Higher Chern-Simons based on (2-)crossed modules

We present higher Chern-Simons theories based on (2-)crossed modules. We start from the generalized differential forms in Generalized Differential Calculus and define the corresponding generalized connections which consist of higher connections. Then we establish the generalized Chern-Simons forms to get the higher Chern-Simons actions. Finally, we develop the higher second Chern forms and Chern-Weil theorems.


Introduction
The purpose of this paper is to construct Higher Chern-Simons (HCS) gauge theories based on (2-)crossed modules.The motivation for this work comes from literature [1][2][3][4] and Soncini and Zucchini's papers [5][6][7][8][9] in which they formulate a 4-dimensional semistrict higher gauge theoretic Chern-Simons (CS) theory.But we only consider the case of the strict higher gauge theories.Besides, our treatments and techniques adopted in constructions are significantly different from that in literature.Within the framework of higher gauge theory [10,11], we construct 2-Chern-Simons (2CS) and 3-Chern-Simons (3CS) gauge theories by applying the Generalized Differential Calculus (GDC) [12].
As an ordinary gauge theory, the CS theory has a wide variety of applications, for instance pure mathematics, string theory, condensed matter physics [40][41][42], and so on.Since the CS theory does not depend on the metric of the underlying spacetime manifold, it is known as a sort of topological field theory [43,44].The CS action is constructed by the connection 1-form on a principle bundle, which is called a "gauge field", and the equation of motion implies the flatness of the connection, i.e. the corresponding curvature 2-form vanishes.Without loss of generality, similar arguments shall be applied to the 2CS and 3CS theories.In carrying out the developments of these HCS theories, the main difficulty is the construction of the 2CS and 3CS actions.The main objects are the 2-connections and 3-connections, which consist of two and three ordinary differential forms valued in higher algebras respectively.We find out that the choice of the generalized differential forms in the GDC seems to be the best choice to our theories.
The GDC has been discussed in a number of literature and they are employed in many different geometrical and physical contexts.For example, in 2002, Guo et al. introduced the generalized topological field theories [45], and found that a direct relation between CS and BF theories can be presented by using the GDC.Besides, they established the generalized Chern-Weil homomorphism for generalized curvature invariant polynomials, and showed that the BF gauge theory can be obtained from the generalized second Chern class.In 2003, they presented a general approach to construct a class of generalized topological field theories with constraints and found that the ordinary BF formulations of general relativity, Yang-Mills theories, and N = 1, 2 chiral super-gravity can be reformulated as generalized topological fields with constraints [46].Recently, the GDC has been developed by Robinson in a number of works [47][48][49][50][51].In the mean time, analogous ideas have been explored in the dual context, and generalized vector fields have been introduced and studied in [52,53].
In the GDC, the algebra and calculus of ordinary differential forms are extended to an algebra and calculus of different type N generalized differential forms.We are concerned in this paper with the type N = 1 and N = 2, which consist of two and three ordinary differential forms, respectively.Based on these facts, it is possible to redefine the 2-connection as a type N = 1 generalized differential form, and the 3-connection as a type N = 2 generalized differential form.The advantage of using the GDC lies in the fact that we can rewrite the higher connection as a kind of generalized connection.
Analogous to higher BF and YM theories, the CS theory will be generalized to the higher counterpart by using the idea of a categorical ladder.As indicated in Table 1, the underlying algebraic structure is promoted from a ordinary group to higher groups.It follows that the Lie algebra-valued connection form shall be replaced by the differential (2-)crossed module-valued connection form.In order to construct the 2CS and 3CS actions, we need to consider the invariant bilinear form on the differential (2-)crossed module.We follow the definition of the invariant form on the differential crossed module in [5,6].Motivated by this construction, we define an invariant form on the differential 2-crossed module.Furthermore, we establish generalized bilinear forms based on the above expansions and establish the generalized CS forms to give the HCS forms and actions.
According to the ordinary CS gauge theories, three questions naturally arise: 1. Whether higher second Chern forms exist? 2. Whether the higher second Chern forms are gauge invariant under higher gauge transformations?3. Whether the higher second Chern forms satisfy higher Chern-Weil theorems?
For the first question, we establish two new second Chern forms called respectively second 2-Chern form and second 3-Chern form, which are constructed under the categorical generalizations.For the second question, we prove that the second 2-Chern form is 2-gauge invariant under the 2-gauge transformation, and the second 3-Chern form is 3-gauge invariant under the 3-gauge transformation.
For the third question, we show that the two new second Chern forms are closed and have topologically invariants, which give 2-and 3-Chern-Weil theorems separately.We list our results (2CS and 3CS) and the known results in Table 2.We note that the higher CS theories are nice generalizations of the CS theory.

