The higher-algebraic skeleton of the superstring -- a case study
aa r X i v : . [ h e p - t h ] F e b THE HIGHER-ALGEBRAIC SKELETON OF THE SUPERSTRING– A CASE STUDY
RAFA L R. SUSZEK
Abstract.
A novel Lie-superalgebraic description of the superstring in the super-Minkowskian back-ground is extracted from the Cartan–Eilenberg super-1-gerbe geometrising the higher gauge field (theGreen–Schwarz super-3-cocycle) that couples to the supercharge carried by the superstring. Thedescription assumes the form of a hierarchy of Lie superalgebras integrable to a hierarchy of Liesupergroups and provides a manifestly supersymmetric model of a family of supermanifolds defin-ing a trivialisation of the super-1-gerbe over the embedded superstring worldsheet. The trivialisation,obtained in a purely topological formulation of the superstring dynamics dual to the standard Nambu–Goto-type one, conforms with the gerbe-theoretic representation of extended sources of higher gaugefields known from previous studies of the σ -model of the bosonic string. Contents
1. Introduction 12. The geometrodynamics of the HP superstring 73. Supersymmetries of the super- σ -model 173.1. Global supersymmetry 173.2. Local supersymmetry of the vacuum 184. A geometrisation of the superbackground – the super-1-gerbe(s) 205. The supersymmetry of the super-1-gerbe(s) 335.1. Higher global supersymmetry 335.2. Higher κ -symmetry & the sLieAlg -skeleton of the vacuum 346. Conclusions & Outlook 49Appendix A. A convention 51References 511. Introduction
The idea to probe spacetime geometry with the dynamics of distributions of charged matter hasbeen around for a long time, cp Refs. [FI29b, FI29a, Wey29, Foc29a, Foc29b, Dir31], yielding an en-hancement of the simple model of a metric spacetime ( M, g ) accessible to a neutral pointlike particle.The enhancement incorporates the ‘higher’ geometry of the gauge fields H ∈ Z p + ( M ) , p ∈ N (and theirnonabelian counterparts) coupling to the respective charges, first neatly packaged by Lubkin [Lub63],Trautman [Tra70] et al. in the structure of fibre bundles, and later generalised as bundle ( p -)gerbes and related objects by Murray et al. [Mur96, MS00, Ste04], cp , in particular, Ref. [Gaj97]. The latterform a descent hierarchy of geometrisations of integral classes in (a suitable refinement of) the de Rhamcohomology of M whose local sections provide us with the (suitably extended) Deligne–Beilinson co-homological data of the gauge fields, employed in the construction of simple models of charge dynamicsand their geometric quantisation already by Alvarez [Alv85] and Gaw¸edzki [Gaw88].In the presence of Killing vector fields K A ∈ Γ ( T M ) , A ∈ , K of the background metric gwhose flows preserve the action functional defining the charge dynamics, the enhancement may takethe form of a deformation or an extension of the Lie algebra of these vector fields. Notable in-stantiations of the former include the algebra [ P µ , P ν ] = qH µν , µ, ν ∈ , { P µ } µ ∈ , to the space of states of a pointlike particle of charge q in a constant electromagneticfield H = H µν d x µ ∧ d x ν ∈ Ω ( Mink ( , )) , and the Poisson germ of the Drinfeld–Jimbo quantum-group structure in the (chiral) Wess–Zumino–Witten model on SU ( ) (with the Cartan 3-form asthe gauge field) in Ref. [Gaw91, Sec. 4]. The latter are amply exemplified by the infinite sequence of For a gentle introduction to the general theory, cp Ref. [Joh02]. An overview was given by Murray in Ref. [Mur10]. xtensions Maxwell n , n ∈ N × of the Poincar´e algebra presented in Refs. [BG10, GK17] as algebraicstructures encoding the rich dynamics of a (possibly backreacting) multipole distribution of chargesin an external electromagnetic field (generalising the pioneering constructions: the kinematical alge-bras of Ref. [BCR70] and the Maxwell algebra of Ref. [Sch72]) and by the Free (super-)Differential-Algebra (FsDA) extensions of the super-Poincar´e algebra considered in Ref. [CdAIPB00] in the con-text of superstring theory in Minkowskian spacetime that build upon the earlier constructions ofRefs. [Gre89, Sie94, BS95]. The supersymmetric extensions, of central relevance to us in what fol-lows, seem to be encompassed by the structure of the Free Lie super-Algebra (FLsA) laid out in therecent study [GKP19]. The common source of the enhancement in the examples listed is an interplaybetween the geometry (topology) of the distribution of charge consistent with its dynamics and theintrinsic cohomology of the gauge field H (tied intimately with the aforementioned higher geometry),the latter being typically assumed to satisfy the strong invariance condition K A ⌟ H ∈ B ● dR ( M ) (ensur-ing quasi-invariance of the lagrangian density) in the case of charge distributions localised on closedsubmanifolds of M , cp , in particular, Refs. [dAGIT89], [GSW10, Cor. 2.2] and [Sus18a, Sec. 3]. The twoare jointly encoded by the (pre)symplectic form of the lagrangian model of dynamics, as given by thefirst-order formalism of Refs. [Gaw72, Kij73, Kij74, KS76, Szc76, KT79], and so a question arises howto isolate information on (the geometry of) a particular classical solution of the dynamics given anenhancement of a reference (neutral) Killing algebra (note that the enhancement captures symmetriesof the entire space of classical solutions). This is the general problem that we tackle in the presentcase study in the framework of supersymmetric dynamics of super- p -branes, to be investigated usingmethods of higher (super)geometry that we review systematically below.The algebraic mechanisms from the previous paragraph have been encountered and employed exten-sively as model-building tools in the setting of (super)field theory with non-linearly realised symmetry[CWZ69, CCWZ69, SS69a, SS69b, ISS71] and supersymmetry [VA72, VA73, IK78, LR79, UZ82, IK82,SW83, FMW83, BW84] (originally contemplated by Schwinger [Sch67] and Wigner [Wei68] in the con-text of effective field theory with chiral symmetries) in which the fibre of the covariant configurationbundle (or the ‘field space’) carries the structure of a homogeneous space G / H of a (super)symmetrygroup G relative to its distinguished closed subgroup H with the tangent Lie algebra h ≡ Lie ( H ) defining a reductive decomposition t ⊕ h = g ≡ Lie ( G ) of the tangent Lie (super)algebra g of G, i.e. ,such that [ h , t ] ⊂ t . Here, the dynamics is modelled in terms of H-basic tensors on G taken from thetensor algebra of the linear space of (G-)left-invariant (LI) (super-)1-forms on the (super)manifold Gand pulled back to the field space G / H along local sections of the principal H-bundle G π G / H ÐÐÐÐ → G / H( π G / H is the quotient map) whose potential nontriviality was accounted for in [Sus19, Sec. 5] and whoselocal sections particularly favoured by physical considerations were analysed at length in the Z / Z -graded setting in [Sus20, Sec. 2]. An in-depth study of the mechanism of spontaneous (super)symmetrybreakdown by a classical solution to the dynamics in this setting, in conjunction with a clever applica-tion of the so-called Inverse Higgs Effect originally discovered by Ivanov and Ogievetsky [IO75], haveprovided us with a reinterpretation of some standard action functionals modelling charge dynamics asGoldstone fields conjugate to distinguished central charges defining extensions of geometric ( i.e. , neu-tral) (super)symmetry algebras [GGT90], and – crucially from the vantage point adopted in the presentpaper – have led to a purely topological reformulation of the Nambu–Goto-type ‘metric’ components ofaction functionals for structureless (as in Ref. [Dir62], cp also Ref. [dAT89]) extended distributions ofcharged matter in, i.a. , Refs. [Wes00, GKW06b, GKW06a, McA10], along the lines of the original ideaof Hughes and Polchinski [HP86] developed by Gauntlett, Itoh and Townsend in Ref. [GIT90].The specific choice, referred to in the last paragraph, of a constructive paradigm of study ofcharge dynamics with the help of the Cartan calculus on a Lie (super)group paves the way fora systematic application of the techniques of Free ( Z / Z -graded) Differential Algebras (FDA) ofRefs. [DF82, vN83, CFG+83]. In the context of interest, these are specialised to an augmentation of thecanonical FDA LI( G ) of LI (super-)1-forms on the Lie (super)group G by the (super-) ( p + ) -formpotential of the relevant gauge field. For these, a finite ladder of integrable (super)central extensions Ð→ a n Ð→ g n + Ð→ g n Ð→ (altogether combining into a generically non -(super)central extension Y g ≡ g N Ð→ g ≡ g of the original Lie (super)algebra g ) is sought that yields a resolution of the gaugefield in the Cartan–Eilenberg (CaE) cohomology CaE p + ( Y G ) ≡ H p + ( Y G ) Y G of the Lie (super)group Y G which integrates Y g . Each rung of the ladder is determined by a (super-)2-cocycle in the decompo-sition of the pullback of the gauge field in terms of elements of LI( G n ) (for G n the Lie (super)group of This is, arguably, most convincingly illustrated in the ‘top-down’ treatment of charge dynamics in Refs. [BG10, GK17]. n ) in conformity with the standard classification of (super)central extensions of a Lie (super)algebra g n by a (super)commutative Lie (super)algebra a n by elements of the group H ( g n , a n ) in the a n -valued cohomology of g n , cp Ref. [Lei75]. While it is not clear a priori that a finite resolution Y g of this kind exists ( cp the Theorem in Ref. [vN83, Sec. 7]), an algorithm devised by de Azc´arragaand collaborators in Ref. [CdAIPB00], which essentially boils down to a systematic reconstruction ofa finite quotient within the FLsA of [GKP19], does produce the desired result in the very special set-ting of the Green–Schwarz-type (GS) super- σ -models of super- p -brane dynamics in super-Minkowskian(super)geometry sMink ( d, ∣ N D d, ) ≡ R d, ∣ N [Cas76, BS81, dAL82, GS84a, GS84b, AETW87] (here, N ∈ N × is the number of supercharges in a Majorana-spinor representation of Cliff ( R d, ) of dimension D d, that generate an N -extended supersymmetry) that we work with in the present paper and onwhich, consequently, we focus henceforth. In fact, it was recently argued by Grasso and McArthur inRefs. [GM18b, GM18a] that these results are essentially unique when viewed as solutions to a coho-mological problem in CaE p + ( Y sMink ( d, ∣ N D d, )) (their argument exploits the assumed triviality ofthe de Rham cohomology of the extension).In order to be able to interpret the extension Y G Ð→ G as a partial geometrisation of the GSsuper- ( p + ) -cocycle H ∈ Z p + ( G ) G in the (standard) sense of Murray, one should put the CaE co-homology of G on the same footing as the underlying de Rham cohomology. That this makes senseis suggested by an old argument due to Rabin and Crane [RC85, Rab87] that essentially explainsthe discrepancy between CaE ● ( sMink ( d, ∣ N D d, )) (for N =
1) and H ● dR ( sMink ( d, ∣ N D d, )) ≡ via the standard duality, from the homology of an orbifold sMink ( d, ∣ N D d, )/ Γ KR ofsMink ( d, ∣ N D d, ) relative to a discrete subgroup Γ KR ⊂ sMink ( d, ∣ N D d, ) that had been encoun-tered previously by Kosteleck´y and Rabin in their study of supersymmetric field theory on the lattice[KR84]. The orbifold has the topological structure of a fibration over its body Mink ( d, ) with com-pact Graßmann-odd fibers [RC85]. The argument led Rabin to postulate that the GS super- σ -modelwith the supertarget sMink ( d, ∣ N D d, ) be interpreted as describing propagation of loop-like distri-butions of supercharge within sMink ( d, ∣ N D d, )/ Γ KR . But then, by a standard argument ( cp , e.g. ,Refs. [DHVW85, DHVW86] and, in particular, [Sus12, Sec. 8.3] and Ref. [Sus13] in which the ideaof a worldvolume orbifold was formalised with reference to the universal gauge principle derived inRefs. [GSW10, GSW13]), one has to incorporate the Γ KR -twisted sector in the superfield theory onthe cover sMink ( d, ∣ N D d, ) , and, indeed, this yields, e.g. , a Graßmann-odd wrapping anomaly in thecanonical picture of [Sus18a, Sec. 4.2] that reproduces the Green extension of the sMink ( d, ∣ N D d, ) superalgebra resolving the GS super-3-cocycle. While vital for internal consistency of our treatment,the last result shows quite explicitly how an enhancement of a neutral Killing algebra in the presenceof a distribution of charged matter combines information on the cohomology of the gauge field and thetopology of the distribution. We may now be more specific in defining our goal: We wish to extracta clearcut signature of the localisation of a classical superstring (and more generally super- p -brane)trajectory in the supertarget from the superalgebraic description of the gauge field that couples to it.To this end, we first need to complete the geometrisation of that field and recall from the extensivestudy of analogous geometrisations in the non- Z / Z -graded setting the higher-geometric representationof extended objects to which the gauge field of the σ -model couples – the D-branes [Pol95], with the It is certainly more natural to associate with the representative H of a class in CaE p + ( G ) a slim Lie ( p + ) -(super)algebra of Baez, Crans and Huerta [BC04, BH11, Hue11], itself a special example of the more general structure(an L ∞ -(super)algebra) encountered in the study of string field theory [Sta92, LS93]. However, to the best of the Author’sknowledge, there do not exist, to date, any explicit constructions of the corresponding integrated structures (the so-calledLie ( p + ) -supergroups) for the known super- p -branes with p > cp below. Exploration of curved supergeometries was pioneered in Refs. [BST86, BST87, BLN+97, dWPPS98, Cla99, MT98,Zho99, AF08, GSW09, FG12, DFGT09]. Attaining a similar goal for physically relevant curved supergeometries with a nontrivial topology, such as, e.g. ,the homogeneous spaces: SU ( , ∣ ) /( SO ( , ) × SO ( )) ≡ s ( AdS × S ) (viewed as the supertarget of the Zhou su-perstring [Zho99]), ( SU ( , ∣ ) × SU ( , ∣ )) /( SO ( , ) × SO ( )) ≡ s ( AdS × S ) (for the Park–Rey superstring [PR99])and SU ( , ∣ )/( SO ( , ) × SO ( )) ≡ s ( AdS × S ) (for the Metsaev–Tseytlin superstring [MT98]), seems to call for anaugmentation of the original organising principle. One natural possibility, suggested by the prime rˆole of the asymptoticcorrespondence between the curved dynamics (its supergeometric data and the lagrangian model) and its flat-superspacecounterpart in the construction of the former, cp Ref. [MT98], is the requirement that the ˙In¨on¨u–Wigner contractionunderlying the asymptotic flattening should lift to the full-fledged geometrisation of the relevant gauge superfield. Theprinciple was laid out in Ref. [Sus18a] (with several no-go results for the Metsaev–Tseytlin superstring in its currentformulation), elaborated and successfully realised for the Zhou superparticle in s ( AdS × S ) in Ref. [Sus18b], and for-malised concisely in Ref. [Sus21]. It is expected to work out in conjunction with the S -expansion scheme put forward inRef. [HS03] and later generalised in Refs. [dAIPV03, IRS06]. This expectation is currently being investigated. oupling encoded in the effective Dirac–Born–Infeld (DBI) lagrangian density [FT85, ACNY87, Lei89].The train of reasoning restated above has laid the foundation for the geometrisation programme,initiated by the Author in Ref. [Sus17], elaborated in Refs. [Sus19, Sus18a, Sus18b, Sus20] and re-cently reviewed in Ref. [Sus21], which sets out to associate with the physically relevant CaE gauge-field super- ( p + ) -cocycles that determine the known GS-type super- σ -models on homogeneous spacesG / H of supersymmetry Lie supergroups G, as well as with the attendant supersymmetric defects[FSW08, RS09, Sus11a, Sus11b], concrete higher-geometric objects of the type conceived by Murray etal. in the non- Z / Z -graded setting (and recently reconsidered in the Z / Z -graded setting by Huerta[Hue20]), and to lift all essential geometric properties of the underlying superfield theories (such as, e.g. , their κ -symmetry) and constitutive relations between them (such as, e.g. , the fundamental asymp-totic relation between the super- p -brane models with the super-AdS m × S m targets and their super-Minkowskian counterparts) to those higher (super)geometries and the associated higher categories.The rationale for the goal thus delineated is an early observation, due to Gaw¸edzki [Gaw88], that thehigher-geometric objects canonically determine, through the so-called cohomological transgression, ageometric (pre)quantisation scheme for the simple charge geometrodynamics of the σ -model. Thus,the existence of the said lifts is to be viewed as a condition of quantum-mechanical consistency of thestructures, properties and relations lifted. In the gerbe-theoretic picture, the D-branes of string theoryare represented by trivialisations of the 1-gerbe of the gauge field of the σ -model over submanifolds ofthe target space M [Gaw99, FW99, CJM02, GR02, Gaw05] – these are described by certain vectorbundles whose connection acquires the interpretation of the (gerbe-twisted) gauge field of the DBItheory.The basic geometric substrate of the principle of descent that lies at the core of Murray’s geometri-sation of the class [ H ] ∈ H p + ( M, Z ) of a gauge field H (assumed integral) [Mur96, Mur10] is a surjective submersion Y M π Y M ÐÐÐ→ M whose total space supports a smooth primitive B ∈ Ω p + ( Y M ) for the pullback of H, i.e. , such that π ∗ Y M H = d B. The ( p + ) -form B can then be viewed as the trivial p -gerbe I ( p ) B of curvature d B and curving B over Y M . The p -gerbe G ( p ) for [ H ] is subsequentlyerected over the nerve of the small category Y [ ] M s ≡ pr / / t ≡ pr / / Y M , defined by the ( π Y M -)fibred square Y [ ] M ≡ Y M × M Y M of the surjective submersion ( cp App. A), as a family G ≡ {G ( p − k ) } k ∈ ,p + of ( p − k ) -gerbes (the l -gerbes with l ∈ { , − } being identified with principal C × -bundles with a compat-ible connection ( l =
0) and connection-preserving isomorphisms between them ( l = − Y [ k + ] M ≡ Y [ k ] M × M Y M , the members of G being subject tovarious coherence constraints. Accordingly, and in keeping with the underlying (super)field-theoreticparadigm in which the (super)field theory over G / H is modelled over G, the point of departure ofthe programme advocated above is the epimorphism of Lie supergroups Y G π Y G ÐÐÐ→
G (alongside theLI primitive for the pullback of the LI gauge field H along π Y G ) returned by the integrable-extensionalgorithm described in the previous paragraph. From this point onwards, one simply turns the crankof Murray’s machine of descent and, recursively, that of de Azc´arraga’s extension procedure, insistingthat all extensions are consistently ad h -equivariant in the latter (a condition essentially built into theFDA techniques employed in the procedure in the guise of the so-called minimal subalgebra [Sul77], cp Ref. [vN83, Sec. 6]), and – in the former – that all secondary surjective submersions that arise inthe process are Lie-supergroup epimorphisms, and that all (connection-preserving) isomorphisms ofprincipal C × -bundles that mark the penultimate stage of the construction and of its sub-constructionsare Lie-supergroup isomorphisms, so that, by the end of the long day, we obtain a ‘bundle p -gerbeobject in the category of Lie supergroups’. The ensuing Cartan–Eilenberg super- p -gerbe G ( p ) stillhas to be descended to the relevant homogeneous space G / H. As demonstrated by Gaw¸edzki, Waldorfand the Author in Refs. [GSW10, GSW13, Sus12, Sus11b, Sus13], this requires that G ( p ) carry a de-scendable H-equivariant structure. One of the crucial features of the advocated geometrisation schemeis that such a structure is inscribed in the very definition of the super- p -gerbe, which lends weightto the claim to naturality of the scheme in the (super)field-theoretic context under consideration.To date, the results of the programme include an explicit construction [Sus17] of the CaE super- p -gerbes over sMink ( d, ∣ D d, ) for the GS super- p -branes with p ∈ { , , } , an extensive study [Sus19] oftheir equivariance properties inspired by the analogy with the purely Graßmann-even WZW σ -model[HM85], and an explicit construction [Sus18b] of a super-0-gerbe over s ( AdS × S ) for the Zhou su-perparticle that provides a constructive application of the principle of ˙In¨on¨u–Wigner contractibilityproposed in Ref. [Sus18a] ( cp the footnote on p. 3). The CaE super- p -gerbes (for a large class of known uper- p -brane species) were also shown [Sus19, Sus20] to carry a canonical and canonically (linearised-)supersymmetric linearised κ -symmetry-equivariant structure, in conformity with an interpretation of κ -symmetry, worked out ibid. , purely in terms of the supertarget geometry. In fact, it is from the latterinterpretation that provides a solution to the problem posed in the present Introduction, and so weconclude the section with a recapitulation of the physical idea behind it.The objective of the present study is to extract a supersymmetric target-space higher-geometricdescription of the fundamental dynamical object of the GS super- σ -model with the supertarget G / H, i.e. , of the (closed) super- p -brane trajectory, from the CaE super- p -gerbe over G associated with theGS super- ( p + ) -cocycle that determines its Wess–Zumino term – all that in the much tractable modelsetting: for the superstring ( p =
