Quantum field theoretic representation of Wilson surfaces: I higher coadjoint orbit theory

This is the first of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov-Kostant-Souriau theory of coadjoint orbits is presented based on the derived geometric framework, which has shown its usefulness in 4--dimensional higher Chern--Simons theory. An original notion of derived coadjoint orbit is put forward. A theory of derived unitary line bundles and Poisson structures on regular derived orbits is constructed. The proper derived counterpart of the Bohr--Sommerfeld quantization condition is then identified. A version of derived prequantization is proposed. The difficulties hindering a full quantization, shared with other approaches to higher quantization, are pinpointed and a possible way--out is suggested. The theory we elaborate provide the geometric underpinning for the field theoretic constructions of the companion paper.


Introduction
Wilson loops were introduced by Wilson in 1974 [1] as a natural set of gauge invariant variables suitable for the description of the non perturbative regime of quantum chromodynamics. Since then, they have been widely employed in lattice gauge theory.
In the loop formulation of gauge theory [2,3], the quantum Hilbert space consists of gauge invariant wave functionals on the gauge field configuration space.
According to a theorem of Giles [4], Wilson loops constitute a basis of the Hilbert space allowing to switch from the gauge field to the loop representation.
Wilson loops are fundamental constitutive elements of a canonical formulation of quantum gravity as a gauge theory, known as loop quantum gravity [5], and their incorporation has led to the very powerful spin network and foam approaches of this latter [6].
Wilson loops are relevant also in condensed matter physics at low energy, specifically in the study of topologically ordered phases of matter described by topological quantum field theories. In models of fractional quantum Hall states as well as lattice models such as Kitaev's toric code [7], fractional braiding statistics between quasiparticles emerges through the correlation function of a pair of Wilson loops forming a Hopf link [8,9].
Wilson loops depend on the topology of the underlying knots and, as shown in Witten's foundational work [10], they can be employed to study knot topology in 3-dimensional Chern-Simons (CS) theory using basic techniques of quantum field theory. CS correlators of Wilson loop operators provide a variety of knot and link invariants.
Higher gauge theory is an generalization of ordinary gauge theory where gauge fields are higher degree forms [11,12]. It is considered to be a promising candidate for the description of the dynamics of the higher dimensional extended objects occurring in supergravity and string theory thought to be the basic constituents of matter and mediators of fundamental interactions (see [13] for an updated general overview). Higher gauge theory is relevant also in spin foam theory [14] and condensed matter physics [15]. 5 Wilson surfaces [16][17][18][19][20], 2-dimensional counterparts of Wilson loops, emerge naturally in theories with higher form gauge fields such as those mentioned in the previous paragraph and are so expected to be relevant in the analysis of various basic aspects of them for reasons analogous to those for which Wilson loops are.

Wilson loops as partition functions
In 4 spacetime dimensions, particle-like excitations cannot braid and have only ordinary bosonic/fermionic statistics. Fractional braiding statistics can still occur through the braiding of either a point-like and a loop-like or two looplike excitations. This has been adequately described through the correlation functions of Wilson loops and surfaces in BF type topological quantum field theories [21][22][23].
Wilson surfaces also should be a basic element of any field theoretic approach to 4-dimensional 2-knot topology [24,25]. Based on Witten's paradigm, it should be possible to study surface knot topology in 4-dimensions computing correlators of Wilson surfaces in an appropriate 4-dimensional version of CS theory using again techniques of quantum field theory [26][27][28].
The aim of the present two-part study is constructing a 2-dimensional topological sigma model whose quantum partition function yields a Wilson surface in strict higher gauge theory on the same lines as the 1-dimensional topological sigma model providing a Wilson loop in ordinary gauge theory. The path we have in mind is described in the next subsections.

Wilson loops as partition functions
The idea of representing a given Wilson loop as the partition function of a 1dimensional sigma model has a long history. In the context of 4-dimensional Yang-Mills theory, this formulation can be traced back to the work of Balachandran et al. [29]. The approach was subsequently developed by Alekseev et al. in [30] and Diakonov and Petrov in [31,32]. More recently, it was applied to the canonical quantization of CS theory by Elitzur et al. in [33]. The functional integral expression of a Wilson loop holds in fact in general for any gauge theory in any dimension. Below, we briefly outline the principles on which this theoretical framework is based. See also [34][35][36] for clear illustrations of the underlying 1. 1 Wilson loops as partition functions 6 theory and some of its most significant applications.
The definition of ordinary Wilson loops in gauge theory is well-known. In a gauge theory with gauge group G, a Wilson loop W R pCq depends on a representation R of G and an oriented loop C in the spacetime manifold M and is given by the gauge invariant trace in R of the holonomy of the gauge field ω along C, (1.1.1) This description of W R pCq is intrinsically quantum mechanical [10]. Expression The description of W R pCq alluded to in the previous paragraph must necessarily be based upon a 1-dimensional field theory on the loop C compatible with the correspondence (1.1.2) and its implications upon quantization. The gauge group G should further act as a symmetry group to account for the gauge invariance of W R pCq. Therefore, it is plausible that this field theory is a 1-dimensional sigma model featuring a G-valued auxiliary bosonic field g coupled to the gauge field ω acting as a background field. The expression of W R pCq we are aiming to should where S R pg, ωq is a gauge-invariant action functional of g and the restriction of ω to C depending upon the representation R given by an integral on C of a local Lagrangian density.

Wilson loops and coadjoint orbits
The quantum system underlying the partition function realization of a Wilson loop can be described quite explicitly as we review below.
We assume that G is a compact semisimple Lie group and that R is an irreducible representation of G. R is uniquely characterized up to equivalence by its highest weight λ and so we write R " R λ . As is well-known, λ P Λ wG`, the lattice of dominant weights of G in the dual space g * of the Lie algebra g of G.
In general, with any element λ P g * there is associated the coadjoint orbit O λ " tAd * γpλq|γ P Gu. O λ is a homogeneous space: O λ " G{G λ , where G λ is the stabilizer subgroup of λ. G is in this way structured as a principal G λ -bundle over O λ . Forms on O λ are thus representable as forms on G which are horizontal and invariant with respect to the multiplicative right G λ -action.
The left multiplicative action of G on itself induces owing to its commutativity with the right G λ -action a G-action on the coadjoint orbit O λ . This action constitutes a primal property of O λ .
In Kirillov-Kostant-Souriau (KKS) theory [37], the coadjoint orbit O λ is promoted to a symplectic manifold by equipping it with the symplectic 2-form where x¨,¨y is the duality pairing of g and g * . In this way, O λ is endowed with a Poisson bracket structure t¨,¨u. ν λ is invariant under the G-action. This latter is actually Hamiltonian. Its moment map q λ : g Ñ C 8 pO λ q is given by q λ pxq " xAd * γpλq, xy (1.2.2) with x P g and satisfies the Poisson bracket tq λ pxq, q λ pyqu " q λ prx, ysq (1.2.3) for x, y P g. As q is a Lie algebra morphism, the Hamiltonian governing the classical system underlying the Wilson loop W R λ pCq is assumably H λ " q λ pς * ωq, (1.2.4) where ς : C Ñ M is the embedding of C in the spacetime manifold M . The quantization of the coadjoint orbit O λ can now be carried in two distinct though related ways. Below, we restrict for simplicity to the case where λ is regular, that is the stabilizer subgroup G λ of λ is a maximal torus of G.
The techniques of geometric quantization [38][39][40] can be applied to the coadjoint orbit O λ if the Bohr-Sommerfeld quantization condition is satisfied requiring that the cohomology class rν λ {2πs P H 2 pO λ , Rq lies in the image of H 2 pO λ , Zq.
This happens precisely when λ P Λ wG , the weight lattice of G, just the situation The Borel-Weil-Bott theorem [41,42] connects the quantization of O λ described above to the representation R λ . It states that H 0 B pO λ , L λ q ‰ 0 precisely when λ P Λ wG`a nd that in that case under the G-action H 0 B pO λ , L λ q is the representation space of the representation R λ .
The methods of functional integral quantization [43] can also be applied to O λ . To this end, we need to begin with an action S λ . This has the standard form S λ " ş C p λ´Hλ q, where λ is a symplectic potential of ν λ satisfying ν λ " d λ , after a conventional overall sign redefinition. Here, g is a G valued field on the closed oriented curve C. Hence, the action S λ is a functional on the mapping space MappC, Gq that generically describes a 1-dimensional sigma model with target space G. Since we are interested in the coadjoint orbit O λ " G{G λ instead than G, S λ , or better its exponentiated form exppiS λ q relevant in functional integral quantization, should rather be a functional on the mapping space MappC, G{G λ q.
This requires that S λ pg 1 , ωq " S λ pg, ωq mod 2πZ for g 1 " gυ with υ P MappC, G λ q resulting again in the condition that λ P Λ wG . The quantization of O λ is now given formally by the partition function Dg exp pi S λ pg, ωqq . (1.2.9) The Wilson loop W R λ pCq equals Z λ pCq up to an overall normalization in accordance with (1.1.4). This identification can be tested in a number of ways by verifying that Z λ pCq enjoys the properties which W R λ pCq does. In fact, as a functional of the gauge field ω, Z λ pCq is gauge invariant. Furthermore, when ω is flat, Z λ pCq is also invariant under smooth variations of the embedding ς as a consequence of certain Schwinger-Dyson relations.
The 1-dimensional sigma model summarily described in the previous paragraph has properties analogous to those of 3-dimensional CS theory. It is in fact a Schwarz type topological quantum field theory. In this paper, we shall call it topological coadjoint orbit (TCO) model for reference.

