4-d semistrict higher Chern-Simons theory I

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding number. We find that the theory admits two seemingly inequivalent canonical quantizations. The first is manifestly topological, it does not require a choice of any additional structure on the spacial 3-fold. The second, more akin to that of ordinary Chern-Simons theory, involves fixing a CR structure on the latter. Correspondingly, we obtain two sets of semistrict higher WZW Ward identities and we find the explicit expressions of two higher versions of the WZW action. We speculate that the model could be used to define 2-knot invariants of 4-folds.


Introduction
Higher gauge theory is a generalisation of ordinary gauge theory where gauge potentials are forms of degree p ≥ 1 and, correspondingly, their gauge curvatures are forms of degree p + 1 ≥ 2. It is thought to govern the dynamics of higherdimensional extended objects. See ref. [1] for a readable, up-to-date review of this subject and extensive referencing.
The origin of higher gauge theory can be traced back to the inception of supergravity. Higher gauge theory has subsequently found application in string theory in the study of D-and M-branes [2][3][4] as well as loop quantum gravity and, in particular, spin foam models [5,6]. Nowadays, the pursuit of higher gauge theory is motivated especially by its potential to provide a Lagrangian description of the N = (2, 0) superconformal 6-dimensional field theory governing the effective dynamics of M5-branes [7].
From a mathematical perspective, higher gauge theory is intimately related to higher algebraic structures, such as 2-categories, 2-groups [8,37] and strong homotopy Lie or L ∞ algebras [9,10] and higher geometrical structures such as gerbes [11,12]. A state of the art exposition of these matters highlighting their manifold relationships to various physical issues can by found in [13].
Higher gauge theory can be formulated as a categorification of ordinary gauge theory by codifying higher gauge symmetry into algebraic structures arising from the categorification of ordinary Lie groups, weak or coherent Lie 2-groups [14][15][16][17].
This approach has been adopted in a large body of literature which would be impossible to summarise rendering full justice to all contributions. We shall limit ourselves to note that until quite recently most studies on the subject were limited to the case where the structure 2-group is strict. Though every coherent 2-group is categorically equivalent to a strict 2-group, categorical equivalence is not a sufficiently fine notion for gauge theory: it does not translate into any viable form of field theoretic equivalence. The study of higher gauge theory with non strict structure 2-group was first undertaken in the very broad context of ∞-Lie theory in refs. [18][19][20]. An alternative approach to the topic was followed in refs. [21,22].

The scope and the plan of this paper
The present paper is devoted to the study of a model of non strict 4-dimensional higher Chern-Simons gauge theory which, in our hope, may have application in the study of 4-dimensional topology just as the ordinary Chern-Simons theory does in 3 dimensions. This paper employs a version of non strict higher gauge theory, called semistrict, first developed by one of the authors in ref. [23], which we shall outline next.
Consider a gauge theory on a space time manifold M whose symmetry is codified by a Lie algebra g. (We shall neglect global issues here.) A connection is then a g-valued 1-form ω ∈ Ω 1 (M, g). A gauge transformation is map γ ∈ Map(M, G), where G is a Lie group integrating g. The gauge transformed connection g ω is then given by where g = Ad γ and σ g = γ −1 dγ. Note now that g ∈ Map(M, Aut(g)) and σ g ∈ Ω 1 (M, g) and that dσ g + 1 2 [σ g , σ g ] = 0, (1.1.2a) In the above relations, any reference to the group G has disappeared: everything is expressed in terms of g-valued forms and Aut(g)-valued maps. In this way, we have dodged the technical task of integrating g to G. In ordinary gauge theory, this problem is not particularly difficult, but its counterpart in semistrict higher gauge theory instead is. The basic proposal of ref. [23] is extending this formulation to a higher gauge theory on M whose symmetry is codified by a semistrict Lie 2-algebra v. Semistrict higher connections and gauge transformations are defined in terms of v-valued forms and Aut(v)-valued maps. An exposition of this framework with new results not originally given in [23] is provided in sect. 2.
The gauge theoretic framework outlined in the previous paragraph has limitations: it can only work in perturbative Lagrangian field theory. Its adequacy for the analysis of parallel transport, a basic problem in gauge theory, is not clear. Further, as it is well-known, relevant non perturbative effects are related to the center Z(G) of G, information about which is lost in Aut(g). It is nevertheless computationally efficient and directly generalisable to semistrict higher gauge theory.
Chern-Simons theory is a 3-dimensional topological field theory of the Schwarz type. (See. ref. [24] for a recent review of the model and exhaustive referencing).
It was first formulated in 1989 by E. Witten in ref. [25]. Witten succeeded to show that many topological knot and link invariants discovered by topologists earlier, such as the HOMFLY and Jones polynomials, could be obtained as correlation function of Wilson loop operators in Chern-Simons theory. He also proved that the Chern-Simons partition function is a topological invariant of the underlying 3-manifold. Multiple connections with the 2-dimensional WZW model were also found [26]. In 1992, Witten also showed that Chern-Simons theory is intimately related to the topological sigma models of both A and B types [27]. This paper is a modest attempt to extend Chern-Simons theory to 4 dimensions in the framework of semistrict higher gauge theory with the hope of achieving a field theoretic expression of 2-knot and link invariants of 4-manifolds and unveiling 3-dimensional higher analogs of WZW theory. In sect. 3, we describe a higher 4-dimensional Chern-Simons model whose symmetry is encoded in a balanced semistrict Lie 2-algebra equipped with a invariant non singular bilinear form. We analyse in detail its gauge invariance and perform its canonical quantization.
Finally in the appendices, we collect various results on 2-groups and Lie 2algebras and their automorphisms which are scattered in the literature in order to define our terminology and notation and for reference throughout the text.