1) in the superspace sMink ( d, ∣ D d, ) ≡ sISO ( d, ∣ D d, )/ Spin ( d, ) . Inthe light of the hitherto discussion, the task boils down to identifying a (higher-)superalgebraic objectrelated to the Lie superalgebra g of the supersymmetry supergroup G ≡ sISO ( d, ∣ D d, ) that capturesa classical ( i.e. , critical) embedding of the superstring worldsheet in the supertarget and, in particular,the supersymmetry that survives such localisation. That the well-posedness of this task is non-obvious isbest illustrated by the discussion of a local (tangential) Graßmann-odd supersymmetry of the GS super- σ -model in the standard NG formulation, aka κ -symmetry, discovered by de Azc´arraga and Lukierski(for the superparticle) in Ref. [dAL83] and subsequently rediscovered and elaborated by Siegel (forthe superstring) in Refs. [Sie83, Sie84], whose existence is tied with the mechanism of restitution ofequibalance of the internal degreees of freedom of both Graßmann parities in the vacuum of the GSsuperfield theory through a removal of fermionic Goldstone modes, consistent with the structure of thesupersymmetry Lie superalgebra of the theory: The symmetry couples the metric and topological termsin the action functional ( i.e. , they are not invariant separately ), and that only for a finely tuned relativenormalisation of the two. It also bracket-generates a (super)algebra whose on-shell closure requiresincorporation of generators of diffeomorphisms of the worldsheet [McA00]. A path to target-space geometrisation of κ -symmetry and field equations of the GS super- σ -model on G / H, and so also towardsa meaningful formulation of the problem of interest, was paved in Refs. [Sus19, Sus20] where a duality –first noted in Ref. [HP86], later elaborated substantially in Ref. [GIT90] and employed in a rederivationof a variety of (super- σ -)models of charge dynamics in Refs. [McA10, Wes00, GKW06b, GKW06a]– was formalised, geometrised and exploited that exists between the original NG formulation of theGS super- σ -model and a purely topological (super)field theory, termed the Hughes–Polchinski (HP)formulation of the GS super- σ -model by the Author, with the superfield space G / H vac associated toanother reductive decomposition g = ( t ⊕ d ) ⊕ h vac , d ⊕ h vac = h with h vac = Lie ( H vac ) that encodesthe spontaneous breakdown H ↘ H vac of the ‘invisible’ gauge symmetry H of the superfield theory.The field space of the new formulation contains additional degrees of freedom, to wit, the bosonicGoldstone fields transverse to (the body of) the vacuum of the GS super- σ -model and modelled on thevector space d ≅ h / h vac . Its topologicality is reflected by the replacement of the original metric termof the NG by the pullback of an H vac -basic LI super- ( p + ) -form on G, itself (a distinguished scalarmultiple of) a volume form Vol ( t ( ) vac ) on a fixed algebraic model t ( ) vac ⊂ t ( ) of the body of the vacuum,to the worldvolume of the super- p -brane along suitably d -augmented sections of the NG formulation.On the higher-geometric side, this means that the dual HP dynamics is entirely determined by a(H vac -equivariant) CaE super- p -gerbe – the tensor product of the original super- p -gerbe for the GSsuper- ( p + ) -cocycle with the trivial one with the curving given by the volume form on t ( ) vac that weshall call, after Ref. [Sus19], the extended Hughes–Polchinski super- p -gerbe over G and denoteas ( λ ∗ p ∈ R × is the scalar mentioned earlier) ̂G ( p ) ≡ G ( p ) ⊗ I ( p ) λ ∗ p Vol ( t ( ) vac ) . (1.1)The removal of the extra Goldstone modes through the Inverse Higgs Effect of Ref. [IO75] puts us backin the original NG formulation. It is realised by imposition of a subset of superfield equations of the HPformulation that can be interpreted as geometric constraints on the tangents of the fields of the modelrestricting the latter to a superdistribution in the tangent sheaf T G of the target supermanifold G,dubbed the HP/NG correspondence superdistribution and denoted as Corr ( sB ( HP ) p,λ ∗ p ) . That all super-field equations geometrise in an analogous manner, as do the gauge-fixing conditions for the ‘invisible’local-symmetry group H vac , altogether giving rise to what was named the HP vacuum superdistribu-tion and denoted as Vac ( sB ( HP ) p,λ ∗ p ) in Ref. [Sus20], is a structural feature of the dual HP formulationthat turns out to be instrumental in resolving the above-posed problem of extraction of super- p -branedata from G ( p ) in a manner that we outline below in the closing paragraph of the Introduction. he (classical) vacuum of the GS super- σ -model in the HP formulation emerges as a sub-super-manifold within G defined as the intersection of the HP local sections of the principal H vac -bundleG Ð→ G / H vac used in the definition of the lagrangean superfield of the theory with an integral su-permanifold of Vac ( sB ( HP ) p,λ ∗ p ) . The existence of a foliation of the sections by such integral leaves callsfor involutivity of Vac ( sB ( HP ) p,λ ∗ p ) that was examined in Ref. [Sus20]. There is yet another superdistri-bution whose regular behaviour is of essence for the consistency of the entire framework, namely, thelimit of the weak derived flag (in the sense of Tanaka [Tan70]) of the projection to Vac ( sB ( HP ) p,λ ∗ p ) ofthe linear span of the set of generators of an enhanced (right) gauge supersymmetry that arises uponrestriction of the superfield of the super- σ -model to Corr ( sB ( HP ) p,λ ∗ p ) . The projection removes the obvi-ous Graßmann-even component modelled on d (reflecting the enhancement h vac ↗ h of the ‘invisible’gauge-symmetry algebra that accompanies the transition between the two formulations) and leaves uswith a superdistribution κ ( sB ( HP ) p,λ ∗ p ) that contains a generic Graßmann-odd component – the latter is the target space-geometric realisation of the κ -symmetry of the GS super- σ -model in the topolog-ical formulation, whence the name κ -symmetry superdistribution given to it in Ref. [Sus20]. Theregularity alluded to above simply means that the limit should stay within Vac ( sB ( HP ) p,λ ∗ p ) , so that itcan be given the interpretation of a gauge supersymmetry of the vacuum , engendered by κ ( sB ( HP ) p,λ ∗ p ) .This happens iff the vacuum superdistribution is involutive, in which case κ ( sB ( HP ) p,λ ∗ p ) is readily seento bracket-generate Vac ( sB ( HP ) p,λ ∗ p ) – the vacuum supermanifold becomes a single orbit of the gauge-symmetry supergroup obtained through integration of the Lie superalgebra vac ( sB ( HP ) p,λ ∗ p ) modellingthe limit. The last fact, taken in conjunction with the higher-geometric interpretation and implemen-tation of gauge symmetries worked out in Refs. [GSW10, GSW13, Sus12, Sus11b, Sus13], leads us tothe following hypothesis of Refs. [Sus20, Sus21]: Upon restriction to the vacuum, the extended HP super- p -gerbe ̂G ( p ) trivialises flatly as ̂G ( p ) ≅ I . In the present paper, we prove the hypothesis for G / H = sISO ( d, ∣ D d, )/ Spin ( d, ) ≡ sMink ( d, ∣ D d, ) and p = σ -model for the superstring (whose physicalcontent and supersymmetry, both global and local ( κ -symmetry), in the dual HP formulation is reviewedfor later reference in Secs. 2 and 3, respectively) as in Ref. [dAT89]. We do that by first lifting the CaEsuper-1-gerbe of Ref. [Sus17, Sec. 5.2], erected directly over the Lie supergroup sMink ( d, ∣ D d, ) , tothe mother Lie supergroup sISO ( d, ∣ D d, ) in Sect. 4 ( Theorem 1 ), and subsequently deriving thetrivialisation 1-isomorphism in Sec. 5.2 (
Theorem 2 ) upon demonstrating briefly the higher-geometricrealisation of global supersymmetry in Sec. 5.1. This yields the most basic solution to the extractionproblem posed at the beginning of the Introduction, which is readily seen by rewriting the trivialisation(symbolically, and for the value λ ∗ = G ( ) ↾ vacuum ≅ I − ( t ( ) vac ) ↾ vacuum , cp Eq. (1.1). The solution has an essential weakness: In consequence of the lack of an obvious Lie-supergroup structure on the vacuum, the 1-isomorphism can only be and is a non-supersymmetricone, and so it exists outside the framework systematically constructed in the first part of the paper(and introduced in the original papers). We strengthen the ‘raw’ result stated in Theorem 2 in thelast part of Sec. 5.2 by passing to the tangent sheaf of the higher-geometric object that representsthe 1-isomorphism and extracting a hierarchy of Lie superalgebras (5.6) associated with the varioussupermanifold components of that object and interrelated analogously but by Lie-superalgebra homo-morphisms (
Theorem 3 ) – this is the structure encoding the supersymmetric supergeometry of thevacuum that we have been after, and it seems appropriate to call it the sLieAlg-skeleton of thevacuum . We conclude our study with one further step in which we integrate the sLieAlg -skeleton toa hierarchy of Lie supergroups (
Theorem 5 ), whereby the sLieGrp-model of the vacuum (5.10)arises.Theorems 2–5 constitute the main results of the present study, consistent with the higher-geometricrepresentations of physical objects charged under the gauge field geometrised, and form a solid basisfor further investigation of geometrisations of gauge fields of the GS super- σ -models for super- p -braneson homogeneous spaces of Lie supergroups that we intend to take up in the future. The concept is very closely related to that of the FLsA of Ref. [GKP19]. . The geometrodynamics of the HP superstring
In this opening section, we recall the definition of the supersymmetric field theory of interest, mod-elling the propagation in Minkowskian spacetime of a loop-like distribution of charges of both Graß-mann parities in equibalance. The definition calls for a supermanifold with the Minkowskian bodyand a supersymmetric de Rham 3-cocycle field that couples to the supercharge current engendered bythe propagating loop. In our presentation, we emphasise, purposefully, the underlying Lie-supergroupstructure and the associated tangential Lie-superalgebra structure.The point of departure of our discussion is the ( d + ) -dimensional Minkowski space (for some d ∈ N × ) R d, ≡ ( R × d + , η ) , η = η ab E a ⊗ E b , ( η ab ) = diag ( − , , , . . . , ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ d times ) with its structure of an abelian Lie group determined by the binary operation (written in terms of theglobal cartesian coordinates { x a } a ∈ ,d ) ∣ m ∣ ∶ R d, × R d, Ð→ R d, ∶ (( x a ) , ( x b )) z→ ( x a + x a ) , with the commutative Lie algebra mink ( d, ) = d ⊕ a = ⟨ P a ⟩ , [ P a , P b ] = E ≡ E a ⊗ P a , E a ( x ) = d x a . To the above, there corresponds the Clifford algebraCliff ( R d, ) = ⟨ Γ a ∣ a ∈ , d ⟩ , { Γ a , Γ b } = η ab that contains the spin group Spin ( d, ) , the universal cover of the connected component SO ( d, ) ofthe identity element of the Lorentz group SO ( d, ) ≡ SO ( R d, ) of R d, , Ð→ Z / Z Ð→ Spin ( d, ) π Spin
ÐÐÐÐ→ SO ( d, ) Ð→ . We pick up a vector space S d, ≅ R × D d, that carries a Majorana-spinor realisation S ∶ Spin ( d, ) Ð→ End ( S d, ) of Spin ( d, ) , assuming the following identities to be satisfied in this realisation: the Fierz identitiesΓ aα ( β Γ a γδ ) = , α, β, γ, δ ∈ , D d, , Γ a ≡ η − ab Γ b , (2.1)and – for the corresponding charge-conjugation operator C ∈ End ( S d, ) – the symmetry relations C T = − C , ( C Γ a ) T = C Γ a ≡ Γ a which, in particular, rule out the possibilities d ∈ { , , } .Given these, we consider the associated super-Poincar´e (super)group, that is the supermanifoldsISO ( d, ∣ D d, ) = ( ̃ ISO ( d, ) ≡ R × d + ⋊ L Spin ( d, ) , O sISO ( d, ∣ D d, ) ≡ C ∞ ( ⋅ , R ) ○ pr ⊗ C ∞ ( ⋅ , R ) ○ pr ⊗ ⋀ ● S d, ) with the crossed product in the definition of its body ̃ ISO ( d, ) determined by the vector realisation L ∶ Spin ( d, ) / / π Spin % % ▲▲▲▲▲▲▲▲▲▲▲▲ End ( R d, ) SO ( d, ) +(cid:11) rrrrrrrrrrrr of the spin group, and the structure sheaf O sISO ( d, ∣ D d, ) written in terms of the structure sheaf C ∞ ( ⋅ , R ) ○ pr ⊗ C ∞ ( ⋅ , R ) ○ pr ≡ O ̃ ISO ( d, ) of the body. The binary operationm ∶ sISO ( d, ∣ D d, ) × sISO ( d, ∣ D d, ) Ð→ sISO ( d, ∣ D d, ) of the Lie-supergroup structure on sISO ( d, ∣ D d, ) is customarily described, in the S -point picture,in terms of the anticommuting generators θ α , α ∈ , D d, of ⋀ ● S d, , the global (cartesian-coordinate)generators x a , a ∈ , d of the structure sheaf of R × d + , and local (Lie-algebra) coordinates φ ab ≡ φ [ ab ] n Spin ( d, ) (their ensemble being identified with a point in the spin group by a mild abuse of thenotation) asm (( θ α , x a , φ bc ) , ( θ α , x a , φ bc )) = ( θ α + S ( φ ) αβ θ β , x a + L ( φ ) ab x b − θ Γ a S ( φ ) θ , ( φ ⋆ φ ) ab ) , (2.2)where θ Γ a S ( φ ) θ ≡ θ α C αβ Γ a βγ S ( φ ) γδ θ δ and where ⋆ represents the standard binary operation on the spin group. In this picture, it is straight-forward to write out coordinate expressions for the basis LI vector fields on the Lie supergroupsISO ( d, ∣ D d, ) , Q α ( θ, x, φ ) = S ( φ ) βα ( ⃗ ∂∂θ β + θ γ C γδ Γ a δβ ∂∂x a ) = ∶ S ( φ ) βα Q β ( θ, x ) ,P a ( θ, x, φ ) = L ( φ ) ba ∂∂x b = ∶ L ( φ ) ba P b ( θ, x ) ,J ab ( θ, x, φ ) = dd t ↾ t = φ ⋆ tφ ab , ( φ ab ) cd = δ ca δ db − δ da δ cb , spanning the tangent sheaf T sISO ( d, ∣ D d, ) of sISO ( d, ∣ D d, ) . These obey the superalgebra { Q α , Q β } = Γ aαβ P a , [ P a , P b ] = , [ Q α , P a ] = , [ J ab , Q α ] = ( Q Γ ab ) α = Γ abβα Q β , [ J ab , P c ] = η bc P a − η ac P b , [ J ab , J cd ] = η ad J bc − η ac J bd + η bc J ad − η bd J ac , expressed in terms of the antisymmetric productsΓ ab = [ Γ a , Γ b ] and called the super-Poincar´e (super)algebra and denoted as ( D = dim siso ( d, ∣ D d, ) − siso ( d, ∣ D d, ) = D d, ⊕ α = ⟨ Q α ⟩ ⊕ d ⊕ a = ⟨ P a ⟩ ⊕ d ⊕ a < b = ⟨ J ab = − J ba ⟩ ≡ D ⊕ A = ⟨ t A ⟩ . When referring symbolically to its (supercommutation) structure equations, we shall write (for homo-geneous generators t A , A ∈ , D of the respective Graßmann parities ∣ t A ∣ ≡ ∣ A ∣ ) [ t A , t B } = f CAB t C = ( − ) ∣ A ∣ ∣ B ∣+ [ t B , t A } . The latter superalgebra is the key ingredient in the alternative (and equivalent) definition of the Liesupergroup sISO ( d, ∣ D d, ) `a la Kostant that identifies the supergroup with the super-Harish–Chandrapair sISO ( d, ∣ D d, ) ≡ (̃ ISO ( d, ) , siso ( d, ∣ D d, )) , with the body Lie group ̃ ISO ( d, ) realised on the Graßmann-odd component siso ( d, ∣ D d, ) ( ) ≡ D d, ⊕ α = ⟨ Q α ⟩ of the Lie superalgebra siso ( d, ∣ D d, ) as ρ ∶ R × d + ⋊ L Spin ( d, ) Ð→ End ( siso ( d, ∣ D d, ) ( ) ) ∶ ( x, φ ) z→ S ( φ ) T ≡ ρ ( x, φ ) . The cotangent sheaf T ∗ sISO ( d, ∣ D d, ) of sISO ( d, ∣ D d, ) , dual to T sISO ( d, ∣ D d, ) (as a O sISO ( d, ∣ D d, ) -module), is globally generated by the duals of the vector fields Q α , P a , J bc , i.e. , the LI super-1-formswith the coordinate presentation q α ( θ, x, φ ) = S ( φ ) − αβ d θ β =∶ S ( φ ) − αβ q β ( θ, x ) ,p a ( θ, x, φ ) = L ( φ ) − ab ( d x b + θ Γ b d θ ) =∶ L ( φ ) − ab p b ( θ, x ) ,j ab ( θ, x, φ ) = L ( φ ) − ac d L ( φ ) cd η − db . Through the ensuing super-Maurer–Cartan equations d q α = − j ab ∧ ( Γ ab q ) α , d p a = q ∧ Γ a q − η bc j ab ∧ p c , d j ab = − η cd j ac ∧ j bd , (2.3) hey generate the Cartan–Eilenberg cochain complex of sISO ( d, ∣ D d, ) , ( Ω ● ( sISO ( d, ∣ D d, )) sISO ( d, ∣ D d, ) ≡ ⟨ q α , p a , j bc ∣ ( α, a ) ∈ , D d, × , d , b < c ∈ , d ⟩ , d ● ≡ d ) . Its cohomology, H ● dR ( sISO ( d, ∣ D d, ) , R ) sISO ( d, ∣ D d, ) ≡ CaE ● ( sISO ( d, ∣ D d, )) , the supersymmetric refinement of the de Rham cohomology of sISO ( d, ∣ D d, ) , is termed the Cartan–Eilenberg cohomology of sISO ( d, ∣ D d, ) . By the Z / Z -graded version of the classic Lie-algebraic result,it is isomorphic with the Chevalley–Eilenberg cohomology of the Lie superalgebra siso ( d, ∣ D d, ) withvalues in the trivial siso ( d, ∣ D d, ) -module R ,CaE ● ( sISO ( d, ∣ D d, )) ≅ H ● ( siso ( d, ∣ D d, ) , R ) , a fact of prime significance for the geometrisation of physically relevant Cartan–Eilenberg (super-)cocyles discussed in Sec. 4. Among nontrivial classes in CaE ● ( sISO ( d, ∣ D d, )) , we find that of the theGreen–Schwarz super-3-cocycle χ ( ) = q ∧ Γ a q ∧ p a (2.4)whose closedness follows directly from the Fierz identities (2.1).Consider, next, the two homogeneous spaces of the super-Poincar´e group associated with the respec-tive reductive decompositions of its tangent Lie superalgebra: siso ( d, ∣ D d, ) = smink ( d, ∣ D d, ) ⊕ spin ( d, ) , [ spin ( d, ) , smink ( d, ∣ D d, )] ⊂ smink ( d, ∣ D d, ) , with smink ( d, ∣ D d, ) = D d, ⊕ α = ⟨ Q α ⟩ ⊕ d ⊕ a = ⟨ P a ⟩ , spin ( d, ) = d ⊕ a < b = ⟨ J ab ⟩ , and siso ( d, ∣ D d, ) = ( smink ( d, ∣ D d, ) ⊕ d ) ⊕ spin ( d, ) vac , [ spin ( d, ) vac , smink ( d, ∣ D d, ) ⊕ d ] ⊂ smink ( d, ∣ D d, ) ⊕ d , with d = ⊕ ( a, ̂ b ) ∈ { , }× ,d ⟨ J a ̂ b ⟩ , spin ( d, ) vac = ⟨ J ⟩ ⊕ ⊕ ̂ a < ̂ b ∈ ,d ⟨ J ̂ a ̂ b ⟩ ≡ spin ( , ) ⊕ spin ( d − ) , coming with the supervector-space projections p ≡ pr ∶ smink ( d, ∣ D d, ) ⊕ spin ( d, ) Ð→ smink ( d, ∣ D d, ) ,p vac ≡ pr ∶ ( smink ( d, ∣ D d, ) ⊕ d ) ⊕ spin ( d, ) vac Ð→ smink ( d, ∣ D d, ) ⊕ d . The former is the super-Minkowski spacesMink ( d, ∣ D d, ) = sISO ( d, ∣ D d, )/ Spin ( d, ) , and the latter issISO ( d, ∣ D d, )/ Spin ( d, ) vac ≡ sMink ( d, ∣ D d, ) × Spin ( d, )/ Spin ( d, ) vac , where Spin ( d, ) vac ≡ Spin ( , ) × Spin ( d − ) . They are bases of the respective principal (super-)bundlesSpin ( d, ) / / sISO ( d, ∣ D d, ) ≡ sMink ( d, ∣ D d, ) ⋊ L,S
Spin ( d, ) π ≡ pr (cid:15) (cid:15) sMink ( d, ∣ D d, ) (2.5) nd Spin ( d, ) vac / / sISO ( d, ∣ D d, ) π vac (cid:15) (cid:15) sISO ( d, ∣ D d, )/ Spin ( d, ) vac , (2.6)on which the Lie supergroup acts (from the left) as [ ℓ ] K ∶ sISO ( d, ∣ D d, ) × sISO ( d, ∣ D d, )/ K Ð→ sISO ( d, ∣ D d, )/ Kin such a manner that [ ℓ ] K ○ ( id sISO ( d, ∣ D d, ) × π K ) = π K ○ ℓ , ℓ ≡ m , where K ∈ { Spin ( d, ) , Spin ( d, ) vac } and ( π Spin ( d, ) , π Spin ( d, ) vac ) = ( π, π vac ) . This is a simple exampleof the general situation described at length in Ref. [Kos77] and, more recently, in Ref. [FLV07]. The Liesupergroup sISO ( d, ∣ D d, ) acquires the interpretation of the supersymmetry group in this context,and its tangent Lie superalgebra becomes the supersymmetry algebra . It acts on itself also fromthe right , ℘ ≡ m ∶ sISO ( d, ∣ D d, ) × sISO ( d, ∣ D d, ) Ð→ sISO ( d, ∣ D d, ) but the latter action does not descend to the homogeneous space.The existence of the surjective submersions (in sMan ) π K can also be used to descend LI ten-sors from the mother Lie supergroup sISO ( d, ∣ D d, ) to the homogeneous space sISO ( d, ∣ D d, )/ Kalong local trivialising sections, an observation behind the long-established model-building techniqueof the so-called nonlinear realisations of (super)symmetry of Refs. [Sch67, Wei68, CWZ69, CCWZ69,SS69a, SS69b, ISS71, VA72, VA73, IK78, LR79, UZ82, IK82, SW83, FMW83, BW84, Wes00, GKW06a,McA00, McA10]. In the case of a covariant tensor T (of rank n ), for the descent to yield a globallydefined object on the homogeneous space, T must be (right-)K-basic. Denote the tangent Lie algebraof the structure group K as k ∈ { spin ( d, ) , spin ( d, ) vac } , with the understanding that k ⊂ m ⊕ k = siso ( d, ∣ D d, ) (where m ∈ { smink ( d, ∣ D d, ) , smink ( d, ∣ D d, ) ⊕ d } is the formerly indicated direct-sumcomplement of k , such that [ k , m ] ⊂ m ), and consider the induced element wise realisation of the bodyLie group ̃ ISO ( d, ) on sISO ( d, ∣ D d, ) by automorphisms in the category sMan of supermanifolds,given by ∣ ℘ ∣ ⋅ ∶ ̃ ISO ( d, ) Ð→ Aut sMan ( sISO ( d, ∣ D d, )) ∶ g z→ ℘ ○ ( id sISO ( d, ∣ D d, ) × ̂ g ) ≡ ∣ ℘ ∣ g , where ̂ g ∈ Hom sMan ( R ∣ , sISO ( d, ∣ D d, )) is the topological point in sISO ( d, ∣ D d, ) corresponding to g ∈ ̃ ISO ( d, ) , whence ∣ ℘ ∣ g ∶ sISO ( d, ∣ D d, ) ≡ sISO ( d, ∣ D d, ) × R ∣ sISO ( d, ∣ Dd, ) × ̂ g ÐÐÐÐÐÐÐÐÐÐÐ→ sISO ( d, ∣ D d, ) × sISO ( d, ∣ D d, ) m ÐÐ→ sISO ( d, ∣ D d, ) , as desired. With these in hand, we can make the concept of K-basicness precise. Thus, a covarianttensor T on sISO ( d, ∣ D d, ) is (right-)K-basic if it is k -horizontal, ∀ ( X ,X ,...,X n ) ∈ siso ( d, ∣ D d, ) × n ∶ ( ∃ i ∈ ,n ∶ X i ∈ k Ô⇒ T ( X , X , . . . , X n ) = ) , and (right-)K-invariant, ∀ ( k,X ) ∈ K × k ∶ ( ∣ ℘ ∣ ∗ k T = T ∧ − L X T = ) . It is now easy to see that the distinguished super-3-cocycle (2.4) is Spin ( d, ) -basic, and so alsoSpin ( d, ) vac -basic, as is the degenerate metric tensor ̂ η = η ab p a ⊗ p b . Indeed, the LI super-1-forms θ ζ L , ζ ∈ , dim m − m are k -horizontal by definition and transform linearly as ∣ ℘ ∣ ∗ k − θ ζ L = ρ ( k ) ζζ ′ θ ζ ′ L10 n consequence of the assumed reductivity of the decomposition siso ( d, ∣ D d, ) = m ⊕ k . Hence, it sufficesto take a linear combination of their tensor products, T = λ ζ ζ ...ζ n θ ζ L ⊗ θ ζ L ⊗ ⋯ ⊗ θ ζ n L , with components λ ζ ζ ...ζ n of a constant ρ ( K ) -invariant tensor as coefficients, ∀ k ∈ K ∶ λ ζ ζ ...ζ n ρ ( k ) ζ ζ ′ ρ ( k ) ζ ζ ′ ⋯ ρ ( k ) ζ n ζ ′ n = λ ζ ′ ζ ′ ...ζ ′ n , to obtain a K-basic tensor T . The ρ ( Spin ( d, )) -invariance of the Minkowski metric is obvious, andthat of Γ a αβ follows straightforwardly from the elementary properties of the generators of the Cliffordalgebra and of the charge-conjugation operator: S ( φ ) − Γ a S ( φ ) = L ( φ ) ab Γ b , C − S ( φ ) T C = S ( φ ) − . (2.7)An example of a Spin ( d, ) vac -basic tensor that is not Spin ( d, ) -basic is provided by the volume super-2-form on the subspace t ( ) vac ∶ = ⟨ P , P ⟩ , that is ( ǫ ab is the totally antisymmetric tensor with ǫ = ( t ( ) vac ) = p ∧ p ≡ ǫ ab p a ∧ p b . Its (right-)Spin ( d, ) vac -invariance is ensured by the fact that ρ ( Spin ( d, ) vac ) restrict to t ( ) vac as uni-modular automorphisms, ∀ φ ≡ ( φ ,φ ) ∈ Spin ( , ) × Spin ( d − ) ∶ det ( ρ ( φ ) ↾ t ( ) vac ) ≡ det ( L ( φ ) ↾ t ( ) vac ) ≡ det L ( φ ) = . Hence, in particular, there exist: a super-3-form H ( ) and a symmetric ( , ) -tensor ̂ η on sMink ( d, ∣ D d, ) ,as well as a super-3-form ̃ H ( ) , a super-2-form υ ( ) and a symmetric ( , ) -tensor ̂̃ η on sISO ( d, ∣ D d, )/ Spin ( d, ) vac such that χ ( ) = π ∗ H ( ) , ̂ η = π ∗ ̂ η ,χ ( ) = π ∗ vac ̃ H ( ) , ̂ η = π ∗ vac ̂̃ η , Vol ( t ( ) vac ) = π ∗ vac υ ( ) . Upon putting together the explicit coordinate presentations of the various LI super-1-forms involvedand identities (2.7), we readily deriveH ( ) = q ∧ Γ a q ∧ p a , ̂ η = η ab p a ⊗ p b . Inspection of the group law (2.2) reveals that the homogeneous space sMink ( d, ∣ D d, ) ⊂ sISO ( d, ∣ D d, ) is, in fact, a Lie sub-supergroup with the binary operationm ∶ sMink ( d, ∣ D d, ) × sMink ( d, ∣ D d, ) Ð→ sMink ( d, ∣ D d, ) admitting the coordinate presentationm (( θ α , x a ) , ( θ α , x a )) = ( θ α + θ α , x a + x a − θ Γ a θ ) , with the corresponding basis LI vector fields Q α ( θ, x ) = ⃗ ∂∂θ α + θ β C βγ Γ a γα ∂∂x a , P a ( θ, x ) = ∂∂x a spanning the super-minkowskian Lie superalgebra smink ( d, ∣ D d, ) = D d, ⊕ α = ⟨ Q α ⟩ ⊕ d ⊕ a = ⟨ P a ⟩ with the structure equations { Q α , Q β } = Γ aαβ P a , [ P a , P b ] = , [ Q α , P a ] = . Clearly, the super-1-forms q α and p a are their (respective) duals, and the descended Green–Schwarzsuper-3-cocycle H ( ) defines a nontrivial class inCaE ● ( sMink ( d, ∣ D d, )) ≅ H ● ( smink ( d, ∣ D d, ) , R ) . As the de Rham cohomology of sMink ( d, ∣ D d, ) is trivial, H ● dR ( sMink ( d, ∣ D d, )) = H ● dR ( Mink ( d, )) = , he Green–Schwarz super-3-cocycle admits a global primitive, albeit only a quasi-invariant one thatcan be chosen in the explicit formB ( ) ( θ, x ) = θ Γ a q ( θ, x ) ∧ p a ( θ, x ) . We shall write β ( ) ∶= π ∗ B ( ) . The hitherto discussion provides us with all the ingredients of the two formulations of the
