Extended Cartan homotopy formula for higher Chern-Simons-Antoniadis-Savvidy theory

The extended Cartan homotopy formula (ECHF) [1, 3, 4, 2] is a widely used tool in mathematics and physics, especially when dealing with Chern-Simons (CS) theory and subspace structures of gauge algebras. For instance, the first well known particular case of the ECHF is the Chern-Weil theorem, which is an important content in differential geometry and topology. Besides, the iterative use of the ECHF gives a subspace separation method, which allows one to (1) separate the CS action in bulk and boundary contributions, and (2) split the Lagrangian in appropriate reflection of the subspace structure of the gauge algebra, systematically. The method is built upon the triangle equation, which is a corollary of the ECHF. It is worth pointing out that, in order to apply the method, one must regard CS forms as particular cases of more general objects known as transgression forms. This gives new insight into the CS theory and is also of physical interest, particularly in the context of gravity[5, 4]. We are thus led to a result: there is a common origin of the Chern-Weil theorem and the triangle equation. The aim of this thesis is to develop the ECHF to higher gauge theory [6, 7, 8, 9, 10, 11], and extend the above results of the CS theory to the higher CS theory [12]. We hope offer a new perspective from which to view the higher gauge theory.

The higher gauge theory is a branch of mathematical physics, which generalizes ordinary gauge theory to higher algebraic structures. Based on the development of this theory, many papers have been written about the higher CS theory. In Refs. [13, 14, 15, 16, 17, 18], Soncini and Zucchini formulated a 4-dimensional semistrict higher gauge theoretic CS theory. Besides, one studied the higher CS theory arising in the AKSZ-formalism based on the $L_{\infty}$-algebra [19, 20, 21, 22]. Moreover, in Ref. [23], the authors constructed low-dimensional 2-Chern-Simons and 3-Chern-Simons gauge theories by applying the generalized differential calculus [24]. Recently, one developed the $(2n+2)$-dimensional higher CS form, namely the higher Chern-Simons-Antoniadis-Savvidy (ChSAS) form [12], which is just the research object of this paper.

In Refs. [25, 26, 27, 28], Antoniadis, Konitopoulos and Savvidy constructed metric independent, gauge invariant, and closed form $\Gamma_{2n+p}$, which is similar to the Pontryagin-Chern form in the ordinary CS theory. The principal significance of $\Gamma_{2n+p}$ is that it can be used in both, odd- and even-dimensional spacetimes. Based on their research, Salgado et al. [3, 29] discussed a particular extended invariant form

where $\langle\cdots\rangle_{\mathcal{g}}$ stands for a multilinear symmetric invariant polynomial $\langle\cdots\rangle_{\mathcal{g}}:\mathcal{g}^{n+1}\longrightarrow\mathbb{R}$ for the Lie algebra $\mathcal{g}$, and $F=dA+A\wedge A$ is the curvature 2-form for the $\mathcal{g}$-valued connection 1-form $A$ and $H=dB+[A,B]$ is the curvature 3-form for the $\mathcal{g}$-valued connection 2-form field $B$. It is straightforward to show that $d\Gamma_{2n+3}=0$ (see Ref. [26]). Then, there is a $(2n+2)$-ChSAS form $\mathfrak{C}^{2n+2}_{ChSAS}$ satisfying $\Gamma_{2n+3}=d\mathfrak{C}^{2n+2}_{ChSAS}$, which can be obtained as a special case of the generalized Chern-Weil theorem.