Equations of motion
The layout of the paper is summarized in the following outline: 1.In section 2, we review the relevant topics of the higher gauge theories.The notations of Lie algebra-valued differential forms and some conventions are presented in subsection 2.1.The 2connection and its 2-gauge transformations are given in 2.2.Similarly, the 3-connection and its 3-gauge transformations are listed in subsection 2.3.2. In section 3, we recall the balanced differential crossed modules and the related invariant forms, which will play major roles in the 2CS theory.Motivated by these arguments, we establish the balanced differential 2-crossed modules and the relational invariant forms, which will play similar roles in the 3CS theory.3.In section 4, we recall the type N generalized differential forms in the GDC and present the type N = 1 and N = 2 generalized p-forms.Then, we develop the two types generalized forms valued in differential crossed modules and 2-crossed modules.4. In section 5, we rewrite the 2-connections as the type N = 1 generalized connections, and the 3-connections as the type N = 2 generalized connections based on the above GDC. 5.In section 6, we present the two main results of this paper.Firstly, in subsection 6.1 we recollect the ordinary second Chern form and Chern-Weil theorem and the CS form.Secondly, in subsection 6.2, we introduce the 2CS form based on the type N = 1 generalized connections, and generalize the second Chern form to the second 2-Chern form satisfying the 2-Chern-Weil theorem.Finally, the similar construction apply to the type N = 2 generalized connections, and we get the 3CS form in subsection 6.3.In addition, we also build the second 3-Chern form and show that it fulfills the 3-Chern-Weil theorem.

Higher connections and gauge transformations
In this section, we shall review the higher connections and associated gauge transformations.Since we focus on the construction of HCS gauge theories, we consider these higher gauge fields (or higher connections) as basic fields of these theories.

Higher algebra valued differential forms
In the ordinary CS gauge theory with gauge group G, the gauge fields are g-valued differential forms.Similarly, the higher gauge fields can be described by higher algebra valued differential forms in the HCS gauge theories.Both Lie crossed modules and Lie 2-crossed modules are the algebraic structures of the HCS gauge theories elaborated in this paper rests.We will not go into all the details about these concepts, however a fuller introduction is given in Appendix Appendix A for serving also the purpose of setting the notations used throughout this paper.Furthermore, see refs.[23,29,[54][55][56][57][58] for an exhaustive exposition of this subject.As for prerequisites, it is enough to be familiar with the following definitions.
A Lie crossed module (H, G; α, ⊲) consists of two Lie groups H and G together with a Lie group action ⊲: G × H −→ H of G on H by automorphisms, and an equivariant target map α : H −→ G satisfying certain identities.The Lie crossed module has an infinitesimal version, (h, g; α, ⊲), called the differential crossed module.In order not to introduce additional notations, we use the same letters α and ⊲ for counterparts in the infinitesimal version of the crossed module.The convention also applies to the Lie 2-crossed modules.
Then, we consider the differential forms valued in these associated algebras as follows.In this paper, we will mostly follow the notations and definitions of [24,59].We denote by Λ k (M, g) the vector space of g-valued differential k-forms on a manifold M over C ∞ (M ).For A ∈ Λ k (M, g), have A = a A a X a with a scalar differential k-form A a and an element X a in g.We define Here, we assume g to be a matrix Lie algebra.Then, we have [X, X ′ ] = XX ′ −X ′ X for each X, X ′ ∈ g.
then there is an identity The convention also applies to h and l.
Besides, for ), Y a } by using (A.15).For more properties for the Lie algebra valued differential forms corresponding to the identities of the differential (2-)crossed module, we refer the reader to [59].

