideas about integration of these Lie ( p + ) -superalgebras to the so-called Lie ( p + ) -supergroups were presented in Ref. [Hue11, Chap. 7] but it seems that the rangeof applicability of the integration method is an open question and – to the best of the Author’sknowledge – no concrete examples of Lie ( p + ) -supergroups of physical (superstring-theoretic)relevance have been constructed explicitly to date (let alone the sought-after super- p -gerbes); ● a physics-guided construction of an extension ̂ G ̂ π ÐÐ→
G that integrates an extension ̂ g ̂ P ÐÐ→ g parametrised by topological charges carried by the super- p -brane and has the fundamentalproperty [̂ π ∗ χ ( p + ) ] CaE , H − basic = ∈ CaE p + (̂ G ) H − basic that no proper sub-superalgebra of ̂ g shares with it – the extension may admit a descriptionin terms of a short exact sequence of Lie supergroups / / Z Z / / ̂ G ̂ π / / G / / integrating a short-exact sequence of the corresponding Lie superalgebras / / z z / / ̂ g ̂ P / / g / / that results from a sequence of supercentral extensions induced, each, by a nontrivial classin the 2nd cohomology group CE ( g int ) H − basic of an intermediate Lie superalgebra g int ( g being one of them) engendered by χ ( p + ) in such a manner that every extension in the sequenceyields a partial, or termwise, trivialisation of the pullback of χ ( p + ) to the Lie supergroupG int (to which g int integrates ) in the corresponding H-basic Cartan–Eilenberg cohomologyCaE ● ( G int ) H − basic (this is the construction, originally due to de Azc´arraga, exploited and A (super-) k -form ω ( k ) on an H-(super)manifold M is termed H-basic if it is H-invariant and H-horizontal, the lattermeaning that contractions of fundamental vector fields on M (induced by the H-action) with ω ( k ) vanish identically. Integrability is essentially controlled by the Tuynman–Wiegerinck criterion, cp Ref. [TW87]. laborated in the super-Minkowskian setting of Ref. [Sus17] where it led to the emergenceof the so-called extended superspacetimes , cp Ref. [CdAIPB00], and in the case of Zhou’ssuper-0-brane in s ( AdS × S ) , the latter being constrained severly by the Green–Schwarz limitof the ˙In¨on¨u–Wigner blow-up s ( AdS × S ) Ð→ sMink ( , ∣ D , ) ), or – alternatively, in thecase of curved supertargets – it may come from an enrichment of the original supersymmetryLie superalgebra g by the said supercharges { Z i } i ∈ ,N , spanning a subspace z = ⊕ Ni = ⟨ Z i ⟩ ⊂ ̂ g with the property [ z , z } ̂ g ∩ g ≠ , as dictated by the physically motivated (˙In¨on¨u–Wigner) asymptotics of ̂ g (this seems to bethe case for the supertargets s ( AdS × S ) and s ( AdS × S ) of super-1-brane (superstring)propagation, cp the tentative proposal of Ref. [Sus18a, Sec. 9]). Remark 6.7.
Whenever the homogeneous space M H is endowed with the structure of a Lie super-group, we may try to restrict the extension procedure to it rather than going via the mother Lie super-group G. This was, indeed, the tactic successfully applied to the super-Minkowskian superbackgroundstreated at length in Ref. [Sus17], where the relevant surjective submersions came from Lie-supergroupextensions of the supertranslation group.The above line of reasoning, based on Lie-supergroup/-superalgebra extensions, leads to a specialisationof the former definitions of a super- p -gerbe, Definition 6.8. A Cartan–Eilenberg super- p -gerbe is a super- p -gerbe object, as described inDef. 6.3, in the category of Lie supergroups, with all constitutive surjective surjections correspondingto Lie-supergroup extensions. ◇ and, in particular, of a super-0- and a super-1-gerbe. Definition 6.9. A Cartan–Eilenberg super-0-gerbe is a super-0-gerbe in the sense of Def. 6.5,such that ● M ≡ G, taken with the left action induced by the supergroup multiplication in G; ● Y M ≡ Y G is endowed with the structure of a Lie supergroup that extends that on G; ● π Y M ≡ π Y G is a Lie-supergroup epimorphism.An isomorphism between Cartan–Eilenberg super-0-gerbes ( Y A G , π Y A G , A ( ) A ) , A ∈ { , } is a super-0-gerbe isomorphism Φ as in Def. 6.5 that is simultaneously a Lie-supergroup isomorphism. ◇ Definition 6.10. A Cartan–Eilenberg super-1-gerbe is a super-1-gerbe in the sense of Def. 6.6,such that ● M ≡ G, taken with the left action induced by the supergroup multiplication in G; ● Y M ≡ Y G (and so also Y [ k ] M ≡ Y [ k ] G , k ∈ { , , } ) as well as L (and so also the variouspullbacks thereof along the canonical projections, and their tensor products) are endowed withthe structure of a Lie supergroup extending that on G; ● π Y M ≡ π Y G and π L are Lie-supergroup epimorphisms, and µ L is a Lie-supergroup isomor-phism establishing an equivalence of the respective extensions.An isomorphism between Cartan–Eilenberg super-1-gerbes ( Y A G , π Y A G , B ( ) A , L A , π L A , A ( ) L A ,µ L A ) is a super-1-gerbe isomorphism Φ = ( YY , G , π YY , G , E, π E , A ( ) E , α E ) as in Def. 6.6 with the fol-lowing properties: ● YY , G is endowed with the structure of a Lie supergroup that extends that on Y , G (theproduct one); ● E is endowed with the structure of a Lie supergroup that extends that on YY , G; ● π YY , G and π E are Lie-supergroup epimorphisms, and α E is a Lie-supergroup isomorphism(for the induced Lie-supergroup structures induced in a canonical manner on its domain andcodomain). The peculiarity of the super-Minkowskian setting is the existence of a Lie supergroup structure on the homogeneousspace itself. As a consequence, one may seek to extend the supersymmetry Lie superalgebra and erect the super- p -gerbedirectly over the supertarget. This idea was put to work in Ref. [Sus17]. While the geometrisation scheme based on Lie-supergroup extensions induced by the super- p -branecharge is largely non-algorithmic and oftentimes driven by physical intuition rather than some obviousmathematical correspondence, we believe – in view of the central rˆole played by the said intuition inthe development of the supersymmetric field theories of interest (starting with the pioneering work ofMetsaev and Tseytlin on superstrings in s ( AdS × S ) ) and their subsequent applications (in particular,in the celebrated but still formally inadequately understood AdS/CFT correspondence), and of the con-creteness of the ensuing supergeometric objects, implying their amenability to a hands-on verificationof such important properties as supersymmetry-equivariance and compatibility with a properly definedand understood κ -symmetry – that it constitutes a structurally most natural and tractable proposalfor a geometrisation of the physically motivated supersymmetry-invariant cohomology, developing in aclose conceptual analogy with the by now much-advanced theory of bundle gerbes behind the bosonic σ -model and related topological field theories. Accordingly, we shall pursue this latter line of thinkingin what follows, leaving the much interesting question regarding a precise relation between the twoapproaches to a future study.There are currently several working examples of the abstract structures recalled above, to wit,the Cartan–Eilenberg super- p -gerbes for p ∈ { , , } of Ref. [Sus17, Sec. 5] over the super-Minkowskispace, associated with the respective Green–Schwarz super- ( p + ) -cocycles H ( p + ) GS (such that χ ( p + ) GS = π ∗ sISO ( d, ∣ ND d, )/ SO ( d, ) H ( p + ) GS , for N ∈ { , } as in Examples 3.11 and 3.12), and the super-0-gerbe ofRef. [Sus18b, Sec. 5] over s ( AdS × S ) , associated with the Zhou super-2-cocycle χ ( ) Zh . There is also thetrivial Cartan–Eilenberg super-1-gerbe of Ref. [Sus18a, Sec. 6], associated with the Metsaev–Tseytlinsuper-3-cocycle χ ( ) MT , but the latter super-1-gerbe was demonstrated not to asymptote to the Green–Schwarz super-1-gerbe over sMink ( , ∣ ) , and so it was argued that the super-3-cocycle itself mayhave to be corrected for the sake of promoting the principle of ˙In¨on¨u–Wigner correspondence betweenthe curved and flat super- σ -models to the rank of a higher-geometric correspondence between therespective super-1-gerbes (a hint as to where to look for a suitable correction was given in Ref. [Sus18a,Sec. 9]). It deserves to be noted at this point that the super-Minkowskian super- p -gerbes referred toabove were proven, in Ref. [Sus19, Sec. 4], to carry a supersymmetric Ad-equivariant structure in perfectconformity with the Graßmann-even intuition.While the global supersymmetry of the super- σ -model, realised by the action of the Lie super-group G and preserving separately each of the two factors in the DF amplitude: the metric one andthe topological one, is accommodated directly in the construction of the super- p -gerbe, any attemptat ‘gerbification’ – in the form of a suitable equivariant structure – of the peculiar local supersym-metry modelled on the supervector space gs ( sB ( HP ) p,λ p ) of Eq. (5.9) in the generic situation or by the κ -symmetry superalgebra gs vac ( sB ( HP ) p,λ p ) in the regular case, and mixing the two factors nontrivially,requires that the higher-geometric discussion be transcribed from the NG formulation over M H tothe HP formulation over the HP section Σ HP of Eq. (3.11) resp. over the HP vacuum foliation Σ HPvac of Def. 4.6. That such a ‘gerbification’ is altogether possible is a consequence of the purely topologicalnature of the DF amplitude in the HP formulation. Cohomological triviality (in the supersymmetricand H vac -basic refinement of the de Rham cohomology) of the new tensorial component d β ( p + ) ( HP ) ofthe superbackground that replaces the supersymmetric and H-basic metric actually renders the liftquite straightforward. The stage for a concretisation of these ideas is set, after Ref. [Sus19, Sec. 6.2], inthe following Definition 6.11.