Green–Schwarz (GS) super- σ -model of the superstring in sMink ( d, ∣ D d, ) . The first of these is the Nambu–Goto (NG) formulation in which we have the theory of (inner-Hom) functorial embeddings ξ ∈ [ Σ , sMink ( d, ∣ D d, )] ≡ Hom sMan ( Σ × − , sMink ( d, ∣ D d, )) of a closed orientable two-dimensional manifold Σ (the worldsheet) in sMink ( d, ∣ D d, ) determinedby the principle of least action applied to the Dirac–Feynman (DF) amplitude (with an obvious inter-pretation of the codomain) A ( NG ) DF ∶ [ Σ , sMink ( d, ∣ D d, )] Ð→ U ( ) ∶ ξ z→ exp [ i ̵ h ( µ ∫ Σ √ det ( ξ ∗ ̂ η ) + ∫ Σ ξ ∗ B ( ) )] , in which µ ∈ R × is a parameter whose numerical value is fixed by the correspondence with the otherformulation stated below, cp Refs. [GS84b, GS84a]. The amplitude is to be evaluated on the Graßmann-odd hyperplanes R ∣ N , N ∈ N × , whereby an N × -indexed family of supersymmetric two-dimensional fieldtheories is obtained, cp Ref. [Fre99]. The other one is the
Hughes–Polchinski (HP) formulation , firstpostulated in [HP86], developed in Refs. [GIT90], applied in Refs. [McA00, Wes00, GKW06a, GKW06b,McA10] and elaborated in [Sus19, Sus20], in which we deal with the theory of (inner-Hom) functorialembeddings ̃ ξ ∈ [ Σ , sISO ( d, ∣ D d, )/ Spin ( d, ) vac ] ≡ Hom sMan ( Σ × − , sISO ( d, ∣ D d, )/ Spin ( d, ) vac ) of the same worldsheet Σ in sISO ( d, ∣ D d, )/ Spin ( d, ) vac determined by the principle of least actionapplied to the DF amplitude A ( HP ) DF ∶ [ Σ , sISO ( d, ∣ D d, )/ Spin ( d, ) vac ] Ð→ U ( ) ∶ ̃ ξ z→ exp [ i ̵ h ∫ Σ ̃ ξ ∗ ( λ υ ( ) + p ∗ B ( ) )] , in which p ≡ pr ∶ sMink ( d, ∣ D d, ) × ( Spin ( d, )/ Spin ( d, ) vac ) Ð→ sMink ( d, ∣ D d, ) and λ ∈ R × is a parameter whose numerical value we establish through a symmetry analysis in Sec. 3.Here, it is presupposed that υ ( ) is nondegenerate on classical field configurations (termed the vacua ofthe theory). In other words, we model the body of the vacuum on t ( ) vac . In virtue of Ref. [Sus19, Prop. 5.3]( cp also Ref. [Sus20, Thm. 3.4] for a more general result), the two formulations become equivalent uponpartial reduction of the latter one through imposition of a subset of its Euler–Lagrange equations. Thisis quite nontrivial as the super- σ -model in the HP formulation is purely topological, unlike its NGcounterpart. A careful analysis of the equivalence between the two formulations sets the stage for allour subsequent higher-geometric considerations, and so we shall now spend some time studying therelevant details in the spirit of Ref. [Sus20]. In so doing, we shall refer to the two pairs: sB ( NG ) = ( sMink ( d, ∣ D d, ) , ̂ η, χ ( ) ) , and sB ( HP ) ,λ = ( sISO ( d, ∣ D d, )/ Spin ( d, ) vac , λ d Vol ( t ( ) vac ) + χ ( ) ≡ ̂ χ ( ) ( λ ) ) as the NG superbackground and
HP superbackground , respectively.The key to understanding the equivalence lies in a patchwise smooth realisation of the two homo-geneous spaces: sISO ( d, ∣ D d, )/ Spin ( d, ) vac and sISO ( d, ∣ D d, )/ Spin ( d, ) within the mother Liesupergroup sISO ( d, ∣ D d, ) by means of judiciously chosen local sections of the respective principalbundles (2.6) and (2.5), along the lines of Refs. [FLV07] and [Sus20]. Let O ∋ e Spin ( d, ) vac be an opensubset of Spin ( d, )/ Spin ( d, ) vac that supports local (normal) coordinates ( φ a ̂ b ) ∶ O ≅ ÐÐ→ V ⊂ d entred on the unital coset e Spin ( d, ) vac ( i.e. , φ a ̂ b ( e Spin ( d, ) vac ) =
0) and a local section of the prin-cipal Spin ( d, ) vac -bundle Spin ( d, ) Ð→ Spin ( d, )/ Spin ( d, ) vac (and so also its local trivialisation),and let { h i } i ∈ I ⊂ Spin ( d, ) , I ∋ h ≡ e ) be such that the translates O i = [∣ l ∣] h i ( O ) , written interms of the induced action [ l ] ∶ Spin ( d, ) × Spin ( d, )/ Spin ( d, ) vac Ð→ Spin ( d, )/ Spin ( d, ) vac of the Lie group Spin ( d, ) on the homogeneous space Spin ( d, )/ Spin ( d, ) vac , satisfying [ l ] ○ ( id Spin ( d, ) × π Spin ( d, )/ Spin ( d, ) vac ) = π Spin ( d, )/ Spin ( d, ) vac ○ l , l ≡ ⋆ , cover Spin ( d, )/ Spin ( d, ) vac , ⋃ i ∈ I O i = Spin ( d, )/ Spin ( d, ) vac . The sets U i ∶= Mink ( d, ) × O i , i ∈ I compose an open cover of the base Mink ( d, ) × Spin ( d, )/ Spin ( d, ) vac over which the principal Spin ( d, ) vac -bundle ̃ ISO ( d, ) Ð→ Mink ( d, ) × Spin ( d, )/ Spin ( d, ) vac triv-ialises. That base is the body of the base sISO ( d, ∣ D d, )/ Spin ( d, ) vac of the principal Spin ( d, ) vac -bundle (2.6), and it is over the U i that we define local sections of the latter. These we take in the(shifted-)exponential form σ vac i = ∣ ℓ ∣ h i ○ e θ α ○ pr ⊗ Q α ⋅ e x a ○ pr ⊗ P a ⋅ e φ a ̂ b ○ pr ⊗ J a ̂ b ○ [∣ ℓ ∣] h − i ∶ U vac i ≡ ( U i , O sISO ( d, ∣ D d, )/ Spin ( d, ) vac ↾ U i ) Ð→ sISO ( d, ∣ D d, ) , (2.8)defined in terms of the left action ∣ ℓ ∣ ⋅ ∶ ̃ ISO ( d, ) Ð→ Aut sMan ( sISO ( d, ∣ D d, )) ∶ g z→ m ○ (̂ g × id sISO ( d, ∣ D d, ) ) ≡ ∣ ℓ ∣ g and the corresponding induced action [∣ ℓ ∣] ⋅ ∶ ̃ ISO ( d, ) Ð→ Aut sMan ( sISO ( d, ∣ D d, )/ Spin ( d, ) vac ) ∶ g z→ [ ℓ ] ○ (̂ g × id sISO ( d, ∣ D d, )/ Spin ( d, ) vac ) ≡ [∣ ℓ ∣] g . They give rise to the
Hughes–Polchinski section Σ HP ∶ = ⊔ i ∈ I V i , V i ∶ = σ vac i ( U vac i ) . Upon choosing a tessellation △ Σ of Σ (composed of plaquettes that make up a set T ⊂ △ Σ , edgesand vertices) subordinate to { U vac i } i ∈ I , ∀ τ ∈ T ∃ i τ ∈ I ∶ ∣̃ ξ ∣( τ ) ⊂ U i τ , we may, next, rewrite the DF amplitude of the above GS super- σ -model in the HP formulation as( ̃ ξ τ ≡ ̃ ξ ↾ τ ) A ( HP ) DF [̃ ξ ] = exp ⎡⎢⎢⎢ ⎣ i ̵ h ∑ τ ∈ T ∫ τ ̃ ξ ∗ τ σ vac ∗ i τ ( λ Vol ( t ( ) vac ) + β ( ) )⎤ ⎥⎥⎥ ⎦ . With the degrees of freedom of the super- σ -model (super)field thus separated into the super-minkowskiansector ( θ α , x a ) and the spin-group sector ( φ a ̂ bi ) (we use the subscript to mark the local coordinateson O i ), the equivalence is completely straightforward to state: Upon expressing the non-dynamicalGoldstone spin-group fields ( φ a ̂ bi ) in A ( HP ) DF in terms of the remaining degrees of freedom ( θ α , x a ) asdictated by the Euler–Lagrange equations of the GS super- σ -model in the HP formulation obtained byvarying the DF amplitude for ̃ ξ in the direction of the spin-group fields ( φ a ̂ bi ) , ̃ ξ ∗ τ σ vac ∗ i τ p ̂ a = , ̂ a ∈ , d , (2.9)we recover the DF amplitude of the NG formulation for p ○ ̃ ξ , written in terms of the global coordinates ( θ α , x a ) on sMink ( d, ∣ D d, ) for a value µ ∗ ≡ µ ( λ ) of the parameter µ determined unquely by λ – this is the so-called Inverse Higgs Effect of Ref. [IO75]. It permits us to restrict our subsequentdiscussion to the HP formulation, with the understanding that conclusions pertinent to the standard The sheaf-theoretic meaning of the σ vac i was given in Ref. [Sus20, Sec. 2]. G formulation of the super- σ -model can be drawn only upon restricting the tangents of the fieldconfigurations σ vac i τ ○ ̃ ξ to the NG/HP correspondence superdistribution
Corr HP ( sB ( HP ) ,λ ) ⊂ T Σ HP , Corr HP ( sB ( HP ) ,λ ) ↾ V i = d ⋂ ̂ a = ker p ̂ a ∩ T V i . Advantages of this approach shall become clear along the way.We conclude the present field-theoretic introduction by deriving the Euler–Lagrange equations ofthe super- σ -model in the HP formulation. To this end, we write the variation V τ ∈ [ τ, T σ vac i τ ( U vac i τ )] ofthe composite embedding X τ ≡ σ vac i τ ○ ̃ ξ τ as V τ = δθ α Q α ( X τ ) + δx a P a ( X τ ) + δφ a ̂ bi τ J a ̂ b ( X τ ) + ∆ Sτ J S ( X τ ) , in which the last term ∆ Sτ J S ( X τ ) ≡ ∆ τ J ( X τ ) + ∆ ̂ a ̂ bτ J ̂ a ̂ b ( X τ ) represents spin ( d, ) vac -vertical corrections that render V τ tangent to the local section σ vac i τ ( U vac i τ ) , cp Ref. [Sus20, Prop. 3.6], and calculate, with the help of the super-Maurer–Cartan equations (2.3), − i ̵ h V ⋅ ⌟ δ log A ( HP ) DF [̃ ξ ] = ∑ τ ∈ T ∫ τ V τ ⌟ ̃ ξ ∗ τ σ vac ∗ i τ ̂ χ ( ) ( λ ) = ∑ τ ∈ T ∫ τ V τ ⌟ ̃ ξ ∗ τ σ vac ∗ i τ ( − λ ǫ ab δ ̂ c ̂ d j a ̂ c ∧ p ̂ d ∧ p b + q ∧ Γ ̂ a q ∧ p ̂ a + η ab q ∧ Γ a ( Dd, − λ Γ ) q ∧ p b ) , so that for ( δθ α , δx a ) =
0, we obtain − i ̵ h V ⋅ ⌟ δ log A ( HP ) DF [̃ ξ ] = − ∑ τ ∈ T ∫ τ λ ǫ ab δ ̂ c ̂ d δφ a ̂ ci τ ̃ ξ ∗ τ σ vac ∗ i τ ( p ̂ d ∧ p b ) . At this stage, we invoke the assumption of nondegeneracy of the volume form Vol ( t ( ) vac ) in the vacuum(which can be viewed as a condition of its partial localisation), implying that the tangent sheaf of thelatter (in sISO ( d, ∣ D d, ) ) is (locally) spanned on the vector fields W α i = ∆ βα i Q β , α ∈ , D d, , W a i = P a ↾ V i + ∆ ̂ ba i P ̂ b + ∆ b ̂ ca i J b ̂ c + ∆ Sa i J S , a ∈ { , } , written in terms of certain (even) sections ∆ βα i , ∆ ̂ ba i , ∆ b ̂ ca i and ∆ Sa i of the structure sheaf of V i tobe determined below, with the spin ( d, ) vac -vertical component correcting the one along d in such away that the sum is in T V i , cp Ref. [Sus20, Prop. 3.6]. Taking the above into account, we obtain theformerly stated Eq. (2.9), or ∆ ̂ ba i = , ( a, ̂ b ) ∈ { , } × , d , which we write as p ̂ a ≈ , ̂ a ∈ , d (2.10)henceforth. Imposition of the latter leaves us with the reduced expression for the variation − i ̵ h V ⋅ ⌟ δ log A ( HP ) DF [̃ ξ ] = η ab ∑ τ ∈ T ∫ τ V τ ⌟ ̃ ξ ∗ τ σ vac ∗ i τ ( q ∧ Γ a ( Dd, − λ Γ ) q ∧ p b ) , from which we readily extract the remaining Euler–Lagrange equations (written in the above notation) P λ q ≈ , P λ ≡ Dd, − λ Γ by setting δx a =
0. If (and only if) λ ∈ { − , } , then the operator P λ is a projector, P ± = P ± of rank rk P ± = D d, , with the additional properties P ± Γ a = Γ a ( D d, − P ± ) , P ± Γ ̂ a = Γ ̂ a P ± , C − P T ± C = D d, − P ± which leads to the emergence of a Lie sub-superalgebra {( Q P ± ) α , ( Q P ± ) β } = ( Γ a P ± ) αβ P a , [ P a , P b ] = , [( Q P ± ) α , P a ] = , J , J ] = , [ J , J ̂ a ̂ b ] = , [ J ̂ a ̂ b , J ̂ c ̂ d ] = δ ̂ a ̂ d J ̂ b ̂ c − δ ̂ a ̂ c J ̂ b ̂ d + δ ̂ b ̂ c J ̂ a ̂ d − δ ̂ b ̂ d J ̂ a ̂ c , [ J , ( Q P ± ) α ] = (( Q P ± ) Γ ) α , [ J ̂ a ̂ b , ( Q P ± ) α ] = (( Q P ± ) Γ ̂ a ̂ b ) α , [ J , P a ] = η a P − η a P , [ J ̂ a ̂ b , P a ] = siso ( d, ∣ D d, ) . The above emphasises the indispensability of the projector P ± for the consis-tency of the field theory under consideration, and so fixes the absolute value of λ , leaving us only theimmaterial choice of its sign, which we declare to be “ + ”, with P ( ) ∶ = P − and ̂ χ ( ) ≡ ̂ χ ( ) ( ) = η ab q ∧ Γ a ( D d, − P ( ) ) q ∧ p b + ǫ ab δ ̂ c ̂ d p ̂ c ∧ p a ∧ j b ̂ d + q ∧ Γ ̂ a q ∧ p ̂ a . Indeed, the projector enforces a reduction of the Graßmann-odd degrees of freedom necessary for therestoration of balance between them and their Graßmann-even counterparts in the vacuum, the latterbeing subject to the (even) Inverse Higgs Constraints (2.10). The Constraints are transmitted untothe Graßmann-odd sector via the anticommutator of supercharges in the supersymmetry superalgebra siso ( d, ∣ D d, ) , and so for the sake of a residual supersymmetry in the vacuum, we need a subspace inthe odd component siso ( d, ∣ D d, ) ( ) ⊂ siso ( d, ∣ D d, ) which the superbracket maps to the survivingGraßmann-even supersymmetries P and P . The ratioBPS ( sB ( HP ) , ) = rk P ( ) D d, ≡ goes under the name of the BPS fraction of the vacuum .Upon fixing a basis { ˘ Q α } α ∈ , Dd, in im P ( ) T ,im P ( ) T = ⟨ Q β P ( ) βα ∣ α ∈ , D d, ⟩ ≡ Dd, ⊕ α = ⟨ ˘ Q α ⟩ , we may write down (local) generators of the the vacuum superdistribution Vac ( sB ( HP ) , ) ⊂ Corr HP ( sB ( HP ) , ) ⊂ T Σ HP within the tangent sheaf T Σ HP of the HP section Σ HP that is determined by the Euler–Lagrangeequations of the super- σ -model, W α i = ˘ Q α ↾ V i , α ∈ , D d, , W a i = P a ↾ V i + ∆ b ̂ ca i J b ̂ c + ∆ Sa i J S , a ∈ { , } . In the light of the physical interpretation of the superdistribution, it is natural to demand involutivity ofthe latter, so that it defines – in virtue of the Frobenius Theorem, Ref. [CCF11, Thm. 6.2.1] – a foliationof the HP supertarget by embedded sub-supermanifolds, identified as the vacua of the supersymmetricfield theory under consideration. Transform the matrices Γ a and Γ S , commuting with P ( ) , into a basisof the Majorana-spinor module adapted to the decomposition of the dual module smink ( d, ∣ D d, ) ( ) into im P ( ) T and its direct-sum complement,im ( D d, − P ( ) ) T ≡ D d, ⊕ ̂ α = Dd, + ⟨ ̂ Q ̂ α ⟩ , whereupon they become block-diagonal, and denote the ensuing vacuum blocks asΓ a ↾ im P ( ) = ∶ ( γ aαβ ) α,β ∈ , Dd, ≡ γ a , det γ a ≠ , γ = − γ , Γ ↾ im P ( ) ≡ − Dd, , Γ ̂ a ̂ b ↾ im P ( ) = ∶ ( γ ̂ a ̂ bαβ ) α,β ∈ , Dd, ≡ γ ̂ a ̂ b . Our discussion of involutivity of Vac ( sB ( HP ) , ) begins with the inspection of the anticommutators { W α i , W β i } = γ αβ ( P − P ) ↾ V i ≡ γ αβ ( W i − W i ) + γ αβ ( ∆ ab i − ∆ ab i ) J a < b from which we read off the constraints∆ ab i = ∆ ab i ≡ ∆ abi , a < b ∈ , d , mplying W a i = P a ↾ V i + ∆ b ̂ ci J b ̂ c + ∆ Si J S . Next, we compute the commutators [ W i , W i ] = ∆ i ( W i − W i ) − ( ∆ ̂ ai + ∆ ̂ ai ) P ̂ a + ( P − P ) ⌟ d ∆ abi J a < b , and infer the constraints∆ ̂ ai = − ∆ ̂ ai ≡ ∆ ̂ ai , ̂ a ∈ , d , ( P − P ) ⌟ d ∆ abi = , a < b ∈ , d . Thus, W a i = P a ↾ V i + ∆ ̂ bi ( J ̂ b − J ̂ b ) + ∆ Si J S , with ( P − P ) ⌟ d ∆ ̂ bi = = ( P − P ) ⌟ d ∆ Si . Finally, we examine the commutators [ W a i , W α i ] = − ∆ i W α + ∆ ̂ b ̂ ci γ ̂ b < ̂ cβα W β − ˘ Q α ⌟ d ∆ bci J b < c , whereby we arrive at the constraints˘ Q α ⌟ d ∆ abi = , a < b ∈ , d . We conclude that an involutive vacuum superdistribution (of the type assumed) is spanned on fields W α i = ˘ Q α ↾ V i , α ∈ , D d, , W a i = P a ↾ V i + ∆ ̂ bi ( J ̂ b − J ̂ b ) + ∆ Si J S , with ( P − P ) ⌟ d ∆ ̂ bi = = ( P − P ) ⌟ d ∆ Si , ˘ Q α ⌟ d ∆ ̂ bi = = ˘ Q α ⌟ d ∆ Si . Note that for the special choice ∆ ̂ bi ≡ ( Ô⇒ ∆ Si ≡ ) the vacuum superdistribution is modelled on the super-minkowskian component of the Lie sub-superalgebra vac ( sB ( HP ) , ) = Dd, ⊕ α = ⟨ ˘ Q α ⟩ ⊕ ⟨ P , P ⟩ ⊕ spin ( d, ) vac (2.11)(the hidden gauge-symmetry algebra spin ( d, ) vac is realised trivially) with the superbrackets { ˘ Q α , ˘ Q β } = γ aαβ P a ≡ γ αβ ( P − P ) , [ P a , P b ] = , [ ˘ Q α , P a ] = , [ J , J ] = , [ J , J ̂ a ̂ b ] = , [ J ̂ a ̂ b , J ̂ c ̂ d ] = δ ̂ a ̂ d J ̂ b ̂ c − δ ̂ a ̂ c J ̂ b ̂ d + δ ̂ b ̂ c J ̂ a ̂ d − δ ̂ b ̂ d J ̂ a ̂ c , [ J , ˘ Q α ] = − ˘ Q α , [ J ̂ a ̂ b , ˘ Q α ] = ( ˘ Q γ ̂ a ̂ b ) α , [ J , P a ] = δ a P + δ a P , [ J ̂ a ̂ b , P a ] = vacuum algebra of the superstring in sMink ( d, ∣ D d, ) . In the next section,we shall interpret the departure from the simple model vac ( sB ( HP ) , ) in the structure of the vacuumsuperdistribution in terms of gauge symmetries of the field theory. In the meantime, we mark thepresence of the above sheaf parameters ∆ ̂ b and ∆ S (with restrictions ∆ ̂ b ↾ V i = ∆ ̂ bi and ∆ S ↾ V i = ∆ Si ,respectively) in our notation asVac ( ∆ ) ( sB ( HP ) , ) ≡ Dd, ⊕ α = ⟨ W α ≡ ˘ Q α ↾ Σ HP ⟩ ⊕ ⊕ a ∈ { , } ⟨ W a ≡ P a ↾ Σ HP + ∆ ̂ b ( J ̂ b − J ̂ b ) + ∆ S J S ⟩ . The disjoint union of integral leaves D i,υ i ⊂ V i (indices υ i from an index set Υ i enumerate the differentleaves within V i ) of the involutive vacuum superdistribution shall be denoted asΣ HPvac = ⊔ i ∈ I ⊔ υ i ∈ Υ i D i,υ i , (2.12)and termed the Hughes–Polchinski vacuum foliation . It is embedded in the HP section, which wewrite as ι vac ∶ Σ HPvac ↪ Σ HP , nd projects to the physical vacuum foliation Σ HPphys vac ≡ ⋃ i ∈ I ⋃ υ i ∈ Υ i π vac ( D i,υ i ) . An alternative interpretation of the vacuum superdistribution and the vacuum algebra shall be givenin the next section. 3.
Supersymmetries of the super- σ -model The principle of supersymmetry lies at the core of the construction of the super- σ -model. Its field-theoretic implementation has its peculiarities that we review below upon identifying the various speciesof supersymmetry present.3.1. Global supersymmetry.
The GS super- σ -model in either formulation has a built-in globalsupersymmetry realised by the respective induced actions [ ℓ ] K under which the integrand of the(metric) volume term is manifestly invariant (being defined in terms of the left invariant super-1-forms p a ), whereas that of the topological term is quasi -invariant, i.e. , invariant up to a total exteriorderivative, as demonstrated explicitly (in the S -point picture) in [ ℓ ] ∗ ( ε,y,ψ ) B ( ) ( θ, x ) = B ( ) ( θ, x ) + d ( S ( ψ ) − ε Γ a θ ( d x a + θ Γ a d θ )) , whence the said invariance of the DF amplitude for Σ closed. The global supersymmetry of the fieldtheory under consideration is reflected in the existence of a siso ( d, ∣ D d, ) -indexed family of Noetherhamiltonians { h X } X ∈ siso ( d, ∣ D d, ) on its space of states. These we derive in the NG formulation inwhich a state is represented by the Cauchy data Ψ ≡ (( θ, x ) ↾ S ≡ ξ ↾ S , P ) (a configuration ξ ↾ S ∈ [ S , sMink ( d, ∣ D d, )] and its LI kinetic momentum P ) of a vacuum localised on an equitemporalslice of the spacetime Σ which we take to be (modelled on) S ⊂ Σ. The presymplectic form of thesuper- σ -model in this formulation readsΩ σ [ Ψ ] = δϑ [ Ψ ] + ∫ S ev ∗ ξ ∗ H ( ) , where ϑ [ Ψ ] = ∫ S Vol ( S ) P a ξ ∗ p a is the canonical (kinetic-)action 1-form on the space(s) of states T ∗ [ S , sMink ( d, ∣ D d, )] , andev ∶ S × [ S , sMink ( d, ∣ D d, )] Ð→ sMink ( d, ∣ D d, ) is the evaluation mapping. The presymplectic form defines a Poisson superbracket on the space ofhamiltonians on the space of states, i.e. , on those sections h of the structure sheaf of the latter thatsatisfy the relation δh = − V h ⌟ Ω σ for some vector field V h , termed hamiltonian – the Poisson superbracket of two such sections: h and h with the respective hamiltonian vector fields V h and V h is given by [ h , h } Ω σ = V h ⌟ V h ⌟ Ω σ . In particular, upon contracting Ω σ with the covariant lift ̃ K X [ Ψ ] ≡ ∫ S Vol ( S ) K X ( ξ ) + ̃ ∆ X [ Ψ ] to T ∗ [ S , sMink ( d, ∣ D d, )] of the suitably spin ( d, ) -vertically corrected right-invariant (RI) vectorfield K X ∈ Γ ( T sMink ( d, ∣ D d, )) , X ∈ siso ( d, ∣ D d, ) on sMink ( d, ∣ D d, ) ≡ sMink ( d, ∣ D d, ) × { } ⊂ sISO ( d, ∣ D d, ) of the form K X = R X ↾ sMink ( d, ∣ D d, ) + Ξ SX J S , X ≡ X A t A ∈ siso ( d, ∣ D d, ) , expressed in terms of the sections Ξ SX of O sISO ( d, ∣ D d, ) ( sMink ( d, ∣ D d, ) of Ref. [Sus20, Prop. 5.1],we obtain the corresponding hamiltonian ̃ K X ⌟ Ω HP =∶ − δh X . Here, covariance is determined by the super-1-form ϑ and expressed by the requirement: − L ̃K X ϑ ! = he relevant basis RI vector fields are R Q α ( θ, x, ) = ⃗ ∂∂θ α − θ β C βγ Γ a γα ∂∂x a ,R P a ( θ, x, ) = ∂∂x a ,R J ab ( θ, x, ) = x c ( η cb ∂∂x a − η ca ∂∂x b ) + Γ αab β θ β ⃗ ∂∂θ α + dd t ↾ t = tφ ab ⋆ X A h A ≡ h X = ∫ S ( Vol ( S ) P a ξ ∗ ( R X ⌟ θ a L ) + ev ∗ ξ ∗ κ X ) , with κ Q α ( θ, x ) = − Γ a αβ θ β ( d x a − θ Γ a d θ ) ,κ P a ( θ, x ) = − θ Γ a d θ ,κ J ab ( θ, x ) = − x c θ Γ ab Γ c d θ . These furnish a realisation of a centrally extended supersymmetry Lie superalgebra siso ( d, ∣ D d, ) within the Poisson superalgebra of observables on the space of states of the super- σ -model, [ h B , h A } Ω HP = f CAB h C + W AB (here, the f CAB are the structure constants of sMink ( d, ∣ D d, ) ), with the components of the wrappinganomaly , W AB = − ∫ S ev ∗ ξ ∗ ( R t A ⌟ d κ t B + f CAB κ t C ) =∶ ∫ S ev ∗ w AB , given by w αβ = d ( a αβ x a ) , w ab = , w aα = d ( − a αβ θ β ) ,w ab cd = d ( − x e θ Γ ab Γ e Γ cd θ ) , w ab c = d ( θ Γ abc θ ) , (3.1) w ab α = d ( − x c ( η ca Γ b − η cb Γ a ) αβ θ β − θ Γ abc θ Γ cαβ θ β ) , cp Ref. [Sus18a]. Clearly, in the trivial super-minkowskian topology, we obtain the result W AB ≡ . Its refinement, to be considered in Sect. 4, is the first step towards geometrisation of the cohomologicalcontent of the super- σ -model.3.2. Local supersymmetry of the vacuum.