Generalized Chern-Weil theorem: Let $(A_{0},B_{0})$ and $(A_{1},B_{1})$ be two generalized connections, which consist of the connection 1-forms $A_{0}$, $A_{1}$ and connection 2-forms $B_{0}$, $B_{1}$, and let $(F_{0},H_{0})$ and $(F_{1},H_{1})$ be their generalized curvatures with $F_{i}=dA_{i}+A_{i}\wedge A_{i}$ and $H_{i}=dB_{i}+[A_{i},B_{i}]$ for $i=0,1$. There are two interpolations $A_{t}=A_{0}+t(A_{1}-A_{0})$ and $B_{t}=B_{0}+t(B_{1}-B_{0})$, $(0\leq t\leq 1)$, and the corresponding curvatures are given by $F_{t}=dA_{t}+A_{t}\wedge A_{t}$ and $H_{t}=dB_{t}+[A_{t},B_{t}]$. Then, the difference $\Gamma^{(1)}_{2n+3}-\Gamma^{(0)}_{2n+3}$ is an exact form

where

is called an Antoniadis-Savvidy transgression form.

For the proofs we refer the reader to Ref. [29]. Following the same procedure followed in the case of the Chern-Simons forms, the $(2n+2)$-ChSAS form can be given explicitly by

by setting $A_{0}=B_{0}=0$, $A_{1}=A$ and $B_{1}=B$ in (1.3), and the equation (1.2) becomes

Consequently, the $(2n+2)$-ChSAS form is regarded as a particular case of the Antoniadis-Savvidy transgression form, and it is analogous to the usual Chern-Simons form, but in even dimensions.

Following Ref. [3], Salgado et al. presented a generalized ECHF to deal with the ChSAS theory. Similar to the usual ECHF, the generalized ECHF can reproduce the generalized Chern-Weil theorem as mentioned above. In addition, there is also a subspace separation method that allows the separation of the ChSAS action into bulk and boundary contributions and the splitting of the bulk Lagrangian into pieces that reflect the particular subspace structure of the gauge algebra. By the same token, this method is built upon the generalized triangle equation

where the exact form of the boundary contribution can be given by

with $A_{t}=A_{0}+t(A_{1}-A_{0})+s(A_{2}-A_{1})$ and $B_{t}=B_{0}+t(B_{1}-B_{0})+s(B_{2}-B_{1})$ by using the generalized ECHF. Thus, we are led to a result: there is also a common origin of the generalized Chern-Weil theorem and the generalized triangle equation.

Based on the higher gauge theory, the ChSAS form has been generalized to the higher partner, i.e., the higher ChSAS form [12], which is given by higher gauge fields. This poses the problem that, analogously to what happens in ordinary and generalized gauge theory, whether the above results are adapted to the case of the higher ChSAS theory? It is the purpose of this paper to construct higher ECHF to reproduce the higher Chern–Weil theorem and give the triangle equation.

This work is organized as follows: in Section 2, we briefly review the principal aspects of the higher ChSAS theory. In Section 3, we construct the higher ECHF and present higher descent equations, which can be seen as a particular case of the former. Then, we show that the higher ECHF can reproduce the higher Chern-Weil theorem and give rise to the triangle equation, mimicking what happens in the ordinary and generalized ECHF. In Section 4, we study the higher Cartan homotopy formula and show that a higher transgression form can be written as the difference of two higher ChSAS forms minus an exact form. Lastly, we finish in Section 5 with conclusions and some considerations on future possible developments.

In this section, we will give a brief exposition of the higher ChSAS theory, and set up notations and terminologies which are needed in subsequent sections. For more details, we refer the reader to Ref. [12].

Consider a principal 2-bundle $E$ over $M$ which is an oriented, compact manifold without boundary, and let the structure Lie 2-group of $E$ be given in terms of the Lie crossed module $(H,G;\bar{\alpha},\bar{\vartriangleright})$ with corresponding differential crossed module $(\mathcal{h},\mathcal{g};\alpha,\vartriangleright)$ (see A for the definitions). A 2-connection on $E$ is a couple $(A,B)$ with $A=\sum\limits_{a}A^{a}X_{a}\in\Omega^{1}(M,\mathcal{g})$ and $B=\sum\limits_{b}B^{b}Y_{b}\in\Omega^{2}(M,\mathcal{h})$, where $A^{a}$ and $B^{b}$ are scalar differential $1$- and $2$-form, and $X_{a}$ and $Y_{b}$ are the Lie algebra generator bases of $\mathcal{g}$ and $\mathcal{h}$, respectively. The corresponding 2-curvature forms $\mathcal{F}\in\Omega^{2}(M,\mathcal{g})$ and $\mathcal{G}\in\Omega^{3}(M,\mathcal{h})$ are given by