Wilson surfaces as partition functions
The natural question arises about the degree to which the above analysis can be extended and adapted to Wilson surfaces. In this paper, we shall consider Wilson surfaces in the simplest version of higher gauge theory, the strict one. As this is the only form of higher gauge theory that we shall deal with, we shall omit the specification 'strict' in the rest of our discussion.
Higher gauge symmetry hinges on Lie group crossed modules. For the purpose of this brief outline, it is sufficient to recall that a Lie group crossed module M consists of two Lie groups G, E together with two structure maps relating them and enjoying certain properties [44,45]. A higher gauge theory with structure crossed module M features a 1-form gauge field ω and a 2-form gauge field Ω valued respectively in the Lie algebras g, e of G, E [46,47].

Our approach to Wilson surfaces
The problem of obtaining a functional integral realizations of a Wilson surface, raised at the end of the previous subsection, has already been tackled in the literature from different perspectives [48][49][50]. In this subsection, we shall outline our approach to the subject firmly framed in higher gauge theory.
As already anticipated, higher gauge symmetry rests on Lie group crossed modules. Our handling of crossed modules is based on the derived set-up originally worked out in refs. [51,52]. It is in essence a superfield formalism providing an efficient way of encoding most of the structural features of crossed modules and proceeds by associating with any crossed module a derived Lie group, a graded Lie group with a structure determined by that of the crossed module. At the infinitesimal level, higher gauge symmetry is described by Lie algebra crossed modules. With any such module there is similarly attached a derived Lie algebra.
The whole derived set-up is compatible with Lie differentiation.
The relevant fields of a higher gauge theory are just crossed module valued inhomogeneous form fields. They can be dealt with in the derived framework in a very elegant and compact manner as derived fields, that is derived Lie group and algebra valued fields. The resulting derived field formalism allows one to cast any higher gauge theory as a derived gauge theory, basically an ordinary gauge theory with the derived group as gauge group. The higher gauge fields and gauge transformations, once expressed in derived form, can then be manipulated very much as their ordinary counterparts. In this manner, by highlighting the close correspondence of the higher to the ordinary setting, the derived formalism enables one to import many ideas and techniques of the latter to the former.
The derived field formalism has been successfully applied in ref. [28] to the formulation of 4-dimensional CS theory, a higher gauge theoretic enhancement of familiar 3-dimensional CS theory. The tight formal relationship of the 4dimensional theory to the 3-dimensional one brought out by the derived design has shown itself to be very useful in the analysis of the properties of the model.
It is reasonable to expect that the derived set-up could be the most appropriate formal framework also for the investigation of Wilson surfaces construed as higher gauge theoretic extension of Wilson loops.
Our approach to the realization of Wilson surfaces as partition functions consists therefore in extending the ordinary geometric or functional integral quantization schemes of the coadjoint orbits reviewed in subsect. 1.2 to a higher crossed module theoretic setting by relying on the derived formal set-up. This however is not simply a matter of a straightforward derived rewriting of these well-established approaches to coadjoint orbit quantization. There are in fact very basic elements of such schemes which do not have any fitting higher counterparts for reasons which we are going to survey momentarily. These issues are inevitably going to come to the surface in some form also in the derived approach.
To the best of our knowledge, there are no obvious counterparts of the notions of coadjoint action and orbit for Lie group crossed modules. Further, there is no fully developed representation theory and no analog of the highest weight theorem for crossed modules. While the derived formulation should provide in principle the definition of these higher objects and describe their properties, the way this is achieved in practice is far from clear.
Lie crossed modules belong to the realm of higher Lie theory. So, the geometric quantization of a derived coadjoint orbit, no matter the way it is conceived, presumably must be formulated in the framework of multisymplectic geometry [53]. Higher geometric quantization of multisymplectic manifolds is however a subject not fully understood yet (see however refs. [54][55][56][57] for a variety of ap-proaches to this issue). Since a 2-dimensional field theory must be the end result of quantization as already noticed, geometric quantization may alternatively be based on a symplectic a loop space [58,59], proceeding via transgression from a finite dimensional 2-plectic space on the lines of ref. [60]. The infinite dimensional geometry involved in this approach is however problematic to deal with.
The uncertainties affecting a workable theory of derived coadjoint orbits render problematic also the construction of the derived TCO sigma model on which the functional integral quantization of one such orbit should be based, if as expected crucial elements of the orbit's geometry are required by the model's formulation.
Higher gauge theory possesses in addition to a gauge symmetry also a gauge for gauge symmetry. The latter should emerge in the derived TCO model as a novel gauge symmetry with non counterpart in the ordinary model and requiring a special handling.
In fact, all the themes discussed above eventually emerge and are dealt with in the derived formulation, but in a novel and unified manner.

Plan of the endeavour
The present endeavour is naturally divided in two parts, henceforth referred to as I and II, of which the present paper is the first. The present paper, which constitutes part I of our endeavour, is devoted to derived KKS theory. In this section, we provide an introductory overview of this subject and an outlook on future developments.
Our presentation of derived KKS theory employs the language of graded differential geometry. The reader is referred to the appendixes for useful background and e.g. ref. [61] for a thorough exposition of this subject. The graded geometric set-up is naturally suited for the description of the higher geometric structures dealt with in this paper. It subsumes the ordinary differential geometric one, but at the same times it enriches and broadens it allowing for a series of non standard constructions which otherwise would not be possible.
The organizing principle of our construction of derived KKS theory is operational calculus, a formal extension of classic Cartan calculus that has found a wide range of applications in differential geometry and topology [62]. The operational framework furnishes indeed an efficient and elegant means of describing the basic geometry of the principal bundles occurring in KKS theory.

Plan of the part I
Paper I is organized in a number of sections and appendixes as follows.
In sect. 3, we survey the basic notions of the derived theory of Lie group crossed modules and review the main results of the derived field formalism used throughout the present work.
In sect. 4, we provide an overview of standard KKS theory and geometric quantization of coadjoint orbits. Our presentation of the subject is unconventional and partial: it relies on an operational description and touches the subject of quantization only marginally. It is however designed in a way that directly points to the derived extension constructed in the following section.
In sect. 5, we construct a derived KKS theory drawing inspiration from ordinary KKS theory as exposed in sect. 4, exploiting the advantages provided by the operational set-up and employing the full power of the derived approach. In particular, we introduce an appropriate definition of derived coadjoint orbit. We further elaborate a suitable notion of derived prequantum line bundle and connection thereof and use a connection's curvature to construct a derived presymplectic structure satisfying the appropriate Bohr-Sommerfeld quantization condition.
We also lay the foundations for derived orbit prequantization. The section ends with the determination the derived counterpart of the classic KKS symplectic structure.
Finally, in the appendixes of sect. A, we collect basic notions of graded geometry and operation theory used throughout the paper.

Overview of derived KKS theory
Since the remarks of subsect. 1.4 are merely qualitative, we provide in this subsection a somewhat more formal introduction to derived KKS theory to more precisely delineate the subject and facilitate the reading of the paper. A more rigorous and complete analysis of the material surveyed below is available in the main body of this paper.
As we indicated earlier, a Lie group crossed module M features a pair of Lie groups E and G, the module's source and target groups. To these there are added an equivariant morphism τ : E Ñ G and an action µ : GˆE Ñ E of G on E by automorphisms, the module's target and action structure maps, with certain natural properties.
The derived Lie group DM of M is a graded Lie group built out of G and the degree shifted variant er1s of the Lie algebra e of E, viz the semidirect product  [51,52] turns out to be quite natural and useful to this end, as DM can be regarded as a principal DM 1 -bundle over DM{ DM 1 , whose basic geometry is aptly described by a DM 1 -operation. In this way, derived maps on DM{ DM 1 of a given kind can be regarded as basic maps on DM of the same kind. The opera- A derived presymplectic structure of this kind belongs to the realm of multisymplectic geometry, since the 3-form component´iB is closed in the usual sense.
They further obey a derived Bohr-Sommerfeld quantization condition, as they arise from connections of L β .
For a suitably non singular derived presymplectic structure´iB of the above type, it is possible to define a natural derived prequantization map. There exists a distinguished subspace DFnc Ah pDMq of the derived Hamiltonian function space DFnc A pDMq closed under derived Poisson bracketing constituted by the prequantizable functions. The prequantization map assigns to any function F P DFnc Ah pDMq a first order differential operator p F acting on a space DΩ 0 h pL β q of 0-form derived sections of L β . The map is such that for any two functions F, G P DFnc Ah pDMq. However, there is no prequantum where σ and Σ are the degree 1-and 2-form components of the Maurer-Cartan form Σ. This is the derived KKS presymplectic structure, the sought for extension of the classic KKS symplectic structure (1.2.1).

Outlook
Though we have gone a long way toward reaching the goal of constructing a higher KKS theory, as essential geometrical underpinning of the realization of Wilson surfaces as partition functions, a few basic issues remain unsolved.
Derived geometric prequantization ostensibly does not admit a prequantum Hilbert space structure with respect to which the operators prequantizing derived Hamiltonian functions are formally Hermitian. This is due to the fact that the manifold on which the would be wave functions should be defined is a non negatively graded one of positive degree. On manifolds of this kind, only Dirac delta distributional integral forms can be integrated [63]. The derived formulation does not yield anything of this kind. The question arises about whether this is an essential impossibility or else new elements can be added to the theory allowing for the construction of a natural prequantum Hilbert space structure.
A definition of a derived analog of polarization, assuming that such a thing exists at all, is still to be achieved. The absence of a prequantum Hilbert space structure precludes in any case going beyond derived geometric prequantization into quantization proper along the familiar lines of ordinary quantization.
We believe that the limitations pointed out above of which derived prequantization suffers indicate that the appropriate geometric quantization of derived KKS theory cannot have some kind of quantum mechanical model, albeit exotic, as its end result but a two dimensional quantum field theory. This is in line with standard expectations to the extent to which derived KKS theory can be regarded as some kind of categorification of the ordinary theory. The derived TCO model studied in paper II is an attempt to concretize these intuitions.
The derived KKS theory formulated in this paper provides the geometric backdrop against which the derived TCO model is built and by virtue of which it has the form it does. In turn, the TCO model furnishes the physical motivation for the elaboration of the derived KKS set-up carried out in the present paper I.
In this section, we review the derived geometric framework originally elaborated in refs. [51,52]. By design, the derived set-up allows reformulating any theory whose symmetry is specified by a Lie group crossed module as one with a symmetry codified by a graded group, the associated derived group, e.g. 4-dimensional higher CS theory [18]. The derived framework also enables one to structure the higher KKS theory worked out in sect. 5 on the model of the standard theory as presented in sect. 4 and lies at the heart of the construction of the higher TCO model in close analogy to the ordinary one in II. By its many virtues, it will employed throughout our endeavour.
The derived set-up belongs to the realm of graded differential geometry in a deeper way than the set-up of employed in standard KKS and TCO theory. In fact, some of the structures featured in it cannot be expressed in the language of ordinary differential geometry in any straightforward if cumbersome way.

