---
abstract: 'A classical theory of general relativity in the $4$-dimensional space time is formulated as a Chern–Weil topological theory. An Einstein–Hilbert gravitational action is shown to be invariant with respect to a novel translation (co-translation) operator up to total derivative, and thus a topological invariant of the second Chern class exists due to the Chern–Weil theorem. With a topological insight, fundamental forms are introduced as a principal bundle of a space time manifold. Canonical quantization of the system is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism. A complete set of the quantum Lagrangian and BRS transformations including auxiliary and ghost fields is given in a self-consistent manner. An appropriate Hilbert space and physical states are introduced into the theory, and the positivity of the physical states and the unitarity of the transition matrix are ensured by the Kugo–Ojima theorem. A non-renormalizability of quantum gravity is reconsidered under the formulation proposed in this study.'
author:
- Yoshimasa
bibliography:
- 'ref.bib'
title: 'General relativity as the four-dimensional Chern–Weil theory'
---

Introduction {#intro}
============

The theory of general relativity is one of the most fundamental theories treating the space time structure of the universe itself, and the correctness of general relativity has been established by many experiments. In particular, new evidence owing to the seminal discovery of gravitational waves was added in 2016[@PhysRevLett.116.241103]. On the other hand, at the microscopic level, nature is described by quantum mechanics. The standard theory of particle physics based on the quantum field theory is shown to be well-established by the discovery of the Higgs boson[@Aad:2012tfa; @Chatrchyan201230]. Thus, our understanding of nature covers a wide range of length scale from the large-scale structure of the universe to the microscopic behavior of sub-atomic elements. However, these two fundamental theories, general relativity and quantum field theory, are not consistent. A construction of a quantum theory of gravity is one of the most fundamental goal of present physics.

Here, the “quantum theory of gravity” is understood as: $1$) a theory which can describe the behavior of ($4$-dimensional) space-time in the region where the uncertainty principle becomes essential ($\approx$ the Planck-length), $2$) a theory which is consistent with well-established general relativity at a large scale, and $3$) a theory which can give experimentally measurable predictions. Immediately after an establishment of general relativity and quantum mechanics in 1920’s, construction of quantum gravity was started on 1930’s. (A history of quantization of general relativity is beyond the scope of this report. For detailed history, see [@Rovelli:2000aw; @doi:10.1142qg] and references therein. .) There are three main-streams of the quantization[@Rovelli:2000aw]: 1) Covariant perturbative approach[@Fierz211]: Following the successful method of the QED, a small fluctuation from the flat Minkowski space is treated as a perturbation, then the Feynman rule of gravitational interaction is derived. This method was slowing down after a discovery of non-renormalizability of thees theories[@'tHooft:1974bx], and it becomes active again after appearing the super-string theoretical approach. 2) Canonical quantization of the metric tensor[@DeWitt:1967yk; @DeWitt:1967ub; @DeWitt:1967uc]: The metric tensor is treated as a dynamical variable and interpreted as an operator, and then it is quantized using the canonical method by requiring the commutation relations. The quantum equation of motion is obtained as the deWitte–Wheeler equation[@DeWitt:1967yk]. This approach is also slowing down because the deWitte–Wheeer equation is not mathematically well-defined, and recently it is renovated as loop-gravity[@rovelli2004] and developments are still continuing. 3) Path-integration quantization: When the path-integration method is simply applied to gravity, non-renormalizable infinities appear as the same as the first approach. A spin-network method[@Rovelli:1995ac] can be categorized as this approach. In despite of the decades of intensive efforts on quantization, a widely accepted theory that satisfies above requirements does not yet exist.

In contrast to the $4$-dimensional case, it is known that quantum gravity exists in the $(1\hspace{-.1em}+\hspace{-.1em}2)$-dimensional space as a Chern–Simons topological theory[@witten1988], that is renormalizable and does not exhibit dynamics[@Witten198846]. We note that $3$-dimensional general relativity does not have any dynamic degree of freedom even at a classical level. In [@Witten198846], Witten mentioned the following two points: firstly the existence of quantum gravity is due to an accidental feature of the $3$-dimensional case in which the invariant quadratic form exists, and secondly the non-renormalizability of quantum gravity in $4$-dimensions is essentially due to the absence of the invariant quadratic that allows us to understand the short-distance limit of space time as trivial zero-energy solutions. Actuary, it is known that a Chern-Simons action can give a topological invariant only in odd-dimensional spaces[@0264-9381-29-13-133001].

While, at first glance, it seems hopeless to construct quantum general relativity using a Chern-Simons form on the $4$-dimensional space time, we have found a novel symmetry, which is referred to as the co-Poincaré symmetry[@doi:10.1063/1.4990708], allows us to construct general relativity as a Chern–Weil theory on a $4$-dimensional space time manifold. This new symmetry is one extended a translation symmetry, and when it is applied on a pure gravitational Lagrangian without a cosmological term, it induces only a total derivative term. In this study, we show that the invariant quadratic can be defined in the $4$-dimensional space time by introducing a Lie algebra of the co-Poincaré group and the Einstein–Hilbert gravitational Lagrangian can be defined as a second Chern class, when there is no cosmological term.

Our approach for quantization is based on the second category “canonical quantization of the metric tensor” in the above list for quantization approaches. In this quantization method, the subject to be quantized is not the space time itself, and thus the space time coordinate $x^\mu$ is not q-number (operator), but c-number[@nakanishi1990covariant; @NakanishiSK2009]. The subject for quantization is a solution of the Einstein equation $g^{(c)}_{\mu\nu}(x)$. In classical general relativity, the geometrical metric tensor $g^{(g)}_{\mu\nu}(x)$ is given by the solution of the classical Einstein equation such as $g^{(g)}_{\mu\nu}(x)=g^{(c)}_{\mu\nu}(x)$, that is nothing other than the Einstein’s equivalent principle. In a quantum level, this relation is not simply fulfilled. The geometrical metric tensor will be given as an expected value of the quantum metric tensor $g^{(g)}_{\mu\nu}(x)=\langle g^{(q)}_{\mu\nu}(x)\rangle$. An outline of our method is as follows; We are starting from the Einstein–Hilbert action of general relativity that has the co-Poincaré symmetry and principal bundle induced by this symmetry. As a result, the fundamental forms can be identified as the spin and surface forms, which will be defined in this article. Based on the fundamental forms, the Nakanishi–Kugo–Ojima covariant quantization is performed, and the complete set of the quantum Lagrangian, equations of motion, BRS transformations and BRS charges for pure gravity is given. As the consequence of quantization, a scattering matrix must fulfill the Kugo–Ojima theorem.

This article is organized as follows: In section II, mathematical preliminaries of differential geometry are introduced in order to explain our terminologies and conventions. Standard formalism of a gravitational Lagrangian and geometrical structure of a principal (Poincaré) bundle are also introduced in this section in contrast to our novel topological approach. New translation operator is introduced in section III. It is shown that an Einstein–Hilbert Lagrangian can be recognized as a second Chern class under co-Poincaé symmetry in this section. With a topological insight given here, appropriate fundamental variables (forms) for a Hamiltonian formalism are introduced at the end of this section. An explicit formulation of canonical quantization of the general relativity using the Nakanishi–Kugo–Ojima formalism[@nakanishi1990covariant] is performed in section IV. Section V is devoted for discussions how to construct an appropriate Hilbert space and physical states on it. In consequence, it is shown that the unitarity of the quantum gravitational S-matrix ensured due to the Kugo–Ojima theorem[@kugo1979local; @Kugo1978459]. A renormalizability of our Cher–Weil general relativity is also discussed in section V. At the end, a summary of this study is given in section VI.

Preliminaries {#prep}
=============

First, standard classical general relativity is geometrically re-formulated in terms of a vierbein formalism according to Ref.[@fre2012gravity; @Kurihara2018].

Differential geometry {#DG}
---------------------

A $4$-dimensional pseudo-Riemannian manifold $(\MM,g)$ with $GL(1,3)$ symmetry is considered. On each coordinate patch $U_p\subset\MM$ around $p\in U_p$, orthonormal coordinate vectors are introduced as $x^\mu$. Accordingly, we take standard base vectors on the tangent space $T_p\MM$ as $\partial_\mu$, and those on the cotangent space $T_p^*\MM$ as $dx^\mu$. An abbreviation $\partial_\bullet:=\partial/\partial x^\bullet$ is used throughout this report. Whole manifold can be covered by such coordinate patch, and thus a tangent space $\TMM=\bigcup_pT_p\MM$ and a cotangent space $\TsMM=\bigcup_pT^*_p\MM$ can be defined on $\MM$. We note that these two base vectors are taken to dual each other such as $dx^\mu\partial_\nu=\delta^\mu_\nu$. The tensor product is defined as an ordered pair of two vectors as $\alpha\otimes\beta$ for vectors $\alpha$ and $\beta$ on the tangent space, and it is a rank-$2$ tensor. Higher rank tensors can be constructed as well. A rank-$p$ tensor space on a manifold $\bullet$ is denoted as $V^p(\bullet)$ in this study. The wedge product is defined as an antisymmetrization of a tensor product as $$\begin{aligned}
\alpha\wedge\beta:=\alpha\otimes\beta-\alpha\otimes\beta~~&\rm{and}&~~
\aaa\wedge\bbb:=\aaa\otimes\bbb-\bbb\otimes\aaa,\end{aligned}$$ for $\alpha,\beta\in V^1(\TMM)$ and $\aaa,\bbb\in V^1(\TsMM)$. We note that the same symbol “$\wedge$” is used as the antisymmetric tensor-product on both tangent and cotangent spaces. A vector on the cotangent space is named a $1$-form, and wedge products of $p$ cotangent-vectors are named a $p$-form. A scalar function is categorized as a $0$-form. A space of $p$-forms is denoted as $\Omega^p(\TsMM)$. The external derivative, that is a map $d:\Omega^p\rightarrow\Omega^{p+1}$, is introduced as a standard manner, e.g. for a form $\aaa:=a_\mu dx^\mu$, $d\aaa:=(\partial_{\mu_1}a_{\mu_2})dx^{\mu_1}\wedge dx^{\mu_2}$ and $dd\aaa=0$. The Einstein’s convention for repeated indices are used throughout this study. When the external derivative is applied on a $0$-form (function) $f(x^\mu)$, it gives a total derivative of a function such as $df(x^\mu)=(\partial_\nu f(x^\mu))dx^\nu$, and it is a $1$-form. Forms are represented using a Fraktur letters “$\AAA,\aaa,\BBB,\bbb,\CCC,\ccc\cdots$” in this study, as already used above.