2-connections and 2-gauge transformations
Given a crossed module O = (H, G; α, ⊲), let O = (h, g; α, ⊲) be the associated differential crossed module.The basic gauge fields of 2-gauge theory are 2-connections valued in the differential crossed module.On the manifold M , a 2-connection (A, B) is given by a g-valued 1-form A ∈ Λ 1 (M, g) and an h-valued 2-form B ∈ Λ 2 (M, h).The curvature (Ω 1 , Ω 2 ) consists of a g-valued fake curvature 2-form and an h-valued 2-curvature 3-form: and call (A, B) fake-flat, if Ω 1 = 0, and flat, if it is fake-flat and Ω 2 = 0. On the other hand, the curvature automatically satisfies the 2-Bianchi Identities: Moreover, there are two kinds of 2-gauge transformations from a 2-connection (A, B) to another one (A ′ , B ′ ): • the first kind of 2-gauge transformation with g ∈ Λ 0 (M, G), • the second kind of 2-gauge transformation with φ ∈ Λ 1 (M, h), If we write the second kind of 2-gauge transformation as then the composition of (2.4) and (2.6) gives a general 2-gauge transformation: Under the general 2-gauge transformation, the associated curvature transforms as follows: There is a basic result that the 2-connection is fake flat on the trivial principal O-2-bundle over M .The 2-curvature 3-form Ω 2 is thus covariant under the general 2-gauge transformation.Proof.According to the definitions of (fake-)flatness and the transformations (2.8), it is straightforward to get (a) and (b).Besides, the proof of (c) is not difficult but is too long to give here.We will give a rigorous proof in Appendix Appendix B.
Starting from the definition of the curvature form, we can also calculate its exterior derivative to obtain the 3-Bianchi Identities: (2.10) For a rigorous derivation of these identities, the reader is referred to [30].
) is also flat, i.e. the flatness is 3-gauge invariant.
(d).The proof of (d) is not difficult but is too long to give here.We will give a rigorous proof in Appendix Appendix B.

Balanced differential (2-)crossed modules with invariant forms
Similar to the role of invariant traces in ordinary CS gauge theory, the invariant form is also an essential ingredient of the construction of the HCS action.Thus, the first key issue for developing the HCS theories is defining the invariant bilinear form on the relevant algebras.
Balanced differential crossed modules.Balanced differential crossed modules play a major role in the construction of the 2CS theory.The notion of balance has arisen in [5], which has non counterpart in ordinary Lie algebra theory.Definition 3.1 (Balanced differential crossed modules).A differential crossed module (h, g; α, ⊲) is said balanced if dim h= dim g.Proposition 3.1.For any non balanced differential crossed module (h, g; α, ⊲), there exists a balanced differential crossed module ( h, g; α, ⊲) minimally extending (h, g; α, ⊲).
Proof.Firstly, let dim g < dim h.Then, there is a balanced differential crossed module ( h, g; α, ⊲) where where w is a vector space such that dim w = dim h − dim g, and the Lie bracket [−, −] in the Lie algebra g is given by Secondly, let dim g > dim h.Then, there is a balanced differential crossed module ( h, g; α, ⊲) where 1. g = g; 2. h = h ⊕ w ′ , where w ′ is a vector space such that dim w ′ = dim g − dim h, and the Lie bracket [−, −] in the Lie algebra h is given by for any Balanced differential 2-crossed modules.Similarly, we can define a balanced differential 2-crossed module in order to develop the invariant form in the 3CS theory.
Proof.Firstly, let dim g < dim l.Then, there is a balanced differential 2-crossed module ( l, h, g; β, α, ⊲, {, }) where 1. l = l and h = h; 2. g = g ⊕ w, where w is a vector space such that dim w = dim l − dim g, and the Lie bracket [−, −] in the Lie algebra g is given by 5. the action ⊲ of g on h and l are given by Secondly, let dim g > dim l.Then, there is a balanced differential crossed module (l, h, g; β, α, ⊲ , {, }) where 1. h = h and g = g; 2. l = l ⊕ v, where v is a vector space such that dim v = dim g − dim l, and the Lie bracket [−, −] in the Lie algebra l is given by 3. the map β : l −→ h is given by 4. α = α; 5. the action ⊲ of g on l is given by Further, the differential 2-crossed module ( l, h, g; β, α, ⊲, {, }) is unique up to isomorphism.
Using the above results, we can always suppose that the differential (2-)crossed module we are dealing with is balanced.We have established the existence of balance, and now we further develop the invariant forms on the differential (2-)crossed modules.
Invariant forms on the differential crossed modules.The study of invariant forms on Lie 2algebras was originally proposed in [5,60] as a way of constructing the semistrict 4-d CS theory.In a similar manner, we will introduce an invariant form on the differential crossed module, which is equivalent to a strict Lie 2-algebra.Definition 3.3.Given a Lie crossed module (H, G; α, ⊲) and the associated differential crossed module (h, g; α, ⊲), a G-invariant form on (h, g; α, ⊲) is defined as a non singular bilinear form −, − g,h : for any g ∈ G, X ∈ g, and Y ∈ h.
Invariant forms on the differential 2-crossed modules.Using a similar argument, we can define an invariant form on the differential 2-crossed module.
for any g ∈ G, X ∈ g, Y, Y ′ ∈ h and Z ∈ l.