Lie group and algebra crossed modules and invariant pairings
Crossed modules encode the symmetry of higher gauge theory both at the finite and the infinitesimal level. A geometric formulation of higher KKS and TCO theory must necessarily set forth from them. In this subsection, we review the theory of Lie group and algebra crossed modules and module morphisms. A more comprehensive treatment complete with detailed definitions and relevant relations is provided in refs. [44,45] and the appendixes of ref. [28]. be presented later. They are defined for any Lie group G. The first is the inner automorphism crossed module of G, Inn G " pG, G, id G , Ad G q. The second is the (finite) coadjoint action crossed module of G, Ad * G " pg * , G, 1 G , Ad G * q, where g is the Lie algebra of G and its dual space g * is viewed as an Abelian group and Crossed module morphisms will occur only occasionally in our analysis, but A crossed module morphism ρ : Inn G 1 Ñ Inn G reduces to a group morphism χ : G 1 Ñ G. A crossed module morphism α : Ad * G 1 Ñ Ad * G is specified similarly by a group morphism λ : G 1 Ñ G and an intertwiner Λ : g 1 * Ñ g * of The structure of infinitesimal Lie crossed module axiomatizes likewise the setup consisting of a Lie algebra g and an ideal e of g equipped with the adjoint action of g. It is therefore the differential version of that of finite Lie crossed is the inner derivation crossed module of g, Inn g " pg, g, id g , ad g q. The second is the (infinitesimal) coadjoint action crossed module of g, Ad * g " pg * , g, 0 g , ad g * q, where g * is regarded as an Abelian algebra and 0 g : g * Ñ g is the vanishing Lie algebra morphism.
A morphism of infinitesimal Lie crossed modules is a map of crossed modules preserving the module structure describing a way such crossed modules are concordant. They constitute therefore the differential counterpart of the morphisms of finite Lie group crossed modules introduced above. More explicitly, a morphism p : m 1 Ñ m of Lie algebra crossed modules consists of two algebra morphisms h : g 1 Ñ g and H : e 1 Ñ e intertwining in the appropriate sense the structure maps t 1 , m 1 , t, m. We shall use often the notation p : m 1 Ñ m " pH, hq to indicate constituent morphisms of the crossed module morphism.
A crossed module morphism r : Inn g 1 Ñ Inn g reduces to an algebra morphism x : g 1 Ñ g. A crossed module morphism a : Ad * g 1 Ñ Ad * g is specified likewise by an algebra morphism l : g 1 Ñ g and an intertwiner L : g 1 * Ñ g * of ad g 1 * to ad g * ˝l .
Lie differentiation plays the same important role in Lie crossed module theory as it does in Lie group theory. With any Lie group crossed module M " pE, G, τ, µq there is associated the Lie algebra crossed module m " pe, g, 9 τ,9µ9q, where e, g are the Lie algebras of Lie groups E, G respectively and the dot notation 9 denotes Lie differentiation along the relevant Lie group, much as a Lie algebra is associated with a Lie group 1 . Similarly, with any Lie group crossed module morphism β : M 1 Ñ M " pΦ, φq there is associated the Lie algebra crossed module morphism 1 Note here that µ has three Lie differentials, 9µ, µ9, 9µ9, according to whether the G, the E and both the G and E arguments are subject to differentiation, respectively [28]. β : m 1 Ñ m " p 9 Φ, 9 φq, just as a Lie algebra morphism is associated with a Lie group morphism.
As examples, we mention that the Lie algebra crossed modules of the Lie group crossed modules Inn G and Ad * G we introduced above for any Lie group G are precisely Inn g and Ad * g, respectively, as expected.
Crossed modules with invariant pairing enter in many higher gauge theoretic constructions. Indeed, invariant pairings play in higher gauge theory a role similar to that of invariant traces in ordinary gauge theory and are basic structural elements of e.g. kinetic terms enjoying the appropriate symmetries. For similar reasons, they appear prominently also in derived KKS and TCO theory.
Following ref. [28], we define an invariant pairing on a Lie algebra crossed module m " pe, g, t, mq as a non singular bilinear form x¨,¨y : gˆe Ñ R enjoying the invariance property xad zpxq, Xy`xx, mpz, Xqy " 0 (3.1.1) for z, x P g, X P e and obeying the symmetry relation xtpXq, Y y " xtpY q, Xy. A Lie group crossed module with invariant pairing is a Lie group crossed module M " pE, G, τ, µq whose associated Lie algebra crossed module m " pe, g, 9 τ,9µ9q is one with invariant pairing. The invariance property however is required to hold not only at the infinitesimal level as in eq. (3.1.1) but also at the finite one, viz xAd apxq, µ9pa, Xqy " xx, Xy for a P G, x P g, X P e.
A Lie algebra crossed module m with invariant pairing is balanced, that is such dim g " dim e, by the non singularity of the pairing. This is not too restrictive.
In fact, any Lie algebra crossed module m can always be trivially extended to a balanced crossed module m c [26]. Similarly a Lie group crossed module M with invariant pairing is balanced, as dim G " dim E. Further, any Lie group crossed module M can always be trivially extended to a balanced crossed module M c .

Derived Lie groups and algebras
The notion of derived Lie group of a Lie group crossed module and the corresponding infinitesimal notion of derived Lie algebra of a Lie algebra crossed module were originally introduced in refs. [51,52].
The formal set-up of derived Lie groups and algebras is an elegant and convenient way of handling certain structural elements of the Lie group and algebra crossed modules appearing in higher gauge theory. It is a compact superfield formalism not unlike the analogous formalisms broadly used in supersymmetric field theories and particularly suited for a higher gauge theoretic setting.
The derived Lie group of a Lie group crossed module does not fully encode this latter, but it only describes an approximation of it in the sense of synthetic geometry. In fact, the target map of the crossed module is not involved in the definition of the derived group, nor could it be because, roughly speaking, the approximation is such to push the range of the target map away out of reach.
Similar considerations apply to the derived Lie algebra of a Lie algebra crossed module. The reader is referred to ref. [51] for a more precise discussion.
Consider a Lie group crossed module M " pE, G, τ, µq. The derived Lie group DM of M is the semidirect product group where er1s is regarded as a G-module through the G-action µ9. DM is therefore a graded Lie group concentrated in degrees 0, 1.

Each element P P DM has the formal representation
Ppαq " e αP p (3.2.2) with α P Rr´1s 2 , where p P G, P P er1s. p, P are called the components of P. In this description, the group operations of DM read as PQpαq " e αpP`µ9pp,Qqq pq, where P, Q P DM are any two group elements with Ppαq " e αP p, Qpαq " e αQ q.
A morphism β : M 1 Ñ M of Lie group crossed modules induces by means of its constituent group morphisms Φ : The notion of derived Lie group has an evident infinitesimal counterpart.
Consider a Lie algebra crossed module m " pe, g, t, mq. The derived Lie algebra of m is the semidirect product algebra where er1s is regarded as a g-module through the g-action m. Dm is so a graded Lie algebra concentrated in degree 0, 1.
Analogously, each element U P Dm has the formal representation with α P Rr´1s, where u P g, U P er1s are the components of U. The Lie bracket read in this set-up as with U, V P Dm any two algebra elements with Upαq " u`αU , Vpαq " v`αV .
A morphism p : m 1 Ñ m of Lie algebra crossed modules induces through its 2 In this paper, α P Rrps has degree p, because its geometrical degree is tacitly considered.
In refs. [51,52], α P Rrps has degree´p, because its algebraic degree is considered instead. In this paper, we work with the geometric degree rather than the algebraic one, as is more natural in a geometrical framework as the one illustrated here. See app. A.1 for more details.
underlying algebra morphisms H : e 1 Ñ e, h : g 1 Ñ g, a Lie algebra morphism Dp : Dm 1 Ñ Dm, analogously to the finite case.
The derived construction introduced above is fully compatible with Lie differentiation. This property is in fact essential for its viability. If M " pE, G, τ, µq is a Lie group crossed module and m " pe, g, 9 τ,9µ9q is its associated Lie algebra crossed module, then Dm is the Lie algebra of DM. Further, if β : M 1 Ñ M is a Lie group crossed module morphism and 9 β : m 1 Ñ m is the corresponding Lie algebra crossed module morphism, then 9 Dβ " D 9 β.