Outlook and open problems
Our study is divided roughly in two parts.
The first part of the paper is devoted to the analysis of the gauge invariance of higher Chern-Simons theory. We find that, analogously to ordinary Chern-Simons theory, the higher Chern-Simons action is invariant under a higher gauge transformation up to a higher winding number only. Full gauge invariance of the quantum theory requires that the winding number be quantized in appropriate units. In all the examples which we have been able to work out in detail, the winding number actually vanishes, but we cannot prove its quantization in general and we are forced to assume it as a working hypothesis. This is a first aspect of the theory that requires further investigation.
The second part of the paper deals with quantization. Several approaches to the problem of quantization are possible in principle. Perturbative quantization based on a straightforward extension of Lorenz gauge fixing involves the choice of a background metric on the base manifold as well as the introduction of Faddeev-Popov ghost and ghost for ghost fields. In the presence of a metric we cannot maintain gauge covariance without resorting to gauge rectifiers whose existence and interpretation is still problematic [23]. We are left with canonical quantization. We find that the theory admits two apparently inequivalent canonical quantizations. We obtain correspondingly two sets of higher WZW Ward identities and we find the explicit expressions of two higher versions of the gauged WZW action.
The canonical quantization of the first kind is manifestly topological in that it does not require a choice of any additional structure on the spacial 3-fold. That of the second kind involves fixing a CR structure on the latter. This is more akin to ordinary Chern-Simons theory's canonical quantization. CR spaces are in fact in many ways the closest 3-dimensional analog of Riemann surfaces. The unitary equivalence of the quantization associated with distinct CR structures is an open problem necessitating a non trivial extension of the analysis of ref. [28].
Furthermore, the relationship between the the topological and CR quantizations remains elusive.
It is necessary to clarify a point on the higher WZW actions emerging in the process of canonically quantizing our higher Chern-Simons theory. They encode the gauge covariance of the relevant wave functionals and, so, are determined by the Ward identities these obey and by a cocycle conditions extending the familiar Polyakov-Wiegmann relation. Presently, however, we have no evidence that they are related to some kind of 3-dimensional sigma model as the ordinary gauged WZW action, although this remains a distinct possibility. In this respect it may be more useful to consider the restriction of the higher Chern-Simons action to flat connection configurations expressed as gauge transformation of the trivial connection on the same lines as [26]. This is left to future work.
The solution of the questions raised in the preceding paragraphs requires a more fundamental theory of higher gauge transformation than that employed in the present paper. Until recently, this was available only for the strict case [16,17].
Promising new results in this direction can be found in ref. [22] .

Semistrict higher gauge theory
In this section, we shall illustrate the local aspects of semistrict higher gauge theory. Since we aim to the construction of higher Chern-Simons gauge theory as a higher counterpart of ordinary one, we neglect bundle theoretic global issues altogether. Part of the material presented here has been already expounded in [23], which the reader is referred to for further details and motivation, but also new results are given.
Before proceeding further, it is useful to recall the general philosophy underlying our approach, which was already alluded to in the introduction. In an ordinary gauge theory with symmetry Lie algebra g, fields are g-valued forms and gauge transformations of fields are expressed in terms of Aut(g)-valued maps and g-valued forms. The theory, at least in its local aspects, can be formulated to a significant extent relying on the Lie algebra g only. In the same way, in our formulation, in a semistrict higher gauge theory with symmetry Lie 2-algebra v, the fields are v-valued forms and gauge transformations of fields are expressed in terms of Aut(v)-valued maps and v-valued forms. The theory, then, is formulated in terms of the Lie 2-algebra v only analogously to the ordinary case.
We present the semistrict theory characterising it as much as possible as a higher version of the ordinary one.
Just as the gauge symmetry of ordinary gauge theory organizes in an infinite dimensional group Gau(N, g), the gauge transformation group, that of semistrict higher gauge theory organizes as an infinite dimensional strict 2-group Gau(N, v), the higher gauge transformation 2-group. The 1-and 2-cells of Gau(N, v) correspond respectively to gauge and gauge for gauge transformations. The notion of gauge for gauge transformation we adopt is however more general than that customarily found in the literature encompassing also transformations of gauge transformations which do not necessarily leave the action on higher gauge con-nections invariant unless further restrictions are imposed.
The basic notions of Lie 2-group and 2-algebra theory are recalled in the appendices, where our notation is also defined. All algebraic structures considered below are real and all fields are smooth, unless otherwise stated.

Lie 2-algebra gauge theory, local aspects
In ordinary as well as higher gauge theory, fields propagate on a fixed d-fold M. To study the local aspects of the theory, we assume that M is diffeomorphic to R d . On such an M, a field of bidegree (m, n) is any element of the space , where E is some vector space.