Let G ( p ) be the super- p -gerbe, as described in Def. 6.3, over M H ≡ G / H with thecurvature given by the descendant H ( p + ) ∈ Z p + ( M H ) G , from the total space G to the base M H ofthe principal H-bundle π G / H ∶ G Ð→ G / H of Eq. (2.4), of the H vac -basic Green–Schwarz super- ( p + ) -cocycle χ ( p + ) defining a class [ χ ( p + ) ] CaE , H vac − basic ∈ CaE p + ( G ) H vac , and let β ( p + ) ( HP ) ∈ Ω p + ( G ) G be theHughes–Polchinski super- ( p + ) -form of Eq. (3.3) that enters definition (3.5) of the Hughes–Polchinski E.g. , the asymptotic correspondence between the curved and flat superbackgrounds, as well as the canonical analysisof the field-theoretic realisation of the supersymmetry algebra, cp Ref. [Sus18a, Sec. 3]. uper- p -brane background and determines, in the usual manner, the trivial Cartan–Eilenberg super- p -gerbe over G, to be denoted as I ( p ) β ( p + ) ( HP ) . The extended Hughes–Polchinski p -gerbe over G isthe product bundle p -gerbe ̂G ( p ) ∶ = π ∗ G / H G ( p ) ⊗ I ( p ) λ p β ( p + ) ( HP ) . Its vacuum restriction is the p -gerbe ι ∗ vac ̂G ( p ) ≡ ̂G ( p ) ↾ Σ HPvac obtained by pullback along the embedding (4.14) of the Hughes–Polchinski vacuum foliation Σ
HPvac ofDef. 4.6 in the Hughes–Polchinski section Σ HP of Eq. (3.11). ◇ Remark 6.12.
As has already been mentioned in passing, the appearance of extensions of supersym-metry algebras, underlying the geometrisation scheme advanced herein, is entirely natural from thepoint of view of the field-theoretic realisation of supersymmetry in the GS super- σ -model. In the HPformulation, this can be seen as follows. The ability to explicitly model the infinitesimal action of thesupersymmetry group on the HP section Σ HP with the help of the vector fields K A paves the way tothe canonical analysis of supersymmetry of the GS super- σ -model (to be understood in the very samefunctorial/ S -point spirit as the DF amplitudes itself). Indeed, upon contracting (the trivial lifts, to themapping supermanifold [ C p , M H vac ] , of) the fundamental vector fields K X ≡ X A K A , X = X A t A ∈ g with the presymplectic form (note that as a result of the topological character of the HP formulationthe presymplectic form depends only on the configuration ̂ ξ C p ≡ ̂ ξ ↾ C p )Ω ( HP ) σ [̂ ξ C p ] = ∫ C p ̂ ξ ∗ C p ( λ p δ B ( p + ) ( HP ) + H ( p + ) vac ) , we readily derive the Noether supersymmetry charges ( ̂ ξ τ ∩ C p ≡ ̂ ξ ↾ τ ∩ C p ) h X [̂ ξ C p ] = ∑ τ ∈ T p + ∫ τ ∩ C p ( σ vac ı τ ○ ̂ ξ τ ∩ C p ) ∗ ( λ p R X ⌟ β ( p + ) ( HP ) + κ ( p ) X ) , X ∈ g , expressing them in terms of the p -forms κ ( p ) X ∈ Ω p ( G ) defined as K X ⌟ χ ( p + ) ≡ R X ⌟ χ ( p + ) = ∶ − d κ ( p ) X . Their existence follows from the assumed quasi-supersymmetry of β ( p + ) , or – in other words – fromglobal supersymmetry of G ( p ) . The Poisson bracket of the charges (associated with the vectors X , X ∈ g ), as determined by the presymplectic form Ω ( HP ) σ , reads { h X , h X } Ω ( HP ) σ [̂ ξ C p ] = h [ X ,X } [̂ ξ C p ] − ∑ τ ∈ T p + ∫ τ ∩ C p ( σ vac ı τ ○ ̂ ξ τ ∩ C p ) ∗ α X ,X ( p ) , with the integrand of the (classical) wrapping anomaly given by α X ,X ( p ) = − L R X κ ( p ) X + κ ( p ) [ X ,X } . The reason why this field-theoretic departure from the underlying supersymmetry Lie superalgebra wasdubbed thus in Ref. [Sus18a, Sec. 3] (strictly speaking, the effect was investigated in the NG formulation)is that it is determined by a de Rham p -cocycle, d α X ,X ( p ) = − − L R X ( R X ⌟ χ ( p + ) ) + d κ ( p ) [ X ,X } = R [ X ,X } ⌟ χ ( p + ) + d κ ( p ) [ X ,X } = , and so it is non-zero solely if the embedding ̂ ξ has a nontrivial monodromy around the Cauchyhypersurface C p , or the embedded worldvolume contains – in the functorial picture – a non-contractiblecycle. In Ref. [Sus18a, Sec. 4 & 5], monodromies around (compactified) Graßmann-odd cycles werepostulated to encode the supercentral extensions of the supertarget G (resp. M H whenever the latteris a Lie supergroup) engendered by the GS super- ( p + ) -cocycle χ ( p + ) that co-determines the super- σ -model. he assumption of existence of a global (super)symmetry in a field theory with some further gauged (super)symmetry and of its lift to the geometrisation of the theory’s topological content impose non-trivial constraints upon the equivariant structure on that geometrisation that incarnates the local(super)symmetry. Indeed, it is natural to demand that the equivariant structure be compatible withthe global (super)symmetry in an obvious manner that we recall after Ref. [Sus19, Sec. 4], where itwas described concisely by The Invariance Postulate and quantified – in the super symmetric setting inhand – through Definition 6.13.
Let M be a (super)manifold with an abelian p -gerbe G ( p ) over it, as describedbefore, and let G and G loc be two Lie (super)groups, with the respective binary operations µ and µ loc and the respective tangent Lie (super)algebras g and g loc , acting on M as λ ∶ G × M Ð→ M , λ loc ∶ G loc × M Ð→ M , the former to be thought of as the global (super)symmetry group and the latter as the local (su-per)symmetry group. Consider the action (super)groupoid G loc ⋉ M ∶ G loc × M λ loc / / pr / / M and form the nerve N ● ( G loc ⋉ M) thereof, ⋯ d ( ) ● / / / / / / / / G × × M d ( ) ● / / / / / / G loc × M d ( ) ● / / / / M , (6.7)with face maps d ( m ) l ∶ G × m loc × M Ð→ G × m − × M , l ∈ , m , m ∈ N × given by the formulæ d ( m ) = pr , ,...,m,m + , d ( m ) m = id G × m − × λ loc , ,d ( m ) i = id G × i − × µ loc × id G × m − − i loc × M , i ∈ , m − . Assume the existence of a G loc -equivariant structure on G ( p ) , understood as a collection of p -gerbe k -isomorphisms Υ ( k ) p , k ∈ , p + N k ( G loc ⋉ M) ≡ G × k loc × M of the nerve,with Υ ( p + ) p subject to a coherence condition over N p + ( G loc ⋉ M) ≡ G × p + × M , that altogether form anatural generalisation of the specific coherent (simplicial) objects explicited in Ref. [Sus19, Sec. 2] for p ∈ { , } ( cp Sec. 7 for an explicit sheaf-cohomological description of its linearisation) We say thatthe equivariant structure is a G -invariantly G loc -equivariant structure on G ( p ) if there exists anaction λ ∶ G × N ( G loc ⋉ M) Ð→ N ( G loc ⋉ M) of G that lifts λ to N ( G loc ⋉ M) ≡ G loc × M (and so canonically induces lifts λ k to the remainingcomponents N k ( G loc ⋉ M) , k > N ● ( G loc ⋉ M) ) in a manner compatible with λ , as expressed by thecommutative diagramG × M λ (cid:15) (cid:15) G × ( G loc × M) λ (cid:15) (cid:15) id G × pr / / id G × λ loc o o G × M λ (cid:15) (cid:15) M G loc × M pr / / λ loc o o M (6.8)and such that each component Υ ( k ) p is G-invariant with respect to the corresponding action λ k in thesense of Def. 6.3. ◇ Cp Ref. [Sus12, Def. 8.6] for a definition of the notion and a comprehensive discussion of its naturalness in andrelevance to the description of gauged symmetries of a field theory. daptation of the above structure to the situation of immediate interest, that is to the descriptionof the enhanced gauge symmetry resp. the κ -symmetry of the globally supersymmetric super- σ -modelin the HP formulation, requires a careful reworking of the original concepts, taking into account thepeculiar nature of the local symmetry, the localisation of the vacuum itself within Σ HP , as well as theimplicit requirement of descent of the higher-geometric structure to the homogeneous space M H vac .Therefore, rather than unpacking the last definition in all generality, we leave it as it stands and,instead, pass directly to the study of a higher-geometric extension of the local supersymmetry of thesuper- σ -model that we are, at long last, ready to address.7. Supersymmetry-invariant gerbification of linearised κ -symmetry In this closing section of the present paper, we seek to investigate a gerbe-theoretic realisation ofthe enhanced local supersymmetry of the GS super- σ -model in the topological HP formulation deter-mined by the extended HP p -gerbe. The supersymmetry is invariably represented by a distinguishedsuperdistribution in the tangent sheaf of the HP section Σ HP – be it the generic enhanced gauge-symmetry superdistribution GS( sB ( HP ) p,λ p ) for the correspondence sector of the field theory, or thelimit κ −∞ ( sB ( HP ) p,λ p ) of the weak derived flag of the κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) (resp.its