Derived field formalism
In this subsection, we shall survey the main spaces of Lie group and algebra crossed module valued fields using a derived field framework.
We assume that the fields propagate on a general non negatively graded manifold X. Later, we shall add the restriction that X is an ordinary orientable and compact manifold, possibly with boundary. To include also differential forms without renouncing to a convenient graded geometric description, the fields will be maps from the shifted tangent bundle T r1sX of X into some graded target manifold T . Below, we denote by MappT r1sX, T q the space of non negative internal degree internal maps from T r1sX into T . The more restricted space MappT r1sX, T q of ordinary maps from T r1sX to T can be also considered and treated much in the same way. See apps. A.1, A.2 for more details.
The fields we shall consider will be valued either in the derived Lie group DM of a Lie group crossed module M " pE, G, τ, µq or in the derived Lie algebra Dm of the associated Lie algebra crossed module m " pe, g, 9 τ,9µ9q (cf. subsect. 3.2).
A more comprehensive treatment of this kind of fields is given in ref. [51].
We consider first DM-valued fields. Fields of this kind are elements of the mapping space MappT r1sX, DMq. If U P MappT r1sX, DMq, then Upαq " e αU u UVpαq " e αpU`µ9pu,V qq uv, Next, we consider first Dm-valued fields. Fields of this kind are elements of the mapping space MappT r1sX, Dmq. If Φ P MappT r1sX, Dmq, then MappT r1sX, Dmq is the virtual Lie algebra of MappT r1sX, DMq. (For an explanation of this terminology, see ref. [51]).
As it turns out, the Dm-valued fields introduced above are not enough for our proposes. One also needs to incorporate fields that are valued in the degree shifted linear spaces Dmrps with p some integer. Together, they constitute the mapping space MappT r1sX, SDmq 4 . If Φ P MappT r1sX, Dmrpsq, then with components φ P MappT r1sX, grpsq, Φ P MappT r1sX, erp`1sq. There is a bilinear bracket that associates with a pair of fields Φ P MappT r1sX, Dmrpsq, Ψ P MappT r1sX, Dmrqsq a field rΦ, Ψs P MappT r1sX, Dmrp`qsq given by MappT r1sX, SDmq becomes in this way a graded Lie algebra. This contains the Lie algebra MappT r1sX, Dmq as its degree 0 subalgebra. 4 Throughout this paper, we denote by SE the degree extended form of a possibly graded vector space E. Explicitly, one has SE " À 8 p"´8 Erps. In refs. [51,52], the very same object is denoted instead as ZE, a notation that we shall employ in later sections with a different meaning.
An adjoint action of MappT r1sX, DMq on the Lie algebra MappT r1sX, Dmq and more generally on the graded Lie algebra MappT r1sX, SDmq is defined. For The adjoint action preserves Lie brackets as in ordinary Lie theory. Indeed, for As is well-known, in the graded geometric formulation we adopt, the nilpotent de Rham differential d is a degree 1 homological vector field on T r1sX, d 2 " 0.
On several occasions, the pull-backs dUU´1, U´1dU P MappT r1sX, Dmr1sq of the Maurer-Cartan forms of DM by a DM field U P MappT r1sX, DMq will enter our considerations. For these, there exist explicit expressions,
Next, let the Lie group crossed module M be equipped with an invariant pairing x¨,¨y (cf. subsect. 3.1). A pairing on the graded Lie algebra MappT r1sX, SDmq is induced in this way: for Φ P MappT r1sX, Dmrpsq, Ψ P MappT r1sX, Dmrqsq pΦ, Ψq " xφ, Ψ y`p´1q pq xψ, Φy. Note that pΦ, Ψq P MappT r1sX, Rrp`q`1sq. The field pairing p¨,¨q therefore has degree 1. p¨,¨q is bilinear. More generally, when scalars with non trivial grading are involved, the left and right brackets p and q behave as if they had respectively degree 0 and 1. For instance, pcΦ, Ψq " cpΦ, Ψq whilst pΦ, Ψcq " p´1q k pΦ, Ψqc if the scalar c has degree k. p¨,¨q is further graded symmetric, p¨,¨q is also non singular.
The field pairing p¨,¨q has several other properties which make it a very natural ingredient in the field theoretic constructions of later sections. First, p¨,¨q is DM- for U P MappT r1sX, DMq. By Lie differentiation, p¨,¨q enjoys also Dm invariance.
This latter, however, admits a graded extension, because of which Second, p¨,¨q is compatible with the derived differential d, i. e. the de Rham vector field d differentiates p¨,¨q through d, dpΦ, Ψq " pdΦ, Ψq`p´1q p pΦ, dΨq. MappT r1sX, SDm 1 q is a differential graded Lie subalgebra of MappT r1sX, SDmq, since it is invariant under the action of d as is evident from (3.3.10).
In concrete field theoretic analyses, one deals with functionals of the relevant derived fields on some compact manifold X. These are given as integrals on T r1sX of certain functions of FunpT r1sXq constructed using the derived fields.
Integration is carried out using the Berezinian X of X.

Ordinary geometric framework as a special case
The geometric framework employed in ordinary gauge theory as well as in the formulation of ordinary KKS theory and the TCO model is in fact a special case of the derived geometric framework. We devote this final subsection to the illustration of this point.
Let G be a Lie group. There exists a unique Lie group crossed module with target group G and trivial source group E " 1, since the target morphism τ and the action map µ in this case can be only the trivial ones. With a harmless abuse of notation, we shall denote this crossed module also by G, since it codifies the Lie group structure of G in a manner equivalent to the usual one. Similarly, for a Lie algebra g, there exists a unique Lie algebra crossed module with target algebra g and trivial source algebra e " 0, since the target morphism t and the action map m again can be only the trivial ones. We shall denote this crossed module also by g, since it provides an equivalent codification of the Lie algebra structure of g. This crossed module reinterpretation of ordinary Lie theory is compatible with Lie differentiation: g is the Lie algebra of the Lie group G if and only if g is the Lie algebra crossed module of the Lie group crossed module G.
We note however that while a Lie algebra g can support an invariant pairing which is not strictly necessary in the ordinary theory but it is essential in the higher derived version. It is also incomplete, because we have deliberately avoided to delve in depth into quantization proper, aware that the problem we aim to eventually solve is to some extent related, although not fully equivalent, to one of quantization of a 2-plectic manifold, a delicate issue that is not completely settled yet in the literature and whose solution lies beyond the scope of the present work

Coadjoint orbits
Let G be a Lie group and g be its Lie algebra. For any element λ P g * of the dual vector space of g, the coadjoint orbit O λ of λ is the submanifold of g * spanned by the coadjoint action of G on λ. Explicitly, O λ " tAd * γpλq|γ P gu, where Ad * denotes the coadjoint representation of G, the dual of the adjoint representation Ad with respect to the canonical duality pairing of g and g * .
where ZG λ is the invariance subgroup of λ in G. Suppose that G is compact and connected. Then, ZG λ always contains a maximal torus T of G. If ZG λ " T, λ is said to be regular. In that case, O λ " G{T. Below, we shall rely heavily on the homogeneous space description of O λ concentrating on the regular case.

Operational description of homogeneous spaces
The fact that coadjoint orbits are instances of homogeneous spaces opens the possibility of an operational formulation of KKS theory. We shall however consider the problem we intend to study from a wider perspective as follows. If G is a Lie group and G 1 is a subgroup of G, then G can be regarded as a principal G 1 -bundle over the homogeneous space G{G 1 . As such, G is amenable to the operational description, from which much information about G{G 1 can be extracted.
For the operational analysis, we need to begin with appropriate coordinates of the shifted tangent bundle T r1sG. By the isomorphism T r1sG » Gˆgr1s, we can use as coordinates of T r1sG variables γ P G and σ P gr1s. They obey where d is the de Rham differential regarded as a homological vector field on T r1sG. σ is therefore identified with the Maurer-Cartan form of G.
The right G 1 -action of G induces an action on T r1sG, which reads as with q P G 1 in terms of the coordinates γ, σ. From here, we can readily read off the action of the derivations of the associated right g 1 -operation of G on γ, σ, As a group, G is characterized also by a left G-action, which we shall write in right form for convenience. In terms of the coordinates γ, σ, it reads as with e P G. Its main feature is the invariance of the Maurer-Cartan form. Below, we shall employ extensively the associated left g-operation. The derivations of this latter act as The right G 1 -and left G-actions commute, as is evident from their coordinate expressions. Consequently, the derivations of the right g 1 -and left g-operations also commute in the graded sense.
Though automorphism symmetry is rarely mentioned in standard presentations of KKS theory, it is nevertheless a feature of the KKS set-up inherent in its bundle theoretic nature. We consider also this aspect in the above wider perspective. The automorphisms of the principal G 1 -bundle G are the fiber preserving invertible maps of G into itself compatible the right G 1 -action. Concretely, they are maps ψ P MappG, G 1 q with certain basicness properties under the right G 1 -action and so naturally constitute a distinguished subgroup Aut G 1 pGq of the infinite dimensional Lie group MappG, G 1 q. Aut G 1 pGq is selected by the condition with q P G 1 , where ψ Rq denotes the pull-back of ψ by the right G 1 -action, i.e.
ψ Rq pγq " ψpγ Rq´1 q. In the right g 1 -operation, so, ψ obeys for x P g 1 , where we view ψ as a map of MappT r1sG, G 1 q relying on the identity The automorphism group Aut G 1 pGq of the G 1 -bundle G acts on G, the action being a left one. In terms of the coordinates γ, σ, the action of an automorphism The second relation follows form the first one and the first relation (4.2.1). Notice that p ψ γq Rq " ψ Rq γ Rq as required by compatibility. By for e P G. In the left g-operation, therefore, for h P g we have j Lh ψψ´1 " 0, j Lh ψψ´1 " 0. Recall that a character of T is a Lie group morphism of T into Up1q, so an element ξ P HompT, Up1qq. With the character ξ, we can associate a unitary line " ξ denotes the equivalence relation on GˆC yielded by the identification pγ, zq " ξ pγq´1, ξpqqzq (4.3.1) with γ P G, z P C and q P T.
A p-form section of L ξ is a map s P MappT r1sG, Crpsq obeying with x P t in the right t-operation, where 9 ξ P Hompt, up1qq denotes the Lie differential of ξ. p-form section of L ξ form a space Ω p pL ξ q.
The operational framework allows for a total space description of Up1q gauge theory, in particular of connections. A unitary connection of the line bundle L ξ is a map a P MappT r1sG, up1qr1sq obeying the second relation being the Bianchi identity. b obeys It acts also on a connection a of L ξ in the familiar manner u a " a´duu´1. The curvature b of a is then gauge invariant