Generalized differential forms valued in differential (2-)crossed modules
In this section, we first recall the generalized differential forms of type N in the framework of GDC [12,[47][48][49][50][51][52].Then we formulate analogous results, which are applicable to the generalized differential forms valued in differential (2-)crossed modules.Since we aim to the construction of 2CS and 3CS gauge theories, we focus on just the types N = 1 and N = 2.

Generalized differential forms
There are N linearly independent minus 1-forms {ξ i }, (i = 1, 2, ..., N ), which assumed to satisfy the ordinary distributive and associative laws of exterior algebra.The exterior product rule is given by where p a is any ordinary p-form on a manifold M of dimension n.In particular, they satisfy a condition of linear independence, ξ 1 ξ 2 ...ξ N = 0 and (ξ i ) 2 = 0.In order to ensure that their exterior derivatives are zero-forms and that d 2 = 0, they are required to satisfy the condition d ξ i = k i , where k i is a constant for any i = 1, ..., N .
A generalized p-form of type N is thus defined as .., i j , ..., i N range and sum over 1, 2, ..., N .It then follows that the generalized forms satisfy the same basic rules of exterior multiplication and differentiation as those which govern the algebra and calculus of ordinary differential forms, apart from p taking the value minus one in standard formulae.For example, these exterior products and derivatives of generalized p-form U and q-form V satisfy the standard rules of exterior algebra and d 2 = 0. From (4.2), we can see that a generalized p-form of type N = 0 is an ordinary differential pform, and the exterior algebra of type N = 0 forms is the ordinary exterior algebra.Further discussion of type N generalized forms can be found in [12].In this paper, we focus on the type N = 1 and N = 2 generalized p-forms.
Type N = 1 generalized p-forms.As in the equation (4.2), the type N = 1 generalized p-form is defined as and the generalized exterior derivative of U is given by where k is a constant.
Type N = 2 generalized p-forms.As in the equation (4.2), the type N = 2 generalized p-form is defined as and the generalized exterior derivative of U is given by with k i for i = 1, 2.