extended variant in Example 5.20) for its vacuum in the regular case of an integrable HP vacuumsuperdistribution Vac ( sB ( HP ) p,λ p ) . Based on the intuition developed in the Graßmann-even setting andrecalled in the previous section, we anticipate the emergence of an equivariant structure of sorts withrespect to the local-symmetry superdistribution on the extended HP p -gerbe. Its precise identificationcalls for a reformulation of Def. 6.13 of a supersymmetric equivariant structure that accomodates thefollowing three facts established previously: ● The local supersymmetry is realised linearly on the HP/NG correspondence sector of the super- σ -model and in the case of the κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) , it restricts further tothe HP vacuum foliation that it envelops, the foliation being engendered by the involutive HPvacuum superdistribution that κ ( sB ( HP ) p,λ p ) bracket-generates. In either situation, integratingany component of the tangential structure to a gauge-symmetry Lie supergroup – even ifthe corresponding model Lie superalgebra should allow it – would a priori be meaningless asthe supergroup could not be realised on Σ HP or Σ HPvac in the standard manner by the sameargument as the one invoked in Sec. 5.1 for the action of the global-supersymmetry group G. ● By the said argument, the global-supersymmetry group G is not , in general, realised on Σ HP orΣ HPvac globally, but does admit a linearised realisation by the global-supersymmetry subspace S HPG of Prop. 5.1, with generators labelled by the Lie superalgebra g , resp. by its vacuum-preserving subspace – the residual global-supersymmetry subspace S HP , vacG of Prop. 5.5, withgenerators labelled by the Lie sub-superalgebra s vac . ● Global structures that arise over the HP section Σ HP , such as, e.g. , the extended HP p -gerbe and any equivariant structure on it, implicitly model corresponding structures on thehomogeneous space M H vac .Putting all these facts together, we conclude that the structure that we are after is a linearisation of astandard (super)group-equivariant structure on the extended HP p -gerbe, compatible with linearisedglobal supersymmetry up to linearised hidden gauge transformations from h vac (a linearisation of thevacuum isotropy group H vac ) – all that in a manifestly H vac -descendable form. Generically, the lineari-sation is not expected to extend beyond the level of the relevant super- p -gerbe 1-isomorphism Υ ( ) p ofEq. (6.5) (or, more accurately, a linearised version thereof), representing the bare local-supersymmetrysuperdistribution and further assumed globally linearised(-G-)supersymmetric, but in the physicallyfavoured regular situation in which the HP vacuum superdistribution is integrable, we should climbwith the vacuum restriction ι ∗ Σ HPvac ̂G ( p ) of the extended HP p -gerbe of Def. 6.11 one level up in thehierarchy defining the standard (super)group-equivariant structure ( i.e. , demand either coherence ofΥ ( ) , or existence of Υ ( ) p for p >
0) and subsequently perform linearisation and check compatibilitywith linearised global supersymmetry up to vacuum gauge symmetry for this extended structure. Aswe approach the task of constructing such a linearised equivariant structure in the intrinsically geo-metric formalism of gerbe theory, an important difference between the two superdistributions comes Of course, we might and – indeed – ought to try to descend the integrated symmetry to the homogeneous space forwhich, however, we should have to devise new tools. We hope to return to this idea in the future. o the fore that effectively rules out the generic structure GS( sB ( HP ) p,λ p ) as a subject of gerbification.Indeed, while the κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) gives rise to an involutive supersymmetrysuperdistribution κ −∞ ( sB ( HP ) p,λ p ) tangent to a family of sub -supermanifolds embedded in Σ HP , and socan be meaningfully restricted to any one of these sub-supermanifolds ( i.e. , the vacuum), as can bethe p -gerbe, the enhanced gauge-symmetry superdistribution GS( sB ( HP ) p,λ p ) acquires its status uponrestriction of field configurations to the non-integrable correspondence superdistribution, which it thengenerically leaves via its weak derived flag that itself does not asymptote to a larger gauge-symmetrysuperdistribution of the underlying field theory. There is – on one hand – no obvious geometric mech-anism of imposing the field-theoretic restriction to the correspondence sector upon the extended HP p -gerbe, and – on the other hand – there is no reason to expect equivariance of the latter without thatrestriction or with respect to the integrable limit of the weak derived flag of GS( sB ( HP ) p,λ p ) on the (sub-)supermanifold of Σ HP that it envelops. Thus, it appears, we are bound to study equivariance of thevacuum restriction ι ∗ Σ HPvac ̂G ( p ) of the extended HP p -gerbe with respect to the vacuum-generating gaugedsupersymmetry realised by κ −∞ ( sB ( HP ) p,λ p ) ≡ Vac ( sB ( HP ) p,λ p ) and modelled on the κ -symmetry superalge-bra gs vac ( sB ( HP ) p,λ p ) ≡ vac ( sB ( HP ) p,λ p ) . In the present section, we treat the lowest layer of such a structurethat, as argued above, accounts for the linear (or supervector-space) structure on gs vac ( sB ( HP ) p,λ p ) , rele-gating the issue of a higher-geometric implementation of the superalgebra structure to a future study.We begin by deriving a linearised version of the constraint of compatibility up to gauge symmetryimposed upon λ through Diag. 6.8. We do that in local coordinates ( i.e. , in the S -point picture) onthe vacuum foliation Σ HPvac = ⊔ i ∈ I Hvac ⊔ υ i ∈ Υ i V vac i,υ i which we write as the disjoint union of integral sub-supermanifolds (or vacua) V vac i,υ i ⊂ V i of the HP vac-uum superdistribution Vac ( sB ( HP ) p,λ p ) (locally labelled by sets Υ i ), the latter being assumed integrablebelow. For the coordinates, we use the shorthand notation ξ i ≡ σ vac i ( χ i ) . In it, the aforementioned realisation, over V vac i,υ i of the residual global-supersymmetry algebra, with thehomogeneous generators ( ∣ S ˘ A ∣ ≡ ∣ ˘ A ∣ ) s vac ≡ S vac ⊕ ˘ A = ⟨ S ˘ A ⟩ ⊂ g , S vac ≡ dim S HP , vacG , and of the local-supersymmetry supervector space (for which we introduce a new symbol to unclutterthe notation) s loc ≡ gs vac ( sB ( HP ) p,λ p ) ≡ p + q + D − δ ⊕ ̃ A = ⟨ V ̃ A ⟩ ⊂ g , with V ̃ A = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ P ̃ A if ̃ A ∈ , pQ ̃ A − p if ̃ A ∈ p + , p + qJ ̃ A − p − q if ̃ A ∈ p + q + , p + q + D − δ , reads s vac × V vac i,υ i ∋ ( X ≡ X ˘ A S ˘ A , ξ i ) z→ X ˘ A K S ˘ A i ( ξ i ) ∈ T V vac i,υ i and s loc × V vac i,υ i ∋ ( Γ ≡ Γ ̃ A V ̃ A , ξ i ) z→ Γ ̃ A T V ̃ A i ( ξ i ) ≡ Γ ̃ A T ̃ A i ( ξ i ) ∈ T V vac i,υ i , We might, in principle, try to impose the restriction in the sheaf-theoretic language but the lack of a (sub-)supermanifold structure behind the restriction would inevitably render such a construction non-canonical. espectively, the latter being trivial in the h vac -sector,Γ ̃ A T ̃ A i ≡ Γ A T A i . Note that the X ˘ A and the Γ ̃ A appear here in the rˆole of global coordinates on the superspaces s vac and s loc , respectively, and so they carry Graßmann parity according to the rule ∣ X ˘ A ∣ = ∣ ˘ A ∣ , ∣ Γ ̃ A ∣ = ∣ ̃ A ∣ . The task in hand now boils down to finding a collection of (smooth) sectionsΛ ̃ C ˘ A ̃ B i ∈ O G ( V vac i,υ i ) , ( ˘ A, ̃ B, ̃ C ) ∈ , S vac × , p + q + D − δ × , υ i ∈ Υ i , i ∈ I H vac (7.1)satifying the identities which by a mild abuse of the notation may be written, up to terms quadraticin X or Γ, as ξ i + X ˘ A K S ˘ A i ( ξ i ) ⌟ d ξ i + Γ ̃ A ( δ ̃ B ̃ A + X C Λ ̃ BC ̃ A i ( ξ i )) T ̃ B i ( ξ i + X ˘ D K S ˘ D i ( ξ i ) ⌟ d ξ i ) ⌟ d ξ i + O ( X , Γ ) = ξ i + Γ ̃ A T ̃ A i ( ξ i ) ⌟ d ξ i + X ˘ C K S ˘ C i ( ξ i + Γ ̃ A T ̃ A i ( ξ i ) ⌟ d ξ i ) ⌟ d ξ i + Γ ̃ A X ˘ C Λ p + q + S ˘ C ̃ A i ( ξ i ) L S ( ξ i ) . Upon expanding the above (dropping terms of order O ( X , Γ ) ) and removing the (arbitrary) coeffi-cients X ˘ C and Γ ̃ A , we arrive at the desired compact equation [ K S ˘ A i , T ̃ B i }( ξ i ) = − Λ C ˘ A ̃ B i ( ξ i ) T C i ( ξ i ) − Λ p + q + S ˘ A ̃ B i ( ξ i ) L S ( ξ i ) , (7.2)to be solved for the Λ ̃ C ˘ A ̃ B i . The existence and uniqueness of the solution follows from
Proposition 7.1.