The GS super- σ -model with the physical supertargetsISO ( d, ∣ D d, )/ K realised within sISO ( d, ∣ D d, ) by means of the local sections has an implicit localsymmetry modelled on the right action of K. In particular, in the NG formulation, we have the largehidden gauge group Spin ( d, ) . Therefore, we anticipate an enhancement of the (tangential) localsymmetry in the HP formulation according to the scheme spin ( d, ) vac ↗ spin ( d, ) upon restriction to the correspondence superdistribution Corr HP ( sB ( HP ) , ) . Inspection of the expression ̂ χ ( ) ↾ Corr HP ( sB ( HP ) , ) = η ab q ∧ Γ a ( D d, − P ( ) ) q ∧ p b immediately corroborates our expectation: The vector fields T a ̂ b ∈ Γ ( T Σ HP ) , T a ̂ b ↾ V i = J a ̂ b ↾ V i + T Sa ̂ b i J S ≡ T a ̂ b i , ( a, ̂ b ) ∈ { , } × , d , with correcting sections T Sa ̂ b i ∈ O Σ HP ( V i ) uniquely determined by the condition T a ̂ b i ∈ Γ ( T V i ) , cp Ref. [Sus20, Prop. 3.6], satisfy ∀ ( a, ̂ b ) ∈ { , } × ,d ∶ T a ̂ b ⌟ ̂ χ ( ) ↾ Corr HP ( sB ( HP ) , ) = . his enhancement does not carry any physically nontrivial information as it merely reflects the residualredundancy of our realisation of the physical supertarget within the mother Lie supergroup. Accord-ingly, we are inclined to fix the hidden gauge by augmenting the set of Euler–Lagrange equationsderived formerly with j a ̂ b ≈ , ( a, ̂ b ) ∈ { , } × , d , so that altogether we arrive at the conjunction of constraints ( D d, − P ( ) ) q ≈ , p ̂ a ≈ , ̂ a ∈ , d , j b ̂ c ≈ , ( b, ̂ c ) ∈ { , } × , d as the definition of the ( hidden ) gauge-fixed vacuum superdistribution Vac hgf ( sB ( HP ) , ) ≡ Vac ( ∆ = ) ( sB ( HP ) , ) = Dd, ⊕ α = ⟨ W α ≡ ˘ Q α ↾ Σ HP ⟩ ⊕ ⊕ a ∈ { , } ⟨ W a ≡ P a ↾ Σ HP ⟩ . Clearly, the hidden gauge-symmetry (sub)distribution spin ( d, ) vac is a symmetry of the above vacuumsuperdistribution, [ spin ( d, ) vac , Vac hgf ( sB ( HP ) , )] ⊂ Vac hgf ( sB ( HP ) , ) , and so the vacuum foliation descends to the supertarget sISO ( d, ∣ D d, )/ Spin ( d, ) vac .But that is not all. Indeed, the very mechanism responsible for the restitution of supersymmetryin the vacuum gives rise to an extra and physically nontrivial local supersymmetry on restriction toCorr HP ( sB ( HP ) , ) , to wit, tangential Graßmann-odd translations along im P ( ) T , ∀ α ∈ , Dd, ∶ W α ⌟ ̂ χ ( ) ↾ Corr HP ( sB ( HP ) , ) = . Furthermore, we readily establish ( W + W ) ⌟ ̂ χ ( ) ↾ Corr HP ( sB ( HP ) , ) = , and so, altogether, we obtain the enhanced gauge-symmetry superdistribution GS ≡ GS ( sB ( HP ) , ) = Dd, ⊕ α = ⟨ W α ⟩ ⊕ ⟨W + W ⟩ ⊕ ⊕ ( a, ̂ b ) ∈ { , } × ,d ⟨T a ̂ b ⟩ ⊂ Corr HP ( sB ( HP ) , ) , modelled on the sub-space gs ≡ gs ( sB ( HP ) , ) = im P ( ) T ⊕ ⟨ P + P ⟩ ⊕ spin ( d, ) , with the component along spin ( d, ) vac realised trivially. Actually, in order to be able to interpret GS as a proper local-symmetry structure of the theory, we should demand that the limit of its weakderived flag, as introduced in Ref. [Sus20, Def. 4.9], stays in the correspondence superdistribution. Thisis, clearly, not the case for GS , and so we are led to extract from it a sub-superdistribution thatsatisfies this extra condition – in this manner, we arrive at the κ -symmetry superdistribution κ ( sB ( HP ) , ) = Dd, ⊕ α = ⟨ W α ⟩ ⊕ ⟨W + W ⟩ . The latter immediately reveals its peculiarity: The limit of its weak derived flag, κ ( sB ( HP ) , ) −∞ = Dd, ⊕ α = ⟨ W α ⟩ ⊕ ⟨W ⟩ ⊕ ⟨W ⟩ is contained not only in Corr HP ( sB ( HP ) , ) , but in the vacuum superdistribution, or, more accurately, κ ( sB ( HP ) , ) −∞ ≡ Vac hgf ( sB ( HP ) , ) i.e. , the κ -symmetry superdistribution is superbracket-generating for the tangent sheaf of the gauged-fixed HP vacuum foliation Σ HPvac , its flows enveloping the integral leaves of the latter. Hence, it makessense to think of κ ( sB ( HP ) , ) −∞ as the gauge (super)symmetry of the vacuum. The generating natureof the Graßmann-odd component κ ( sB ( HP ) , ) ( ) ≡ Dd, ⊕ α = ⟨ W α ⟩ ⊂ κ ( sB ( HP ) , ) on it its name – the square root of (the chiral half of) the vacuum – in Ref. [Sus20]. It is the HPcounterpart of the odd gauge symmetry of the super- σ -model in the NG formulation, known underthe name of κ -symmetry, that was originally found and studied by de Azc´arraga and Lukierski inRefs. [dAL82, dAL83] in the setting of the super- σ -model of superparticle dynamics, and subsequentlyrediscovered and elaborated by Siegel in Ref. [Sie83] and, in the two-dimensional setting, in Ref. [Sie84].In the present context, the Lie superalgebra vac ( sB ( HP ) , ) acquires the interpretation of the gauge-symmetry algebra of the vacuum, confirmed trivially by its inclusion in the kernel of the suitablyrestricted presymplectic form Ω HP = ∑ τ ∈ T ∫ S ∩ τ ev ∗ σ vac ∗ i τ ̂ χ ( ) of the GS super- σ -model in the HP formulation (in which the space of states is parametrised by con-figurations ̃ ξ ↾ S ). The crucial feature of the gauge supersymmetry modelled by vac ( sB ( HP ) , ) is itstarget space-geometric nature, to be contrasted with the mixed target-space/worldsheet and hencesomewhat obscure nature of its NG predecessor, cp Refs. [McA00, GKW06a] – this, in conjunctionwith the geometrisation of the superfield equations, makes the HP formulation perfectly suited for afully fledged higher-geometric analysis that we carry out in the remainder of this paper. We readily con-vince ourselves that it integrates to a Lie sub-supergroup of the supersymmetry group sISO ( d, ∣ D d, ) represented by the super-Harish–Chandra pairsISO ( d, ∣ D d, ) vac = ( Mink ( d, ) vac ⋊ Spin ( d, ) vac , vac ( sB ( HP ) , )) ⊂ sISO ( d, ∣ D d, ) with Mink ( d, ) vac = { ( x a ) ∈ Mink ( d, ) ∣ ∀ ̂ a ∈ ,d ∶ x ̂ a = } and with the action of the body Lie group on the Lie superalgebra vac ( sB ( HP ) , ) inherited fromsISO ( d, ∣ D d, ) . We shall, henceforth, refer to this Lie supergroup by the name of the κ -symmetrygroup of the superstring in sMink ( d, ∣ D d, ) . Its action on sISO ( d, ∣ D d, ) (by right translations)splits the latter into orbits – the integral leaves of the (integrable) superdistribution vac ( sB ( HP ) , ) ⊂ T sISO ( d, ∣ D d, ) , the very ones whose ‘intersections’ with the V i , i ∈ I model the vacua within themother Lie supergroup.4. A geometrisation of the superbackground – the super-1-gerbe(s)
Behind every two-dimensional (super-) σ -model, there is a 1-gerbe. A 1-gerbe is a geometrisation,proposed by Murray in Ref. [Mur96] and recalled exhaustively in Ref. [Sus17, Sec. 2] ( cp also Ref. [Hue20]for a recent rendering in the Z / Z -graded setting), of (the cohomology class of) a de Rham 3-cocyclethat couples to the charge current defined by the embedding of the worldsheet Σ in the (super)target.This higher-geometric object gives a rigorous meaning to the topological term in the DF amplitude ( cp Ref. [Gaw88]), determines a prequantisation of the field theory through cohomological transgression( ibid. ) and naturally encodes information on its (pre)quantisable global symmetries and their gauging( cp Refs. [GR03, GSW08, GSW11, GSW10, Sus12, Sus13, GSW13]). More generally, the bicategoryof 1-gerbes provides us with geometric and cohomological data necessary for a quantum-mechanicallyconsistent definition of a poly-phase σ -model in which various phases are separated by edges of aso-called defect localised on a graph in Σ, cp Refs. [FSW08, RS09, RS11], and distinguished 1-cellsof the bicategory model dualities between equivalent σ -models (with, sometimes, different targets), cp Refs. [Sus11a, Sus11b]. The Reader is urged to consult Refs. [Sus17, Sus19, Sus20] for a review ofapplications of gerbe theory in the field-theoretic setting of interest. Below, we merely recapitulate thelogic behind the construction of the (super-)1-gerbe for the GS super- σ -model and present details ofthe construction itself.A prerequisite of a meaningful geometrisation of the 3-form component of the (super)backgroundof the loop’s propagation is the identification of the cohomology relevant to the field theory in hand.The discussion carried out in the preceding sections leaves no room for doubt – the cohomology to beconsidered in the setting of the super- σ -model is the supersymmetric refinement of the standard deRham cohomology of the supertarget. Owing to the inherent noncompactness of the supersymmetrygroup sMink ( d, ∣ D d, ) (and sISO ( d, ∣ D d, ) , for that matter), the latter cohomology differs from thestandard de Rham cohomology (trivial for sMink ( d, ∣ D d, ) ), and the GS super-3-cocycle of interesthappens to define a nontrivial class in CaE ( sMink ( d, ∣ D d, )) ≡ H ( sMink ( d, ∣ D d, )) sMink ( d, ∣ D d, ) (lifting to a nontrivial class in CaE ( sISO ( d, ∣ D d, )) ). A non-pragmatic rationale for a geometrisation f the Cartan–Eilenberg cohomology of the supertarget sMink ( d, ∣ D d, ) , as stated – after Rabin andCrane, cp Refs. [RC85, Rab87] – in Ref. [Sus17, Sec. 3], is a topologisation of the said cohomology as thedual of the singular homology of an orbifold sMink ( d, ∣ D d, )/ Γ KR of the super-Minkowski space by thenatural action of the Kosteleck´y–Rabin discrete supersymmetry group Γ KR of Ref. [KR84], generatedby integer Graßmann-odd translations (in the S -point picture, and for a suitable choice of the Majoranarepresentation of Cliff ( R d, ) ). Accordingly, the GS super- σ -model ought to be interpreted as an effectivedescription of standard loop dynamics in sMink ( d, ∣ D d, )/ Γ KR . Prior to investigating the consequencesof the latter idea, we pause to decode its meaning and present a concrete realisation, in a semi-heuristicapproach in which we gloss over any ( e.g. , topological) subtleties encountered along the way. Thus, wechange the hitherto (Kostant’s) perspective on supermanifolds and present sMink ( d, ∣ D d, ) – uponinvoking [Bat80, Def.-Cor.] ( cp also Refs. [Gaw77, Bat79]) – as (a direct limit N → ∞ of) a nestedfamily, indexed by N × ∋ N , of DeWitt’s skeletons given essentially by (‘soul’) vector bundlesSkel N ( sMink ( d, ∣ D d, )) ≡ d ⊕ a = ( R ⊕ E ( N ) ⊕ k = ⋀ k R × N ) ⊕ D d, ⊕ α = E ( N − ) ⊕ l = ⋀ l + R × N Ð→ d ⊕ a = R ≡ R × d + of rank 2 N − ( d + + D d, ) − d − R × d + ≡ Mink ( d, ) , cp Ref. [DeW84, Sec. 2.1]. Practicallyspeaking, this amounts to realising the global coordinate generators ( θ α , x a ) of the structure sheaf O sMink ( d, ∣ D d, ) in the N th skeleton as (functional) linear combinations of elements of a basis { } ∪ { e i ∧ e i ∧ ⋯ ∧ e i m } i < i < ... < i m ∈ ,N, m ∈ ,N of the corresponding exterior algebra ⋀ ● R × N as θ α ( N ) = E ( N − ) ∑ l = θ αi i ...i l + e i ∧ e i ∧ ⋯ ∧ e i l + , x a ( N ) = x a + E ( N ) ∑ k = x ai i ...i k e i ∧ e i ∧ ⋯ ∧ e i k This presentation seems naturally compatible with Freed’s identification of the super- σ -model map-ping space [ Σ , sMink ( d, ∣ D d, )] as the appropriate inner-Hom functor – indeed, we may think of themorphisms from [ Σ , sMink ( d, ∣ D d, )]( R ∣ N ) as probing the N th skeleton. Now, the Rabin–Crane ar-gument at level N refers to the subgroup Γ ( N ) KR ⊂ Skel N ( sMink ( d, ∣ D d, )) of the N th skeleton (withrespect to super-minkowskian multiplication, realised in terms of the exterior product) generated byodd vectors ν α ( N ) = E ( N − ) ∑ l = n αi i ...i l + e i ∧ e i ∧ ⋯ ∧ e i l + , α ∈ , D d, with all (pure-soul) coefficients in Z ∋ n αi i ...i l + . An example of the orbifold Skel N ( sMink ( d, ∣ D d, ))/ Γ ( N ) KR whose homology is readily seen to encode the CaE cohomology of sMink ( d, ∣ D d, ) (owing to the poly-nomial character of the binary operation of this Lie supergroup in the S -point picture) was explicitlyconstructed in Ref. [RC85, App.], and the crucial observation of a generic nature is that it has compact odd (and even) fibre directions. The nested character of the of DeWitt’s skeletal presentation impliesthat the latter observation carries over to the direct limit, and so in view of our earlier remark on theinterpretation of Freed’s prescription in the present context, it becomes clear that we should allow formonodromies of the embedding fields of both parities ( i.e. , the so-called twisted sector) in the super- σ -model with the supertarget sMink ( d, ∣ D d, ) when modelling the field theory with the Rabin–Craneorbifold sMink ( d, ∣ D d, )/ Γ KR as the supertarget. Taking into account the top line in Eq. (3.1), we areled to consider a supercentral wrapping-charge extension Ð→ R d ∣ D d, Ð→ ̃ smink ( d, ∣ D d, ) Ð→ smink ( d, ∣ D d, ) Ð→ of the original supersymmetry algebra smink ( d, ∣ D d, ) with the supervector space structure ̃ smink ( d, ∣ D d, ) = ( D d, ⊕ α = ⟨̃ Q α ⟩ ⊕ d ⊕ a = ⟨̃ P a ⟩) ⊕ ( D d, ⊕ β = ⟨ Z β ⟩ ⊕ d ⊕ b = ⟨ R b ⟩) ≡ smink ( d, ∣ D d, ) ⊕ R d ∣ D d, In a suitable Majorana representation of the Clifford algebra with integer-valued matrices of the generators, cp Ref. [KR84]. nd with the Lie-superalgebra structure determined by the relations {̃ Q α , ̃ Q β } = Γ aαβ (̃ P a + η ab R b ) , [̃ P a , ̃ P b ] = , [̃ Q α , ̃ P a ] = Γ a αβ Z β , [̃ Q α , R a ] = , [̃ P a , R b ] = , {̃ Q α , Z β } = , [̃ P a , Z α ] = , [ R a , R b ] = , { Z α , Z β } = , [ R a , Z α ] = . We note, parenthetically, that considerations similar to ours were employed in Ref. [dAGIT89] in aderivation of central topological charges associated with the (even) wrapping states of the super- p -brane. It is to be emphasised, though, that neither the Rabin–Crane argument, nor the monodromy inthe Graßmann-odd directions and the attendant subtlety of the twisted sector were considered in thatwork.The Graßmann-even ( d -vector) component of the extension is trivial – it can be removed by thesimple redefinition ̃ P a z→ ̃ P a + η ab R b that leaves us with the irreducible Graßmann-odd extension Ð→ R ∣ D d, Ð→ Y smink ( d, ∣ D d, ) π Y smink ( d, ∣ Dd, ) ÐÐÐÐÐÐÐÐÐÐ→ smink ( d, ∣ D d, ) Ð→ (4.1)with the supervector-space structure Y smink ( d, ∣ D d, ) = ( D d, ⊕ α = ⟨ Y Q α ⟩ ⊕ d ⊕ a = ⟨ Y P a ⟩) ⊕ D d, ⊕ β = ⟨ Z β ⟩ ≡ smink ( d, ∣ D d, ) ⊕ R ∣ D d, and the Lie superbracket { Y Q α , Y Q β } = Γ aαβ Y P a , [ Y P a , Y P b ] = , [ Y Q α , Y P a ] = Γ a αβ Z β , { Y Q α , Z β } = , [ Y P a , Z α ] = , { Z α , Z β } = , alongside a decoupled abelian algebra R × d , ̃ smink ( d, ∣ D d, ) ≅ Y smink ( d, ∣ D d, ) ⊕ R × d . The odd extension, which is none other than the Green superalgebra of Ref. [Gre89], has an attrac-tive cohomological feature, to wit, the pullback of the nontrivial 3-cocycle H ( ) to Y smink ( d, ∣ D d, ) trivialises. Indeed, denote the super-1-forms dual to the new generators Z α as z α to obtain π ∗ Y smink ( d, ∣ D d, ) H ( ) = ̂ δz α ∧ π ∗ Y smink ( d, ∣ D d, ) q α = ̂ δ ( z α ∧ π ∗ Y smink ( d, ∣ D d, ) q α ) . This is a manifestation of a Z / Z -graded variant of the classic Lie-algebraic phenomenon: For any Lie(super)algebra g , classes in the second group H ( g , a ) of the Chevalley–Eilenberg cohomology of g with values in a trivial (super)module a are in a one-to-one correspondence with equivalence classesof (super)central extensions Ð→ a Ð→ ̃ g Ð→ g Ð→ of g by a , cp Ref. [Sus17]. Now, the supervector spaceΩ ( sMink ( d, ∣ D d, )) sMink ( d, ∣ D d, ) = D d, ⊕ α = ⟨ q α ⟩ ⊕ d ⊕ a = ⟨ p a ⟩ carries a natural structure of a smink ( d, ∣ D d, ) -module determined by the action − L ⋅ ∶ smink ( d, ∣ D d, ) × Ω ( sMink ( d, ∣ D d, )) sMink ( d, ∣ D d, ) Ð→ Ω ( sMink ( d, ∣ D d, )) sMink ( d, ∣ D d, ) ∶ ( X, ω ) z→ − L X ω , This need not be so on the level of the associated Lie supergroup, cp Ref. [CdAIPB00, Sec. 2.3.1], but we shall notpursue this point in what follows. In the present paper, the super-forms appear in a (seemingly) double rˆole: as sections of the sheaf of superdifferentialforms on the Lie supergroup (regarded as a supermanifold) and as (super-)forms on its tangent Lie superalgebra. We usethe same symbol(s) for both rˆoles, which, however, should not lead to confusion as it is always clear from the contextwhich rˆole is currently under consideration (in particular, we reserve the symbol ̂ δ for the coboundary operator of theLie-superalgebra cohomology). nd its Graßmann-odd subspace ( Ω ( sMink ( d, ∣ D d, )) sMink ( d, ∣ D d, ) ) ( ) ≡ D d, ⊕ α = ⟨ q α ⟩ is a trivial submodule. Accordingly, the GS super-3-cocycleH ( ) ≡ ( p a ∧ Γ a αβ q β ) ∧ q α (4.2)acquires the interpretation of a nontrivial ( Ω ( sMink ( d, ∣ D d, )) sMink ( d, ∣ D d, ) ) ( ) -valued super-2-cocycleon smink ( d, ∣ D d, ) , and as such it gives rise to the supercentral extension (4.1). The idea of trivi-alising the CaE super- ( p + ) -cocycles that codefine the Green–Schwarz-type super- σ -models of the(half-BPS) super-minkowskian super- p -branes through the above Lie-superalgebraic mechanism (nec-essarily stepwise for p >
1) was invented by de Azc´arraga et al. in Ref. [CdAIPB00]. Its adapta-tion to, interpretation and elaboration in the higher-geometric context under consideration consti-tutes the foundation of the geometrisation programme advanced by the Author in a series of papers[Sus17, Sus19, Sus18a, Sus18b, Sus20, Sus21] that we turn to next.The Lie-superalgebra extension (4.1) integrates to a supercentral Lie-supergroup extension Ð→ R ∣ D d, Ð→ Y sMink ( d, ∣ D d, ) Ð→ sMink ( d, ∣ D d, ) Ð→ with the supermanifold structure Y sMink ( d, ∣ D d, ) = sMink ( d, ∣ D d, ) × R ∣ D d, ∋ ( θ α , x a , ξ β ) and the Lie-supergroup structure determined by the binary operation Y m ∶ Y sMink ( d, ∣ D d, ) × Y sMink ( d, ∣ D d, ) Ð→ Y sMink ( d, ∣ D d, ) with the coordinate presentation Y m (( θ α , x a , ξ β ) , ( θ α , x a , ξ β )) = ( θ α + θ α , x a + x a − θ Γ a θ , ξ β + ξ β + Γ b βγ θ γ x b − ( θ Γ b θ ) Γ bβγ ( θ γ + θ γ )) , ensuring the desired left-invariance of the super-1-form z α ( θ, x, ξ ) = d ξ α − Γ a αβ θ β ( d x a + θ Γ a d θ ) . The corresponding basis LI vector fields are Y Q α ( θ, x, ξ ) = ⃗ ∂∂θ α + Γ aαβ θ β ∂∂x a + Γ a αβ θ β Γ aγδ θ γ ⃗ ∂∂ξ δ , Y P a ( θ, x, ξ ) = ∂∂x a + Γ a αβ θ β ⃗ ∂∂ξ α ,Z α ( θ, x, ξ ) = ⃗ ∂∂ξ α . The idea of Ref. [Sus17] was to take the epimorphism π Y sMink ( d, ∣ D d, ) ≡ pr ∶ Y sMink ( d, ∣ D d, ) ≡ sMink ( d, ∣ D d, ) × R ∣ D d, Ð→ sMink ( d, ∣ D d, ) in the category sLieGrp of Lie supergroups, together with the LI primitive Y B ( ) ∶= z α ∧ π ∗ Y sMink ( d, ∣ D d, ) q α of the CaE super-3-cocycle H ( ) on its total space Y sMink ( d, ∣ D d, ) , dY B ( ) = π ∗ Y sMink ( d, ∣ D d, ) H ( ) , as the point of departure ( i.e. , the surjective submersion and the curving, respectively) of the standardgeometrisation procedure due to Murray, and subsequently bring the procedure to completion within sLieGrp .Thus, as the next step, we consider the fibred-product Lie supergroup Y [ ] sMink ( d, ∣ D d, ) ≡ Y sMink ( d, ∣ D d, ) × sMink ( d, ∣ D d, ) Y sMink ( d, ∣ D d, ) ∶= Y sMink ( d, ∣ D d, ) π sMink ( d, ∣ Dd, ) × π sMink ( d, ∣ Dd, ) Y sMink ( d, ∣ D d, ) Note that the cartesian product is not that in sLieGrp . Our convention on fibred products in the category sMan is given in App. A. ith the binary operation inherited from the cartesian product Y sMink ( d, ∣ D d, ) × Y sMink ( d, ∣ D d, ) of Lie supergroups through restriction. It admits global coordinates Y [ ] sMink ( d, ∣ D d, ) ∋ (( θ, x, ξ ) , ( θ, x, ξ )) . On its tangent Lie superalgebra Y [ ] smink ( d, ∣ D d, ) ≡ Y smink ( d, ∣ D d, ) ⊕ smink ( d, ∣ D d, ) Y smink ( d, ∣ D d, ) = D d, ⊕ α = ⟨( Y Q α , Y Q α )⟩ ⊕ d ⊕ a = ⟨( Y P a , Y P a )⟩ ⊕ D d, ⊕ β = ⟨( Z β , )⟩ ⊕ D d, ⊕ γ = ⟨( , Z γ )⟩ , endowed with (the restriction of) the standard direct-sum superbracket, to be denoted as [ ⋅ , ⋅ } ⊕ , wefind the nontrivial super-2-cocycle F ( ) = ( pr ∗ − pr ∗ ) Y B ( ) , with the coordinate presentationF ( ) (( θ, x, ξ ) , ( θ, x, ξ )) = d θ α ∧ d ( ξ α − ξ α ) . In virtue of the formerly invoked correspondence, the super-2-cocycle determines a central extension Ð→ R Ð→ l π l ÐÐ→ Y [ ] smink ( d, ∣ D d, ) Ð→ with the supervector-space structure l ≅ Y [ ] smink ( d, ∣ D d, ) ⊕ R ∋ ( X, r ) with respect to which π l ≡ pr ∶ Y [ ] smink ( d, ∣ D d, ) ⊕ R Ð→ Y [ ] smink ( d, ∣ D d, ) , and with the Lie superbracket [( X , r ) , ( X, r )} F ( ) = ([ X , X } ⊕ , X ⌟ X ⌟ F ( ) ) , cp Ref. [Sus17, App. C]. Thus, we have l = D d, ⊕ α = ⟨ L Q α ⟩ ⊕ d ⊕ a = ⟨ L P a ⟩ ⊕ D d, ⊕ β = ⟨ L Z β ( ) ⟩ ⊕ D d, ⊕ γ = ⟨ L Z γ ( ) ⟩ ⊕ ⟨Z⟩ with the structure equations (in which we drop the subscript F ( ) on the superbrackets) { L Q α , L Q β } = Γ aαβ L P a , [ L P a , L P b ] = , [ L Q α , L P a ] = Γ a αβ ( L Z β ( ) + L Z β ( ) ) , − { L Q α , L Z β ( ) } = δ βα Z = { L Q α , L Z β ( ) } , [ L P a , L Z β ( n ) ] = , { L Z α ( m ) , L Z β ( n ) } = , [ L Q α , Z ] = , [ L P a , Z ] = , [ L Z α ( n ) , Z ] = , [ Z , Z ] = . Upon denoting the super-1-form dual to the central generator Z as ζ , we readily establish the identity ̂ δζ = π ∗ l F ( ) . As before, the Lie-superalgebra extension integrates to a central Lie-supergroup extension – this time,we obtain Ð→ C × Ð→ L Ð→ Y [ ] sMink ( d, ∣ D d, ) Ð→ with the supermanifold structure L = Y [ ] sMink ( d, ∣ D d, ) × C × ∋ (( θ α , x a , ξ β ) , ( θ α , x a , ξ β ) , z ) and the Lie-supergroup structure determined by the binary operation L m ∶ L × L Ð→ L with the coordinate presentation L m ((( θ , x , ξ , ) , ( θ , x , ξ , ) , z ) , (( θ , x , ξ , ) , ( θ , x , ξ , ) , z )) = ( Y m (( θ , x , ξ , ) , ( θ , x , ξ , )) , Y m (( θ , x , ξ , ) , ( θ , x , ξ , )) , e i θ α ( ξ , − ξ , ) α ⋅ z ⋅ z ) uch that the super-1-form ζ (( θ, x, ξ ) , ( θ, x, ξ ) , z ) = i d zz + θ α d ( ξ α − ξ α ) = ∶ i d zz + a (( θ, x, ξ ) , ( θ, x, ξ )) is LI. The extension has the structure of a (trivial) principal C × -bundle π L ≡ pr ∶ L ≡ Y [ ] sMink ( d, ∣ D d, ) × C × Ð→ Y [ ] sMink ( d, ∣ D d, ) with an obvious fibrewise action of the structure group C × and with the LI principal connectionsuper-1-form A ( ) L ≡ ζ of curvature F ( ) , d A ( ) L = π ∗ L F ( ) . Finally, we consider the fibred-product Lie supergroup Y [ ] sMink ( d, ∣ D d, ) ≡ Y sMink ( d, ∣ D d, ) × sMink ( d, ∣ D d, ) Y sMink ( d, ∣ D d, ) × sMink ( d, ∣ D d, ) Y sMink ( d, ∣ D d, ) (defined analogously to the fibred square Y [ ] sMink ( d, ∣ D d, ) ) and, over it, the pullback bundles π pr ∗ i,j L ≡ pr ∶ pr ∗ i,j L ≡ Y [ ] sMink ( d, ∣ D d, ) pr i,j × π L L Ð→ Y [ ] sMink ( d, ∣ D d, ) endowed with the Lie-supergroup structure obtained, through restriction, from the product one on Y [ ] sMink ( d, ∣ D d, ) × L ⊃ Y [ ] sMink ( d, ∣ D d, ) × pr i,j L . On the fibred cube Y [ ] sMink ( d, ∣ D d, ) , wehave coordinates (( θ, x, ξ ) , ( θ, x, ξ ) , ( θ, x, ξ )) ≡ ( y , y , y ) , and so for the pullback bundles, we obtain coordinatespr ∗ i,j L ∋ (( y , y , y ) , ( y i , y j , z )) ≡ ( y , , , ( y i,j , z )) . In these, the induced binary operation L i,j m ∶ pr ∗ i,j L × pr ∗ i,j L Ð→ pr ∗ i,j L reads L i,j m (( y , , , ( y i,j , z )) , ( y , , , ( y i,j , z ))) = (( Y m ( y , y ) , Y m ( y , y ) , Y m ( y , y )) , L m (( y i,j , z ) , ( y i,j , z ))) . Out of the first two pullback bundles, pr ∗ , L and pr ∗ , L , we form the tensor-product principal C × -bundle [ π L ○ pr ] ∶ pr ∗ , L ⊗ pr ∗ , L Ð→ Y [ ] sMink ( d, ∣ D d, ) , defined as the associated bundlepr ∗ , L ⊗ pr ∗ , L = ( pr ∗ , L pr × pr pr ∗ , L )/ C × , with the projection to the base (written out in coordinates) [ π L ○ pr ](( y , , , ( y , , )) ⊗ ( y , , , ( y , , z ))) = y , , . Here, we are quotienting out the ‘diagonal’ action λ ∶ C × × ( pr ∗ , L pr × pr pr ∗ , L ) Ð→ pr ∗ , L pr × pr pr ∗ , L with the coordinate presentation λ ( z, ( y , , , ( y , , z )) , ( y , , , ( y , , z ))) = (( y , , , ( y , , z ⋅ z )) , ( y , , , ( y , , z ⋅ z − ))) . The tensor-product bundle inherits a natural Lie-supergroup structure from (the restricted productone on) pr ∗ , L pr × pr pr ∗ , L , [ L , , m ] ∶ ( pr ∗ , L ⊗ pr ∗ , L ) × ( pr ∗ , L ⊗ pr ∗ , L ) Ð→ pr ∗ , L ⊗ pr ∗ , L , with the coordinate presentation [ L , , m ](( y , , , ( y , , )) ⊗ ( y , , , ( y , , z )) , ( y , , , ( y , , )) ⊗ ( y , , , ( y , , z ))) Formally, we perform the quotienting in the body and subsequently take the sub-sheaf composed of λ -invariantsections in the structure sheaf of pr ∗ , L × Y [ ] sMink ( d, ∣ D d, ) pr ∗ , L as the structure sheaf of the quotient supermanifold. L , m (( y , , , ( y , , )) , ( y , , , ( y , , ))) ⊗ L , m (( y , , , ( y , , z )) , ( y , , , ( y , , z ))) . At this stage, it suffices to compare the base components of the principal connection super-1-forms onpr ∗ , L ⊗ pr ∗ , L and pr ∗ , L , ( pr ∗ , a + pr ∗ , a )( y , , ) = pr ∗ , a ( y , , ) , to infer the existence of a connection-preserving isomorphism of principal C × -bundles µ L ∶ pr ∗ , L ⊗ pr ∗ , L ≅ ÐÐ→ pr ∗ , L with the coordinate presentation µ L (( y , , , ( y , , )) ⊗ ( y , , , ( y , , z ))) = ( y , , , ( y , , z )) . We reserve the suggestive (symbolic) notation µ L ≡ for an isomorphism of the above trivial form. The isomorphism satisfies the coherence (groupoid)identity pr ∗ , , µ L ○ ( pr ∗ , , µ L ⊗ id pr ∗ , L ) = pr ∗ , , µ L ○ ( id pr ∗ , L ⊗ pr ∗ , , µ L ) over the quadruple fibred product Y [ ] sMink ( d, ∣ D d, ) , the latter being equipped with the canonicalprojections pr i,j,k ∶ Y [ ] sMink ( d, ∣ D d, ) Ð→ Y [ ] sMink ( d, ∣ D d, ) , ( i, j, k ) ∈ {( , , ) , ( , , ) , ( , , ) , ( , , )} and pr m,n ∶ Y [ ] sMink ( d, ∣ D d, ) Ð→ Y [ ] sMink ( d, ∣ D d, ) , ( m, n ) ∈ {( , ) , ( , )} (definedin an obvious manner). Clearly, µ L is also a Lie-supergroup isomorphism, µ L ○ [ L , , m ] = L , m ○ ( µ L × µ L ) . The 1-gerbe G ( ) GS ∶ = ( Y sMink ( d, ∣ D d, ) , π Y sMink ( d, ∣ D d, ) , Y B ( ) , L , π L , A ( ) L , µ L ) was named the Green–Schwarz super-1-gerbe over sMink ( d, ∣ D d, ) in Ref. [Sus17, Def. 5.9]. It isan example of a Cartan–Eilenberg super-1-gerbe (over the Lie supergroup sMink ( d, ∣ D d, ) ), thatis a distinguished 1-gerbe in the category of Lie supergroups. Its existence and equivariance properties,the latter to be discussed at length in Sec. 5, are markers of a quantum-mechanical consistency of theGS super- σ -model of the superstring in sMink ( d, ∣ D d, ) . As it stands, the super-1-gerbe is naturallyassociated with the NG formulation that can be phrased in terms of the differential-geometric data ofthe homogeneous space sMink ( d, ∣ D d, ) of the mother supersymmetry group sISO ( d, ∣ D d, ) exclu-sively. Instrumental in its construction is the ‘accidental’ Lie-supergroup structure on this particularhomogeneous space. In the case of a generic homogeneous space G / H of a supersymmetry Lie super-group G associated with a Lie subgroup H ⊂ ∣ G ∣ of its body ∣ G ∣ , the geometrisation scheme exemplifiedabove, making use of the relations between the Cartan–Eilenberg cohomology of the Lie supergroupand the Chevalley–Eilenberg cohomology of its tangent Lie superalgebra and the interpretation of thedistinguished second cohomology group of the latter, has to start on G and only in the end descend toG / H. The cohomology to be geometrised under such circumstances is CaE ● ( G ) further restricted toH-basic super-forms, and – in the light of the findings of Refs. [GSW10, Sus11b, Sus12, GSW13, Sus13]– the CaE super-1-gerbe that we seek to erect over G has to carry a descendable H-equivariant struc-ture . The advantage of carrying out the geometrisation over G is that it yields a higher-geometricobject that can be restricted directly to the HP section Σ HP and used there in a rigorous study of thedual HP formulation upon incorporating a suitable correction coming from the LI volume super-2-formVol ( t ( ) vac ) . In order to attain the same goal in the super-minkowskian setting under consideration, weneed to walk the path laid out above in the reverse direction, that is we must lift the GS super-1-gerbe tosISO ( d, ∣ D d, ) . An obvious thing to do would be to pull back G ( ) GS along the canonical projection π ofEq. (2.5) – this operation was actually employed in the original studies of supersymmetry-equivarianceof the GS super-1-gerbe of the superstring in sMink ( d, ∣ D d, ) that was reported in Ref. [Sus19]. Theobvious drawback of this idea is that it depends on the supermanifold morphism π that is not aLie-supergroup homomorphism, and so the lift takes us out of the category of CaE super-1-gerbesthat seems to be the most adequate one for our physically motivated purposes. Below, we propose analternative construction that effectively circumnavigates the obstacle encoutered on our way towards This notion shall be recalled and illustrated in Sec. 5 gerbification’ of the super-minkowskian GS super- σ -model in the supergeometrically largely tractableHP formulation.The idea that we wish to pursue now consists in lifting the supercentral extension Y smink ( d, ∣ D d, ) of the super-minkowskian Lie superalgebra equivariantly to an extension of the full supersymmetry Liesuperalgebra siso ( d, ∣ D d, ) . Taking a closer look at the lift (2.4) of the GS super-3-cocycle (4.2), χ ( ) ≡ ( p a ∧ Γ a αβ q β ) ∧ q α , and the precise relation between the lifted LI super-1-forms q α in it and the q α that we previouslyidentified as the basis of the (trivial) smink ( d, ∣ D d, ) -module defining the extension, q α ( θ, x, φ ) = S ( φ ) − αβ q β ( θ, x ) , (4.3)we are readily led to postulate the lift Y siso ( d, ∣ D d, ) = ( D d, ⊕ α = ⟨ Y Q α ⟩ ⊕ d ⊕ a = ⟨ Y P a ⟩ ⊕ D d, ⊕ β = ⟨ Z β ⟩) ⊕ d ⊕ b < c = ⟨ Y J bc ⟩ ≡ Y smink ( d, ∣ D d, ) ⊕ spin ( d, ) in the form of a spin ( d, ) -module Lie superalgebra with superbrackets { Y Q α , Y Q β } = Γ aαβ Y P a , [ Y P a , Y P b ] = , [ Y Q α , Y P a ] = Γ a αβ Z β , { Y Q α , Z β } = , [ Y P a , Z α ] = , { Z α , Z β } = , [ Y J ab , Y Q α ] = Γ abβα Y Q β , [ Y J ab , Y P c ] = η bc Y P a − η ac Y P b , [ Y J ab , Z α ] = − Γ abαβ Z β , [ Y J ab , Y J cd ] = η ad Y J bc − η ac Y J bd + η bc Y J ad − η bd Y J ac , that extends the analogous structure on the equivariant lift siso ( d, ∣ D d, ) of smink ( d, ∣ D d, ) . Thus,the extra generators Z α transform under spin ( d, ) as spinors, in conformity with Eq. (4.3). The onlycomponents of the ensuing super-Jacobiator that are not trivially null ( e.g. , because of being identicalwith their un-extended counterparts) readsJac ( Y Q α , Y Q β , Y Q γ ) =
3! Γ a ( αβ Γ aγ ) δ Z δ , sJac ( Y Q α , Y P a , Y J bc ) = (( Γ a Γ bc ) ( αβ ) − η ab Γ c αβ + η ac Γ b αβ ) Z β , sJac ( Z α , Y J ab , Y J cd ) = ( Γ ab Γ cd − Γ cd Γ ab − η ad Γ bc + η ac Γ bd − η bc Γ ad + η bd Γ ac ) αβ Z β . The first of these vanishes in virtue of the Fierz identities (2.1). Next, we compute ( Γ a Γ bc ) T = Γ cb Γ a , and use it to perform the reductionΓ a Γ bc + ( Γ a Γ bc ) T − η ab Γ c + η ac Γ b = , which implies that the second component of the super-Jacobiator is zero. Finally, the third componentcan be rewritten assJac ( Z α , Y J ab , Y J cd ) ≡ − q α ( sJac ( Y Q β , Y J ab , Y J cd )) Z β = , which concludes the proof of existence of a Lie-superalgebra structure on Y siso ( d, ∣ D d, ) with thesuperbracket postulated above. Denote as π Y siso ( d, ∣ D d, ) ∶ Y siso ( d, ∣ D d, ) Ð→ siso ( d, ∣ D d, ) the Lie-superalgebra epimorphism obtained by linearly extending the assignment π Y siso ( d, ∣ D d, ) ∶ ( Y Q α , Y P a , Z β , Y J bc ) z→ ( Q α , P a , , J bc ) , and let the duals of the generators Z α be z α . We then obtain the desired identity ̂ δ ( z α ∧ π ∗ Y siso ( d, ∣ D d, ) q α ) = π ∗ Y siso ( d, ∣ D d, ) χ ( ) . Next, we readily enhance the Lie superalgebra Y siso ( d, ∣ D d, ) to a super-Harish–Chandra pair ( i.e. ,to a Lie supergroup) Y sISO ( d, ∣ D d, ) = ( ̃ ISO ( d, ) , Y siso ( d, ∣ D d, )) ith the body Lie group ̃ ISO ( d, ) realised on the Graßmann-odd component Y siso ( d, ∣ D d, ) ( ) ≡ D d, ⊕ α = ⟨ Y Q α ⟩ ⊕ D d, ⊕ β = ⟨ Z β ⟩ of Y siso ( d, ∣ D d, ) as Y ρ ∶ R × d + ⋊ L Spin ( d, ) Ð→ End ( Y siso ( d, ∣ D d, ) ( ) ) ∶ ( x, φ ) z→ S ( φ ) T ⊕ S ( φ ) − ≡ Y ρ ( x, φ ) . Thus, π Y siso ( d, ∣ D d, ) integrates to a Lie-supergroup epimorphism π Y sISO ( d, ∣ D d, ) = pr × id Spin ( d, ) ∶ Y sISO ( d, ∣ D d, ) ≡ Y sMink ( d, ∣ D d, ) ⋊ L,S,S − T Spin ( d, ) Ð→ sISO ( d, ∣ D d, ) , and we have a coordinate description of the Lie supergroup Y sMink ( d, ∣ D d, ) ⋊ L,S,S − T Spin ( d, ) ∋ ( θ α , x a , ξ β , φ bc ) in which the binary operation Y m ∶ Y sISO ( d, ∣ D d, ) × Y sISO ( d, ∣ D d, ) Ð→ Y sISO ( d, ∣ D d, ) takes the form Y m (( θ α , x a , ξ β , φ bc ) , ( θ α , x a , ξ β , φ bc )) = ( θ α + S ( φ ) αγ θ γ , x a + L ( φ ) ad x d − θ Γ a S ( φ ) θ ,ξ β + ξ γ S ( φ ) − γβ + Γ d βγ θ γ L ( φ ) de x e − ( θ Γ d S ( φ ) θ ) Γ dβγ ( θ γ + S ( φ ) γδ θ δ ) , ( φ ⋆ φ ) bc ) . The LI super-1-forms z α admit the explicit coordinate presentation z α ( θ, x, ξ, φ ) = z β ( θ, x, ξ ) S ( φ ) βα and we arrive at the anticipated identity π ∗ Y sISO ( d, ∣ D d, ) χ ( ) = d ( z α ∧ π ∗ Y sISO ( d, ∣ D d, ) q α ) , whence also the choice of the curving Y β ( ) ∶= z α ∧ π ∗ Y sISO ( d, ∣ D d, ) q α of the CaE super-1-gerbe over sISO ( d, ∣ D d, ) under reconstruction. The canonical projection π ofDiag. (2.5) lifts to the extensions as the supermanifold morphism Y π ≡ pr ∶ Y sISO ( d, ∣ D d, ) ≡ Y sMink ( d, ∣ D d, ) × Spin ( d, ) Ð→ Y sMink ( d, ∣ D d, ) with the property π ○ π Y sISO ( d, ∣ D d, ) = π Y sMink ( d, ∣ D d, ) ○ Y π , and we establish the descent relation Y β ( ) ≡ Y π ∗ Y B ( ) . On the level of the underlying supervector spaces, we have the corresponding linear maps Y p ≡ pr ∶ Y smink ( d, ∣ D d, ) ⊕ spin ( d, ) Ð→ Y smink ( d, ∣ D d, ) trivially satisfying the identity π Y smink ( d, ∣ D d, ) ○ Y p = p ○ π Y siso ( d, ∣ D d, ) , and the corresponding relation Y β ( ) ≡ Y p ∗ Y B ( ) between super-2-forms on the respective Lie superalgebras.From this point onwards, the construction proceeds along the same lines as for sMink ( d, ∣ D d, ) .Thus, we take the fibred-square Lie super group Y [ ] sISO ( d, ∣ D d, ) ≡ Y sISO ( d, ∣ D d, ) × sISO ( d, ∣ D d, ) Y sISO ( d, ∣ D d, ) ≅ Y [ ] sMink ( d, ∣ D d, ) ⋊ L,S,S − T × S − T Spin ( d, ) , endowed with the canonical projection Y [ ] π ≡ Y π × Y π ∶ Y sISO ( d, ∣ D d, ) × sISO ( d, ∣ D d, ) Y sISO ( d, ∣ D d, ) → Y sMink ( d, ∣ D d, ) × sMink ( d, ∣ D d, ) Y sMink ( d, ∣ D d, ) , and its tangent Lie superalgebra Y [ ] siso ( d, ∣ D d, ) ≡ Y siso ( d, ∣ D d, ) ⊕ siso ( d, ∣ D d, ) Y siso ( d, ∣ D d, ) = D d, ⊕ α = ⟨( Y Q α , Y Q α )⟩ ⊕ d ⊕ a = ⟨( Y P a , Y P a )⟩ ⊕ D d, ⊕ β = ⟨( Z β , )⟩ ⊕ D d, ⊕ γ = ⟨( , Z γ )⟩ ⊕ d ⊕ b < c = ⟨( Y J bc , Y J bc )⟩ ≅ Y [ ] smink ( d, ∣ D d, ) ⊕ spin ( d, ) , coming with the supervector-space projection Y [ ] p ≡ Y p ⊕ Y p ≡ pr ∶ Y [ ] smink ( d, ∣ D d, ) ⊕ spin ( d, ) Ð→ Y [ ] smink ( d, ∣ D d, ) . The nontrivial super-2-cocycle F ( ) = ( pr ∗ − pr ∗ ) Y β ( ) ≡ Y [ ] p ∗ F ( ) on Y [ ] siso ( d, ∣ D d, ) engenders a central extension Ð→ R Ð→ ̃ l π ̃ l ÐÐ→ Y [ ] siso ( d, ∣ D d, ) Ð→ (4.4)with the supervector-space structure ̃ l = ( D d, ⊕ α = ⟨ ̃ L Q α ⟩ ⊕ d ⊕ a = ⟨ ̃ L P a ⟩ ⊕ D d, ⊕ β = ⟨ ̃ L Z β ( ) ⟩ ⊕ D d, ⊕ γ = ⟨ ̃ L Z γ ( ) ⟩) ⊕ ⟨Z⟩ ⊕ d ⊕ b < c = ⟨ ̃ L J bc ⟩ ≡ l ⊕ spin ( d, ) with respect to which π ̃ l ≡ π l ⊕ id spin ( d, ) ∶ l ⊕ spin ( d, ) Ð→ Y [ ] siso ( d, ∣ D d, ) , and with the Lie superbracket { ̃ L Q α , ̃ L Q β } = Γ aαβ ̃ L P a , [ ̃ L P a , ̃ L P b ] = , [ ̃ L Q α , ̃ L P a ] = Γ a αβ ( ̃ L Z β ( ) + ̃ L Z β ( ) ) , − { ̃ L Q α , ̃ L Z β ( ) } = δ βα Z = { ̃ L Q α , ̃ L Z β ( ) } , [ ̃ L P a , ̃ L Z α ( m ) ] = , { ̃ L Z α ( m ) , ̃ L Z β ( n ) } = , [ ̃ L Q α , Z ] = , [ ̃ L P a , Z ] = , [ ̃ L Z α ( m ) , Z ] = , [ Z , Z ] = , [ ̃ L J ab , ̃ L Q α ] = Γ abβα ̃ L Q β , [ ̃ L J ab , ̃ L P c ] = η bc ̃ L P a − η ac ̃ L P b , [ ̃ L J ab , ̃ L Z α ( m ) ] = − Γ abαβ ̃ L Z β ( m ) , [ ̃ L J ab , Z ] = , [ ̃ L J ab , ̃ L J cd ] = η ad ̃ L J bc − η ac ̃ L J bd + η bc ̃ L J ad − η bd ̃ L J ac . With ζ denoting the super-1-form dual to Z and ̃ L p ≡ pr ∶ l ⊕ spin ( d, ) Ð→ l , we obtain, similarly as before, ̂ δζ = π ∗ ̃ l F ( ) ≡ ̃ L p ∗ π ∗ l F ( ) . The above Lie-superalgebra extension integrates to a central Lie-supergroup extension Ð→ C × Ð→ ̃ L π ̃ L ÐÐÐ→ Y [ ] sISO ( d, ∣ D d, ) Ð→ with the supermanifold structure ̃ L = Y [ ] sISO ( d, ∣ D d, ) × C × ≅ L × Spin ( d, ) for which π ̃ L ≡ π L × id Spin ( d, ) ∶ ̃ L Ð→ Y [ ] sMink ( d, ∣ D d, ) × Spin ( d, ) ≡ Y [ ] sISO ( d, ∣ D d, ) , and with the Lie-supergroup structure determined by the binary operation ̃ L m ∶ ̃ L × ̃ L Ð→ ̃ L with the coordinate presentation ̃ L m ((( θ , x , ξ , , φ ) , ( θ , x , ξ , , φ ) , z ) , (( θ , x , ξ , , φ ) , ( θ , x , ξ , , φ ) , z )) ( Y m (( θ , x , ξ , , φ ) , ( θ , x , ξ , , φ )) , Y m (( θ , x , ξ , , φ ) , ( θ , x , ξ , , φ )) , e i θ α ( ξ , − ξ , ) β S ( φ ) − β α ⋅ z ⋅ z ) . Its form ensures left-invariance of the super-1-form ζ (( θ, x, ξ , φ ) , ( θ, x, ξ , φ ) , z ) = i d zz + θ α d ( ξ α − ξ α ) ≡ i d zz + Y [ ] π ∗ a (( θ, x, ξ , φ ) , ( θ, x, ξ , φ )) . Writing ̃ L π ≡ pr ∶ L × Spin ( d, ) Ð→ L , with π L ○ ̃ L π = π ○ π ̃ L , we obtain ζ ≡ ̃ L π ∗ ζ . Once again, we end up with the structure of a (trivial) principal C × -bundle π ̃ L ∶ ̃ L Ð→ Y [ ] sISO ( d, ∣ D d, ) with the LI principal connection super-1-form A ( ) ̃ L ≡ ζ of curvature F ( ) , d A ( ) ̃ L = π ∗ ̃ L F ( ) . A reasoning fully analogous to the one presented in the case of L leads to the trivial groupoid structure µ ̃ L ≡ ∶ pr ∗ , ̃ L ⊗ pr ∗ , ̃ L ≅ ÐÐ→ pr ∗ , ̃ L that isomorphically maps the Lie-supergroup structure on its domain to the one on its codomain. Alsothe latter admits a Lie-superalgebraic description, which we state hereunder for later reference. Itsreconstruction starts with the self-explanatory definition of the pullback Lie superalgebraspr ∗ i,j ̃ l ≡ Y [ ] siso ( d, ∣ D d, ) pr i,j ⊕ π ̃ l ̃ l pr / / pr (cid:15) (cid:15) ̃ l π ̃ l (cid:15) (cid:15) Y [ ] siso ( d, ∣ D d, ) pr i,j / / Y [ ] siso ( d, ∣ D d, ) , ( i, j ) ∈ {( , ) , ( , ) , ( , )} , with the respective basespr ∗ , ̃ l = D d, ⊕ α = ⟨(( Y Q α , Y Q α , Y Q α ) , ̃ L Q α ) ≡ ̃ L Q ( , ) α ⟩ ⊕ d ⊕ a = ⟨(( Y P a , Y P a , Y P a ) , ̃ L P a ) ≡ ̃ L P ( , ) a ⟩ ⊕ D d, ⊕ β = ⟨(( Z β , , ) , ̃ L Z β ( ) ) ≡ ̃ L Z ( , ) β ( ) ⟩ ⊕ D d, ⊕ γ = ⟨(( , Z γ , ) , ̃ L Z γ ( ) ) ≡ ̃ L Z ( , ) γ ( ) ⟩ ⊕ D d, ⊕ δ = ⟨(( , , Z δ ) , ) ≡ ̃ L Z ( , ) δ ( ) ⟩ ⊕ ⟨(( , , ) , Z ) ≡ Z ( , ) ⟩ ⊕ d ⊕ b < c = ⟨(( Y J bc , Y J bc , Y J bc ) , ̃ L J bc ) ≡ ̃ L J ( , ) bc ⟩ , pr ∗ , ̃ l = D d, ⊕ α = ⟨(( Y Q α , Y Q α , Y Q α ) , ̃ L Q α ) ≡ ̃ L Q ( , ) α ⟩ ⊕ d ⊕ a = ⟨(( Y P a , Y P a , Y P a ) , ̃ L P a ) ≡ ̃ L P ( , ) a ⟩ ⊕ D d, ⊕ β = ⟨(( , Z β , ) , ̃ L Z β ( ) ) ≡ ̃ L Z ( , ) β ( ) ⟩ ⊕ D d, ⊕ γ = ⟨(( , , Z γ ) , ̃ L Z γ ( ) ) ≡ ̃ L Z ( , ) γ ( ) ⟩ ⊕ D d, ⊕ δ = ⟨(( Z δ , , ) , ) ≡ ̃ L Z ( , ) δ ( ) ⟩ ⊕ ⟨(( , , ) , Z ) ≡ Z ( , ) ⟩ d ⊕ b < c = ⟨(( Y J bc , Y J bc , Y J bc ) , ̃ L J bc ) ≡ ̃ L J ( , ) bc ⟩ , pr ∗ , ̃ l = D d, ⊕ α = ⟨(( Y Q α , Y Q α , Y Q α ) , ̃ L Q α ) ≡ ̃ L Q ( , ) α ⟩ ⊕ d ⊕ a = ⟨(( Y P a , Y P a , Y P a ) , ̃ L P a ) ≡ ̃ L P ( , ) a ⟩ ⊕ D d, ⊕ β = ⟨(( Z β , , ) , ̃ L Z β ( ) ) ≡ ̃ L Z ( , ) β ( ) ⟩ ⊕ D d, ⊕ δ = ⟨(( , , Z γ ) , ̃ L Z γ ( ) ) ≡ ̃ L Z ( , ) β ( ) ⟩ ⊕ D d, ⊕ δ = ⟨(( , Z δ , ) , ) ≡ ̃ L Z ( , ) β ( ) ⟩ ⊕ ⟨(( , , ) , Z ) ≡ Z ( , ) ⟩ ⊕ d ⊕ b < c = ⟨(( Y J bc , Y J bc , Y J bc ) , ̃ L J bc ) ≡ ̃ L J ( , ) bc ⟩ and the superbracket obtained from the direct-sum one on Y [ ] siso ( d, ∣ D d, ) ⊕ ̃ l through restriction.Finally, we form the ‘tensor product’ of the first two,pr ∗ , ̃ l ⊗ pr ∗ , ̃ l ≡ ( pr ∗ , ̃ l pr ⊕ pr pr ∗ , ̃ l )/ ∼ R , by identifying the generators ( Z ( , ) , ) ∼ R ( , Z ( , ) ) , so thatpr ∗ , ̃ l ⊗ pr ∗ , ̃ l = D d, ⊕ α = ⟨( ̃ L Q ( , ) α , ̃ L Q ( , ) α ) ≡ ̃ L Q ( , , ) α ⟩ ⊕ d ⊕ a = ⟨( ̃ L P ( , ) a , ̃ L P ( , ) a ) ≡ ̃ L P ( , , ) a ⟩ ⊕ D d, ⊕ β = ⟨( ̃ L Z ( , ) β ( ) , ̃ L Z ( , ) β ( ) ) ≡ ̃ L Z ( , , ) β ( , ) ⟩ ⊕ D d, ⊕ γ = ⟨( ̃ L Z ( , ) γ ( ) , ̃ L Z ( , ) γ ( ) ) ≡ ̃ L Z ( , , ) γ ( , ) ⟩ ⊕ D d, ⊕ δ = ⟨( ̃ L Z ( , ) δ ( ) , ̃ L Z ( , ) δ ( ) ) ≡ ̃ L Z ( , , ) δ ( , ) ⟩ ⊕ ⟨ [(Z ( , ) , )] ∼ R ≡ Z , , ⟩ ⊕ d ⊕ b < c = ⟨(( ̃ L J ( , ) bc , ̃ L J ( , ) bc ) ≡ ̃ L J ( , , ) bc ⟩ , with the superbracket (defined as the projection of the restricted direct-sum superbracket on pr ∗ , ̃ l ⊕ Y [ ] siso ( d, ∣ D d, ) pr ∗ , ̃ l , computed for arbitrary representatives of the equivalence classes of arguments, back to the quo-tient) { ̃ L Q ( , , ) α , ̃ L Q ( , , ) β } = Γ aαβ ̃ L P ( , , ) a , [ ̃ L P ( , , ) a , ̃ L P ( , , ) b ] = , [ ̃ L Q ( , , ) α , ̃ L P ( , , ) a ] = Γ a αβ ( ̃ L Z ( , , ) β ( , ) + ̃ L Z ( , , ) β ( , ) + ̃ L Z ( , , ) β ( , ) ) , − { ̃ L Q ( , , ) α , ̃ L Z ( , , ) β ( , ) } = δ βα Z ( , , ) = { ̃ L Q ( , , ) α , ̃ L Z ( , , ) β ( , ) } , { ̃ L Q ( , , ) α , ̃ L Z ( , , ) β ( , ) } = , [ ̃ L P ( , , ) a , ̃ L Z ( , , ) α ( m,n ) ] = , { ̃ L Z ( , , ) α ( m,n ) , ̃ L Z ( , , ) β ( r,s ) } = , [ ̃ L Q ( , , ) α , Z ( , , ) ] = , [ ̃ L P ( , , ) a , Z ( , , ) ] = , [ ̃ L Z ( , , ) α ( m,n ) , Z ( , , ) ] = , [ Z ( , , ) , Z ( , , ) ] = , [ ̃ L J ( , , ) ab , ̃ L Q ( , , ) α ] = Γ abβα ̃ L Q ( , , ) β , [ ̃ L J ( , , ) ab , ̃ L P ( , , ) c ] = η bc ̃ L P ( , , ) a − η ac ̃ L P ( , , ) b , [ ̃ L J ( , , ) ab , ̃ L Z ( , , ) α ( m,n ) ] = − Γ abαβ ̃ L Z ( , , ) β ( m,n ) , [ ̃ L J ( , , ) ab , Z ( , , ) ] = , [ ̃ L J ( , , ) ab , ̃ L J ( , , ) cd ] = η ad ̃ L J ( , , ) bc − η ac ̃ L J ( , , ) bd + η bc ̃ L J ( , , ) ad − η bd ̃ L J ( , , ) ac . Comparing the above with the superbracket of pr ∗ , ̃ l , we infer the existence of a Lie-superalgebraisomorphism µ ̃ l ∶ pr ∗ , ̃ l ⊗ pr ∗ , ̃ l ≅ ÐÐ→ pr ∗ , ̃ l iven by the unique linear extension of the assignment ( ̃ L Q ( , , ) α , ̃ L P ( , , ) a , ̃ L Z ( , , ) β ( , ) , ̃ L Z ( , , ) γ ( , ) , ̃ L Z ( , , ) δ ( , ) , Z , , , ̃ L J ( , , ) bc ) z→ ( ̃ L Q ( , ) α , ̃ L P ( , ) a , ̃ L Z ( , ) β ( ) , ̃ L Z ( , ) γ ( ) , ̃ L Z ( , ) δ ( ) , Z , , ̃ L J ( , ) bc ) . This is the Lie-superalgebraic counterpart of the groupoid structure µ ̃ L , its triviality being reflectedin the identity µ ̃ l ( Z , , ) = Z , , which we encode in the same notation: µ ̃ l ≡ as the one used for the trivial µ ̃ L .By the end of the long day, we conclude that Theorem 1.