Type N = 1 generalized forms valued in differential crossed modules
Similar to ordinary algebra-valued differential forms, we define a generalized form valued in the differential crossed module, which consists of a pair of ordinary algebra-valued differential forms.Definition 4.1.Given a differential crossed module (h, g; α, ⊲) and an n-dimension manifold M , a type N = 1 generalized p-form valued in (h, g; α, ⊲) is given by ) According to the GDC in subsection 4.1 and the higher gauge theory in the derived formulation considered by Zucchini in [5][6][7][8], we can define the generalized exterior product and derivative for the above generalized p-forms valued in the differential crossed modules.Definition 4.2.Given a generalized p-form A and a q-form B valued in a differential crossed module (h, g; α, ⊲), B ∈ Λ q+1 (M, h), the generalized exterior product is given by and the generalized exterior derivative of A is given by where k is a constant.
The important point to note here is that the equations (4.13) and (4.14) are induced by (4.6) and (4.7), respectively.This definition provides a computing method for higher algebra-valued differential forms.It is easy to show that Furthermore, it can be straightforwardly verified that and that Encouraged by these constructions of generalized differential form, we can define a generalized bilinear forms ≪ −, − ≫ for the higher algebra-valued differential forms.Given a pair of generalized forms of type N = 1, (A, B) of the term (4.12), define From the property of the non singular G-invariant bilinear form on (h, g; α, ⊲), one can apparently note that ≪ −, − ≫ is a symmetric G-invariant bilinear form.This structure will lead us to some unexpected and novel results in the construction of 2CS theory.

Type N = 2 generalized forms valued in differential 2-crossed modules
We have already defined the differential crossed module-valued generalized forms of type N = 1, and along this approach we will establish a type N = 2 generalized forms valued differential 2-crossed modules.
Definition 4.3.Given a differential 2-crossed module (l, h, g; β, α, ⊲, {, }) and a n-dimension manifold M , a type N = 2 generalized p-form valued in (l, h, g; β, α, ⊲, {, }) is given by In the same measure, we can define the generalized exterior product and derivative for the above generalized p-forms valued in the differential 2-crossed modules.Definition 4.4.Given a generalized p-form A and a q-form B valued in a differential 2-crossed module (l, h, g; β, α, ⊲, {, }), with q b ∈ Λ q (M, g), b i ∈ Λ q+1 (M, h) and q+2 b ∈ Λ q+2 (M, l) for i = 1, 2, the generalized exterior product is given by which is induced by (4.9), and the generalized exterior derivative of A is given by which is induced by (4.10), and where k i is a constant and i, j = 1, 2, i = j.
In this definition, the equations (4.21) and (4.22) are induced by (4.9) and (4.10), respectively.Consequently, we infer that and that By the same token, we define a generalized bilinear form ≪ −, − ≫ for these differential 2-crossed module-valued generalized forms.Given a pair of generalized forms of type N = 2, (A, B), of the term (4.20), we define From the property of the non singular G-invariant bilinear form on (l, h, g; β, α, ⊲, {, }), one can also apparently note that ≪ −, − ≫ is a G-invariant bilinear form.This structure will be useful in the construction of 3CS theory.

Generalized connections
In this section, we present a generalized formulation of the higher gauge theory based on the subsections 4.2 and 4.3.Under the framework of the GDC, it is easy to show the close relationship of higher to ordinary gauge theory.Based on the above argument, it allows so to many import ideas and techniques of the CS gauge theory to the higher CS gauge theories in the next section.

Higher Chern-Simons theory
In this section, we first briefly remind the ordinary CS gauge theory and the relevant Chern-Weil theorem.Then we will construct the 2CS and 3CS gauge theories based on the generalized connections, and generalize these associated discussions in the ordinary CS theory to the HCS.