Ordinary gauge theory
In an ordinary gauge theory with structure Lie algebra g (cf. app. A.3), fields are generally drawn from the spaces Ω m (M, g[n]). The main field of the gauge theory is the connection ω, which is a bidegree (1, 0) field. ω is characterized by its curvature f , the bidegree (2, 0) field given by The connection ω is flat if the curvature f = 0.
The covariant derivative of a field φ is given by the well-known expression and satisfies the standard Ricci identity The Bianchi identity (2.1.2) obeyed by f can be written compactly as

Semistrict higher gauge theory
In semistrict higher gauge theory with structure Lie 2-algebra v (cf. app. analogous to the Bianchi identity (2.1.2) of ordinary gauge theory. The connection (ω, Ω ω ) is flat if the curvature components f = 0 and F f = 0.
Let (φ, Φ φ ) be a field doublet of bidegree (p, q). The covariant derivative doublet of (φ, Φ φ ) is the field doublet (Dφ, DΦ φ ) of bidegree (p + 1, q) given by 1 The sign (−1) p+q is conventional, since the relative sign of φ, Φ φ cannot be fixed in any natural manner. From (2.1.8), we deduce easily the appropriate version of the Ricci identities, The explicit appearance of the connection component ω in the right hand side of (2.1.9b) is a consequence of the presence of a term quadratic in ω in (2.1.8b).
The above definition of covariant differentiation is yielded by the request that the Bianchi identities (2.1.7) be expressed as the vanishing of the covariant deriva- as it is the case for the Bianchi identity of ordinary gauge theory, eq. (2.1.5).

The 2-group of higher gauge transformations
Just as gauge transformations play a basic role in ordinary gauge theory, higher gauge transformations play a similar basic role in higher gauge theory. In this section, following the approach of ref. [23] already outlined in the introduction, we shall review the main properties of higher gauge transformations highlighting the way they generalize ordinary ones. To this end, we shall slightly extend the notion of the latter.

Ordinary gauge transformations
In ordinary gauge theory, symmetry is codified in a Lie algebra g. A gauge transformation is a pair of: 1. a map g ∈ Map(M, Aut(g)) (cf. app. A.6), 2. a flat connection σ g , related to g through the condition . We shall denote the gauge transformation by (g, σ g ) or simply by g, having in mind that now σ g is not determined by g but participates with g in the transformation. Further, we shall denote by Gau(M, g) the set of all such extended gauge transformations.
The definition of gauge transformation given above is more general than the one commonly quoted in the literature. If G is a Lie group exponentiating g and γ ∈ Map(M, G), then the pair (Ad γ, γ −1 dγ) is a gauge transformation in the sense just defined. However, not every gauge transformation (g, σ g ) is of this form.

Ordinary gauge transformation group
Gau(M, g) is an infinite dimensional Lie group, the (extended) gauge transformation group of the theory. The composition and inversion and the unit of Gau(M, g) are defined by the relations

Higher gauge transformations
In semistrict higher gauge theory, symmetry is codified in a Lie 2-algebra v.
A higher 1-gauge transformation consists of the following data.
(cf. app. A.3) g, σ g , Σ g , τ g are required to satisfy a number of relations. If g = (g 0 , g 1 , g 2 ) (cf. app. A.6), then one has hold. In the following, we are going to denote a 1-gauge transformation such as the above as (g, σ g , Σ g , τ g ) or simply as g. Again, in so doing, we are not implying that σ g , Σ g , τ g are determined by g, but only that they are the partners of g in the gauge transformation. We shall denote the set of all higher 1-gauge The above definition of higher gauge transformation is at first glance a bit mysterious and needs to be justified. It is the minimal extension of the ordinary notion to the higher setting. When the Lie algebra g is replaced by the Lie 2-algebra v, g turns from an Aut(g)-valued map into Aut(v)-valued one and the flat connection σ g gets promoted to a flat connection doublet (σ g , Σ g ), as is natural. This leads immediately to eqs. (2.2.4). The reason for introducing the further datum τ g satisfying (2.2.6) is not as evident and must be explained.
For an ordinary gauge transformation (g, σ g ) the Maurer-Cartan equation it is sufficient that σ g is flat. Showing this involves crucially the use of the Jacobi identity of the Lie algebra g. When we pass to a Lie 2-algebra v, that identity is no longer available. For this reason, we must introduce the new datum τ g and modify the naive relations g −1 0 dg 0 (π) = [σ g , π], g −1 1 dg 1 (Π) = [σ g , Π], as indicated in (2.2.6a), (2.2.6b). In fact, if τ g vanished, for the Maurer-Cartan equations the flatness relations (2.2.4) would not suffice by themselves: one would need to add an extra purely algebraic condition on the flat connection doublet (σ g , Σ g ), ] = 0, which does not fit naturally into our higher gauge theoretic set-up. Once we allow for τ g , however, this condition becomes a differential consistency relation satisfied by τ g , viz (2.2.5). This latter deserves therefore to be called "2-Maurer-Cartan equation".
In semistrict higher gauge theory, one has in addition gauge for gauge symmetry. For any two 1-gauge transformations g, h ∈ Gau 1 (M, v), a higher 2-gauge transformation from g to h consists of the following data. F , A F are required to satisfy the relations, In the following, we are going to denote a 2-gauge transformation like the above as (F, A F ), meaning that A F is the partner of F in the transformation, or simply as F . We shall also write F : g ⇒ h to indicate its source and target. We shall denote the set of all 2-gauge transformations F : g ⇒ h by Gau 2 (M, v)(g, h) and that of all 2-gauge transformations F by Gau 2 (M, v).
The above definition of 2-gauge transformation is again a bit puzzling and needs to be justified. Suppose we ask what the most natural class of deformations of a 1-gauge transformation (g, σ g , Σ g , τ g ) which preserve its being such and can be expressed in terms of elementary fields is. As g, h ∈ Map(M, Aut 1 (v)), it is reasonable to demand that g, h are the source and the target of some F ∈ Map(M, Aut 2 (v))(g, h). Granting this, the only remaining deformational field datum is an element A F ∈ Ω 1 (M, v 1 ) turning σ g into σ h = σ g − ∂A F . We take A F v 1 -rather than v 0 -valued in order to be able to employ it to deform Σ g into