The “[*dot-product*]{}” between a form and vector is represented using the standard coordinate as $\aaa\cdot{\bm b}=\sum_{\mu_1<\cdots<\mu_p}\alpha_{\mu_1\cdots \mu_p}\beta^{\mu_1\cdots \mu_p}\in\R$, where $$\begin{aligned}
\aaa&=&\sum_{\mu_1<\cdots<\mu_p}\alpha_{\mu_1\cdots \mu_p}
dx^{\mu_1}\wedge\cdots\wedge dx^{\mu_p}\in\Omega^p\left(\TsMM\right),\\
{\bm b}&=&\sum_{\mu_1<\cdots<\mu_p}\beta^{\mu_1\cdots \mu_p}
\partial_{\mu_1}\otimes\cdots\otimes\partial_{\mu_p}\in V^p\left(\TMM\right).\end{aligned}$$ A result of dot-product is independent of choice of the coordinate system, when dual bases are used on $\TMM$ and $\TsMM$.

General relativity is constructed on a $4$-dimensional (pseudo) Riemannian manifold $\MM$ with a metric tensor $g_{\mu\nu}$. A line element can be expressed as $$\begin{aligned}
ds^2&=&g_{\mu_1\mu_2}(x)dx^{\mu_1}\otimes dx^{\mu_2}.\end{aligned}$$ Because the affine connection $\Gamma^\rho_{~\mu\nu}$ is not a tensor, it is always possible to find a frame in which the affine connection vanishes as $\Gamma^\rho_{~\mu\nu}(p)=0$ at any point $p\in\MM$. A local manifold with a vanishing affine connection has a Poincaré symmetry $ISO(1,3)=SO(1,3)\ltimes T{(4)}$, where $T{(4)}$ is a group of $4$-dimensional translations. This local manifold at $p$ is referred to as the local Lorentz manifold and is denoted by $\M_p$. On $\M_p$, a vanishing gravity, such as $\partial_\rho g_{\mu\nu}=0$, is not required in general. A fiber bundle $\M:=\bigcup_p\M_p$ is regarded as the principal bundle on $\MM$. The metric tensor $diag[{\bm \eta}]=(1,-1,-1,-1)$ is employed in this study. Tangent space $\TM$ and cotangent spaces $\TsM$ can be respectively introduced as well as those on $\MM$. Local coordinate vectors, and standard base-vectors on $\TM$ and $\TsM$, are represented as $x^a$, $\partial_a$ and $\aaa^a$, respectively. A base manifold of vectors and forms, $\MM$ or $\M$, is distinguished by Greek suffices (on $\MM$) or Roman suffices (on $\M$) in our representation. A $SO(1,3)$ invariant coordinate-transformation $\Lambda$ forms an automorphism group referred to as the Lorentz transformation, and it is represented as; $$\begin{aligned}
\Lambda:V^p(\TM)\rightarrow V^p(\TM):
{\bm \alpha}\mapsto\widetilde{\bm \alpha}={\bm \Lambda}^p{\bm \alpha},&~~~&
\left[{\bm \Lambda}^p{\bm \alpha}\right]^{a_1\cdots a_p}:=
\left[{\bm \Lambda}\right]^{a_1}_{~b_1}\cdots\left[{\bm \Lambda}\right]^{a_p}_{~b_p}
\left[{\bm \alpha}\right]^{b_1\cdots b_p},\end{aligned}$$ where ${\bm \Lambda}$ is the Lorentz transformation matrix. A notation $[\bullet]$ shows a component of a tensor, form, or matrix $\bullet$, and it is used throughout this study. A map $\Varepsilon:\MM\rightarrow\M:{\bm g}\mapsto{\bm \eta}$ is a kind of general coordinate-transformations (diffeomorphism), and is represented using a standard coordinate on any chart of $\MM$ as; $$\begin{aligned}
\eta^{ab}&=&\Varepsilon^a_{\mu_1}(x^\mu)\Varepsilon^b_{\mu_2}(x^\mu)g^{\mu_1\mu_2}(x^\mu),\end{aligned}$$ where $\eta^{ab}=[{\bm \eta}]^{ab}$ and $g^{\mu\nu}=[{\bm g}]^{\mu\nu}$ are the metric tensors on $\M$ and $\MM$, respectively. A function $\Varepsilon^a_{\mu}(x^\mu)=[{\bm \Varepsilon}(x^\mu)]^a_{\mu}$ is referred to as the vierbein. We note that the diffeomorphism map $\Varepsilon$ induces isomorphism $\TMM\cong \TM$, and thus a form $\aaa\in\Omega^p(\TsM)$ and its pull-back $\Varepsilon^*[\aaa]\in\Omega^p(\TsMM)$ are identified each other and denoted simply $\aaa\in\Omega^p$ in this study. Orthogonal base vectors can be obtained respectively from those on $\TMM$ and $\TsMM$ as $\partial_a:=\Varepsilon_a^\mu\partial_\mu$ and $\eee^a:=\Varepsilon^a_\mu dx^\mu$, where $\Varepsilon_a^\mu:=[{\bm \Varepsilon}^{-1}]_a^\mu$. They are dual bases each other such as $
\eee^a\cdot\partial_b=\Varepsilon^a_{\mu_1}\Varepsilon_b^{\mu_2}dx^{\mu_1}\partial_{\mu_2}
=\Varepsilon^a_{\mu}\Varepsilon_b^{\mu}=\delta^a_b.
$ A base vector $\eee^a$ is referred to as the virbein form, which is $\eee\in\Omega^1\cap V^1(\TM)$. (We have to avoid a representation “$dx^a=\Varepsilon^a_\mu dx^\mu$” because it is not closed $d\left(\Varepsilon^a_\mu dx^\mu\right)\neq0$.) A spin connection $\www\in\sss\ooo(1,3)$ can be introduced such as $\www^t\eta+\eta\www=0$, where $\www^t$ is a transpose matrix of $\www$. The spin connection $\www$ is a connection-valued $1$-form on $\TsMM$ and is referred to as the spin form hereafter. The spin form can be represented using a standard coordinate as $\www^{ab}=\omega^{~a}_{\mu~c}\eta^{cb}dx^\mu$ where $\omega^{~a}_{\mu~c}:=[\www]^{~a}_{\mu~c}$, and it is anti-symmetric as $\www^{ab}=-\www^{ba}$. The spin form is not a Lorentz tensor, and its Lorentz transformation is given as; $$\begin{aligned}
\widetilde{\www}&=&{\bm \Lambda}^{-1}\www{\bm \Lambda}+{\bm \Lambda}^{-1}d{\bm \Lambda}.\label{wLT}\end{aligned}$$ A $SO(1,3)$ covariant derivative $\td$ can be represented using the spin form as; $$\begin{aligned}
\td\aaa^{a_1\cdots a_p}&=&
d \aaa^{a_1\cdots a_p}+\www^{a_1}_{~~b}\wedge\aaa^{ba_2\cdots a_p}+\cdots+
\www^{a_p}_{~~b}\wedge\aaa^{a_1\cdots a_{p-1}b}.\end{aligned}$$ A direct calculation can show that $d_\www \aaa^{a_1\cdots a_p}$ is transformed as a rank-$p$ tensor under the $SO(1,3)$ transformation (\[wLT\]).

