4-d Chern-Simons Theory: Higher Gauge Symmetry and Holographic Aspects
aa r X i v : . [ h e p - t h ] J a n Roberto Zucchini
Dipartimento di Fisica ed Astronomia,Università di Bologna,I.N.F.N., sezione di Bologna,viale Berti Pichat, 6/2Bologna, ItalyEmail: [email protected] , [email protected] Abstract:
We present and study a 4–d Chern–Simons (CS) model whose gaugesymmetry is encoded in a balanced Lie group crossed module. Using the derivedformal set–up recently found, the model can be formulated in a way that inmany respects closely parallels that of the familiar 3–d CS one. In spite of theseformal resemblance, the gauge invariance properties of the 4–d CS model differconsiderably. The 4–d CS action is fully gauge invariant if the underlying base4–fold has no boundary. When it does, the action is gauge variant, the gaugevariation being a boundary term. If certain boundary conditions are imposed onthe gauge fields and gauge transformations, level quantization can then occur.In the canonical formulation of the theory, it is found that, depending againon boundary conditions, the 4–d CS model is characterized by surface chargesobeying a non trivial Poisson bracket algebra. This is a higher counterpart of thefamiliar WZNW current algebra arising in the 3–d model. 4–d CS theory thusexhibits rich holographic properties. The covariant Schroedinger quantization ofthe 4–d CS model is performed. A preliminary analysis of 4–d CS edge fieldtheory is also provided. The toric and Abelian projected models are described insome detail.MSC: 81T13 81T20 81T45 1 ontents1
Introduction Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Lie crossed modules and invariant pairings Lie group and algebra crossed modules . . . . . . . . . . . . . . . . 122.2
Crossed modules with invariant pairing . . . . . . . . . . . . . . . . 152.3
Crossed submodules and isotropy . . . . . . . . . . . . . . . . . . . 23 Higher gauge theory in the derived formulation Derived Lie groups and algebras . . . . . . . . . . . . . . . . . . . . 323.2
Derived superfield formulation . . . . . . . . . . . . . . . . . . . . . 363.3
Higher gauge theory in the derived formulation . . . . . . . . . . . . 403.4
The derived functional framework of higher gauge theory . . . . . . 443.5
Derived description of non trivial higher principal bundles . . . . . . . 46 . . . . . . . . . . . . . . . . . . . . . . . 554.2 Gauge invariance of the 4–d Chern–Simons model . . . . . . . . . . 594.3
Level quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4
Global issues in 4–d Chern–Simons theory . . . . . . . . . . . . . . . 674.5
Canonical formulation . . . . . . . . . . . . . . . . . . . . . . . . . 704.6
Surface charges and holography . . . . . . . . . . . . . . . . . . . . 804.7
Toward the edge field theory of 4–d Chern–Simons theory . . . . . . 834.8
Covariant Schroedinger quantization . . . . . . . . . . . . . . . . . 89 Sample applications The toric 4–dimensional CS model . . . . . . . . . . . . . . . . . . 955.2
The Abelian projection model . . . . . . . . . . . . . . . . . . . . . 972
Appendixes
A.1
Basic definitions and identities of crossed module theory . . . . . . . 104A.2
Lie differentiation of crossed modules . . . . . . . . . . . . . . . . . 107A.3
Crossed modules with invariant pairing . . . . . . . . . . . . . . . . 109A.4
Proof of the decomposition theorem . . . . . . . . . . . . . . . . . . 111A.5
Basic results of Cartan–Weyl theory . . . . . . . . . . . . . . . . . . 1123
Introduction
There are several reasons why the formulation and study of 4-dimensional higherChern–Simons (CS) theory is an interesting and worthwhile endeavour. Somehave to do with physics, other with mathematics.4–dimensional BF theory is an instance of 4–dimensional CS theory. Thoughas a topological quantum field theory (TQFT) it involves no metric and possessesno local degrees of freedom, it yields however general relativity when the basicfields are suitably expressed in terms of or related to metric data. This has beendone in two independent ways [1,2] which differ by the choice of the gauge group.A 4-dimensional gravitational CS model was worked out in ref. [3] by Kaluza–Klein compactification of a 5-dimensional (anti) de Sitter gravitational CS model.Specific instances of 4–dimensional CS theory have appeared as topological sectorsof CS modified gravity [4, 5] and string based cosmological models [6] describingaxion–like fields and their coupling to gauge fields.In refs. [7–10] the three types of solutions of the Yang–Baxter equation,rational, trigonometric and elliptic, and their properties were obtained from a 4–dimensional CS model compactified down to 2 dimensions. The model is definedon a 4–fold of the form Σ ˆ C , where Σ is a 2–fold and C is a Riemann surface,and involves a background meromorphic (1,0)–form on C . So, it is only partiallytopological, the 4-dimensional diffeomorphism symmetry being broken to the 2–dimensional one.There exist two main quite different approaches to the construction of CStype TQFTs: the algebraic–topological approach and the differential–geometricalapproach. These frameworks are related. It is possible to identify a TQFT de-fined in the former combinatorial approach and another TQFT defined in thelatter continuum one if the partition functions of these TQFTs can be provento be equal for appropriate assignments of input data. This correspondence iswell understood in three dimensions. The Turaev–Viro–Barrett–Westbury model[11, 12] is a combinatorial model of 3–dimensional quantum gravity with cos-mological constant known to be equivalent to 3–dimensional BF theory with4osmological term [13–15] when their underlying Lie groups are the same and thequantum group parameter and cosmological constant are properly related. TheReshetikhin–Turaev model [16] is a 3–dimensional combinatorial TQFT believedto be equivalent 3–dimensional CS TQFT [17, 18] under similar conditions. TheDijkgraaf–Witten model [19] is another combinatorial TQFT that can be relatedto a 3–dimensional Abelian BF theory (being in fact viewable as a special case ofa general Turaev–Viro–Barrett-Westbury construction). The correspondence isnot as well understood in its details in four dimensions. The 4–dimensional coun-terpart of the Turaev–Viro–Barrett–Westbury model is the Crane–Yetter–Brodamodel [20–22], which is identified with 4–dimensional BF theory with cosmolog-ical term for equal underlying Lie groups and related parameters [23]. Likewisethe Yetter model [24], a 4-dimensional higher analogue of the Dijkgraaf–Wittenmodel, can be related to a 4-dimensional Abelian BF theory. All 4–dimensionalgeometrically defined continuum TQFT mentioned above are again instances of4–dimensional CS models.At low energy, topologically ordered phases of matter are described by TQFTs.In 3–dimensional spacetime, many fractional quantum Hall states as well as lat-tice models such as Kitaev’s toric code model can be explained by suitable CSmodels [25–27]. In this cases, fractional braiding statistics between quasiparti-cles emerges through the correlation functions of a pair of Wilson loops forminga Hopf link. It is expected that fractional braiding statistics has a 4–dimensionalspacetime analog. Since particle–like excitations do not braid and have only or-dinary bosonic/fermionic statistics in this case, fractional statistics can only arisefrom the braiding of either a point–like and a loop–like or two loop–like exci-tations. This has been adequately described through BF type TQFTs [28–30]through the correlation functions of Wilson loops and surfaces, pointing again to4–dimensional CS theory.Wilson lines [31] are relevant in the study of confinement in quantum chro-modynamics, loop quantum gravity, symmetry breaking in string theory and, aswe have just seen, condensed matter physics. They depend on the topology ofthe underlying knots and, as shown in Witten’s foundational work [17], they5an be used to study knot topology in 3-dimensional CS theory using basic tech-niques of quantum field theory. CS correlators of Wilson line operators provideknot and link invariants. The 4-dimensional counterpart of Wilson lines, Wilsonsurfaces, are expected to be relevant in the analysis of non perturbative featuresof higher form gauge theory and quantum gravity. They also should be a basicelement of any field theoretic approach to 4-dimensional 2–knot topology [32].Based on Witten’s paradigm, it should be possible to study surface knot topologyin 4–dimensions computing correlators of Wilson surfaces in an appropriate 4–dimensional version of CS theory using again techniques of quantum field theory[33–38].The holographic principle [39,40] has emerged as one of the most far reachingtheoretical ideas in the last two decades. The first known occurrence of holog-raphy in quantum field theory is the correspondence discovered by Witten [17]between 3–dimensional CS theory with gauge group G as the bulk field theoryand the 2–dimensional Wess–Zumino–Novikov–Witten (WZNW) model [41, 42]with target group G as the boundary field theory, manifesting itself as an equiva-lence between the space of quantum states of the CS model on a 2–fold S and thespace of conformal blocks of the WZNW model on S . The holographic principlehas also allowed for an effective 3–dimensional CS description of edge states isthe fractional quantum Hall effect [43, 44]. Little is known, to the best of ourknowledge on the holographic features of 4–dimensional CS theory. Outline of the paper
The above considerations support our claim that the study of 4–dimensional CStheory is a worthy undertaking. In this paper, following the differential–geometricapproach to TQFT, we shall formulate a 4–dimensional CS theory as a certainkind of strict higher gauge theory. In fact, as ordinary CS theory exists only inodd dimensional manifolds, 4–dimensional CS theory must be necessarily built inthe framework of higher gauge theory (see [45] for a review). Such theory has beenalready considered in refs. [46–48] also in the semistrict case on 4-folds withoutboundary. Here, we shall study it on 4-folds with boundary. As we shall see, it6s precisely in this case that the theory exhibits its most interesting holographicfeatures. Below, we outline briefly the construction of the 4–dimensional CSmodel we have carried out.The higher gauge symmetry of our 4–dimensional CS model is encoded by aLie group crossed module. Various relevant topics of crossed module theory arediscussed in sect. 2. Here it is enough to recall that a Lie group crossed module M consists of two Lie groups E , G together with a Lie group action µ : G ˆ E Ñ E of G on E by automorphisms and an equivariant target map τ : E Ñ G satisfyinga certain identity [49, 50].On general grounds, in order to construct the kinetic term of the Lagrangianof a field theory, a non singular bilinear pairing invariant under the symmetries ofthe theory is required. In the 4–dimensional CS model, a crossed module M withinvariant pairing x¨ , ¨y : g ˆ e Ñ R carries out this task. In fact, invariant pairingsin higher gauge theory play a role similar to that of invariant traces in ordinaryone. An invariant pairing selects further isotropic crossed submodules of M as adistinguished subclass of submodules. These correspond to standard choices oflinear boundary conditions for CS gauge fields and gauge transformations.In sect. 3, we introduce the formal set–up of derived Lie groups and algebrasoriginally worked out in refs. [51, 52]. This is essentially a superfield formalismproviding an elegant and convenient way of handling certain structural elementsof a Lie group crossed module by organizing them as functions of a formal oddvariable ¯ α P R r´ s . The derived Lie group D M of a crossed module M consistsof superfields of the form P p ¯ α q “ e ¯ αP p , where p P G , P P e r s , with certain groupoperations determined by M . Its Lie algebra D m , the derived Lie algebra, consistssimilarly of superfields of the form U p ¯ α q “ u ` ¯ αU , where u P g , U P e r s , withthe associated Lie bracket.The relevant fields of the 4–dimensional CS model are based on a 4–fold M .They are crossed module valued inhomogeneous form fields. More formally, theyare maps from the shifted tangent bundle T r s M of M into either the derivedgroup D M or the derived algebra D m or a shifted version of this. They are thusrepresentable as derived superfields, functions of a formal odd variable α P R r s .7he derived superfield formalism is particularly suited for our 4–dimensionalCS model because of its compactness and capability of presenting it as an ordinaryCS theory with an exotic graded gauge group, the derived group. Indeed, bymaking evident the close relationship of higher to ordinary gauge theory, it allowsimporting many ideas and techniques of the latter to the former. In particular,the gauge fields and the gauge transformations of the 4–dimensional CS modelcan be treated in this fashion in the derived set–up.In sect. 4, we present the 4–dimensional CS model as a strict higher gaugetheoretic model using a the derived superfield formalism. In this way, if M is thegauge crossed module, the gauge field of the model is a superfield of the form Ω p α q “ ω ´ αΩ, (1.1.1)where ω P Map p T r s M, g r sq , Ω P Map p T r s M, e r sq . ω , Ω represent the usual1– and 2–form components of a gauge 2–connection in higher gauge theory. The4–dimensional CS action reads as CS p Ω q “ k π ż T r s M ̺ M ` Ω , d Ω ` r Ω , Ω s ˘ , (1.1.2)where k is a constant, p¨ , ¨q is a certain degree 1 invariant pairing derived by theinvariant pairing x¨ , ¨y and d is a certain degree 1 nilpotent differential extendingthe de Rham differential d . CS is formally identical to the 3–dimensional CSaction. It is 4–dimensional however because of the 1 unit of degree provided bythe pairing p¨ , ¨q . CS can be expressed explicitly in components as CS p ω, Ω q “ k π ż T r s M ̺ M @ dω ` r ω, ω s ´ τ p Ω q , Ω D ´ k π ż T r sB M ̺ B M x ω, Ω y . (1.1.3)This can be described as a generalized BF theory with boundary term and cos-mological term determined by the Lie differential τ of the target map τ of thecrossed module M .The higher gauge symmetry of the 4–dimensional CS model can also be de-scribed in the derived set–up. Analogously to a gauge field, a gauge transforma-8ion is a superfield expression of the form U p α q “ e αU u, (1.1.4)where u P Map p T r s M, G q , U P Map p T r s M, e r sq . u , U represent the usual 0–and 1–form components of a 1–gauge transformation in higher gauge theory. U acts on the higher gauge field Ω as Ω U “ Ad U ´ p Ω q ` U ´ dU . (1.1.5)This gauge transformation action is formally identical to that of ordinary gaugetheory, but it yields when expressed explicitly in components the usual gaugetransformation relations of higher gauge theory.In spite of the formal resemblance of 4– and 3–dimensional CS theory whenthe derived formulation is used, the invariance properties of the higher CS modeldiffer in several important aspects from those of the ordinary CS one, especially inrelation to the effect of a boundary in the base manifold. Unlike its 3–dimensionalcounterpart, the 4–dimensional CS action CS is fully gauge invariant if the 4–fold M has no boundary. When M does have a boundary, the action is nolonger invariant, but the gauge variation is just a boundary term. The gaugeinvariance of the 4–dimensional theory depend therefore in a decisive way on thekind of boundary conditions which are imposed on the gauge fields Ω and gaugetransformations U . These are discussed in sect. 4.To quantize higher CS theory, proceeding as in the ordinary case, one shouldpresumably allow for the broadest gauge symmetry leaving the Boltzmann weight exp p i CS q invariant possibly restricting the value of the CS level k as appropriate.On a 4–fold M with no boundary, the theory is fully gauge invariant and sothere are no restrictions on either the gauge symmetry or the level. On a 4–fold M with boundary, one should impose on the relevant higher gauge fields Ω andtransformations U the weakest possible boundary conditions capable of renderingthe gauge variation an integer multiple of π . These are also discussed in sect.4. Depending also on crossed module M and the invariant pairing x¨ , ¨y , levelquantization can occur. 9n sect. 4, a canonical analysis a la Dirac of the 4–dimensional CS model iscarried out. The close relationship of the canonical formulations of 4– and 3–dimensional CS theory is again especially evident in the derived framework. Theresults of the ordinary theory generalize to the higher one, but in a non trivialway. The Hamiltonian generators of the gauge symmetry which through theirweak vanishing define the physical phase space of flat gauge fields of the theoryare determined. They are first class only if appropriate boundary conditionsare obeyed by the gauge fields and the gauge transformations. It is found that,depending again on these boundary conditions, there exist generically surfacecharges obeying a non trivial Poisson bracket algebra. This is a higher counterpartof the familiar WZNW current algebra arising in the corresponding canonicalanalysis of 3–dimensional CS model.Gauge theories defined on manifolds with boundaries may exhibit emergentboundary degrees of freedom called edge modes. In fact, boundaries normallybreak gauge invariance transforming in this way gauge degrees of freedom intophysical ones. In sect. 4, we outline a canonical theory of the edge modes of4–dimensional CS theory and their physical symmetries, extending the corre-sponding analysis of the 3-dimensional theory [53–57].In sect. 4, we finally show how the covariant Schroedinger quantization of4–dimensional CS theory can be carried out on the lines of the 3–dimensionalcase [58]. Among other things, we obtain the higher analogue of the WZNWWard identities obeyed by the wave functionals and an expression of the higherWZNW action, which turns out to be fully topological.In sect. 5, we finally illustrate a few field theoretic models which are interestingnon trivial instances of 4–dimensional CS theory, in particular the toric and theAbelian projection models. Outlook
There remain to ascertain to what extent the 4–dimensional CS theory presentedin this paper is capable to reproduce various ’disguised’ CS model that haveappeared in the literature and yield new interesting ones.10ur analysis of the edge sector of 4–dimensional CS theory is still incomplete.Although we have identified the edge fields and the physical edge symmetry group,there remain basic problems to be solved such as the lack of a Lagrangian andHamiltonian description of the dynamics of edge fields and the identification ofthe edge modes observables. A viable edge field theory of the 4–dimensional CSmodel is a topic certainly deserving an in depth study.In this paper, we have not discussed the incorporation of Wilson surfaces in thetheory. This can be done in the standard framework of strict higher gauge theoryalong the lines of refs. [34, 35]. There is however another so far unexploredrout to dealing with this problem. In a series of papers [59–62] (see also ref.[63] for a review) a geometrical action capable to compute Wilson lines in 3–dimensional CS theory was obtained and studied. Given the formal similarityof 4– and 3–dimensional CS in the derived formulation, it is conceivable that aformally analogous geometrical action may be found that is capable of computingWilson surfaces in 4–dimensional CS theory. This is a promising line of inquirydeserving to be pursued.All the above matters are left for future work [64].11
Lie crossed modules and invariant pairings
Lie group and algebra crossed modules constitute the type of algebraic structureon which the higher CS theory elaborated in this paper rests. In this section,we review this topic, with no pretence of mathematical rigour or completeness,dwelling only on those points which are relevant in the following analysis. Thetheory of crossed modules is best formulated in a categorical framework. However,we shall not insist on the categorical features of these algebraic structures. Seeref. [49, 50] for an exhaustive exposition of this subject.The subject matter covered in this section is disparate and has been gatheredmainly for later reference. Subsect. 2.1 is a review of the theory of Lie groupand algebra crossed modules serving also the purpose of set the notation usedthroughout this paper. The material of subsect. 2.2 is required mostly fromsubsect. 4.1 onwards. The material of subsect. 2.3 is used primarily in subsects.4.2, 4.3, 4.6.
Lie group and algebra crossed modules
Crossed modules encode the symmetry of higher gauge theory both at the finiteand the infinitesimal level. Our path toward 4–dimensional CS theory mustnecessarily start with them. In this subsection, we review the theory of Lie groupand algebra crossed modules and module morphisms. The precise definitions andproperties of crossed modules are collected in app. A.1.
Lie group crossed modules
The structure of finite Lie crossed module abstracts and extends the set–upconsisting of a Lie group G and a normal Lie subgroup E of G acted upon by G by conjugation. A Lie group crossed module M consists indeed of two Lie groups E , G together with a Lie group action µ : G ˆ E Ñ E of G on E by automorphismsand an equivariant Lie group map τ : E Ñ G rendering µ compatible with adjointaction of E (cf. eqs. (A.1.1), (A.1.2)). E , G and τ , µ are called the source andtarget groups and the target and action structure maps of M , respectively. Below,we shall write M “ p E , G , τ, µ q to specify the crossed module through its data.12 morphism of finite Lie crossed modules is a map of crossed modules preserv-ing the module structure expressing a relationship of sameness or likeness of themodules involved. More explicitly, a morphism β : M Ñ M of Lie group crossedmodules consists of two Lie group morphisms φ : G Ñ G and Φ : E Ñ E inter-twining in the appropriate sense the structure maps τ , µ , τ , µ (cf. eqs. (A.1.3),(A.1.4)). We shall normally write β : M Ñ M “ p Φ, φ q to indicate constituentmorphisms of the crossed module morphism.Taking the direct product of the relevant constituent data in the Lie groupcategory, it is possible to construct the direct product M ˆ M of two Lie groupcrossed modules M , M and the direct product β ˆ β of two Lie group crossedmodule morphisms β , β in straightforward fashion. Complicated crossed mod-ules and module morphisms can sometimes be analyzed by factorizing them intodirect products of simpler modules and module morphisms.There exist many examples of Lie group crossed modules and crossed modulemorphisms. In particular, Lie groups and automorphisms, representations andcentral extensions of Lie groups can be described as instances of Lie group crossedmodules. Lie group morphisms can be employed to construct morphisms of suchcrossed modules. There are two basic model crossed modules to which a broadrange of crossed modules entering in the formulation of higher CS theory canbe related to. They are defined for any Lie group G . The first is the innerautomorphism crossed module of G , INN G “ p G , G , id G , Ad G q . The second is the(finite) coadjoint action crossed module of G , AD ∗ G “ p g ∗ , G , G , Ad G ∗ q , where g is the Lie algebra of G and its dual space g ∗ is viewed as an Abelian group.A crossed module morphism ρ : INN G Ñ INN G reduces to a group morphism χ : G Ñ G . A crossed module morphism α : AD ∗ G Ñ AD ∗ G is specified by agroup morphism λ : G Ñ G and an intertwiner Λ : g ∗ Ñ g ∗ of Ad G ∗ to Ad G ∗ ˝ λ . Lie algebra crossed modules
The structure of infinitesimal Lie crossed module axiomatizes likewise the set–up consisting of a Lie algebra g and a Lie ideal e of g equipped with the adjointaction of g . It is therefore the differential version of that of finite Lie crossedmodule. A Lie algebra crossed module m consists so of two Lie algebras e , g m : g ˆ e Ñ e of g on e by derivations and anequivariant Lie algebra map t : e Ñ g making m compatible with adjoint actionof e (cf. eqs. (A.1.7), (A.1.8)). e , g and t , m are called the source and targetalgebras and the target and action structure maps of m , respectively. Below, weshall write m “ p e , g , t, m q to specify the crossed module through its data.A morphism of infinitesimal Lie crossed modules is a map of crossed modulespreserving the module structure describing a way such crossed modules are con-gruent. It is therefore the differential version of that of morphism of finite Liecrossed module. More explicitly, a morphism p : m Ñ m of Lie algebra crossedmodules consists of two Lie algebra morphisms h : g Ñ g and H : e Ñ e inter-twining in the appropriate sense the structure maps t , m , t , m (cf. eqs. (A.1.9),(A.1.10)). We shall use often the notation p : m Ñ m “ p H, h q to indicateconstituent morphisms of the crossed module morphism.Similarly to the Lie group case, taking the direct sum of the relevant con-stituent data in the Lie algebra category, it is possible to define the direct sum m ‘ m of two Lie algebra crossed modules m , m and direct sum p ‘ p of twoLie algebra crossed module morphisms p , p in obvious fashion. These notionsanswer at the differential level to those of direct products of finite crossed mod-ules and module morphisms. They allow to analyze crossed modules and modulemorphisms by decomposing them as direct sums of more elementary modules andmodule morphisms as we shall see in particular in subsect. 2.2 below.Many examples of Lie algebra crossed modules and crossed module morphismsare also available. They pair with the basic examples of Lie group crossed mod-ules and crossed module morphisms recalled above. Ordinary Lie algebras andderivations, representations and central extensions of Lie algebras can be de-scribed as instances of Lie algebra crossed modules and Lie algebra morphismscan be assembled variously to construct morphisms of such crossed modules. Inparticular, there are two basic model crossed modules defined for any Lie al-gebra g corresponding to the inner automorphism and coadjoint action crossedmodules introduced above. The first is the inner derivation crossed module of g , INN g “ p g , g , id g , ad g q . The second is the (infinitesimal) coadjoint action crossed14odule of g , AD ∗ g “ p g ∗ , g , g , ad g ∗ q , where g ∗ is regarded as an Abelian algebra.A crossed module morphism r : INN g Ñ INN g reduces to an algebra morphism x : g Ñ g . A crossed module morphism a : AD ∗ g Ñ AD ∗ g is specified by analgebra morphism l : g Ñ g and an intertwiner L : g ∗ Ñ g ∗ of ad g ∗ to ad g ∗ ˝ l .Lie differentiation plays the same important role in Lie crossed module theoryas it does in Lie group theory. With any Lie group crossed module M “ p E , G , τ, µ q there is associated the Lie algebra crossed module m “ p e , g , τ, µ q , where e , g arethe Lie algebras of Lie groups E , G respectively and the dot notation denotesLie differentiation along the relevant Lie group (cf. app. A.2) for more details),much as a Lie algebra is associated with a Lie group. Similarly, with any Liegroup crossed module morphism β : M Ñ M “ p Φ, φ q there is associated the Liealgebra crossed module morphism β : m Ñ m “ p Φ, φ q , just as a Lie algebramorphism is associated with a Lie group morphism.As examples, we mention that the Lie algebra crossed modules of the Liegroup 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
