Higher Chern-Simons-Antoniadis-Savvidy forms based on crossed modules

We present higher Chern-Simons-Antoniadis-Savvidy (ChSAS) forms based on crossed modules. We start from introducing a generalized multilineal symmetric invariant polynomial for the differential crossed modules and constructing a metric independent, higher gauge invariant, and closed form using the higher curvature forms. Then, we establish the higher Chern-Weil theorem and prove that the higher ChSAS forms is a special case of this theorem. Finally, we get the link of two independent higher ChSAS theories.


Introduction
Higher gauge theory [1][2][3][4][5][6] is a generalization of ordinary gauge theory, where the underlying algebraic structure is promoted from an ordinary group to a higher group.And the associated gauge potentials and their gauge curvatures are generalized to higher degree forms valued in higher algebras.In this paper, we need only consider the case of 2-gauge theory.In conventional gauge theory, the connection 1-form equips curves with holonomies in the gauge group G. While, in the 2-gauge theory, there is a 2-connection consisting of the differential 1-form and 2-form, which can be used to equip surfaces with a new kind of surface holonomy, given by elements of another group H.More precisely, the algebraic structure that replaces the gauge group in 2-gauge theory is a crossed module (H, G; α, ⊲), as described in section 2 below.
In the ordinary Chern-Simons gauge theory, the Chern-Simons action is constructed by the connection 1-form A on a principle G-bundle, and the equation of motion implies the flatness of the connection, i.e., the corresponding curvature 2-form F = dA + A ∧ A vanishes.The associated (2n + 2)-Pontryagin-Chern form P 2n+2 is a higher order polynomial of the curvature form which is given by where g is the Lie algebra of G and • • • g stands for a multilinear symmetric invariant polynomial • • • g : g n+1 −→ R (see Ref. [25]).It is straightforward to show that P 2n+2 is a closed form, i.e., dP 2n+2 = 0.By the Poincaré lemma, there exists a (2n + 1)-form C 2n+1 locally such that P 2n+2 = dC 2n+1 , which is called a Chern-Simons form.Besides, the explicit expression for the Chern-Simons form can be obtained as a special case of the Chern-Weil theorem.
Chern-Weil theorem: For two g-valued connection 1-forms A 1 and A 0 and the corresponding curvatures F 1 and F 0 , there is an interpolation A t = A 0 +t(A 1 −A 0 ), (0 ≤ t ≤ 1), and the corresponding curvature is F t = dA t + A t ∧ A t .Then, we have where is called a transgression (2n + 1)-form.
Setting A 0 = 0 and 3), we get Recently, Antoniadis, Konitopoulos and Savvidy introduced a closed invariant form similar to the Pontryagin-Chern form (1.1) in the context of the so-called extended gauge theory, denoted by Γ 2n+p with p = 3, 4, 6, 8 (see Refs. [26][27][28][29]).Since dΓ 2n+p = 0, one can have Γ 2n+p = dC ChSAS , where C (2n+p−1) ChSAS is just a ChSAS (2n + p − 1)-form.And the ChSAS form can be expressed explicitly in terms of a higher order polynomial of the curvature forms.As with the standard Chern-Simons form, the ChSAS form is background-free, quasi-invariant and only locally defined.Later, P. Salgado and S. Salgado found that the extended invariant form Γ 2n+p can also be constructed from an algebraic structure known as free differential algebra in Refs.[30,31].
Especially, in Refs.[25,32], Salgado et al. studied a particular extended invariant form which is given by where H = dB + [A, B] is the 3-form field-strength for the 2-form field B valued in g.It is simple to prove that dΓ 2n+3 = 0, seeing the proof in Ref. [27].Then, there is a (2n + 2)-ChSAS form C 2n+2

ChSAS
satisfying Γ 2n+3 = dC 2n+2 ChSAS .Besides, they found that the explicit expression for C 2n+2 ChSAS can be obtained as a special case of the generalized Chern-Weil theorem.
Setting A 0 = B 0 = 0, A 1 = A and B 1 = B in (1.7), we can get the (2n + 2)-ChSAS form From a higher gauge theory of view, it is natural to ask a question: whether the above results are adapted to the case of 2-connections?It is the purpose of this paper to show that the extended invariant form (1.5) is well adapted to generalize the higher Chern-Simons considered in Ref. [13].
This paper is organized as follows.In section 2, we briefly review the relevant topics of the 2gauge theories and introduce the Lie algebra-valued differential forms and some conventions presented in this paper.In section 3, we define a generalized multilinear symmetric invariant polynomial for the differential crossed modules.Then, we construct a higher invariant form consisting of the higher curvatures and find the associated higher ChSAS form.In section 4, we extend the generalized Chern-Weil theorem to the higher Chern-Weil theorem and prove that the results in the section 3 can be obtained as a special case of the higher Chern-Weil theorem.In section 5, we consider a kind of higher transgression gauge field theory where the lagrangian is a higher AST form.Finally, we know that two independent higher ChSAS theories have closed links under some boundary conditions.