Adopt the hitherto notation, and in particular that of Props. 3.6 and 5.1. Thereexists a canonical realisation of the residual global-supersymmetry algebra s vac on s loc × Σ HPvac , s vac × ( s loc × Σ HPvac ) ⊃ g × ( s loc × V vac i,υ i ) Ð→ T ( s loc × V vac i,υ i ) ⊂ T ( s loc × Σ HPvac )∶ ( X ≡ X ˘ A S ˘ A , ( Γ ≡ Γ ̃ A t ̃ A , ξ i )) z→ X ˘ A K S ˘ A i ( ξ i ) + Γ ̃ B X ˘ C Λ ̃ A ˘ C ̃ B i ( ξ i ) ⃗ ∂∂ Γ ̃ A , compatible with the realisation of the local-supersymmetry supervector space s loc on V vac i,υ i by the limit κ −∞ ( sB ( HP ) p,λ p ) of the weak derived flag of the κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) and that of s vac by the residual global-supersymmetry subspace S HP , vacG with a trivial linearised correcting ( h vac -)gaugetransformation. It is determined by the sections Λ ̃ CA ̃ B i ∈ O G ( V vac i,υ i ) of Eq. (7.1) of the form Λ C ˘ A ̃ B i = −[ K S ˘ A i , T ̃ B i } ⌟ θ C L ≡ − ˘Ξ S ˘ A i f CS ̃ B , Λ p + q + SA ̃ B i ≡ , in which S ∈ , D − δ, C ∈ , p + q and the sections ˘Ξ SA i ∈ O G ( V vac i,υ i ) are uniquely defined – in the spiritof Prop. 5.1 – by the formula K S ˘ A i = R S ˘ A ↾ V vac i,υi + ˘Ξ S ˘ A i L S . Proof.
Follows straightforwardly from an adaptation of the proof of Prop. 5.4 to the integrable sub-superdistribution Vac ( sB ( HP ) p,λ p ) ⊂ T Σ HP , and from the identity T C i ⌟ θ A L ( ξ i ) = δ AC + T SC i δ AS , cp Eq. (3.12), taken together with the calculation [ K S ˘ A i , T ̃ B i } = ((− ) ∣ ˘ A ∣ ⋅ ∣ ̃ B ∣ + L ̃ B ⌟ d ˘Ξ S ˘ A i + ˘Ξ T ˘ A i L T ⌟ d T S ̃ B i + (− ) ∣ ˘ A ∣ ⋅ ∣ ̃ B ∣ + T T ̃ B i L T ⌟ d ˘Ξ S ˘ A i + R S ˘ A ⌟ d T S ̃ B i + ˘Ξ U ˘ A i T V ̃ B i f SUV ) L S + ˘Ξ S ˘ A i f ̃ CS ̃ B L ̃ C in which relation (5.4) has been used. (cid:3) Prior to lifting the linearised realisation of the residual supersymmetry on s loc × Σ HPvac to the extendedHP p -gerbe, we pause to reformulate the fundamental identity (7.2) in a manner reflecting its ac-tual meaning (as a condition of compatibility of the two actions), and with view to its subsequentapplications. Thus, we have roposition 7.2. Adopt the notation of Prop. 7.1 and consider the fundamental vector fields for thelinearised action of G on s loc × Σ HPvac , with the local coordinate presentation ˘ K ˘ A ( Γ , ξ i ) ∶ = K S ˘ A ( ξ i ) + (− ) ∣ ˘ A ∣ ⋅ ∣ ̃ B ∣ Γ ̃ B Λ ̃ C ˘ A ̃ B i ( ξ i ) ⃗ ∂∂ Γ ̃ C , spanning a subspace in the tangent sheaf T ( s loc × Σ HPvac ) to be denoted as ̂ S HP , vacG ⊂ Γ ( T ( s loc × Σ HPvac )) , alongside those generating the limit κ −∞ ( sB ( HP ) p,λ p ) of the weak derived flag of the κ -symmetry superdis-tribution κ ( sB ( HP ) p,λ p ) , now regarded as s loc -linear vector fields on s loc × Σ HPvac as per ̂ T ( Γ , ξ i ) ∶ = Γ ̃ A T ̃ A ( ξ i ) . (7.3) These satisfy the commutation relations [ ˘ K ˘ A , ̂ T ]( Γ , ξ i ) = + O ( Γ ) , expressing compatibility of the two actions (in the linear order).Proof. Follows from Prop. 7.1. (cid:3)
In the remainder of the present paper, we establish a higher-geometric lift of the structure identifiedabove. To this end, we first derive, from a detailed analysis of the restriction of a standard group-equivariant structure (in the Graßmann-even setting) to an infinitesimal vicinity of the group unit,the appropriate notion of a s loc -equivariant structure, to be understood as a consistent realisation ofthe local-symmetry superdistribution κ −∞ ( sB ( HP ) p,λ p ) on the vacuum restriction of the extended HP p -gerbe. The underlying supergeometric setting will be the one delineated in Def. 6.13. Given theinfinitesimal and hence local nature of the realisation sought after, we are free to employ the sheaf-theoretic description of p -gerbes and the associated k -isomorphisms, which will prove particularlyconvenient.Thus, consider a p -gerbe G ( p ) of curvaturecurv (G ( p ) ) ≡ χ ( p + ) represented by a ( p + ) -cocycle B ( p + ) ∈ Ker D ( p + ) ⊂ ˇCD p + (O M , D( p + ) ● ) , D ( p + ) B ( p + ) = . and assume existence of a descendable G loc -equivariant structure on G ( p ) , i.e. , one whose 1-isomorphismΥ ( ) p ∶ λ ∗ loc G ( p ) ≅ ÐÐ→ pr ∗ G ( p ) has ̺ − θ L ( p + ) ≡ . A necessary condition for that to be the case is the horizontality of the curvature with respect to thefundamental vector fields K λ loc ⋅ , K λ loc Γ ⌟ χ ( p + ) = , Γ ∈ g loc . (7.4)Note that the fundamental vector fields induce g loc -linear vector fields on g loc × M as per ̂K λ loc ∶ g loc × M ∋ ( Γ , m ) z→ K λ loc Γ ( m ) ∈ T m M . As we intend to investigate a linearisation of the above structure, we restrict our considerations togroup elements of the form e Γ from an infinitesimal (contractible) neighbourhood O e of the groupunit, e ∈ G loc , for which the G loc -equivariant structure can be presented (locally) as a collection of ( p + − l ) -cochains P ( p + − l ) ( l ) ∈ ˇCD p + − l ({O e } × l × O M , D( p + ) ● ) , l ∈ , p + λ ∗ loc B ( p + ) − pr ∗ B ( p + ) + D ( p ) P ( p ) ( ) ≡ d ( ) ∗ B ( p + ) − d ( ) ∗ B ( p + ) + D ( p ) P ( p ) ( ) = , k + ∑ r = ( − ) k + − r d ( k + ) ∗ r P ( p + − k ) ( k ) + D ( p − k ) P ( p − k ) ( k + ) = , k ∈ , p + ∑ s = ( − ) p + − s d ( p + ) ∗ s P ( ) ( p + ) = , to be evaluated on ( e Γ , m ) , ( e Γ k + , e Γ k , . . . , e Γ , m ) and ( e Γ p + , e Γ p + , . . . , e Γ , m ) , respectively. In ther´egime of ‘small’ Γ l , in which we may – by a mild abuse of the notation – write e Γ l = e + Γ l + O ( Γ l ) , weexpand the various cochains appearing in the above equations in the powers of the Γ l and keep onlyterms at most linear in (any of) the Γ l , in particular ( cp Eq. (7.3)), ( λ ∗ loc B ( p + ) − pr ∗ B ( p + ) )( e Γ , m ) = − L ̂ K λ loc pr ∗ B ( p + ) ( Γ , m ) + O ( Γ ) (and similarly for the pullbacks of the P ( p + − l ) ( l ) along the respective d ( l + ) l + ) and P ( p + − l ) ( l ) ( e Γ l , e Γ l − , . . . , e Γ , m ) = ∶ P ( p + − l ) ( l ) , ( m ) + ∑ lr = P ( p + − l ) ( l ) , ( r ) ( Γ r , m ) + O ( Γ r Γ s ) , P ( p + − l ) ( l ) , ( r ) ( Γ r , m ) ≡ Γ ̃ Ar P ( p + − l ) ( l ) , ( r )̃ A ( m ) . When substituted to the equations, these yield their linearisation (written in the natural coordinateson the s × l loc × O M introduced before): − L ̂ K λ loc pr ∗ B ( p + ) ( Γ , m ) + D ( p ) P ( p ) ( ) , ( m ) + D ( p ) P ( p ) ( ) , ( ) ( Γ , m ) = , P ( p ) ( ) , ( m ) + − L ̂ K λ loc pr ∗ P ( p ) ( ) , ( Γ , m ) + D ( p − ) P ( p − ) ( ) , ( m ) + ∑ r ∈ { , } D ( p − ) P ( p − ) ( ) , ( r ) ( Γ r , m ) = , P ( p − ) ( ) , ( ) ( Γ , m ) − P ( p − ) ( ) , ( ) ( Γ , m ) − − L ̂ K λ loc P ( p − ) ( ) , ( Γ , m ) + D ( p − ) P ( p − ) ( ) , ( m ) + ∑ r ∈ { , , } D ( p − ) P ( p − ) ( ) , ( r ) ( Γ r , m ) = , ⋯ , p + ∑ s = ( − ) p + − s ( s − ∑ r = P ( ) ( p + ) , ( r ) ( Γ r , m ) + p + ∑ r = s + P ( ) ( p + ) , ( r ) ( Γ r + , m ) + P ( ) ( p + ) , ( s ) ( Γ s + + Γ s , m )) + p + ∑ r = P ( ) ( p + ) , ( r ) ( Γ r , m ) + ( − ) p + p + ∑ r = P ( ) ( p + ) , ( r ) ( Γ r + , m ) + + ( − ) p P ( ) ( p + ) , ( m ) + ( − ) p + − L ̂ K λ loc pr ∗ P ( ) ( p + ) , ( Γ , m ) = l (and so, in particular, for Γ l ≡ p -gerbe, − L ̂ K λ loc pr ∗ B ( p + ) ( Γ , m ) + D ( p ) P ( p ) ( ) , ( ) ( Γ , m ) = , and a system of coupled equations involving all the P ( p + − l ) ( l ) , and the P ( p + − l ) ( l ) , ( r ) with ( l, r ) ≠ ( , ) .The latter is always solvable , and so it is but an artifact of our sheaf-theoretic description. Thus, bythe end of the day, we are left with a single equation (written in terms of the p -cochain ̂ P ( p ) ≡ P ( p ) ( ) , ( ) ) − L ̂ K λ loc pr ∗ B ( p + ) + D ( p ) ̂ P ( p ) = . This equation has an obvious higher-geometric interpretation which leads us directly to the desired
Definition 7.3.