Crossed modules with invariant pairing are an essential ingredient of the con-struction of 4–dimensional CS actions. Indeed, invariant pairings in higher gaugetheory play a role similar to that of invariant traces in ordinary gauge theory. Weintroduce and discuss this topic in this subsection.On general grounds, in order to construct the kinetic term of the Lagrangianof a field theory, a non singular bilinear pairing is required. Further, when thefield theory is characterized by certain symmetries, the same symmetries must beenjoyed by the pairing, which so is in addition invariant.The field content of 4–dimensional CS gauge theory whose symmetry is de-scribed infinitesimally by a Lie algebra crossed module m “ p e , g , t, m q comprisesa g –valued 1–form gauge field ω and an e –valued 2–form gauge field Ω . Thebilinear pairing entering in the kinetic term of the gauge fields, thus, must bedefined on either g ˆ g or g ˆ e or e ˆ e . The CS kinetic term, which must be15f derivative order in order the field equations to be equivalent to the flatnessconditions for the gauge fields, can take thus three forms K “ x dω, ω y , K “ x dω, Ω y , K “ x dΩ, Ω y , (2.2.1)These are a 3–, 4–, 5–form yielding a 3–, 4– and 5–dimensional CS model respec-tively. As we are interested in a 4–dimensional one, it is the second form of thepairing that is relevant for us. So, the pairing will be a non singular bilinear form x¨ , ¨y : g ˆ e Ñ R .The infinitesimal higher gauge symmetry described by m is ultimately codifiedin the adjoint action of g on itself and the module action m of g on e . The kineticterm will so have the required invariance properties if the pairing x¨ , ¨y obeys x ad z p x q , X y ` x x, m p z, X qy “ (2.2.2)for z, x P g , X P e . It remains to clarify how the pairing x¨ , ¨y behaves with regardto the module target map t . This boils down to find an appropriate requirementfor the difference x t p X q , Y y ´ x t p Y q , X y with for X, Y P e . The minimal choiceavoiding introducing further structures consists in demanding that x t p X q , Y y “ x t p Y q , X y . (2.2.3) Lie algebra crossed modules with invariant pairing
A Lie algebra crossed module with invariant pairing is Lie algebra crossedmodule m “ p e , g , t, m q equipped with a non singular bilinear form x¨ , ¨y : g ˆ e Ñ R enjoying properties (2.2.2), (2.2.3).A crossed module m with invariant pairing is balanced, which means that dim g “ dim e , because of the non singularity of the pairing. This is not as stronga restriction as it may appear at first sight. It can be shown that any Lie algebracrossed module m can always be trivially extended to a balanced crossed module m , for which depending on cases one has either e “ e ‘ p and g “ g or e “ e and g “ g ‘ q for suitable Abelian Lie algebras p , q .A morphism p : m Ñ m “ p H, h q of Lie algebra crossed modules with invari-ant pairings x¨ , ¨y , x¨ , ¨y is naturally defined as a crossed module morphism that16reserves the pairings (cf. eq. (A.3.3)). Such a morphism describes a strongerform of sameness or likeness of the crossed modules concerning not only theiralgebraic structures but involving also to their invariant pairings.We shall now explore the implications of having an invariant pairing structureattached to the crossed module. Core and residue of a crossed module with invariant pairing If m is a Lie algebra crossed module, then ker t is a central ideal of e and ran t is an ideal of g . Using these properties, one can show that with m there arecanonically associated two further Lie algebra crossed modules.The first one, which we shall call the core of m in the following, is the crossedmodule C m “ p e { ker t, ran t, t C , m C q , where t C p X ` ker t q “ t p X q , (2.2.4) m C p x, X ` ker t q “ m p x, X q ` ker t (2.2.5)for x P ran t , X P e . It can be verified that the structure maps t C and m C arewell defined and satisfy the required properties (A.1.7), (A.1.8). The core crossedmodule C m is characterizing by the invertibility of t C .The second one, which we shall call the residue of m , is the crossed module R m “ p ker t, g { ran t, t R , m R q , where t R p X q “ ran t, (2.2.6) m R p x ` ran t, X q “ m p x, X q (2.2.7)for x P g , X P ker t . Again, it can be verified that the structure maps t R and m R are well defined and satisfy the properties (A.1.7), (A.1.8). The characterizingproperty of the residue crossed module R m is the vanishing of t R .If m is in addition equipped with an invariant pairing x¨ , ¨y , then C m and R m are equipped with induced invariant pairing x¨ , ¨y C and x¨ , ¨y R . For C m , we have x x, X ` ker t y C “ x x, X y (2.2.8)where x P ran t , X P e . It is straightforward to check that x¨ , ¨y C is well defined17nd obeys conditions (A.3.1), (A.3.2). For R m , we have similarly x x ` ran t, X y R “ x x, X y (2.2.9)where x P g , X P ker t . Again, x¨ , ¨y R is well defined and satisfies properties(A.3.1), (A.3.2).In subsect. 2.1, we introduced two basic model Lie algebra crossed modules,the inner derivation crossed module INN g and the infinitesimal coadjoint actioncrossed module AD ∗ g of a Lie algebra g . They are evidently both balanced andthey can both be equipped with invariant pairings, as we shall show momentarily.They are indeed prototypical crossed modules with these properties.The inner derivation crossed module of g is INN g “ p g , g , id g , ad g q . INN g carries no canonical invariant pairing, but any ad g invariant symmetric non sin-gular pairing of g can be used as one. INN g is characterized by the followingproperty. If m “ p e , g , t, m q is a Lie algebra crossed module with invariant pairingsuch that t is invertible, then there is an invariant pairing on INN g such that m is isomorphic to INN g , the isomorphism i of m onto INN g being given bythe pair p t, id g q and the invariant pairing on INN g being related to that of m by x x, t ´ p X qy INN “ x x, X y .The infinitesimal coadjoint action crossed module of g we consider next is AD ∗ g “ p g ∗ , g , g , ad g ∗ q . Unlike the inner derivation crossed module discussedabove, AD ∗ g carries a natural invariant pairing, the duality pairing of g and g ∗ . AD ∗ g enjoys the following property. If m “ p e , g , t, m q is a Lie algebra crossedmodule with invariant pairing with t vanishing, then m is isomorphic to AD ∗ g ,the isomorphism j of INN g onto m being given by the pair p J, id g q , where J is thelinear isomorphism of g ∗ onto e such that x x, J p X qy “ x x, X y AD ∗ with x¨ , ¨y AD ∗ the duality pairing of g and g ∗ . The decomposition theorem
Consider again a generic Lie algebra crossed module m “ p e , g , t, m q withinvariant pairing, The characterizing property of C m is t C being a linear isomor-phism. This makes C m isomorphic to the crossed module INN ran t equippedwith a suitable invariant pairing. Likewise, the characterizing property of R m is18 R vanishing. R m is in this way isomorphic to the crossed module AD ∗ p g { ran t q with the canonical invariant pairing.The following decomposition theorem is key to understanding relevant aspectsof the gauge symmetry of 4–dimensional CS theory studied later on. Considera balanced crossed module m “ p e , g , t, m q with invariant pairing. Suppose thatthere exists an ideal h of g such that ran t X h “ and g » ran t ‘ h . (2.2.10)Then, m decomposes as m » C m ‘ R m . (2.2.11)in the category of crossed modules with invariant pairing. The proof of thetheorem requires in an essential way the use of the invariant pairing (cf. app.A.4). By the isomorphisms noticed earlier, we could write (2.2.11) as m » INN ran t ‘ AD ∗ p g { ran t q . (2.2.12)This means that in the analysis of the following sections, under weak conditions,we can assume that the relevant Lie algebra crossed module m with invariantpairing is of the form INN g c ‘ AD ∗ g r for certain Lie algebras g c , g r with thedirect summands equipped respectively with the appropriate and the canonicalinvariant pairings. Lie group crossed modules with invariant pairing
The subtlest features of 4–dimensional CS theory emerge when the underlyinghigher gauge symmetry is considered at the finite level through the appropriateLie group crossed module. Invariant pairings are naturally defined only on Liealgebra crossed modules. It is possible however to attach an invariant pairing alsoto a Lie group crossed module by endowing the associated Lie algebra crossedmodule with one. However, upon doing so, it is necessary to strengthen theinvariance condition of the pairing by requiring invariance to hold at the finiteand not only infinitesimal level.We shall thus define a Lie group crossed module with invariant pairing as19 crossed module M “ p E , G , τ, µ q such that the associated Lie algebra crossedmodule m “ p e , g , τ, µ q (cf. app. A.2) is a crossed module with invariant pairing x¨ , ¨y enjoying the property that x Ad a p x q , µ p a, X qy “ x x, X y (2.2.13)for a P G , x P g , X P e (cf. apps. A.2, A.3) Notice that (2.2.13) implies (2.2.2)with m “ µ via Lie differentiation with respect to a , while (2.2.3) holds with t “ τ .Again, the non singularity of x¨ , ¨y implies that M is balanced, dim E “ dim G .Analogously to the Lie algebra case, it can be shown that any Lie group crossedmodule M can always be trivially extended to a balanced crossed module M , witheither E “ E ˆ P and G “ G or E “ E and G “ G ˆ Q for suitable Abelian Liegroups P , Q , depending on cases.For a Lie group crossed module M “ p E , G , τ, µ q with invariant pairing x¨ , ¨y ,identity (2.2.13) implies the relation x x, µ p y, A qy “ x y, µ p x, A ´ qy , (2.2.14)where x, y P g and A P E , under mild assumptions on the Lie group E . Specifical-ly, (2.2.14) holds when E is connected and also when E is not connected in theconnected component of the identity of E and in any connected component of E where it holds for at least one element. (2.2.14) holds also when τ is invertiblewith no restrictions on E . Property (2.2.14) in a sense completes (2.2.13). Weshall call the crossed module M fine if (2.2.14) holds. The seemingly technicalcondition plays in fact an important role in the analysis of the gauge invarianceof 4–dimensional CS theory, as we shall see in due course.If M “ p E , G , τ, µ q is a Lie group crossed module with invariant pairing suchthat a direct sum decomposition of g of the form (2.2.10) is available, then thedirect sum decomposition (2.2.11) of the associated Lie algebra crossed module m “ p e , g , τ, µ q into its core and residue C m , R m holds. One does not expect acorresponding direct product factorization of the module M to occur in analogy towhat happens in the akin setting of Lie group theory. It is nevertheless instructive20o examine this issue is some detail.Similarly to the Lie algebra case, it is possible to canonically associate with M two Lie group crossed modules with invariant pairing, its core and residue crossedmodules, relying on the properties that ker τ is a central normal subgroup E and ran τ is a normal subgroup of G , in analogy with the Lie algebra case.The core of M is the crossed module C M “ p E { ker τ, ran τ, τ C , µ C q , where τ C p A ker τ q “ τ p A q , (2.2.15) µ C p a, A ker τ q “ µ p a, A q ker τ (2.2.16)for a P ran τ , A P E . It can be verified that the structure maps τ C and µ C are well defined and satisfy the required properties (A.1.1), (A.1.2). The corecrossed module C M is characterized by the invertibility of τ C . The Lie algebracrossed module associated with C M is precisely the core C m of m defined in(2.2.6), (2.2.7). The invariant pairing x¨ , ¨y of m provides C m with the invariantpairing x¨ , ¨y C defined by eq. (2.2.8). x¨ , ¨y C in turn satisfies property (A.3.5) as aconsequence of x¨ , ¨y doing so. C M is in this way a crossed module with invariantpairing. C M is fine, even if M is not, as τ C is invertible.The residue of M is the crossed module R M “ p ker τ, G { ran τ, τ R , µ R q , where τ R p A q “ ran τ, (2.2.17) µ R p a ran τ, A q “ µ p a, A q (2.2.18)for a P G , A P ker τ . Again, it can be verified that the structure maps τ R and µ R are well defined and satisfy the properties (A.1.1), (A.1.2). The residue crossedmodule R M is characterized by the vanishing of τ R . The Lie algebra crossedmodule associated with R M is precisely the residue R m of m defined in (2.2.8),(2.2.9). The invariant pairing x¨ , ¨y of m provides R m with the invariant pairing x¨ , ¨y R defined by eq. (2.2.9) and x¨ , ¨y R obeys (A.3.5) since x¨ , ¨y does. So, R M too is a crossed module with invariant pairing. R M is not fine in general, but itis fine if M is.In subsect. 2.1, we introduced two basic model Lie groups crossed modules,21he inner automorphism crossed module INN G and the finite coadjoint actioncrossed module AD ∗ G of a Lie group G . Since their associated Lie algebra crossedmodules are respectively INN g and AD ∗ g , they are both balanced and they canboth presumably be equipped with the invariant pairings of these latter.The inner automorphism crossed module of G is INN G “ p G , G , id G , Ad G q .The Lie algebra crossed module associated with INN G is the inner derivationcrossed module of INN g which we discussed earlier. If g is equipped with a Ad G –invariant symmetric non singular pairing, INN g gets endowed with an invariantpairing obeying (A.3.5). In this way, INN G becomes a crossed module with invari-ant pairing. Since id G is trivially invertible, INN G is fine. INN G is characterizedby the following property. If M “ p E , G , τ, µ q is a crossed module with invariantpairing such that τ is invertible, then there is an invariant pairing on INN G suchthat M is isomorphic to INN G . The isomorphism ξ of M onto INN G is given bythe pair p τ, id G q . Its Lie differential ξ is precisely the isomorphism i of m onto INN g , which we described in the Lie algebra case. Note that if M is a crossedmodule with generic τ , then its core C M is isomorphic to INN ran τ .The finite coadjoint action crossed module of G is AD ∗ G “ p g ∗ , G , G , Ad G ∗ q .The Lie algebra crossed module associated with AD ∗ G is the infinitesimal coad-joint action crossed module of AD ∗ g which we also discussed above. AD ∗ G isendowed with a natural invariant pairing obeying (A.3.5), since AD ∗ g is equippedwith the Ad G –invariant duality pairing of g and g ∗ . AD ∗ G is fine, since g ∗ isclearly connected. Contrary to what one may expect, if M “ p E , G , τ, µ q is acrossed module with invariant pairing with τ vanishing, then M is not necessarilyisomorphic to AD ∗ G . The isomorphism j of INN g onto m we defined above,given by the pair p J, id G q with J the linear isomorphism of g ∗ onto e such that x x, J p X qy “ x x, X y AD ∗ , does not integrate in general to an isomorphism η of INN G onto M , because J does not integrate to an isomorphism of g ∗ onto E . Asa consequence, when M is a crossed module with generic τ , its residue R M isgenerally not isomorphic to AD ∗ p G { ran τ q .From the above discussion, it is now not surprising that M fi C M ˆ R M ingeneral for the reason that the Lie algebra isomorphisms h : g Ñ ran τ ‘p g { ran τ q H : e Ñ p e { ker τ q‘ ker τ underlying the crossed module isomorphism (2.2.11)(cf. app. A.4) in general cannot be lifted to corresponding Lie group isomor-phisms φ : G Ñ ran τ ˆ p G { ran τ q and Φ : E Ñ p E { ker τ q ˆ ker τ unless all Liegroups involved are connected and simply connected. Crossed submodules and isotropy
The concept of crossed submodule of a Lie group crossed module answers to thefamiliar concept of subgroup of a Lie group and similarly in the Lie algebra case.In this subsection, we shall define and study submodules of a given crossed moduleand their associated normalizer and Weyl crossed modules. When the crossedmodule is equipped with an invariant pairing, isotropic crossed submodules canbe considered and constitute a distinguished subclass of submodules with specialfeatures. Before proceeding to detailing the properties of these substructures,however, we shall explain in simple elementary terms why they are relevant inthe construction of 4–dimensional CS theory.In 4–dimensional CS theory, the higher gauge field Ω and infinitesimal gaugetransformations Θ are non homogeneous forms on the underlying 4–fold valuedin the Lie algebra crossed module m of the symmetry Lie group crossed module M (cf. subsect. 3.3).When the base manifold has a boundary, as we generally assume in 4–dimen-sional CS theory, boundary conditions must be imposed on the gauge field Ω andinfinitesimal gauge transformations Θ ensuring that the variational problem iswell–posed on one hand (cf. subsect. 4.1) and that gauge invariance is preservedon the other (cf. subsect. 4.2). The boundary condition on Ω must be suchto make the boundary integral yielded by the variation of the CS action vanish.Since the boundary integrand involves the invariant pairing x¨ , ¨y of the crossedmodule m , this can be achieved by demanding the field Ω to be valued on theboundary in a crossed submodule m of m isotropic with respect to x¨ , ¨y . Requiringthis boundary condition to be gauge invariant forces also the transformations Θ to be valued in m on the boundary. Such particularly simple choice of boundaryconditions, which we shall generally adopt in the following, is called isotropic23inear for evident reasons.Infinitesimal gauge transformations form a Lie algebra. In particular, the Liebracket r Θ , Θ s is defined for any gauge transformation pair Θ , Θ . In the canon-ical formulation of 4–dimensional CS theory (cf. subsect. 4.5), the action of thegauge transformations Θ is generated by Hamiltonian functionals Q p Θ q of thegauge field Ω (not explicitly shown here). The functionals Q p Θ q span under Pois-son bracketing a centrally extended representation of the gauge transformationLie algebra. Explicitly, t Q p Θ q , Q p Θ qu “ Q pr Θ , Θ sq ` κ ¨ c p Θ , Θ q , (2.3.1)where c is a certain 2–cocycle of the gauge transformation Lie algebra.The physical higher gauge field phase space is defined by the constraints Q p Θ q « , (2.3.2)with Θ any gauge transformation, according to naive Dirac theory. However, the Q p Θ q do not form a first class set of phase space functionals because of the centralterm appearing in the right hand side of (2.3.1). Furthermore, the vanishing ofthe Q p Θ q by itself does not select the functional submanifold of flat gauge fields Ω , as one would like to in CS theory.Both the 2–cocycle c p Θ , Θ q in (2.3.1) and the obstructing term of Q p Θ q pre-venting (2.3.2) from singling out the flat gauge field space are given by certainboundary integrals. The above problems can therefore be solved by imposingsuitable boundary conditions on the gauge field Ω and gauge transformations Θ making those integral expressions vanish. Since the integrands involve the in-variant pairing x¨ , ¨y of the crossed module m , this can be achieved by requiringboth Ω and the Θ to be valued on the boundary in a crossed submodule m of m isotropic with respect to x¨ , ¨y , i.e. by imposing again isotropic linear boundaryconditions. When Ω and the Θ obey these conditions, as we suppose, (2.3.2)becomes a set of genuine flatness enforcing first class constraints.If Θ is a gauge transformation such that t Q p Θ q , Q p Θ qu « (2.3.3)24or any gauge transformation Θ obeying the isotropic linear boundary condition,then Q p Θ q represents a physical symmetry surface charge. Comparing (2.3.3)to (2.3.1), we see that for this to be the case, Θ must itself obey a boundarycondition ensuring that r Θ , Θ s satisfies the isotropic linear boundary conditionand c p Θ , Θ q “ for any such Θ . This condition consists so in requiring Θ tobe valued on the boundary in the orthogonal normalizer crossed module ON m of m , that is the maximal crossed submodule of m that normalizes m in theappropriate sense and is orthogonal to m with respect to the invariant pairing x¨ , ¨y . However, as Q p Θ q is weakly invariant under the shift Θ Ñ Θ ` Θ with Θ obeying the isotropic linear boundary condition by virtue of (2.3.2), Θ iseffectively valued on the orthogonal Weyl crossed module OW m “ ON m { m of m . The boundary condition is therefore called orthogonal Weyl linear.Having motivated the consideration of crossed submodules, in particular iso-tropic ones, in 4–dimensional CS theory, we now proceed to describe these struc-tures in more precise terms. Albeit the above discussion has been carried outmostly at the infinitesimal level, we shall examine first the finite case. Lie group crossed submodules and their normalizer and Weyl modules
The notion of crossed submodule of a finite Lie crossed module is analogousto that of subgroup of a Lie group. A crossed submodule M of a Lie groupcrossed module M is indeed a substructure of M which is itself a Lie group crossedmodule. More formally, given two Lie group crossed modules M “ p E , G , τ, µ q , M “ p E , G , τ , µ q , M is a submodule of M if E , G are Lie subgroups of E , G and τ , µ are the restrictions of τ , µ to E , G ˆ E , respectively.The concept of normalizer crossed module of a submodule of a finite Liecrossed module corresponds in turn to that of normalizer group of a subgroupof a Lie group. The normalizer N M of a crossed submodule M of a Lie groupcrossed module M is the largest crossed submodule of M normalizing M (cf. app.A.1). N M can be described rather explicitly as follows. Let M “ p E , G , τ, µ q , M “ p E , G , τ , µ q . The normalizer of G in G , N G , is the set of all elements a P G such that aba ´ , a ´ ba P G for b P G . Similarly, the µ –normalizer of E in G , µ N E , is the set of all elements a P G such that µ p a, B q , µ p a ´ , B q P E for25 P E . The µ –transporter of G into E in E , µ T G E is the set of all elements A P E such that µ p b, A q A ´ , A ´ µ p b, A q P E for b P G . N G and µ N E turnout to be Lie subgroups of G and likewise µ T G E a Lie subgroup of E . Then, N M “ p µ T G E , N G X µ N E , τ N , µ N q , where τ N , µ N are the restrictions of τ , µ to µ T G E , N G X µ N E ˆ µ T G E , respectively. It can be verified that thestructure maps τ N , µ N are well defined and satisfy the required properties (A.1.1),(A.1.2). N M is so a Lie group crossed module. By construction, N M is a crossedsubmodule of M containing M as a crossed submodule and normalizing it and ismaximal with these properties.The Weyl crossed module of a submodule of a finite Lie crossed module ismuch like the Weyl group of a subgroup of a Lie group. For a crossed submodule M of a Lie group crossed module M , this is just the quotient W M “ N M { M removing from the normalizer crossed module N M of M the trivially normalizingsubmodule M . Specifically, let again M “ p E , G , τ, µ q , M “ p E , G , τ , µ q . Itcan be shown that G , E are normal Lie subgroups of N G X µ N E , µ T G E ,respectively. We have then W M “ p µ T G E { E , N G X µ N E { G , τ W , µ W q , wherethe structure maps τ W , µ W , are given by τ W p A E q “ τ p A q G , (2.3.4) µ W p a G , A E q “ µ p a, A q E (2.3.5)for a P N G X µ N E , A P µ T G E . It can be verified that the structure maps τ W , µ W are well defined and obey relations (A.1.1), (A.1.2). W M is therefore a Liegroup crossed module as required.There are plenty of examples of crossed submodules of Lie group crossedmodules, in particular of the model ones described in subsect. 2.1. They areillustrated next, but before doing so we introduce some notation. Let G be afixed Lie group H , K be Lie subgroups of G . We denote by r H , K s the commutatorsubgroup of H , K in G . We denote further by T H K the transporter of H into K ,the largest subgroup L of G such that r H , L s is contained in K . This coincideswith the Ad –transporter Ad T H K as previously defined.Let G be a Lie group, H be a subgroup of G and K be a normal subgroup26f H . Then, INN H K “ p K , H , id H | K , Ad H | H ˆ K q is a Lie group crossed module,called inner H –automorphism crossed module of K , and is a crossed submoduleof INN G “ p G , G , id G , Ad G q , the inner automorphism crossed module of G . Thenormalizer crossed module of INN H K is a crossed module of the same kind, viz N INN H K “ INN N H X N K T H K . It is easily verified that T H K is a normal sub-group of N H X N K as required. The Weyl crossed module of INN H K is there-fore W INN H K “ p T K , H , N K , H , id N K , H | T K , H , Ad N K , H | N K , H ˆ T K , H q , where T K , H “ T H K { K , N K , H “ N H X N K { H . (Here, id N K , H | T K , H , Ad N K , H | N K , H ˆ T K , H denote abusively thenatural map T K , H Ñ N K , H and the maps induced by Ad N K , H upon compositionwith it, respectively.)Let G be a Lie group and H , K be subgroups of G with H Ď N K . Then, AD H ∗ K “ p k , H , H | k , Ad H ∗ | H ˆ k q , k Ď g ∗ being the annihilator of k , is a Liegroup crossed module, called the finite coadjoint H –action crossed module of K ,and is a crossed submodule of AD ∗ G “ p g ∗ , G , G , Ad G ∗ q , the finite coadjointaction crossed module of G . If H , K are connected, the normalizer crossed moduleof AD H ∗ K is a crossed module of the same kind, as N AD H ∗ K “ AD N H X N K ∗ r H , K s .Again, it can be straightforwardly verified that N H X N K Ď N r H , K s as required.Above, connectedness is assumed only for the sake of simplicity. The Weyl crossedmodule of AD H ∗ K then is W AD H ∗ K “ p t K , H , N K , H , N K , H | t K , H , Ad N K , H ∗ | N K , H ˆ t K , H q ,where t K , H “ r h , k s { k , N K , H “ N H X N K { H . Lie algebra crossed submodules and their normalizer and Weyl modules