The GS super-1-gerbe over sMink ( d, ∣ D d, ) canonically lifts ( Spin ( d, ) -equivariantly)to a CaE super-1-gerbe over sISO ( d, ∣ D d, ) . The resulting CaE super-1-gerbe ̃ G ( ) GS ∶= ( Y sISO ( d, ∣ D d, ) , π Y sISO ( d, ∣ D d, ) , Y β ( ) , ̃ L , π ̃ L , A ( ) ̃ L , µ ̃ L ) ≡ π ∗ G ( ) GS , a distinguished (Spin ( d, ) -equivariant) pullback of G ( ) GS , shall be called the lifted Green–Schwarzsuper-1-gerbe over sISO ( d, ∣ D d, ) . It constitutes the point of departure of a full-fledged ‘gerbifi-cation’ of the GS super- σ -model in the purely topological HP formulation that we shall carry out inwhat follows. Its first step consists in extending ̃ G ( ) GS by the trivial CaE super-1-gerbe I ( ) ( t ( ) vac ) ≡ ( sISO ( d, ∣ D d, ) , id sISO ( d, ∣ D d, ) , ( t ( ) vac ) , sISO ( d, ∣ D d, ) × C × , pr , pr ∗ ϑ C × , ) over the supersymmetry group sISO ( d, ∣ D d, ) associated with the LI super-2-form 2Vol ( t ( ) vac ) (fea-turing as its curving). Above, the total space sISO ( d, ∣ D d, ) × C × ∋ ( θ α , x a , z ) of the trivial principal C × -bundle pr ∶ sISO ( d, ∣ D d, ) × C × Ð→ sISO ( d, ∣ D d, ) carries the product Lie-supergroup structure and comes equipped with the trivial principal connectionsuper-1-form pr ∗ ϑ C × with the coordinate presentationpr ∗ ϑ C × (( θ, x ) , z ) ≡ ϑ C × ( z ) ∶= i d zz , manifestly LI with respect to the said Lie-supergroup structure. The tensor product ̃ G ( ) GS ⊗ I ( ) ( t ( ) vac ) = ( Y sISO ( d, ∣ D d, ) , π Y sISO ( d, ∣ D d, ) , Y β ( ) + Y π ∗ Vol ( t ( ) vac ) ≡ ̂ Y β ( ) , ̃ L , π ̃ L , A ( ) ̃ L , µ ̃ L ) ≡ ̂ G ( ) HP is also a CaE super-1-gerbe, to be referred to – after Ref. [Sus19, Def. 6.5], but taking into accountits supersymmetry established above – the extended Hughes–Polchinski super-1-gerbe over sISO ( d, ∣ D d, ) . The 1-gerbe over Σ HP with restrictions ̂ G ( ) HP ↾ V i over the components V i of thatsupermanifold shall be denoted as ̂ G ( ) Σ HP ≡ ⊔ i ∈ I ̂ G ( ) HP ↾ V i . In the remainder of the present paper, we investigate at great length structural properties of its vacuumrestriction G ( ) vac ≡ ι ∗ vac ̂ G ( ) Σ HP ≡ ̂ G ( ) Σ HP ↾ Σ HPvac =∶ ( Y Σ HPvac , π Y Σ HPvac , ̂ Y β ( ) vac , ̃ L vac , π ̃ L vac , A ( ) ̃ L vac , µ ̃ L vac ) , with view to understanding the quantum-mechanical aspect of the vacuum of the GS super- σ -modeland of its global and local supersymmetry, as encoded by the (super-)gerbe theory of the field theoryof interest. . The supersymmetry of the super-1-gerbe(s)
Prequantisable symmetries of the (super-) σ -model have specific gerbe-theoretic manifestations thatensure the existence of their consistent lift to the Hilbert space of the (super)field theory. These havebeen known for quite some time from the extensive study of the subject carried out in the non- Z / Z -graded geometric category. They fall into the two classes, mentioned previously, with a fundamentalydifferent ontological status and, accordingly, a different higher-geometric implementation, to wit, theglobal and the local symmetries that we discuss in sequence hereunder.5.1. Higher global supersymmetry.
We begin with global symmetries that set in correspondence inequivalent field configurations. In the non- Z / Z -graded setting, these are represented by families of1-gerbe 1-isomorphisms indexed by elements of the symmetry group that identify the 1-gerbe G ( ) of the σ -model as invariant under the element-wise realisation of the group, a fact established firmlyin Refs. [Sus11a, Sus11b]. More specifically, given a Lie group G of those isometries of the target M whose action on fields of the σ -model induced from its action λ ⋅ ∶ G × M Ð→ M ∶ ( g, m ) z→ λ g ( m ) on the target preserves the DF amplitude, we demand the existence of 1-isomorphismsΦ g ∶ λ ∗ g G ( ) ≅ ÐÐ→ G ( ) , g ∈ G . In the case of a homogeneous space G / K of a supersymmetry Lie supergroup G (relative to its Liesubgroup K), this simple scenario requires, in general, a straightforward sheaf-theoretic adaptationthat separately takes into account invariance under the element-wise action [∣ ℓ ∣] K ⋅ ∶ ∣ G ∣ Ð→ Aut sMan ( G / K ) ∶ g z→ [ ℓ ] K ○ (̂ g × id G / K ) of the body Lie group ∣ G ∣ , ∣ Φ ∣ g ∶ [∣ ℓ ∣] K ∗ g G ( ) ≅ ÐÐ→ G ( ) , g ∈ ∣ G ∣ , and that under the element-wise tangential action of the Lie superalgebra g of the Lie supergroup G, d Φ X ∶ − L K X G ( ) ≅ ÐÐ→ I ( ) , X ∈ g . Here, − L K X G ( ) is a super-1-gerbe obtained from G ( ) by Lie-differentiating local data of the latter inthe direction of the fundamental vector field K X for the induced action [ ℓ ] K of G on G / K, and I ( ) is the flat trivial super-1-gerbe with a null curving. In the super-minkowskian setting, we may readilyput the components of the structure sheaf of the supersymmetry supergroup sISO ( d, ∣ D d, ) of bothparities on the same footing by using the global generators θ α of the Graßmann-odd component ofthat sheaf, alongside the remaining coordinates ( x a , φ bc ) , and demand the existence of 1-isomorphismsthat we write, in a self-explanatory notation , asΦ ( ε,y,ψ ) ∶ [ ℓ ] Spin ( d, ) ∗ ( ε,y,ψ ) G ( ) GS ≅ ÐÐ→ G ( ) GS , ( ε, y, ψ ) ∈ sISO ( d, ∣ D d, ) . Actually, we shall go one step further and consider, instead, the corresponding 1-isomorphisms ̃ Φ ( ε,y,ψ ) ∶ ℓ ∗ ( ε,y,ψ ) ̃ G ( ) GS ≅ ÐÐ→ ̃ G ( ) GS , ( ε, y, ψ ) ∈ sISO ( d, ∣ D d, ) . for its lift. In so doing, we get a chance to appreciate the structural merits of the geometrisation schemeadopted in which the implementation of the global supersymmetry is seen to essentially trivialise. Thus,we take as the surjective submersion of the pullback super-1-gerbe ℓ ∗ ( ε,y,ψ ) ̃ G ( ) GS the very same one asfor ̃ G ( ) GS – that this makes sense follows from the identity π Y sISO ( d, ∣ D d, ) ○ ̂ ℓ ( ε,y,ψ ) = ℓ ( ε,y,ψ ) ○ π ℓ ∗( ε,y,ψ ) Y sISO ( d, ∣ D d, ) , π ℓ ∗( ε,y,ψ ) Y sISO ( d, ∣ D d, ) ≡ π Y sISO ( d, ∣ D d, ) , written for ̂ ℓ ( ε,y,ψ ) ≡ Y ℓ ( ε,y, ,ψ ) ≡ Y ℓ (( ε, y, , ψ ) , ⋅ ) and ensured by the equivariance of π Y sISO ( d, ∣ D d, ) .For this choice of the surjective submersion, we find ̂ ℓ ∗ ( ε,y,ψ ) Y β ( ) = Y β ( ) , That is, in the coordinate picture, in which [ ℓ ] Spin ( d, )( ε,y,ψ ) ≡ [ ℓ ] Spin ( d, ) (( ε, y, ψ ) , ⋅ ) . nd so we infer that Y β ( ) is the curving of the pullback super-1-gerbe. Continuing along these lines, wetake ̃ L as the principal C × -bundle of the pullback super-1-gerbe, a choice legitimised by the identity π ̃ L ○ ̂ ℓ [ ]( ε,y,ψ ) = Y [ ] ℓ ( ε,y, ,ψ ) ○ π ̂ ℓ × ∗( ε,y,ψ ) ̃ L , π ̂ ℓ × ∗( ε,y,ψ ) ̃ L ≡ π ̃ L in which ̂ ℓ [ ]( ε,y,ψ ) ≡ L ℓ ( ε,y, ,ψ, ) ≡ L ℓ (( ε, y, , ψ, ) , ⋅ ) . The left-invariance of A ( ) ̃ L , ̂ ℓ [ ] ∗ ( ε,y,ψ ) A ( ) ̃ L = A ( ) ̃ L , now permits us to take A ( ) ̃ L as the principal C × -connection on the pullback principal C × -bundle. Theconstruction is consistently completed by taking ̂ ℓ [ ] ∗ ( ε,y,ψ ) µ ̃ L ≡ µ ̃ L as the groupoid structure of the pullback super-1-gerbe (for an obvious definition of ̂ ℓ [ ]( ε,y,ψ ) ). Alto-gether, then, we obtain ℓ ∗ ( ε,y,ψ ) ̃ G ( ) GS ≡ ̃ G ( ) GS , whence also ̃ Φ ( ε,y,ψ ) ≡ id ̃ G ( ) GS . Given the nature of the trivial correction I ( ) ( t ( ) vac ) , we ultimately obtain 1-isomorphisms ̂ Φ ( ε,y,ψ ) ≡ id ̂ G ( ) GS ∶ ℓ ∗ ( ε,y,ψ ) ̂ G ( ) GS ≅ ÐÐ→ ̂ G ( ) GS , ( ε, y, ψ ) ∈ sISO ( d, ∣ D d, ) . This is the anticipated higher-geometric realisation of the global supersymmetry of the GS super- σ -model in the HP formulation.5.2. Higher κ -symmetry & the sLieAlg-skeleton of the vacuum. Next, we pass to local symme-tries that relate ( gauge -) equivalent field configurations, or – according to the passive interpretation ofsymmetry – different (and equivalent) coordinate descriptions of a given field configuration, and signalreducibility of the set of degrees of freedom of the field theory to those charting the space of orbits ofthe action of the gauge group. Whenever an action λ of a group G of global symmetries of a σ -modelwith the target M is rendered local, or gauged, the ensuing (gauged) field theory effectively describesa σ -model on the orbispace M / G, or actually descends to the quotient manifold if the latter exists, cp Ref. [Sus12, Sec. 8] and Ref. [GSW13, Sec. 9], which happens, e.g. , when λ is free and proper. Aquantum-mechanical consistency of the descent of the σ -model to the orbispace calls for the existenceof a G-equivariant structure on the associated 1-gerbe G ( ) (of, say, curvature curv ( G ( ) ) ∈ Z ( M ) ),to arise over the nerve N ● ( G ⋉ M ) ≡ G ● × M ⋯ d ( )● / / / / / / / / G × × M d ( )● / / / / / / G × M d ( )● / / / / M (5.1)of the action (Lie) groupoid G ⋉ M , i.e. , a simplicial manifold with face maps (written for x ∈ M, g, g k ∈ G , k ∈ , m with m ∈ N × ) d ( ) ( g, x ) = x ≡ pr ( g, x ) , d ( ) ( g, x ) = λ g ( x ) ,d ( m ) ( g m , g m − , . . . , g , x ) = ( g m − , g m − , . . . , g , x ) ,d ( m ) m ( g m , g m − , . . . , g , x ) = ( g m , g m − , . . . , g , ℓ g ( x )) ,d ( m ) i ( g m , g m − , . . . , g , x ) = ( g m , g m − , . . . , g m + − i , g m + − i ⋅ g m − i , g m − − i , . . . , g , x ) , i ∈ , m − , cp Ref. [GSW10]. The first component of the structure is a 1-isomorphismΥ ∶ d ( ) ∗ G ( ) ≅ ÐÐ→ d ( ) ∗ G ( ) ⊗ I ( ) ̺ θ L of 1-gerbes over the arrow manifold G × M of G ⋉ M , written in terms of the 2-form ̺ θ L = pr ∗ κ A ∧ pr ∗ θ A L − pr ∗ ( K A ⌟ κ B ) pr ∗ ( θ A L ∧ θ B L ) ∈ Ω ( G × M ) n whose definition K A ≡ K t A , the θ A L are components of the LI g -valued Maurer–Cartan 1-form θ L = θ A L ⊗ t A ∈ Ω ( G ) ⊗ g associated with the generators { t A } A ∈ , dim g of the Lie algebra g , and the κ A are 1-forms on M satisfying the identities K A ⌟ curv ( G ( ) ) = − d κ A , as required for G to be a symmetry of the σ -model in the first place. The second component is a2-isomorphism ( d ( ) ○ d ( ) ) ∗ G ( ) d ( )∗ Υ / / d ( )∗ Υ (cid:15) (cid:15) ( d ( ) ○ d ( ) ) ∗ G ( ) ⊗ I ( ) d ( )∗ ̺ θ L d ( )∗ Υ ⊗ id I( ) d ( )∗ ̺θ L (cid:15) (cid:15) γ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ r z ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ ( d ( ) ○ d ( ) ) ∗ G ( ) ⊗ I ( ) d ( )∗ ̺ θ L ( d ( ) ○ d ( ) ) ∗ G ( ) ⊗ I ( ) d ( )∗ ̺ θ L + d ( )∗ ̺ θ L between the 1-isomorphisms over G × × M , satisfying, over G × × M , the coherence condition d ( ) ∗ γ ● ( id ( d ( ) ○ d ( ) ) ∗ Υ ○ d ( ) ∗ γ ) = d ( ) ∗ γ ● (( d ( ) ∗ γ ⊗ id id I( )( d ( ) ○ d ( ) )∗ ̺θ L ) ○ id ( d ( ) ○ d ( ) ) ∗ Υ ) in which ○ and ● are the horizontal and vertical compositions of 1-gerbe 2-isomorphisms, respectively.Thus, altogether, a 1-gerbe with a G-equivariant structure relative to the 2-form ̺ θ L is the triple ( G ( ) , Υ , γ ) as defined and constrained by the conditions of coherence above. The principle of descent for λ suchthat M / G is a smooth manifold is now encoded in the equivalence Grb ∇ ( M / G ) ≅ Grb ∇ ( M ) G ̺ θ L ≡ (5.2)between the bicategory Grb ∇ ( M / G ) of 1-gerbes (with a connective structure) over M / G and thebicategory
Grb ∇ ( M ) G ̺ θ L ≡ of 1-gerbes (with a connective structure) over M with a G-equivariantstructure relative to ̺ θ L ≡ cp Ref. [GSW10, Thm. 5.3]. Generically, the 2-form ̺ θ L does not vanish,and then a fairly complex construction of Refs. [GSW10, Sus12, GSW13, Sus13] has to be carried outto formulate loop dynamics with the global symmetry G gauged. The construction employs a bundle P G × λ M associated with a principal G-bundle P G over Σ and endowed with the Crittenden connectioninduced from that on P G and with an action of the gauge group Γ ( Ad P G ) of global sections of theadjoint bundle Ad P G ≡ P G × Ad G. It goes well beyond the classic minimal-coupling scheme. The loopdynamics descends to M / G if the latter exists as a smooth manifold, or is taken to model it otherwise.It may also happen that the 1-gerbe of the σ -model carries a G-equivariant structure relative to thevanishing 2-form ̺ θ L ≡
0, in which case the 1-gerbe, and with it the σ -model, directly descends toresp. models loop dynamics on the orbispace M / G. This is the very special situation that we encouterbelow.Bearing in mind that the existence of a gerbe-theoretic realisation of a symmetry is requisite for itsquantum-mechanical consistency, we shall, now, put the κ -symmetry of Sec. 3.2 in the above framework.In trying to do that, though, we stumble upon a peculiarity of the symmetry that takes us out ofthe standard scheme: The symmetry is realised in its full and integrated form only after impositionof the Euler–Lagrange equations of the super- σ -model, i.e. , it is a gauge symmetry of the vacuum.Therefore, when looking for a higher-geometric signature of κ -symmetry, we should investigate thevacuum restriction G ( ) vac of the extended HP super-1-gerbe. Taking into account the higher-geometricinterpretation of gauge symmetries, we seek to establish a sISO ( d, ∣ D d, ) vac -equivariant structure onthe latter. The very definition of the κ -symmetry superdistribution (and of the limit of its weak derivedflag) makes it obvious that the structure, if present, is descendable – indeed, G ( ) vac is a flat 1-gerbe. But For a general supertarget G / H vac realised patchwise within G by means of local sections of the principal H vac -bundle G Ð → G / H vac , there is yet another problematic peculiarity that we encounter, to wit, the symmetry seems to bewell-defined ( on G) only in its infinitesimal (tangential) form due to the intrinsic ambiguities of the patchwise realisationover intersections of elements of the trivialising cover of G / H vac . hen the correspondence (5.2) in conjunction with our earlier description of the leaves of Σ HPvac as fullorbits of sISO ( d, ∣ D d, ) vac leads us to posit, as a hypothesis to be verified, the nullity of G ( ) vac , that isthe existence of a 1-isomorphism τ ∶ G ( ) vac ≅ ÐÐ→ I ( ) , in which I ( ) is to be understood as (the pullback of) the unique 1-gerbe over the 0-dimensionalorbispace of a leaf of the lifted vacuum foliation with respect to the action of the κ -symmetry group.This hypothesis was first formulated (without a mention of the lift) in Ref. [Sus20, Rem. 7.15], cp alsoRef. [Sus21, Sec. 7]. Below, we prove it directly (essentially in the de Rham cohomology) and in a moremanifestly supersymmetric procedure, suggested by the track of thought delineated in Ref. [Sus21], inwhich we stay in the tangent sheaf of a leaf of Σ HPvac and exploit the Lie-superalgebra structure on itsmodel vac ( sB ( HP ) , ) .We begin our investigation on the leaf D i,υ i ⊂ Σ HPvac ∩ V i of the vacuum superdistribution (2.12),embedded in the superdomain with the previously introduced local coordinates ( θ α , x a , φ b ̂ ci ) . We have(keeping the pullbacks by ι vac implicit to unburden the notation) ̂ χ ( ) ↾ D i,υi = , and so we pass to consider the curving of the extended HP super-1-gerbe restricted to Y D i,υ i ≡ Y sISO ( d, ∣ D d, ) ↾ D i,υi ∋ ( θ α , x a , ξ β , φ b ̂ ci ) , whereby we find ̂ Y β ( ) ( θ, x, ξ, φ i ) = d ξ α ∧ d θ α + L ( φ i ) − a L ( φ i ) − b d x a ∧ d x b + ̃ ∆ ( θ, x, φ i ) , with ̃ ∆ ( θ, x, φ i ) = L ( φ i ) − a L ( φ i ) − b ( d x a ∧ θ Γ b d θ − d x b ∧ θ Γ a d θ + θ Γ a d θ ∧ θ Γ b d θ ) − p a ( θ, x, φ i ) ∧ S ( φ i ) − θ Γ a q ( θ, x, φ i ) . However, on D i,υ i , where Γ q ↾ D i,υi = − Γ q ↾ D i,υi , we obtain the identity L ( φ i ) − a L ( φ i ) − b θ Γ a d θ ∧ θ Γ b d θ = , and so also ̃ ∆ ( θ, x, φ i ) = , whence also (for the totally skew tensor ǫ ab = − ǫ ba with ǫ = ̂ Y β ( ) ( θ, x, ξ, φ i ) = d ( θ α d ξ α + ǫ ab L ( φ i ) − ac L ( φ i ) − bd x c d x d ) . (5.3)Indeed, the identities d L ( φ i ) − ab = − j ac ( θ, x, φ i ) η cd L ( φ i ) − db that obtain on D i,υ i yield the desired result d ( ǫ ab L ( φ i ) − ac L ( φ i ) − bd x c d x d ) − L ( φ i ) − a L ( φ i ) − b d x a ∧ d x b = . Following the standard gerbe-theoretic procedure, we erect a trivial principal C × -bundle over D i,υ i , π E i,υi ≡ pr ∶ E i,υ i ≡ Y D i,υ i × C × Ð→ Y D i,υ i , and endow it with the principal connection super-1-form ( ( θ α , x a , φ b ̂ ci , ξ β , z ) ∈ Y D i,υ i × C × ) A ( ) E i,υi ( θ, x, ξ, φ i , z ) = i d zz − θ α d ξ α − ǫ ab L ( φ i ) − ac L ( φ i ) − bd x c d x d = ∶ i d zz + A i,υ i ( θ, x, ξ, φ i ) . The bundle may subsequently be pulled back to Y [ ] D i,υ i ≡ Y D i,υ i × D i,υi Y D i,υ i along the canonicalprojections pr n ∶ Y [ ] D i,υ i Ð→ Y D i,υ i , n ∈ { , } , whereupon we obtain the two (trivial) principal C × -bundles π pr ∗ n E i,υi ≡ pr ∶ pr ∗ n E i,υ i ≡ Y [ ] D i,υ i pr n × π E i,υi E i,υ i Ð→ Y [ ] D i,υ i , n ∈ { , } ith ̂ pr n ≡ pr ∶ pr ∗ n E i,υ i Ð→ E i,υ i and with the respective principal connection super-1-forms ̂ pr ∗ n A ( ) E i,υi ≡ pr ∗ A ( ) E i,υi . Next, we tensor the second of these bundles, pr ∗ E i,υ i , with the restriction of the principal C × -bundle ̃ L of the extended HP super-1-gerbe to Y [ ] D i,υ i and look for a connection-preserving principal C × -bundle isomorphism α E i,υi ∶ ̃ L ↾ Y [ ] D i,υi ⊗ pr ∗ E i,υ i ≅ ÐÐ→ pr ∗ E i,υ i . Direct comparison of the base components of the respective connection super-1-forms, ( Y [ ] π ∗ a + pr ∗ A i,υ i )(( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i )) = pr ∗ A i,υ i (( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i )) , indicates that we may take the isomorphism in the trivial form, with the coordinate presentation α E i,υi ((( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i ) , ) ⊗ (( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i , z ))) = (( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i ) , ( θ, x, ξ , φ i , z )) , or, symbolically, α E i,υi ≡ . This is automatically compatible with the (trivial) groupoid structure on (the fibres of) ̃ L ↾ Y [ ] D i,υi ,and so we conclude that the quadruple τ i,υ i = ( E i,υ i , π E i,υi , A ( ) E i,υi , α E i,υi ) defines a trivialisation τ i,υ i ∶ G ( ) vac ↾ D i,υi ≅ ÐÐ→ I ( ) ↾ D i,υi . Combining the local trivialisations over the entire vacuum foliation gives us the sought-after globaltrivialisation τ ≡ ( E , π E , A ( ) E , α E ) ∶ G ( ) vac ≅ ÐÐ→ I ( ) , τ ≡ ⊔ i ∈ I ⊔ υ i ∈ Υ i τ i,υ i . (5.4)Of course, ultimately, we want to make statements about the extended HP super-1-gerbe descended to the physical vacuum foliation Σ HPphys vac . For the results of the above analysis to descend to thehomogeneous space sISO ( d, ∣ D d, )/ Spin ( d, ) vac ⊃ Σ HPphys vac , we need essentially to equip τ with a descendable Spin ( d, ) vac -equivariant structure. Luckily, our construction provides us with such a struc-ture, namely – the trivial one. Indeed, upon invoking the relation between the local sections σ vac i ofEq. (2.8) over nonempty intersections U vac ij ≡ U vac i ∩U vac j of superdomains U vac i (with global coordinates ( θ α , x a , ξ β , φ b ̂ ci ) ) and U vac j (with global coordinates ( θ α , x a , ξ β , φ b ̂ cj ) ), σ vac j ↾ U vac ij = ∣ ℘ ∣ h ij ○ pr ( σ vac i ↾ U vac ij ) , expressed in terms of the transition maps h ij ∶ O ij Ð→ Spin ( d, ) vac of the principal Spin ( d, ) vac -bundle Spin ( d, ) Ð→ Spin ( d, )/ Spin ( d, ) vac (inherited by that of Eq. (2.6)), we readily verify thedesired gluing property for the base components of the relevant restrictions A ( ) E i,υi and A ( ) E j,υj of theprincipal C × -connection A ( ) E on E ,A j,υ j ( θ, x, ξ, φ j ) = A i,υ i ( θ, x, ξ, φ i ) , which follows from the block-diagonal structure of the matrix L ( h ) for h ∈ Spin ( d, ) vac with respectto the decomposition mink ( d, ) = ⟨ P , P ⟩ ⊕ ( ⊕ d ̂ a = ⟨ P ̂ a ⟩) . We conclude that Theorem 2.