Let M be a manifold and G loc a Lie group (with Lie algebra g loc ) realised on it by(local) diffeomorphisms, as in Def. 6.13, that induce a local-symmetry distribution S g loc ⊂ T M Indeed, it admits the trivial (zero) solution. panned by the fundamental vector fields K λ loc ⋅ ∶ g loc Ð→ Γ ( T M) ∶ Γ z→ K Γ , and so giving rise to a g loc -linear vector field on g loc × M , ̂K λ loc ∶ g loc × M ∋ ( Γ , m ) z→ K Γ ( m ) ∈ T ( Γ ,m ) ( g loc × M) . (7.5)Let, next, G ( p ) be a p -gerbe over M presented by a ˇCech–Deligne ( p + ) -cocycle B ( p + ) ∈ Ker D ( p + ) ⊂ ˇCD p + (O M , D( p + ) ● ) associated with an open cover O M of M . A ( descendable ) g loc -equivariantstructure on G ( p ) is a p -gerbe 1-isomorphism ̂ Λ g loc ∶ − L ̂ K λ loc pr ∗ G ( p ) ≅ ÐÐ→ I ( p ) over g loc × M , written for the trivial p -gerbe I ( p ) over g loc × M with a vanishing global curving and a p -gerbe − L ̂ K λ loc pr ∗ G ( p ) over the same base with local (sheaf-cohomological) data over the open cover { g loc } × O M obtained from the pullback of the local data B ( p + ) of G ( p ) to it along pr by taking theirLie derivative along ̂K λ loc (component-wise) – we call this latter p -gerbe the Lie derivative of the p -gerbe pr ∗ G ( p ) along the vector field ̂K λ loc .We extend the above definition verbatim to the category of supermanifolds, whereby the notion ofa (descendable) g loc -equivariant structure on a p -gerbe over a supermanifold (and, in particular, on asuper- p -gerbe) for g loc a Lie superalgebra arises. ◇ Remark 7.4.
The only thing that requires a word of justification is the claim that − L ̂ K λ loc pr ∗ B ( p + ) is a ˇCech–Deligne ( p + ) -cocycle over { g loc } × O M whenever B ( p + ) is one over O M . This followsstraightforwardly from Prop. 6.1.Note also that it is merely the vector-space structure on g loc that is relevant to the above definition asa algebraic model of the local-symmetry distribution S g loc . This is to be kept in mind when generalisingthe definition to the supergeometric setting.We have Proposition 7.5.
Adopt the notation of Def. 7.3 and Prop. 6.1. There exists a canonical descendable g loc -equivariant structure on every p -gerbe G ( p ) with a S g loc -horizontal curvature. The local data ̂ P ( p ) ∈ ˇCD p ({ g loc } × O M , D( p + ) ● ) of the structure associated with the open cover O M of M are determinedby those of the p -gerbe, B ( p + ) , as ̂ P ( p ) = − ̂K λ loc ⌟ pr ∗ B ( p + ) . (7.6) Consequently, we may write the p -gerbe 1-isomorphism of the g loc -equivariant structure as ̂ Λ g loc ≡ − ̂K λ loc ⌟ pr ∗ G ( p ) . Proof.
For a compact proof, write the Deligne coboundary operator in an obvious shorthand notationas D ( k ) ≡ D ( k )( ● , ● ) ,D ( k )( ● , ● ) ≡ ( D ( k )( m,n ) ≡ d ( n ) + ( − ) n + ˇ δ ( m ) ) ( m,n ) ∈ N × ({ − } ∪ N ) m + n = k ≡ d ( ● )( k ) + ( − ) ● + ˇ δ ( ● )( k ) cp Eq. (6.1). In this notation, we readily compute − L ̂ K λ loc pr ∗ B ( p + ) ≡ ̂K λ loc ⌟ d ( ● )( p + ) pr ∗ B ( p + ) + d ( ● )( p ) ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) = ̂K λ loc ⌟ pr ∗ ( χ ( p + ) , D ( p + ) B ( p + ) ) − ̂K λ loc ⌟ pr ∗ ( − ) ● + ˇ δ ( ● )( p + ) B ( p + ) + d ( ● )( p ) ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) = d ( ● )( p ) ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) + ̂K λ loc ⌟ pr ∗ ( − ) ● ˇ δ ( ● )( p + ) B ( p + ) ≡ d ( ● )( p ) ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) + ( − ) ● + ˇ δ ( ● )( p ) ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) ≡ D ( p ) ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) , whence the thesis of the proposition follows. (cid:3) emark 7.6. Structures of the kind considered above, or – indeed – their full-fledged Lie-algebraicextensions induced, through linearisation, from group-equivariant structures on the bi-category of 1-gerbes, appeared for the first time in the context of the gauging of global σ -model symmetries inRef. [GSW10, Sec. 7.2]. We shall encounter them again when we take up the issue of (globally linearised-supersymmetric) equivariance with respect to a realisation of the κ -symmetry Lie superalgebra in aforthcoming paper [Sus20].As a corollary to the above, we may state our first physical higher-(super)geometric result: Proposition 7.7.
Adopt the hitherto notation, in particular that of Def. 6.11 and Prop. 7.2, and as-sume integrability of the vacuum superdistribution
Vac ( sB ( HP ) p,λ p ) of Def. 4.3. There exists a canonical gs vac ( sB ( HP ) p,λ p ) -equivariant structure, in the sense of Def. 7.3, on the vacuum restriction ι ∗ vac ̂G ( p ) of theextended Hughes–Polchinski p -gerbe ̂G ( p ) . It takes the form ̂ Λ gs vac ( sB ( HP ) p,λp ) = − ̂T ⌟ pr ∗ ι ∗ vac ̂G ( p ) (as introduced in Prop. 7.5), where ̂T is the vector field on gs vac ( sB ( HP ) p,λ p ) × Σ HPvac given by Eq. (7.3) .Proof.
Follows from an adaptation of Prop. 7.5. (cid:3)
Next, we investigate compatibility of the above (canonical) g loc -equivariant structure on G ( p ) with itsassumed linearised G-symmetry, to be defined next.As the first step towards this goal, we adapt the result of our linearisation procedure to the case ofa global symmetry, as recapitulated on p. 71. A moment’s thought about the nature of the differencebetween the gerbe-theoretic manifestations of a global symmetry and a local one convinces us of theadequacy of the following Definition 7.8.
Let G be a Lie group with a Lie algebra g , and let M be a manifold on whichG is realised by (local) diffeomorphisms, as in Def. 6.13, that induce a global-symmetry subspace S G ⊂ Γ (T M) of Def. 5.2, spanned by the fundamental vector fields K λ ⋅ ∶ g Ð→ Γ ( T M) ∶ X z→ K λX . A p -gerbe G ( p ) over M is termed a g -invariant p -gerbe if there exists a family of p -gerbe 1-isomorphisms { ̃ Λ X ∶ − L K λX G ( p ) ≅ ÐÐ→ I ( p ) } X ∈ g over M , written for the trivial p -gerbe I ( p ) over M with a vanishing global curving and for the Liederivative − L K λX G ( p ) of G ( p ) along K λX , as described in Def. 7.3.Given two p -gerbes G ( p ) A , A ∈ { , } over a common base M that are g -symmetric, with therespective families of 1-isomorphisms { ̃ Λ AX ∶ − L K λX G ( p ) A ≅ ÐÐ→ I ( p ) } X ∈ g , and a p -gerbe 1-isomorphism Φ ∶ G ( p ) ≅ ÐÐ→ G ( p ) between them, we call the latter a g -invariant p -gerbe 1-isomorphism if there exists a family of p -gerbe 2-isomorphism − L K λX G ( p ) − L K λX Φ / / ̃ Λ X (cid:15) (cid:15) − L K λX G ( p ) ̃ Λ X (cid:15) (cid:15) ψ X tttttttttttttttttttt u } tttttttttttttttttttt I ( p ) I ( p ) I ( p ) , X ∈ g , written for the corresponding p -gerbe 1-isomorphisms − L K λX Φ with local (sheaf-cohomological) dataover an open cover O M of M common to all three: G ( p ) , G ( p ) and Φ obtained from the corresponding ocal data of Φ by taking their Lie derivative along K λX – we call the p -gerbe 1-isomorphism thus formedthe Lie derivative of the p -gerbe 1-isomorphim Φ along the vector field K λX .We extend the above definition verbatim to the category of supermanifolds, whereby the notions ofa g -invariant structure on a p -gerbe over a supermanifold (and, in particular, on a super- p -gerbe), aswell as that of a g -invariant 1-isomorphism between such p -gerbes arise for a Lie superalgebra g . ◇ Remark 7.9.