As it might be expected, there are infinitesimal counterparts of the Lie grouptheoretic notions introduced above. Albeit it is not difficult to guess them, wereport them below for completeness.The concept of crossed submodule of a infinitesimal Lie crossed module isinspired by that of subalgebra of a Lie algebra. A crossed submodule m of aLie algebra crossed module m is indeed a substructure of m which is itself aLie algebra crossed module. Specifically, given two Lie algebra crossed modules m “ p e , g , t, m q , m “ p e , g , t , m q , m is a submodule of m if e , g are Liesubalgebras of e , g and t , m are the restrictions of t , m to e , g ˆ e , respectively.The notion of normalizer crossed module of a submodule of a infinitesimal Lie27rossed module correlates with that of normalizer algebra of a subalgebra of aLie algebra as expected. The normalizer N m of a crossed submodule m of a Liealgebra crossed module m is the largest crossed submodule of m normalizing m (cf. app. A.1). Explicitly, N m can be specified as follows. Let m “ p e , g , t, m q , m “ p e , g , t , m q . The normalizer of g in g , N g , is the set of all elements u P g such that r u, v s P g for v P g . Similarly, the m –normalizer of e in g , m N e , isthe set of all elements u P g such that m p u, V q P e for V P e . The m –transporterof g into e in e , m T g e is the set of all elements U P e such that m p v, U q P e for v P g . N g and m N e turn out to be Lie subalgebras of g and likewise m T g e a Lie subalgebra of e . Then, N m “ p m T g e , N g X m N e , t N , m N q , where t N , m N are the restrictions of t , m to m T g e , N g X m N e ˆ m T g e , respectively.It can be verified that the structure maps t N , m N are well defined and satisfythe required properties (A.1.7), (A.1.8). N m so turns out to be a Lie algebracrossed module. By the way we have designed it, N m is a crossed submodule of m containing m as a crossed submodule and normalizing it and is maximal withthese properties.The Weyl crossed module of a submodule of a infinitesimal Lie crossed moduleis now conceived similarly to the Weyl algebra of a subalgebra of a Lie algebra.For a crossed submodule m of a Lie algebra crossed module m , this is just thequotient W m “ N m { m removing from the normalizer crossed module N m of m the trivially normalizing submodule m . Concretely, let again m “ p e , g , t, m q , m “ p e , g , t , m q . It can be shown that g , e are Lie ideals of N g X m N e , m T g e , respectively. We have then W m “ p m T g e { e , N g X m N e { g , t W , m W q ,where the structure maps t W , m W , are given by t W p U ` e q “ t p U q ` g , (2.3.6) m W p u ` g , U ` e q “ m p u, U q ` e (2.3.7)for u P N g X m N e , U P m T g e . It can be verified that the structure maps t W , m W are well defined and obey relations (A.1.1), (A.1.2). W m is therefore a Liealgebra crossed module as required.We present next a class of examples of crossed submodules of the model Lie28lgebra crossed modules of subsect. 2.1, matching those introduced above inthe finite case, after recalling a few basic notions of Lie theory. Let g be a Liealgebra and h , k be Lie subalgebras of g . We denote by r h , k s the commutatorsubalgebra of h , k in g . We denote further by T h k the transporter of h into k , thatis the largest subalgebra l of g such that r h , l s is contained in k . This is just the ad –transporter ad T h k we defined earlier.Let g be a Lie algebra, h be a subalgebra of g and k be an ideal of h . Then, INN h k “ p k , h , id h | k , ad h | h ˆ k q turns out to be a Lie algebra crossed module, calledthe inner h –derivation crossed module of k , and is a crossed submodule of INN g “ p g , g , id g , ad g q , the inner derivation crossed module of g . The normalizer crossedmodule of INN h k is a crossed module of the same kind as it, since we have N INN h k “ INN N h X N k T h k , T h k being an ideal of N h X N k . The Weyl crossedmodule of INN h k is hence W INN h k “ p t k , h , n k , h , id n k , h | t k , h , ad n k , h | n k , h ˆ t k , h q , where t k , h “ T h k { k , n k , h “ N h X N k { h . (Here, id n k , h | t k , h , ad n k , h | n k , h ˆ t k , h denote abusively asbefore the natural map t k , h Ñ n k , h and the map induced by ad n k , h upon compositionwith it, respectively.)Next, let g be a Lie algebra and h , k be subalgebras of g such that h Ď N k .Then, AD h ∗ k “ p k , h , h | k , ad h ∗ | h ˆ k q , k Ď g ∗ being again the annihilator of k , isa Lie algebra crossed module, the infinitesimal coadjoint h –action crossed moduleof k , and is a crossed submodule of AD ∗ g “ p g ∗ , g , g , ad g ∗ q , the infinitesimalcoadjoint action crossed module of g . The normalizer crossed module of AD h ∗ k is again a crossed module of the same kind, NAD h ∗ k “ AD N h X N k ∗ r h , k s , whereone has N h X N k Ď N r h , k s . The Weyl crossed module of AD h ∗ k is in this wayfound to be W AD h ∗ k “ p t k , h , n k , h , n k , h | t k , h , ad n k , h ∗ | n k , h ˆ t k , h q , where t k , h “ r h , k s { k , n k , h “ N h X N k { h .The above constructions are of course designed to be compatible with Liedifferentiation. If M , M are Lie group crossed modules with associated Lie algebracrossed modules m , m and M is a crossed submodule of M , then m is a crossedsubmodule of m . Further, the Lie algebra crossed modules of the normalizer andWeyl crossed modules N M and W M of M are precisely the normalizer and Weylcrossed modules N m and W m of m . 29econsider the model examples INN H K Ď INN G , AD H ∗ K Ă AD ∗ G of crossedsubmodules described earlier in this subsection defined for a Lie group G andsubgroups H , K of G satisfying the stated conditions. Then, the Lie algebracrossed module of INN H K , AD H ∗ K are INN h k , AD h ∗ k , respectively, as it mightbe expected. The orthogonal case
We now consider the case where an ambient Lie algebra crossed module isequipped with an invariant pairing (cf. subsect. 2.2). This allows us to considerisotropic crossed submodules.Let m “ p e , g , t, m q be a Lie algebra crossed module with invariant pairing x¨ , ¨y (cf. subsect. 2.2) and let m “ p e , g , t , m q be a crossed submodule of m . m is said to be isotropic if e Ď g , where K denotes orthogonal complement withrespect to x¨ , ¨y . m is said to be Lagrangian if it is maximally isotropic, i.e. if e “ g . When m is isotropic, dim g ` dim e ď dim g “ dim e , the bound beingsaturated when m is Lagrangian.When m is an isotropic submodule, it is possible to define the orthogonalnormalizer ON m of m refining the normalizer N m . ON m is the orthogonalcomplement of m in N m with respect to the pairing x¨ , ¨y in the appropriate sense.More formally, ON m “ p m T g e X g , N g X m N e X e , t ON , m ON q , where t ON , m ON are the restrictions of t , m to m T g e X g , N g X m N e X e ˆ m T g e X g ,respectively. ON m turns out to be a crossed submodule of N m and so m itself.Further, ON m contains m as a submodule. The orthogonal Weyl crossed moduleis the quotient module OW m “ ON m { m . ON m reduces to m and OW m vanishes when m is Lagrangian.Suppose that M is a Lie group crossed module with invariant pairing x¨ , ¨y (cf.subsect. 2.2) and that M is a Lie group crossed submodule of M and that m , m are the Lie algebra crossed modules of M , M , respectively. M is said to be anisotropic submodule of M if m is an isotropic submodule of m in the sense definedabove and similarly in the Lagrangian case.When M is an isotropic crossed submodule of M , it is possible under certainconditions to define an orthogonal normalizer ON M of M refining the normalizer30 M introduced above. ON M is a crossed submodule of N M having ON m asassociated Lie algebra crossed module and containing M as a crossed submodule.Because of its defining properties, ON M normalizes M and so the orthogonalWeyl crossed module OW M “ ON M { M of M can be defined. We shall notinvestigate here the precise conditions ensuring the existence of ON M . In thefollowing we shall tacitly assume that they are satisfied. For instance, if G , E are connected a choice of ON M exists.We now examine whether the model crossed submodules introduced aboveturn out to be isotropic in the presence of an invariant pairing.Let g be a Lie algebra equipped with an invariant symmetric non singularbilinear form x¨ , ¨y . With this extra datum, the inner derivation crossed module INN g “ p g , g , id g , ad g q of g is a Lie algebra crossed module with invariant pairing(cf. subsect. 2.2). Let h be a subalgebra of g and k be an ideal of h such that k Ď h K . Then, the inner h –derivation crossed module INN h k “ p k , h , id h | k , ad h | h ˆ k q of k which we defined above is an isotropic submodule of INN g . The orthogonalnormalizer crossed module of INN h k is ONINN h k “ INN N h X N k X k K p T h k X h K q andis once more of the same kind. The orthogonal Weyl crossed module of INN h k is OW INN h k “ p ot k , h , on k , h , id on k , h | ot k , h , ad on k , h | on k , h ˆ ot k , h q , where ot k , h “ T h k X h K { k , on k , h “ N h X N k X k K { h . We remark here that under the stated hypotheses thesubalgebra h is isotropic in g . We also note that INN h k is Lagrangian preciselywhen k “ h K .Next, let g be a Lie algebra. Then, the duality pairing x¨ , ¨y of g , g ∗ ren-ders the infinitesimal coadjoint action crossed module AD ∗ g “ p g ∗ , g , g , ad g ∗ q of g a Lie algebra crossed module with invariant pairing (cf. subsect. 2.2).Let h , k be subalgebras of g with h Ď k . Then, h Ď N k and the infinitesimalcoadjoint h –action crossed module AD h ∗ k “ p k , h , h | k , ad h ∗ | h ˆ k q of k whichwe defined above is an isotropic submodule of AD ∗ g . The orthogonal nor-malizer crossed module of AD h ∗ k is ON AD h ∗ k “ AD N h X k ∗ pr h , k s ` h q , and sois of the same kind too. The orthogonal Weyl crossed module of AD h ∗ k is OW AD h ∗ k “ p ot k , h , on k , h , on k , h | ot k , h , ad on k , h ∗ | on k , h ˆ ot k , h q with ot k , h “ pr h , k s ` h q { k , on k , h “ N h X k { h . We note that AD h ∗ k is Lagrangian precisely when k “ h .31 Higher gauge theory in the derived formulation
In this section, we formulate higher gauge theory in a novel derived formal frame-work worked out in refs. [51, 52], that we shall adopt in the construction of 4–dimensional CS theory. The framework has the distinguished merit of showingthat the relationship of higher to ordinary gauge theory is much closer than itwas hitherto thought. It also provides an elegant graded geometric set–up for themanipulation of crossed module valued non homogeneous forms. There are twoversions of the derived set up, the ordinary and the internal . The former, suit-able for the conventional formulation of 4–dimensional CS theory studied in thispaper, is illustrated in this section. The latter, required in the Batalin–Vilkovisky(BV) [69, 70] formulation, will be presented elsewhere. Derived Lie groups and algebras
The derived Lie group of a Lie group crossed module and the correspondinginfinitesimal notion of derived Lie algebra of a Lie algebra crossed module wereoriginally introduced in refs. [51,52]. In the 4–dimensional CS theory we present,they pay a role analogous to that of the gauge group of ordinary gauge theory.Before proceeding to the illustration of this topic a few introductory commentsare useful. The formal set–up of derived Lie groups and algebras is an elegantand convenient way of handling certain structural elements of the Lie group andalgebra crossed modules entering in the formulation of higher gauge theory. Inpractice, it is a kind of superfield formalism not dissimilar to the analogous for-malisms broadly used in supersymmetric field theories. It is particularly suitedfor applications to 4–dimensional CS theory because of its compactness and ca-pability of presenting it as an ordinary CS theory with an exotic graded gauge We recall briefly the difference between ordinary and internal maps. Consider a pair X , Y of graded manifolds and a map ϕ from X to Y . When expressed in terms of local body andsoul coordinates x a and ξ r of X , the components of one such function with respect to localbody and soul coordinates y i and η h of Y are polynomials in the ξ r with coefficients which aresmooth functions of the x a . If ϕ is an ordinary map, these coefficient have degree If instead ϕ is internal, the may have non zero degree. M “ p E , G , τ, µ q . The derived Lie group D M of M consists of the internal maps from R r´ s to the semidirect product E ¸ µ G of the Lie groups E and G with respect to the G –action µ of the form P p ¯ α q “ e ¯ αP p (3.1.1)with ¯ α P R r´ s , where p P G , P P e r s with the following operations. For any P , Q P D M with P p ¯ α q “ e ¯ αP p , Q p ¯ α q “ e ¯ αQ q , one has PQ p ¯ α q “ e ¯ α p P ` µ p p,Q qq pq, (3.1.2) P ´ p ¯ α q “ e ´ ¯ αµ p p ´ ,P q p ´ . (3.1.3) D M is a graded Lie group. The graded Lie group isomorphism D M » e r s ¸ µ G (3.1.4)holds, where e is regarded as an Abelian Lie group and e r s ¸ µ G denotes thesemidirect product of the Lie groups e r s and G with respect to the G –action µ . The operator D has nice functorial properties. A morphism β : M Ñ M ofLie group crossed modules induces through its constituent Lie group morphisms Φ : E Ñ E , φ : G Ñ G a Lie group morphism D β : D M Ñ D M . Further, if M , M are Lie group crossed modules, then D p M ˆ M q “ D M ˆ D M .33he notion of derived Lie group has an infinitesimal counterpart. Considera Lie algebra crossed module m “ p e , g , t, m q . The derived Lie algebra D m of m consists of the internal maps from R r´ s to the semidirect product e ¸ m g of theLie algebras e and g with respect to the g –action m of the form U p ¯ α q “ u ` ¯ αU, (3.1.5)with ¯ α P R r´ s , where u P g , U P e r s with the obvious linear structure and thefollowing Lie bracket. For any U , V P D m such that U p ¯ α q “ u ` ¯ αU , V p ¯ α q “ v ` ¯ αV , one has r U , V sp ¯ α q “ r u, v s ` ¯ α p m p u, V q ´ m p v, U qq . (3.1.6) D m is a graded Lie algebra. The graded Lie algebra isomorphism D m » e r s ¸ m g (3.1.7)holds, where e is regarded as an Abelian Lie algebra and e r s ¸ m g denotes thesemidirect product of the Lie algebras e r s and g with respect to the g –action m .As in the finite case above, the operator D has functorial properties. A morphism p : m Ñ m of Lie algebra crossed modules induces via its underlying Lie algebramorphisms H : e Ñ e , h : g Ñ g , a Lie algebra morphism D p : D m Ñ D m .Further, if m , m are Lie algebra crossed modules, D p m ‘ m q “ D m ‘ D m .The derived set–up introduced above is fully compatible with Lie differenti-ation. If M “ p E , G , τ, µ q is a Lie group crossed module and m “ p e , g , τ, µ q isits associated Lie algebra crossed module (cf. subsect. 2.1), then D m is the Liealgebra of D M . Further, if β : M Ñ M is a Lie group crossed module morphismand β : m Ñ m is the corresponding Lie algebra crossed module morphism, then D β “ D β .The derived set–up is also consistent with the submodule structure of theunderlying crossed module (cf. subsect. 2.3). If M is a Lie group crossed moduleand M is a crossed submodule of M , then D M is Lie subgroup of D M . Further,if the Lie group E of M is connected, then the normalizer and Weyl crossedmodules N M and W M of M satisfy D N M “ N D M and D W M “ W D M ,34here ND M and W D M “ ND M { D M are the normalizer and Weyl Lie groupsof D M . The reason why E is required to be connected is that N D M normalizesonly the Lie algebra e and so only the connected component of E in E upon Lieintegration. Likewise, if m is a Lie algebra crossed module and m is a crossedsubmodule of m , then D m is Lie subalgebra of D m . Further, the normalizer andWeyl crossed modules N m and W m of m have the property that D N m “ ND m and DW m “ W D m , where ND m and W D m “ ND m { D m are the normalizerand Weyl Lie algebras of D m .Suppose that m is a Lie algebra crossed module with invariant pairing x¨ , ¨y (cf. subsect. 2.2. Then, D m is equipped with an induced symmetric non singularbilinear form p¨ , ¨q : D m ˆ D m Ñ R r´ s defined by p U , V q “ x u, σ V y ` x v, σ U y (3.1.8)for any U , V P D m , where σ : e r s Ñ R r´ s b e is the natural –fold sus-pension map. σ serves the purpose of identifying D m with the internal mapspace Map p ∗ , D m q , where ∗ is the singleton, since Map p ∗ , v r p sq is isomorphic to R r´ p s b v for any vector space v . The identification in turn is necessary for theright hand side of (3.1.8) to make sense. The pairing p¨ , ¨q is invariant as pr W , U s , V q ` p U , r W , V sq “ (3.1.9)for U , V , W P D m .Let m be a crossed submodule of m . Then, p¨ , ¨q restricts to a symmetricbilinear form p¨ , ¨q : D m ˆ D m Ñ R r´ s , which however is not non singularany longer in general. m is an isotropic crossed submodule of m (cf. subsect.2.3) precisely when D m is an isotropic subalgebra of D m . In that case, dim D m ď dim D m “ dim g “ dim e , the bound being saturated when m is Lagrangian.If m is an isotropic crossed submodule of m , the orthogonal normalizer andWeyl crossed modules ON m and OW m of m turn out to be DON m “ OND m and D OW m “ OW D m respectively, where OND m “ N D m X D m and OW D m “ OND m { D m are the orthogonal normalizer and Weyl Lie algebrasof D m , D m denoting the orthogonal complement of D m .35 .2 Derived superfield formulation
In this subsection, we shall survey the main spaces of Lie group and algebracrossed module valued fields using a derived superfield formulation. This allowsfor a very compact geometrically transparent formulation of 4–dimensional CStheory studied in later sections.We assume that the fields propagate on a general manifold X . Later, we shalladd the restriction that X is orientable and compact, possibly with boundary.To include also differential forms, using however a convenient graded geometricdescription, the fields will be maps from the shifted tangent bundle T r s X of X into some graded target manifold T . Below, we denote by Map p T r s X, T q thespace of ordinary maps from T r s X into T . The broader space of internal mapsfrom T r s X to T required to incorporate ghost-like fields in a BV set–up can alsobe considered, though we shall not do so in this paper.The fields we shall consider will be valued either in the derived Lie group D M of a Lie group crossed module M “ p E , G , τ, µ q or in the derived Lie algebra D m of the associated Lie algebra crossed module m “ p e , g , τ, µ q (cf. subsects. 2.1)and 3.1). A more comprehensive treatment of this kind of fields is provided inref. [51].We consider first D M –valued fields. Fields of this kind are elements of themapping space Map p T r s X, D M q . If U P Map p T r s X, D M q , then U p α q “ e αU u (3.2.1)with α P R r s , where u P Map p T r s X, G q , U P Map p T r s X, e r sq . u , U are thecomponents of U . Map p T r s X, D M q has a Lie group structure induced by thatof D M : if U P Map p T r s X, D M q , V P Map p T r s X, D M q , then UV p α q “ e α p U ` µ p u,V qq uv, U ´ p α q “ e ´ αµ p u ´ ,U qq u ´ . (3.2.2)Next, we consider first D m –valued fields. Fields of this kind are elements ofthe mapping space Map p T r s X, D m q . If Φ P Map p T r s X, D m q , then Φ p α q “ φ ` αΦ (3.2.3)36ith α P R r s , where φ P Map p T r s X, g q , Φ P Map p T r s X, e r sq . Again, φ , Φ arethe components of Φ . Map p T r s X, D m q has a Lie algebra structure induced bythat of D m : if Φ P Map p T r s X, D m q , Ψ P Map p T r s X, D m q , then r Φ , Ψ sp α q “ r φ, ψ s ` α p µ p φ, Ψ q ´ µ p ψ, Φ qq . (3.2.4) Map p T r s X, D m q is the virtual Lie algebra of Map p T r s X, D M q . (For an expla-nation of this terminology, see ref. [51]).As it turns out, the D m –valued fields introduced above are not enough forour proposes. One also needs to incorporate fields that are valued in the degreeshifted linear spaces D m r p s with p an integer. If Φ P Map p T r s X, D m r p sq , then Φ p α q “ φ ` p´ q p αΦ (3.2.5)with components φ P Map p T r s X, g r p sq , Φ P Map p T r s X, e r p ` sq . There is abilinear bracket that associates with a pair of fields Φ P Map p T r s X, D m r p sq , Ψ P Map p T r s X, D m r q sq a field r Φ , Ψ s P Map p T r s X, D m r p ` q sq given by r Φ , Ψ sp α q “ r φ, ψ s ` p´ q p ` q α p µ p φ, Ψ q ´ p´ q pq µ p ψ, Φ qq (3.2.6)Setting Z D m “ À p “´8 D m r p s , Map p T r s X, ZD m q is a graded Lie algebra. Thiscontains the Lie algebra Map p T r s X, D m q as its degree subalgebra.An adjoint action of Map p T r s X, D M q on the Lie algebra Map p T r s X, D m q and more generally on the graded Lie algebra Map p T r s X, ZD m q is defined. For U P Map p T r s X, D M q , Φ P Map p T r s X, D m r p sq , one has Ad U p Φ qp α q “ Ad u p φ q ` p´ q p α p µ p u, Φ q ´ µ p Ad u p φ q , U qq , (3.2.7) Ad U ´ p Φ qp α q “ Ad u ´ p φ q ` p´ q p αµ p u ´ , Φ ` µ p φ, U qq . (3.2.8)The adjoint action preserves Lie brackets as in ordinary Lie theory. Indeed, for U P Map p T r s X, D M q , Φ P Map p T r s X, D m r p sq , Ψ P Map p T r s X, D m r q sq , r Ad U p Φ q , Ad U p Ψ qs “ Ad U pr Φ , Ψ sq . (3.2.9)As is well–known, in the graded geometric formulation we adopt, the nilpotent37e Rham differential d is a degree homological vector field on T r s X , d “ . d induces a natural degree derived differential d on the graded vector space Map p T r s X, Z m q . Concisely, d “ d ` τ d { dα . In more ore explicit terms, for Φ P Map p T r s X, D m r p sq , the field d Φ P Map p T r s X, D m r p ` sq reads asd Φ p α q “ dφ ` p´ q p τ p Φ q ` p´ q p ` αdΦ. (3.2.10)It can be straightforwardly verified that shown thatd r Φ , Ψ s “ r d Φ , Ψ s ` p´ q p r Φ , d Ψ s (3.2.11)for Φ P Map p T r s X, D m r p sq , Ψ P Map p T r s X, D m r q sq and thatd “ . (3.2.12)In this way, Map p T r s X, ZD m q becomes a differential graded Lie algebra.On several occasions, the pull–back dUU ´ , U ´ dU P Map p T r s X, D m r sq ofthe Maurer–Cartan forms of D M by a D M field U P Map p T r s X, D M q will enterour considerations. For these, there exits explicit expressions, dUU ´ p α q “ duu ´ ` τ p U q´ α ` dU ` r U, U s ´ µ p duu ´ ` τ p U q , U q ˘ , (3.2.13) U ´ dU p α q “ Ad u ´ ` duu ´ ` τ p U q ˘ ´ αµ ` u ´ , dU ` r U, U s ˘ . (3.2.14)By the relation d “ d ` τ d { dα , (3.2.13), (3.2.14) follow from (3.2.1) and the varia-tional identities δ e αX e ´ αX “ exp p α ad X q´ α ad X δ p αX q , e ´ αX δ e αX “ ´ exp p´ α ad X q α ad X δ p αX q with δ “ τ d { dα , owing to the nilpotence of α .Next, we assume that the Lie group crossed module M is equipped with aninvariant pairing x¨ , ¨y . A pairing on the graded Lie algebra Map p T r s X, Z m q isinduced in this way: for Φ P Map p T r s X, D m r p sq , Ψ P Map p T r s X, D m r q sqp Φ , Ψ q “ x φ, Ψ y ` p´ q pq x ψ, Φ y . (3.2.15)Note that p Φ , Ψ q P Map p T r s X, R r p ` q ` sq . The field pairing p¨ , ¨q therefore hasdegree . p¨ , ¨q is bilinear. More generally, when scalars with non trivial grading38re involved, the left and right brackets p and q behave as if they had respectivelydegree and . For instance, p c Φ , Ψ q “ c p Φ , Ψ q whilst p Φ , Ψ c q “ p´ q k p Φ , Ψ q c ifthe scalar c has degree k . p¨ , ¨q is further graded symmetric, p Φ , Ψ q “ p´ q pq p Ψ , Φ q . (3.2.16) p¨ , ¨q is also non singular.The field pairing p¨ , ¨q has several other properties which make it a very naturalingredient in the field theoretic constructions of later sections. First, p¨ , ¨q is D M invariant. If Φ P Map p T r s X, D m r p sq , Ψ P Map p T r s X, D m r q sq , we have p Ad U p Φ q , Ad U p Ψ qq “ p Φ , Ψ q (3.2.17)for U P Map p T r s X, D M q . By Lie differentiation, p¨ , ¨q enjoys also D m invariance.This latter, however, admits a graded extension, because of which pr Ξ , Φ s , Ψ q ` p´ q pr p Φ , r Ξ , Ψ sq “ , (3.2.18)for Ξ P Map p T r s M, D m r r sq .Second, p¨ , ¨q is compatible with the derived differential d, i. e. the de Rhamvector field d differentiates p¨ , ¨q through d, d p Φ , Ψ q “ p d Φ , Ψ q ` p´ q p p Φ , d Ψ q . (3.2.19)Let M , M be Lie group crossed modules with associated Lie algebra crossedmodules m , m . Suppose that M is a submodule of M and that, consequently, m is a submodule of m (cf. subsect. 2.3). As D M is a Lie subgroup of D M , Map p T r s X, D M q is a Lie subgroup of Map p T r s X, D M q . Similarly, as D m is aLie subalgebra of D m , Map p T r s X, D m q is a Lie subalgebra of Map p T r s X, D m q .What is more, Map p T r s X, ZD m q is a differential graded Lie subalgebra of Map p T r s X, ZD m q , since it is invariant under the action of d as is evident from(3.2.10). If M is also equipped with invariant pairing x¨ , ¨y with respect to which M is isotropic (cf. subsect. 2.3), then m is isotropic and thus the Lie algebra Map p T r s X, ZD m q is isotropic, that is p Φ , Ψ q “ for Φ P Map p T r s X, D m r p sq , Ψ P Map p T r s X, D m r q sq . 39 .3 Higher gauge theory in the derived formulation
In this subsection, we present a formulation of higher gauge theory based onthe derived superfield formalism of subsect. 3.2. The framework we are goingto devise has the virtue of showing the close relationship of higher to ordinarygauge theory and allows so to import many ideas and techniques of the latter tothe former. The benefits of this approach will become evident in sect. 4 below,where 4–dimensional CS theory is worked out.In higher gauge theory, one should specify to begin with a Lie group crossedmodule M and a higher principal M –bundle P on some base manifold X . Inthis general setting, higher gauge fields and gauge transformations consist incollections of local Lie valued map and form data organized respectively as nonAbelian differential cocycles and cocycle morphisms [65, 66]. For the scope of thepresent paper, this level of generality is not necessary. It is enough that P be thetrivial M –bundle for which gauge fields and gauge transformations turn out to bemaps and forms globally defined on X . We shall however come back to this topicin greater detail in subsect. 3.5 below.The basic field of higher gauge theory is the gauge field, which in the derivedframework is a map Ω P Map p T r s X, D m r sq . In components, this reads as Ω p α q “ ω ´ αΩ, (3.3.1)where ω P Map p T r s X, g r sq , Ω P Map p T r s X, e r sq (cf. eq. (3.2.5)). ω , Ω arenothing but the familiar – and –form gauge fields of higher gauge theory.The higher gauge field Ω is characterized by its curvature Φ defined Φ “ d Ω ` r Ω , Ω s , (3.3.2)where the Lie bracket r¨ , ¨s and the differential d are defined by (3.2.6) and(3.2.10), respectively. The expression of Φ is otherwise formally identical tothat the curvature of a gauge field in ordinary gauge theory. By construction, Φ P Map p T r s X, D m r sq . Expressed in components, Φ reads as Φ p α q “ φ ` αΦ, (3.3.3)40here φ P Map p T r s X, g r sq , Φ P Map p T r s X, e r sq . φ , Φ are just the usual highergauge theoretic – and –form curvatures. They are expressible in terms of ω , Ω through the familiar relations φ “ dω ` r ω, ω s ´ τ p Ω q , (3.3.4) Φ “ dΩ ` µ p ω, Ω q . (3.3.5)The higher curvature Φ satisfies the higher Bianchi identityd Φ ` r Ω , Φ s “ (3.3.6)which follows from (3.3.2) in the usual way. This turns into a pair of Bianchiidentities for the curvature components φ , Φ , viz dφ ` r ω, φ s ` τ p Φ q “ , (3.3.7) dΦ ` µ p ω, Φ q ´ µ p φ, Ω q “ . (3.3.8)A higher gauge transformation is codified in a derived Lie group valued map U P Map p T r s X, D M q . U acts on the higher gauge field Ω as Ω U “ Ad U ´ p Ω q ` U ´ dU (3.3.9)where the adjoint action and pulled–back Maurer–Cartan form of U in the righthand side are defined in eqs. (3.2.8) and (3.2.14), respectively. Again, in thederived formulation the higher gauge transformation action is formally identicalto that of ordinary gauge theory. The higher curvature transforms as Φ U “ Ad U ´ p Φ q , (3.3.10)as expected. The gauge transformation U can be expressed in components as U p α q “ e αU u (3.3.11)with u P Map p T r s X, G q , U P Map p T r s X, e r sq according to (3.2.1). In terms ofthese, using systematically relations (3.2.8), (3.2.14), it is possible to write downthe gauge transform of the higher gauge field components ω , Ω ,41 u,U “ Ad u ´ ` ω ` duu ´ ` τ p U q ˘ , (3.3.12) Ω u,U “ µ ` u ´ , Ω ` µ p ω, U q ` dU ` r U, U s ˘ , (3.3.13)as well as those of the higher curvature components φ , Φ , φ u,U “ Ad u ´ p φ q , (3.3.14) Φ u,U “ µ ` u ´ , Φ ` µ p φ, U q ˘ . (3.3.15)These relations are the well–known gauge transformation expressions of the 1–and 2–form gauge fields and the 2– and 3–form curvatures of higher gauge theory.An infinitesimal higher gauge transformation is a derived Lie algebra valuedmap Θ P Map p T r s X, D m q . The Θ variation of the higher gauge field Ω is δ Θ Ω “ d Θ ` r Ω , Θ s , (3.3.16)where as before the Lie bracket r¨ , ¨s and the differential d are given by (3.2.6) and(3.2.10), respectively. In the derived formulation, the infinitesimal higher gaugetransformation action is again formally identical to that of ordinary gauge theory,in particular it still is the linearized form of its finite counterpart. As expected,so, the Θ variation of the higher curvature reads as δ Θ Φ “ r Φ , Θ s . (3.3.17)The gauge transformation Θ can be expressed in components as Θ p α q “ θ ` αΘ (3.3.18)with θ P Map p T r s X, g q , Θ P Map p T r s X, e r sq according to (3.2.1). In terms ofthese, exploiting relations (3.2.8), (3.2.14), we can write down the infinitesimalgauge transform of the higher gauge field components ω , Ω , δ θ,Θ ω “ dθ ` r ω, θ s ` τ p Θ q , (3.3.19) δ θ,Θ Ω “ dΘ ` µ p ω, Θ q ´ µ p θ, Ω q , (3.3.20)as well as those of the higher curvature components φ , Φ ,42 θ,Θ φ “ r φ, θ s , (3.3.21) δ θ,Θ Φ “ µ p φ, Θ q ´ µ p θ, Φ q . (3.3.22)Again, these are the infinitesimal gauge variation expressions of the 1– and 2–formgauge fields and the 2– and 3–form curvatures in higher gauge theory.A gauge transformation U is special if its components u , U have the form u “ τ p A q , (3.3.23) U “ ´ dAA ´ ´ µ p ω, A q (3.3.24)where A P Map p T r s X, E q . Note the dependence on the underlying gauge field Ω . Its action on the higher gauge field components ω , Ω is particularly simple, ω u,U “ ω. (3.3.25) Ω u,U “ Ω ` µ p φ, A ´ q . (3.3.26)Therefore, ω is invariant. Ω is not except for when the curvature component φ vanishes. The requirement φ “ is known in higher gauge theory as vanishingfake curvature condition. A special gauge transformation is one related to thetrivial gauge transformation by a gauge for gauge transformation. The latteris codified in the group valued map A .A special infinitesimal gauge transformation Θ has components θ , Θ given by θ “ τ p Ξ q , (3.3.27) Θ “ ´ dΞ ´ µ p ω, Ξ q (3.3.28)where Ξ P Map p T r s X, e q . In keeping with (3.3.25), (3.3.26), the correspondingvariations of the higher gauge field components ω , Ω are δ θ,Θ ω “ , (3.3.29) δ θ,Θ Ω “ ´ µ p φ, Ξ q , (3.3.30)with Ω invariant for φ “ .We introduce some notations that will be used frequently in the following.43he field space of higher gauge theory consists of the higher gauge field manifold C M p X q “ Map p T r s X, D m r sq . The higher gauge transformations constitute aninfinite dimensional Lie group G M p X q “ Map p T r s X, D M q acting on C M p X q ac-cording to (3.3.9). Similarly the infinitesimal higher gauge transformations makeup an infinite dimensional Lie algebra G M p X q “ Map p T r s X, D m q acting varia-tionally on C M p X q through (3.3.16). The special gauge transformations form asubgroup G M p X, ω q of G M p X q depending on an assigned gauge field Ω through itscomponent ω with associated Lie algebra G M p X, ω q . The derived functional framework of higher gauge theory