A torsion $2$-form $\TTT^a$ can be defined using the covariant derivative as $\TTT^a:=\td\eee^a=d\eee^a+\www^a_{~b}\wedge\eee^b$. Hereafter, dummy [**Roman**]{}-indices are often abbreviate to a dot (or an asterisk), and this notation is used throughout this study. Dummy [**Greek**]{}-indices are not abbreviated in contrast. In this notation, a curvature $2$-form is written as $\RRR^{ab}:=d\www^{ab}+\www^a_{~\bcdot}\wedge\www^{\bcdot b}$. When multiple dots appear in an expression, pairing must be a left-to-right order at both upper and lower indices, e.g., a $4$-dimensional volume form can be written as $\vvv:=\frac{1}{4!}\epsilon_{\cdott\cdott}\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot=\frac{1}{4!}\epsilon_{abcd}\eee^{a}\wedge\eee^{b}\wedge\eee^{c}\wedge\eee^{d}$, where $\epsilon_{abcd}$ is a completely antisymmetric tensor (the Levi Civita tensor) whose component is $\epsilon_{0123}=1$ on $\TM$. We note that there are two constant-valued tensors on $\TM$, the metric tensor ${\bm \eta}$ and the Levi Civita tensol ${\bm \epsilon}$, and they are not a constant valued on $\TMM$. A $2$-dimensional surface form is defined as $\SSS_{ab}:=\frac{1}{2}\epsilon_{ab\cdott}\eee^\bcdot\wedge\eee^\bcdot$ which is a $2$-dimensional surface perpendicular to both $\eee^a$ and $\eee^b$. While the spin form $\www^{ab}$ is not a $SO(1,3)$ tensor-valued form, the curvature form $\RRR^{ab}$ is a Lorentz tensor-valued $2$-form. More precisely, $\RRR^{ab}$ is a rank-$2$ tensor on $\TM$ with respect to Roman indices, and is $2$-from on $\TsM$ (and also on $\TsMM$), such as $\RRR\in\Omega^2\cap V^2(\TM)$. Using this notation, the Bianchi identities can be expressed as; $$\begin{aligned}
\td\TTT^a=\RRR^{a\bcdot}\wedge\eee^{\bcdot}\hspace{.1em}\eta_{\bcdott},&~~~&
~\td\RRR^{ab}=0.\label{bianchi1}\end{aligned}$$ A volume form $\vvv$ which is $GL(1,3)$ (and also $SO(1,3)$) invariant can written as; $$\begin{aligned}
\vvv&=&\frac{1}{4!}\epsilon_{\cdott\cdott}\Varepsilon^\bcdot_{\mu_1}\Varepsilon^\bcdot_{\mu_2}
\Varepsilon^\bcdot_{\mu_3}\Varepsilon^\bcdot_{\mu_4}
dx^{\mu_1}\wedge dx^{\mu_2}\wedge dx^{\mu_3}\wedge dx^{\mu_4}
=\sqrt{-det[{\bm g}]}dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}.\end{aligned}$$ We note that $\sqrt{-det[{\bm g}]}=-det[{\bm \Varepsilon}]$.

In addition to that, a new operator is introduced for shorter representations. This is a map that transfers a rank-$p$ tensor field to a rank-$(n\hspace{-.1em}-\hspace{-.1em}p)$ tensor field defined on an $n$-dimensional space time manifold such that $$\begin{aligned}
\overline{\aaa^{a_1\cdots a_p}}&=&\overline{\aaa}_{a_1\cdots a_{n-p}}=~~
\frac{1}{p!}\epsilon_{b_1\cdots b_p a_1\cdots a_{n-p}}
\aaa^{b_1\cdots b_p},\\
\overline{\aaa_{a_1\cdots a_p}}&=&\overline{\aaa}^{a_1\cdots a_{n-p}}=-\frac{1}{p!}
\epsilon^{a_1\cdots a_{n-p} b_1\cdots b_p}
\aaa_{b_1\cdots b_p},\end{aligned}$$ where $\aaa\in\Omega^p$ and $\epsilon$ is a completely anti-symmetric tensor in $n$-dimensions. It follows the relation $\overline{\overline{\aaa}}=\aaa$, when $\aaa$ is anti-symmetric with respect to all indices. The surface form can be expressed using this expression as $\SSS_{ab}=\overline{\eee^a\wedge\eee^b}$. We note that this operator is totally different from the Hodge-$*$ operator. In contrast with that the Hodge-$*$ operator maps a $p$-form to $(n\hspace{-.2em}-\hspace{-.2em}p)$-form, this operator, which is referred to as the [*bar-dual operator*]{} in this study, maps a rank-$p$ tensor to a rank-$(n\hspace{-.2em}-\hspace{-.2em}p)$ tensor with keeping a rank of a form as $\bar{\bullet}:\Omega^q\cap V^p(\TM)\rightarrow\Omega^q\cap V^{n-p}(\TM)$. We note that the Hodge-$*$ operator is not appropriate for constructing general relativity because the metric tensor is hidden in its definition.

Lagrangian formalism {#lagrange}
--------------------

The Lagrangian form of gravity that reproduces the Einstein equation without matter fields can be written as $$\begin{aligned}
\LLL_G(\vomega,\eee)
&=&\frac{1}{2}\SSS_\cdott\wedge\left(
\RRR^\cdott-\frac{\Lambda}{3!}\overline{\SSS}^\cdott
\right),
\label{Lagrangian44}\end{aligned}$$ where $\Lambda$ is the cosmological constant. While this Lagrangian form is invariant under the general coordinate transformation and the local ${SO}(1,3)$ transformation, it is not invariant under a local translation. We will discuss this point in detail later in a next section. On the local Lorentz manifold, number of independent components of the Lagrangian form is ten, of which, six components are from the curvature $2$-form and four components are from the vierbein forms. These degrees of freedom (d.o.f.) correspond to the total number of d.o.f$.$ from the Poincarè group. The Lagrangian form is now treated as a functional with two independent forms, $\{\vomega^{ab}, \eee^c\}$, and is taken to vary independently. This method is commonly known as the Palatini method[@Palatini2008; @ferreris]. The gravitational action is introduced as the integration of the Lagrangian such that $$\begin{aligned}
\I_G
&=&\frac{c^4}{4\pi G}\int_{\Sigma}\LLL_G.\label{gravact}\end{aligned}$$ The integration region $\Sigma$ is taken to be an entire space of $\MM$, for instance.

The equations of motion with respect to the two independent functions (spin and vierbein forms) can be obtained by requiring a stationary condition for the variation of the action for each form separately. From the variation with respect to the spin form, an equation of motion, $$\begin{aligned}
\TTT^a&=&d\eee^a+
\vomega^{a}_{~\bcdot}\wedge\eee^\bcdot=0,\label{torsion less}\end{aligned}$$ can be obtained, which is referred to as the torsion less condition. The torsion less condition is obtained from the solution of the equation of motion, rather than being an independent constraint on the manifold. This equation of motion includes six independent equations, which is the same as the number of the independent components of the spin form. Therefore, the spin form can be uniquely determined from (\[torsion less\]) when the vierbein forms are given. Next, taking the variation with respect to the vierbein form, one can obtain an equation of motion as; $$\begin{aligned}
\epsilon_{a\cdott\bcdot}\left(
\frac{1}{2}\RRR^\cdott\wedge\eee^\bcdot
-\frac{\Lambda}{3!}\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot
\right)&=&0.\label{EoM2}\end{aligned}$$ This is the Einstein equation without any matter fields. The curvature and vierbein forms can be uniquely determined by simultaneously solving the equations of motion (\[torsion less\]) and (\[EoM2\]) as a quotient set of ${GL}(1,3)$ diffeomorphism. The Planck unit ($c=\hbar=4\pi G=1$) is employed hereafter.

Form-tensor duality {#TcoT}
-------------------

Classical general relativity can be represented simply and clearly in a covariant manner using differential forms as shown in previous section. This representation is referred to as the cotangent representation because differential forms are defined on a cotangent bundle $\TsM$. While the cotangent representation is suitable for a formal discussion due to its simplicity of the expression, it is not convenient for concrete calculations. On the other hand, a tensor representation on a tangent bundle $\TM$ is suitable for concrete calculations, and it is referred to as the tangent representation, hereafter. A relation between these two representations is discussed in this section.

A scalar function valued $p$-form $\aaa\in\Omega^p\otimes V^0(\TM)$ is considered here. The dual tensor ${\bm A}$ of the form $\aaa$ is introduced through a map $\V:\Omega^p\rightarrow V^p:\aaa\mapsto \AA$. A form $\aaa$ can be expanded using a standard base on $\TsM$ as; $$\begin{aligned}
\aaa
&=:&\sum_{a_1<\cdots<a_p}
\alpha_{a_1\cdots a_p}\hspace{.1em}\eee^{a_1}\wedge\cdots\wedge\eee^{a_p}
=\frac{1}{p!}\alpha_{a_1\cdots a_p}\hspace{.1em}\eee^{a_1}\wedge\cdots\wedge\eee^{a_p}.\end{aligned}$$ We note that $\alpha_{a_1\cdots a_p}$ is completely antisymmetric with respect to its lower suffixes. A rank-$p$ tensor $\AA\in V^p$ can be introduced using these expansion coefficients as; $$\begin{aligned}
\AA&:=&
\frac{1}{p!}\alpha^{a_1\cdots a_p}\hspace{.1em}
\partial_{a_1}\wedge\cdots\wedge\partial_{a_p},\end{aligned}$$ where $\alpha^{a_1\cdots a_p}:=\eta^{a_1b_1}\cdots\eta^{a_pb_p}\alpha_{b_1\cdots b_p}$. We note that symmetric components of the tensor $\AA$ does not appear due to antisymmetry of the coefficient $\alpha^{a_1\cdots a_p}$. This map uniquely gives the dual tensor independently from a choice of coordinate system. A dot product between the form and its dual vector is given as $$\begin{aligned}
\aaa\cdot\AA&=&\sum_{a_1<\cdots<a_p}\left(\alpha^{a_1\cdots a_p}\right)^2\in\R,\end{aligned}$$ which is the length of the vector $\AA$, and thus it is $SO(1,3)$ invariant.