As in the case of the Lie derivative of a p -gerbe, the existence of the Lie derivative − L K λX Φ for a given p -gerbe 1-isomorphism Φ is ensured by the commutativity of the Lie derivativewith the Deligne coboundary operator ( cp Prop. 6.1).Furthermore, in analogy with Def. 7.3, it is only the vector-space structure on g that enters thedefinition as an algebraic model for the global-symmetry subspace S G .Thus, by the assumed global supersymmetry of the GS super- p -gerbe G ( p ) of Def. 6.3 and in view ofthe manifestly supersymmetric definition (3.3) of the HP super- ( p + ) -form, we obtain a g -indexedfamily of p -gerbe 1-isomorphismsΛ X ∶ − L K X ̂G ( p ) ≅ ÐÐ→ I ( p ) , X ∈ g defined for the extended HP p -gerbe G ( p ) of Def. 6.11. Those associated with generators of the residualglobal-supersymmetry subalgebra s vac ⊂ g of Prop. 5.5 are readily seen to descend to its vacuumrestriction (being induced by vector fields tangent to the vacuum), and so we obtain a family of p -gerbe 1-isomorphisms ˘Λ ˘ A ∶ − L K S ˘ A ι ∗ vac ̂G ( p ) ≅ ÐÐ→ I ( p ) , ˘ A ∈ , S vac . These will prove instrumental in understanding s vac -invariance of the previously established canoni-cal gs vac ( sB ( HP ) p,λ p ) -equivariant structure on ι ∗ vac ̂G ( p ) as they form the basis for the application of thefollowing Proposition 7.10.
Let M , G and G loc all be as in Def. 6.13, and let G ( p ) be a p -gerbe over M thatwe assume g -invariant in the sense of Def. 7.8, with the basis p -gerbe 1-isomorphisms ̃ Λ A ∶ − L K λA G ( p ) ≅ ÐÐ→ I ( p ) , A ∈ , dim g . Fix a vector field ̂V ∈ Γ (T ( g loc × M)) . If there exists a lift of the global-symmetry subspace S G to a g loc -linear subspace ̂ S G ⊂ Γ (T ( g loc × M)) , spanned on the lifts ˘ K λA , A ∈ , dim g of the respective vectorfields K λA and defined as in Prop. 7.2, with the property ∀ A ∈ , dim g ∶ [ ˘ K λA , ̂V] = , then the Lie derivative − L ̂ V pr ∗ G ( ) of the pullback of G ( p ) to g loc × M along the canonical projection,understood as in Def. 7.3, is canonically g -invariant, with the corresponding p -gerbe 1-isomorphisms ˘Λ ̂ V A ∶ − L ˘ K λA ( − L ̂ V pr ∗ G ( p ) ) ≅ ÐÐ→ I ( p ) given by the formula ˘Λ ̂ V A = − L ̂ V pr ∗ ̃ Λ A , to be understood in the spirit of Def. 7.3 and Prop. 6.1.Proof. Adopt the previously introduced notation. Let B ( p + ) ∈ Ker D ( p + ) be a ˇCech–Deligne ( p + ) -cocycle presenting G ( p ) for an open cover O M of M over which – for every value A ∈ , dim g –there exists, by assumption, a p -cochain P ( p ) A ∈ ˇCD p (O M , D( p + ) ● ) that represents the g -invariantstructure, − L K λA B ( p + ) = − D ( p ) P ( p ) A . We might have to vary the choice of the open cover as A ranges over 1 , dim g , which – however – would not affectthe validity of our sheaf-cohomological argument, and we might ultimately take their common refinement. nvoking the definitions (and the properties that follow therefrom) of the Lie derivatives (of p -gerbes)involved and using the defining relation between K λA and (its lift ) ˘ K λA , as well as the explicit form ofthe induced cover { g loc } × O M of g loc × M , we readily calculate − L ˘ K λA − L ̂ V pr ∗ B ( p + ) = − L [ ˘ K λA , ̂ V ] pr ∗ B ( p + ) + − L ̂ V − L ˘ K λA pr ∗ B ( p + ) ≡ − L ̂ V pr ∗ − L K λA B ( p + ) = − − L ̂ V pr ∗ D ( p ) P ( p ) A = − D ( p ) − L ̂ V pr ∗ P ( p ) A , and thus identify the p -cochain − L ̂ V pr ∗ P ( p ) A ∈ ˇCD p ({ g loc } × O M , D( p + ) ● ) as a local presentation of the component, associated with the generator τ A ∈ g , of the (canonical) g -invariant structure on − L ̂ V pr ∗ G ( ) predicted by the proposition. (cid:3) Again, we obtain a physically relevant corollary
Proposition 7.11.
Adopt the hitherto notation, in particular that of Def. 6.11 and Props. 5.5 and 7.2,and assume integrability of the vacuum superdistribution
Vac ( sB ( HP ) p,λ p ) of Def. 4.3. The Lie derivative − L ̂ T pr ∗ ι ∗ vac ̂G ( p ) of the pullback of the vacuum restriction of the extended Hughes–Polchinski p -gerbe tothe supermanifold gs vac ( sB ( HP ) p,λ p ) × Σ HPvac along the vector field ̂T on the latter defined in Eq. (7.3) iscanonically s vac -invariant, with the corresponding p -gerbe 1-isomorphisms ˘Λ ˘ A ∶ − L ˘ K ˘ A ( − L ̂ T pr ∗ ι ∗ vac ̂G ( p ) ) ≅ ÐÐ→ I ( p ) given by the formula ˘Λ ˘ A = − L ̂ T pr ∗ Λ ˘ A in terms of the p -gerbe 1-isomorphisms Λ ˘ A ∶ − L K S ˘ A ι ∗ vac ̂G ( p ) ≅ ÐÐ→ I ( p ) , ˘ A ∈ , S vac whose existence is ensured by s vac -invariance of ι ∗ vac ̂G ( p ) .Proof. Follows from an adaptation of Prop. 7.10, taken in conjunction with Prop. 7.2. (cid:3)
So far, we have been dealing with independent linearisations of a G loc -equivariant structure on a p -gerbe and of a realisation of the global-symmetry group G on it (by p -gerbe 1-isomorphisms). Wemay now finally come to the context of immediate interest in which both linearisations are combinedin a cohrent manner, that is to a gerbe-theoretic rendering of the compatibility scenario laid out inDef. 6.13. Our goal is to acquire tools that enable us to verify canonical s vac -invariance of the canonical gs vac ( sB ( HP ) p,λ p ) -equivariant structure on the vacuum restriction of the extended HP p -gerbe.As previously, we work out the requisite intuitions in the Graßmann-even setting first. Proposition 7.12.
Adopt the hitherto notation, and in particular that of Defs. 6.13, 7.3 and 7.8, aswell as that of Prop. 7.10. Let M , G and G loc be all as in Def. 6.13, with S G the global-symmetrysubspace of Def. 5.2 and S g loc ⊂ T M the local-symmetry distribution introduced in Def. 7.3, and let G ( p ) (for p > ) be a p -gerbe over M that we assume g -invariant in the sense of Def. 7.8, with thebasis p -gerbe 1-isomorphisms ̃ Λ A ∶ − L K λA G ( p ) ≅ ÐÐ→ I ( p ) , A ∈ , dim g , and suppose that the curvature χ ( p + ) of G ( p ) is S g loc -horizontal, as expressed in Eq. (7.4) . If thereexists a lift of S G to a g loc -linear subspace ̂ S G ⊂ Γ (T ( g loc × M)) , spanned on the lifts ˘ K λA , A ∈ , dim g of the respective vector fields K λA and defined as in Prop. 7.2, with the property ∀ A ∈ , dim g ∶ [ ˘ K λA , ̂K λ loc ] = , (7.7) expressed in terms of the vector field ̂K λ loc of Eq. (7.5) , then the canonical g loc -equivariant struc-ture (7.6) on G ( p ) is canonically g -invariant in the sense of Def. 7.8. The corresponding p -gerbe -isomorphisms − L ˘ K λA ( − L ̂ K λ loc pr ∗ G ( p ) ) − L ˘ K λA ̂ Λ g loc / / ˘Λ ̂ K λ loc A (cid:15) (cid:15) I ( p ) I ( p ) (cid:15) (cid:15) ψ A ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ s { ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ I ( p ) I ( p ) I ( p ) , A ∈ , dim g can be written in the form ψ A = ̂K λ loc ⌟ pr ∗ ̃ Λ A . Proof.