The Lie 2-algebra of infinitesimal higher gauge transformations
In higher gauge theory, as in ordinary gauge theory, many aspects of gauge symmetry are often conveniently studied by switching to the infinitesimal form of gauge transformation.

Ordinary infinitesimal gauge transformations
Consider again an ordinary gauge theory with symmetry Lie algebra g. An where u, v ∈ gau(M, g). In (2.3.3a), the brackets in the right hand side are those of aut(g) thought of as holding pointwise on M (cf. eq. (A.7.2)).

Adjoint type infinitesimal gauge transformations
With any s ∈ Ω 0 (M, g), there is associated an element ad M s ∈ gau(M, g) by the adjoint of s. In (2.3.4a), the adjoint operator in the right hand side is that of g holding pointwise on M (cf. eq. (A.7.3)).

Ordinary gauge transformation exponential map
Infinitesimal gauge transformations can be exponentiated to finite ones. The exponential map exp ⋄ : gau(M, g) → Gau(M, g) is given by where u ∈ gau(M, g). In (2.3.5a), the exponentiation in the right hand side is that of aut(g) thought of as holding pointwise on M.

Higher infinitesimal gauge transformations
Consider next a higher gauge theory with symmetry Lie 2-algebra v. 2. a linearized flat connection doublet (σ u ,Σ u ), u,σ u ,Σ u ,τ u are required to satisfy the relations stemming from (2.2.6) by linearization. If u = (u 0 , u 1 , u 2 ) (cf. app. A.7), then these read In the following, we shall denote the infinitesimal 1-gauge transformation as (u,σ u ,Σ u ,τ u ), indicating as usualσ u ,Σ u ,τ u as the partners of u in the gauge transformation data, or simply as u. We shall denote the set of all infinitesimal The gauge for gauge symmetry of semistrict higher gauge theory also has an infinitesimal version. An infinitesimal higher 2-gauge transformation is a linearized version of a 2-gauge transformation. Expansion around the unit transformation I i to first order shows that it consists of the data 1. a map P ∈ Map(M, aut 1 (v)); 2. an elementȦ P ∈ Ω 1 (M, v 1 ).
There are no further relations these objects must obey. We shall denote the infinitesimal 2-gauge transformation as (P,Ȧ P ), indicatingȦ P as the partner of P in the gauge transformation, or simply as P . We shall denote the set of all infinitesimal higher 2-gauge transformations by gau 1 (M, v).  The strict Lie 2-algebra gau(M, v) can also be described as a differential Lie crossed module. The two underlying Lie algebras are gau 0 (M, v) and gau 1 (M, v).

Higher infinitesimal gauge transformation
The differential Lie crossed module Lie brackets, target map and action are given by the expressions

Adjoint type higher infinitesimal gauge transformations
For any s ∈ Ω 0 (M, v 0 ), an element ad M s ∈ gau 0 (M, v), is defined, the adjoint of s. In ( the adjoints of s, t and S, respectively. In Higher gauge transformation exponential map.
Infinitesimal Lie 2-algebra gauge transformations can be exponentiated to finite ones. The exponential map exp ⋄ :

Orthogonal gauge transformations
In the higher Chern-Simons theory that we are going to construct later, one of the basic datum is an invariant form on the relevant algebra.

Ordinary orthogonal gauge transformations
We consider an ordinary gauge theory with symmetry Lie algebra g equipped with an invariant bilinear symmetric form (·, ·) (cf. app. A.9). A gauge transfor- We shall denote by OGau(M, g) the set of all orthogonal elements g ∈ Gau(M, g).

OGau(M, g) is an infinite dimensional Lie subgroup of the gauge Lie group
Gau(M, g).

Ordinary infinitesimal orthogonal gauge transformations
An infinitesimal gauge transformation (u,σ u ) of gau(M, g) is accordingly orthogonal if u is pointwise orthogonal, We let ogau(M, g) be the set of all orthogonal elements u ∈ gau(M, g). ogau(M, g) is an infinite dimensional Lie subalgebra of the gauge Lie algebra gau(M, g).
ogau(M, g) is also the Lie algebra of the orthogonal gauge Lie group OGau(M, g).