An external derivative of $\aaa\in\Omega^p\otimes V^0(\TM)$ can be written as; $$\begin{aligned}
d\aaa&=&\frac{1}{p!}\left(\partial_{a_0}\alpha_{a_1\cdots a_p}\right)\eee^{a_0}
\wedge\eee^{a_1}\wedge\cdots\wedge\eee^{a_p}+\frac{1}{(p-1)!}\alpha_{a_1\cdots a_p}d\eee^{a_1}\wedge\cdots\wedge\eee^{a_p},\\
&=&\frac{1}{(p+1)!}\alpha_{a_0a_1\cdots a_p}\eee^{a_0}\wedge\eee^{a_1}\wedge\cdots\wedge\eee^{a_p},\end{aligned}$$ where $$\begin{aligned}
\alpha_{a_0a_1\cdots a_p}&=&(p+1)\left(\partial_{a_0}\alpha_{a_1\cdots a_p}+p\hspace{.1em}\alpha_{ba_2\cdots a_p}(\partial_{a_0}\Varepsilon^{b}_\nu)\Varepsilon^\nu_{a_1}\right), \end{aligned}$$ and thus $d\aaa$ is a $(p\hspace{-.2em}+\hspace{-.2em}1)$-form and its dual is a rank-($p\hspace{-.2em}+\hspace{-.2em}1$) tensor. On the other hand, an external derivative for the dual vector $\AA$ can be obtained as $$\begin{aligned}
d\AA&=&\frac{1}{p!}\left(\partial_{a_0}\alpha^{a_1\cdots a_p}\right)\hspace{.1em}
\eee^{a_0}\hspace{-.1em}\wedge\partial_{a_1}\wedge\cdots\wedge\partial_{a_p}\\&~&+~
\frac{1}{(p-1)!}\alpha^{ba_2\cdots a_p}
\left(\partial_{a_0}\Varepsilon^\nu_b\right)\Varepsilon^{a_1}_\nu\hspace{.1em}\eee^{a_0}\hspace{-.1em}\wedge\partial_{a_1}
\wedge\partial_{a_2}\wedge\cdots\wedge\partial_{a_p}.\end{aligned}$$ Therefore, one can confirm a diagram that; $$\begin{aligned}
\begin{array}{ccc}
\aaa&\longrightarrow^{\hspace{-1.2em}\V}&\AA \\
d\downarrow~&~&d\downarrow~\\
%d\aaa&\longleftarrow^{\hspace{-1.2em}\V^{-1}}&d\AA\label{diag}
d\aaa&\longrightarrow^{\hspace{-1.2em}\V      }&d\AA\label{diag}
\end{array}\end{aligned}$$ This diagram can be straightforwardly extended for a rank-$q$ tensor valued $p$-forms applying it to each tensor component. E.g., the curvature $2$-form $\RRR^{ab}\in\Omega^2\cap V^2(\TM)$ can be expanded as; $$\begin{aligned}
\RRR^{ab}&=&\frac{1}{2}R^{ab}_{~~c_1c_2}\eee^{c_1}\wedge\eee^{c_2},\end{aligned}$$ where $R^{ab}_{~~c_1c_2}$ is the equivalent to the standard Riemann curvature tensor. The Einstein equation on the cotangent bundle can be translated into that on the tangent bundle as; $$\begin{aligned}
\epsilon_{a\cdott\bcdot}\RRR^\bcdott\wedge\eee^\bcdot=0 &\longleftrightarrow^{\hspace{-1.2em}\V}&~
G^{ab}\hspace{.2em}\partial_a\otimes\partial_b=0,\end{aligned}$$ where $G^{ab}$ is a rank-$2$ tensor referred to as the Einstein tensor defined as; $$\begin{aligned}
G^{ab}&:=&R^{ab}-\frac{1}{2}\eta^{ab}R,\end{aligned}$$ and $R^{ab}=\Ri^{\bcdot b}_{~~\bcdot \ast}\eta^{a\ast}$ is a Ricci tensor, and $R=\Ri^{\bcdot\ast}_{~~\bcdot\ast}$ is a scalar curvature. We note that both $R^{ab}$ and $\eta^{ab}$ (and thus the Einstein tensor $G^{ab}$) are symmetric tensor. While two representations looked to have different degrees of freedom (d.o.f$.$) at first glance, it is not the case. A tangent-bundle representation has $4\times(4+1)/2-4=6$ d.o.f$.$ because the Einstein tensor is symmetric and divergenceless under the $SO(1,3)$ covariant derivative. On the other hand, a cotangent-bundle representation of the Einstein equation is given by a vector-valued three-form, and thus it can be written as; $$\begin{aligned}
\epsilon_{a\cdott\bcdot}\RRR^\bcdott\wedge\eee^\bcdot=\frac{1}{3!}[R\hspace{-.2em}E_a]_{\bcdot\bcdott}
\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot,~(a=0,1,2,3),\end{aligned}$$ where $[R\hspace{-.2em}E_a]_{bcd}$ is an expansion coefficient of the form $\epsilon_{a\cdott\bcdot}\RRR^\bcdott\wedge\eee^\bcdot$, and $R\hspace{-.2em}E_a$ is completely anti-symmetric rank-three tensor. There are four such forms according to $a=0,1,2,3$ and they are not independent each other, and it forms a completely antisymmetric rank-$4$ tensor as $[R\hspace{-.2em}E]_{abcd}:=[R\hspace{-.2em}E_a]_{bcd}$. The d.o.f$.$ of $R\hspace{-.2em}E$ is $4\times(4-1)/2=6$, and thus it is the same as that in the tangent-representation. On the other hand, a total d.o.f$.$ of Poincaré group is ten. Other four d.o.f$.$ out of total ten d.o.f$.$ are fixed due to a torsion-less equation, which has four independent constraints on the spin form.

A functional derivative with respect to a form can be understand as follows: A 1-forms $\aaa^{ab}:=\alpha^{~ab}_{\bcdot}\eee^\bcdot\in\Omega^1\otimes V^2(\TM)$ and a functional of that $\fff(\aaa^{a_1a_2})$ are considered as an example. On the tangent space $\TM$, a functional derivative with respect to a tensor function can be understand as an operation such as; $$\begin{aligned}
\left(\frac{\delta~~~~}{\delta\alpha^{~b_1b_2}_{b_3}}\right)\fff\left(\alpha^{~a_1a_2}_{\bcdot}\eee^\bcdot\right)
&=&
\fff\left(\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}\eee^{b_3}\right),\end{aligned}$$ where $\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}=\delta^{a_1}_{b_1}\delta^{a_2}_{b_2}-\delta^{a_2}_{b_1}\delta^{a_1}_{b_2}$. This operator often appears in the quantum field theory, and it is independent of a choice of coordinate systems. A functional derivative on the cotangent space $\TsM$ can be defined as; $$\begin{aligned}
\frac{\delta\fff(\aaa^{a_1a_2})}{\delta\aaa^{b_1b_2}}&:=&
\left(\iota_{\bcdot}\frac{\delta~~~~}{\delta\alpha^{~b_1b_2}_\bcdot}\right)\fff\left(\alpha^{~a_1a_2}_{\bcdot}\eee^\bcdot\right)
=
\fff\left(\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}\right),\end{aligned}$$ where $\iota_a$ is a contraction with respect to a base vector $\partial_a$ such as $\iota_a\eee^b=\delta^b_a$. This is also a coordinate independent when dual bases are used. It can be extended straightforwardly to more general cases such as $\aaa\in\Omega^p\otimes V^q$. Direct calculations can show that a formal operation, e.g., $$\begin{aligned}
\frac{\delta~~~~}{\delta\aaa^{a_1a_2a_3}}\left(\frac{1}{3!}\aaa^{\bcdot\cdott}\wedge\bbb_{\bcdot\cdott}\right)&=&
\bbb_{a_1a_2a_3},\end{aligned}$$ gives a consistent result with that in the tangent representation. We note that $\left(\iota_{b_2}\otimes\iota_{b_1}\right)\eee^{a_1}\wedge\eee^{a_2}=\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}$.

Poincaé bundle {#poincare}
--------------

The general relativity described above can be seen as a geometric theory of the manifold with a global ${GL}(1,3)$ symmetry equipped with the local Poincaré bundle as the principal bundle. Let us introduce the generators $P_a$ for $\ttt{(4)}$ and $J_{ab}$ for $\sss\ooo(1,3)$. The commutation relations of the standard Lie algebra of $\iii\sss\ooo(1,3)$ are given as follows; $$\begin{aligned}
\left[P_a,P_b\right]&=&0,\label{PJcr1}\\
\left[J_{ab},P_c\right]&=&-\eta_{ac}P_b+\eta_{bc}P_a,\label{PJcr2}\\
\left[J_{ab},J_{cd}\right]&=&
-\eta_{ac}J_{bd}+\eta_{bc}J_{ad}
-\eta_{bd}J_{ac}+\eta_{ad}J_{bc}.\label{PJcr3}\end{aligned}$$ According to these generators, the principal (Poincaré) connection can be introduced as follows; $$\begin{aligned}
\AAA_0&=&\{J_\cdott, P_\bcdot\}\times\{\vomega^\cdott,\eee^\bcdot\}
=J_\cdott\vomega^\cdott+ P_\bcdot\eee^\bcdot.\label{PCoF}\end{aligned}$$ A curvature $2$-form based on the principal connection can be expressed as follows; $$\begin{aligned}
\FFF_0&=&d\AAA_0+\AAA_0\wedge\AAA_0
= J_\cdott\RRR^\cdott+ P_\bcdot\TTT^\bcdot.\label{PCaF}\end{aligned}$$

New topological formulation {#topol}
===========================

So far we introduced a standard formalism of general relativity in terms of differential geometry. While the vierbein form can be considered as a gauge field of a local translation symmetry, general relativity cannot be interpreted as a $ISO(1,3) $ gauge theory, because the Lagrangian does not keep $ISO(1,3)$ symmetry. In this section, we introduce a new translation operator, under which the Lagrangian is invariant up to a total derivative. Under this novel symmetry, an Einstein–Hilbert Lagrangian is interpreted as the topological invariant of the second Chern class. The mathematical details in this section can be found in Ref.[@doi:10.1063/1.4990708].