Adopt the previously introduced notation. Let B ( p + ) ∈ Ker D ( p + ) be a ˇCech–Deligne ( p + ) -cocycle presenting G ( p ) for an open cover O M of M over which there exist ● P ( p ) A ∈ ˇCD p (O M , D( p + ) ● ) , A ∈ , dim g – a local presentation of the g -invariant structure associated with a basis { τ A } A ∈ , dim g of g , − L K λA B ( p + ) = − D ( p ) P ( p ) A ; ● ̂ P ( p ) ∈ ˇCD p ({ g loc } × O M , D( p + ) ● ) – a local presentation of the g loc -equivariant structure, − L ̂ K λ loc pr ∗ B ( p + ) = − D ( p ) ̂ P ( p ) . We compute, along similar lines as in the proof of Prop. 7.10 and using Eqs. (7.6) and (7.7), alongsideProp. 6.1, − L ˘ K λA ̂ P ( p ) − − L ̂ K λ loc pr ∗ P ( p ) A ≡ − − L ˘ K λA ( ̂K λ loc ⌟ pr ∗ B ( p + ) ) − − L ̂ K λ loc pr ∗ P ( p ) A = − [ ˘ K λA , ̂K λ loc ] ⌟ pr ∗ B ( p + ) − ̂K λ loc ⌟ pr ∗ ( − L K λA B ( p + ) ) − − L ̂ K λ loc pr ∗ P ( p ) A = ̂K λ loc ⌟ D ( p ) pr ∗ P ( p ) A − − L ̂ K λ loc pr ∗ P ( p ) A = − D ( p − ) ( ̂K λ loc ⌟ pr ∗ P ( p ) A ) and extract from that computation the definition ̂K λ loc ⌟ pr ∗ P ( p ) A ∈ ˇCD p − ({ g loc } × O M , D( p + ) ● ) , A ∈ , dim g of a family of ( p − ) -cochains that compose a local presentation of the (canonical) g -invariant 2-isomorphisms from the thesis of the proposition. (cid:3) Our hitherto efforts are crowned by the following theorem in which the various results are put together.
Theorem 7.13.
Adopt the hitherto notation, and in particular that of Props. 7.7, 7.11 and 7.12.Thus, let G be the supersymmetry group (with the tangent Lie superalgebra g ) and H vac the vac-uum isotropy group of a Green–Schwarz super- σ -model for the super- p -brane in the Hughes–Polchinskiformulation, let Σ HPvac be the vacuum foliation of Def. 4.6 of the Hughes–Polchinski section Σ HP ofEq. (3.11) determined by an integrable Hughes–Polchinski vacuum superdistribution Vac ( sB ( HP ) p,λ p ) ofDef. 4.3, and let s vac and s loc ≡ gs vac ( sB ( HP ) p,λ p ) be the Lie sub-superalgebras of g that model – re-spectively – the residual global-supersymmetry supspace of Prop. 5.5 and the limit κ −∞ ( sB ( HP ) p,λ p ) of theweak derived flag of the κ -symmetry superdistribution κ ( sB ( HP ) p,λ p ) within Vac ( sB ( HP ) p,λ p ) . There exists acanonical and canonically s vac -invariant s loc -equivariant structure on the vacuum restriction ι ∗ vac ̂G ( p ) Cp the comment on p. 92. In the super-Minkowskian setting, we should consider the sub-superalgebra smink ( d, ∣ ND d, ) ⊂ siso ( d, ∣ ND d, ) as the model for the global-supersymmetry supervector space due to the very nature of the existing constructions ( i.e. ,extensions), taking as the point of departure that Lie supergroup, and not the super-Poincar´e group. f the extended Hughes–Polchinski p -gerbe ̂G ( p ) of Def 6.11. The canonical structure consists of the p -gerbe 1-isomorphisms ̂ Λ s loc ≡ − ̂T ⌟ pr ∗ ι ∗ vac ̂G ( p ) ∶ − L ̂ T pr ∗ ι ∗ vac ̂G ( p ) ≅ ÐÐ→ I ( p ) , ˘Λ ˘ A ≡ − L ̂ T pr ∗ Λ ˘ A ∶ − L ˘ K ˘ A ( − L ̂ T pr ∗ ι ∗ vac ̂G ( p ) ) ≅ ÐÐ→ I ( p ) , ˘ A ∈ , S vac , the latter written in terms of the p -gerbe 1-isomorphisms Λ ˘ A ∶ − L K S ˘ A ι ∗ vac ̂G ( p ) ≅ ÐÐ→ I ( p ) encoding the assumed s vac -invariance of ι ∗ vac ̂G ( p ) , and of the p -gerbe 2-isomorphisms ̂T ⌟ pr ∗ Λ ˘ A ≡ ψ A ≡ − L K S ˘ A ( − L ̂ T pr ∗ ι ∗ vac ̂G ( p ) ) − L ˘ K ˘ A ̂ Λ s loc ( ( ˘Λ ˘ A ψ A (cid:11) (cid:19) I ( p ) , ˘ A ∈ , S vac . Proof.
Follows directly from Props. 7.7, 7.11 and 7.12. (cid:3)
Remark 7.14.
Clearly, our results can be repeated verbatim in the situation described in Example5.20 if we simply replace the κ -symmetry superdistribution appearing above with the extended one.The statement of the theorem concludes our study of linearised equivariance of the vacuum restrictionof the extended Hughes–Polchinski p -gerbe of the Green–Schwarz super- σ -model (in the topologicalHughes–Polchinski formulation) with respect to the vacuum-generating gauged supersymmetry (aka κ -symmetry), compatible with the residual global supersymmetry. Remark 7.15.
The above symmetry analysis, while far from complete, traces a line of reasoningthat we are tempted to pursue speculatively, not least because of the hints as to directions of furtherdevelopment that it may provide. Thus, it is to be noted that the existence of a full-fledged
Lie su-peralgebra -equivariant structure on the vacuum restriction ι ∗ vac ̂G ( p ) of the extended Hughes–Polchinski p -gerbe would essentially imply – in virtue of the standard interpretation of an equivariant struc-ture, corroborated in Refs. [GSW10, Sus12, GSW13, Sus13] in the (1-)gerbe-theoretic context – that ι ∗ vac ̂G ( p ) descends to the orbispace of the ‘action’ of the symmetry structure, i.e. , of κ −∞ ( sB ( HP ) p,λ p ) , onthe vacuum Hughes–Polchinski section. But, then, the vacuum foliation is actually generated by theflows of the latter superdistribution, which means – in the S -point picture – that each of its leaves, orvacua, can be retracted to any one of its points. Consequently, ι ∗ vac ̂G ( p ) would be supersymmetrically1-isomorphic to. . . the trivial p -gerbe with zero curving – the unique p -gerbe over a point. This, inturn, would infer a vacuum trivialisation ι ∗ vac π ∗ G / H G ( p ) ≅ I ( p ) − λ p ι ∗ vac β ( p + ) ( HP ) . Trivialisations of that kind have long been recognised as hallmarks of the presence of a defect in thefield theory, (D-)branes and bi-branes of string theory being examples of such structures. In the lightof the above, we are led to consider a potential correspondence between vacua of the Green–Schwarzsuper- σ -model for (a homogeneous space of) a given super-Harish–Chandra pair and (homogeneousspaces of) sub-super-Harish–Chandra pairs thereof with a ( p + ) -dimensional body over which thesuper- p -gerbe of the super- σ -model trivisalises in the manner suggested above. We shall certainlycontemplate this attractive idea in the future. . Conclusions & Outlook
In the present paper, we have established a simple and complete geometric interpretation of thetangential gauge supersymmetry, also known as κ -symmetry, of a large class of Green–Schwarz super- σ -models for the super- p -brane functorially embedded in a reductive homogeneous space of a Lie super-group, and lifted the supersymmetry to the higher-geometric structure associated with the topologicalcomponent of the super- σ -model superbackground – the extended Hughes–Polchinski p -gerbe – in theform of a linearised equivariant structure on the latter, compatible with the global supersymmetrypresent. The gauge supersymmetry has acquired the interpretation of an odd-generated superdistribu-tion (bracket-)generating – through its weak derived flag – the tangent sheaf of the (classical) vacuum ofthe field theory (resp. its chiral component, cp Example 5.20) and thus enveloping the vacuum, cp Sec. 5and in particular Thm. 5.10, whence also the name given to it in the title of the paper – the squareroot of the vacuum . Its gerbification as a tangential equivariant structure, consistent with the find-ings of previous studies on gerbe-theoretic realisations of gauge symmetries in the two-dimensionalnon-linear bosonic σ -model, has been proven to possess a canonical form, canonically invariant withrespect to the residual global supersymmetry of the vacuum, cp Thm. 7.13. These results have beenobtained through an in-depth analysis of the universal phenomenon of enhancement of gauge (su-per)symmetry in the topological Hughes–Polchinski formulation of the Green–Schwarz super- σ -modelthat occurs in the physical correspondence sector thereof, introduced in Sec. 3 with direct reference tothe correspondence superdistribution of Def. 3.10, in which the field theory is (classically) dual to thestandard Nambu–Goto formulation of the super- σ -model, the conditions for the duality having beenworked out in the present paper in greater generality than heretofore, cp Thm. 3.4, on the basis of arigorous supergeometric description of the supertarget(s) using Kostant’s