Adjoint type ordinary orthogonal infinitesimal gauge transformations
For s ∈ Ω 0 (M, g), the adjoint type infinitesimal gauge transformation ad M s ∈

Ordinary gauge transformation exponential and orthogonality
The

Higher orthogonal gauge transformations
We consider now a semistrict higher gauge theory having as symmetry algebra a balanced Lie 2-algebra v equipped with an invariant bilinear form (·, ·) (cf. We shall denote by OGau 1 (M, v) the set of all orthogonal elements g ∈ Gau 1 (M, v).
An invariant form (·, ·) can be seen as a special invariant symmetric bilinear form on the direct sum v 0 ⊕ v 1 with non vanishing off-diagonal elements only.
From this perspective, the relationship to the ordinary case is more evident.
Condition (2.4.1) is at first glance a bit mysterious, but it emerges naturally in many contexts and is a necessary condition for orthogonal symmetry invariance in higher Chern-Simons theory.
being two 1-gauge transformations, is said orthogonal if both g, h are orthogonal. .

Higher gauge transformation exponential and orthogonality
The

4-d higher gauge theoretic Chern-Simons theory
In this section, we shall construct and analyse a 4-dimensional semistrict analog of the standard Chern-Simons theory [25]. .Beside providing a potentially interesting example of higher gauge theory, our construction, if it turns out successful, may furnish a basic field theoretic framework for the study of 4-dimensional topology.
Below, we assume tacitly that manifold on which fields are defined is oriented and that the fields satisfy asymptotic or boundary conditions allowing for the convergence of the integration and integration by parts.

The gauge transformation action
In ordinary gauge theory the construction of gauge invariant action functionals requires a prior definition of a gauge transformation action on gauge connections.
In the same way, in semistrict higher gauge theory the construction of higher gauge invariant action functionals is possible upon defining a higher gauge transformation action on connection doublets. This is the topic of this subsection. We follow here the formulation of ref. [23].
In the familiar geometrical formulation of ordinary gauge theory, the basic geometrical datum is a principal G-bundle P on a manifold N. Connections are g-valued 1-forms on P satisfying the so called Ehrensmann conditions. Fields are horizontal and equivariant g-valued forms on P . Gauge transformations are automorphisms of P projecting to the identity id N on N. The gauge transformation action is then defined in terms of the pull-back action of automorphisms on connections and fields. Because of the way we have have formulated the theory of gauge transformation in subsect. 2.2, this type of approach is not immediately extendable to higher gauge theory. We proceed therefore in an alternative way closer in spirit to the physical approach to gauge symmetry.

Gauge transformation action in ordinary gauge theory
In ordinary gauge theory with symmetry Lie algebra g, gauge transformation action is a left action of the gauge transformation group Gau(N, g) on connections ω and fields φ compatible with covariant differentiation (cf. eq. (2.1.3)), in the sense that for any gauge transformation g ∈ Gau(N, g) This requirement essentially determines the gauge transformation action. The gauge transform g ω of the connection ω is Further, the gauge transform g φ of the field φ reads as 3), one has as required that . The gauge transform g f of the curvature f of ω (cf. eq. (2.1.1)) is The gauge variation δ u f of f reads similarly as For the infinitesimal gauge transformation (u,σ u ) = (ad s, ds), (3.1.6), (3.1.7) take the well-known adjoint form.

BRST cohomology in ordinary gauge theory
In standard gauge theory, gauge symmetry is most efficiently analyzed concentrating on infinitesimal gauge transformation of the adjoint type. This is codified by a bidegree (0, 1) ghost field c through the ghost degree 1 infinitesimal gauge transformation w ∈ gau(M, g) [1] given by w = − ad M c andσ w = dc (cf. eqs. (2.3.4)) and is implemented by the odd BRST operator s = δ w . By (3.1.6), then, . We can make s nilpotent by suitably defining the variation sc of c. As by (3.1.8) by a simple computation we can enforce s 2 ω = 0 by setting (ω, Ω ω ) and field doublets (φ, Φ φ ) compatible with covariant differentiation (cf. eqs. (2.1.8)). The straightforward generalization of (3.1.1) to the higher setting, however cannot be made to hold unless a natural restriction on the curvature of the connection doublet is imposed. Through selection by way of selfconsistency, a coherent definition of the gauge transformation action can be worked out [23].
The gauge transform ( g ω, g Ω ω ) of (ω, Ω ω ) is found to be being the bidegree of (φ, Φ φ ). We observe that the action (3.1.15) explicitly depends on and cannot be defined without the prior assignment of a connection doublet. Under the action (3.1.14), (3.1.15), one has now from which it emerges that (3.1.14) holds provided the restriction f = 0 on the curvature of the connection doublet, known as vanishing fake curvature condition in the literature, is imposed.
The gauge variation (δ u f, δ u F f ) of (f, F f ) reads similarly as This in turn induces an action of P on an infinitesimal 1-gauge transformation u ∈ gau 0 (M, v) given by BRST cohomology in semistrict higher gauge theory In semistrict higher gauge theory, analogously to ordinary gauge theory, higher gauge symmetry is most efficiently analyzed concentrating on higher infinitesimal gauge transformation of the adjoint type. Infinitesimal higher 1-gauge transformation is codified by a bidegree (0, 1) ghost field doublet of (c, C c ) through the ghost degree 1 infinitesimal 1-gauge transformation w ∈ gau 0 (M, v) [1] given by   to suitably define also the variation sΓ of Γ . To this end, we note that In conclusion s is nilpotent as desired provided we restrict to connection doublets (ω, Ω ω ) such that f = 0. We note here that the ghost sector here is not pure, as the BRST variation sC c explicitly depends on the connection component ω.
For completeness, we report the BRST variation of curvature doublet (f, F f ) of (ω, Ω ω ), which by ( We expect BRST cohomology to play the same basic role in semistrict higher gauge theory, which it does in ordinary gauge theory.
The higher orthogonal case