CS gauge theory and Chern-Weil theorem
Let us consider a principle bundle P (M, G) on a 3-dimensional manifold M with a gauge group G and the associated Lie algebra g.Locally, a gauge field is a connection 1-form A ∈ Λ 1 (M, g) that transforms under the gauge transformation like so that the corresponding curvature 2-form F = dA + A ∧ A ∈ Λ 2 (M, g) transforms like Besides, they satisfy the Bianchi Identity, dF + A ∧ [,] F = 0.
In order to consider their topological properties, let us first briefly remind the curvature invariant symmetric polynomial on the bundle P (M, G), which is called the second Chern form, where −, − g is a G-invariant symmetric non-degenerate bilinear form, satisfying the invariance condition for any X, X ′ , X ′′ ∈ g.Under the gauge transformation (6.1), it is not difficult to verify that the second Chern form is gauge invariant Take an action of the form and the action is gauge invariant due to (6.5).However, the action gives completely trivial equations.
To see this, simply compute, having It is apparent from the above equation that δS = 0 for all A.
Next we recall the Chern-Weil theorem for the second Chern form [45] and give a rigorous proof.This theorem will be generalized to the 2CS and 3CS gauge theories in subsection 6.2 and 6.3, respectively.And these proofs of the new Chern-Weil theorems can be completed by the method analogous to that used as follows.Theorem 6.1 (Chern-Weil theorem).The second Chern form P (F, F ) satisfies 1).P (F, F ) is closed, i.e. dP (F ) = 0; 2).P (F, F ) has topologically invariant integrals, namely satisfying the Chern-Weil homomorphism formula: with where A 0 and A 1 are two connection 1-forms, F 0 and F 1 the corresponding curvature 2-forms, (6.10) the interpolation between A 0 and A 1 , (6.11) the curvature of this interpolation.
2).It is easy to see that Hence by using the Bianchi Identity D t F t = 0. Thus = dQ(A 0 , A 1 ).(6.15)This theorem shows that P (F 1 , F 1 ) and P (F 0 , F 0 ) differ by an exact form.In other wards, their integrals over 4-d manifolds without boundary give the same results, and Q(A 0 , A 1 ) is called the secondary topological class.
In particular, we consider A 1 = A and A 0 = 0 in (6.8), then where Q CS is given by which is called the Chern-Simons form.Then the Chern-Simons action on a general 3-dimensional manifold M is defined by with κ is the coupling constant, sometimes called the level of the theory.This action is topological because it does not depend on a choice of metric on M .It is easy to show that the equation of motion is the condition of flatness for the connection:

2CS gauge theory and 2-Chern-Weil theorem
In this section, we construct a 2CS gauge theory based on the type N = 1 generalized connection theory.This construction was inspired by [6,60].As ordinary CS gauge theory exists only in odd dimensional manifolds, the 2CS gauge theory exists in 4-dimensional manifolds and is built in the framework of the 2-gauge theory [10].The 2-gauge symmetry of this theory is encoded by a Lie crossed module.
Consider a Lie crossed module O = (H, G; α, ⊲) and the associated differential crossed module O = (h, g; α, ⊲).We assume that M is an oriented, compact manifold without boundary.No further restrictions are imposed.Let (A, B) be a 2-connection on the principle 2-bundle P (M, O).The 2CS action is defined by a type N = 1 generalized connection A = A + Bξ as follows By using (4.13), (4.15) with k = −1 and (4.18), a straightforward computation gives Using (3.15), we have By substituting (6.22) into (6.21),we obtain with F = dA + A ∧ A. So, the 2CS theory can be described as a generalized BF theory with a cosmological term determined by the map α.
Varying the action S 2CS with respect to A and B, we obtain the equations of motion: Clearly, the solution of these equations gives a flat 2-connection (A, B) on the trivial principal O-2bundle over M , analogously to the standard CS theory.Let us generalize the second Chern form (6.5) to a new one with the 2-curvature (Ω 1 , Ω 2 ), which is called the second 2-Chern form.Lemma 6.1.
2).Differentiating both Ω t 1 and Ω t 2 with respect to t gives d dt Then On the other hand, we have by using (3.15), and By comparing (6.38), (6.39) and (6.40), we obtain and integrating the both sides of this equation of t between the limits 0 and 1, we have This shows that P (Ω 1 1 , Ω 1 2 ) and P (Ω 0 1 , Ω 0 2 ) differ by an exact form.In other wards, their integrals over 5-d manifolds without boundary give the same results, and we call Q(A 1 , A 0 , B 1 , B 0 ) the secondary 2-topological class.
In particular, consider A 0 = 0, B 0 = 0, A 1 = A and B 1 = B in (6.30), and we have We call Q 2CS the 2-Chern-Simons form, having Then we obtain the 2CS action on a 4-dimensional manifold M For these reasons, by its analogy to the standard CS theory and as implied by its given name, the present model can be legitimately considered a 2CS gauge theory.