which automatically satisfy the 2-Bianchi Identities,

Call $(A,B)$ fake-flat, if $\mathcal{F}=0$, and flat, if it is fake-flat and $\mathcal{G}=0$. Moreover, there is a general 2-gauge transformation for the 2-connection $(A,B)$,

with $g\in G$ and $\phi\in\Omega^{1}(M,\mathcal{h})$. The corresponding curvatures transform as follows

In the above expression, we have mostly followed the notations and conventions of [12]. We denote the vector space of $\mathcal{g}$-valued differential $k$-forms on $M$ over $C^{\infty}(M)$ by $\Omega^{k}(M,\mathcal{g})$. The convention also applies to $\mathcal{h}$. We define the exterior differential and wedge product of Lie algebra valued differential forms to be

for $A_{1}=\sum\limits_{a}A^{a}_{1}X_{a}\in\Omega^{k_{1}}(M,\mathcal{g})$, $A_{2}=\sum\limits_{a}A^{a}_{2}X_{a}\in\Omega^{k_{2}}(M,\mathcal{g})$.

Imitating the extended invariant form (1.1), there exists a higher invariant form

where $\langle\cdots\rangle_{\mathcal{g}\mathcal{h}}$ stands for a generalized multilinear symmetric invariant polynomial for the differential crossed modules $(\mathcal{h},\mathcal{g};\alpha,\vartriangleright)$, whose definition is given by B. It is easy to check that this form is gauge invariant under the higher gauge transformation (2.4) and (2.5). Besides, it is closed, i.e., $d\langle\mathcal{F}^{n},\mathcal{G}\rangle_{\mathcal{g}\mathcal{h}}=0$, by direct computation of the derivative. According to the Poincaré lemma, this implies that $\mathcal{P}_{2n+3}$ can be locally written as an exterior differential of a certain $(2n+2)$-form, which can be given by the higher Chern-Weil theorem.

Higher Chern-Weil theorem: 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

for $0\leq t\leq 1$ and their curvatures are given by

Then, have

where

is called a 2-Antoniadis-Savvidy transgression form, which is a higher analog of the Antoniadis-Savvidy transgression form (1.3).

Setting $A_{0}=B_{0}=0$ and $A_{1}=A$, $B_{1}=B$ in (2.15), one can get the 2-Chern-Simons-Antoniadis-Savvidy (2ChSAS) form

which satisfies (2.14) becoming

Particularly, for $n=1$, there is the same result in [23], where the author computed $Q_{2CS}$ as the $4$-d CS form

We conclude similarly that the generalized Chern-Weil theorem can be generalized to the higher partner, whose particular case gives the explicit expression of the 2ChSAS form. In other wards, the 2ChSAS form is regard as a particular case of the 2-Antoniadis-Savvidy transgression form, which is a more general object similar to the generalized transgression form (1.3). This conclusion makes a preparation of the consideration of a subspace separation method for the 2ChSAS theory in section 3.

From what has already been given for (generalized) CS theory, we obtain that the (generalized) ECHF can reproduce the (generalized) Chern-Weil theorem and triangle equation [2, 3]. These results encourage us to develop further the higher ECHF for the 2ChSAS theory [12]. And we will show that it can also reproduce the higher Chern-Weil theorem and give the corresponding higher triangle equation. The latter states that we can find a subspace separation method to allow for a deeper understanding of the 2ChSAS Lagrangian too.