Co-translation operator {#newsym}
-----------------------

As pointed out by Witten[@Witten198846], there is no invariant quadratic-form, and the second Chern class $\mathrm{Tr}[\FFF_0\wedge\FFF_0]$ does not match with the $4$-dimensional Einstein–Hilbert action. The reason why the general relativity on the $(1\hspace{-.2em}+\hspace{-.2em}2)$-dimensional spacetime can be understood as the Chern–Simons theory is that there exists the ${ISO}(1,2)$ non-degenerate invariant, $\epsilon_{\cdott\bcdot}P^\bcdot J^\cdott$. In contrast with the $4$-dimensional case, the Einstein–Hilbert action in ($1\hspace{-.2em}+\hspace{-.2em}2$)-dimension is translation invariant[@Witten198846], and the translation only induces a total derivative on the Lagrangian form[@0264-9381-29-13-133001].

To realize the same trick in the $4$-dimensional theory, we introduce a new operator defined as; $$\begin{aligned}
P_{ab}&=&P_a\iota_b,\end{aligned}$$ where $\iota_a$ is a contraction with respect to the vector field $\xi^a\in\Omega^0\cap V^1(\TM)$. More precisely, it is defined as $$\begin{aligned}
\iota_a=\iota_{\xi^a},&~~~&\xi^a=\eta^{ab}\Varepsilon^{\mu}_{b}\partial_\mu=.\label{tfv}\end{aligned}$$ This operator is referred to as co-translation hereafter. The Lie algebra associated with the co-translation can be expressed as follows: $$\begin{aligned}
\left[P_{ab},P_{cd}\right]&=&0,\label{PJcr1-2}\\
\left[J_{ab},P_{cd}\right]&=&-\eta_{ac}P_{bd}+\eta_{bc}P_{ad},\label{PJcr2-2}\end{aligned}$$ and (\[PJcr3\]). The invariant quadratics corresponding to the Lie algebra are now found to be $$\begin{aligned}
%\langle J_{ab},P_{cd} \rangle&=&-\epsilon_{acbd}+\epsilon_{bcad},\label{iq1}\\
\langle J_{ab},P_{cd} \rangle&=&\epsilon_{abcd},\label{iq1}\\
\langle J_{ab},J_{cd}\rangle&=&\langle P_{ab},P_{cd}\rangle~=~0.\label{iq2}\end{aligned}$$ The principal connection can be obtained as; $$\begin{aligned}
\AAA&=&\{J_\cdott, P^\cdott\}\times\{\vomega^\cdott,\SSS_\cdott\}
= J_\cdott\vomega^\cdott+ P^\cdott\SSS_\cdott,\label{Pcon}\end{aligned}$$ where the lowering and raising of the Roman indices are done using the Lorentz metric. This connection is referred as co-Poincaré connection hereafter.

We note that the second term of (\[Pcon\]) is also a one-form because the contraction maps $p$-forms to $(p\hspace{-.2em}-\hspace{-.2em}1)$-forms. The curvature $2$-form due to the co-Poincaré connection is given by $$\begin{aligned}
\FFF&=&d\AAA+\AAA\wedge\AAA
= J_\cdott\RRR^\cdott+ P^\cdott\td\SSS_\cdott~.\label{c2f}\end{aligned}$$ The second Chern class can be constructed from the invariant quadratics (\[iq1\]), (\[iq2\]), and the curvature $2$-from (\[c2f\]) using a following method: We consider the curvature $2$-form embedded in a $5$-dimensional manifold $\Sigma_5\subset\M_5\in\left(\M_4\otimes\R\right)$, and the integration boundary is $\partial\Sigma_5:=\Sigma_4\subset\Sigma_5$. A manifold $\Sigma_5$ has no boundary other than $\Sigma_4$. An inclusion map $\varphi:\M_4\rightarrow\M_5$ maps the non-singular $(4\hspace{-.2em}\times\hspace{-.2em}4)$-matrix (rank-$2$ tensor) ${\bm T}_4$ to the non-singular $(5\hspace{-.2em}\times\hspace{-.2em}5)$-matrix (rank-$2$ tensor) ${\bm T}_5$ as; $$\begin{aligned}
{\bm T}_5&:=&
 \left[
    \begin{array}{cc}
      {\bm T}_4 & 0\\
      0 & 1 
    \end{array}
  \right].\end{aligned}$$ Standard base vectors on $\Sigma_4$ are transformed as $\varphi[(x^0,x^1,x^2,x^3)]=(x^0,x^1,x^2,x^3,1)$ on any trivial chart. A derivative of the map $D\varphi$ acts on ${\bm T}_4$ as; $$\begin{aligned}
D\varphi[{\bm T}_4]&=&\left(\varphi[{\bm T}_4+{\bm \tau}]-\varphi[{\bm T}_4]\right){\bm \tau}^{-1}=
 \left[
    \begin{array}{cc}
      {\bm I}_4 & 0\\
      0 & 0 
    \end{array}
  \right],\end{aligned}$$ where ${\bm \tau}$ and ${\bm \tau}^{-1}$ are respectively any non-singular $(4\hspace{-.2em}\times\hspace{-.2em}4)$-matrix and its inverse, and ${\bm I}_4$ is a $(4\hspace{-.2em}\times\hspace{-.2em}4)$-identity matrix. Because the map $D\varphi$ keeps a rank of a matrix, the map $\varphi$ is immersion, and thus $D\varphi$ is a map on the tangent bundle such as $D\varphi:\TM_4\rightarrow\TM_5$.

The second Chern class can be express as; $$\begin{aligned}
\frac{1}{4}\int_{\Sigma_5}\mathrm{Tr}\left[
\FFF\wedge\FFF
\right]
&=&\frac{1}{2}\int_{\Sigma_5}
\td\SSS_\cdott\wedge\RRR^\cdott
=\frac{1}{2}\int_{\Sigma_5}\td\left(\SSS_\cdott\wedge\RRR^\cdott\right)
,\label{csaction1}\end{aligned}$$ where the Bianchi identity (\[bianchi1\]) is used. Now the covariant derivative $\td(\bullet)$ can be replaced by the external derivative $d(\bullet)$ because a form $\SSS_\cdott\wedge\RRR^\cdott$ is a Lorentz scalar. Thus, the action integral can be obtained in the $4$-dimensional manifold $\M_4$ as the second Chern class as; $$\begin{aligned}
(\ref{csaction1})&=&\frac{1}{2}\int_{\Sigma_5}d\left(\SSS_\cdott\wedge\RRR^\cdott\right)
=\int_{\partial\Sigma_5=\Sigma_4}\LLL_{CW}\label{csaction0},\end{aligned}$$ where $$\begin{aligned}
\LLL_{CW}&=&\frac{1}{2}\SSS_\cdott\wedge\RRR^\cdott,\label{csaction}\end{aligned}$$ and it is nothing more than the Einstein action (\[gravact\]) in the $4$-dimensional space time without the cosmological term. This Lagrangian, $\LLL_{CW}$, is referred to as the Chern-Weil Lagrangian in this study because this topological invariant can be understood as the result of a Chern-Weil theory[@zbMATH03077491; @doi:10.1063/1.4990708]. In conclusion, we have confirmed that the Einstein–Hilbert gravitational action in the $4$-dimensional space time is the topological invariant of the second Chern class. Our formalism can be categorized as a topological theory. (Here, the topological theory does not mean the “topological field theory” whose correlation function is independent from the metric tensor. In this report, the “topological theory” means the theory with conserved topological (characteristic) classes. ) Because several theories which have an invariant characteristic class are know and investigated intensively, a brief explanation of the topological theory is summarized in Appendix \[apBF\]. Next the co-translation $\delta_{CT}$ of the Lagrangian is considered. A co-translation operator defined as $\delta_{CT}=\varepsilon^{1/2}\xi^a\delta_T\iota_a$ induces a transformation on the fundamental forms as follows: $$\begin{aligned}
\delta_{CT}\left(\eee^a\wedge\eee^b\right)&=&\varepsilon\left\{
\xi^a\wedge\td\xi^b-\xi^b\wedge\td\xi^a\right\},\label{dctS}\\
\delta_{CT} \www^{ab}&=&0,\label{dctW}\end{aligned}$$ where $\varepsilon\in\R^+$ is (positive real) infinitesimal parameter, $\delta_T$ is the translation operator which induces $
\delta_T \eee^a=\varepsilon\td\xi^a.
$ Here $\xi^a\in\Omega^0\cap V^1(\TM)$ is a vector introduced in (\[tfv\]). A result of the co-translation operator on the Lagrangian form can be obtained as follows: $$\begin{aligned}
\delta_{CT}\LLL_{CW}
&=&\frac{1}{4}d\left(\varepsilon\epsilon_{\cdott\cdott}
\RRR^\cdott\xi^\bcdot\wedge\xi^\bcdot\right),\label{sft}\end{aligned}$$ which is a total derivative. A relation (\[sft\]) does not depend on a choice of $\xi$’s. In conclusion, it is confirmed that the Einstein–Hilbert Lagrangian without neither the cosmological constant nor matter fields are invariant under the co-translation up to the total derivative.

Due to the co-translation symmetry, the $4$-dimensional general relativity can have a topological invariant of a second Chern class when there is no cosmological term. We note that the cosmological term cannot appear in the action under the invariant quadratic of (\[iq2\]), because a term $\vvv=\SSS_\cdott\wedge\overline{\SSS}^\cdott$ has no co-translation symmetry. To implement the cosmological term by means of the second Chern class, we have to use an invariant quadratic such as $\langle P_{ab},P_{cd}\rangle\neq0$. In reality, for orthogonal groups in $4$-dimension, there are two isomorphisms such as ${SO}(4)\simeq{SU}(2)\times{SU}(2)$, and ${SO}(2,2)\simeq{SL}(2,R)\times{SL}(2,L)$. The second splitting induces the invariant quadratic with $\langle P_{ab},P_{cd}\rangle\neq0$. However, one must use a complex representation for the second splitting in the case of the ${SO}(1,3)$ group. It is known that the complexified theory may lead to difficulties in the construction of the quantum Hilbert space[@MenaMarugan:1994sj; @Witten:2003mb; @Oda:2003iu] after the quantization of the theory. On the other hand, the three-dimensional theory has two sets of [*real*]{} invariant-quadratics[@Witten198846]. Therefore, the gravitational theory in the $(1+2)$-dimension can treat both de Sitter and anti-de Sitter spaces as the topological theory.