In field theoretic analysis, one deals with functionals of the relevant higher gaugefield on some compact manifold X . These are given as integrals on the shiftedtangent bundle T r s X of X of certain functions of Fun p T r s X q constructed usingthe gauge field. Integration is carried out using the Berezinian ̺ X of X .In higher gauge theory, the relevant field manifold is the higher gauge fieldspace C M p X q “ Map p T r s X, D m r sq introduced in subsect. 3.3. The field func-tionals we will consider belong to the graded algebra O ∗ M p X q “ Fun p T r s C M p X qq . O ∗ M p X q is a complex, its differential being the canonical homological vectorfield δ of T r s C M p X q . In more conventional terms, the field functionals alge-bra O ∗ M p X q envisaged here is that of non homogeneous differential forms on thespace C M p X q and δ is the corresponding de Rham differential. Below, we shallset F M p X q “ O M p X q for convenience.The graded algebra F ∗ p X q “ Fun p T r s X q is also involved in the considera-tions below via the spaces Map p T r s X, D m r p sq . It must be kept in mind here thatthe grading of O ∗ M p X q is distinct from that of F ∗ p X q . We adopt the conventionby which ΦΨ “ p´ q pq ` jk Ψ Φ for Φ P F p p X q b O j M p X q , Ψ P F q p X q b O k M p X q .The differential of the higher gauge field Ω P C M p X qq can be formally viewedas a special element δ Ω P Map p T r s X, D m r sq b O M p X q . If F P F M p X q is agiven field functional, its differential δF can be written as δF “ ż T r s X ̺ X ˆ δ Ω , δ F δ Ω ˙ (3.4.1)44here δ F { δ Ω P Map p T r d X ´ s X, D m r sq b F M p X q with d X “ dim X , because ofthe non singularity of the field pairing p¨ , ¨q . This relation defines the functionalderivative δ F { δ Ω . We can write relation (3.4.1) formally as δF “ ż T r s X ̺ X ˆ δ Ω , δδ Ω ˙ F . (3.4.2)This relation defines in turn δ { δ Ω as a formally D m r d X ´ s valued functionaldifferentiation operator. It is not difficult to verify that δ { δ Ω satisfies the Leibnizproperty as required.Let V M p X q “ Vect p C M p X qq denote the Lie algebra of the functional vectorfields of C M p X q . A vector field V P V M p X q can then be expressed as V “ ż T r s X ̺ X ˆ V , δδ Ω ˙ , (3.4.3)where V P Map p T r s X, D m r sq b F M p X q . The contraction operator associatedwith V , ι V , is characterized by the property that ι V ż T r s X ̺ X p δ Ω , Ξ q “ ż T r s X ̺ X p V , Ξ q . (3.4.4)for Ξ P Map p T r s X, D m r d X ´ sq .On account of (3.3.1), the differential δ Ω of the higher gauge field Ω introducedabove has the component expression δ Ω p α q “ δ ω ´ α δ Ω (3.4.5)Correspondingly, the functional differentiation operator δ { δ Ω defined throughrelation (3.4.1) can be written in components as δδ Ω p α q “ δδΩ ` p´ q d X α δδω (3.4.6)where δ { δΩ , δ { δω are g r d X ´ s , e r d X ´ s valued functional differentiation op-erators, respectively. Using these expressions and employing relations (3.2.15),(3.4.1), (3.4.3), it is possible to obtain component expressions of the functionalderivative δ F { δ Ω of a functional F P F M p X q as well as that of a vector field V P V M p X q .Above, we did not define the precise content of the functional algebra O ∗ M p X q .45oing so is a technical task beyond the scope of this paper and moreover isnot required by the formal developments of later sections. There are howevervariations of the derived functional framework expounded above based on mod-ifications of that content. First, one could replace the smooth function space F ∗ p X q “ Fun p T r s X q by its distributional extension F ∗ p X q “ Fun p T r s X q .This would lead a larger functional algebra O ∗ M p X q of functionals whose inte-gral expressions allow for distributions in addition to smooth functions. Second,one could add the field theoretic constraint of locality, obtaining the local ver-sions O ∗ M loc p X q and O ∗ M loc p X q of the previous two functional algebras. Let usrecall how these are defined. By pointwise local smooth functional ψ we mean anelement of F ∗ p X q b O ∗ M p X q that can be expressed at each point of T r s X as apolynomial in the higher gauge field Ω and its differential δ Ω and a finite numberof derivatives of Ω but none of δ Ω . Similarly, by pointwise local distributionalfunctional ψ we mean an element of F ∗ p X q b O ∗ M p X q that can be expressed as ψ “ ř i ψ i δ X i , where the ψ i are pointwise local smooth functionals and the δ X i are Dirac distributions supported on certain submanifolds X i of X . An element Ψ P O ∗ M loc p X q is one that can be expressed as an integral over T r s X of somepointwise local functional ψ . Likewise, an element Ψ P O ∗ M loc p X q is one thatcan be expressed as an integral over T r s X of some pointwise local distributionalfunctional ψ . Derived description of non trivial higher principal bundles
In this subsection, we shall show that the derived set–up can be used to describehigher gauge theory on a non trivial higher principal M –bundle P for any Liegroup crossed module M “ p E , G , τ, µ q . Albeit we shall not encounter this situa-tion, it is still important to examine this issue in order to ascertain whether thederived formulation is capable to efficiently handle this more general case.In what follows, we shall describe higher gauge fields and gauge transforma-tions by certain local data and their global definedness through matching data.For this reason, we pick an open covering t O i u of the base manifold X of the bun-dle P . For conciseness, we shall denote by O ij “ O i X O j , O ijk “ O i X O j X O k ,46tc. the non empty intersections of the covering’s opens.At a most basic level, a higher gauge field Ω is a collection t Ω i u of local maps Ω i P Map p T r s O i , D m r sq . Ω i can expanded in components Ω i p α q “ ω i ´ αΩ i , (3.5.1)in keeping with (3.2.5), where ω i P Map p T r s O i , g r sq , Ω i P Map p T r s O i , e r sq .In order the local gauge field data Ω i to describe a globally defined entity, thedata must match on any double intersection O ij in a way governed by a collection F of local matching data t F ij u , where F ij P Map p T r s O ij , D M q . In accordancewith (3.2.1), F ij has the component structure F ij p α q “ e αF ij f ij , (3.5.2)where f ij P Map p T r s O ij , G q , F ij P Map p T r s O ij , e r sq . The matching of gaugefield data Ω i then read as Ω i “ F ij Ω j “ Ad F ij p Ω j q ´ d F ij F ij ´ . (3.5.3)Note the formal analogy of these relations to the corresponding one of ordinarygauge theory. Using (3.5.1), (3.5.2), eq. (3.5.3) takes the component form ω i “ Ad f ij p ω j q ´ df ij f ij ´ ´ τ p F ij q , (3.5.4) Ω i “ µ p f ij , Ω j q ´ dF ij ´ r F ij , F ij s ´ µ p ω i , F ij q . (3.5.5)We recover in this way the well–known gluing relations of higher gauge fieldcomponents in higher gauge theory [65, 66].Consistency of the matching of local gauge field data Ω i on the triple inter-sections O ijk does not require simply that F ik “ F ij F jk , as is the case in ordinarygauge theory, but more generally that F ik “ H ijk F ij F jk , (3.5.6)where H ijk P Map p T r s O ijk , D M q such that H ijk Ω i “ Ad H ijk p Ω i q ´ d H ijk H ijk ´ “ Ω i . (3.5.7)47gain, these identities read more explicit in components. Let H ijk p α q “ e αH ijk h ijk , (3.5.8)where h ijk P Map p T r s O ijk , G q , H ijk P Map p T r s O ijk , e r sq . Using (3.5.2), (3.5.8),relations (3.5.6) take the form f ik “ h ijk f ij f jk , (3.5.9) F ik “ H ijk ` µ p h ijk , F ij ` µ p f ij , F jk qq . (3.5.10)Using (3.5.8) once more, conditions (3.5.7) become h ijk ,H ijk ω i “ Ad h ijk p ω i q ´ dh ijk h ijk ´ ´ τ p H ijk q “ ω i , (3.5.11) h ijk ,H ijk Ω i “ µ p h ijk , Ω i q ´ dH ijk ´ r H ijk , H ijk s ´ µ p ω i , H ijk q “ Ω i . (3.5.12)In refs. [65, 66], it is shown that the higher gauge field Ω being fake flat, dω i ` r ω i , ω i s ´ τ p Ω i q “ , (3.5.13)is a necessary and sufficient condition for the well–definedness of higher holonomies.It can be checked that this requirement is compatible with the matching relations(3.5.4), (3.5.5). In this case, the data h ijk , H ijk obeying (3.5.11), (3.5.12) are ofthe special gauge transformation form of eqs. (3.3.23), (3.3.24), h ijk “ τ p T ijk q , (3.5.14) H ijk “ ´ µ p ω i , T ijk q ´ dT ijk T ijk ´ , (3.5.15)where T ijk P Map p T r s O ijk , E q . Relations (3.5.9), (3.5.10) then get f ik “ τ p T ijk q f ij f jk , (3.5.16) F ik “ Ad T ijk p F ij ` µ p f ij , F jk qq ´ µ p ω i , T ijk q ´ dT ijk T ijk ´ . (3.5.17)The collection H “ t H ijk u of Lie valued data must itself satisfy a set of con-sistency conditions on the quadruple intersections O ijkl , H ikl H ijk “ H ijl F ij H jkl F ij ´ . (3.5.18)48n account of (3.5.2), (3.5.8), the component form of (3.5.18) read as h ikl h ijk “ h ijl f ij h jkl f ij ´ , (3.5.19) H ikl ` µ p h ikl , H ijk q “ H ijl ` µ p h ijl f ij , H jkl q (3.5.20) ` µ p h ijl , F ij q ´ µ p h ikl h ijk , F ij q . A straightforward calculation shows that when the data h ijk , H ijk have the specialform (3.5.14), (3.5.15) of refs. [65, 66], conditions (3.5.19), (3.5.20) reduce into τ p T ikl T ijk q “ τ p T ijl µ p f ij , T jkl qq , (3.5.21) µ p ω i , T ikl T ijk q “ µ p ω i , T ijl µ p f ij , T jkl qq . (3.5.22)Both of these are fulfilled if T ikl T ijk “ T ijl µ p f ij , T jkl q . (3.5.23)The collection of Lie valued data f “ t f ij u , T “ t T ijk u obeying (3.5.16),(3.5.23) defines a non Abelian cocycle [67]. It describes the background higherprincipal M –bundle P on X supporting the higher gauge fields.The collection of Lie valued data ω “ t ω i u , Ω “ t Ω i u , f “ t f ij u , F “ t F ij u , T “ t T ijk u obeying (3.5.4), (3.5.5), (3.5.16), (3.5.17), (3.5.23), constitutes a nonAbelian differential cocycle [68]. It encodes a higher gauge field as a globallydefined 2–connection of the bundle P .At a basic level, a higher gauge transformation U is a collection t U i u of Lievalued mappings U i P Map p T r s O i , D M q . In components, these maps read as U i p α q “ e αU i u i , (3.5.24)by (3.2.1), where u i P Map p T r s O i , G q , U i P Map p T r s O i , e r sq . U acts on a highergauge field Ω yielding a gauge field U Ω locally given by U Ω i “ U i Ω i “ Ad U i p Ω i q ´ d U i U i ´ . (3.5.25)Note that this gauge transformation action is related to the one defined in (3.3.9)by U Ω “ Ω U ´ . We use here the left form of the action to comply with the most49ommon convention. In components, (3.5.25) reads u,U ω i “ Ad u i p ω i q ´ du i u i ´ ´ τ p U i q , (3.5.26) u,U Ω i “ µ p u i , Ω i q ´ dU i ´ r U i , U i s ´ µ p u.U ω i , U i q , (3.5.27)the local component form of higher gauge transformation of refs. [65, 66].The local data of a higher gauge transformation U must satisfy certain match-ing relations implied by the gauge transform U Ω of a higher gauge field Ω beingitself a gauge field. In this regard, one must keep in mind that the matching datacollection F depends on the underlying gauge field Ω , as F is constrained by rela-tions (3.5.6) in which the data collection H depending on Ω via condition (3.5.7)appears. So, the matching data collections U F , U H of the transformed gauge field U Ω are generally different from the corresponding collections F , H of the givengauge field Ω . The fact that U Ω obeys the same kind of matching relations as Ω ,viz (3.5.3), entails that the local data U i satisfy relations of the form U i “ A ij U F ij U j F ij ´ (3.5.28)on the double intersections O ij , where A ij P Map p T r s O ij , D M q such that A ij U Ω i “ Ad A ij p U Ω i q ´ d A ij A ij ´ “ U Ω i . (3.5.29)Again, these identities read more explicit in components. Let A ij p α q “ e αA ij a ij , (3.5.30)where a ij P Map p T r s O ij , G q , A ij P Map p T r s O ij , e r sq . Using (3.5.2), (3.5.24),(3.5.30), conditions (3.5.29) lead to u i “ a ij u,U f ij u j f ij ´ , (3.5.31) U i “ µ p a ij u,U f ij , U j q ` A ij ´ µ p u i , F ij q ` µ p a ij , u,U F ij q . (3.5.32)Using (3.5.30) again, conditions (3.5.29) get a ij ,A ij u,U ω i “ Ad a ij p u,U ω i q ´ da ij a ij ´ ´ τ p A ij q “ u,U ω i , (3.5.33) a ij ,A ij u,U Ω i “ µ p a ij , u,U Ω i q ´ dA ij ´ r A ij , A ij s ´ µ p u,U ω i , A ij q “ u,U Ω i . (3.5.34)50n the formulation of refs. [65, 66] in which the gauge field Ω is fake flat, the data a ij , A ij obeying (3.5.33), (3.5.34) are of special gauge transformation type of eqs.(3.3.23), (3.3.24)) a ij “ τ p B ij q , (3.5.35) A ij “ ´ µ p u,U ω i , B ij q ´ dB ij B ij ´ . (3.5.36)The matching relations (3.5.31), (3.5.32) then read as u i “ τ p B ij q u,U f ij u j f ij ´ , (3.5.37) U i “ Ad B ij p µ p u,U f ij , U j q ` u,U F ij q ´ µ p u i , F ij q (3.5.38) ´ µ p u,U ω i , B ij q ´ dB ij B ij ´ . Relations (3.5.28) can be used to express the gauge transformed matchingdata U F ij in terms of the original data F ij and the gauge transformation data U i . Similarly, relations (3.5.31), (3.5.32) allow to write the gauge transformedmatching data components u,U f ij , u,U F ij through the matching data components f ij , F ij and the gauge transformation data components u i , U i . This is a simpleexercise that we leave to the reader.The collection A “ t A ijk u of Lie valued data must itself satisfy a set of con-sistency conditions on the triple intersections O ijk , A ik “ U i H ijk U i ´ A ij U F ij A jk U F ij ´ H ijk ´ . (3.5.39)By (3.5.2), (3.5.8), (3.5.24), (3.5.30), the component form of (3.5.39) read as a ik “ u i h ijk u i ´ a ij u,U f ij a jk u,U f ij ´ u,U h ijk ´ , (3.5.40) A ik “ U i ´ µ p u i h ijk u i ´ , U i q ` µ p u i h ijk u i ´ a ij , u,U F ij q (3.5.41) ´ µ p u i h ijk u i ´ a ij u,U f ij a jk u,U f ij ´ , u,U F ij q ` µ p u i h ijk u i ´ , A ij q` µ p u i h ijk u i ´ a ij u,U f ij , A jk q ` µ p u i , H ijk q´ µ p u i h ijk u i ´ a ij u,U f ij a jk u,U f ij ´ u,U h ijk ´ , u,U H ijk q . Ω is fake flat and that the data h ijk , H ijk and a ij , A ij obeying (3.5.11),(3.5.12) and (3.5.33), (3.5.34) are of special gauge transformation type (3.5.14),(3.5.15) and (3.5.35), (3.5.36), respectively. Then, a straightforward calculationshows that conditions (3.5.40), (3.5.41) reduce to τ p B ik q “ τ p µ p u i , T ijk q B ij µ p u,U f ij , B jk q u,U T ijk ´ q , (3.5.42) µ p u,U ω i , B ik q ` dB ik B ik ´ (3.5.43) “ µ p u,U ω i , µ p u i , T ijk q B ij µ p u,U f ij , B jk q u,U T ijk ´ q` d p µ p u i , T ijk q B ij µ p u,U f ij , B jk q u,U T ijk ´ qˆ p µ p u i , T ijk q B ij µ p u,U f ij , B jk q u,U T ijk ´ q ´ Both of these are fulfilled if B ik “ µ p u i , T ijk q B ij µ p u,U f ij , B jk q u,U T ijk ´ (3.5.44)This is almost the property such data are required to have in the formulation ofrefs. [65, 66]. We shall come back to this point momentarily.Relations (3.5.39) combined with the (3.5.22) can be used to express the gaugetransformed matching data U H ijk in terms of the original data H ijk and the gaugetransformation data U i . Similarly, relations (3.5.40), (3.5.41) together with the(3.5.31), (3.5.32) allow to write the gauge transformed matching data components u,U h ijk , u,U H ijk through the given matching data components h ijk , H ijk , and thegauge transformation data components u i , U i . This is again left to the reader.The collection of Lie valued data u “ t U i u , U “ t U i u , B “ t B ij u obeying(3.5.37), (3.5.38), (3.5.44) is an equivalence of the non Abelian differential cocyclepair ω “ t ω i u , Ω “ t Ω i u , f “ t f ij u , F “ t F ij u , T “ t T ijk u , u,U ω “ t u,U ω i u , u,U Ω “ t u,U Ω i u , u,U f “ t u,U f ij u , u,U F “ t u,U F ij u , u,U T “ t u,U T ijk u [68].As we have seen above, the collection of Lie valued data f “ t f ij u , T “ t T ijk u obeying (3.5.16), (3.5.23) define a non Abelian cocycle describing the backgroundhigher principal M –bundle structure for the gauge fields. It is natural to require52hat such data be gauge invariant u,U f ij “ f ij , (3.5.45) u,U T ijk “ T ijk . (3.5.46)This is in keeping with the analogous requirement imposed on the matching data f ij in ordinary principal bundle theory. It ensures that the M –bundle structureconstitutes the background for the gauge transformations too. With (3.5.45),(3.5.46) holding, conditions (3.5.44) take the form they have in the framework ofrefs. [65, 66].In conclusion, the derived set–up can be used to describe rather compactlya higher principal M –bundle and the gauge fields and gauge transformations itsupports. It further renders manifest the formal analogy of higher to ordinaryprincipal bundles, gauge fields and gauge transformations. However, to makecertain implicit constraints explicit, it is unavoidable to resort to a componentanalysis.We end this subsection with a discussion shedding light upon a basic differenceexisting between the nature of higher gauge field and gauge transformations in anon trivial higher principal bundle background and of their ordinary counterparts.This diversity is responsible for making the construction of 4–dimensional CStheory in such a background problematic. (More on this in subsect. 4.4.)For a full global description of the higher gauge field Ω , its local data Ω i arenot sufficient. The data Ω i match through certain data F ij obeying consistencyconditions involving further data H ijk depending in turn on the Ω i (cf. eqs.(3.5.3), (3.5.6), (3.5.7), (3.5.18)). Thus, it is not possible to organize all thesedata in a hierarchy and consider in particular the F ij as those encoding a fixedhigher principal bundle structure independent and preexisting any gauge fieldsuperimposed to it, as is the case in ordinary gauge theory. Only the components f ij and h ijk of F ij , H ijk can be assumed to be independent from the gauge fielddata Ω i and therefore be ascribed to a fixed bundle background. The components F ij and H ijk conversely cannot and might be considered as part of the data of a2–connection on the same footing as the components ω i , Ω i of the Ω i .53imilarly, for a full global description of a higher gauge transformation U ,its local data U i are not enough. The data U i match through the data F ij associated with some assigned gauge field data Ω i and further data A ij obeyingcompatibility conditions involving the Ω i , U i and also H ijk (cf. eqs. (3.5.28),(3.5.30), (3.5.39)). Again, it is not possible to organize all these data in a hierarchyand consider in particular the U i as those representing a gauge transformationstanding independently from the gauge fields it acts on, as in ordinary gaugetheory. Only the components a ij of A ij can be assumed to be independent fromthe gauge field and gauge transformation data Ω i and U i . The components A ij instead cannot and might considered part of the data defining an equivalence oftwo 2–connections on a par as the components u i , U i of the U i .To allow for the well–definedness of higher holonomies, the gauge field compo-nents ω i , Ω i are required to obey the vanishing fake curvature condition (3.5.13).The components h ijk , H ijk and a ij , A ij then have the structure shown in (3.5.14),(3.5.15) and (3.5.42), (3.5.43) with conditions (3.5.23) and (3.5.44) satisfied. Con-ditions (3.5.45), (3.5.46) are further imposed. However, again, the matching dataare or obey condition depending on the gauge field data.54 In this section, we introduce and study the –dimensional higher CS model, whichis the main topic of this paper, focusing in particular on its gauge symmetries.We illustrate further its Hamiltonian formulation.As we shall see, 4–dimensional CS theory exhibits its most interesting fea-tures when the underlying –fold has a boundary. This fact highlights its richholographic properties, which we shall describe in great detail.As already anticipated in sect. 2, we shall work in a graded geometric settingwhere forms are ordinary maps from the shifted tangent bundle T r s X of a rele-vant manifold X to some target graded manifold. Integration will be implementedthrough the Berezinian ̺ X of X .The basic algebraic structures the model are a Lie group crossed module M “ p E , G , τ, µ q and the associated Lie algebra crossed module m “ p e , g , τ, µ q (cf. subsect. 2.1). We assume that M is equipped with an invariant pairing x¨ , ¨y and that M is fine and that the conditions sufficient for the direct sumdecomposition (2.2.11) to hold are verified (cf. subsect. 2.2), though some of ourresults do not hinge on this restriction.All the fields occurring in the theory are valued either in the derived Lie group D M of M or in the derived Lie algebra D m of m (cf. subsect. 3.1). The derivedsuperfield formalism of subsect. 3.2 is employed throughout. The field pairing p¨ , ¨q induced by x¨ , ¨y (cf. eq. (3.2.15)) is used systematically in the construction.The higher gauge theoretic framework of subsect 3.3 conjoined with the derivedfunctional framework of subsect. 3.4 allow for a particularly geometrically intu-itive formulation highlighting the close relationship of the higher CS theory tothe ordinary one. In this subsection, we present the –dimensional higher CS model, which is themain topic of this paper. In component form, this model first appeared in [46]and was further studied in [47, 48] on –folds without boundary.55elow, we assume that M is an oriented, compact –fold, possibly with bound-ary. No further restrictions are imposed.The action of –dimensional CS theory is CS p Ω q “ k π ż T r s M ̺ M ` Ω , d Ω ` r Ω , Ω s ˘ , (4.1.1)where Ω P C M p M q is a higher gauge field (cf. subsect. 3.3) and k is a constant,the CS level. Formally, the expression of the higher CS action put forth here isidentical to that of familiar CS theory with the pairing p¨ , ¨q in place of the usualLie algebraic trace. However, since the latter has degree rather than , theLagrangian has degree instead than . It is precisely for this reason that thepresent higher CS theory works in dimensions.Expressed through the components ω , Ω of the higher gauge field Ω , the –dimensional CS action CS reads as CS p ω, Ω q “ k π ż T r s M ̺ M @ dω ` r ω, ω s ´ τ p Ω q , Ω D ´ k π ż T r sB M ̺ B M x ω, Ω y . (4.1.2)The boundary contribution, absent in (4.1.1), is yielded by an integration byparts. So, higher CS theory can be described as as a generalized BF theory withboundary term and cosmological term determined by the Lie differential τ ofthe target map τ . This way of regarding it is however somewhat reductive. –dimensional CS theory is characterized by a higher gauge symmetry which placesit safely in the realm of higher gauge theory. We shall analyze this matter ingreater detail in subsect. 4.2 below.The variation of the 4–dimensional CS action CS under a variation of thehigher gauge field Ω is given by δCS “ k π ż T r s M ̺ M p δ Ω , Φ q ` k π ż T r sB M ̺ B M p δ Ω , Ω q , (4.1.3)where Φ is the higher gauge curvature defined in eq. (3.3.2). (Here and below,the variational operator δ is defined as in subsect. 3.4.) If a suitable boundary56ondition is imposed on Ω which makes the boundary term in (4.1.3) vanish,rendering CS differentiable in the sense of refs. [72, 73], the field equations read Φ “ . (4.1.4)These can be written in terms of the components ω , Ω using relations (3.3.4),(3.3.5). In this way, as in ordinary CS theory, the higher CS field equationsenforce the flatness of Ω .If no boundary condition is imposed, the 4–dimensional CS action CS belongsto the distributional extension F M loc p M q of the local smooth functional space F M loc p M q , as the variation δCS of CS given in (4.1.3) contains a boundary term Γ “ ´ k π ż T r sB M ̺ B M p δ Ω , Ω q , (4.1.5)that cannot be turned into a legitimate bulk one using Stokes’ theorem (cf. sub-sect. 3.4). Imposing an appropriate boundary condition eliminates the offendingboundary contribution and makes CS belong to F M loc p M q with a well definedvariational problem leading to the field equations (4.1.4).The choice of the appropriate boundary condition to be prescribed to thehigher gauge field Ω depends on the type of physics the higher CS theory ismeant to describe. To analyze this matter in full generality within the scope oflocal field theory, we proceed as follows.First, to have available the broadest possible range of boundary conditions, weallow for the addition to the action CS of a local boundary term ∆ CS independentfrom any boundary background field. The resulting modified CS action is then CS “ CS ` ∆ CS . (4.1.6)The inclusion of ∆ CS provides the variation δCS of CS with a boundary contri-bution that is added to the problematic boundary contribution Γ yielded by δCS .Note that the two boundary contributions cannot cancel out since the former is δ -exact in O ∗ M loc p M q while the latter is not.Second, we impose a local boundary condition on the higher gauge field Ω .The most general such condition is specified by a local functional submanifold L
57f the boundary higher gauge field space C M pB M q , that is one defined by meansof local constraints in C M pB M q , and takes the form Ω | T r sB M P L . (4.1.7)The boundary condition must by such to completely cancel the boundary contri-bution to the variation δCS of CS .The above two step procedure ensures that CS does indeed belong to lo-cal smooth field functional space F M loc p M q , making the associated variationalproblem well defined, as we now show. The assumed qualifications of the bound-ary term ∆ CS guarantee the existence of a boundary local smooth functional ∆ CS B P F M loc pB M q independent from any boundary background field such that ∆ CS p Ω q “ ∆ CS B p Ω B q| Ω B “ Ω | T r sB M . (Here and below we denote boundary fields andfield functionals thereof by a subscript B for clarity.) The boundary contributionto the variation δCS of CS in (4.1.3), Γ , is similarly related to the boundary localfunctional –form Γ B P O M loc pB M q , Γ B “ ´ k π ż T r sB M ̺ B M p δ B Ω B , Ω B q , (4.1.8)as Γ p Ω q “ Γ B p Ω B q| Ω B “ Ω | T r sB M . Hence, the total boundary contribution to δCS isgoing to vanish if ∆ CS and L are such that Γ B ´ δ B ∆ CS B “ in L . (4.1.9)This can be achieved by suitably adjusting either ∆ CS B or L or both.A functional submanifold L of C M pB M q will be called admissible if it can beemployed to define a viable boundary condition. The boundary term ∆ CS aswell as the associated modified action CS are fixed once a choice of one suchsubmanifold L is made. We shall denote them as ∆ CS L and CS L when it isnecessary to indicate such dependence. The problem of classifying the possiblechoices of boundary conditions reduces in this way to that of classifying theadmissible submanifolds.In the spirit of the covariant canonical approach (see e. g. ref. [71] for astandard review), the boundary functional 1–form Γ B given in eq. (4.1.8) provides58he expression of the appropriate symplectic potential of the higher CS model fieldspace C M pB M q . The associated symplectic form Υ B P O M loc pB M q thus is Υ B “ δ B Γ B “ k π ż T r sB M ̺ B M p δ B Ω B , δ B Ω B q . (4.1.10)The admissible submanifolds L of C M pB M q which describe the higher CS theoryboundary conditions then constitute a distinguish subset of isotropic submanifoldsto Υ , that is the submanifolds L such that Υ B “ in L (4.1.11)In terms of the gauge field components ω , Ω the symplectic form reads Υ B “ k π ż T r sB M ̺ B M x δ B ω B , δ B Ω B y . (4.1.12)The local boundary condition classification problem is therefore similar enoughto that of the description of isotropic submanifolds in ordinary Hamiltonian me-chanics. We shall not tackle this issue in full generality and for the time beingcontent ourselves with a basic class of such conditions.With any isotropic submodule M of the Lie group crossed module M , (cf.subsect. 2.3), there is associated the submanifold C M pB M q of C M pB M q . By virtueof the isotropy M , C M pB M q is a local submanifold of C M pB M q such that Γ B “ on C M pB M q , hence an admissible submanifold of C M pB M q with ∆ CS C M pB M q “ and CS C M pB M q “ CS . C M pB M q thus defines a special choice of boundary condition for4–dimensional CS theory. In the following we shall refer mostly to this kind ofboundary condition, which we shall call isotropic linear of type M for reference.The most permissive isotropic linear boundary condition is that for which M isLagrangian. This will be called Lagrangian linear of type M . Gauge invariance of the 4–d Chern–Simons model