3CS gauge theory and 3-Chern-Weil theorem
In this section, we construct a 3CS gauge theory based on a similar heuristic argument with the 2CS gauge theory.Similar arguments apply to the type N = 2 generalized connection, and the 3CS gauge theory exists in 5-dimensional manifolds and is built in the framework of the 3-gauge theory.Besides, the 3-gauge symmetry of this theory is encoded by a Lie 2-crossed module.
Taking the variational derivative of the action S 3CS , we obtain the equations of motion: ) ) Then the solution of these equations gives a flat 3-connection (A, B, C) on the trivial principal 3-bundle, analogously to standard CS gauge theory.Similar considerations about Chern form apply to the 3CS gauge theory.Let us first introduce the second 3-Chern form defined by the 3-curvature (Ω 1 , Ω 2 , Ω 3 ) The lemma follows will tell us the second 3-Chern form is 3-gauge invariant.Lemma 6.2.P (Ω 1 , Ω 2 , Ω 3 ) = 2 Ω 1 , Ω 3 g,l + Ω 2 , Ω 2 h is 3-gauge invariant under the general 3-gauge transformation (2.16).
Proof.Under the general 3-gauge transformation (2.16), have For the first section, have and for the second section, by using (3.19), (3.15) and (3.22).Then The same manipulations work apply to the second 3-Chern form, having and varying this action with respect to A, B and C gives by using (3.20)- (3.22) and (3.19).Note that ) B, δA by using (3.19) and (3.22).Substitute (6.60)-(6.62) to (6.59) to get (6.63) It is apparent from the above equation that δS = 0 for all (A, B, C) by using the 3-Bianchi Identities (2.10).Then the action also gives completely trivial equations.
2).P (Ω 1 , Ω 2 , Ω 3 ) has topologically invariant integral.Namely, it satisfies the 3-Chern-Weil homomorphism formula: where (A 0 , B 0 , C 0 ) and 3 ) the corresponding 3-curvatures, .66) Proof.1).Using the 3-Bianchi Identities (2.10), have 2).Differentiating Ω t 1 , Ω t 2 and Ω t 3 with respect to t gives Then, have On the other hand, we have and integrating the both sides of the this equation of t between the limits 0 and 1, we have There is an alternative proof of 2) given in Appendix Appendix C.
In particular, consider A 0 = 0, B 0 = 0, C 0 = 0, A 1 = A, B 1 = B,and C 1 = C in (6.64), and we have with We call Q 3CS the 3-Chern-Simons form.Then we obtain the 3CS action on the 5-dimensional manifold For these reasons, by its analogy to the standard CS theory and as implied by its given name, the present model can be legitimately considered a 3CS gauge theory.

Conclusion and outlook
In this article, we constructed the generalized differential forms valued in the differential (2-)crossed modules by using the GDC, and developed the generalized connections which consist of the higher connections.Based on the ordinary CS gauge theory, we established the 2CS and 3CS gauge theories.Finally, we generalized the second Chern form to second 2-Chern form and 3-Chern form, which satisfy the corresponding 2-Chern-Weil theorem and 3-Chern-Weil theorem, respectively.
Likewise, the higher CS gauge theories may be studied using a certain kind of functional integral quantization which exists in the ordinary CS gauge theory.We leave the argument for future work.

) where p a and p+1 a 1 a
are, respectively, ordinary p-and (p + 1)-forms and p can take integer values from −1 to n with −= 0. Then the generalized exterior product of the generalized p-form U in (4.5) and a generalized q-form V = q b + q+1 b ξ is given by

a 1 a = − 2 a
are, respectively, ordinary p-, (p + 1)-and (p + 2)-forms, and p can take integer values from −2 to n with −= 0 and ξ 12 = ξ 1 ∧ ξ 2 .Then the generalized exterior product of the generalized p-form U in (4.8) and a generalized q-form V = q b + q+1 b i ξ i + q+2 b ξ 12 is given by

Table 1 :
The mathematical structures of (higher) CS theories

Table 2 :
The (higher) CS gauge theories G and actions ⊲ of G on G by conjugation, and on H, L by automorphisms, as well as a G-equivariant smooth function {−, −} : H ×H −→ L, called the Peiffer lifting.If (H, G; α, ⊲) α −→