approach to the theory of Liesupergroups, cp Sec. 2. Instrumental in the analysis has been the derivation and subsequent geometri-sation of the Euler–Lagrange equations of the topological super- σ -model, cp Prop. 4.2 and Def. 4.3,and an exhaustive study of the algebraic conditions of integrability, global supersymmetry and de-scendability of the vacuum superdistribution determined by these equations in Def. 4.3. The abstractconsiderations have been illustrated on a large number of concrete examples of super- σ -models witha single topological charge: the Green–Schwarz super- p -branes in the super-Minkowskian backgroundand the super-1-branes in s ( AdS n × S n ) for n ∈ { , , } .While essentially resolving the issue of the geometric nature of κ -symmetry, the study reported in thepresent paper leaves us with several follow-up questions and challenges. The first obvious one is a Lie-superalgebraic classification of super- σ -models with an integrable vacuum superdistribution generatedby its κ -symmetry superdistribution in terms of Lie sub-superalgebras of (physically motivated) Liesuperalgebras associated with a pair of projectors defining the vacuum subspace t vac ⊂ t in the direct-sum complement of the isotropy subalgebra h of the mother supersymmetry algebra g , cp Thm. 5.10.This question is intimately related to the issue of existence of minimal spinors in a given metric geometry (∣ G / H ∣ , g ) of Sec. 2 ( cp , in particular, Remark 3.13). Further self-consistency conditions are anticipatedto follow from imposition of the requirement of existence of a full-fledged (residual) supersymmetry-invariant Lie superalgebra -equivariant structure on the vacuum restriction of the Hughes–Polchinski p -gerbe – its derivation and detailed investigation should, therefore, be undertaken next. In its course,we should most certainly keep in mind and elaborate the attractive idea formulated in Rem. 7.15. In thiscontext, we are fortunate to have at our disposal a rich pool of field-theoretic/supergeometric exampleswith a hands-on Lie-superalgebraic and (local-)coordinate descriptions – this richness ought to beexploited towards further advancement of our understanding of the intricate nature of the vacuum ofthe super- σ -model in the format of a case-by-case study that is certain to provide us with new insights,just as it has done so far, and to yield concrete results for field theories with a potential applicationin the modelling of realistic strongly coupled systems of the QCD-type. Thus, in particular, the studyreported and motivated herein is hoped to serve – in the long run – the outstanding goal of elucidatingthe still (mathematically) elusive AdS/CFT correspondence. The differential-supergeometric and Lie-superalgebraic discussion of the super- σ -model and its vacuum elaborated in the present work does notseal the fate of field theories and field configurations that depart from the neat scenario of a regular( i.e. , integrable) vacuum superdistribution, and it does not tell the story of super- σ -models that donot satisfy the set of constraints imposed in the derivation of the Euler–Lagrange field equations inSec. 4 – it would certainly be both interesting and useful to study these departures in greater detail,with view to extending and generalising the results of our work. The example of the super-0-brane inthe Zhou superbackground of s ( AdS × S ) treated in the Appendix, taken in conjunction with thesupergerbe-theoretic results for this particular superbackground obtained in Ref. [Sus18b], seems to be perfect point of departure of a quest thus oriented. Finally, reaching out beyond the compass of thepresent work, there is the fascinating question of general field-theoretic and geometric consequences ofthe duality between the dynamical Nambu–Goto formulation of the (super-) σ -model and the purely topological Hughes–Polchinski one. Among other things, one could envisage its application as a newpotent tool in the by now fairly advanced study of T-duality – another symmetry entangling the metricand topological degrees of freedom in the standard Nambu–Goto formulation – where it is expected tolead to a unified topological (that is gerbe-theoretic) description of this loop-mechanical duality and, inthis manner, pave the way to a systematic description and construction of T-folds ( via gerbe-theoreticT-duality gauge defects). We hope to return to these issues in the future. ppendix A. The Zhou super-0-brane in s ( AdS × S ) . We describe the relevant superbackground along the lines of Examples 3.11–3.16. ● The mother super-Harish–Chandra pair:SU ( , ∣ ) ≡ ( SO ( , ) × SO ( ) , su ( , ∣ ) ) , as in Example 3.14; ● The (infinitesimal) T e Ad H -invariance of the vacuum splitting: follows by the same argumentas in Example 3.12 due to the identical structure of and relations between the commutators [ d , t ( ) vac ] and [ d , e ( ) ] ; ● The homogeneous spaces: the NG ones ( AdS × S ) = SU ( , ∣ ) /( SO ( , ) × SO ( )) , H = SO ( , ) × SO ( ) , h ≡ so ( , ) ⊕ so ( ) = ⟨ J ⟩ ⊕ ⟨ J ⟩ , with the body as in Example 3.14 and the HP oneSU ( , ∣ ) / SO ( ) , H vac = SO ( ) , h vac = ⟨ J ⟩ , d = ⟨ J ⟩ , t ( ) vac = ⟨ P ⟩ , d − t ( ) vac = ⟨ P ⟩ ⊊ e ( ) ; ● The exponential superparametrisation(s): σ vac0 ( θ α ′ α ′′ I , x a , φ ) = e θ α ′ α ′′ I ⊗ Q α ′ α ′′ I ⋅ e x a ⊗ P a ⋅ e φ ⊗ J ; ● The superbackgrounds: the NG one sB ( NG ) = ( s ( AdS × S ) , η ab θ a L ⊗ θ b L , i Σ L ∧ ( C ⊗ σ ) Σ L + θ ∧ θ ≡ χ ( ) Zh ) ,θ L = Σ α ′ α ′′ I L ⊗ Q α ′ α ′′ I + θ a L ⊗ P a + θ ⊗ J + θ ⊗ J with a supersymmetric global predecessor of the curving β ( ) Zh = − θ on SU ( , ∣ ) that does not descend to s ( AdS × S ) , and the HP one sB ( HP ) ,λ = ( SU ( , ∣ ) / SO ( ) , χ ( ) Zh + λ ( Σ L ∧ ( C γ ⊗ ) Σ L − θ ∧ θ ) ≡ ̂ χ ( ) Zh ) , ⟨ P ⟩ ⊥ η ⟨ P ⟩ ; ● The Body-Localisation Constraints: θ a ′′ L ≈ , a ′′ ∈ { , } . (A.1)The logarithmic variation of the DF amplitude of the corresponding GS super- σ -model in the HPformulation, computed as in Sec. 4, takes the explicit form − i δ δ ̂ ξ log A ( HP ) , ,λ DF [ ξ ] = ∑ τ ∈ T ∫ τ ( σ vac ı τ ○ ̂ ξ τ ) ∗ [ δθ α ′ α ′′ Iι τ ( C γ ⊗ ) α ′ α ′′ Iβ ′ β ′′ J ( λ − i γ ⊗ σ ) β ′ β ′′ Jγ ′ γ ′′ K Σ γ ′ γ ′′ K L (A.2) + δx ι τ θ + δx ι τ ( λ θ − θ ) − λ δφ ι τ θ ] , from which we read off – for λ ∈ {− , } – the definition of the possible projectors P ( ) ± = ± i γ ⊗ σ with properties ( γ ⊗ ) P ( ) ± = P ( ) ± ( γ ⊗ ) , ( γ ̂ a ⊗ ) P ( ) ± = ( − P ( ) ± ) ( γ ̂ a ⊗ ) , ( C ⊗ ) P ( ) ( C ⊗ ) − = P ( ) T , nd so upon choosing P ( ) ≡ P ( ) + = + i γ ⊗ σ , also the EL equations (( − P ( ) ) Σ L , θ , θ − θ ) ≈ , to be augmented with the BLCs (A.1). The last of the EL equations indicates that the body of thevacuum is actually ‘(gauge-)tilted’ in the direction of J relative to the ‘canonical’ direction P .Altogether, we deduce the HP vacuum superdistribution with restrictionsVac ( SU ( , ∣ ) / SO ( ) , ̂ χ ( ) Zh ) ↾ V i = ⊕ α = ⟨ T α i ⟩ ⊕ ⟨ T i + T i ⟩ . At this stage, it suffices to compute the supercommutators { P ( ) γ ′ γ ′′ Kα ′ α ′′ I Q γ ′ γ ′′ K , P ( ) δ ′ δ ′′ Lβ ′ β ′′ J Q δ ′ δ ′′ L } = P ( ) γ ′ γ ′′ Kβ ′ β ′′ J (( C γ ⊗ ) α ′ α ′′ Iγ ′ γ ′′ K P − i ( C ⊗ σ ) α ′ α ′′ Iγ ′ γ ′′ K J ) ≡ ( C γ ⊗ ) α ′ α ′′ Iγ ′ γ ′′ K ( T + ( P ( ) − ) J ) γ ′ γ ′′ Kδ ′ δ ′′ L P ( ) δ ′ δ ′′ Lβ ′ β ′′ J = (( C γ ⊗ ) P ( ) ) α ′ α ′′ Iβ ′ β ′′ J T and [ T , T ] = = [ J , J ] , [ J , T ] = , [ T , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = (( i ̃ γ ′ γ ⊗ σ + γ ⊗ ) ⋅ P ( ) ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ≡ (( i ̃ γ ′ ⊗ ) ⋅ ( − P ( ) ) ⋅ P ( ) ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J = , [ J , P ( ) β ′ β ′′ Jα ′ α ′′ I Q β ′ β ′′ J ] = ( γ ⊗ ) β ′ β ′′ Jα ′ α ′′ I P ( ) γ ′ γ ′′ Kβ ′ β ′′ J Q γ ′ γ ′′ K to conclude that it is an SO ( ) -descendable integrable superdistribution associated with the Lie super-algebra vac ( SU ( , ∣ ) / SO ( ) , ̂ χ ( ) Zh ) = ⊕ α = ⟨ Q α ⟩ ⊕ ⟨ P + J ⟩ ⊕ ⟨ J ⟩ . Another look at Eq. 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