The super-1-gerbe of the GS super- σ -model in the HP formulation descended from theextended HP super-1-gerbe to the supertarget sISO ( d, ∣ D d, )/ Spin ( d, ) vac trivialises upon restrictionto the vacuum of the (super)field theory. he main principle underlying the scheme of geometrisation proposed in Ref. [Sus17] (and recalled inSec. 4) is the invariance of all structures under consideration with respect to the global supersymmetry enforced through restriction of the standard constructions (of 1-gerbes, their 1- and 2-isomorphisms) dueto Murray et al. to the category of Lie supergroups. There seems to be no obvious way of implementingthis principle in the last construction that leads up to Theorem 2. – quite simply because there is nonatural Lie-supergroup structure on the vacuum of the super- σ -model. Rather than trying to save theday, at least partially, by imposing invariance with respect to a residual global supersymmetry of thevacuum (an idea that we leave for a future study), we rectify the present situation by passing to thetangent sheaves of the various supermanifolds entering the definition of the vacuum restriction G ( ) vac of the extended HP super-1-gerbe of the superstring and read off the Lie-superalgebraic trace of thetrivialisation (5.4).The point of departure of our analysis is the concise restatement of a faithful Lie-superalgebraicmodel of the composite diagram (cid:15) (cid:15) s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ µ ̃ L ≡ ∶ pr ∗ , ̃ L ⊗ pr ∗ , ̃ L ≅ Ð→ pr ∗ , ̃ L (cid:15) (cid:15) C × / / ̃ L , A ( ) ̃ L π ̃ L (cid:15) (cid:15) R ∣ D d, (cid:15) (cid:15) Y [ ] sISO ( d, ∣ D d, ) pr , - - pr , / / pr , Y [ ] sISO ( d, ∣ D d, ) pr / / pr / / (cid:15) (cid:15) Y sISO ( d, ∣ D d, ) , ̂ Y β ( ) π Y sISO ( d, ∣ Dd, ) (cid:15) (cid:15) sISO ( d, ∣ D d, ) , ̂ χ ( ) o o in sMan (actually, in sLieGrp ), decorated with the relevant CaE data (as well as indicators of thevarious supercentral extensions), in the form (cid:15) (cid:15) s s ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ µ ̃ l ≡ ∶ pr ∗ , ̃ l ⊗ pr ∗ , ̃ l ≅ Ð→ pr ∗ , ̃ l (cid:15) (cid:15) R / / ̃ l , ζ π ̃ l (cid:15) (cid:15) R ∣ D d, (cid:15) (cid:15) Y [ ] siso ( d, ∣ D d, ) pr , - - pr , / / pr , Y [ ] siso ( d, ∣ D d, ) pr / / pr / / (cid:15) (cid:15) Y siso ( d, ∣ D d, ) , ̂ Y β ( ) π Y siso ( d, ∣ Dd, ) (cid:15) (cid:15) siso ( d, ∣ D d, ) , ̂ χ ( ) o o , Cp Ref. [Sus20, Prop. 5.5]. o be referred to as the sLieAlg-skeleton of the supermanifold diagram. Now, given the supermanifolddiagram describing the trivialisation of the extended HP super-1-gerbe over the vacuum foliation, (cid:15) (cid:15) α E ≡ ∶ ̃ L vac ⊗ pr ∗ E ≅ ÐÐ→ pr ∗ E (cid:15) (cid:15) C × / / E , A ( ) E π E (cid:15) (cid:15) Y [ ] Σ HPvac pr / / pr / / Y Σ HPvac , ̂ Y β ( ) vac π Y ΣHPvac (cid:15) (cid:15) / / Σ HPvac , ̂ χ ( ) vac ≡ , (5.5)and the Lie-superalgebra monomorphism Ð→ vac ( sB ( HP ) , ) vac ÐÐÐ→ siso ( d, ∣ D d, ) at its root, regarded as a sLieAlg -skeleton of the embeddingΣ HPvac ⊃ D i,υ i ↪ sISO ( d, ∣ D d, ) , we may enquire as to the existence of a consistent extension & & ▲▲▲▲▲▲▲▲▲▲▲▲ y y ssssssssssss R ∣ ∆ d, (cid:15) (cid:15) (cid:31) (cid:127) / / R ∣ D d, (cid:15) (cid:15) Y vac ( sB ( HP ) , ) (cid:31) (cid:127) Y vac / / π Y vac ( sB ( HP ) , ) (cid:15) (cid:15) Y siso ( d, ∣ D d, ) π Y siso ( d, ∣ Dd, ) (cid:15) (cid:15) vac ( sB ( HP ) , ) $ $ ❏❏❏❏❏❏❏❏❏❏❏❏ (cid:31) (cid:127) vac / / siso ( d, ∣ D d, ) y y sssssssssssss of the latter, written for some sub-superspace R ∣ ∆ d, of R ∣ D d, (with ∆ d, ≤ D d, ) and for a Liesuperalgebra Y vac ( sB ( HP ) , ) to be established together with a Lie-superalgebra monomorphism Y vac and the extension π Y vac ( sB ( HP ) , ) , and such that there exists a sLieAlg -skeleton of Diag. (5.5) of the orm (cid:15) (cid:15) α e ≡ ∶ Y [ ] ∗ vac ̃ l ⊗ pr ∗ e ≅ ÐÐ→ pr ∗ e (cid:15) (cid:15) R / / e , ˘ ζ E π e (cid:15) (cid:15) Y [ ] vac ( sB ( HP ) , ) pr / / pr / / Y vac ( sB ( HP ) , ) , ̂ Y β ( ) vac π Y vac ( sB ( HP ) , ) (cid:15) (cid:15) / / vac ( sB ( HP ) , ) , ̂ χ ( ) vac ≡ Ð→ R Ð→ e π e ÐÐ→ Y vac ( sB ( HP ) , ) Ð→ (5.7)is a central extension determined by the super-2-cocycle ̂ Y β ( ) vac = Y ∗ vac ̂ Y β ( ) and such that the super-1-form ˘ ζ E on e dual to the central generator given as the image of 1 ∈ R in Y vac ( sB ( HP ) , ) trivialises the pullback of ̂ Y β ( ) vac along π e , ̂ δ ˘ ζ E = − π ∗ e ̂ Y β ( ) vac , (5.8)and in which Y [ ] ∗ vac ̃ l is a central extension Ð→ R Ð→ Y [ ] ∗ vac ̃ l π Y [ ] ∗ vac ̃ l ≡ pr ÐÐÐÐÐÐÐÐÐ→ Y [ ] vac ( sB ( HP ) , ) Ð→ consistent with that of Eq. (4.4) in the sense expressed by the diagram (cid:15) (cid:15) R x x rrrrrrrrrrrr & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ Y [ ] ∗ vac ̃ l (cid:31) (cid:127) ̃ L vac / / π Y [ ] ∗ vac ̃ l (cid:15) (cid:15) ̃ l π ̃ l (cid:15) (cid:15) Y [ ] vac ( sB ( HP ) , ) % % ▲▲▲▲▲▲▲▲▲▲▲▲▲ (cid:31) (cid:127) Y [ ] vac / / Y [ ] siso ( d, ∣ D d, ) x x rrrrrrrrrrrrrr , with the Lie-superalgebra monomorphism ̃ L vac . A constructive positive answer to the question thusposed is laid out below.The structure of the first extension, Y vac ( sB ( HP ) , ) , is readily read off from the commutator [ Y Q α , Y P a ] of Y siso ( d, ∣ D d, ) : Upon restricting to the generators Y ˘ Q α spanning im P ( ) T within the sub-superspace ⊕ D d, α = ⟨ Y Q α ⟩ and to the Y P a spanning the lift of t ( ) vac , we are naturally restricted to the subspaceim P ( ) ≡ ⟨ P ( ) αβ Z β ∣ α ∈ , D d, ⟩ ⊂ D d, ⊕ α = ⟨ Z α ⟩ . enote its basis as { ˘ Z α } α ∈ , Dd, to postulate Y vac ( sB ( HP ) , ) ∶= ( Dd, ⊕ α = ⟨ Y ˘ Q α ⟩ ⊕ ⟨ Y P , Y P ⟩ ⊕ Dd, ⊕ β = ⟨ ˘ Z β ⟩) ⊕ ( ⟨ Y J ⟩ ⊕ d ⊕ ̂ a < ̂ b = ⟨ Y J ̂ a ̂ b ⟩) ≡ Y smink ( d, ∣ D d, ) vac ⊕ spin ( d, ) vac (5.9)with the Lie-superalgebra structure induced by the restriction of the superbracket of Y siso ( d, ∣ D d, ) .We confirm the self-consistency of the postulate by inspecting the brackets [ Y J , ( P ( ) Z ) α ] = − Γ αβ ( P ( ) Z ) β and [ Y J ̂ a ̂ b , ( P ( ) Z ) α ] = − Γ ̂ a ̂ bαβ ( P ( ) Z ) β . Thus, we have ∆ d, ≡ D d, and π Y vac ( sB ( HP ) , ) ≡ π Y siso ( d, ∣ D d, ) ↾ Y vac ( sB ( HP ) , ) . On the new Lie superalgebra, we find the super-2-form ̂ Y β ( ) vac = ˘ z α ∧ π ∗ Y vac ( sB ( HP ) , ) ˘ q α + π ∗ Y vac ( sB ( HP ) , ) ( p ∧ p ) , written in terms of the duals ˘ z α of the ˘ Z α and the duals ˘ q α of the ˘ Q α . The super-2-form satifies theidentity ̂ δ ̂ Y β ( ) vac = ∗ vac ̂ χ ( ) ≡ , and so it determines a central extension (5.7) with the supervector-space structure e = Dd, ⊕ α = ⟨ E ˘ Q α ⟩ ⊕ ⟨ E P , E P ⟩ ⊕ Dd, ⊕ β = ⟨ E ˘ Z β ⟩ ⊕ ⟨ ˘ Z ⟩ ⊕ ⟨ E J ⟩ ⊕ d ⊕ ̂ a < ̂ b = ⟨ E J ̂ a ̂ b ⟩ ≅ ( Y smink ( d, ∣ D d, ) vac ⊕ R ) ⊕ spin ( d, ) vac and the associated projection π e ≡ pr ⊕ id spin ( d, ) vac ∶ ( Y smink ( d, ∣ D d, ) vac ⊕ R ) ⊕ spin ( d, ) vac Ð→ Y smink ( d, ∣ D d, ) vac ⊕ spin ( d, ) vac , and with the superbrackets { E ˘ Q α , E ˘ Q β } = γ aαβ E P a , [ E P , E P ] = Z , [ E ˘ Q α , E P a ] = γ aαβ E ˘ Z β , { E ˘ Q α , E ˘ Z β } = − δ βα ˘ Z , [ E P a , E ˘ Z α ] = , { E ˘ Z α , E ˘ Z β } = , [ E ˘ Q α , ˘ Z ] = , [ E P a , ˘ Z ] = , [ ˘ Z , ˘ Z ] = , [ E J , E ˘ Q α ] = γ βα E ˘ Q β , [ E J ̂ a ̂ b , E ˘ Q α ] = γ ̂ a ̂ bβα E ˘ Q β , [ E J , E P a ] = δ a E P + δ a P , [ E J ̂ a ̂ b , E P c ] = , [ E J , E ˘ Z α ] = − γ αβ E ˘ Z β , [ E J ̂ a ̂ b , E ˘ Z α ] = − γ ̂ a ̂ bαβ E ˘ Z β , [ E J , ˘ Z ] = , [ E J ̂ a ̂ b , ˘ Z ] = , [ E J , E J ̂ a ̂ b ] = , [ E J ̂ a ̂ b , E J ̂ c ̂ d ] = δ ̂ a ̂ d E J ̂ b ̂ c − δ ̂ a ̂ c E J ̂ b ̂ d + δ ̂ b ̂ c E J ̂ a ̂ d − δ ̂ b ̂ d E J ̂ a ̂ c . Let ˘ ζ E be the dual of ˘ Z . Clearly, it satisfies the desired relation (5.8), and so the first stage of theconstruction is complete.In the next step, we form the pullback Lie superalgebra Y [ ] ∗ vac ̃ l ≡ Y [ ] vac ( sB ( HP ) , ) Y [ ] vac ⊕ π ̃ l ̃ l ith π Y [ ] ∗ vac ̃ l ≡ pr ∶ Y [ ] ∗ vac ̃ l Ð→ Y [ ] vac ( sB ( HP ) , ) , ̃ L vac ≡ pr ∶ Y [ ] ∗ vac ̃ l Ð→ ̃ l and with the basis Y [ ] ∗ vac ̃ l = Dd, ⊕ α = ⟨(( Y ˘ Q α , Y ˘ Q α ) , ̃ L ˘ Q α ) ≡ ̃ L vac ˘ Q α ⟩ ⊕ ⊕ a ∈ { , } ⟨(( Y P a , Y P a ) , ̃ L P a ) ≡ ̃ L vac P a ⟩ ⊕ Dd, ⊕ β = ⟨(( ˘ Z β , ) , ̃ L ˘ Z β ( ) ) ≡ ̃ L vac ˘ Z β ( ) ⟩ ⊕ Dd, ⊕ γ = ⟨(( , ˘ Z γ ) , ̃ L ˘ Z γ ( ) ) ≡ ̃ L vac ˘ Z γ ( ) ⟩ ⊕ ⟨(( , ) , Z ) ≡ Z vac ⟩ ⊕ ⟨(( Y J , Y J ) , ̃ L J ) ≡ ̃ L vac J ⟩ ⊕ d ⊕ ̂ a < ̂ b = ⟨(( Y J ̂ a ̂ b , Y J ̂ a ̂ b ) , ̃ L J ̂ a ̂ b ) ≡ ̃ L vac J ̂ a ̂ b ⟩ and the superbracket obtained from the direct-sum one on Y [ ] vac ( sB ( HP ) , ) ⊕ ̃ l through restriction.With this Lie superalgebra in hand, we may, at long last, finish the construction of the sLieAlg -skeletonof Diag. (5.5). To this end, consider the pullback Lie superalgebraspr ∗ n e ≡ Y [ ] vac ( sB ( HP ) , ) pr n ⊕ π e e , n ∈ { , } with the respective basespr ∗ e = Dd, ⊕ α = ⟨(( Y ˘ Q α , Y ˘ Q α ) , E ˘ Q α ) ≡ E ˘ Q ( ) α ⟩ ⊕ ⊕ a ∈ { , } ⟨(( Y P a , Y P a ) , E P a ) ≡ E P ( ) a ⟩ ⊕ Dd, ⊕ β = ⟨(( ˘ Z β , ) , E ˘ Z β ) ≡ E ˘ Z ( ) β ( ) ⟩ ⊕ Dd, ⊕ γ = ⟨(( , ˘ Z γ ) , ) ≡ E ˘ Z ( ) γ ( ) ⟩ ⊕ ⟨(( , ) , ˘ Z ) ≡ E ˘ Z ( ) ⟩ ⊕ ⟨(( Y J , Y J ) , E J ) ≡ E J ( ) ⟩ ⊕ d ⊕ ̂ b < ̂ c = ⟨(( Y J ̂ b ̂ c , Y J ̂ b ̂ c ) , E J ̂ b ̂ c ) ≡ E J ( )̂ b ̂ c ⟩ , pr ∗ e = Dd, ⊕ α = ⟨(( Y ˘ Q α , Y ˘ Q α ) , E ˘ Q α ) ≡ E ˘ Q ( ) α ⟩ ⊕ ⊕ a ∈ { , } ⟨(( Y P a , Y P a ) , E P a ) ≡ E P ( ) a ⟩ ⊕ Dd, ⊕ β = ⟨(( ˘ Z γ , ) , ) ≡ E ˘ Z ( ) β ( ) ⟩ ⊕ Dd, ⊕ γ = ⟨(( , ˘ Z β ) , E ˘ Z β ) ≡ E ˘ Z ( ) γ ( ) ⟩ ⊕ ⟨(( , ) , ˘ Z ) ≡ E ˘ Z ( ) ⟩ ⊕ ⟨(( Y J , Y J ) , E J ) ≡ E J ( ) ⟩ ⊕ d ⊕ ̂ b < ̂ c = ⟨(( Y J ̂ b ̂ c , Y J ̂ b ̂ c ) , E J ̂ b ̂ c ) ≡ E J ( )̂ b ̂ c ⟩ . and superbrackets induced from the direct-sum ones, and form the ‘tensor-product’ Lie superalgebra Y [ ] ∗ vac ̃ l ⊗ pr ∗ e ≡ ( Y [ ] ∗ vac ̃ l π Y [ ] ∗ vac ̃ l ⊕ pr pr ∗ e )/ ∼ R , based on the identification ( Z vac , ) ∼ R ( , E ˘ Z ( ) ) . The latter is the supervector space Y [ ] ∗ vac ̃ l ⊗ pr ∗ e = Dd, ⊕ α = ⟨( ̃ L vac ˘ Q α , E ˘ Q ( ) α ) ≡ ˘ Q ⊗ α ⟩ ⊕ ⊕ a ∈ { , } ⟨( ̃ L vac P a , E P ( ) a ) ≡ P ⊗ a ⟩ ⊕ Dd, ⊕ β = ⟨( ̃ L vac ˘ Z β ( ) , E ˘ Z ( ) β ( ) ) ≡ ˘ Z ⊗ β ( ) ⟩ ⊕ Dd, ⊕ γ = ⟨( ̃ L vac ˘ Z γ ( ) , E ˘ Z ( ) γ ( ) ) ≡ ˘ Z ⊗ γ ( ) ⟩ ⊕ ⟨[( Z vac , )] ∼ R ≡ Z ⊗ ⟩ ⊕ ⟨( ̃ L vac J , E J ( ) ) ≡ J ⊗ ⟩ ⊕ d ⊕ ̂ b < ̂ c = ⟨( ̃ L vac J ̂ b ̂ c , E J ( )̂ b ̂ c ) ≡ J ⊗ ̂ b ̂ c ⟩ endowed with the superbracket { ˘ Q ⊗ α , ˘ Q ⊗ β } = γ aαβ P ⊗ a , [ P ⊗ , P ⊗ ] = Z ⊗ , [ ˘ Q ⊗ α , P ⊗ a ] = γ a αβ ( ˘ Z ⊗ β ( ) + ˘ Z ⊗ β ( ) ) , { ˘ Q ⊗ α , ˘ Z ⊗ β ( ) } = − δ βα ˘ Z ⊗ , { ˘ Q ⊗ α , ˘ Z ⊗ β ( ) } = , P ⊗ a , ˘ Z ⊗ α ( m ) ] = , { ˘ Z ⊗ α ( m ) , ˘ Z ⊗ β ( n ) } = , [ ˘ Q ⊗ α , ˘ Z ⊗ ] = , [ P ⊗ a , ˘ Z ⊗ ] = , [ ˘ Z ⊗ α ( m ) , ˘ Z ⊗ ] = , [ ˘ Z ⊗ , ˘ Z ⊗ ] = , [ J ⊗ , ˘ Q ⊗ α ] = γ βα ˘ Q ⊗ β , [ J ⊗ ̂ a ̂ b , ˘ Q ⊗ α ] = γ ̂ a ̂ bβα ˘ Q ⊗ β , [ J ⊗ , P ⊗ c ] = δ c P ⊗ + δ c P ⊗ , [ J ⊗ ̂ a ̂ b , P ⊗ c ] = , [ J ⊗ , ˘ Z ⊗ α ( m ) ] = − γ αβ ˘ Z ⊗ β ( m ) , [ J ⊗ ̂ a ̂ b , ˘ Z ⊗ α ( m ) ] = − γ ̂ a ̂ bαβ ˘ Z ⊗ β ( m ) , [ J ⊗ , ˘ Z ⊗ ] = , [ J ⊗ ̂ a ̂ b , ˘ Z ⊗ ] = , [ J ⊗ , J ⊗ ̂ a ̂ b ] = , [ J ⊗ ̂ a ̂ b , J ⊗ ̂ c ̂ d ] = δ ̂ a ̂ d J ⊗ ̂ b ̂ c − δ ̂ a ̂ c J ⊗ ̂ b ̂ d + δ ̂ b ̂ c J ⊗ ̂ a ̂ d − δ ̂ b ̂ d J ⊗ ̂ a ̂ c . Comparison of the above with the superbracket of pr ∗ e reveals the existence of the sought-after Lie-superalgebra isomorphism α ̃ e ≡ ∶ Y [ ] ∗ vac ̃ l ⊗ pr ∗ e ≅ ÐÐ→ pr ∗ e given by the unique linear extension of the assignment ( ˘ Q ⊗ α , P ⊗ a , ˘ Z ⊗ β ( ) , ˘ Z ⊗ γ ( ) , Z ⊗ , J ⊗ , J ⊗ ̂ b ̂ c ) z→ ( E ˘ Q ( ) α , E P ( ) a , E ˘ Z ( ) β ( ) , E ˘ Z ( ) γ ( ) , E ˘ Z ( ) , E J ( ) , E J ( )̂ b ̂ c ) . We summarise our findings in
Theorem 3.
The null trivialisation τ of the descended super-1-gerbe of the GS super- σ -model in theHP formulation stated in Theorem 2. admits a sLieAlg -skeleton. The sLieAlg -skeleton determines a formal setting in which we may quite naturally address thequestion of existence of a sLieGrp-model of the vacuum, by which we mean a diagram (cid:15) (cid:15) α E ≡ ∶ Y [ ] J ∗ vac ̃ L ⊗ pr ∗ E ≅ ÐÐ→ pr ∗ E (cid:15) (cid:15) C × / / E , ˘ A ( ) E π E (cid:15) (cid:15) Y [ ] sISO ( d, ∣ D d, ) vac pr / / pr / / Y sISO ( d, ∣ D d, ) vac , ˘ β ( ) π Y sISO ( d, ∣ Dd, ) vac (cid:15) (cid:15) / / sISO ( d, ∣ D d, ) vac , ˘ χ ( ) ≡ sLieGrp (decorated by constitutive CaE data) that integrates Diag. (5.6) in a self-explanatorymanner, elaborated below. This is the anticipated Lie-supergroup structure that we were unable toassociate directly with the vacuum foliation in Diag. (5.5) – it seems fitting to call it the extended κ -symmetry group of the superstring in sMink ( d, ∣ D d, ) . Here, we start by passing to the (sub-)supermanifold description of the Lie sub-supergroup sISO ( d, ∣ D d, ) vac ⊂ sISO ( d, ∣ D d, ) as the locusof the coordinate equations ( − P ( ) ) αβ θ β = , α ∈ , D d, , x ̂ a = , ̂ a ∈ , d ,φ b ̂ c = , ( b, ̂ c ) ∈ { , } × , d . Using an adapted basis in siso ( d, ∣ D d, ) obtained through completion of the one for vac ( sB ( HP ) , ) given in Eq. (2.11), we thus obtain an embedding J vac ∶ sISO ( d, ∣ D d, ) vac ≡ sMink ( d, ∣ D d, ) vac ⋊ L,S
Spin ( d, ) vac ↪ sISO ( d, ∣ D d, ) , ith sMink ( d, ∣ D d, ) vac ⊂ sMink ( d, ∣ D d, ) defined by the top-line equations above. The embeddingadmits the explicit coordinate description J vac ( ˘ θ α , ˘ x a , ˘ φ S ) = ( ˘ θ α , , ˘ x a , , ˘ φ S , ) , where the zeros correspond to the nullified coordinates on ker P ( ) , the x ̂ a and the φ b ̂ c , respectively.The binary operation m vac on the embedded Lie supergroup is inherited, through restriction, fromthat on sISO ( d, ∣ D d, ) ,m vac ≡ J − ○ m ○ ( J vac × J vac ) ∶ sISO ( d, ∣ D d, ) vac × sISO ( d, ∣ D d, ) vac Ð→ sISO ( d, ∣ D d, ) vac , and reads, in coordinates,m vac (( ˘ θ α , ˘ x a , ˘ φ S ) , ( ˘ θ α , ˘ x a , ˘ φ S )) = ( ˘ θ α + ˘ S ( ˘ φ ) αβ ˘ θ β , ˘ x a + ˘ L ( ˘ φ ) ab ˘ x b − ˘ θ γ a ˘ S ( ˘ φ ) ˘ θ , ( ˘ φ ⋆ ˘ φ ) S ) , where we have used the notation ˘ S ( ˘ φ ) and ˘ L ( ˘ φ ) for the ‘vacuum’ blocks of the block-diagonal(in the adapted basis) matrices S ( ˘ φ , ) and L ( ˘ φ , ) , respectively. In the next step, we take thesub-supermanifold Y sMink ( d, ∣ D d, ) vac ≡ sMink ( d, ∣ D d, ) vac × R ∣ Dd, of Y sMink ( d, ∣ D d, ) with the second cartesian factor defined by the coordinate equations ξ β ( − P ( ) ) βα = , α ∈ , D d, , and use it to write the desired embedding Y J vac ∶ Y sISO ( d, ∣ D d, ) vac ≡ Y sMink ( d, ∣ D d, ) vac ⋊ L,S,S − T Spin ( d, ) vac ↪ Y sISO ( d, ∣ D d, ) in adapted coordinates ( cp Eq. (5.9)) as Y J vac ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) = ( ˘ θ α , , ˘ x a , , ˘ ξ β , , ˘ φ S , ) . The sub-supermanifold submerses surjectively onto sISO ( d, ∣ D d, ) vac , π Y sISO ( d, ∣ D d, ) vac ∶ Y sISO ( d, ∣ D d, ) vac Ð→ sISO ( d, ∣ D d, ) vac , as π Y sISO ( d, ∣ D d, ) vac ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) = ( ˘ θ α , ˘ x a , ˘ φ S ) . The inherited binary operation Y m vac ≡ Y J − ○ Y m ○ ( Y J vac × Y J vac ) ∶ Y sISO ( d, ∣ D d, ) vac × Y sISO ( d, ∣ D d, ) vac Ð→ Y sISO ( d, ∣ D d, ) vac uses the same objects ˘ S and ˘ L as m vac , Y m vac (( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) , ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S )) = ( ˘ θ α + ˘ S ( ˘ φ ) αβ ˘ θ β , ˘ x a + ˘ L ( ˘ φ ) ab ˘ x b − ˘ θ γ a ˘ S ( ˘ φ ) ˘ θ , ˘ ξ β + ˘ ξ γ ˘ S ( ˘ φ ) − γβ + γ d βγ ˘ θ γ ˘ L ( φ ) de ˘ x e , ( ˘ φ ⋆ ˘ φ ) S ) , (5.11)and gives rise, as usual, to the left regular action Y ˘ ℓ ≡ Y m vac . of the Lie supergroup Y sISO ( d, ∣ D d, ) vac on itself. In the coordinate presentation of Y m vac , we havetaken into account the identity γ a ⊗ γ a = Y m trilinear in the Graßmann-odd coordinatesaccordingly.On the Lie supergroup Y sISO ( d, ∣ D d, ) vac , we find the standard Y vac ( sB ( HP ) , ) -valued LI Maurer–Cartan super-1-form˘ θ L = π ∗ Y sISO ( d, ∣ D d, ) vac ˘ q α ⊗ Y ˘ Q α + π ∗ Y sISO ( d, ∣ D d, ) vac ˘ p a ⊗ Y P a + ˘ z β ⊗ ˘ Z β + π ∗ Y sISO ( d, ∣ D d, ) vac ˘ j S ⊗ Y J S whose components along Y smink ( d, ∣ D d, ) , with the coordinate presentations (derived in a procedureanalogous to the one leading to the formulæ for their counterparts on Y sISO ( d, ∣ D d, ) )˘ q α ( ˘ θ, ˘ x, ˘ φ ) = ˘ S ( ˘ φ ) − αβ d ˘ θ β =∶ ˘ S ( ˘ φ ) − αβ ˘ q α ( ˘ θ, ˘ x ) , ˘ p a ( ˘ θ, ˘ x, ˘ φ ) = ˘ L ( ˘ φ ) − ab ( d ˘ x b + ˘ θ γ b d ˘ θ ) =∶ ˘ L ( ˘ φ ) − ab ˘ p b ( ˘ θ, ˘ x ) , z β ( ˘ θ, ˘ x, ˘ ξ, ˘ φ ) = ( d ˘ ξ β − γ a βγ ˘ θ γ d ˘ x a ) ˘ S ( ˘ φ ) βα =∶ ˘ z β ( ˘ θ, ˘ x, ˘ ξ ) ˘ S ( ˘ φ ) βα , enter the definition of the LI super-2-cocycle˘ β ( ) = π ∗ Y sISO ( d, ∣ D d, ) vac ˘ q α ∧ ˘ z α + π ∗ Y sISO ( d, ∣ D d, ) vac ( ˘ p ∧ ˘ p ) = π ∗ Y sISO ( d, ∣ D d, ) vac pr ∗ ˘ q α ∧ pr ∗ ˘ z α + π ∗ Y sISO ( d, ∣ D d, ) vac pr ∗ ( ˘ p ∧ ˘ p ) , whose final (descended) form follows from the unimodularity of ρ Spin ( d, ) vac ↾ t ( ) vac , ( ˘ p ∧ ˘ p )( ˘ θ, ˘ x, ˘ φ ) = ( ˘ p ∧ ˘ p )( ˘ θ, ˘ x ) . Following the logic of the geometrisation programme, we seek to associate with the latter a centralextension Ð→ C × Ð→ E ≡ Y sISO ( d, ∣ D d, ) vac × C × π E ≡ pr ÐÐÐÐÐ→ Y sISO ( d, ∣ D d, ) vac Ð→ (5.12)integrating the formerly obtained Lie-superalgebra extension (5.7). To this end, we compute the non-LIprimitive of ˘ β ( ) , ˘ β ( ) ( ˘ θ, ˘ x, ˘ ξ, ˘ φ ) = d ( ˘ θ α d ˘ ξ α + ǫ ab ˘ x a d ˘ x b ) , and define the principal C × -connection super-1-form˘ A ( ) E ∈ Ω ( E ) on the total space E ∋ ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S , ˘ z ) of the principal C × -bundle π E ∶ E Ð→ Y sISO ( d, ∣ D d, ) vac explicitly as ˘ A ( ) E ( ˘ θ, ˘ x, ˘ ξ, ˘ φ, ˘ z ) = i d ˘ z ˘ z − ˘ θ α d ˘ ξ α − ǫ ab ˘ x a d ˘ x b =∶ i d ˘ z ˘ z + ˘A ( ˘ θ, ˘ x, ˘ ξ, ˘ φ ) , determining the Lie-supergroup structure on E through imposition of the usual demand that theprimitive − ˘ A ( ) E of the pullback π ∗ E ˘ β ( ) of the super-2-cocycle ˘ β ( ) be LI. From the direct computation(carried out for ( ˘ ε α , ˘ y a , ˘ ζ β , ˘ ψ S ) ∈ Y sISO ( d, ∣ D d, ) vac ) of Y ˘ ℓ ∗ ( ˘ ε, ˘ y, ˘ ζ, ˘ ψ ) ˘A ( ˘ θ, ˘ x, ˘ ξ, ˘ φ ) = ˘A ( ˘ θ, ˘ x, ˘ ξ, ˘ φ ) + d ( ˘ ξ ˘ S ( ˘ ψ ) − ˘ ε − ǫ ab ˘ y a ( ˘ L ( ˘ ψ ) bc ˘ x c − ˘ ε γ b ˘ S ( ˘ ψ ) ˘ θ ) − ǫ ab ˘ ε γ a ˘ S ( ˘ ψ ) ˘ θ ˘ L ( ˘ ψ ) bc ˘ x c ) , using the identities ǫ ab γ a ⊗ γ b = , ǫ ab γ a = − γ b , and hence leading to ( Y ˘ ℓ ∗ ( ˘ ε, ˘ y, ˘ ζ, ˘ ψ ) − id ∗ Y sISO ( d, ∣ D d, ) vac ) ˘A ( ˘ θ, ˘ x, ˘ ξ, ˘ φ ) = d ( ˘ ξ ˘ S ( ˘ ψ ) − ˘ ε − ǫ ab ˘ y a ˘ L ( ˘ ψ ) bc ˘ x c + ( ˘ y a + ˘ L ( ˘ ψ ) ab ˘ x b ) ˘ ε γ a ˘ S ( ˘ ψ ) ˘ θ )) , we read off the candidate for the binary operation:Em vac ∶ E × E Ð→ Ein the coordinate formEm vac (( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S , ˘ z ) , ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S , ˘ z )) = ( Y m vac (( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) , ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S )) , e i ( ˘ ξ ˘ S ( ˘ φ ) − ˘ θ − ǫ ab ˘ x a ˘ L ( ˘ φ ) bc ˘ x c + ( ˘ x a + ˘ L ( ˘ φ ) ab ˘ x b ) ˘ θ γ a ˘ S ( ˘ φ ) ˘ θ ) ⋅ ˘ z ⋅ ˘ z ) . Through inspection, we readily prove roposition 4. The supermanifold E , together with the supermanifold morphism Em vac defined aboveas the binary operation and the pair of supermanifold morphisms EInv vac ∶ E Ð→ E , E ε vac ∶ R ∣ Ð→ E with the coordinate presentations EInv vac ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S , ˘ z ) = ( − ˘ S ( ˘ φ ) − αγ ˘ θ γ , − ˘ L ( ˘ φ ) − ab ˘ x b , ( − ˘ ξ γ + ˘ x a γ a γδ ˘ θ δ ) ˘ S ( ˘ φ ) γβ , − ˘ φ S , e i ˘ ξ ˘ θ ⋅ ˘ z − ) , E ε vac ( ● ) = ( , , , , ) as the inverse and the unit, respectively, is a Lie supergroup that centrally extends Y sISO ( d, ∣ D d, ) vac as in Eq. (5.12) . At this stage, it remains to establish the existence of the trivial
Lie-supergroup isomorphism α E ≡ ∶ Y [ ] J ∗ vac ̃ L ⊗ pr ∗ E ≅ ÐÐ→ pr ∗ Eover the fibred square Y [ ] sISO ( d, ∣ D d, ) vac ≡ Y sISO ( d, ∣ D d, ) vac × sISO ( d, ∣ D d, ) vac Y sISO ( d, ∣ D d, ) vac of the surjective submersion Y sISO ( d, ∣ D d, ) vac (endowed with the Lie-supergroup structure inducedfrom the product one on Y sISO ( d, ∣ D d, ) vac × Y sISO ( d, ∣ D d, ) vac through restriction) and check thatit is simultaneously a connection-preserving principal C × -bundle isomorphism. Here, Y [ ] J ∗ vac ̃ L is theLie sub-supergroup of the product Lie supergroup Y [ ] sISO ( d, ∣ D d, ) vac × ̃ L whose support is theprincipal C × -bundle π Y [ ] J ∗ vac ̃ L ≡ pr ∶ Y [ ] J ∗ vac ̃ L ≡ Y [ ] sISO ( d, ∣ D d, ) vac Y [ ] J vac × π ̃ L ̃ L Ð→ Y [ ] sISO ( d, ∣ D d, ) vac with coordinates ((( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) , ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S )) , (( ˘ θ α , , ˘ x a , , ˘ ξ β , , ˘ φ S , ) , ( ˘ θ α , , ˘ x a , , ˘ ξ β , , ˘ φ S , ) , z )) ∈ Y [ ] J ∗ vac ̃ L , and the tensor product in Y [ ] J ∗ vac ̃ L ⊗ pr ∗ E is defined analogously to the one on p. 25. Comparing thebase components of the principal C × -connection super-1-forms of the two principal C × -bundles to berelated by α E ≡ , Y [ ] π ∗ a (( ˘ θ α , , ˘ x a , , ˘ ξ β , , ˘ φ S , ) , ( ˘ θ α , , ˘ x a , , ˘ ξ β , , ˘ φ S , )) + ˘A ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) − ˘A ( ˘ θ α , ˘ x a , ˘ ξ β , ˘ φ S ) = , we conclude that the bundles are related by the connection-preserving isomorphism indicated. Wemerely need to check if the latter is a Lie-supergroup homomorphism. That this is, indeed, the casefollows from the equalitye i ˘ θ α ( ˘ ξ , β − ˘ ξ , β ) ˘ S ( ˘ φ ) − βα ⋅ e i ( ˘ ξ , ˘ S ( ˘ φ ) − ˘ θ − ǫ ab ˘ x a ˘ L ( ˘ φ ) bc ˘ x c + ( ˘ x a + ˘ L ( ˘ φ ) ac ˘ x c ) ˘ θ γ a ˘ S ( ˘ φ ) ˘ θ ) = e i ( ˘ ξ , ˘ S ( ˘ φ ) − ˘ θ − ǫ ab ˘ x a ˘ L ( ˘ φ ) bc ˘ x c + ( ˘ x a + ˘ L ( ˘ φ ) ac ˘ x c ) ˘ θ γ a ˘ S ( ˘ φ ) ˘ θ ) of the ‘phase’ factors in – on the one hand (the left-hand side of the equality sign) – the α E -image ofthe product of the ((( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn ) , ( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn )) , (( ˘ θ αn , , ˘ x an , , ˘ ξ n, β , , ˘ φ Sn , ) , ( ˘ θ αn , , ˘ x an , , ˘ ξ n, β , , ˘ φ Sn , ) , )) ⊗ ((( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn ) , ( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn )) , ( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn , ˘ z n )) ∈ Y [ ] J ∗ vac ̃ L ⊗ pr ∗ Ewith n ∈ { , } , and – on the other hand (the right-hand side of the equality sign) – the product oftheir α E -images ((( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn ) , ( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn )) , ( ˘ θ αn , ˘ x an , ˘ ξ n, β , ˘ φ Sn , ˘ z n )) ∈ pr ∗ E . Thus, altogether, we arrive at
Theorem 5.