Hamiltonian formalism {#Hamilton}
---------------------

For the sake of the quantization of the theory, we rewrite the classical gravitation theory in the Hamiltonian formalism in a completely covariant way. The cosmological constant is set to be zero hereafter, because our interest is in the formulation of general relativity as a Chern–Weil theory. The inclusion of the cosmological constant in the Hamiltonian formalism is straightforward[@Kurihara2018], if a topological aspect is ignored. Based on discussions in a previous section, the covariant Hamiltonian formalism is introduced as follows: In the geometrical quantization method, the principal bundle is identified as the phase space of a dynamical system, and thus $\vomega$ can be identified as the first canonical variable. Then, the second canonical variable $\MMM$ (the canonical momentum) is introduced as; $$\begin{aligned}
\MMM_{ab}=\frac{\delta{\LLL}_G}{\delta\left(d\vomega^{ab}\right)}=\SSS_{ab}.\end{aligned}$$ Therefore, the two fundamental forms $\{\vomega,\MMM=\SSS\}$ form a phase space (fundamental forms), consistent with the previous observation. The Hamiltonian density can be obtained using the standard method as; $$\begin{aligned}
\HHH_G&=&
\frac{1}{2}\MMM_\cdott\wedge d\vomega^\cdott-
{\LLL}_G
=
-\frac{1}{2}\SSS_\cdott\wedge
\vomega^{\bcdot}_{~\ast}\wedge\vomega^{\ast\bcdot}.\label{HG}\end{aligned}$$ We note that this Hamiltonian density is $not$ the local ${SO}(1,3)$ invariant because it depends on the spin form itself, which is not the Lorentz tensor. It is natural because the gravitational field can be eliminated at any point of the global manifold due to the Einsteins equivalence principle. Therefore, the energy of the gravitational field depends on the local coordinate. For example, the energy of the gravitational wave is well-defined only in the asymptotically flat space time. A quantity $\iota_a\HHH_G$ can be understood as an energy flow with respect to the direction of the vector $\xi^a$ in the flat asymptotic frame[^1].

The first canonical equations can then be obtained as; $$\begin{aligned}
\frac{\delta\HHH_G}{\delta\MMM_{ab}}&=&
-\epsilon_{\cdott\cdott}\left(\epsilon_{ab\bcdot\ast}\eee^\ast\right)^{-1}\wedge\eee^\bcdot\wedge\www^\bcdot_{~\ast}\wedge\www^{\ast\bcdot}=
d\vomega^{ab},\label{domega}\end{aligned}$$ where $\delta\bullet/\delta\SSS=(\delta\bullet/\delta\eee)(\delta\SSS/\delta\eee)^{-1}$ is used. Thus we can obtain $$\begin{aligned}
-\epsilon_{a\cdott\bcdot}\eee^\bcdot\wedge\www^\bcdot_{~\ast}\wedge\www^{\ast\bcdot}&=&
\epsilon_{a\cdott\bcdot}\eee^\bcdot\wedge d\vomega^{\cdott}.\label{canonicaleq1}\end{aligned}$$ The second canonical equations can then be obtained as; $$\begin{aligned}
\frac{\delta\HHH_G}{\delta\vomega^{ab}}&=&-d\MMM_{ab},\label{canonicaleq2}\end{aligned}$$ The first equation (\[canonicaleq1\]) gives the equation of motion, $
\epsilon_{a\cdott\bcdot}\RRR^\cdott\wedge\eee^\bcdot=0,
$ which simply leads to the Einstein equation without matter fields and the cosmological constant, and the second equation (\[canonicaleq2\]) leads to the torsion less condition, such as $\td\SSS_{a\bcdot}\wedge\eee^\bcdot=2\TTT^\bcdot\wedge\SSS_{a\bcdot}=0$, as expected.

The Poisson bracket can be introduced in the covariant formalism as; $$\begin{aligned}
\left\{\aaa,\bbb\right\}_\mathrm{PB}&=&
\frac{\delta\aaa}{\delta\vomega^\cdott}\wedge\frac{\delta\bbb}{\delta\SSS_\cdott}-
\frac{\delta\bbb}{\delta\vomega^\cdott}\wedge\frac{\delta\aaa}{\delta\SSS_\cdott}\end{aligned}$$ where $\aaa$ and $\bbb$ are arbitrary forms. The Poisson brackets for the fundamental forms become $$\begin{aligned}
\left\{\vomega^{ab},\vomega^{cd}\right\}_\mathrm{PB}&=&
\left\{\SSS_{ab},\SSS_{cd}\right\}_\mathrm{PB}~=~0,\label{pb1}\\
\left\{\vomega^{a_1a_2},\SSS_{b_1b_2}\right\}_\mathrm{PB}&=&
\delta^{[a_1}_{b_1}\delta^{a_2]}_{b_2}.\label{pb2}\end{aligned}$$ One can confirm easily that $$\begin{aligned}
\epsilon_{a\cdott\bcdot}\left\{\vomega^\cdott,\HHH_G\right\}_\mathrm{PB}\wedge\eee^\bcdot&=&
-\epsilon_{a\cdott\bcdot}\vomega^\bcdot_{~\ast}\wedge\vomega^{\ast\bcdot}\wedge\eee^\bcdot
=\epsilon_{a\cdott\bcdot}d\vomega^\cdott\wedge\eee^\bcdot,\\
%
\left\{\SSS_{ab},\HHH_G\right\}_\mathrm{PB}&=&
-\left(-\eta_{b\bcdot}\vomega^\cdott\wedge\SSS_{\bcdot a}\right)
=d\SSS_{ab},\end{aligned}$$ where the Einstein equation and torsion less condition are used. The Hamiltonian form can be understood as a generator of a total derivative of a given form.

Canonical quantization {#canonical}
======================

Quantum field theory of general relativity under the Kugo–Ojima formalism[@Kugo1978459; @kugo1979local] discussed by Nakanishi in a series of papers[@Nakanishi:1977gt; @Nakanishi:1978zx; @Nakanishi:1978np; @Nakanishi:1978rt; @Nakanishi:1979ff; @Nakanishi:1979rr; @Nakanishi:1979rq; @Nakanishi:1979um; @Nakanishi:1980rf; @Nakanishi:1980db; @Nakanishi:1980yf; @Nakanishi:1980hp; @Nakanishi:1980kf; @Nakanishi:1981fj] and summarized in reports[@nakanishi1983manifestly; @nakanishi1990covariant]. It is self consistent, physically and mathematically rigorous, and very beautiful theory. We follow their method to quantize our topological theory in this section. While discussions in this section are highly technical, they are important to understand that our quantum general relativity is mathematically well-defined.

BRS-transformation {#6-1}
------------------

The Diracs procedure of the canonical quantization for the field theories with restraint conditions, such as the Yang–Mills gauge theory, require to add supplementary terms to kill a unphysical degree of freedom, e.g. gauge fixing and Faddeev–Popov ghost terms. After killing unphysical [d.o.f$.$]{}, still a symmetry remains, i.e. the BRS-symmetry, in the Lagrangian. A systematic way to perform quantization for such a restraint system has been established by Nakanishi[@Nakanishi01061966], and Kugo and Ojima[@Kugo1978459; @kugo1979local]. The BRS-transformation of the general relativity is introduced according to their recipe[@Nakanishi01061978; @Nakanishi01071978].

Necessary auxiliary and Faddeev–Popov (anti-)ghost fields are introduced as;

-   auxiliary field:    $\beta_{\mu~b}^{~a}(x)$,

-   ghost fields:       $\chi_{~b}^a(x)$ and $\chi_\mu(x)$,

-   anti-ghost field:  $\tilde{\chi}_{\mu~b}^{~a}(x)$,

where $x\in\MM$. These fields are assumed to be Hermitian functions (operators). Here $\tilde{\bullet}$ is used to represent an anti-ghost, because a standard representation $\bar{\bullet}$ is already used for another operator. Another reason why $\bar{\bullet}$ is not used for the anti-ghost field here is that a relation between ghost and anti-ghost is not a particle–anti-particle relation, but are independent Hermitian-fields each other. The ghost field is considered to be factorized to two parts, a local Lorentz part $\chi^a_{~b}$ and a global part $\chi_\mu$, but the anti-ghost is not. The b-field and (anti-)ghost fields are assumed to be the global vector with respect to Greek indices, however not local Lorentz tensor with respect to Roman indices. Their Lorentz transformation is assumed to be the same as the spin connection. Roman indices can be brought-up or -down by $\eta^{ab}$ or $\eta_{ab}$, e.g. $\beta_\mu^{~ab}=\beta_{\mu~\bcdot}^{~a}\eta^{\bcdot b}$ and so on. Here, ghost and anti-ghost fields are assumed to anti-commute as in the quantization of the usual field theory. Moreover, those are anti-symmetric under an exchange between two Roman indices.