Semistrict higher Chern-Simons theory
In this section, we shall describe in detail Lie 2-algebra Chern-Simons theory.
To highlight the way in which the model generalizes ordinary Chern-Simons theory [25], we first review this latter using the gauge theoretic framework developed in the previous section.

Ordinary Chern-Simons Theory
The basic algebraic datum of ordinary Chern-Simons theory is a Lie algebra g equipped with an invariant symmetric form (·, ·) (cf. app. A.9). The topological background is a compact oriented 3-fold N. The field content consists in a g- connection ω on N. The classical action functional reads where the curvature f is given by (2.1.1). The classical field equations are (cf. eq. (2.1.1)) and entail that the connection ω is flat. We shall denote this classical field theory by CS 1 (N, g) or simply CS 1 .
Let X be any manifold. In gauge theory, the de Rham complex Ω * (X) contains the special subcomplex Ω g * (X) formed by those forms that are polynomials in one or more connections ω a and their differentials dω a . In turn, Ω g * (X) includes the subcomplex Ω ginv * (X) of the elements invariant under the action (3.1.2) of the orthogonal gauge transformation group OGau(X, g). For any g-connection ω on X, a form L 1 ∈ Ω 3 (X),

2.3)
formally identical to the Lagrangian density of the CS 1 action is defined. While L 1 ∈ Ω g 3 (X), one has L 1 ∈ Ω ginv 3 (X), since, as is well-known, for g ∈ OGau(X, g). It is a standard result of gauge theory that where C 1 ∈ Ω 4 (X) is the curvature bilinear Clearly, C 1 ∈ Ω g 4 (X). Unlike L 1 , however, C 1 is invariant under OGau(X, g), Thus, C 1 ∈ Ω ginv 4 (X) as well. By for g ∈ OGau (N, g), where the anomaly Q 1 (g) is given by (3.2.11) Q 1 (g) is in fact simply related to the CS 1 functional itself, The independence of Q 1 (g) from the connection ω implies so that the field equations (3.2.2) are gauge invariant. Indeed this follows directly and independently from eq. (3.1.5).
From (3.2.11), the anomaly density is the form q 1 ∈ Ω 3 (N) Note that, since σ g is a connection, q 1 ∈ Ω g 3 (N). From By (2.2.1), the anomaly density q 1 can be cast as This relation indicates that with q 1 there is associated a special Chevalley-Eilenberg cochain χ 1 ∈ CE 3 (g), in CE(g), then q 1 is exact in Ω g * (N). In order the anomaly Q 1 (g) to be non vanishing, so, it is necessary that H CE 3 (g) = 0. This is the case if g is semisimple.
Since Q 1 (g) vanishes for any gauge transformation g continuously connected with the identity i, CS 1 is annihilated by the BRST operator s (cf. eq. (3.1.8)), only if the value of κ 1 is such that κ 1 Q 1 (g) ∈ 2πZ for all g ∈ OGau(N, g). For g = u(n) and (·, ·) = − tr fund ( · · ) this is achieved if where k ∈ Z is an integer called level.

Semistrict higher Chern-Simons theory
After reviewing ordinary Chern-Simons theory, we introduce the semistrict   1.6b)). They imply that the connection doublet (ω, Ω ω ) is flat, analogously to standard CS theory. We shall denote this classical field theory by CS 2 (N, v) or simply CS 2 .
Let X be any manifold. In semistrict gauge theory, in analogy to ordinary gauge theory, the de Rham complex Ω * (X) contains the special subcomplex Ω v * (X) formed by those forms that are polynomials in the components of one or more connection doublets (ω a , Ω a ) and their differentials (dω a , dΩ a ). In turn, Ω v * (X) includes the subcomplex Ω vinv * (X) of the elements invariant under the action (3.1.14) of the orthogonal 1-gauge transformation group OGau 1 (X, v). For any v-connection doublet (ω, Ω ω ) on X, a form L 2 ∈ Ω 4 (X) formally identical to the Lagrangian density of the CS 2 action is defined. While L 2 ∈ Ω v 4 (X), one has L 2 ∈ Ω vinv 4 (X), since for g ∈ OGau 1 (X, v). Similarly to standard gauge theory, one has