The sLieAlg -skeleton of the null trivialisation τ of the super-1-gerbe of the GS super- σ -model in the HP formulation from theorem 3. integrates to a sLieGrp -model. Theorems 2-5 attest to the veracity of our expectation with regard to (the triviality of) the vacuumrestriction of the extended HP super-1-gerbe over sISO ( d, ∣ D d, ) . We conclude our study by demon-strating how this fact implies the existence of a descendable sISO ( d, ∣ D d, ) vac -equivariant structureon the restriction. In so doing, we localise our analysis over a single leaf D i,υ i of the vacuum fo-liation within Σ HPvac , with the understanding that the mechanisms discovered in its course descend o the physical vacuum in Σ HPphys vac . Moreover, we exclude the ‘hidden’ gauge-symmetry subgroupSpin ( d, ) vac ⊂ sISO ( d, ∣ D d, ) vac from our discussion as the latter is an artifact of the realisation ofthe physical supertarget sISO ( d, ∣ D d, )/ Spin ( d, ) vac in the mother Lie supergroup sISO ( d, ∣ D d, ) ,with the corresponding equivariance explicitly built into the construction of the (extended) super-1-gerbe. This leaves us with the visible κ -symmetry group κ vis ≡ sMink ( d, ∣ D d, ) vac vis ÐÐÐ→ sISO ( d, ∣ D d, ) of the superstring in Mink ( d, ∣ D d, ) (embedded in an obvious manner in the mother supersymmetrygroup) as the Lie supergroup for which we are to establish an equivariant structure on G ( ) vac ↾ D i,υi .The point of departure of our considerations is the action (super)groupoid κ i,υ i ≡ κ vis × D i,υ i pr / / λ vis ≡ ℘○ ( Inv ○ vis × id D i,υi ) / / D i,υ i ≡ κ i,υ i , with the target map λ vis purposefully turned into a left action, so that we may directly employ theconstruction (5.1) introduced previously. Over the arrow supermanifold of this category, we set up apair of surjective submersions π f ∗ Y D i,υi ≡ pr ∶ f ∗ Y D i,υ i ≡ κ i,υ i f × π Y sISO ( d, ∣ Dd, ) Y D i,υ i Ð→ κ i,υ i , f ∈ { λ vis , pr } with ̂ f ≡ pr ∶ f ∗ Y D i,υ i Ð→ Y D i,υ i , and demand the existence of a principal C × -bundle π F ∶ F Ð→ λ ∗ vis Y D i,υ i pr × pr pr ∗ Y D i,υ i ≡ Y λ vis D i,υ i over the κ i,υ i -fibred product of the two, with a principal C × -connection super-1-form A ( ) F ∈ Ω ( F ) satisfying the identity d A ( ) F = π ∗ F ( pr ∗ ○ ̂ pr ∗ − pr ∗ ○ ̂ λ ∗ vis ) ̂ Y β ( ) , and such that there exists, over λ ∗ vis Y D i,υ i pr × pr pr ∗ Y D i,υ i pr × pr λ ∗ vis Y D i,υ i pr × pr pr ∗ Y D i,υ i ≡ Y λ vis λ vis D i,υ i , a connection-preserving isomorphism α F ∶ pr ∗ , ̂ λ × ∗ vis ̃ L ⊗ pr ∗ , F ≅ ÐÐ→ pr ∗ , F ⊗ pr ∗ , ̂ pr × ∗ ̃ L of the principal C × -bundles obtained by tensoring, in the manner discussed earlier, the pullbacks of thebundles π ̂ f × ∗ ̃ L ≡ pr ∶ ̂ f × ∗ ̃ L ≡ Y ff D i,υ i ̂ f × × π ̃ L ̃ L Ð→ Y ff D i,υ i ≡ f ∗ Y D i,υ i × κ i,υi f ∗ Y D i,υ i given by π pr ∗ , ̂ λ × ∗ vis ̃ L ≡ pr ∶ pr ∗ , ̂ λ × ∗ vis ̃ L ≡ Y λ vis λ vis D i,υ i pr , × pr ̂ λ × ∗ vis ̃ L Ð→ Y λ vis λ vis D i,υ i (5.13)(for f = λ vis ) and π pr ∗ , ̂ pr × ∗ ̃ L ≡ pr ∶ pr ∗ , ̂ pr × ∗ ̃ L ≡ Y λ vis λ vis D i,υ i pr , × pr ̂ pr × ∗ ̃ L Ð→ Y λ vis λ vis D i,υ i (for f = pr ), respectively, with the pullback bundles π pr ∗ i,j F ≡ pr ∶ pr ∗ i,j F ≡ Y λ vis λ vis D i,υ i pr i,j × π F F Ð→ Y λ vis λ vis D i,υ i , ( i, j ) ∈ {( , ) , ( , )} . Using Eq. (5.3), we readily find ( pr ∗ ○ ̂ pr ∗ − pr ∗ ○ ̂ λ ∗ vis ) ̂ Y β ( ) = d [( pr ∗ ○ ̂ λ ∗ vis − pr ∗ ○ ̂ pr ∗ ) A i,υ i ] , and so we postulate F ≡ Y λ vis λ vis D i,υ i × C × ∋ (((( ˘ ε, ˘ y ) , ( θ, x, φ )) , ( θ − S ( φ ) ˘ ε, x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε, ξ , φ )) , ((( ˘ ε, ˘ y ) , ( θ, x, φ )) , ( θ, x, ξ , φ )) , z ) ≡ ϕ ith π F ≡ pr and A ( ) F ( ϕ ) = i d zz + A i,υ i ( θ − S ( φ ) ˘ ε, x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε, ξ , φ ) − A i,υ i ( θ, x, ξ , φ ) = ∶ i d zz + π ∗ F a i,υ i ( ϕ ) . Inspection of the base components of the relevant principal C × -connection super-1-forms,a (( θ − S ( φ ) ˘ ε, x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε, ξ ) , ( θ − S ( φ ) ˘ ε, x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε, ξ )) + π ∗ F a i,υ i ( ϕ ) = π ∗ F a i,υ i ( ϕ ) + a (( θ, x, ξ ) , ( θ, x, ξ )) , written for ϕ A = (((( ˘ ε, ˘ y ) , ( θ, x, φ )) , ( θ − S ( φ ) ˘ ε, x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε, ξ A − , φ )) , ((( ˘ ε, ˘ y ) , ( θ, x, φ )) , ( θ, x, ξ A , φ )) , z ) , A ∈ { , } , reveals that we may take α F in the trivial form α F ≡ , automatically compatible with µ ̃ L ≡ . Thus, we obtain dataΥ vis ≡ ( Y λ vis D i,υ i , id Y λ vis2 D i,υi , F , π F , A ( ) F , α F ) of the 1-isomorphism Υ vis ∶ λ ∗ vis G ( ) vac ≅ ÐÐ→ pr ∗ G ( ) vac of the (descendable) κ vis -equivariant structure sought after.In order to complete the construction, we move to κ i,υ i ≡ κ × × D i,υ i , with its face maps d ( ) = pr , , d ( ) = id κ vis × λ vis , d ( ) = m × id D i,υi to κ i,υ i , and define, over the fibred products π Y λ vis2; i D i,υi ≡ pr ∶ Y λ vis i D i,υ i ≡ ( κ i,υ i d ( ) i × pr λ ∗ vis Y D i,υ i ) pr × pr ( κ i,υ i d ( ) i × pr pr ∗ Y D i,υ i ) Ð→ κ i,υ i d ( ) i × pr λ ∗ vis Y D i,υ i , the respective principal C × -bundles π Y λ vis2; i D i,υi pr × × π F F ≡ pr ∶ Y λ vis i D i,υ i pr × × π F F Ð→ Y λ vis i D i,υ i . We identify ι ≡ ( pr , ( pr , ○ pr , pr ○ pr )) ∶ d ( ) ∗ pr ∗ Y D i,υ i ≅ ÐÐ→ d ( ) ∗ λ ∗ vis Y D i,υ i ,ι ≡ ( pr , (( m × id D i,υi ) ○ pr , pr ○ pr )) ∶ d ( ) ∗ λ ∗ vis Y D i,υ i ≅ ÐÐ→ d ( ) ∗ λ ∗ vis Y D i,υ i ,ι ≡ ( pr , (( m × id D i,υi ) ○ pr , pr ○ pr )) ∶ d ( ) ∗ pr ∗ Y D i,υ i ≅ ÐÐ→ d ( ) ∗ pr ∗ Y D i,υ i , and define (in an obvious shorthand notation) Y λ vis D i,υ i ≡ d ( ) ∗ λ ∗ vis Y D i,υ i × κ i,υi d ( ) ∗ pr ∗ Y D i,υ i × κ i,υi d ( ) ∗ pr ∗ Y D i,υ i ≅ ≡ id × ι × id ÐÐÐÐÐÐÐ→ d ( ) ∗ λ ∗ vis Y D i,υ i × κ i,υi d ( ) ∗ λ ∗ vis Y D i,υ i × κ i,υi d ( ) ∗ pr ∗ Y D i,υ i ≅ ≡ ι × id × ι ÐÐÐÐÐÐÐ→ d ( ) ∗ λ ∗ vis Y D i,υ i × κ i,υi d ( ) ∗ λ ∗ vis Y D i,υ i × κ i,υi d ( ) ∗ pr ∗ Y D i,υ i , in terms of which we may writepr ∗ , ( Y λ vis D i,υ i pr × × π F F ) ≡ Y λ vis D i,υ i pr , × pr ( Y λ vis D i,υ i pr × × π F F ) pr ÐÐÐ→ Y λ vis D i,υ i , ( pr , (○ ≅ )) ∗ ( Y λ vis D i,υ i pr × × π F F ) ≡ Y λ vis D i,υ i pr , ( ○ ≅ ) × pr ( Y λ vis D i,υ i pr × × π F F ) pr ÐÐÐ→ Y λ vis D i,υ i Coordinate expressions for the identification mappings can be read off directly from Ref. [Sus19, Sec. 4.2]. nd ( pr , (○ ≅ ○ ≅ )) ∗ ( Y λ vis D i,υ i pr × × π F F ) ≡ Y λ vis D i,υ i pr , ( ○ ≅ ○ ≅ ) × pr ( Y λ vis D i,υ i pr × × π F F ) pr ÐÐÐ→ Y λ vis D i,υ i . Subsequently, we form the tensor product of the former two,pr ∶ pr ∗ , ( Y λ vis D i,υ i × pr × F ) ⊗ pr ∗ , ( Y λ vis D i,υ i × pr × F ) Ð→ Y λ vis D i,υ i , and identify it as the principal C × -bundle of the product 1-isomorphism d ( ) ∗ Υ vis ○ d ( ) ∗ Υ vis , to be re-lated to the latter one, in which we recognise the principal C × -bundle of d ( ) ∗ Υ vis . The relation is read-ily established through comparison of the base components of the respective principal C × -connectionsuper-1-forms. For that purpose, write k , ∶ = (( ˘ ε , ˘ y ) , ( ˘ ε , ˘ y ) , ( θ, x, φ )) ∈ κ i,υ i and consider (( k , , ((( ˘ ε , ˘ y ) , ( θ − S ( φ ) ˘ ε , x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε , φ )) , ( θ − S ( φ ) ( ˘ ε + ˘ ε ) , x − L ( φ ) ( ˘ y + ˘ y + ˘ ε Γ ˘ ε ) + θ Γ S ( φ ) ( ˘ ε + ˘ ε ) , ξ , φ ))) , ( k , , ((( ˘ ε , ˘ y ) , ( θ − S ( φ ) ˘ ε , x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε , φ )) , ( θ − S ( φ ) ˘ ε , x − L ( φ ) ˘ y + θ Γ S ( φ ) ˘ ε , φ, ξ , φ ))) , ( k , , ((( ˘ ε , ˘ y ) , ( θ, x, φ )) , ( θ, x, ξ , φ )))) ≡ (( k , , ϕ ) , ( k , , ϕ ) , ( k , , ϕ )) ∈ Y λ vis D i,υ i . The connection super-1-forms now compare as ( pr ∗ , pr × ∗ + ≅ ∗ pr ∗ , pr × ∗ ) a i,υ i (( k , , ϕ ) , ( k , , ϕ ) , ( k , , ϕ )) = ≅ ∗ ≅ ∗ pr ∗ , pr × ∗ a i,υ i (( k , , ϕ ) , ( k , , ϕ ) , ( k , , ϕ )) , whence also the choice β ≡ ∶ pr ∗ , ( Y λ vis D i,υ i × pr × F ) ⊗ pr ∗ , ( Y λ vis D i,υ i × pr × F ) ≅ ÐÐ→ pr ∗ , ( Y λ vis D i,υ i × pr × F ) , trivially compatible with α F ≡ . The ensuing 2-isomorphism γ vis ≡ ( Y λ vis D i,υ i , id Y λ vis22 D i,υi , β ) ∶ d ( ) ∗ Υ vis ○ d ( ) ∗ Υ vis ≅ ÔÔ⇒ d ( ) ∗ Υ vis of the κ vis -equivariant structure under reconstruction is manifestly (and trivially) coherent. We sum-marise the results of our check in Theorem 5.
The vacuum restriction of the extended super-1-gerbe of the GS super- σ -model in the HPformulation from theorem 3. carries a canonical descendable κ vis -structure. The last result – a consequence of the trivialisation mechanism discussed earlier that we dissectedabove for the sake of illustration that may prove useful in geometrically more involved circumstances –completes our systematic investigation of the higher-geometric and -algebraic content of the super- σ -model of the superstring in sMink ( d, ∣ D d, ) in the purely topological HP formulation in which thatcontent becomes particularly manifest and structured. It leaves us with a fairly complete picture ofthe (classical) vacuum of the theory and its global and local supersymmetries, alongside their verynatural (super)gerbe-theoretic realisations with Lie-superalgebraic and -supergroup ‘skeleta’. We hopeto return to the line of research drawn hereabove in the future.6. Conclusions & Outlook
In the present paper, we have associated with the classical vacuum of the super- σ -model for the su-perstring in the super-Minkowski spacetime ( i.e. , with the embedded superstring worldsheet) a higherLie-superalgebraic object – the sLieAlg -skeleton of Theorem 3. – that models, in the category ofLie superalgebras, the tangent sheaf of the null trivialisation of the vacuum restriction of an extendedsuper-1-gerbe geometrising, through an adaptation of the general scheme of Ref. [Mur96] to the super-geometric setting proposed in Ref. [Sus17], the topological action functional of the super- σ -model inthe Hughes–Polchinski formulation consistently with the supersymmetries present. The geometrisationhas been obtained, in Theorem 1., as an equivariant lift of the one originally constructed in Ref. [Sus17]over the physical supertarget sMink ( d, ∣ D d, ) to the full supersymmetry group sISO ( d, ∣ D d, ) , and he said trivialisation, postulated in Ref. [Sus20, Sus21] and proven as Theorem 2., can be viewed asa higher-geometric manifestation of the nature of the vacuum, which is that of an integral leaf of aninvolutive superdistribution (over the extended supertarget sISO ( d, ∣ D d, ) ) of its tangential gaugesupersymmetries, the latter being bracket-generated by the κ -symmetry superdistribution of Sec. 3.2and modelled on the Lie sub-superalgebra vac ( sB ( HP ) , ) of the supersymmetry algebra siso ( d, ∣ D d, ) of the super- σ -model given in Eq. (2.11). The sLieAlg -skeleton has been demonstrated to integrate toa higher-geometric object – the sLieGrp -model of Theorem 4. – in the category of Lie supergroupsthat acquires the interpretation of the higher gauge supersymmetry group of the vacuum. The state-ment of null trivialisation of the vacuum-restricted super-1-gerbe has been shown, in Theorem 5., tostrengthen the statement of descendable equivariance of that super-1-gerbe with respect to the extended κ -symmetry group of the superstring sISO ( d, ∣ D d, ) vac ⊂ sISO ( d, ∣ D d, ) at the root of the sLieGrp -model, anticipated, already in Ref. [Sus19] and in a more structured form in Ref. [Sus20], on the basisof the interpretation of the latter supergroup as the gauge supersymmetry group of the vacuum, cp Refs. [GSW10, GSW13]. In the light of the long-established interpretation of the higher-geometric ob-jects associated with the cohomological content of the (super-) σ -model as structures encoding, throughthe transgression mechanism of Refs. [Gaw88, Sus11a], the pre-quantisation of the (super)field theoryunder consideration, the findings of the present paper are to be understood as novel markers of quantum-mechanical coherence of the super- σ -model and, simultaneously, as strong and nontrivial evidence forthe internal consistency of the gerbe-theoretic approach to Green–Schwarz-type super- σ -models ad-vanced in Ref. [Sus17] and developed in the series of papers [Sus19, Sus18a, Sus18b, Sus20, Sus21]that followed. First and foremost, though, they provide us with a realisation of the goal set up in theIntroduction, which consists in extracting a higher (super-)algebraic representation of the fundamentalobject of the super-field theory under consideration – the superstring (trajectory/current) – from thegeometrisation of the background gauge field obtained through an extension of the supersymmetryalgebra.The geometrisation of the supersymmetrically invariant cohomological content of the super-mink-owskian super- σ -model reviewed and elaborated in the present paper can be understood, in the spiritof Refs. [RC85, Rab87], as standard geometrisation, `a la Murray, of the cohomological content of asuper- σ -model with an orbifold of the supermanifold sMink ( d, ∣ D d, ) with respect to a natural actionof the discrete Kosteleck´y–Rabin group Γ KR ⊂ sMink ( d, ∣ D d, ) as the supertarget, the orbifold havinga highly nontrivial topology, also in the Graßmann-odd fibre. This remark disperses the illusion oftriviality of the constructions considered which may arise as a consequence of the topological trivialityof (the body of) the apparent supertarget sMink ( d, ∣ D d, ) . It legitimises the present choice of thesuperstring supergeometry as the one in which the novel phenomena entailed by the Z / Z -grading ofthe target geometry and the supersymmetry of the dynamics that takes place in it and captured by thediscrepancy between the de Rham cohomology and its physically favoured supersymmetric refinementare most neatly and tractably separated from the standard ones known from the study of σ -modelswith topologically nontrivial targets. That said, it is only natural, and very well justified from thephysical point of view, to look for analogons of the structures and mechanisms reported herein insuperstring superbackgrounds with topologically nontrivial curved supertargets. The results for thefamily of superbackgrounds over the AdS p × S q obtained in Refs. [Sus18a, Sus18b] and Ref. [Sus20]provide a firm basis for such developments.The key idea of the paper, which boils down to associating a particular diagram in the categoryof Lie superalgebras decorated with (and determined by) Chevalley–Eilenberg cohomological data andintegrable to the corresponding diagram in the category of Lie supergroups, to the vacuum of the super- σ -model through a supersymmetrically invariant trivialisation of the super-1-gerbe that geometrisesthe relevant Cartan–Eilenberg super-3-cocycle upon restriction to the embedded (vacuum) superstringworldsheet, also admits an obvious generalisation to other species of BPS states encountered in su-perstring theory. Indeed, there are two independent sources of such structures, of a different physicalstatus, that we may derive from the known super- ( p + ) -cocycles that define Green–Schwarz-typesuper- σ -models for super- p -branes, to wit, ● trivialisations of the super- p -gerbes geometrising the super- ( p + ) -cocycles over the embeddedsuper- p -brane vacua in the Hughes–Polchinski formulation, with curvatures given by the volumesuper- ( p + ) -forms of the vacua (as in the present paper, in which p = ● arbitrary modules of the same super- p -gerbes arising over sub-supermanifolds within the re-spective supertargets endowed with actions of subgroups of the supersymmetry groups anddefining (Dirichlet) boundary conditions in the super- p -brane super- σ -models. he latter class naturally extends to that of supersymmetric super- p -gerbe bimodules associated withworldvolume defects in these super- σ -models. We shall study their higher Lie-superalgebraic incarnationat length in an upcoming paper. Acknowledgements:
The Author gratefully acknowledges the hospitality extended to him, in Sep-tember 2020, by the Erwin Schr¨odinger International Institute for Mathematics and Physics where thiswork was brought to completion. He is also thankful to the Organisers of the Thematic Programme“Higher Structures and Field Theory”, and to Thomas Strobl in particular, for the inspiring atmosphereof the meeting, for their interest in his research reported herein, and for financial support during theperiod of his sojourn in Vienna.
Appendix A. A convention
Let M , M and N be supermanifolds, and let ϕ n ∶ M n Ð→ N , n ∈ { , } be supermanifoldmorphisms of which (at least) one is a surjective submersion. We then define the fibred product of M and M over N as the supermanifold M × N M ≡ M ϕ × ϕ M embedded in the cartesian product M × M ( cp Ref. [Vor14, Sec. 2.4.9]) and described by the com-mutative diagram M × N M / / pr (cid:15) (cid:15) M ϕ (cid:15) (cid:15) M ϕ / / N . Its existence is ensured by Ref. [Keß19, Prop. 3.2.11].
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