The BRS-transformation, denoted as $\delBRS[\bullet]$ in this report, satisfies four rules stated in Ref.[@Nakanishi01071978]. First the BRS-transformation of the coordinate vector on $\MM$ should respect the general linear-transformation as; $$\begin{aligned}
\delBRS\left[x^\mu\right]&=&\chi^\mu.\end{aligned}$$ In addition to that, we require the postulate given in [@Nakanishi01071978] such as; $$\begin{aligned}
\delBRS\left[\partial_\mu X\right]&=&\partial_\mu\delBRS\left[X\right]
-\left(\partial_\mu\delBRS\left[x^\nu\right]\right)\partial_\nu X,\end{aligned}$$ where $X$ is any field defined on $T\MM$. For differential forms on $T^*\MM$, the BRS-transformation acts as; $$\begin{aligned}
\delBRS\left[dx^\mu\right]&=&\left(\partial_\nu\delBRS\left[x^\mu\right]\right)dx^\nu
=d\left(\delBRS\left[x^\mu\right]\right)
=~d\chi^\mu.\label{deltaBRSd}\end{aligned}$$ Therefore, the BRS-transformation and external derivative are commute each other[@Nakanishi01071978], i.e., $$\begin{aligned}
\left[\delBRS,d\right]\bullet&=&\delBRS\left[d\bullet\right]-
d\left(\delBRS\left[\bullet\right]\right)=~0.\label{dBRS}\end{aligned}$$ The BRS-transformation of above fields can be defined as; $$\begin{aligned}
\left\{
\begin{array}{cl}
\delBRS\left[\beta_{\mu~b}^{~a}\right]&=~0\\
\delBRS\left[\chi_\mu\right]&=~-
g_{\mu\rho}\left(
\partial^\rho\chi^\nu+
\partial^\nu\chi^\rho
\right)\chi_\nu\label{BRSchi}\\
\delBRS\left[\chi^{a}_{~b}\right]&=~\chi^a_{~\bcdot}\chi^\bcdot_{~b}\\
\delBRS\left[\tilde{\chi}_{\mu~b}^{~a}\right]&=~i\beta_{\mu~b}^{~a}
\end{array}
\right.\end{aligned}$$ The BRS transformation is assumed to follow the Leibniz rule as; $$\begin{aligned}
\delBRS\left[XY\right]&=&\delBRS\left[X\right]Y+\epsilon_XX\delBRS\left[X\right],\label{Leib}\end{aligned}$$ where the signature $\epsilon_X=-1$ when $X\in\{\chi_{~b}^a(x),\chi_\mu(x), \tilde{\chi}_{\mu~b}^{~a}(x)\}$, and $\epsilon_X=+1$ otherwise. The BRS-transformation of vierbein and spin connection are defined as; $$\begin{aligned}
\delBRS\left[\Varepsilon^a_\mu\right]&=&
\Varepsilon^{~\bcdot}_\mu~\chi_{~\bcdot}^{a}
-\left(\partial_\mu\chi^{\nu}\right)\Varepsilon^a_\nu,\\
\delBRS\left[\omega^{~ab}_{\mu}\right]&=&
\omega_{\mu}^{~a\bcdot}~\chi_{~\bcdot}^{b}+
\omega_{\mu}^{~\bcdot b}~\chi_{~\bcdot}^{a}-
\partial_\mu\chi^{a}_{~\bcdot}~\eta^{b\bcdot}
-\left(
\partial_\mu\chi^\nu
\right)\omega^{~ab}_{\nu}.\end{aligned}$$ The BRS-transformation of the vierbein inverse $\Varepsilon^\mu_a$ can be obtained from $
\delBRS\left[\Varepsilon^a_\mu~\Varepsilon_a^\nu\right]=
\delBRS\left[\delta_\mu^\nu\right]~=~0,
$ as; $$\begin{aligned}
\delBRS\left[\Varepsilon_a^\mu\right]&=&-\chi^{\bcdot}_{~a}~\Varepsilon_{\bcdot}^{\mu}
+\Varepsilon_a^\nu\left(\partial_\nu\chi^{\mu}\right).\end{aligned}$$ Accordingly, the BRS-transformation of the metric tensor and its inverse are obtained from above relations as; $$\begin{aligned}
\delBRS\left[g_{\mu\nu}\right]&=&\delBRS\left[
\eta_\cdott
\Varepsilon^\bcdot_\mu
\Varepsilon^\bcdot_\nu
\right]=
-g_{\mu\rho}\partial_\nu\chi^\rho
-g_{\nu\rho}\partial_\mu\chi^\rho,\label{BRSg}\\
\delBRS\left[g^{\mu\nu}\right]&=&\delBRS\left[
\eta^\cdott
\Varepsilon_\bcdot^\mu
\Varepsilon_\bcdot^\nu
\right]=
g^{\mu\rho}\partial_\rho\chi^\nu+
g^{\rho\nu}\partial_\rho\chi^\mu.\label{BRSgi}\end{aligned}$$ Anti-symmetry of the ghost filed $\chi_{ab}=-\chi_{ba}$ and symmetry of the metric tensor are used here. The BRS-transformations of the vierbein and spin forms become $$\begin{aligned}
\delBRS\left[\eee^a\right]&=&
\chi_{~\bcdot}^{a}\eee^\bcdot,\\
\delBRS\left[\vomega^{ab}\right]
&=&\vomega^{a\bcdot}~\chi_{~\bcdot}^{b}+\vomega^{\bcdot b}~\chi_{~\bcdot}^{a}-d\chi^{a}_{~\bcdot}~\eta^{b\bcdot},\end{aligned}$$ follows from above results. In response to above quantities, corresponding one-forms are introduced as; $$\begin{aligned}
\left\{
\begin{array}{lll}
\bbb^{ab}&=~\beta_{\mu}^{~ab}~\partial_\nu\chi^\mu dx^\nu&=~
\beta_{\mu}^{~ab}~d\chi^\mu,\label{b-form}\\
\ccc^{a}~&=~\chi^a_{~\bcdot}~\eee^\bcdot&=~\chi^a_{~\bcdot}~\Varepsilon^\bcdot_\mu~dx^\mu,\\
\tilde{\ccc}^{ab}&=~\tilde{\chi}_{\mu}^{~ab}\partial_\nu\chi^\mu dx^\nu&=~
\tilde{\chi}_{\mu}^{~ab}~d\chi^\mu,\label{g-form}
\end{array}
\right.\end{aligned}$$ which are named as “[*b-form*]{}” and “[*$($anti-$)$ghost-form*]{}”, respectively. Note that $\bbb^{ab}$ and $\ccc^{a}$ ($\tilde{\ccc}^{ab}$) are anti-commutable (commutable), respectively. The BRS-transformations of those forms are given by $$\begin{aligned}
\left\{
\begin{array}{ll}
\delBRS\left[\bbb^{ab}\right]&=
%~\delBRS\left[\ccc\right]~=
~\delBRS\left[\ccc^a\right]~=~0,\\
\delBRS\left[\tilde{\ccc}^{ab}\right]&=~i\bbb^{ab}.
\end{array}
\right.\end{aligned}$$ Moreover, the BRS-transformation of the vierbein form can be represented as; $$\begin{aligned}
\delBRS\left[\eee^a\right]&=&\ccc^a.\end{aligned}$$ One can easily confirm that the BRS-transformation is nilpotent for all forms and fields defined above by direct calculations. A proof of nilpotency for all necessary forms are given in Appendix \[ap1\].

Quantum Lagrangian {#6-2}
------------------

We start from the Lagrangian of the classical gravity given in (\[Lagrangian44\]). The quantum Lagrangian can be obtained by adding gauge-fixing and Faddeev–Popov ghost terms to the Lagrangian for the classical gravity, i.e. $$\begin{aligned}
\LLL_{QG}&=&\LLL_G+\LLL_{GF}+\LLL_{FP},\label{QLGF}\end{aligned}$$ where $\LLL_{GF}$ is referred to as the gauge-fixing Lagrangian form, and $\LLL_{FP}$ is as the Faddeev–Popov Lagrangian form. The quantum Lagrangian form must satisfy following conditions:

1.  invariant under the general linear transformation,

2.  invariant under the local Lorentz transformation,

3.  invariant under the BRS-transformation, and

4.  nilpotent under the BRS-transformation.

Requirements $3$ and $4$ imply that the Lagrangian form must be the BRS-null object, i.e., $\delBRS\left[\LLL_{QG}\right]=0$. It can be confirmed that the classical part of the Lagrangian form $\LLL_G$ retains all conditions.

The gauge-fixing Lagrangian form is determined to obtain a desired gauge-fixing condition. Here we require the de Donder condition for the gauge fixing, which can be represented using the metric tensor as $\partial_\mu\left(\sqrt{-g}~g^{\mu\nu}\right)=0$. An alternative representation is also possible as $\Gamma^\lambda_{~\mu\nu}~g^{\mu\nu}=0$. Correspondent representations in the vierbein formalism are given as $d\SSS_{ab}=0$, which is identical to $\vomega^a_{~\bcdot}\wedge\eee^\bcdot=0$, if the torsion less condition is assumed.

A simplest candidate of the gauge-fixing form, which give the de Donder condition as a Euler-Lagrange equation with respect to variation of the b-form is given as; $$\begin{aligned}
\LLL_{GF}&=&
-\frac{1}{2}\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}
\right)\wedge\SSS_\cdott,\label{LGF}\end{aligned}$$ where $\alpha\in\R$ is a gauge fixing parameter. In a practical sense, the $\alpha$-term is vanished autonomously, i.e., $\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\wedge\SSS_\cdott=0$, due to the anti-commutation of the b-form. However, it is left here to discuss how the gauge-fixing term works. The term $d\bbb^{ab}+\alpha\bbb^a_{~\bcdot}\wedge\bbb^{\bcdot b}$ itself (without multiplying the surface form) retains Lorentz invariance, when $\alpha=1$. It can be confirmed as follows: The first two terms $d\bbb^{ab}+\bbb^a_{~\bcdot}\wedge\bbb^{\bcdot b}$ is a Lorentz tensor, because it has the same structure as the curvature form using the b-form instead of the spin from. Therefore, a $4$-form such as $\left(d\bbb^\cdott+\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\SSS_\cdott$ is local Lorentz invariant.