In this subsection, we analyze in some detail the gauge symmetry of the 4–dimensional CS model introduced in subsect. 4.1. In spite of the formal resem-blance of higher to ordinary CS theory when the derived formulation is used, the59nvariance properties of the 4–dimensional CS model differ in several importantaspects from those of the 3–dimensional one, especially in relation to the effectof a boundary in the base manifold.For a higher gauge transformation U P G M p M q (cf. subsect. 3.3), the 4–dimensional CS action (4.1.1) varies as CS p Ω U q “ CS p Ω q ` A p Ω; U q (4.2.1)for Ω P C M p M q , where A p Ω; U q is given by A p Ω; U q “ ´ k π ż T r s M ̺ M ` dUU ´ , r dUU ´ , dUU ´ s ˘ ` k π ż T r sB M ̺ B M ` Ω , dUU ´ ˘ (4.2.2)The gauge variation term A p Ω; U q is formally identical to that of ordinary CStheory: a bulk WZNW–like term plus a boundary term.The real nature of the gauge variation term (4.2.2) emerges when it is ex-pressed through the components ω , Ω of the higher gauge field Ω and u , U of thehigher gauge transformation U . Using (3.2.13), it can be verified that the bulkterm is exact and hence reduces to a boundary term, yielding the expression A p ω, Ω ; u, U q“ k π ż T r sB M ̺ B M ” @ τ p U q , dU ` r U, U s D ´ @ duu ´ ` τ p U q , dU ` r U, U s D ` @ ω, dU ` r U, U s ´ µ p duu ´ ` τ p U q , U q D ´ @ duu ´ ` τ p U q , Ω D ı . (4.2.3)In particular, A “ identically if B M “ { . In this sense, in the higher theory, thegauge non invariance of the CS action is ’holographic’ in nature. This propertydistinguishes 4–dimensional CS theory from its 3–dimensional counterpart.When a boundary term ∆ CS is added to the basic higher CS action CS , amodified action CS is obtained (cf. eq. (4.1.6)). ∆ CS is generally non invariantunder gauge transformation. One hence has ∆ CS p Ω U q “ ∆ CS p Ω q ` ∆ A p Ω , U q (4.2.4)60here ∆ A is a boundary gauge variation. The modified gauge variation A , thevariation of the modified action CS , is therefore given by A “ A ` ∆ A . (4.2.5)Depending on the form of ∆ CS , A may differ considerably from A .The gauge invariance properties of 4–dimensional CS theory depend to a largeextent on the kind of boundary condition one imposes on the higher gauge fields Ω to render the CS variational problem well–defined.As we have explained in subsect. 4.1, in 4–dimensional CS theory a choice ofboundary condition is specified by an admissible submanifold L of C M pB M q . Theboundary condition requires that Ω satisfies Ω | T r sB M P L . With the boundarycondition, further, there is associated a boundary term ∆ CS L that is to be addedto the basic CS action CS yielding the appropriate variationally well–behavedmodified CS action CS L (cf. eq. (4.1.6)).When a certain boundary condition is prescribed for the higher gauge fields Ω a corresponding boundary condition must be imposed to the higher gaugetransformations U : they must preserve L . The boundary condition can thereforebe expressed as the requirement that U | T r sB M P I L (4.2.6)where I L is the invariance subgroup of L in G M pB M q , the subgroup formed bythe boundary gauge transformations U B P G M pB M q such that L U B “ L .Since the modified action CS L results from adding the boundary term ∆ CS L to the basic CS action CS according to (4.1.6), the boundary gauge variation ∆ A L is added to the basic gauge variation term A to yield the modified gaugevariation term A L given by (4.2.5). As we shall see momentarily, the expression A L may take a simpler form when the boundary conditions obeyed by both thegauge fields and transformations are taken into account.For the isotropic linear boundary condition of type M (cf. subsect. 4.1),where M is an isotropic crossed submodule of M , more detailed information canbe provided. In this case, the condition is specified by the admissible submanifold61 M pB M q of C M pB M q . The precise content of the invariance subgroup I C M pB M q of C M pB M q is not straightforward to describe in simple terms, but it is not difficult toidentify a broad distinguished subgroup I N C M pB M q of I C M pB M q . I N C M pB M q consistsof the boundary gauge transformations U B P G N M pB M q satisfying d B u B u B´ ` τ p U B q “ mod Map p T r sB M, g r sq , (4.2.7) d B U B ` r U B , U B s “ mod Map p T r sB M, e r sq , (4.2.8)where N M is the normalizer crossed module of M (cf. subsect. 2.3). I N C M pB M q being contained in I C M pB M q follows from (3.3.12), (3.3.13) and the defining prop-erties of G N M pB M q . It can be further shown that I N C M pB M q contains G M pB M q asa subgroup and that I N C M pB M q “ I C M pB M q if the groups G , E are connected.Since the boundary term ∆ CS C M pB M q “ identically for the isotropic lin-ear boundary condition, the modified action CS C M pB M q and the associated gaugevariation term A C M pB M q are equal to their basic counterparts CS and A , respec-tively. If U is a higher gauge transformation obeying the boundary condition U | T r sB M P I N C M pB M q , the gauge variation (4.2.3) takes the CS form A p ω, Ω ; u, U q “ k π ż T r sB M ̺ B M @ τ p U q , dU ` r U, U s D , (4.2.9)by the isotropy of M . A p ω, Ω ; u, U q is so independent from the higher gauge fieldcomponents ω , Ω . Note that A p ω, Ω ; u, U q “ when U | T r sB M P G M pB M q . Level quantization
To quantize 4–dimensional CS theory, one should allow for the widest gauge sym-metry leaving the Boltzmann exponential exp p i CS q invariant possibly restrictingthe value of the CS level k . In ordinary CS theory, this permits the incorporationof large gauge transformation in the symmetry, when the CS level is suitablyquantized. One wonders if something similar happens in our higher setting.By (4.2.3), when the boundary B M of M is empty, the higher CS theory enjoysfull higher gauge symmetry and there is no problem. When B M is non empty,one should impose on the relevant higher gauge fields Ω and transformations U A an integer multiple of π . Given the varied form such conditions can take,here we can only examine basic examples.If an isotropic linear boundary condition of type M is implemented, where M is an isotropic crossed submodule of M , the higher gauge fields Ω P C M p M q and gauge transformations U P G M p M q must satisfy Ω | T r sB M P C M pB M q and U | T r sB M P I C M pB M q (cf. subsects. 4.1, 4.2). We identified a subgroup I N C M pB M q of I C M pB M q essentially exhausting it formed by the boundary gauge transformations U B P G N M pB M q obeying (4.2.7), (4.2.8). For the gauge transformations U suchthat U | T r sB M P I N C M pB M q , the gauge variations A has the simple CS form (4.2.9).This however neither vanishes nor enjoys any quantization property a priori. Weare thus forced to consider a more restrictive boundary condition for the U . Anoption is replacing the invariance subgroup I N C M pB M q by its orthogonal subgroup I ON C M pB M q , where ON M is the orthogonal normalizer crossed module of M (cf.subsect. 2.3). I ON C M pB M q is constituted by the boundary gauge transformations U B P G ON M pB M q satisfying (4.2.7), (4.2.8). For the gauge transformations U suchthat U | T r sB M P I ON C M pB M q , the gauge variation A takes the form A p ω, Ω ; u, U q “ ´ k π ż T r sB M ̺ B M x τ p U q , r U, U sy . (4.3.1)This can be roughly viewed as a kind of winding number of a Lie group valuedmap, since by (4.2.7) τ p U q can be identified with duu ´ on the boundary T r sB M up to a term belonging to Map p T r sB M, g r sq .As explained in subsect. 2.2, under weak assumptions the Lie algebra crossedmodule with invariant pairing m is isomorphic to the direct sum of the Lie algebracrossed module INN ran τ with a suitable invariant pairing and AD ∗ p g { ran τ q withcanonical invariant pairing. Having this in mind, we are going to find out whichform the expression (4.3.1) of the gauge variation A takes in the cases where theLie group crossed module M is either INN G with a suitable invariant pairingor AD ∗ G with the canonical duality pairing, where G is a Lie group, and M is an isotropic crossed submodule of these for which the orthogonal normalizercrossed module ON M exists, such as the submodules INN H K or AD H ∗ K studied63n subsect. 2.3 with H , K suitable connected Lie subgroups of G .We consider first the case where M “ INN G and M “ INN H K . We have then E “ G and τ p X q “ X for X P g . The less restrictive boundary conditions on thegauge transformation U are those for which the gauge variation A is an integertimes π . The weakest conditions one can envisage are as follows. The boundarygauge transformations U B P I ON C INN
H K pB M q such that there is a boundary gaugetransformation V B P G INN H K pB M q obeying d B v B v B´ ` V B “ ´ Ad u B´ p d B u B u B´ ` U B q (4.3.2)form a distinguished subgroup I ON ∗C INN
H K pB M q of I ON C INN
H K pB M q . For a transfor-mation U B P I ON C INN
H K pB M q , one has ´ π ż T r sB M ̺ B M x U B , r U B , U B sy“ ż T r sB M ̺ B M π @ d B ˜ u B ˜ u B´ , r d B ˜ u B ˜ u B´ , d B ˜ u B ˜ u B´ s D (4.3.3)where ˜ u B “ u B v B . If the closed form π x κ, r κ, κ sy of G , where κ is the Maurer–Cartan form, is a representative of an integer cohomology class, the above ex-pression takes integer values time π . Let us assume this is indeed the case. Byvirtue of (4.3.1), if U is gauge transformation obeying the boundary condition U | T r sB M P I ON ∗C INN
H K pB M q , A takes the form A “ k π ż T r sB M ̺ B M @ d ˜ u ˜ u ´ , “ d ˜ u ˜ u ´ , d ˜ u ˜ u ´ ‰D . (4.3.4)where ˜ u “ uv , v being an extension to a neighborhood of B M of the component v B a boundary gauge transformation V B P G INN H K pB M q satisfying condition (4.3.2)with U B “ U | T r sB M . If the level k is an integer, A is integer values time π asdesired, much as in ordinary CS theory. Level quantization so occurs.We consider next the case where M “ AD ∗ G . We have then E “ g ∗ , viewedas an Abelian group and τ p X q “ for X P g ∗ . By (4.3.1), A then vanishes, A “ . (4.3.5)In this case, level quantization of course does not occur.64he isotropic linear boundary conditions considered above serve the purposeof rendering the CS variational problem well–defined and gauge covariant. Byvirtue of their origin, they suit the perturbative semiclassical limit k Ñ 8 inwhich k can be considered as a continuous parameter regardless its integrality.Below, we envisage other types of boundary conditions are appropriate for theopposite non perturbative quantum finite k regime.The boundary condition we shall study is best expressed in components. Werequire that the higher gauge fields Ω P C M p M q to be fake flat on the boundary φ “ dω ` r ω, ω s ´ τ p Ω q “ on T r sB M (4.3.6)(cf. subsect. (3.3)). We require further that the allowed gauge transformations U P G M p M q are special on the boundary, that is of the form u “ τ p B q , (4.3.7) U “ ´ dBB ´ ´ µ p ω, B q on T r sB M, (4.3.8)where B P Map p T r sB M, E q (cf. eqs. (3.3.23), (3.3.24)). Note that the Ω arenot fake flat in general, as the fake flatness condition (4.3.6) is required to holdonly on the boundary B M of M . Likewise, the U are not special in general, asthey are required to be of the form (3.3.23), (3.3.24) only on B M . Finally noticethat by (3.3.25), (3.3.26) we have ω u,U “ ω , Ω u,U “ Ω on T r sB M . These gaugetransformations so leave the boundary values of the gauge field components fixed.The boundary fake flatness condition (4.3.6) can be enforced by adding to CSaction CS a boundary term of the form ∆ CS p Ω , Λ q “ k π ż T r sB M ̺ B M x φ, Λ y , (4.3.9)where Λ P Map p T r sB M, e r sq is an auxiliary boundary field. By (2.2.13), (3.3.14)and (4.3.7), ∆ CS is invariant under any gauge transformation U with the bound-ary form (4.3.7), (4.3.8) provided Λ transforms as Λ U “ Ad B ´ p Λ q . (4.3.10)When the relevant higher gauge field ω , Ω and the gauge transformation u ,65 obey the boundary conditions (4.3.6) and (4.3.7),(4.3.8), the gauge variationterm (4.2.3) takes the form A “ k π ż T r sB M ̺ B M @ τ p dBB ´ q , “ dBB ´ , dBB ´ ‰D . (4.3.11) . This can again viewed as a kind of the winding number of a Lie group valuedmap. A is a homotopy invariant, as one might expect. Indeed, under a variation δB of B , the variation of the integrand in the right hand side of (4.3.11) is d x τ p δBB ´ q , r dBB ´ , dBB ´ sy entailing that δ A “ .For reasons we explained earlier, we are going to obtain the form the expression(4.3.11) of the gauge variation A takes when the Lie group crossed module M iseither INN G with a suitable invariant pairing or AD ∗ G with the canonical dualitypairing, where G is a Lie group.We consider first the case where M “ INN G for which E “ G and τ p X q “ XX P g . By (4.3.11), A takes the form A “ k π ż T r sB M ̺ B M @ dBB ´ , “ dBB ´ , dBB ´ ‰D . (4.3.12)Again, if the closed form π x κ, r κ, κ sy of G with κ the Maurer–Cartan form isa representative of an integer cohomology class, as we assume presently, A takesinteger values time π provided the level k is integer. Level quantization oncemore occurs.We consider next the case where M “ AD ∗ G . We have then E “ g ∗ , viewedas an Abelian group and τ p X q “ for X P g ∗ . By (4.3.11), A then vanishes A “ . (4.3.13)In this case, again level quantization does not occur.The calculations carried out above show that level quantization, when itoccurs, is a boundary effect. This remarkable property markedly distinguisheshigher CS theory from its ordinary counterpart. The full expression of A contains in the integrand a term ´ x φ, µ p ω, B qy which vanishesby (4.3.6) and a further term xr ω, ω s , µ p ω, B qy ´ @ ω, µ ` r ω, ω s , B ´ ˘D which vanish by (2.2.14)which in turn holds by the assumed fineness of the crossed module M . .4 Global issues in 4–d Chern–Simons theory
In this subsection, we examine the issue whether it is possible to give a reasonabledefinition of 4–dimensional CS theory on a non trivial higher principal bundle.We refer the reader to subsect. 3.5 for a preliminary discussion of this matter.The components ω , Ω of the higher gauge field Ω are only locally defined whenthe underlying higher principal bundle is non trivial. Consequently also the 4–dimensional CS Lagrangian density is only locally defined and formula (4.1.1)giving the CS action is unusable. This is only the first of a number of subtlepoints which must be settled before attempting a definition of 4–dimensional CStheory on a non trivial background. We leave a more thorough analysis of theseissues for future work and here we shall limit ourselves to tackle the problem froma different more elementary perspective.We look for an expression of the 4–dimensional CS action CS on a trivialhigher gauge principal bundle that can be sensibly extended also on a non trivialone. To this end, we try to adapt a strategy that has shown itself to be successfulin the familiar 3–dimensional case. We write the gauge field Ω as the sum of abackground gauge field Ω and a deviation W , viz Ω “ Ω ` W . (4.4.1)We assume furthermore that Ω is flat Φ “ d Ω ` “ Ω , Ω ‰ “ . (4.4.2)This is not done only for mathematical convenience, but also because it allows fora more precise characterization of the CS action CS p Ω q , as shown momentarily.We also assume that Ω obeys an isotropic linear boundary condition (cf. subsect.4.1) and require that W also does. In this way, Ω will satisfy it too.The Lagrangian of 4–dimensional CS theory can now expressed as ` Ω , d Ω ` r Ω , Ω s ˘ “ ` Ω , d Ω ` r Ω , Ω s ˘ ` ` W , DW ` r W , W s ˘ ´ d ` Ω , W ˘ , (4.4.3)67here we have conventionally set D “ d ` ad Ω . (4.4.4)Upon integration on T r s M , the last term in the right hand side of (4.4.3) givesa vanishing contribution because of the isotropic boundary conditions obeyed byboth Ω and W . From (4.1.1), we find so that CS p Ω q “ CS p Ω q ` k π ż T r s M ̺ M ` W , DW ` r W , W s ˘ . (4.4.5)We now concentrate on the background CS action CS p Ω q .Denote by δ a variation with respect to the background gauge field Ω respect-ing both the flatness requirement (4.4.2) and the given isotropic linear boundarycondition. Then, since the flatness condition (4.4.2) coincides with the CS fieldequation (cf. subsect. 4.1), we have δCS p Ω q “ . (4.4.6) CS p Ω q is therefore constant on each connected component of the space of flatbackground gauge fields Ω . If M has no boundary, CS p Ω q is also fully gaugeinvariant. In such a case, CS p Ω q represent a locally constant function on themoduli space of flat gauge fields Ω . If conversely M has a boundary, then CS p Ω q ,or more precisely e i CS p Ω q , is a section of a flat unitary line bundle on the modulispace whose matching data are defined by the exponentiated gauge variation(4.2.9).When M has no boundary, CS p Ω q can be evaluated by a method borrowedonce more from the 3–dimensional case. Suppose that the 4–fold M is the bound-ary of a 5–fold Ă M . We extend the background gauge field Ω to a gauge field r Ω on Ă M such that r Ω | T r s M “ Ω . Since r d `r Ω , r d r Ω ` “r Ω , r Ω ‰˘ “ ` r Φ , r Φ ˘ , (4.4.7)where r Φ is the curvature of r Ω defined according to (3.3.2), we have CS p Ω q “ k I Ă M p r Ω q , (4.4.8)68here I Ă M p r Ω q is given by I Ă M p r Ω q “ π ż T r s Ă M ̺ Ă M `r Φ , r Φ ˘ . (4.4.9)The value of I Ă M p r Ω q does not depend on the choice of r Ω as r δI Ă M p r Ω q “ π ż T r sB Ă M ̺ Ă M `r δ r Ω , r Φ ˘ “ π ż T r s M ̺ M ` δ Ω , Φ ˘ “ (4.4.10)by (4.4.2). The value of I Ă M p r Ω q does not depend also on the choice of Ă M because ` r Φ , r Φ ˘ is exact by (4.4.7). It is interesting to notice here that quadratic curvaturepolynomial π ` r Φ , r Φ ˘ is formally analogous the familiar Chern 4–form. It hashowever degree 5. Expressed through the curvature components r φ , r Φ , I Ă M p r ω, r Ω q reads in fact as I Ă M p r ω, r Ω q “ π ż T r s Ă M ̺ Ă M @ r φ, r Φ D . (4.4.11)Suppose now that the background principal bundle is non trivial. We pickagain a background gauge field specified, as explained in subsect. (3.5), by acollection of local data Ω i . The Ω i relate via matching data F ij as in (3.5.3). The F ij adapt in turn via the consistency data H ijk as in (3.5.6) with the H ijk obeyingconditions (3.5.7) and (3.5.18). Because of (3.5.3), the integrand of the secondterm in the right hand side of (4.4.5) will be globally defined if the local data W i of the deviation field match according to W i “ Ad F ij p W j q . (4.4.12)Integration on T r s M is then possible. In this respect, the context is formallysimilar to that of the 3–dimensional theory. If (4.4.12) holds, the local dataof the gauge field are Ω i “ Ω i ` W i . The corresponding matching data F ij and consistency data H ijk so equal their background counterparts F ij and H ijk .However, the data H ijk will obey (3.5.7) only if Ad H ijk p W i q “ W i . (4.4.13)This is a constraint on the deviation data W i whose implementation in the clas-69ical as well quantum theory is problematic. Alternatively, we can disregard(4.4.13) giving up (3.5.7), but then the gauge field data Ω i no longer can beconsidered as specifying a 2–connection.Leaving aside these issues, when the background principal bundle is non trivialother problems arise with regard to the proper definition of the background CSaction CS p Ω q using the procedure outlined above valid for a base 4–fold M withno boundary. To begin with, we have to extend the background bundle and 2–connection structure on M , given by the data Ω i , F ij , H ijk , to one on the chosen5–fold Ă M , given by the data r Ω i , r F ij , r H ijk . Assuming that this is indeed possible,the extended higher curvature data r Φ i match as r Φ i “ Ad r F ij p r Φ j q . (4.4.14)By virtue of this, the integrand in the right hand side (4.4.9) is globally definedand its integration over T r s Ă M can be carried out. The problem arising hereis that the quadratic curvature polynomial π ` r Φ , r Φ ˘ has no a priori integralityproperties and so the value of I Ă M p r Ω q depends in principle on the choice of theextending 5–fold Ă M by an amount that does not vanish modulo π Z . The quan-tization of the level k as integer is of no avail here in sharp contrast with whathappens in the corresponding 3–dimensional setting.We conclude this subsection with one more remark pointing to a further prob-lem. To allow for the well–definedness of higher holonomies, in turn necessary forthe incorporation of Wilson surfaces in 4–dimensional CS theory, the gauge fieldcomponents ω i , Ω i are required to satisfy the vanishing fake curvature condition(3.5.13). The fake flatness condition is however one the field equations of the CSmodel. So, it should emerge from the classical variational problem and shouldnot be assumed from the onset. Canonical formulation
In this subsection, we shall illustrate the canonical analysis of the 4–dimensionalCS model introduced and studied in the previous subsections. The close relation-ship of the canonical formulations of 4– and 3–dimensional CS theory is again70specially evident in the derived framework. We shall describe the phase spaceof the model in the derived set–up and obtain compact derived expressions ofits Poisson bracket. We shall further identify the model’s phase space constraintmanifold as the vanishing higher curvature locus and describe the reduced phasespace and its Poisson bracket. The results of the ordinary theory generalize how-ever to the higher one only up to a certain extent, which we shall make precisein due course.To carry out the canonical analysis, we assume that M “ R ˆ S , where S is an oriented compact –fold possibly with boundary, viewing the Cartesianfactors R and S respectively as a time axis and a space manifold. M of course isnot compact, as we assumed earlier, making it necessary imposing integrabilityconditions on fields to have a finite action integral. Alternatively, when S iscompact, one may compactify R into the circle S requiring fields to be periodic.In the canonical formulation, it is natural to rely on a hybrid geometricalframework whereby the function algebra Fun p T r s M q of M is viewed as the alge-bra Map p T r s R , Fun p T r s S qq of maps from the shifted tangent bundle T r s R of R into the internal function algebra Fun p T r s S q of S . Proceeding in this way,a generic derived superfield field Ψ P Map p T r s M, D m r p sqq decomposes as Ψ “ dt Ψ t ` Ψ S , (4.5.1)where Ψ t P Map p R , Map p T r s S, D m r p ´ sqq , Ψ S P Map p R , Map p T r s S, D m r p sqq and t and dt denote conventionally the base and fiber coordinates of T r s R .Similarly, the differential d of T r s M decomposes asd “ dtd t ` d S (4.5.2)in terms of the differential d S of T r s S , where d t “ d { dt and both d and d S aredefined according to (3.2.10).A higher gauge field Ω P C M p M q can so be expressed in terms of components Ω t P Map p R , Map p T r s S, D m r sqq , Ω S P Map p R , Map p T r s S, D m r sqq in accor-dance with (4.5.1). Its curvature Φ can be similarly decomposed in components Φ t P Map p R , Map p T r s S, D m r sqq , Φ S P Map p R , Map p T r s S, D m r sqq . Geome-71rically, Ω S P Map p R , C M p S qq is to be regarded as a time dependent higher gaugefield on S . Φ S is then identified with the curvature of Ω S , since Φ S is given by(3.3.2) in terms of d S and Ω S .Expressed in terms of the higher gauge field components Ω t , Ω S , the 4–dimensional CS action (4.1.1) takes the form CS p Ω t , Ω S q “ k π ż R dt ż T r s S rp d t Ω S , Ω S q ` p Ω t , Φ S qs` k π ż R dt ż T r sB S p Ω t , Ω S q . (4.5.3)It is natural to interpret the component Ω t as a Lagrange multiplier implementingthe vanishing curvature constraint Φ S “ . (4.5.4)upon variation of the action CS . However, CS is not differentiable with respectto Ω t in the sense established in refs. [72, 73] because of the presence of theboundary term.Naively, it would seem that the problem could be solved by requiring that Ω t | T r sB S “ . (4.5.5)A similar boundary condition was imposed in ref. [74] to cope with the analo-gous issue arising in the canonical formulation of 3–dimensional CS theory. Thequestion of the stability of a boundary condition of this sort under gauge trans-formation is however quite different in the ordinary and higher cases. In theordinary theory, the condition is preserved by gauge transformations which aretime independent on the boundary, which constitute a tractable subgroup of thefull gauge group. In the higher theory, the condition (4.5.5) is preserved by gaugetransformations obeying a complicated boundary condition involving also Ω S ,as emerges by inspection of the component expressions the transformations ofeqs. (3.3.12), (3.3.13), leaving doubts about the eventual viability of the wholeapproach.It seems more natural to resort to a boundary condition of the isotropic linear72ind introduced in subsect. 