Poisson structures on a regular homogeneous space
In KKS theory, the prequantization of a coadjoint orbit requires that its symplectic structure matches the curvature of a suitable prequantum line bundle. In this subsection, working from a broader perspective as done before, we study the presymplectic structures on a regular homogeneous space G{T of the kind con- for x P t, and have the property that there is a vector field P f P VectpGq with   We can then expect that the Poisson bracket structure t¨,¨u a will exhibit left symmetry properties. In fact, the left G-action is Hamiltonian. The Hamiltonian function q a phq P Fun a pT r1sGq corresponding the Lie algebra element h P g is q a phq "´ij Lh a. The Hamiltonian property of q a phq follows from the left invariance of a and the commutativity of the right Tand left G-actions, which imply that q a phq satisfies (4.4.1) and (4.4.2) with P qaphq " S Lh . Crucially, the map q a : g Ñ Fun a pT r1sGq is equivariant and constitutes a representation of g, as for h, k P g, l Lh q a pkq " q a prh, ksq " tq a phq, q a pkqu a , (4.4.7) a relation that characterizes q a as the moment map of the left G-symmetry.
The Poisson bracket structure t¨,¨u a has simple gauge invariance properties.
Since the connection a is restricted to be left invariant by We conclude this subsection by observing that the Poisson bracket structure t¨,¨u a in general is not induced by a genuine symplectic structure on G{T unless the right T-action vertical vector fields S Rx with x P g are the only vector fields V P VectpGq such that j V b " 0. In KKS theory, this is the situation customarily considered because in such a case Fun a pG{Tq " FunpG{Tq and standard geometric prequantization is possible.

Prequantization
The datum on which prequantization is based is the Poisson structure of the regular homogeneous space G{T associated with a unitary connection a of the line bundle L ξ of a character ξ of T. Prequantization requires that the underlying presymplectic structure´ib is a symplectic one. We therefore assume that the curvature b of a is non singular, so that j V b " 0 with V P VectpGq only if V " S Rx for some x P g.
Prequantization takes its start by setting for f, g P Fun a pG{Tq. The basic requirement of prequantization is thus met.
By virtue of (4.4.6), for h P g the operator p q a phq associated with the left G-symmetry moment map q a phq takes the form p q a phq " il Lh , for h, k P g, as expected.
The space Ω 0 pL ξ q of 0-form sections of L ξ has a Hilbert structure: for any two sections s, t P Ω 0 pL ξ q xs, ty " G{T p´ibq n s * t, The basic data of the KKS construction are a compact semisimple Lie group G and a regular element λ P g * of the dual of the Lie algebra g of G. So, the coadjoint orbit O λ " G{T for some maximal torus T of G. We can so rely on the results we obtained in the previous subsections.
The Lie algebra t of T is a maximal toral subalgebra of g. By restricting λ to t, we obtain an element λ P t * denoted in the same way for simplicity. If λ{2π P Λ G * , the dual of the integral lattice Λ G of t, then there exists a character ξ λ P HompT, Up1qq given by ξ λ pe t q " e ixλ,ty (4.6.1) for t P t. With λ, we can thus associate a unitary line bundle L λ :" L ξ λ using the construction of subsect. 4.3.
L λ is equipped with a canonical unitary connection a λ "´ixλ, σy.  The whole above structure has natural invariance properties under the automorphism group action of the principal T-bundle G. With any automorphism ψ P Aut T pGq, there is associated a map u λ P MappT r1sG, Up1qq by (4.6.5) As dψψ´1 P MappT r1sG, tr1sq represents an element of Ω 1 pG, tq with periods in the integral lattice Λ G and λ P Λ G * , u λ is singlevalued as required. By (4.2.7), u λ obeys relations (4.3.6) and so is a gauge transformation, u λ P GaupL λ q. A group morphism ψ Ñ u λ from the automorphism group Aut T pGq of G to the gauge transformation group GaupL λ q of L λ is established by (4.6.5) in this way.
For an automorphism ψ P Aut T pGq, let ψ a λ be the connection given by (4.6.2) with σ replaced by its transform ψ σ (cf. eqs. (4.2.8)). Then, ψ a λ " u λ a λ where the expression in the right hand side is given by (4.3.8). Consequently, one has By the above findings, it appears that L λ is a prequantum line bundle on G{T.
A prequantum Hilbert space structure is then defined as described in subsect. 4.5.
To get the quantum Hilbert space, one needs a polarization. This is obtained by endowing G{T with a complex structure J realized as an integrable complex splitting of the complexified tangent bundle T C pG{Tq. Because of the left Ginvariance of the whole geometric set-up, it is enough to provide the splitting at the identity coset T of G{T. Since T CT pG{Tq » pg a tq b C, the complexification of the orthogonal complement of t with respect to the Cartan form of g, we may choose pg a tq 1,0 " À αP∆`e α , pg a tq 0,1 " À αP∆`e´α , where ∆`is a set of positive roots of g and the e α are the root spaces of g b C. Further, J is compatible with the symplectic structure´ib λ in the sense that´ib λ p¨, J¨q constitutes a left G-invariant Kaehler metric on G{T whose Kaehler form is precisely´ib λ . In this way, once endowed with the complex structure J, G{T is a Kaehler manifold.
L λ then turns out to be a holomorphic line bundle on G{T. With the above polarization of G{T available, it is possible to define the quantum Hilbert space H λ as the space of square integrable holomorphic sections of L λ , 6.6) provided that this latter is non vanishing. According to the Borel-Weil-Bott theorem [41,42], H 0 B pG{T, L λ q is non zero precisely when λ P Λ wG`, the lattice of dominant weights of G. The highest weight theorem establishes a one-to-one correspondence between Λ wG`a nd the set of equivalence classes of irreducible representations of G. The theorem provides in this way the grounding for the relation between coadjoint orbit geometric quantization and partition function description of Wilson loops to be discussed in II.
In this section, working systematically within the derived framework of sect. 3, we shall elaborate a derived KKS theory on the model of the standard KKS theory expounded in sect. 4. In ref. [28], we showed that higher 4-dimensional CS theory admits a derived formulation that very closely parallels that of its familiar ordinary 3-dimensional counterpart. The idea is to follow the same path and formulate higher KKS theory by reproducing the constructions of its ordinary counterpart in a derived perspective. This approach will lead us far but it will not solve all the problems. The decisive step of geometric quantization, the construction of a prequantum Hilbert space and a polarization, remains elusive even though the immediate reasons for this seem clear. The content of this section is mostly technical. The nature of the topics covered makes this essentially unavoidable. We collect anyway the relevant component identities at the end of each subsection, to make a first reading easier.

Derived coadjoint orbits and regular elements
Our construction of derived KKS theory must necessarily begin with the definition of the notions of derived coadjoint orbit and regularity. As anticipated, we shall do so by taking the corresponding notions of ordinary KKS theory as a model (cf. subsect. 4.1) and using firmly the derived set-up of sect. 3 as our framework.
In ordinary KKS theory, a coadjoint orbit always refers to some element λ P g * of the dual space of the Lie algebra g of the given Lie group G. In derived KKS theory, the relevant Lie group crossed module M " pE, G, τ, µq is as a rule equipped with an invariant pairing x¨,¨y (cf. subsect. 3.1). In this way, e, as a vector space, can be identified with g * . Further, by (3.1.3), the G-action of e, µ9, is dual with respect to the pairing to the adjoint G-action of g, Ad. It is so sensible that a derived coadjoint orbit should refer to some element Λ P e. It could do so even in the absence of an invariant pairing, µ9 playing the role of the coadjoint action.
With the above considerations in mind, we introduce a few pertinent concepts of crossed module theory. Let M " pE, G, τ, µq be a Lie group crossed module and let M 1 " pE 1 , G 1 q be a crossed submodule of M. We can associate with M 1 two subgroups of G and one of E. The centralizer ZG 1 of G 1 is the subgroup of G constituted by those elements a P G such that aba´1 " b for b P G 1 . Similarly, the µ-centralizer of µ ZE 1 of E 1 is the subgroup of G of the elements a P G such that µpa, Bq " B for B P E 1 . Finally, the µ-trivializer µ FG 1 of G 1 is the Lie subgroup of E formed by those elements A P E obeying µpb, Aq " A for b P G 1 . It can be easily verified that the structure ZM 1 " pµ FG 1 , ZG 1 X µ ZE 1 q is a crossed submodule of M, that we call the centralizer crossed module of M 1 .
The case we have in mind features a crossed module M and a crossed submodule M 1 of M that is a submodule of its centralizer ZM 1 . This property implies restrictively that both E 1 and G 1 are Abelian and that the G 1 -action on E 1 is trivial.
We now have at our disposal all the elements required for the definition of the derived coadjoint orbit of an element Λ P e. We denote by E Λ and G Λ the subgroups of E and G constituted by the elements of the form e xΛ and e y 9 τ pΛq with x, y P R, respectively. Then, the structure M Λ " pE Λ , G Λ q is a crossed submodule of M, the 1-parameter submodule generated by Λ. The centralizer crossed module of M Λ , which we shall call the centralizer crossed submodule of Λ in M for clarity, is ZM Λ " pZE Λ , µ ZE Λ q, where ZE Λ and µ ZE Λ are here the subgroups of E and G of the elements A P E and a P G such that Ad ApΛq " Λ and µ9pa, Λq " Λ, respectively. As M Λ is a crossed submodule of ZM Λ , this is a case of the kind considered in the previous paragraph.
The derived KKS coadjoint orbit of Λ is the derived homogeneous space in Inn G N is the crossed module Inn Z G Λ pN X Z G Λq, where Z G Λ is the subgroup of G of the elements a P G such that Ad apΛq " Λ. If G is compact and T is a maximal torus of G, the maximal toral crossed submodule of T in Inn G N is the crossed module Inn T pN X Tq. Λ is regular precisely when Z G Λ " T for some maximal torus T, i.e. when Λ is regular as an element of g in the usual sense.
With any central extension 1 / / C ι / / Q π / / G / / 1 of Lie groups, there is associated a Lie group crossed module CpQ π / / Gq " pQ, G, π, αq, where the action α is given by αpa, Aq " σpaqAσpaq´1 for a P G, A P Q with σ : G Ñ Q a section of the projection π, i.e. π˝σ " id G , whose choice is immaterial. If Λ P q, there is an induced Lie group central extension 1 and σ Z G Λ are the subgroups of Q and G of the elements A P Q and a P G such that Ad ApΛq " Λ and Ad σpaqpΛq " Λ, respectively. The centralizer crossed submodule of Λ in CpQ π / / Gq is so the crossed module CpZ Q Λ π / / σ Z G Λq. If G is compact and T is a maximal torus of G, there is an induced group sequence