2.23)
where C 2 ∈ Ω 5 (X) is the curvature bilinear Clearly, C 2 ∈ Ω v 5 (X). Unlike L 2 , however, C 2 is invariant under OGau 1 (X, v), of C 2 under arbitrary variations variations δω, δΩ ω of ω, Ω ω is given by where the 5-form in the right hand side is OGau 1 (X, v) invariant It follows that, although C 2 is not necessarily exact in Ω vinv * (X), its variation δC 2 always is. This property characterizes then L 2 as the Chern-Simons form of a higher characteristic class [C 2 ] inv .
The CS 2 action is not invariant under the OGau 1 (N, v) action (3.1.14). In fact, from (3.2.22), analogously to ordinary Chern-Simons theory, one has CS 2 ( g ω, g Ω ω ) = CS 2 (ω, Ω ω ) − κ 2 Q 2 (g) (3.2.28) for g ∈ OGau 1 (N, v), where the anomaly Q 2 (g) is given by Q 2 (g) is in fact simply related to the CS 2 action itself, From (3.2.29), the anomaly density is the form q 2 ∈ Ω 4 (N) infinitesimal variation of g necessarily vanishes. In analogy to ordinary Chern-Simons theory, Q 2 (g) represents a higher winding number of the higher gauge transformation g.
By using (2.2.4b), the anomaly density q 2 can be cast as With q 2 there is therefore associated a special higher Chevalley-Eilenberg cochain

2.37)
where k ∈ Z is an integer, which we shall call level as in the ordinary theory.
An important issue of the theory is the classification of the admissible pairs (N, v). We cannot provide any solution of it presently. This is also related to the fact that the integrability of a semistrict Lie 2-algebra v to a semistrict Lie 2group V is not guaranteed in general. In the canonical quantization of semistrict higher Chern-Simons theory carried out in the next subsections, we assume as a working hypothesis that v is a balanced Lie 2-algebra with invariant form such that (N, v) is admissible for a sufficiently ample class of closed 4-folds N.

Canonical quantization
In this section, we shall briefly review the canonical quantization of ordinary Chern-Simons theory and then pass to that of the semistrict higher Chern-Simons theory.
To carry out the canonical quantization of a field theory, we restrict to the case

Ordinary Chern-Simons theory
In the CS 1 (R × M, g) theory, the g-connection ω decomposes as where ω t ∈ Ω h 0 (R × M, g), ω s ∈ Ω h 1 (R × M, g). The curvature f of ω splits as , in similar fashion (cf. eqs. (2.1.1)). ω s is itself a g-connection and f s is the associated curvature. The CS 1 action (3.2.1) reads then as The field equations read then as where D s denotes the covariant differentiation operator associated with the connection ω s defined according to (2.1.3) and ω t is treated as a bidegree (0, 0) field.
The momenta ξ t , ξ s canonically conjugate to ω t , ω s can easily be read off from (3.3.3). In virtue of the linear isomorphisms g ∨ ≃ g induced by the bilinear form Ordinary Chern-Simons theory is therefore constrained. This requires the application of Dirac's quantization algorithm.

3.3) is
We quantize CS 1 (R × M, g) by replacing the classical field ω s satisfying the

Semistrict higher Chern-Simons theory
The canonical quantization of semistrict higher Chern-Simons theory proceeds on the same lines as the ordinary case. The structural similarities and differences of the two models should be evident to the reader.
This requires once more the application of Dirac's quantization algorithm. Its implementation turns out to be straightforward.
The canonical Hamiltonian implied by (3.3.19) is The primary constraints stemming from relations (3.3.21a)-(3.3.21d) are where s s ω s , s s Ω ωs are given by

Choice of polarization and Ward identities
To build a representation of the operator algebra yielded by canonical quantization, we must choose a polarization, a maximal integrable distribution on the classical phase space, the restriction of the Dirac symplectic form to which vanishes. The polarization must be gauge invariant by consistency.
Henceforth, we shall make reference exclusively to the space manifold M. We shall thus suppress the index s throughout as it is no longer necessary lightening in this way the notation.
An expression of S W ZW 2 (g, ω 100 , ω 001 , Ω ω 101 ) fulfilling relation (3.4.38) holding when M is the boundary of a 4-fold B and g can be extended to an element of

Examples
We present a few examples to illustrate the higher Chern-Simons theory developed in subsect. 3.2.
Let (ω, Ω ω ) be a connection doublet and (f, F f ) be its curvature doublet.

A.1 Strict 2-groups
The theory of strict 2-groups is formulated most elegantly in the language of higher category theory [37]. Here, we shall limit ourselves to providing the basic definitions and properties.
Ordinary groups.
We recall first the familiar definition of group.
A group (in delooped form) consists of the following set of data: 1. a set of 1-cells G; 2. a composition law of 1-cells • : G × G → G; 3. a inversion law of 1-cells −1• : G → G; 4. a distinguished unit 1-cell 1 ∈ G These are required to satisfy the following axioms.
Here and in the following, a, b, c, · · · ∈ V 1 , A, B, C, · · · ∈ V 2 , where V 2 denotes the set of all 2-cells. For clarity, we often denote A ∈ V 2 (a, b) as A : a ⇒ b. All identities involving the vertical composition and inversion hold whenever defined.
V is in fact a one-object strict 2-category in which all 1-morphisms are invertible and all 2-morphisms are both horizontal and vertical invertible, a one-object strict 2-groupoid.
is an ordinary group and (V 1 , V 2 , · , −1 · , 1 − ) is a groupoid. Viewing this as a category V having V 1 , V 2 as its collection of objects and morphisms, • : V ×V → V and −1• : V → V are both functors and V turns out to be a strict monoidal category in which every morphism is invertible and every object has a strict inverse.