4.1. We thus demand that the higher gauge field Ω P C M p M q satisfies the requirement that Ω | T r sB M P C M pB M q , where M is anisotropic submodule of M . When Ω is expressed in terms of the components Ω t , Ω S , the boundary condition constrains Ω t | T r sB S , Ω S | T r sB S to be D m r s , D m r s valued respectively, making the problematic boundary term in the right hand sideof (4.5.3) vanish. For Ω S , the condition can be cast transparently as Ω S | T r sB S P Map p R , C M pB S qq . (4.5.6)Next, we examine the issue of higher gauge symmetry. In the hybrid geometri-cal framework we are employing here, a higher gauge transformation U P G M p M q factorizes as U “ U t U S , where U t P Map p T r s R , Map p T r s S, D M qq is a gaugetransformation of the form U p α q “ e αdtU t with U t P Map p R , Map p T r s S, e r sqq and U S P Map p R , Map p T r s S, D M qq .The isotropic linear boundary condition which we have imposed on the highergauge field Ω , viz Ω | T r sB M P C M pB M q , is stable under the gauge transformations U P G M p M q which satisfy the boundary condition U | T r sB M P I N C M pB M q , where I N C M pB M q is the invariance subgroup of C M pB M q introduced and studied in sub-sect. 4.2. When U is expressed in terms of its components U t , U S as indicatedabove, this condition can be written suggestively as U S | T r sB S P Map p R , I N C M pB S q q , (4.5.7)There is however a further restriction involving both U t and U S following from(4.2.7), (4.2.8). It ensures that the boundary condition obeyed by Ω t is stableunder gauge transformation. For fixed U S , this restriction may fail to be satisfiedby any U t unless U S is further delimited. For this reason, in (4.5.7) it may benecessary to replace the invariance subgroup I N C M pB S q with a proper subgroupof it. However, in the canonical set–up illustrated below both Ω t and U t do notappear and we may therefore disregard this extra limitation.In canonical theory, we replace the higher gauge field component Ω S witha time independent gauge field Ω viewed as a point of an ambient functionalphase space C M p S q , where we suppress the subscript S for notational simplicity.73imilarly, we replace the higher gauge transformation component U S with a timeindependent gauge transformation U of an ambient phase space gauge group G M p S q acting on Ω according the familiar prescription (3.3.9). No boundaryconditions on either Ω or U are imposed at this stage.The physical phase space C M ph p S q is the functional subspace of the ambientphase space C M p S q defined by the flatness constraint corresponding to (4.5.4) Φ « , (4.5.8)where Φ is the curvature of Ω defined according (3.3.2). C M p S q is invariant un-der the action of the ambient gauge transformation group G M p S q . The reducedphysical phase space r C M ph p S q is the quotient of C M ph p S q by G M p S q , r C M ph p S q “ C M ph p S q{ G M p S q . (4.5.9)All this is rather formal, since the above quotient turns out to be singular. Ittherefore calls for a more precise formulation of the symplectic structure of C M p S q ,which we provide below.The short action term in the right hand side of (4.5.3) indicates the appropri-ate expression of the symplectic potential Γ P O M p S q as the –form Γ “ k π ż T r s S ̺ S p Ω , δ Ω q (4.5.10)(cf. eq. (4.1.8)). The symplectic –form Υ P O M p S q yielded by Γ is Υ “ δΓ “ k π ż T r s S ̺ S p δ Ω , δ Ω q (4.5.11)(cf. eq. (4.1.10)). The non singularity of Υ follows from that of the field pair-ing p¨ , ¨q . The higher gauge field manifold C M p S q is in this way equipped withthe appropriate symplectic structure. Our task now is expressing the associatedPoisson bracket.For any functional F P F M p S q , the Hamiltonian vector field H F P V M p S q of F is characterized by the property that ι H F Υ ` δF “ . (4.5.12)74rom (3.4.1), H F is given by relation (3.4.3) with V replaced by H F “ πk δ F δ Ω . (4.5.13)The field functional algebra F M p S q is so equipped with the Poisson bracket t F , G u “ ι H F δG “ πk ż T r s S ̺ S ˆ δ F δ Ω , δ G δ Ω ˙ (4.5.14)for F , G P F M p S q . The basic Poisson bracket of the theory is in this way "ż T r s S ̺ S p Ω , Σ q , ż T r s S ̺ S p Ω , Σ q * “ πk ż T r s S ̺ S p Σ , Σ q (4.5.15)with Σ , Σ P Map p T r s S, D m r sq .We consider next the field functionals Q p Θ q “ k π „ż T r s S ̺ S p Φ , Θ q ´ ż T r sB S ̺ B S p Ω , Θ q . (4.5.16)where Θ P G M p S q is an infinitesimal gauge transformation and Φ is the curvatureof higher gauge field Ω given by (3.3.2) as before. The boundary term is addedto render Q p Θ q differentiable in the sense established in ref. [72, 73], as is evidentby writing Q p Θ q in the form Q p Θ q “ k π ż T r s S ̺ S “ p Ω , d Θ q ` pr Ω , Ω s , Θ q ‰ (4.5.17) Q p Θ q is the Hamiltonian of Θ , " Q p Θ q , ż T r s S ̺ S p Ω , Σ q * “ ż T r s S ̺ S p δ Θ Ω , Σ q . (4.5.18)for Σ P Map p T r s S, D m r sq , where the gauge variation δ Θ Ω is given by (3.3.16).Under Poisson bracketing, the Hamiltonians Q p Θ q form a centrally extended rep-resentation of the gauge transformation Lie algebra. Specifically, we have t Q p Θ q , Q p Θ qu “ Q pr Θ , Θ sq ` k π c p Θ , Θ q (4.5.19)with Θ , Θ P G M p S q , where c is the –cocycle c p Θ , Θ q “ ż T r sB S ̺ B S p Θ , d Θ q . (4.5.20)75he Poisson bracket relation (4.5.19) describes a higher 3–dimensional currentalgebra analogous to the 2–dimensional current algebra appearing in the canonicalformulation of ordinary CS theory. More on this in the next subsection.We now write the above results in terms of the components ω , Ω of the highergauge field Ω for the sake of concreteness. The symplectic form Υ defined in eq.(4.5.11), has a simple component expression, Υ “ k π ż T r s S ̺ S x δ ω, δ Ω y , (4.5.21)which shows that ω , Ω are canonical conjugate fields. The component expressionof the Poisson bracket (4.5.14) takes so the familiar canonical form t F , G u “ πk ż T r s S ̺ S „B δ F δΩ , δ G δω F ´ B δ G δΩ , δ F δω F . (4.5.22)The basic Poisson bracket (4.5.15) reads in this way as "ż T r s S ̺ S x ω, Σ y , ż T r s S ̺ S x σ, Ω y * “ πk ż T r s S ̺ S x σ, Σ y (4.5.23)for σ P Map p T r s S, g r sq , Σ P Map p T r s S, e r sq .To write down the component expressions of the gauge transformation Hamil-tonians, we need the components φ , Φ of the higher gauge curvature Φ given by(3.3.4), (3.3.5). From (4.5.16), the Hamiltonian of an infinitesimal higher gaugetransformation of components θ , Θ is Q p θ, Θ q “ k π "ż T r s S ̺ S rx φ, Θ y ` x θ, Φ ys´ ż T r sB S ̺ B S rx ω, Θ y ` x θ, Ω ys * . (4.5.24)The component form of the Hamiltonian relation (4.5.18) is then " Q p θ, Θ q , ż T r s S ̺ S x ω, Σ y * “ ż T r s S ̺ S x δ θ,Θ ω, Σ y , (4.5.25) " Q p θ, Θ q , ż T r s S ̺ S x σ, Ω y * “ ż T r s S ̺ S x σ, δ θ,Θ Ω y , (4.5.26)where the gauge variations δ θ,Θ ω , δ θ,Θ Ω are given by (3.3.19), (3.3.20).76n components, the Q generator Poisson bracket (4.5.19) reads as t Q p θ, Θ q , Q p θ , Θ qu “ Q pr θ, θ s , µ p θ, Θ q ´ µ p θ , Θ qq ` k π c p θ, Θ ; θ , Θ q (4.5.27)and that of the occurring –cocycle (4.5.20) as c p θ, Θ ; θ , Θ q “ ´ ż T r sB S ̺ B S rx dθ, Θ y ´ x dθ , Θ y ` x τ p Θ q , Θ ys . (4.5.28)As we have already stated, the physical phase space C M ph p S q of higher CStheory is the functional hypersurface in the ambient phase space C M p S q definedby the flatness condition Φ « (cf. eq. (4.5.8)). As the bulk contribution to theHamiltonian functionals Q p Θ q is proportional to Φ (cf. eq. (4.5.16)), it seemsplausible that the constraint may be expressed through the weak constraints Q p Θ q « (4.5.29)with Θ P G M p S q , in analogy to ordinary CS theory. There are a number of prob-lems with this approach. First, the Q p Θ q contain also a boundary proportionalto Ω , making the use of (4.5.29) as definition of the physical phase space doubt-ful. Second, since the Q p Θ q generate infinitesimal gauge transformations (cf. eq.(4.5.18)), they should be first class functionals, while they are not because of the2–cocycle c p Θ , Θ q appearing in the Poisson bracket relations (4.5.19).Both the term proportional to Ω in the Q p Θ q and the 2–cocycle c p Θ , Θ q are supported on the boundary B S of S . They could be removed by imposingappropriate boundary conditions on Ω and Θ . Requiring that Ω P C M p S q obeys Ω | T r sB S P C M pB S q (4.5.30)and that Θ P G M pB S q satisfies Θ | T r sB S P G M pB S q , (4.5.31)where M is some isotropic submodule of the crossed module M , eliminates at oncethe unwanted term in Q p Θ q and the 2–cocycle c p Θ , Θ q , rendering the Q p Θ q hon-est first class functionals defining the physical phase space via (4.5.29) as desired.77ote that (4.5.30) precisely answers to the boundary condition (4.5.6) discussedearlier, while (4.5.31) is compatible with the boundary condition (4.5.7), since G M pB M q Ď I N C M pB M q . Below, we shall so refer to the boundary conditioned phasespace C M , M p S q Ă C M p S q formed by those gauge fields Ω which satisfy (4.5.30) andsimilarly to the boundary conditioned gauge algebra G M , M p S q Ă G M p S q formedby those infinitesimal gauge transformations Θ which satisfy (4.5.31). Before pro-ceeding further, we notice that while the boundary conditions (4.5.31) is essen-tially mandated by the requirement of first classness of the functionals Q p Θ q , theboundary condition (4.5.30) could be weakened by requiring less restrictively that Ω | T r sB S P C ON M pB S q , where ON M is the orthogonal normalizer of the isotropiccrossed submodule M (cf. subsect. 2.3). We shall come back to this point insubsect. 4.6.All field functionals F P F M p S q we consider are defined on the full phase space C M p S q containing all higher gauge fields Ω obeying no preassigned boundary con-dition. The boundary condition (4.5.30) is implemented by restricting the func-tionals to the conditioned phase space C M , M p S q . The calculation of the relevantPoisson brackets is correspondingly performed employing the unrestricted phasespace canonical framework described above. The boundary condition (4.5.30)is imposed at the end of the calculation. Doing so before that may lead to in-consistencies.On the basis of the above analysis of boundary conditions of gauge fields andgauge transformations, it appears that the physical phase space may be describedas the submanifold C M , M phys p S q of C M , M p S q defined by the weak constraints L « (4.5.32)with L P F M , M triv p S q , F M , M triv p S q being the ideal of F M p S q generated by theHamiltonians Q p Θ q with Θ P G M , M p S q . F M , M triv p S q codifies the infinitesimalhigher gauge symmetry action associated with the Lie subalgebra G M , M p S q . Thereduced physical phase space r C M , M phys p S q is the quotient of C M , M phys p S q by thisgauge symmetry. The physical field functional algebra is the algebra of field func-tionals on r C M , M phys p S q . As is well known, r C M , M phys p S q is a complicated non local78bject that is problematic to describe in local field theory. Moreover, by (3.3.29),(3.3.30), special gauge transformations in G M , M p S q are inert on the flat gaugefields Ω which constitute C M , M phys p S q . Thus, r C M , M ph p S q is also a singular mani-fold. Consequently, also the field functional algebra of r C M , M ph p S q is problematicto describe. A proper treatment of r C M , M ph p S q and its field functionals in localfield theory requires the full apparatus of BRST–BV theory for reducible gaugesymmetries. For the time being, it is enough to adopt a more modest stance andproceed as follows.Since these Hamiltonians Q p Θ q with Θ P G M , M p S q obey a Poisson algebraof the form (4.5.19) with vanishing central extension by the isotropy of M , theconstraints (4.5.32) are first class, as indeed t L , M u « (4.5.33)for L , M P F M , M triv p S q . A physical field functional r F is represented by a gaugeinvariant functional F P F M p S q , that is one such that t F , L u « (4.5.34)for L P F M , M triv p S q . The representative F is not unique however being modifiableby the addition of any functional ∆ F P F M , M triv p S q , so that F , F ` ∆ F can beconsidered as physically equivalent. Let us denote by F M , M inv p S q the subalgebraof F M p S q of the functionals F satisfying (4.5.34). Then, the algebra of physicalfunctionals is the quotient Ă F M , M phys p S q “ F M , M inv p S q{ F M , M triv p S q . (4.5.35) Ă F M , M phys p S q supports the induced Poisson bracket. If r F , r G P Ă F M , M phys p S q arephysical functionals represented by gauge invariant functionals F , G P F M , M inv p S q ,then t r F , r G u is represented by the gauge invariant functional t F , G u .The above has a mathematical formalization. Let I M , M p S q be the ideal of F M p S q generated by the Hamiltonians Q p Θ q with Θ P G M , M p S q . I M , M p S q is aPoisson subalgebra of F M p S q , but not a Poisson ideal. We consider so the Poissonnormalizer N P I M , M p S q of I M , M p S q , the set of all functionals F P F M p S q such79hat t F , L u P I M , M p S q for all L P I M , M p S q . N P I M , M p S q is both a subalgebraand a Poisson subalgebra of F M p S q . The Poisson Weyl algebra of I M , M p S q W P I M , M p S q “ N P I M , M p S q{ I M , M p S q (4.5.36)is then defined. W P I M , M p S q is both an algebra and a Poisson algebra with theinduced Poisson bracket t F ` I M , M p S q , G ` I M , M p S qu “ t F , G u ` I M , M p S q . (4.5.37)It should be apparent that I M , M p S q , N P I M , M p S q and W P I M , M p S q correspondrespectively to F M , M triv p S q , F M , M inv p S q and Ă F M , M phys p S q in the previous moreconventional characterization.The constraints (4.5.29) are not independent though, as we show next. Re-call that a special infinitesimal gauge transformation Θ ∗ is an infinitesimal gaugetransformation depending on the underlying gauge field Ω that in component formreads as in (3.3.27), (3.3.29) for some map Ξ P Map p T r s S, e q . Such a Θ ∗ musthence be regarded as a functional of Ω and Ξ . Θ ∗ will obey the boundary condition(4.5.31), if Ω and Ξ satisfy respectively the boundary condition (4.5.30), readingin components as ω | T r sB S P Map p T r sB S, g r sq , Ω | T r sB S P Map p T r sB S, e r sq ,and Ξ | T r sB S P Map p T r sB S, e q . Using the component expression (4.5.24) of Q p Θ ∗ q and the Bianchi identity (3.3.7), it is found that Q p Θ ∗ q “ (4.5.38)when Ω and Ξ are restricted as indicated above. Consequently, Q p Θ ∗ q “ strongly. For varying Ξ , the (4.5.38) represent a set of relations obeyed by the Q p Θ q for general infinitesimal gauge transformations Θ showing their non inde-pendence. The higher gauge symmetry is so reducible signaling a higher gaugefor gauge symmetry of the theory. Surface charges and holography
In this subsection, we analyze one of the most interesting holographic propertiesof higher CS theory: the existence of surface charges obeying under suitable80onditions a non trivial Poisson bracket algebra that is a higher counterpart ofthe familiar Kac–Moody current algebra.The canonical framework of subsect. 4.5 turns out to be particularly suitedfor this purpose. For any infinitesimal gauge transformation Θ P G M , M p S q , theHamiltonian Q p Θ q P I M , M p S q , where I M , M p S q is the constraint ideal. For ageneric infinitesimal gauge transformation Θ P G M p S q , though, the Hamiltonian Q p Θ q R N P I M , M p S q , N P I M , M p S q being the Poisson normalizer of I M , M p S q , aswe have t Q p Θ q , Q p Θ qu R I M , M p S q for Θ P G M , M p S q in general. In order Q p Θ q P N P I M , M p S q , the gauge transformation Θ must be suitably restricted and sincewhat distinguishes the gauge transformations Θ P G M , M p S q is only their obeyingthe boundary condition (4.5.31), it is a boundary condition that presumably hasto be imposed on Θ . A detailed analysis shows indeed that Q p Θ q P N P I M , M p S q if Θ P G M , ON M p S q , where ON M is the orthogonal normalizer of the crossedsubmodule M (cf. subsect. 2.3) and G M , ON M p S q is the subalgebra of G M p S q ofthe infinitesimal gauge transformations Θ satisfying the boundary condition Θ | T r sB S P G ON M pB S q (4.6.1)analogously to the subalgebra G M , M p S q . To see this, we note that ON m being acrossed submodule of N m ensures that for Θ P G M , M p S q , r Θ , Θ s P G M , M p S q by(3.2.6). Further, by the expression (4.5.20) of the 2–cocycle c , ON m being theorthogonal complement of m in N m and (3.2.10) imply that for Θ P G M , M p S q , c p Θ , Θ q “ . It follows by virtue of(4.5.19) that t Q p Θ q , Q p Θ qu P I M , M p S q , asrequired.Since m is a crossed submodule of ON m , for Θ P G M , ON M p S q , Θ P G M , M p S q one has Θ ` Θ P G M , ON M p S q . The identity Q p Θ ` Θ q “ Q p Θ q ` Q p Θ q showsthen that Q p Θ ` Θ q and Q p Θ q are equivalent modulo I M , M p S q and so definethe same element Q p Θ ` G M , M p S qq P W P I M , M p S q , the reduced gauge invariantfunctional algebra, as explained in subsect. 4.5.Remarkably, the reduced Hamiltonians Q p Θ ` G M , M p S qq P W P I M , M p S q with Θ P G M , ON M p S q form a subalgebra of the reduced Poisson algebra W P I M , M p S q ,as we now show. Pick Θ , Θ P G M , ON M p S q . ON m being a crossed submodule of81 m implies by (3.2.6) that G M , M p S q is an ideal of the Lie algebra G M , ON M p S q .So, the Lie bracket r Θ , Θ s depends on the choice of Θ , Θ mod G M , M p S q onlymod G M , M p S q . Further, as ON m is the orthogonal complement of m in N m , c p Θ , Θ q is independent from the choice of Θ , Θ mod G M , M p S q and may bedenoted as c p Θ ` G M , M p S q , Θ ` G M , M p S qq . By virtue of (4.5.19), the Poissonbracket of Q p Θ ` G M , M p S qq , Q p Θ ` G M , M p S qq thus read t Q p Θ ` G M , M p S qq , Q p Θ ` G M , M p S qqu (4.6.2) “ Q pr Θ ` G M , M p S q , Θ ` G M , M p S qsq ` k π c p Θ ` G M , M p S q , Θ ` G M , M p S qq . This is the Poisson subalgebra of W P I M , M p S q sought for.As G M , M p S q is an ideal of the Lie algebra G M , ON M p S q , the quotient Lie algebra G M , ON M p S q{ G M , M p S q is defined. The reduced Hamiltonians Q p Θ ` G M , M p S qq areparametrized by the cosets Θ ` G M , M p S q P G M , ON M p S q{ G M , M p S q . By (4.6.2),the resulting map Q : G M , ON M p S q{ G M , M p S q Ñ W P I M , M p S q is a Lie algebramorphism. This morphism is projective because of the central extension term.The Lie algebra G M , ON M p S q{ G M , M p S q is non trivial only if M is isotropic butnot Lagrangian. In fact, when M is Lagrangian, one has ON m “ m and hence G M , ON M p S q “ G M , M p S q . The same thus holds for the reduced Hamiltonians Q p Θ ` G M , M p S qq and their Poisson bracket algebra (4.6.2),From (4.5.17), it is apparent that on the constraint submanifold, where Φ « ,the Hamiltonians Q p Θ ` G M , M p S qq reduce to the surface term supported on B S .For this reason, the Q p Θ ` G M , M p S qq are identified with the surface charges ofhigher CS theory. The nature of these charges, in particular their non triviality,depends on the boundary conditions imposed on the higher gauge field Ω .If we required the gauge field Ω to obey the boundary condition (4.5.30),the surface charges Q p Θ ` G M , M p S qq would vanish since the crossed submodule ON m is the orthogonal complement of m in N m . If we want as we do the Q p Θ ` G M , M p S qq to be non trivial a less severe boundary condition is required.We have already anticipated in subsect. 4.5 that in order the Hamiltonians Q p Θ q with Θ P G M , M p S q to define through the weak constraints Q p Θ q « the flat82igher gauge functional submanifold Φ « , it is enough to require that Ω | T r sB S P C ON M pB S q (4.6.3)This boundary condition is weaker than (4.5.30) and subsumes it. Further, itis invariant under the infinitesimal gauge transformation action of G M , M p S q (cf.eqs. (3.3.19), (3.3.20)) and when the Lie group G in M is connected also underthe finite gauge transformation action of G M , M p S q (cf. eqs. (3.3.12), (3.3.13)),where G M , M p S q is the subgroup of G M p S q of the gauge transformations U suchthat U | T r sB S P G M pB S q . Finally, it makes the boundary term of the Q p Θ q with Θ P G M , M p S q vanish identically ensuring that the constraints Q p Θ q « areequivalent to Φ « as required.The Poisson algebra (4.6.2) bears striking formal similarities to the 2–dimen-sional current algebra known also as Kac–Moody algebra in mathematics [75],which occurs also in ordinary CS theory in an analogous context [76–79]. Thestructure of 2–cocycle c given in eq. (4.5.20) shows this rather clearly. So, (4.6.2)can be considered a higher current algebra hinged on a Lie algebra crossed modulerather than an ordinary Lie algebra.The occurrence of a non trivial Poisson algebra of surface charges is a holo-graphic feature that 4–dimensional CS theory shares not only with its familiar3–dimensional counterpart but also with other 4–dimensional theories, notablyelectrodynamics and general relativity. (See ref. [80] for a review.) 4–dimensionalCS theory might so provide an ideal testing ground for studying holography in 4dimensions. Toward the edge field theory of 4–d Chern–Simons theory
Gauge theories, including topological ones, on manifolds with boundaries nor-mally exhibit emergent boundary degrees of freedom called edge fields. In thissubsection, we outline a canonical theory of the edge modes of 4–dimensional CStheory and their physical symmetries, extending the corresponding analysis ofthe 3-dimensional theory [53–57]. Although there still remain basic issues to beclarified, as discussed shortly, it is already possible to shed light on some of its83ain features. A more in–depth analysis will be provided elsewhere [64].We follow the method originally worked out in ref. [56]. In the canonicalframework of 4–dimensional CS theory, where the underlying 4–fold M “ R ˆ S with S a 3–fold, it is possible to construct an extended phase space P M p S q , whichcomprises extra degrees of freedom localized at the boundary besides the inte-rior ones. Edge fields are in this way added to the original bulk gauge fields.Gauge invariance dictates the nature of the edge modes and the form of theirPoisson brackets by the requirement that P M p S q be invariant under the group G M p S q gauge transformations of the original system, inclusive of those which areeffective at the boundary, on the physical shell. In addition, the edge fields areacted upon by another set of Hamiltonian transformations. This form an infinitedimensional boundary symmetry group K M pB S q emerging as a consequence ofthe original gauge invariance. The boundary symmetry and gauge transforma-tions reciprocally commute. Therefore, the charges generating K M pB S q are gaugeinvariant, i.e. physical boundary observables.The way gauge invariance is implemented in the extended phase space is abit subtle. The bulk and edge symplectic 2–forms, Υ and Υ B , are not separatelyinvariant under bulk gauge transformations. The gauge variation of the formeris however a boundary term which is cancelled by that of the latter. For a mech-anism like this to work out for a given bulk field content, gauge transformationprescription and symplectic structure, the edge field content and its gauge trans-formation properties and symplectic structure must be suitably adjusted. Thereis no a priori guarantee that this is possible at all, but happily it is in our case.We shall now describe the above construction in more precise terms. We utilizethe derived formalism of subsect. 3.3 for convenience and frame our analysis inthe covariant canonical theory [71]. On the 3–fold S with boundary B S , theinterior fields are just the bulk gauge fields Ω P C M p S q already considered. Theboundary fields comprise the edge gauge fields Ω B P C M pB S q and Stueckelbergfields H B P G M pB S q , boundary gauge transformations promoted to dynamicaledge fields. The extended phase space P M p S q is the submanifold of the productfield manifold C M p S q ˆ C M pB S q ˆ G M pB S q defined by the condition Ω | T r sB S “ Ω B U P G M p S q , a bulk gauge field Ω transformsas in (1.1.5), while a edge gauge field Ω B and Stueckelberg field H B transform as Ω B U “ Ad U ´ p Ω B q ` U ´ dU , (4.7.1) H B U “ U ´ H B , (4.7.2)where U is tacitly restricted T r sB S . Notice that the expression of Ω B U is dictatedby the gauge covariance of the compatibility requirement Ω | T r sB S “ Ω B .As we found in subsect. 4.5, the bulk field symplectic 2–form Υ and Pois-son brackets t¨ , ¨u are given by expressions (4.5.11) and (4.5.15). The edge fieldsymplectic form Υ B and Poisson brackets t¨ , ¨u B cannot be assigned independentlybeing determined by the requirement of the full gauge invariance of the totalsymplectic 2–form Υ tot “ Υ ` Υ B and Poisson brackets t¨ , ¨u tot thereof on the shell(4.5.8). This can be stated more precisely as follows.Let gau : P M p S q ˆ G M p S q Ñ P M p S q , pr : P M p S q ˆ G M p S q Ñ P M p S q be the gau-ge transformation action and projection maps, respectively. On the shell (4.5.8),the pull–backs gau ∗ Υ , pr ∗ Υ of the bulk symplectic 2–form Υ are found to differby a boundary term. A straightforward analysis shows however that the edgesymplectic 2–form Υ B can be defined such that gau ∗ Υ ´ pr ∗ Υ « ´ gau ∗ Υ B ` pr ∗ Υ B .Setting Υ tot “ Υ ` Υ B , one has therefore gau ∗ Υ tot « pr ∗ Υ tot . (4.7.3)In this sense, Υ tot is gauge invariant.The above result can be understood intuitively as follows. A Stueckelbergfield H B P G M pB S q can be extended non uniquely in the interior of S to a field H P G M p S q . If H is an extension of H B and U P G M p S q is a gauge transformation,then H U “ U ´ H is an extension of the gauge transform H B U of H B . We view H U as the gauge transform of H , in keeping with (4.7.2). If Ω P C M p S q is a bulk gaugefield, its gauge transform Ω H depends on the extension H used, but for a fixedchoice of H it is gauge invariant. The 2–form Υ tot obtained from Υ by replacing Ω with Ω H in (4.5.11) is by construction gauge invariant. On the shell (4.5.8),85 tot turns out to equal the sum of Υ and a boundary term Υ B , depending on theedge fields Ω B , H B but not on the chosen extension H of H B , which is precisely theedge symplectic 2–form.The procedure described in the previous paragraph provides a practical wayof computing the edge symplectic 2–form Υ B . The expression of Υ B that we findreads as Υ B “ k π ż T r sB S ̺ B S “` δ B H B H B´ , δ B Ω B ˘ (4.7.4) ´ ` d p δ B H B H B´ q , δ B H B H B´ ˘ ´ ` Ω B , r δ B H B H B´ , δ B H B H B´ s ˘‰ . From this expression, the edge Poisson brackets t¨ , ¨u B can be determined. Inparticular, it is found that edge Poisson brackets of functionals of the Stueckelbergfield H B only Poisson commute.As we found in subsect. 4.5, the bulk gauge transformation action on bulkfields is Hamiltonian. For an infinitesimal gauge transformation Θ P G M p S q , theassociated bulk Hamiltonian functional Q p Θ q is given by eq. (4.5.16) and thegauge transformation action on bulk fields can be expressed through the Poissonbracket (4.5.18). Remarkably, the bulk gauge transformation action on edgefields is Hamiltonian as well. For the gauge transformation Θ , the associatededge Hamiltonian functional Q B p Θ q reads as Q B p Θ q “ k π ż T r sB S ̺ B S p Ω B , Θ q . (4.7.5)and the gauge transformation action on edge fields can be cast as t Q B p Θ q , F B u B “ δ Θ F B , (4.7.6)where F B is an edge field functionals and δ Θ denotes variation with respect theinfinitesimal form of the gauge transformation (4.7.1), (4.7.2).The bulk Hamiltonian functionals Q p Θ q the Poisson bracket algebra (4.5.19)featuring a central extension with G M p S q –2–cocycle c given by (4.5.20). The edgeHamiltonian functionals Q B p Θ q obey a totally similar Poisson bracket algebra, t Q B p Θ q , Q B p Θ qu B “ Q B pr Θ , Θ sq ´ k π c p Θ , Θ q , (4.7.7)86ith the same central extension up to sign.For an infinitesimal gauge transformation Θ , the total bulk plus edge Hamil-tonian functional is Q tot p Θ q “ Q p Θ q ` Q B p Θ q “ ż T r s S ̺ S p Φ , Θ q . (4.7.8)By virtue of (4.5.19), (4.7.7), the Hamiltonian functionals Q tot p Θ q obey a center-less Poisson bracket algebra, t Q tot p Θ q , Q tot p Θ qu tot “ Q tot pr Θ , Θ sq . (4.7.9)Thanks to (4.7.8) and (4.7.9), the physical on–shell condition (4.5.8) can hencebe consistently cast in the form Q tot p Θ q « (4.7.10)with non need to impose any boundary conditions on either the bulk gauge field Ω or Θ , as we were forced to in subsect. 4.5.As we outlined at the beginning of this subsection, the extended phase spaceenjoys a second physical non gauge surface symmetry. We now describe it ingreater detail. A surface transformation is specified by an element T B P G M pB S q .It acts on the bulk gauge fields Ω trivially and on the edge fields as follows, Ω B T B “ Ω B , (4.7.11) H B T B “ H B T B . (4.7.12)As for the bulk gauge symmetry, the form of Ω B T B is dictated by the surfacesymmetry covariance of the compatibility requirement Ω | T r sB S “ Ω B .The surface symmetry turns out to be Hamiltonian. For an infinitesimalsurface transformation Λ B P G M pB S q , the associated edge Hamiltonian functional C B p Λ B q has the simple form C B p Λ B q “ ´ k π ż T r sB S ̺ B S ` Ω B H B , Λ B ˘ (4.7.13)and the surface transformation action on edge fields takes the expected form t C B p Λ B q , F B u B “ δ Λ B F B , (4.7.14)87here F B is an edge field functionals and δ Λ B denotes variation with respect theinfinitesimal form of the transformations (4.7.11), (4.7.12).The surface edge Hamiltonian functionals C B p Λ B q obey a Poisson bracket al-gebra formally identical to that of the gauge edge Hamiltonian functionals Q B p Θ q shown in eq. (4.7.7). One has indeed, t C B p Λ B q , C B p Λ B1 qu B “ C B pr Λ B , Λ B1 sq ´ k π c p Λ B , Λ B1 q (4.7.15)where the G M pB S q –2–cocycle c is given again by expression (4.5.20) with Θ , Θ replaced by Λ B , Λ B1 throughout.An important property of the surface Hamiltonians C B p Λ B q is their Poissoncommuting with the gauge Hamiltonians Q B p Θ q , t Q B p Θ q , C B p Λ B qu B “ . (4.7.16)That a relation like the above must hold is evident also from the fact that C B p Λ B q is defined in (4.7.13) through the gauge invariant combination Ω B H B . Relation(4.7.16) proves further the physical nature of the surface symmetry. The C B p Λ B q are the associated charges. Surface symmetry is therefore infinite dimensional.By (4.7.15), the surface charges are in involution if the infinitesimal surfacetransformations Λ B P G M pB S q are restricted in such a way to make the centralterm vanish. An inspection of (4.5.20) shows readily that there are several waysin which this can be achieved. For instance, we may require that Λ B P G M pB S q ,where M is an isotropic crossed submodule of M (cf. subsect. 2.3), or that Λ B P G M cl pB S q , the Lie subalgebra of G M pB S q spanned by those elements Λ B which satisfy the equation d Λ B “ .The above account of the edge sector of 4–dimensional CS theory is still in-complete. There remains a basic problem to be solved: the lack of a Lagrangianand Hamiltonian formulation describing the dynamics of edge fields, if any. Infact, the corresponding analysis for 3–dimensional CS theory (see e.g. [57] fora discussion of this point) shows that the edge dynamics of topological gaugetheories may be non trivial as that of non topological ones. Whether this is thecase also for out 4–dimensional model is an issue deserving further investigation.88 .8 Covariant Schroedinger quantization