the elements
A P A such that σpaqAσpaq´1 " A for a P T. This sequence is generally not exact, as π may fail to be onto. It is if σ can be chosen such that σ : T Ñ Q is a group morphism for one and so all maximal tori T. In that case, the maximal toral crossed submodule of T in CpQ π / / Gq is the crossed module Cpσ F Q T π / / Tq. Λ is regular precisely when σ Z G Λ " T for some maximal torus T. This being so depends ultimately on the form of the projection π.
A Lie group crossed module Dpρq " pV, G, 1 G , ρq can be associated with the data consisting of a Lie group G, a vector space V regarded as an Abelian group, the trivial morphism 1 G of V into G and a representation ρ of G in V. In this case, the centralizer crossed submodule of an element Λ P v is non trivial only if its invariance subgroup Inv G,ρ Λ is non trivial and in the compact case Λ is regular only provided that the representation ρ is trivial. 2) due to the special form of the derived differential d (cf. eq.

Operational description of derived homogeneous spaces
(3.3.10)), which allows Q´1dQ to be non vanishing.
In the operational set-up associated with the right DM 1 -action, Γ behaves as with X P Dm 1 . By virtue of (5.2.1), Σ then satisfies

Target kernel symmetry
The target kernel symmetry of a derived homogeneous space is the counterpart of the left symmetry of an ordinary homogeneous space described in subsect. 4.2.
Its properties however turn out to be a bit more involved. The symmetry is the topic of this subsection.
Every Lie group crossed module M " pE, G, τ, µq is characterized as such by a canonical crossed submodule, the target kernel crossed module M τ " pker τ, Gq.
With this there is associated a distinguished symmetry of the homogeneous space DM{ DM 1 of the kind studied in subsect. 5.2.
As a manifold, DM is endowed with a left DM τ -action, which we shall write in right form for convenience. This has a fairly simple expression in terms derived coordinates Γ, Σ. On the base coordinate Γ, the action reads as with E P DM τ . The action on the fiber coordinate Σ is trivial The left DM τ -action manifestly commutes with the right DM 1 -action of eqs. The associated derived automorphism symmetry is the subject of this subsection.
The full content of the derived automorphism group Aut DM 1 pDMq is specified by the requirement that automorphisms Ψ P Aut DM 1 pDMq transform under the right DM 1 -action of DM according to  l Rx,X ψψ´1 "´x`Ad ψpxq, (5.4.13) l Rx,X Ψ "´9µ9px, Ψ q´X`µ9pψ, Xq. Having defined the notion of character, we consider next a compact Lie group crossed module M " pE, G, τ, µq (cf. subsect. 5.1) and a maximal toral crossed submodule J " pH, Tq of M equipped with a character β " pΞ, ξq.
Consider the complex vector space The coordinate Z P DC has therefore a component expansion of the form with α P Rr´1s, where z P C, Z P Cr1s. Conjugation in DC is defined such that Z * has components z * , Z * . DC has further a natural field structure with WZpαq " wz`αpzW`wZq, Z´1pαq " z´1´αz´2Z. where Ξ " ξ˝τ .
We introduce next the vector bundle on DM{ DJ defined as where " β is the equivalence relation pΓ, Zq " β pΓQ, ρ β pQ´1qpZqq The unitary connections of L β form an affine space ConnpL β q.
When a connection A of L β is assigned, the derived covariant differential of a p-form section S of L β can be defined, By virtue of (5.5.12), (5.5.13) and (5.5.14), (5.5.15), d A S is a p`1-form section of L β as required. The derived Ricci identity holds, As in earlier similar instances, for concreteness and completeness, we express the above relations in terms of the components of the fields involved. This will also exemplify how of the derived framework effectively encodes non trivial higher gauge theoretic relations. The uninterested reader can skip directly to the paragraph following eq. (5.5.57).
Consider a p-form section S of L β with components s, S. In terms of these latter, the horizontality and covariance conditions (5.5.12), (5.5.13) take the form j Rx,X s " 0, (5.5.28) j Rx,X S " 0, (5.5.29) l Rx,X s " 9 ξpxqs, (5.5.30) l Rx,X S " 9 ξpxqS`p´1q p 9 ΞpXqs, (5.5.31) where x, X are the components of the Lie algebra element X.
A connection A of L β can similarly be expressed through its components a, A. By (5.5.14), (5.5.15), a, A obey j Rx,X a "´9 ξpxq, We conclude this subsection with an examination of basic differential topological issues concerning derived unitary line bundles. Is there a topological classification of such bundles based on a cohomological characterization of their curvature analogous to that of ordinary line bundle? This is a far reaching question whose full answer lies beyond the scope of the present work. We shall limit ourselves to the following considerations.
We start with the following premises. First, the derived line bundle L β of a character β of J is a graded vector bundle on the graded manifold DM{ DJ.
Second, a unitary connection A of L β is a non ordinary DInn up1qr1s-valued internal map on the graded manifold T r1s DM with special properties. Similarly, the curvature B of A is a non ordinary DInn up1qr2s-valued internal map on T r1s DM. The bundle theoretic set-up we are concerned with here, therefore, is a non standard one. This renders the application of standard differential topological results problematic. We may try to tackle the issue anyway from the derived standpoint on which our whole approach is based.
The characters β of the maximal toral crossed submodule J are in one-toone correspondence with the characters ξ of the maximal torus T of G. These constitute a lattice, the dual lattice Λ G * of the integer lattice Λ G of t. The derived line bundles L β of the characters β thus form a discrete family of vector bundles on DM{ DJ organized as a kind of lattice isomorphic in some sense to Λ G * .
Below, we shall consider derived cohomology, that is the cohomology of the complex of Inn Up1q-valued forms on DM{ DJ and the derived differential d, and so the terms 'closed' and 'exact' will tacitly refer to such complex. We saw earlier that, due to ( In customary de Rham cohomology, B is so exact whilst b is not even closed.
Hence, the derived Chern class cpL β q cannot be straightforwardly framed in de Rham cohomology.
A relationship of the theory of derived line bundles with connection to the theory of twisted Hermitian line bundles of bundle gerbes proposed by C. Rogers in ref. [53] is conceivable but remains to be elucidated. The point is that our approach to derived line bundles, based as it is on an operational set-up, is essentially a total space one while the Rogers's is a base space one. Clarifying the nature of the relations between the two standpoints, if a relation does exist, would require further work. where S RX is the vertical vector field of the DJ-action corresponding to X.
Next, we consider the set Vect iA pDMq of all vector fields V P VectpDMq leaving B invariant, that is satisfying It is straightforward to verify that Vect iA pDMq is a Lie subalgebra of VectpDMq and that Vect kA pDMq is a Lie ideal of Vect iA pDMq. We can thus construct the quotient Lie algebra Vect qA pDMq " Vect iA pDMq{ Vect kA pDMq.
Let Vect v pDMq be the Lie subalgebra of VectpDMq of the vertical vector fields S RX with X P Dj. Vect v pDMq is a Lie subalgebra of Vect iA pDMq, as l RX B " 0 for X P Dj by (5.5.19). The Lie derivatives l S RX are thus derivations of the Lie algebra Vect iA pDMq preserving the Lie ideal Vect kA pDMq. They so induce derivations l qS RX of the Lie algebra Vect qA pDMq defined in the previous paragraph. Vect bA pDMq " Ş XPDj ker l qS RX is consequently a Lie subalgebra of Vect qA pDMq. Since we aim eventually to construct a structure on the derived homogeneous space DM{ DJ, Vect bA pDMq is the relevant vector field algebra.
Explicitly, Vect bA pDMq consists of the vector fields V P VectpDMq obeying (5.6.1) and defined modulo vector fields V 1 P VectpDMq obeying (5.6.2) with the property that l S RX V obeys (5.6.2) for all X P Dj.
The derived real line is the real vector space DR " R ' Rr1s. DR has properties completely analogous to those of the derived complex line DC studied earlier in subsect. 5.5, in particular it is a field (cf. eqs. (5.5.2), (5.5.2)). The space MappT r1s DM, DRrpsq of degree p derived R-valued functions is then available. It has a component reduction analogous to that of its complex counterpart (cf. eq. (5.5.10)) and is acted upon by the derived differential d (cf. eq. (5.5.11)).
Below, we shall concentrate on the space DFncpDMq " MappT r1s DM, DRr0sq of derived functions of DM. An element F P DFncpDMq is therefore a derived field: Fpαq " f`αF with f P MappT r1s DM, Rr0sq, F P MappT r1s DM, Rr1sq.
DFncpDMq has a natural algebra structure induced by the field structure of DR. There exists a special mapping : DFncpDMq Ñ DFncpDMq defined componentwise as Fpαq " f . is an algebra morphism. Its range DFnc pDMq is therefore a subalgebra of DFncpDMq, which by construction is isomorphic to MappT r1s DM, Rr0sq. In what follows, the functions of DFnc pDMq will be called short and DFnc pDMq will be referred to as the short subalgebra.
By the isomorphism DFnc pDMq » MappT r1s DM, Rr0sq noticed above, for any F P DFnc pDMq and V P VectpDMq, the product FV " f V is defined, rendering VectpDMq a DFnc pDMq-module. This restricted module structure is responsible for the special features of the derived Poisson theoretic construction carried out below.