A.2 Strict 2-groups and crossed modules
Strict 2-groups are intimately related to crossed modules. A crossed module [38] consists in the following elements.
Conversely, with any strict 2-group V there is associated a crossed module (G, H), as follows.

A.3 Lie 2-algebras
In this appendix, we review the notion of Lie 2-algebra, which is basic in the present work. Again, Lie 2-algebras have an elegant categorical formulation [8]. Here, we shall present them as 2-term L ∞ algebras, which is an equivalent computationally efficient description.

Ordinary Lie algebras
A Lie 2-algebra consists of the following set of data: 1. a vector space g; 2. a linear map [·, ·] : g ∧ g → g; This is required to satisfy the following axiom: where π is given by is the 1 step degree shifted dual of g, assumed to have degree 0. It is immediately verified that (A.3.1) is equivalent to the familiar Jacobi identity.

Lie algebra Chevalley-Eilenberg cohomology
The Chevalley-Eilenberg algebra CE(g) of g is the graded commutative algebra S(g ∨ [1]) ≃ * g ∨ generated by g ∨ [1], the 1 step degree shifted dual of g. The Chevalley-Eilenberg differential Q CE(g) is the degree 1 differential defined by It is immediately verified that Q CE(g) is nilpotent, as a consequence of (A.3.1). (CE(g), Q CE(g) ) is so a cochain complex. Its cohomology H CE * (g) is the Chevalley-Eilenberg cohomology, also known as Lie algebra cohomology, of g.

Lie 2-algebras
A Lie 2-algebra consists of the following set of data: 1. a pair of vector spaces on the same field v 0 , v 1 ;

a linear map
These are required to satisfy the following axioms: where π and Π are given by π = π a ⊗ e a , (A.3.6a) , [·, ·, ·]) to emphasize its underlying structure.

Lie 2-algebra Chevalley-Eilenberg cohomology
Similarly to ordinary Lie algebras, the Chevalley-Eilenberg algebra CE(v) of v is the graded commutative algebra S(v 0 . The Chevalley-Eilenberg differential Q CE(v) is the degree 1 differential defined by Just as crossed modules provide an equivalent description of strict 2-groups, differential Lie crossed modules furnish an alternative description of strict Lie 2-algebras.
A differential Lie crossed module [40] consists in the following elements.  With any Lie group G, there is associated a Lie algebra g. g is the tangent space to G at 1. The brackets of g are defined by the relations

A.6 The Lie 2-algebra automorphism group
Automorphisms of a Lie algebra or a Lie 2-algebra provide structural information and play a basic role in gauge and semistrict higher gauge theory as formulated in this paper.

The derivation Lie 2-Lie algebra
Let v be a Lie 2-algebra. The derivation strict Lie 2-Lie algebra aut(v) of v is described as follows.
An element of α of aut 0 (v), a 1-derivation, consists of three mappings.
Relations (A.7.6) ensure that the basic relations (A.3.5) are satisfied by the above boundary and brackets.
In more precise terms, the following propositions hold.
Let v be a Lie 2-algebra such that dim v 0 < dim v 1 . Then, there is a balanced Lie 2-algebra with the following properties.
We assume below that g is a Lie algebra with invariant form (·, ·).
The orthogonal automorphisms of a Lie algebra with invariant form A automorphism φ ∈ Aut(g) is said orthogonal if (φ(x), φ(y)) = (x, y), (A.9.2) for any x, y ∈ g. We shall denote by OAut(g) the subset of all orthogonal elements φ ∈ Aut(g). OAut(g) is a Lie subgroup of the Lie group Aut(g).

The orthogonal derivations of a Lie algebra with invariant form
A derivation α ∈ aut(g) is said orthogonal if (α(x), y) + (x, α(y)) = 0, (A.9.3) for any x, y ∈ g. We shall denote by oaut(g) the subset of all orthogonal elements α ∈ aut(g). oaut(g) is a Lie subalgebra of the Lie algebra aut(g). oaut(g) is the Lie algebra of the Lie group OAut(g).

Adjoint action and orthogonality in Lie algebras with invariant form
For any x ∈ g, the derivation ad x ∈ aut(g) is orthogonal, ad x ∈ oaut(g) (cf. eq. (A.7.3)). This is an immediate consequence of (A.9.1).

Exponential map and orthogonality in Lie algebras with invariant form
The exponential map exp • : oaut(g) → OAut(g) of oaut(g) is simply the restriction of the exponential map exp • : aut(g) → Aut(g) of aut(g) to oaut(g). In particular, the orthogonal exponential is still computed by the expression (A.7.4).

Invariant forms on balanced Lie 2-algebras
Let v be a balanced Lie 2-algebra. An invariant form on v is a non singular bilinear mapping (·, ·) : v 0 × v 1 → R enjoying the following properties. for any x, y ∈ v 0 , X, Y ∈ v 1 .
We assume below that v is a balanced Lie 2-algebra equipped with an invariant form (·, ·).

Exponential map and orthogonality in balanced algebras with invariant form
The exponential map exp • : oaut(v) → OAut(v) of oaut(v) is simply the restriction of the exponential map exp • : aut(v) → Aut(v) of aut(v) to oaut(v).
In particular, the orthogonal exponential is still computed by the expressions (A.7.11).