Let us consider a family of 2-connections $\{(A_{i},B_{i}),i=0,\cdots,p+1\}$ on the principal 2-bundle $E$ over the $(2n+2)$-dimensional manifold $M$. Let $T_{p+1}$ be a $(p+1)$-dimensional oriented simplex smoothly parametrized by a set of variables $\{t^{i},i=0,\cdots,p+1\}$, which must satisfy the constraints

The simplex and its boundary are denoted by

where the symbol “ $\hat{}$ ” over $A_{i},B_{i}$ indicates that $A_{i},B_{i}$ is deleted from the sequence $(A_{0},B_{0};\cdots;A_{p+1},B_{p+1})$. It follows immediately that the convex combination $(A_{t},B_{t})$ with

transforms as a gauge 2-connection in the same way as every individual $(A_{i},B_{i})$ does.

Apart from the usual antiderivation $d$ with respect to $x$, there is another antiderivation $d_{t}$ with respect to the parameter $t$. Besides, there is also a homotopy derivation $l_{t}$ given by

i.e., decreasing the degree in $dx$ by one and increasing the degree in $dt$ by one. It satisfies Leibniz’s rule as well as $d$ and $d_{t}$. The three operators $d$, $d_{t}$ and $l_{t}$ define a graded algebra satisfying

For $f(l_{t})$ a polynomial in $l_{t}$, it can be verified the relationship

by using (3.6). Taking $f(l_{t})=e^{l_{t}}$, as given by its Taylor expansion, we have

from (3.7). These results go back to the work of Zumino et al. [1].

Let $\Pi$ be a polynomial in the forms $\{A_{t},B_{t},\mathcal{F}_{t},\mathcal{G}_{t},d_{t}A_{t},d_{t}B_{t},d_{t}\mathcal{F}_{t},d_{t}\mathcal{G}_{t}\}$, where $\mathcal{F}_{t}$, $\mathcal{G}_{t}$ are the corresponding curvature forms

The action of $l_{t}$ is defined by

which derives the relations

Then, let Eq.(3.8) act on the polynomial $\Pi$, having

Expanding both sides of this equation, we obtain

which is just the higher ECHF in differential form. Integrating this equation over $T_{p+q+1}$ in the space of parameters $\{t^{i}\}$, we obtain the integral form of the higher ECHF

where $q$ is the degree of $\Pi$ on $T_{p+q+1}$. If $\Pi$ is an $m$-form on $M$, then we have $p\leq m$. About the convention for this “incomplete” integration, see Ref. [1] for more details.

Let us consider the polynomial

which has the three following properties,

• 1.

  $\Pi$ is $M$-closed, i.e., $d\Pi=0$;

• 2.

  $\Pi$ is a $0$-form on $T$, i.e., $q=0$;

• 3.

  $\Pi$ is a $(2n+3)$-form on $M$, i.e., $m=2n+3$.

Thus, we have $p$ with valued in $0,\cdots,2n+3$ and the higher ECHF (3.14) reduces in this case to

which is also called a set of higher descent equations.

In the following subsection, we will study two cases $p=0$ and $p=1$ of the Eq. (3.16). As what we have anticipated, the higher ECHF is the common origin of the higher Chern-Weil theorem and the triangle equation.

In Ref. [4], one study the relation between the transgression forms and the CS forms by using the Cartan homotopy formula. In this section, we adopt the same technique to investigate the 2ChSAS theory.

It is known that the ECHF includes as a special case the ordinary Cartan homotopy formula. The same reasoning applies to the higher ECHF (3.13), thus we consider the particular case $p=0$

which we call a higher Cartan homotopy formula in differential, and its integral form is given by

The higher Chern-Simons theorem will be reproduced if we still take $\Pi=\langle\mathcal{F}^{n}_{t},\mathcal{G}_{t}\rangle_{\mathcal{g}\mathcal{h}}$ in the above equation. This brings us back to the discussion of the subsection 3.1.

Our task now is to consider

where

It is clear that $d\Pi=\langle\mathcal{F}^{n}_{t},\mathcal{G}_{t}\rangle_{\mathcal{g}\mathcal{h}}$ and $\Pi$ is a $0$-form on the simplex $T_{1}$, i.e., $q=0$.

In this case, the equation (4.2) reads

Evidently, one can observe that

and