Again, as we aim eventually to obtain a structure on DM{ DJ, we restrict the range of derived functions we consider to the subset DFnc b pDMq of DFncpDMq of the basic ones. This is formed by the derived functions F P DFncpDMq obeying DFnc b pDMq is a subalgebra of DFnc b pDMq, its short subalgebra.
Suppose that F P DFnc b pDMq and P F P VectpDMq obey the equation Then, as is straightforward enough to check, P F P Vect iA pDMq, P F is determined by eq. (5.6.5) mod Vect kA pDMq and l S RX P F P Vect kA pDMq for all X P Dj.
Thus, P F P Vect bA pDMq and as such P F is uniquely determined by F. If the vector field P F exists, F is said to be a Hamiltonian derived function and P F is called the Hamiltonian vector field of F as (5.6.5) is clearly a derived extension of (4.4.2), the relation defining Hamiltonian functions and vector fields in the standard theory. We denote by DFnc A pDMq the set of the Hamiltonian functions of DFnc b pDMq and by Vect A pDMq that of the Hamiltonian vector fields of Vect bA pDMq. We denote similarly by DFnc A pDMq the set of the short functions of DFnc A pDMq and by Vect A pDMq that of the associated vector fields of Vect A pDMq.
DFnc A pDMq is only a vector subspace of DFnc b pDMq, while DFnc A pDMq is a subalgebra of DFnc b pDMq. The reason for this somewhat surprising difference in their algebraic properties can be ultimately traced back to the fact that VectpDMq is not a DFncpDMq-module but only a DFncpDMq-module. We shall come back to this point momentarily.
It can also be seen that Vect A pDMq is a Lie subalgebra of Vect bA pDMq and Mimicking a similar construction of Poisson theory, we may define a bracket t¨,¨u A : DFnc A pDMqˆDFnc A pDMq Ñ DFnc A pDMq as follows. It can be shown that if F, G are Hamiltonian derived functions and P F , P G are the associated Hamiltonian vector fields, then´ij P G j P F B is also a Hamiltonian function and rP F , P G s is the associated Hamiltonian vector field. Taking relation (4.4.3) of the standard theory as a model, we then define the derived Poisson bracket tF, Gu A "´ij P G j P F B. (5.6.6) This bracket is bilinear and antisymmetric. Unlike its ordinary counterpart, however, it fails to satisfy the Jacobi identity. One has indeed for F, G, H P DFnc A pDMq, the Jacobiator in the right hand side being given by xF, G, Hy A " idj P H j P G j P F B. (5.6.8) Note that, even though the derived Lie group DM is a non negatively graded manifold and´iB is a degree 2 derived function, xF, G, Hy A is generally non vanishing because the degree 3 component B of B is not necessarily annihilated by j P H j P G j P F . By virtue of (5.6.5) and d-exactness, however, the Hamiltonian vector field P xF,G,Hy A of xF, G, Hy A vanishes and so xF, G, Hy A is central, txF, G, Hy A ,¨u A " 0. (5.6.9) DFnc A pDMq equipped with the bracket t¨,¨u A can so be described as a twisted Lie algebra. The connection A is called simple if the Jacobiator vanishes.
It is remarkable that the twisted Lie bracket t¨,¨u A just introduced restricts to a honest Poisson bracket on the short subalgebra DFnc A pDMq Ă DFnc A pDMq.
Indeed, on DFnc A pDMq the bracket t¨,¨u A turns out to be Leibniz in both arguments and the Jacobiator x¨,¨,¨y A to vanish identically. As a Lie algebra, DFnc A pDMq is a Lie ideal of DFnc A pDMq. With this we mean that the Lie bracket tF, Gu A is short when at least one of its two arguments is and that the Jacobiator xF, G, Hy A vanishes when at least one of its three arguments is short.
The function space DFnc A pDMq together with the short function algebra For F, G P DFnc A pDMq, the Lie bracket tF, Gu A has a component expression of the form tF, Gu A pαq " tf, gu a,A`α tF, Gu a,A . The notation used here is merely suggestive and should be handled with care, since in general tf, gu a,A , tF, Gu a,A both depend on f, F , g, G, as the derived expression (5.6.6) gives tf, gu a,A "´ij P g,G j P f,F b, (5.6.12) tF, Gu a,A "´ij P g,G j P f,F B. For F, G, H P DFnc A pDMq, the Jacobiator xF, G, Hy A has the component structure xF, G, Hy A pαq " xf, g, hy a,A`α xF, G, Hy a,A , where again the notation used is merely suggestive for reasons already explained. The Jacobiator compo-nents xf, g, hy a,A , xF, G, Hy a,A are given by xf, g, hy a,A "´ij P h,H j P g,G j P f,F B, (5.6.14) xF, G, Hy a,A " idj P h,H j P g,G j P f,F B. (5.6.15) Notice that they depend only on the degree 3 components B of B. A term of the form idj P h,H j P g,G j P f,F b does not appear in the right hand side of (5.6.14), since it vanishes identically for grading reasons.
The notion of derived Poisson structure worked out above is reminiscent of that of pre-2-symplectic Lie 2-algebra proposed by Rogers in [64]. Roughly, the latter is obtained from the former by replacing the derived gauge field and curvature A and B by their degree 2 and 3 components A and B, respectively.
We consider the set Vect iA pDMq of all vector fields V P VectpDMq satisfying The Hamiltonian functions are the elements of F P DFnc b1 pDMq for which there exists a vector field P F P Vect bA , necessarily unique, such that We denote by DFnc * A1 pDMq the subspace of DFnc b1 pDMq they form and by DFnc * A pDMq " DFnc b0 pDMq ' DFnc * A1 pDMq the corresponding subspace of DFnc b pDMq.
We finally introduce a unary, a binary and a trinary bracket t¨u A , t¨,¨u A , t¨,¨,¨u A on DFnc * A pDMq whose only non zero instances are tf u A " df, (5.6.19) tF, Gu A "´ij P G j P F B, (5.6.20) tF, G, Hu A " ij P H j P G j P F B (5.6.21) with f P DFnc b0 pDMq, F, G, H DFnc * A1 pDMq. By means of a´1 degree shift placing DFnc b0 pDMq, DFnc * A1 pDMq respectively in degree´1, 0, the graded vector space DFnc * A pDMq equipped with the brackets t¨u A , t¨,¨u A , t¨,¨,¨u A is a semistrict Lie 2-algebra [45].
By (5.6.11), we have a mapping λ : DFnc A pDMq Ñ DFnc * A1 pDMq given by with H, K P Dm τ . From (5.5.14), (5.5.15)  In subsect. 5.6, we exhibited a mapping λ : DFnc A pDMq Ñ DFnc * A1 pDMq connecting the Hamiltonian derived and pre-2-plectic function spaces preserving the bracket structure in the appropriate sense (cf eqs. (5.6.22), (5.6.23)). λ also relates the derived and homotopy moment map data in a consistent manner, This clarifies again the relationship between our constructions and those of ref.

Derived prequantization
In derived KKS theory, prequantization is implemented along lines analogous to those of ordinary KKS prequantization as reviewed in subsect. 4.5. Derived prequantization is however more involved than its ordinary counterpart for a number of reasons. In this subsection, we illustrate its construction and the problems which affect it.
The foundation on which derived prequantization rests is the derived Poisson structure of the regular derived homogeneous space DM{ DJ associated with a unitary connection A of the derived line bundle L β of a character β of J introduced and studied in subsect. 5.6. This however is not a genuine Poisson structure.
The derived Poisson bracket t¨,¨u A has a generally non trivial Jacobiator and the Hamiltonian derived function space DFnc A pDMq is not an algebra. It is therefore necessary to impose a suitable condition on A and to restrict the range of Hamiltonian functions to an appropriate subspace DFnc Ah pDMq of the space DFnc A pDMq. Upon doing so, t¨,¨u A becomes a genuine Lie bracket as required by prequantization. It can then be shown that with any function F P DFnc Ah pDMq there is associated an endomorphism p F of a certain subspace DΩ 0 h pL β q of the space DΩ 0 pL β q of 0-form sections of L β such that for F, G P DFnc Ah pDMq, one has r p F, p Gs " i { tF, Gu A . L β gets so interpreted as the derived prequantum line bundle.
In this way, a derived KKS prequantization is defined associating an operator with any suitably restricted Hamiltonian derived function such that the resulting operator commutator structure is fully compatible with the derived Poisson bracket structure. However, there apparently is no natural derived prequantum Hilbert space structure on DM{ DJ with respect to which the operators yielded by derived prequantization are formally Hermitian. This seems to be in line with the findings of several higher prequantization schemes available in the literature, see in particular refs. [54,56,57].
Derived prequantization requires the underlying derived presymplectic structure to be symplectic as in the ordinary setting. We thus assume that the curvature B of the connection A is non singular. In this way, if V P VectpDMq and j V B " 0, then V " S RX for some X P Dj pointwise on DM.
For any Hamiltonian derived function F P DFnc A pDMq and derived 0-form section S P DΩ 0 pL β q, we set where P F is the Hamiltonian vector field of F (cf. eq. (5.6.5)). It can be readily verified that r FS P DΩ 0 pL β q using (5.5.12), (5.5.13) and (5.6.3), (5.6.4). r F is clearly an endomorphism of the vector space DΩ 0 pL β q. By the structural analogy of eq. (5.8.1) to relation (4.5.1) providing the prequantization of Hamiltonian functions in ordinary KKS theory, r F can naively be thought of as the derived prequantization of F. Unfortunately, we have r r F, r Gs ‰ i Č tF, Gu A for F, G P DFnc A pDMq in general 6 . Our simpleminded derived prequantization approach requires therefore appropriate modifications in order to be viable.
We assume in what follows that the connection A of L β of 1-form type, that is with the property that j V j W A " 0 for any two vector fields V, W P VectpDMq.
The curvature B of A is then of 2-form type, i.e. one has j V j W j Z B " 0 for any V, W, Z P VectpDMq. Furthermore, A is both simple and strict (cf. subsects.