In this final subsection, we study the covariant Schroedinger quantization of 4–dimensional CS theory. Although there still are points requiring clarification anda more in–depth analysis, in particular in connection to the edge field theory ofthe model discussed in subsect. 4.7, it is still possible to elucidate its outlines toa considerable extent.The covariant Schroedinger quantization scheme of 4–dimensional CS theoryis based on its covariant phase space. This is the space C M pB M q of boundarygauge field configurations ω B , Ω B . A straightforward analysis totally analogousto that of subsect. 4.6 shows that the symplectic form of Υ B , the associatedPoisson bracket t¨ , ¨u B , the Hamiltonians Q B p θ B , Θ B q of the boundary infinitesimalgauge transformations θ B , Θ B , their Poisson action and Poisson bracket algebraare given respectively by relations (4.5.21), (4.5.22), (4.5.23), (4.5.24), (4.5.25),(4.5.26) and (4.5.27) with S “ B M , B S “ H , ω , Ω replaced by ω B , Ω B , θ , Θ by θ B , Θ B and c “ . In this case, so, the Q B Poisson algebra features no centralextension.The Hilbert space of 4–dimensional CS theory consists of complex wave func-tionals on C M pB M q obeying a polarization condition, that is annihilated by thevector fields on C M pB M q belonging to an integrable Lagrangian distribution of T C M pB M q . There are two obvious choices of the distribution. The first is spannedby the vector fields δ { δΩ B and produces wave functionals Ψ p ω B q of ω B . The secondis generated by the vector fields δ { δω B and leads to wave functionals Ψ p Ω B q of Ω B .Although these two alternatives are defined in a seemingly symmetrical manner,only the first one is viable once gauge transformation is implemented. In fact,inspection of (3.3.12), (3.3.13) shows that under gauge transformation ω B doesnot mix with Ω B whilst Ω B does with ω B .In the following, we therefore consider only the first choice of polarization. Inthis canonical quantum set–up, the Hilbert space inner product reads as x Ψ , Ψ y “ ż D ω B Ψ p ω B q ∗ Ψ p ω B q , (4.8.1)where D ω B is a suitable formal functional measure. Further, the operators p ω B ,89 Ω ω quantizing ω B , Ω ω take the familiar form p ω B “ ω B ¨ , (4.8.2) p Ω B “ ´ πik δδω B . (4.8.3)They are formally selfadjoint with respect to the inner product structure (4.8.1).The infinitesimal gauge transformation Hamiltonians Q B p θ B , Θ B q constitute aset of first class covariant phase space functionals. Since the Q B p θ B , Θ B q are linearin the curvature components φ B , Φ B , the physical phase space is defined by theconstraints Q B p θ B , Θ B q « . At the quantum level, the constraints translate intoa set of linear conditions the wave functionals must satisfy, p Q B p θ B , Θ B q Ψ “ . (4.8.4)Here, the p Q B p θ B , Θ B q are operators quantizing the phase functionals Q B p θ B , Θ B q .The quantization must be such that the commutator algebra “ p Q B p θ B , Θ B q , p Q B p θ B1 , Θ B1 q ‰ “ i p Q B pr θ B , θ B1 s , µ p θ B , Θ B1 q ´ µ p θ B1 , Θ B qq (4.8.5)is obeyed in conformity with (4.5.27). This guarantees in particular the consis-tency of the conditions (4.8.1).Conditions (4.8.1) imply that the wave functional Ψ satisfy a pair of functionaldifferential equations, ż T r sB M ̺ B M B dω B ` r ω B , ω B s ` πik τ ´ δδω B ¯ , Θ B F Ψ p ω B q “ , (4.8.6) ż T r sB M ̺ B M B θ B , ´ πik „ d δδω B ` µ ´ ω B , δδω B ¯F Ψ p ω B q “ . (4.8.7)On account of (3.3.19), these identities imply that δ θ B ,Θ B Ψ p ω B q “ ik π ż T r sB M ̺ B M @ dω B ` r ω B , ω B s , Θ B D Ψ p ω B q . (4.8.8)Therefore, the variation of Ψ under a finite boundary gauge transformation u B , U B is given by a multiplicative factor Ψ p ω B u B ,U B q “ exp p i W Z B p u B , U B ; ω B qq Ψ p ω B q . (4.8.9)90y the very structure of this relation, the functional W Z B p u B , U B ; ω B q appearingin it is a U p q –valued cocycle for the boundary gauge transformation action onthe degree 1 boundary gauge field component, W Z B p u B v B , U B ` µ p u B , V B q ; ω B q“ W Z B p u B , U B ; ω B q ` W Z B p v B , V B ; ω B u B ,U B q mod π Z . (4.8.10)To reproduce the infinitesimal variation (4.8.8), W Z B p u B , U B ; ω B q must furthersatisfy the normalization condition δ θ B ,Θ B W Z B p u B , U B ; r ω B q “ k π ż T r sB M ̺ B M @ dω B ` r ω B , ω B s , Θ B D , (4.8.11)where the tilde indicates that δ θ B ,Θ B is inert on ω B . Properties (4.8.10), (4.8.11)determine the cocycle W Z B up to a trivial cocycle, W Z B p u B , U B ; ω B q “ k π ż T r sB M ̺ B M ” @ τ p U B q , dU B ` r U B , U B s D ` x ω B , r U B , U B sy ` @ dω B ` r ω B , ω B s , U B D ı ` K B p ω B u B ,U B q ´ K B p ω B q , (4.8.12)where K B p ω B q is a local boundary functional which cannot be determined in thepresent method. We expect K B to be generated by quantum effects as we shalldiscuss in greater detail momentarily.At this point, it is important to remark that the above Schroedinger quantiza-tion scheme of 4–dimensional CS theory mirrors closely the Bargmann one usedin ordinary 3–dimensional CS theory [58]. In particular, relations (4.8.6), (4.8.7)are the higher counterpart of the familiar WZNW Ward identities, the cocycle W Z B appearing in (4.8.9) is a higher gauged WZNW functional and the cocyclerelation (4.8.10) is just a higher version of the Polyakov–Wiegmann identity [81].However, unlike its ordinary counterpart, the WZNW functional W Z B is fullytopological, being independent on any background metric structure. Further, itdepends only on the second component of the underlying gauge transformation, U B , but not on the first one, u B .The issue of the cohomological triviality of the cocycle functional W Z B , theproperty that W Z B p ω B q “ S B p ω B u B ,U B q ´ S B p ω B q mod π Z for a local boundary91unctional S B p ω B q , is relevant. When it occurs, the modified wave functional Ψ p ω B q “ e i S B p ω B q Ψ p ω B q (4.8.13)is fully gauge invariant so that W Z B1 “ mod π Z . (4.8.14)It is interesting to illustrate this point by some examples. Consider the casewhere M “ INN G is the inner automorphism crossed module of a Lie group G with an invariant symmetric non singular bilinear form x¨ , ¨y on g . Then, a simplecalculation shows that the WZNW functional W Z B can be cast as W Z B p u B , U B ; ω B q “ k π ż T r sB M ̺ B M @ du B u B´ , “ du B u B´ , du B u B´ ‰D ` CS B p ω B u B ,U B q ´ CS B p ω B q ` K B p ω B u B ,U B q ´ K B p ω B q , (4.8.15)where CS B p ω B q is the boundary CS action CS B p ω B q “ k π ż T r sB M ̺ B M @ ω B , dω B ` r ω B , ω B s D . (4.8.16)If G is a compact semisimple Lie group, x¨ , ¨y is the suitably normalized Killingform of g and k is an integer, then the first term in the right hand side of (4.8.15)vanishes mod π Z and so W Z B is cohomologically trivial. When M “ AD ∗ G isthe coadjoint action crossed module of G with the canonical duality pairing or ageneric crossed module, W Z B is cohomologically non trivial.In 4–dimensional CS theory on a 4–fold M , a wave functional Ψ M p ω B q isyielded by path integration over all gauge field configuration ω , Ω such that ω | T r sB M “ ω B . Formally, one has Ψ M p ω B q “ ż ω | T r sB M “ ω B D ω D Ω e i CS p ω,Ω q , (4.8.17)leaving aside such relevant issues such as normalization and gauge fixing. Theconsistent quantization of the theory requires however that the CS action CS employed be differentiable in the sense of refs. [72, 73] under the boundary con-dition enforced. 92s explained in subsect. 4.1, the variation δCS of the CS action CS , given byeq. (4.1.3), exhibits a boundary contribution showing that CS is not differentiableas it is. To obtain a differentiable CS action, it is necessary to i q replace CS bya modified the CS action CS obtained by adding to CS a suitable boundaryterm ∆ CS as in eq. (4.1.6) and ii q impose a boundary condition on the gaugefield components ω , Ω such that the variation δCS of CS is given by the bulkcontribution in the right hand side of eq. (4.1.3) only, once the boundary conditionis enforced. For the boundary condition used in (4.8.17), the expression of theappropriate boundary term ∆ CS is readily found, ∆ CS p ω, Ω q “ k π ż T r sB M ̺ B M x ω, Ω y . (4.8.18)With this choice, the modified CS action CS is given by the right hand side of(4.1.2) with the boundary term removed. The boundary part of the variation δCS of CS then turns out to be δCS boundary “ k π ş T r sB M ̺ B M x δ ω, Ω y . Thisvanishes when the boundary condition ω | T r sB M “ ω B with assigned ω B is impos-ed, rendering CS differentiable as required. Here, it is appropriate to recall thatthe above procedure has a well–known counterpart in 3–dimensional CS theory.In that case, however, the boundary term depends on the choice of a conformalstructure on the 2–dimensional boundary, since the two boundary gauge field 1–form components are canonically conjugated [58]. In the present case, conversely,since ω is canonically conjugate to Ω , there is no need to introduce a new structurein the theory and so the boundary term is fully topological.In sect. 4.2, we found that the gauge variation A of the modified CS action CS is given by (4.2.5) in terms of the gauge variations A and ∆ A of the CS action CS and the boundary term ∆ CS . Using relations (4.2.3) for A and computing ∆ CS from (4.8.18) employing (3.3.12), (3.3.13), it is straightforward to obtain A p ω ; u, U q “ k π ż T r sB M ̺ B M ” @ τ p U q , dU ` r U, U s D ` x ω, r U, U sy ` @ dω ` r ω, ω s , U D ı . (4.8.19)Comparing (4.8.12) and (4.8.19), we find that A p ω ; u, U q reproduces the firstterm in the right hand side of (4.8.12) when u | T r sB M “ u B , U | T r sB M “ U B ,93 | T r sB M “ ω B . On account of (4.8.17), this shows that such term is the fullclassical contribution to W Z B . The remaining terms, therefore, if they arise atall, are of a quantum nature. 94 Sample applications
In this section, we illustrate a few field theoretic models which are interesting in-stances of 4–dimensional CS theory: the toric and the Abelian projection models.Here, our aim is showing by direct construction that 4–dimensional CS theorycan find explicit realizations related to various areas of theoretical research on onehand and prepare the ground for a more systematic study of the models presentedto appear in future work on the other. So, this last section should also providean outlook for perspective applications of the theory which we have developed.
The toric 4–dimensional CS model
Dijkgraaf–Witten theory [19] is known to classify symmetry protected topologi-cal phases without fermions in low dimension. 4–dimensional Dijkgraaf–Wittentheory in turn has a continuum description in terms of toric 4–dimensional CStheory [82–84].The Lie group crossed module of the toric model is the toric crossed module M “ p T , T , ς, ̟ q , where T is a torus, that is a ompact connected Abelian group, ς : T Ñ T is an endomorphism and ̟ : T ˆ T Ñ T is the trivial action of T onitself. The associated Lie algebra crossed module is m “ p t , t , ς, ̟ q .The torus T can be represented as the quotient t { Λ , where Λ is the integral lat-tice of T defined by the property that e l “ T for l P t . ς being an endomorphismof T entails that ς : Λ Ñ Λ is a lattice endomorphism.Let x¨ , ¨y : t ˆ t Ñ R be a symmetric non singular bilinear form on t . Sincethe Lie algebra t is Abelian and the action ̟ is trivial, the form x¨ , ¨y satisfiesthe invariance property (2.2.2) trivially. x¨ , ¨y satisfies the symmetry property(2.2.3) if the endomorphism ς is symmetric with respect to x¨ , ¨y , which we assumehenceforth to be the case.It is natural to suppose that the form x¨ , ¨y restricts to a lattice bilinear form x¨ , ¨y : Λ ˆ Λ Ñ Z , which we denote by the same symbol for the sake of simplicity. ς : Λ Ñ Λ is then a symmetric lattice endomorphism.The toric higher gauge field components are ω P Ω p M, t q , Ω P Ω p M, t q . The95oric CS action is CS p ω, Ω q “ k π ż M @ dω ´ ς p Ω q , Ω D ´ k π ż B M x ω, Ω y (5.1.1)A toric gauge transformation consists of a map u P Map p M, T q , U P Ω p M, t q .Its action on the toric gauge field is according to eqs. (3.3.12), (3.3.13), ω u,U “ ω ` duu ´ ` ς p U q , (5.1.2) Ω u,U “ Ω ` dU. (5.1.3)These transformations are gauge symmetries only if B M is empty as we have seen.If we identify t » R r and Λ » Z r for some integer r , then we have x x, y y “ ř ri,j “ K ij x i y j , (5.1.4)where K is an r ˆ r matrix of the form K ij “ n i δ ij (5.1.5)with n i P Z , n i ą . The endomorphism ς is similarly expressed as ς p x q “ ř rj “ s ij x j , (5.1.6)where s is a certain r ˆ r matrix. Requiring that is a symmetric lattice endomor-phism leads to the property that s ij P Z and the condition n i s ij ´ n j s ji “ . (5.1.7)In the toric models of ref. [82, 83], one has n i s ij “ ´ p ij lcm p n i , n j q ´ χ i p ii n i δ ij , (5.1.8)where p ij is a symmetric integer matrix and χ i “ or according to whether n i is even or odd respectively. It is immediately checked that the matrix t furnishedby (5.1.8) is integer and satisfies (5.1.7).The Lie group crossed submodules of M , which as we have seen in sect. 4are the basic datum of linear boundary conditions, have a simple structure in96oric CS theory. The most general one is of the form M “ p V , U , ς | V , ̟ | U ˆ V q where U , V are subgroups of T such that ς p V q Ď U . The corresponding Liealgebra crossed submodule is therefore m “ p v , u , ς | v , ̟ | u ˆ v q . Thus, m isotropicif v Ď u K and Lagrangian if v “ u K . The associated orthogonal normalizerand Weyl crossed modules are respectively ON m “ p u K , v K , ς | u K , ̟ | v K ˆ u K q and OW m “ p u K { v , v K { u , ς | u K { v , ̟ | v K { u ˆ u K { v q . The Abelian projection model
Abelian projection [85] is a theoretical framework for investigating the proper-ties of confining gauge theories. It consists in a gauge choice reducing the gaugesymmetry from a non Abelian group to a maximal Abelian subgroup. Abeliangauge fields emerge then from the non Abelian background and with these mag-netic monopoles presumably responsible for confinement. In this subsection, weshow how a kind of Abelian projection can be implemented in 4–dimensional CStheory, even though no Higgs field is provisioned by it, leaving to future work theexploration of possible physical applications. Cartan–Weyl theory of semisimpleLie algebras is used throughout. See app. A.5 for a brief review of some of basicfacts and notation used.We begin by showing that we can associate a toric 4–dimensional CS model(cf. subsect. 5.1) to a maximal torus F of a compact semisimple Lie group E whose Lie algebra e is endowed with an invariant symmetric non singular bilinearform. The model’s toric crossed module M E “ p T , T , ς, ̟ q is defined as follows.The torus T is just U p q r , where r is the rank of E . Below, we shall view theelements T as ordered r -uples p e x α q α P Π ` of U p q elements indexed by a set Π ` ofsimple positive roots of e , where x α P i R . The target map ς is given by ς ` p e x α q α P Π ` ˘ “ ` e ř β P Π ` C βα x β ˘ α P Π ` , (5.2.1)where C αβ is the Cartan matrix of e defined by (A.5.3). Note that ς is welldefined because C is a matrix with integer entries. The action map ̟ is trivial. M E can be further equipped with an invariant pairing. Writing the elements of t as ordered r -uples p x α q α P Π ` of u p q elements indexed by Π ` analogously to the97nite case, this reads @ p x α q α P Π ` , p y α q α P Π ` D “ ř α P Π ` κ α x α y α , (5.2.2)where κ α is the inverse half lengths square of the root α defined by (A.5.4). Thesymmetry property (2.2.3) is fulfilled as can be easily checked upon noticing that ς ` p ix α q α P Π ` ˘ “ ` i ř β P Π ` C βα x β ˘ α P Π ` and using the identity κ α C βα “ κ β C αβ .In the above formal framework, the components of a toric higher gauge fieldare ordered r –tuples p ω α q α P Π ` , p Ω α q α P Π ` with ω α P Ω p M, i R q , Ω α P Ω p M, i R q .The CS toric model action (5.1.1) then reads explicitly as CS ` p ω α q α P Π ` , p Ω α q α P Π ` ˘ “ k π ż M ř α P Π ` κ α ´ dω α ´ ř β P Π ` C βα Ω β ¯ Ω α ´ k π ż B M ř α P Π ` κ α ω α Ω α . (5.2.3)The components of a toric higher gauge transformation are similarly ordered r –tuples ` e f α ˘ α P Π ` , p U α q α P Π ` with f α P Ω p Ă M , i R q , U α P Ω p M, i R q . Here, thefunctions f α are generally multivalued and thus properly defined on the universalcovering Ă M of M . The integrality condition πi ż c df α P Z , (5.2.4)where c is a 1–cycle of M must be satisfied in order e f α to be a well defined elementof Map p M, U p qq . In accordance with (5.1.2), (5.1.3). the gauge transformedgauge field p ω p e f q , p U q α q α P Π ` , p Ω p e f q , p U q α q α P Π ` reads as ω p e f q , p U q α “ ω α ` df α ` ř β P Π ` C βα U β , (5.2.5) Ω p e f q , p U q α “ Ω α ` dU α . (5.2.6)Again, these transformations are gauge symmetries only if B M is empty.The Abelian projection associates a CS toric model of the type just describedwith a CS model based on a Lie group crossed module M “ p E , G , τ, µ q whosesource group E is a compact semisimple Lie group and a choice of a maximaltorus F of E . Its explicit construction goes through a few steps detailed next.98he source e of the Lie algebra crossed module m “ p e , g , τ, µ q is a compactsemisimple Lie algebra. Hence, z p e q “ . So, since ker τ is a central ideal of e , wehave ker τ “ .Suppose M is equipped with an invariant pairing x¨ , ¨y . Then, m is balancedso that dim e “ dim g . Since ker τ “ as seen above, we have ran τ “ g . τ istherefore a Lie algebra isomorphism and e » g . g is consequently also a compactsemisimple Lie algebra. τ being a Lie algebra isomorphism allows us to define a distinguished invariantsymmetric non singular bilinear of the Lie algebra e , namely x X, Y y τ “ x τ p X q , Y y , (5.2.7)which we shall tacitly employ in what follows.A higher gauge field ω , Ω induces a toric higher gauge field p ω α q α P Π ` , p Ω α q α P Π ` ω α “ x ω, H α y , (5.2.8) Ω α “ x τ p H α ∗ q , Ω y , (5.2.9)where H α , H α ∗ P i f are the root and weight Cartan subalgebra generators asso-ciated with the roots and weights α , α ∗ . We shall call p ω α q α P Π ` , p Ω α q α P Π ` theAbelian projection of ω , Ω . In turn, with any toric higher gauge field p ω α q α P Π ` , p Ω α q α P Π ` there is associated a higher gauge field ω , Ω of the form ω “ ř α P Π ` ω α τ p H α ∗ _ q , (5.2.10) Ω “ ř α P Π ` Ω α H α _ , (5.2.11)where H α _ , H α ∗ _ P i f are the coroot and coweight Cartan subalgebra generatorsassociated with the coroots and coweights α _ , α ∗ _ . We shall call a gauge field ofthis form Abelian projected. It is immediately verified that the Abelian projec-tion of the Abelian projected gauge field corresponding to the toric gauge field p ω α q α P Π ` , p Ω α q α P Π ` equals this latter. See app. A.5 for some technical details.The CS action of an Abelian projected higher gauge field ω , Ω equals preciselythe toric CS action of the underlying toric higher gauge field p ω α q α P Π ` , p Ω α q α P Π ` CS p ω, Ω q “ CS ` p ω α q α P Π ` , p Ω α q α P Π ` ˘ . (5.2.12)The Abelian projection involves a reduction of the higher gauge symmetrysimilarly to ordinary gauge theory. The residual gauge symmetry is describedby the normalizer crossed module N M F of a certain crossed submodule M F of M depending on the maximal torus F (cf. subsect. 2.3).The characteristic crossed module of F is M F “ p F , τ p F q , τ | F , µ | τ p F qˆ F q . M F is a crossed submodule of M . M F is toric as F , τ p F q are maximal tori of E , G ,respectively, and µ | τ p F qˆ F is trivial. It is simple to show that the normalizer crossedmodule of M F is N M F “ p N F , µ N F , τ | N F , µ | µ N F ˆ N F q . The Weyl crossed module of M F so turns out to be W M F “ p N F { F , µ N F { τ p F q , τ | N F { F , µ | µ N F { τ p F qˆ N F { F q . W M F isa finite discrete crossed module. Indeed, N F { F “ W E , the familiar Lie theoreticWeyl group of E . Furthermore, since it turns out that µ N F “ N τ p F q , as isstraightforwardly shown, µ N F { τ p F q “ W G , the Weyl group of G . Notice that W G » W E , since g » e .As already mentioned, N M F is the crossed module of the residual higher gaugesymmetry left over by the Abelian projection. Therefore, to implement the pro-jection, we restrict to the subgroup of N M F –valued higher gauge transformations,that is, by what found in the previous paragraph, the gauge transformations u , U , with u P Map p M, µ N F q and U P Ω p M, f q . u , U have for this reason a specialform. Since µ N F is a disjoint union of finitely many cosets of τ p F q , on eachconnected component of M , one has u “ u c a, (5.2.13)where u c P Map p M, τ p F qq and a P µ N F is constant. u c can thus be expressed as u c “ e ř α P Π ` f α τ p H α ∗ _ q (5.2.14)with f α P Ω p Ă M , i R q through the coweight generators H α ∗ _ P i f . As before, the f α are generally multivalued functions. The well-definedness of the exponentialin the right hand side of (5.2.14) requires that100 πi ż c ř α P Π ` df α H α ∗ _ P Λ f (5.2.15)for any 1–cycle c of M , where Λ f is the integral lattice of f . Condition (5.2.15) iscompatible with (5.2.4) since the integral lattice Λ f is a sublattice of the coweightlattice Λ cw f . However, the integer values which the periods of df α can take arerestricted unless Λ f “ Λ cw f , which happens when the center Z p E q of E is trivial,as Z p E q “ Λ cw f { Λ f . Finally, U can be expanded as U “ ř α P Π ` U α H α _ . (5.2.16)with U α P Ω p M, i R q in terms of the coroot generators H α _ P i f . In this way,we can associate with an N M F –valued gauge transformation u , U a toric gaugetransformation ` e f α ˘ α P Π ` , p U α q α P Π ` satisfying (5.2.15), the Abelian projection of u , U . This can be defined alternatively through df α “ x duu ´ , H α y , (5.2.17) U α “ x τ p H α ∗ q , U y , (5.2.18)analogously to (5.2.8), (5.2.9) projecting on the root and weight generators H α , H α ∗ P i f . Viceversa, with any toric gauge transformation ` e f α ˘ α P Π ` , p U α q α P Π ` satisfying (5.2.15), we can associated an Abelian projected N M F –gauge transfor-mation u c , U through (5.2.14), (5.2.16). Note that in performing the Abelianprojection of a N M F –valued gauge transformation u , U all the information aboutthe discrete factor a P µ N F appearing in (5.2.13) is lost. Correspondingly, forthe Abelian projected gauge transformation u c , U yielded by a toric gauge trans-formation we have u c P Map p M, τ p F qq only.For a P µ N F , the µ p a, ¨q are automorphisms of the Lie subalgebra f . Thefindings of the previous paragraph indicate that their action on the root, coroot,weight and coweight lattices Λ r f , Λ cr f , Λ w f , Λ cw f of f may be relevant to the analysisof the gauge invariance of Abelian projection. One finds µ p a, H α q “ ř β P Π ` χ βα p a q H β , (5.2.19)101 p a, H α _ q “ ř β P Π ` χ _ βα p a q H β _ , (5.2.20) µ p a, H α ∗ q “ ř β P Π ` χ _ αβ p a ´ q H β ∗ , (5.2.21) µ p a, H α ∗ _ q “ ř β P Π ` χ αβ p a ´ q H β ∗ _ , (5.2.22)where χ : µ N F Ñ GL p r, Q q , χ _ : µ N F Ñ GL p r, Q q are certain group morphisms.In fact, χ _ , χ are simply related χ p a q “ κχ _ p a q κ ´ , (5.2.23)where κ P GL p r, R q is the diagonal matrix κ αβ “ κ α δ αβ , (5.2.24) κ α being defined by eq. (A.5.4). These relations follow form observing that the µ p a, ¨q are automorphisms of both the integral lattice Λ f of f and its dual lattice Λ f ∗ and that the root and coroot generators span Λ f ∗ , Λ f over Q , since the rootand coroot lattices Λ r f , Λ cr f are sublattices of Λ f ∗ , Λ f respectively. Note also that,as Aut p F q is a discrete group, χ _ p a q “ χ p a q “ r for a P τ p F q . So, recallingthat µ N F { τ p F q » W E , we have induced group morphisms χ : W E Ñ GL p r, Q q , χ _ : W E Ñ GL p r, Q q , which we denote by the same symbol for simplicity.Let u , U be a N M F –valued higher gauge transformation with associated toricgauge transformation ` e f α ˘ α P Π ` , p U α q α P Π ` and discrete factor a as defined by eqs.(5.2.13), (5.2.14), (5.2.16). If ω , Ω is a higher gauge field and ω u,U , Ω u,U is thegauge transformed gauge field, the toric higher gauge fields p ω α q α P Π ` , p Ω α q α P Π ` and p ω u,U α q α P Π ` , p Ω u,U α q α P Π ` associated to ω , Ω and ω u,U , Ω u,U by Abelianprojection according to eqs. (5.2.8), (5.2.9) are related as ω u,U α “ ř β P Π ` χ βα p a q ω p e f q , p U q β , (5.2.25) Ω u,U α “ ř β P Π ` χ _ αβ p a ´ q Ω p e f q , p U q β , (5.2.26)where in accordance with eqs. (5.2.5), (5.2.6) p ω p e f q , p U q α q α P Π ` , p Ω p e f q , p U q α q α P Π ` is the toric gauge transform of p ω α q α P Π ` , p Ω α q α P Π ` . Correspondingly, if ω , Ω is an Abelian projected higher gauge field and ω u,U , Ω u,U is again the gauge102ransformed gauge field, then ω u,U , Ω u,U is also Abelian projected and the toricgauge fields p ω α q α P Π ` , p Ω α q α P Π ` and p ω u,U α q α P Π ` , p Ω u,U α q α P Π ` underlying ω , Ω and ω u,U , Ω u,U in eqs. (5.2.10), (5.2.11) are again related as in (5.2.25), (5.2.26).Hence, the toric gauge transformation ` e f α ˘ α P Π ` , p U α q α P Π ` induced by thegauge transformation u , U does not exhaust its action. There is a residual finitediscrete W E action. This constitutes an extra Weyl group symmetry of the toricCS action (5.2.3) for Abelian projected gauge fields ω , Ω in addition to the torichigher gauge symmetry (for B M “ H ).103 Appendixes
The following appendixes collect basic results and identities used repeatedly inthe main body of the paper and provided also the proofs of a few basic statementsrelevant in our analysis, which are original to the best of out knowledge.