2-gauge theory
Let us begin by giving a short review of the 2-gauge theory.For additional information on the topic, see for example Refs.[1][2][3][4].
Firstly, the relevant algebraic tool involved in the description of this higher theory is a Lie crossed module (H, G; ᾱ, ⊲), which consists of two Lie groups H and G together with a smooth morphism ᾱ : H −→ G and a smooth left action ⊲ of G on H by automorphisms such that for each g ∈ G and h, h ′ ∈ H.For the crossed module (H, G; ᾱ, ⊲), the corresponding infinitesimal version is the differential crossed module (h, g; α, ⊲), which consists of two Lie algebras g :=Lie G and h :=Lie H together with a Lie algebra map α : h −→ g and a left action ⊲ of g on h by derivations such that for each X ∈ g and Y, Y ′ ∈ h.Thanks to the fact that the higher gauge fields are viewed as higher differential forms valued in the higher algebras, we consider the algebra-valued differential forms and introduce the component notations.Let Λ k (M, g) be the vector space of g-valued differential k-forms on the 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 a basis X a for g.Here, we consider matrix Lie algebras, and have [X, for Then there is an identity The convention also applies to h.
Besides, given a g-valued connection 1-form A, one can define a 'covariant derivative' D on a g-valued differential form A ′ and an h-valued differential form B, respectively, (2.7) Then, we consider the basic gauge fields of 2-gauge theory, which are given by 2-connections.Given a crossed module (H, G; ᾱ, ⊲) with the associated differential crossed module (h, g; α, ⊲), the 2-connection (A, B) is described by a g-valued 1-form A and an h-valued 2-form B. The corresponding g-valued fake curvature 2-form and h-valued 2-curvature 3-form are given by which automatically satisfy the 2-Bianchi Identities: ) We call (A, B) fake-flat, if F = 0, and flat, if it is fake-flat and G = 0.Moreover, there is a general 2-gauge transformation for the 2-connection (A, B): with g ∈ G and φ ∈ Λ 1 (M, h).The corresponding curvatures transform as follows: (2.13)

Higher invariant and ChSAS forms
Based on the 2-gauge theory, we now turn to the construction of the higher invariant forms and find the explicit expression of the associated ChSAS forms.A key issue for the construction is the establishment of the higher order polynomial of the higher curvature forms.
Inspired by the invariant polynomial • • • g for the Lie algebra g, we can define a generalized multilinear symmetric invariant polynomial for the differential crossed modules (h, g; α, ⊲): The symmetry implies that and the invariance states clearly that for each g ∈ G, which can be given by taking g as an infinitesimal transformation and using the identity (3.2).In the case of n = 1, (3.1) becomes a bilinear form •, • gh : g × h −→ R in [13,20].For example, there is a crossed module (T, T ; ᾱ, ⊲), where T is a compact connected abelian group, ᾱ : T −→ T is an endomorphism and ⊲ : T ×T −→ T is the trivial action of T on itself.The associated differential crossed module is (t, t; α, ⊲), where the algebra t is abelian and the action ⊲ is trivial.Let •, • t be a symmetric non singular bilinear form on t and the endomorphism α be symmetric with respect to this pairing.See [20] for details.Moreover, there is a Poincaré 2-group (H = R 3 , G = SO(2, 1); ᾱ, ⊲), where ᾱ is trivial and ⊲ is the representation of SO(2, 1) on R 3 .The corresponding Poincaré 2-algebra is (h = R 3 , g = so(2, 1); α, ⊲) with α(P a ) = 0 for P a ∈ h and J i ⊲ P a = ǫ ia b P b , where {J i , i = 0, 1, 2} and {P a , a = 0, 1, 2} are the generators of so(2, 1) and R 3 , respectively.The commutation relation is [J i , J j ] = ǫ ijk J k , and ǫ ijk is the Levi-Civita symbol.Then, (3.1) can be given by for the case of n = 1, where η = diag(−, +, +).It is easy to prove that the pairing satisfies (3.2) and (3.3).The above algebra with the pairing (3.6) has been used for generalizing (2+1)-dimensional pure gravity with vanishing cosmological constant to the higher gauge theory formalism in [33].More examples can be found in the works of Baez, Schreiber [34] and Zucchini [20].Generally, for the differential crossed modules (h, g; α, ⊲), where g is a matrix Lie algebra, the generalized multilinear symmetric invariant polynomial can be given by Besides, the equation (3.2) essentially boils down to where {A i , i = 1, • • • , n} is a set of g-valued differential forms and B is an h-valued differential form.Thus, imitating the extended invariant form (1.5), we can construct a higher invariant form based on the 2-gauge field theory where F and G are the higher curvature forms (2.8).It is possible to prove that the higher pairing form is also 2-gauge invariant under the 2-gauge transformation (2.11) and (2.12) by direct computations.Additionally, using eqs.(3.3), (3.8) and the 2-Bianchi-Identities (2.9)-(2.10),we have which means that P 2n+3 is a closed form.By the Poincaré lemma, P 2n+3 can be locally written as an exterior differential of a certain (2n + 2)-form.Actually, this (2n + 2)-form potential is a categorical generalization of the ChSAS form in (1.8).We find the certain (2n + 2)-form by using a kind of variational approach in [25].Since the variation δP 2n+3 is given by .12) Following Ref. [25], we can introduce a one-parameter family of the higher potentials and strengths through the parameter t, 0 ≤ t ≤ 1: Choosing a variation of the form δ = δt(∂/∂t), we have δA t = δtA and δB t = δtB.From (3.12), we get which we call a "2-Chern-Simons-Antoniadis-Savvidy" (2ChSAS) form.From the above facts, we know that the extension in Ref. [25] is applicable equally to the higher gauge field theory.

Higher Chern-Weil theorem
In this subsection, we generalize the AST form (1.7) and the generalized Chern-Weil theorem based on the 2-gauge theory.Let (A 0 , B 0 ) and (A 1 , B 1 ) be two 2-connections, and the corresponding curvature forms are given by for i = 0, 1. Define the interpolations between the two connections as for 0 ≤ t ≤ 1 and their curvatures are given by By direct computations, we have where D t θ = dθ + A t ∧ [,] θ and D t Φ = dΦ + A t ∧ ⊲ Φ.Then, the difference between P