5.6, 5.7).
A derived function F P DFnc b pDMq is said to be pure if j V F " 0 for any vector field V P VectpDMq. The pure derived functions constitute a subalgebra DFnc bh pDMq of DFnc b pDMq. Similarly, a 0-form section S P DΩ 0 pL β q is said to be pure if j V S " 0 for any vector field V P VectpDMq. The pure 0-from sections form a subspace DΩ 0 h pL β q of DΩ 0 pL β q. Let DFnc Ah pDMq " DFnc A pDMqXDFnc bh pDMq be the space of pure Hamiltonian derived functions. For any function pair F, G P DFnc Ah pDMq, one has tF, Gu A P DFnc Ah pDMq. Further, the restriction of the twisted Lie bracket t¨,¨u A to DFnc Ah pDMq is a genuine Lie bracket, since the connection A is simple and so the Jacobiator x¨,¨,¨y A of t¨,¨u A vanishes identically (cf. eqs. (5.6.7), (5.6.8)).
It can be readily shown that r FS P DΩ 0 h pL β q for any F P DFnc Ah pDMq and S P DΩ 0 h pL β q as a consequence of A being of 1-form type. Let p F be the restriction of r F to DΩ 0 h pL β q. Then, p F is an endomorphism of the vector space DΩ 0 h pL β q. Furthermore, the commutation relation holds for any two functions F, G P DFnc Ah pDMq. Note that (5.8.3) is consistent with the Jacobi property of the endomorphism commutator because t¨,¨u A is a Lie 6 We have in fact that The terms in the right hand side beyond the first one do not vanish in general. A relation analogous to the above one holds also in ordinary KKS theory, where however the extra terms can be shown to vanish identically for grading reasons.  [63]). Unfortunately, the integrand in the right hand side of eq. (5.8.6) does not have this property. In conclusion, the prequantum Hilbert structure of ordinary KKS theory does not have a viable derived analog.

5.9
Derived KKS theory in the regular case The curvature B Λ of the connection A Λ is the object that interests us most because of its eventual relation to the derived KKS symplectic structure. It is as expected a derived enhancement of the curvature of the ordinary KKS theory 7 To the reader's benefit, we explain how M c is defined. One chooses two Abelian Lie groups N, H such that dim E`dim N " dim G`dim H, one of which can be assumed to be trivial.
One then sets E c " EˆN and G c " GˆH and defines τ c : E c Ñ G c and µ c : G cˆEc Ñ E c by τ c pAˆKq " τ pAqˆ1 H and µ c paˆk, AˆKq " µpa, AqˆK with a P G, k P H, A P E, A mapping Ψ Ñ U Λ from the automorphism group Aut DJ pDMq of DM to the gauge transformation group GaupL Λ q of L Λ is established by (5.9.8), (5.9.9).
As is easily checked, this mapping is a group morphism as a consequence of the identity of ZM Λ and J.
We concluded this subsection by noting that the connection A Λ is of 1-form type as is immediate to realize by inspection of (5.9.2), (5.9.3). Since its curvature B Λ is also non singular as already noticed, the incomplete derived prequantization procedure described in subsect. 5.8 can be implemented in the present case.

Conclusions
In this section, we have gone a long way toward a complete and satisfactory formulation of higher KKS theory. We have done so in the derived framework, that seems to be ideally suited for this purpose. As we have seen, derived prequantization suffers certain limitation which we have detailed. We believe that the reason for this is ultimately that the appropriate geometric quantization of derived KKS theory, whatever form it takes, cannot have some kind of quantum mechanical model, albeit exotic, as its end result but a two dimensional quantum field theory.
This seems to give further support to standard expectations that derived KKS theory can be regarded as some kind of categorification of the ordinary theory.
The TCO Model we elaborate in paper II is an attempt to concretize the above intuitions.

A Appendixes
The following appendixes review certain nonstandard notions whose knowledge is tacitly assumed throughout the paper. Our presentation has no pretence of rigour and completeness and serves mainly the purpose of facilitating the reading.
A.1 Generalities on the graded geometric set-up In this paper, we adhere to a graded geometric perspective. This provides an especially natural language for the description the higher geometric structures encountered along the way. The reader is referred to e.g. ref. [61] for a readable introduction to graded differential geometry and its applications. Here, we shall limit ourselves to discuss a few conceptual issues.
Adopting the most common standpoint of the physical literature, we shall describe a graded manifold X using local coordinates. As well-known, these divide into body and soul coordinates x a and ξ r characterized by integer degrees.
The coordinates should be regarded as formal parameters. The topology and geometry of X is encoded in the transition functions relating sets of coordinates x a , ξ r andx a ,ξ r . These are the degree 0 smooth functions f a pxq, f r r 1 ...rp pxq appearing in the coordinate change relations At several points in our analysis the distinction between ordinary and internal functions and maps, albeit technical, will play an important role. To the reader's convenience, we recall briefly the difference between these two types of maps. Let X and Y be graded manifolds and ϕ : Y Ñ X a smooth map. In terms of local coordinates x a , ξ r and y i , η h of X and Y , ϕ has an expression of the form ϕ a py, ηq " finite. We distinguish the ordinary and internal map sets by employing the generic notation Fun and Map for the former and Fun and Map for the latter. In sect.
3, the derived set-up is formulated mostly in the internal case, since internal functions provide a broader function range. The ordinary case can be treated anyway essentially in the same way.
We can associate a clone manifold X`" Mapp * , Xq with any graded manifold X, where * is the singleton manifold. Inspection of (A.1.3), (A.1.4) for the case where Y " * reveals that X`provides a description of the 'range' of any assigned local coordinate system x a , ξ r of X, since the points of the range are in bijective correspondence with a subset of maps of X`. Working with X`furnishes in this manner an index free way of encoding coordinate expressions. While this does not constitute by itself a sufficient reason for a systematic application of cloning in general, it is when X is endowed with a linear structure that the usefulness of cloning becomes apparent as we illustrate next.
Let E be a negatively graded vector space. Then, the dual E * of E is a positively graded vector space. The coordinate functions of E, as elements of E * , have thus positive degrees. With any basis x a of E, there is associated a full set of coordinate functions of E, namely the basis x a of E * dual to x a . A basis change x a Þ Ñx a in E induces a linear transformation x a Þ Ñx a . In this way, upon treating the x a as formal parameters, E can be viewed as a positively graded manifold.
So, while E is negatively graded as a vector space, it is positively graded as a manifold. Because of this sign mismatch between vector and geometric grading, the geometric structure of E cannot be directly described through the vector structure. However, since the clone manifold E`of E is naturally a positively graded vector space with a vector grading content matching the geometric one of E 8 , the geometric properties of E can be expressed in principle as vector properties of E`in a very convenient and natural manner. Similar considerations apply to negatively graded vector bundles.
Though the difference between a vector space or bundle E and its clone Em atters if one is to have the correct grading matching as explained above, E and E`may be harmlessly confused in practice in most instances. We decided so, albeit with some hesitation, that keeping the distinction between E and Em anifest may burden notation unnecessarily. Throughout this paper, therefore, we shall not discriminate notationally between E and E`. It will be clear from context which is which. In our analysis, we shall mainly focus on graded geometric aspects and correspondingly it will be the clone space that will be tacitly considered. At any rate, the reader not interested in fine technical points such as this and content with mere formal manipulations can ignore cloning altogether.

A.2 Differential forms as graded functions
Differential forms on a manifold X can be described as graded functions on the shifted tangent bundle T r1sX. We have indeed a graded algebra isomorphism Ω * pXq » FunpT r1sXq. This well-known property will be extensively exploited in the present work. In the graded geometric framework we are adopting, it is definitely more natural to treat forms as graded functions. This standpoint has also the advantage of extending the range of manipulations which can be performed with forms. We shall also consider internal differential forms and deal with these as internal functions exploiting the isomorphism Ω * pXq » FunpT r1sXq. 8 The difference between E and E`can be illustrated somewhat more explicitly as follows.
Let E 0 the ungraded vector space underlying E and x 0 a the basis of E 0 corresponding to a basis x a of E. Then, E consists of the linear combinations c a x a with c a P R, while E`of the formal expressions x a b x 0 a . elements x P f and the degree 1 de Rham differential d of P . Since B " P {F, the operational set-up allows for an elegant description of the differential geometry of B. The function algebra FunpT r1sBq of T r1sB can be identified with the basic subalgebra of FunpT r1sP q, the joint kernel of the derivations j x and l x with x P f. Similarly, the mapping space MappT r1sB, Xq can be identified with a basic subspace of MappT r1sP, Xq.
The operational approach, can be adapted straightforwardly to the case where the principal bundle P , its base B and its structure group F are positively graded manifolds. One important aspect distinguishing an operational set-up of graded manifolds from an ordinary one is that by consistency the underlying graded algebra is the internal function algebra FunpT r1sP q rather than the ordinary function algebra FunpT r1sP q, because of the graded nature of f. Besides, the whole formal apparatus works much in the same way.