A.1
Basic definitions and identities of crossed module theory
In this appendix, we collect a number of basic definitions and relations which areassumed and used throughout the main text of the paper. This will also allowus to set our notation. A part of this material is fairly standard [49], the rest isoriginal to the best of our knowledge.
Lie group crossed modules and module morphisms
A Lie group crossed module M consists of two Lie groups E and G together withLie group morphisms τ : E Ñ G and µ : G Ñ Aut p E q such that τ p µ p a, A qq “ aτ p A q a ´ , (A.1.1) µ p τ p A q , B q “ ABA ´ (A.1.2)for a P G , A, B P E , where here and below we view µ : G ˆ E Ñ E for con-venience. τ and µ are called the target and action maps and (A.1.1), (A.1.2)are called equivariance and Peiffer properties, respectively. As a rule, we write M “ p E , G , τ, µ q to specify the crossed module through its constituent data.A morphism β : M Ñ M of Lie group crossed modules consists of two Liegroup morphisms φ : G Ñ G and Φ : E Ñ E with the property that τ p Φ p A qq “ φ p τ p A qq . (A.1.3) Φ p µ p a, A qq “ µ p φ p a q , Φ p A qq (A.1.4)for a P G , A P E . The morphism β is an isomorphism precisely when Φ , φ areboth isomorphisms. We normally write β : M Ñ M “ p Φ, φ q to indicate theconstituent morphisms of the crossed module morphism.There are obvious notions of direct product M ˆ M of two Lie group crossed104odules M , M and direct product β ˆ β of two Lie group crossed modulemorphisms β , β consisting in taking the direct product of the correspondingconstituent data in the Lie group category.Lie group crossed modules and morphisms thereof with the direct productoperation constitute a monoidal category. Lie group crossed submodules
Let M , M be Lie group crossed modules. M is a submodule of M if E , G are Liesubgroups of E , G and τ , µ are restrictions of τ , µ , respectively or, equivalently, ifthere are inclusion Lie group morphisms E Ď E , G Ď G which are the componentsof an inclusion Lie group crossed module morphism M Ď M .Let M , M be crossed submodules of a Lie group crossed module M with M asubmodule of M . M is said to normalize M if the following conditions are met.For a P G , b P G , one has aba ´ P G . For a P G , B P E , one has µ p a, B q P E .Finally, for b P G , A P E , one has µ p b, A q A ´ P E .If M normalize M , it is possible to define the quotient crossed module M { M .By the condition listed in the previous paragraph, G , E are normal Lie subgroupsof G , E respectively, making it possible to construct the quotient Lie groups G { G , E { E . Then, M { M “ p E { E , G { G , τ M , µ M q , where τ M , µ M are thestructures maps defined by τ M p A E q “ τ p A q G , (A.1.5) µ M p a G , A E q “ µ p a, A q E (A.1.6)for a P G , A P E . It can be verified that τ M , µ M are well defined and obeyrelations (A.1.1), (A.1.2).Just as the notions of Lie algebra and algebra morphism and Lie subalgebra arethe infinitesimal counterpart of those of Lie group and group morphism and Liesubgroup, so the concepts of Lie algebra crossed module and module morphismand Lie algebra crossed submodule are the infinitesimal counterpart of those of Liegroup crossed module and module morphism and Lie group crossed submodule. Lie algebra crossed modules and module morphisms
105 Lie algebra crossed module m consists of two Lie algebras e and g together withLie algebra morphisms t : e Ñ g and m : g Ñ Der p e q such that t p m p u, U qq “ r u, t p U qs , (A.1.7) m p t p U q , V q “ r U, V s (A.1.8)for u P g , U, V P e , where here and below we view m : g ˆ e Ñ e for convenience. t and m are called the target and action maps and (A.1.7), (A.1.8) are calledequivariance and Peiffer properties, respectively, in analogy to the group case.Again, we write m “ p e , g , t, m q to identify the crossed module through its definingelements.A morphism p : m Ñ m of Lie algebra crossed modules consists of two Liealgebra morphisms H : e Ñ e and h : g Ñ g with the property that t p H p U qq “ h p t p U qq . (A.1.9) H p m p u, U qq “ m p h p u q , H p U qq . (A.1.10)for u P g , U P e . The morphism p is said to be an isomorphism if and only if H , h are both isomorphisms. Again, we write as a rule p : m Ñ m “ p H, h q tospecify the crossed module morphism by means of its defining morphisms.Similarly to the Lie group case, there are obvious notions of direct sum m ‘ m of two Lie algebra crossed modules m , m and direct sum p ‘ p of two Liealgebra crossed module morphisms p , p consisting in taking the direct sum ofthe corresponding constituent data in the Lie algebra category.Lie algebra crossed modules and morphisms thereof with the direct sum op-eration constitute a monoidal category. Lie algebra crossed submodules
Let m , m be Lie algebra crossed modules. m is a submodule of m if e , g are Lie subalgebras of e , g and t , m are restrictions of t , m , respectively or,equivalently, if there are inclusion Lie algebra morphisms e Ď e , g Ď g which arethe components of an inclusion Lie algebra crossed module morphism m Ď m .Let m , m be crossed submodules of a Lie algebra crossed module m with m
106 submodule of m . m is said to normalize m if the following conditions are met.For u P g , v P g , one has r u, v s P g . For u P g , V P e , one has m p u, V q P e .Finally, for v P g , U P e , one has m p v, U q P e .If m normalize m , it is possible to define the quotient crossed module m { m .By the condition listed in the previous paragraph, g , e are Lie ideals of g , e respectively, making it possible to construct the quotient Lie algebras g { g , e { e .Then, m { m “ p e { e , g { g , t m , m m q , where t m , m m are the structures maps t m p U ` e q “ t p U q ` g , (A.1.11) m m p u ` g , U ` e q “ m p u, U q ` e (A.1.12)for u P g , U P e . It can be verified that t m , m m are well defined and obeyrelations (A.1.1), (A.1.2). A.2
Lie differentiation of crossed modules
From what shown in app.A.1, it is apparent that the Lie algebra crossed modulecategory is the infinitesimal counterpart of the Lie group crossed module one. Asit might be expected, they are related by Lie differentiation.As a convention, whenever a Lie group theoretic structure and S and a Liealgebra theoretic structure s denoted by the same letter appear in a given context,it is tacitly assumed that s is yielded by S via Lie differentiation, unless otherwisestated. Lie differentiation
Let M “ p E , G , τ, µ q be a Lie group crossed module. With the structure map τ : E Ñ G , there is associated its Lie differential τ : e Ñ g . Likewise, with thestructure map µ : G ˆ E Ñ E , there are associated three distinct Lie differentials,namely µ : G ˆ e Ñ e , µ : g ˆ E Ñ e and µ : g ˆ e Ñ e . Since τ is a Lie groupmorphism, τ is a Lie algebra morphism. Similarly, since µ encodes a Lie groupmorphism µ : G Ñ Aut p E q , µ and µ encode respectively a Lie group morphism µ : G Ñ Aut p e q and Lie algebra morphism µ : g Ñ Der p e q . The interpretationof µ is less obvious: as it turns out, µ : g Ñ C Ad1 p E , e q is a linear morphism of g –cocycles of E on e . The precise definitionand main properties of these objects are provided below.Let β : M Ñ M “ p Φ, φ q be a Lie group crossed module morphism. The Liedifferentials of the Lie group morphisms Φ : E Ñ E , φ : G Ñ G are then Liealgebra morphisms Φ : e Ñ e , φ : g Ñ g .Given a Lie group crossed module M “ p E , G , τ, µ q , the data m “ p e , g , τ, µ q define a Lie algebra crossed module. m is in this way associated with M muchas a Lie algebra is associated with a Lie group. Similarly, given a Lie groupcrossed module morphism β : M Ñ M “ p Φ, φ q , the data β : m Ñ m “ p Φ, φ q define a Lie algebra crossed module morphism. Again, β is associated with β just as a Lie algebra morphism is associated with a Lie group morphism. TheLie algebra crossed module associated with the direct product M ˆ M of twoLie group crossed modules M , M is the direct sum m ‘ m of the associatedLie algebra crossed modules m , m . Similarly, the Lie algebra crossed modulemorphism associated with the direct product β ˆ β of two Lie group crossedmodule morphisms β , β is the direct sum β ‘ β of the associated Lie algebracrossed module morphisms β , β .The map that associates with each Lie group crossed module M its Lie algebracrossed module m and with each Lie group crossed module morphism β : M Ñ M its Lie algebra crossed module morphism β : m Ñ m is a functor of the Lie groupinto the Lie algebra crossed module monoidal category.Let M , M , be Lie group crossed modules with associated Lie algebra crossedmodules m , m . If M is a crossed submodule of M , then m is a crossed submoduleof m . Next, let M be a Lie group crossed module with Lie algebra crossedmodule m . If M is a crossed submodule of M normalizing M , then m is crossedsubmodule of m normalizing m . Moreover, the Lie algebra crossed module of thequotient crossed module M { M is precisely m { m . Basic Lie theoretic identities
The relevant differentiated structure mappings τ : e Ñ g , µ : G ˆ e Ñ e , µ : g ˆ E Ñ e and µ : g ˆ e Ñ e of a Lie group crossed module M “ p E , G , τ, µ q whichwe introduced above satisfy a host of identities often used in detailed calculations108nd analyses. µ obeys the following algebraic identities: µ p ab, X q “ µ p a, µ p b, X qq , (A.2.1) µ p a, µ p x, X qq “ µ p Ad a p x q , µ p a, X qq , (A.2.2)where a, b P G , x P g , X P e . µ in turn satisfies the following relations: τ p µ p x, A qq “ x ´ Ad τ p A qp x q , (A.2.3) µ p τ p X q , A q “ X ´ Ad A p X q , (A.2.4) µ pr x, y s , A q “ µ p x, µ p y, A qq ´ µ p y, µ p x, A qq ´ r µ p x, A q , µ p y, A qs , (A.2.5) µ p x, AB q “ µ p x, A q ` Ad A p µ p x, B qq , (A.2.6) µ p a, µ p x, A qq “ µ p Ad a p x q , µ p a, A qq , (A.2.7)where a, b P G , A, B P E , x, y P g , X P e .The following variational identities hold: δµ p a, A q µ p a, A q ´ “ µ p a, µ p a ´ δa, A q ` δAA ´ q , (A.2.8) δµ p a, X q “ µ p a, µ p a ´ δa, X q ` δX q , (A.2.9) δ µ p x, A q “ µ p δx, A q ` µ p x, δAA ´ q ´ r µ p x, A q , δAA ´ s , (A.2.10)where a P G , A P E , x P g , X P e . A.3
Crossed modules with invariant pairing
In this appendix, we provide the definition and main properties of Lie group andalgebra crossed modules with invariant pairing used in the main text. We alsoprovide details on isotropic crossed submodules.
Lie algebra crossed modules with invariant pairing
A Lie algebra crossed module with invariant pairing is a Lie algebra crossedmodule m “ p e , g , t, m q endowed with a non singular bilinear map x¨ , ¨y : g ˆ e Ñ R enjoying the properties that x ad z p x q , X y ` x x, m p z, X qy “ (A.3.1)109or z, x P g , X P e and that x t p X q , Y y “ x t p Y q , X y (A.3.2)for X, Y P e . The non singularity of x¨ , ¨y implies that m is balanced, dim e “ dim g .A morphism p : m Ñ m “ p H, h q of Lie algebra crossed modules with invari-ant pairing is a crossed module morphism that respects the pairing, that is x h p x q , H p X qy “ x x, X y (A.3.3)for x P g , X P e .If m , m are Lie algebra crossed modules with invariant pairing, then theirdirect sum m “ m ‘ m is a crossed module with the invariant pairing x x ‘ x , X ‘ X y “ x x , X y ` x x , X y , (A.3.4)where x P g , X P e , x P g , X P e .Lie algebra crossed modules with invariant pairing and morphisms thereofwith the direct sum operation constitute a monoidal category that is a sub-category of the monoidal category of Lie algebra crossed modules and modulemorphisms. Lie group crossed modules with invariant pairing
A Lie group crossed module with invariant pairing is a crossed module M “p E , G , τ, µ q such that the associated Lie algebra crossed module m “ p e , g , τ, µ q (cf. app. A.2) is a crossed module with invariant pairing x¨ , ¨y satisfying x Ad a p x q , µ p a, X qy “ x x, X y (A.3.5)for a P G , x P g , X P e . Note that (A.3.5) implies (A.3.1) with m “ µ throughLie differentiation with respect to a . Again, the non singularity of x¨ , ¨y impliesthat M is balanced, dim E “ dim G .A morphism β : M Ñ M of Lie group crossed modules with invariant pairing isa morphism of the underlying crossed modules such that the induced Lie algebracrossed module morphism β : m Ñ m is a morphism of crossed modules with110nvariant pairing as defined earlier (cf. eq. (A.3.3)).If M , M are Lie group crossed modules with invariant pairing, then theirdirect product M “ M ˆ M is a crossed module with the invariant pairing, sincethe associated Lie algebra crossed module m “ m ‘ m is endowed with theinvariant pairing (A.3.4) satisfying (A.3.5).Lie group crossed modules with invariant pairing and morphisms thereof withthe direct product operation constitute a monoidal category that is a subcategoryof the monoidal category of Lie group crossed modules and module morphisms. Fine crossed modules
Let M “ p E , G , τ, µ q be a Lie group crossed module with invariant pairing x¨ , ¨y . M is said to be fine if for x, y P g and A P E one has x x, µ p y, A qy “ x y, µ p x, A ´ qy . (A.3.6)This property is a sense dual to (A.3.5). M is fine under mild assumptions on theLie group E . In particular, M is fine when E is connected and also when E is notconnected in the connected component of the identity of E and in any connectedcomponent of E where it holds for at least one element. M is fine also when τ isinvertible with no restrictions on E . A.4
Proof of the decomposition theorem
In this appendix, we provide a sketch of the proof of the decomposition theorem(2.2.11) of a Lie algebra crossed module with invariant pairing m satisfying thehypothesis (2.2.10). The theorem states the isomorphism m » C m ‘ R m , (A.4.1)where C m , R m are the thee core and residue of m , the Lie algebra crossed mod-ules with invariant pairing defined by (2.2.4), (2.2.5), (2.2.8) and (2.2.6), (2.2.7),(2.2.9), respectively.By the assumption (2.2.10), we have the Lie algebra direct sum decomposition g “ ran t ‘ h , (A.4.2)111here h is an ideal of g with ran t X h “ . The projector π : g Ñ ran t associatedwith the decomposition is a Lie algebra morphism.The duality pairing of g and e established by the invariant pairing x¨ , ¨y entailsthe Lie algebra direct sum decomposition e “ ker t ‘ h K , (A.4.3)where h K , the orthogonal complement of h with respect to the pairing, an idealof e with ker t X h K “ . The projector Π : g Ñ ker t associated with thedecomposition is again a Lie algebra morphism.We define mappings h : g Ñ ran t ‘ p g { ran t q and H : e Ñ p e { ker t q ‘ ker t by h p x q “ π p x q ‘ pp ´ π qp x q ` ran t q , (A.4.4) H p X q “ pp ´ Π qp X q ` ker t q ‘ Π p X q (A.4.5)with x P g , X P e . h , H are the components of Lie of a Lie algebra crossedmodule isomorphism p : m Ñ C m ‘ R m Indeed, as it is straightforward to verify, h , H are Lie algebra isomorphisms. Further, by virtue of the relations, π p t p X qq “ t pp ´ Π qp X qq , (A.4.6) p ´ π qp t p X qq “ , (A.4.7) Π p m p x, X qq “ m pp ´ π qp x q , Π p X qq , (A.4.8) p ´ Π qp m p x, X qq “ m p π p x q , p ´ Π qp X qq , (A.4.9) h , H obey the required conditions (A.1.9), (A.1.10). Property (A.3.3) is imme-diately checked. A.5
Basic results of Cartan–Weyl theory
In this appendix, we review the basic notions of the Cartan–Weyl theory of liealgebras used in sect. 5. A standard reference is [86].Let E be a compact semisimple Lie group and F a maximal torus of E . Then, e is a compact semisimple Lie algebra and f is a maximal toroidal Lie subalgebraof e . The integer r “ dim f is the rank of E .112he structure of the Lie algebra e is best analyzed by complexification. Welet e C “ C b e and f C “ C b f , the Cartan subalgebra of e C . Then, e C has thevector space direct sum decomposition e C “ À α P ∆ e α ‘ f C , (A.5.1)where ∆ Ă f C ∗ is set of roots of e C , the eigenvalues of ad f C , and the e α are theroot subspaces, the associated eigenspaces of ad f C . The e α can be shown to beall 1–dimensional. e C admits an invariant symmetric non singular bilinear pairing x¨ , ¨y K uniqueup to normalization in each simple component of e C . x¨ , ¨y K restricts to a nonsingular pairing on f C . Through x¨ , ¨y K , each element κ P f C ∗ is then identifiedwith a unique generator H κ P f C . An symmetric non singular bilinear pairing x¨ , ¨y K on f C ∗ , defined by x κ, λ y K “ x H κ , H λ y K for κ, λ P f C ∗ , is so induced.The roots α P ∆ are therefore identified with generators H α of f C . With the α ,there are further associated the coroots α _ P f C given by α _ “ α {x α, α y K , whichin turns are identified with generators H α _ of f C . It can be shown that H α , H α _ P i f . Unlike the H α , however, the H α _ do not depend on the normalization of x¨ , ¨y K .Besides these, there exist normalized generators X α P e α such that the basic Liebrackets of e C read as r H α _ , X ˘ α s “ ˘ X ˘ α , r X α , X ´ α s “ H α _ (A.5.2)The root set ∆ is spanned over Z by a set of positive simple roots Π ` Ă ∆ .Note that | Π ` | “ r . The Cartan matrix C αβ “ x α, β y K x α, α y K “ x α _ , β _ y K x β _ , β _ y K , α, β P Π ` (A.5.3)is independent from the normalization of the invariant pairing x¨ , ¨y K . C is aninvertible r ˆ r matrix with integer entries in the range ´ , ´ , ´ , , completelycodifying e as a Lie algebra. In particular, C determines essentially all the ratiosof the normalization dependent simple root inverse half lengths squares κ α “ x α, α y K “ x α _ , α _ y K , α P Π ` . (A.5.4)113or non orthogonal roots α, β P Π ` , κ α { κ β “ C αβ { C αβ “ , , , { , { depend-ing on cases.The real subspace i f Ă f C is characterized by six lattices:• the root lattice Λ r f , the lattice of i f generated by H α with α P Π ` ,• the coroot lattice Λ cr f , the lattice of i f generated by H α _ with α P Π ` ,• the integral lattice Λ f , the set of all X P i f such that e πiX “ E ,and their dual lattices with respect to x¨ , ¨y K :• the weight lattice Λ w f “ Λ cr f ∗ ,• the coweight lattice Λ cw f “ Λ r f ∗ ,• the dual integral lattice Λ f ∗ .The weight and coweight lattices can also be defined by through generators. Thesimple roots α P Π ` can be paired with the fundamental weights and coweights α ∗ , α ∗ _ P f C ∗ defined by x α ∗ , β _ y K “ x α ∗ _ , β y K “ δ αβ . (A.5.5)The weight and coweight lattices Λ w f , Λ cw f are then the lattices of i f generated by H α ∗ , H α ∗ _ with α P Π ` , respectively. Explicitly, we have H α ∗ “ ř β P Π ` C ´ βα H β , H α ∗ _ “ ř β P Π ` C ´ αβ H β _ . It is known that Λ cr f Ď Λ f Ď Λ cw f and Λ r f Ď Λ f ∗ Ď Λ w f .While the lattices Λ cr f , Λ cw f Λ r f , Λ w f depend only on the Lie algebra e and aretherefore the same for all Lie groups E which share e as their Lie algebra, thelattices Λ f , Λ f ∗ do depend on E . In this regard, one has Z p E q » Λ cw f { Λ f » Λ f ∗ { Λ r f and π p E q » Λ f { Λ cr f » Λ w f { Λ f ∗ . 114cknowledgements. The author thanks the organizers of the Erwin SchroedingerInstitute Program “Higher Structures and Field Theory” whose seminars anddiscussion sessions have been source of much inspiration for him. He thanks inparticular Thomas Strobl for inviting him to the Program and for the interestshown in his work. The author further acknowledges financial support from INFNResearch Agency under the provisions of the agreement between University ofBologna and INFN. 115 eferences [1] S. W. MacDowell and F. Mansouri, Unified geometric theory of gravity and super-gravity ,Phys. Rev. Lett. (1977) 739, Erratum: Phys. Rev. Lett. (1977) 1376.[2] J. F. Plebanski, On the separation of Einsteinian substructures ,J. Math. Phys. (1977) 2511.[3] I. Morales, B. Neves, Z. Oporto and O. Piguet Chern–Simons gravity in fourdimensions ,Eur. Phys. J. C (2017) no.2, 87, [ arXiv:1701.03642 [gr-qc] ].[4] S. M. Carroll, G. B. Field and R. Jackiw, Limits on a Lorentz and parity violatingmodification of electrodynamics ,Phys. Rev. D (1990) 1231.[5] D. Grumiller and N. Yunes, How do black holes spin in Chern-Simons modifiedgravity? ,Phys. Rev. D (2008) 044015, [ arXiv:0711.1868 [gr-qc] ].[6] G. Shiu, W. Staessens and F. Ye, Large field inflation from axion mixing ,JHEP (2015) 026, [ arXiv:1503.02965 [hep-th] ].[7] K. Costello,
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