Searching for the Feddeev–Popov ghost-Lagrangian, we note that the gauge-fixing Lagrangian can be represented as; $$\begin{aligned}
\LLL_{GF}&=&\frac{i}{2}
\delBRS\left[d\tilde{\ccc}^\cdott+\alpha
\tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}
\right]\wedge\SSS_\cdott.\end{aligned}$$ Therefore, if the Faddeev–Popov form is taken as; $$\begin{aligned}
\LLL_{FP}&=&-\frac{i}{2}\left(
d\tilde{\ccc}^\cdott+
\alpha
\tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}
\right)\wedge\delBRS\left[\SSS_\cdott\right],\end{aligned}$$ a sum of gauge-fixing and Faddeev–Popov terms is BRS-null, because anti-ghost and surface forms are nilpotent. As the result, the Faddeev–Popov Lagrangian form can be obtained as; $$\begin{aligned}
\LLL_{FP}&=&
-\frac{i}{2}\left(d\tilde{\ccc}^\cdott+
\alpha
\tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right)
\wedge\left(\epsilon_{\cdott c_1c_2}\chi^{c_1}_{~c_3}
\eee^{c_3}\wedge\eee^{c_2}
\right),\nonumber\\
&=&-\frac{i}{2}\left(
d\tilde{\ccc}^\cdott+
\alpha
\tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right)
\wedge\ccc^\star\wedge\overline{\eee}_{\star\cdott}.\label{LFP}\end{aligned}$$ The Faddeev–Popov Lagrangian form is [*not*]{} Lorentz invariant, even if $\alpha=1$, and the $\alpha$-term in $\LLL_{FP}$ is not vanished after multiplying $\delBRS\left[\SSS_{ab}\right]$. Therefore, the Faddeev–Popov Lagrangian form can fix the Lorentz invariance. Finally the quantum Lagrangian form, (\[QLGF\]), (\[LGF\]), and (\[LFP\]) with any value of $\alpha$, satisfies all four requirements given above. According to the particle-physics terminologies, it is referred to as Landau-gauge when $\alpha=0$, and Feynman-gauge when $\alpha=1$, respectively.

An equation of motion can be obtained as the Euler–Lagrange equation under the variational principle. All relevant equations of motion are derived in Appendix \[app2\].

Commutation relations
---------------------

From the quantum Lagrangian (\[QLGF\]), the spin form $\vomega^{ab}$, and forms $\bbb^a, \tilde{\ccc}^{ab}$ defined by (\[g-form\]), the formal expressions of the commutation relation can be singled out as $$\begin{aligned}
\left[\widehat{\vomega}^{ab}(x),\widehat{\SSS}_{cd}(y)\right]&=&-i\delta^{(4)}(x-y)
\delta^{[a}_{c}\delta^{b]}_{d},\label{CR1}\\
\left[\widehat{\bbb}^{ab}(x),\widehat{\SSS}_{cd}(y)\right]&=&-i\delta^{(4)}(x-y)
\delta^{[a}_{c}\delta^{b]}_{d},\label{CR2}\\
\left\{\widehat{\tilde\ccc}^{ab}(x),(\widehat{\ccc^\bcdot\wedge\overline{\eee}}_{\bcdot cd})(y)\right\}&=&-i\delta^{(4)}(x-y)
\delta^{[a}_{c}\delta^{b]}_{d}.\label{CR3}\end{aligned}$$ where $\widehat{\bullet}$ is denoted the operator corresponding to the field or form $\bullet$, hereafter. We note that $\delta\LLL_{QG}/\delta\bbb^{ab}=\SSS_{ab}$. We dont need to treat the form $\ccc^a$ as the canonical variable. For later discussions, above commutation relations are rewritten by means of original fields instead of forms. First the commutation relations (\[CR1\]) and (\[CR2\]) are given by $$\begin{aligned}
\left[\widehat{\omega}_{\mu~\bcdot}^{~a}(x)\eta^{\bcdot b},\widehat{\SSS}_{cd}(y)\right]dx^\mu&=&-i\delta^{(3)}(x-y)
\delta^{[a}_{c}\delta^{b]}_{d},\label{fCR1}\\
\left[\widehat{\beta}_{\nu~\bcdot}^{~a}(x)\eta^{\bcdot b}\partial_\mu\widehat{\chi}^\nu,
\widehat{\SSS}_{cd}(y)\right]dx^\mu&=&-i\delta^{(3)}(x-y)
\delta^{[a}_{c}\delta^{b]}_{d}.\label{fCR2}\end{aligned}$$ They are still formal expressions with omitting a $2$-dimensional integration measure in the right-hand side in order to avoid long expressions. Since these expressions are enough for following discussions, we use these formal expressions. To obtain the field-representation for (\[CR3\]), some manipulations are necessary. The left had side of the (\[CR3\]) can be expressed as $$\begin{aligned}
(\ref{CR3})&=&
\epsilon_{cde_1e_2}~\widehat{\chi}^{e_1}_{~~\bcdot}~\epsilon^{\bcdot e_2 f_1f_2}\left\{
\widehat{\tilde{\chi}}_{\nu~\ast}^{~a}\eta^{\ast b}\partial_\mu\widehat{\chi}^\nu,\widehat{\SSS}_{f_1f_2}
\right\}dx^\mu.\end{aligned}$$ Let us introduce matrix expressions as $$\begin{aligned}
\widehat{\bm{\epsilon\chi\bar{\epsilon}}}&:=&
\epsilon_{ab e_1e_2}~\widehat{\chi}^{e_2}_{~~\bcdot}~\epsilon^{\bcdot e_1cd},\\
\widehat{\{\tilde{\bm\chi},\bm\SSS\}}&:=&
\left\{
\widehat{\tilde{\chi}}_{\nu~\bcdot}^{~a}\eta^{\bcdot b}\partial_\mu\widehat{\chi}^\nu
,\widehat{\SSS}_{cd}
\right\}dx^\mu.\end{aligned}$$ Matrices $\widehat{\bm{\epsilon\chi\bar{\epsilon}}}$ and $\widehat{\{\tilde{\bm\chi},\bm\SSS\}}$ are $4\times4$ matrices, whose elements are again $4\times4$ matrices. The commutation relation (\[CR3\]) can be express as $$\begin{aligned}
\widehat{\bm{\epsilon\chi\bar{\epsilon}}}\bcdot\widehat{\{\tilde{\bm\chi},\bm\SSS\}}&=&
-i\delta^{(3)}(x-y)~\bm{\mathrm I},\label{CR3-2}\end{aligned}$$ where $\bm{\mathrm I}$ is a matrix expression of $\delta^{[a}_c\delta^{b]}_d$ that is also a $4\times4$ matrix of $4\times4$ matrices. If each element of matrix $\widehat{\bm{\epsilon\chi\bar{\epsilon}}}$ is invertible, one can obtain the commutation relation of $\widehat{\{\tilde{\bm\chi},\bm\SSS\}}$ by applying the inverse matrix on (\[CR3-2\]). In reality, the matrix is invertible except a diagonal part of the matrix. While the diagonal part shows $
\left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}\right]_{aa}=\bm{0}
$, where $\bm{0}$ is $4\times4$ zero matrix, the diagonal part is not necessary for the inversion because the diagonal part of $\widehat{\{\tilde{\bm\chi},\bm\SSS\}}$ is also zero-matrix due to the antisymmetric of the surface form. Therefore, one can get a matrix inverse as; $$\begin{aligned}
\left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}^{-1}\right]_{IJ}&=&
\left\{
\begin{array}{cl}
\left(\left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}\right]_{JI}\right)^{-1}&(I\neq J)
\\
\bm{0}&(I=J),
\end{array}
\right.\end{aligned}$$ where $I$ and $J$ are indices for each $4\times4$ matrix. Thus the commutation relation can be obtained as; $$\begin{aligned}
\widehat{\{\tilde{\bm\chi},\bm\SSS\}}&=&
-i\delta^{(3)}(x-y)
\widehat{\bm{\epsilon\chi\bar{\epsilon}}}^{-1}, \label{fCR3}\end{aligned}$$ as the matrix representation. Each element of the matrix $\left[\widehat{\bm{\epsilon\chi\bar{\epsilon}}}^{-1}\right]_{IJ}$ is one of $\pm\left(\chi^a_{~b}\right)^{-1}$ or zero.

In summary, three non-zero commutation-relations (\[fCR1\]), (\[fCR2\]) and (\[fCR3\]) are obtained. All other commutation relations are zero.

BRS-charge {#BRSCharge}
----------

The Noether charge associated with the BRS-transformation can be constructed according to the recipe presented in [@Kurihara2018]. The conserved Noether charge can be given by $$\begin{aligned}
\QQQ_\xi&=&\frac{1}{2}\left(
\iota_\xi\vomega^\cdott
\right)\SSS_\cdott,\end{aligned}$$ where $\xi^\mu$ a vector on $T\MM$. First, taking the auxiliary field as the vector field $\xi^\mu$, we get a conserved charge as; $$\begin{aligned}
\QQQ_\bbb&:=&
\frac{1}{2}\left(\iota_{\bbb}\www^\cdott\right)\SSS_\cdott=
\frac{1}{2}g^{\mu_1\mu_2}{\beta}^{~a_1}_{\mu_3~a_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1},\end{aligned}$$ and its BRS-transformation is; $$\begin{aligned}
\widehat{\QQQ}_\bbb&:=&\delBRS\left[\QQQ_\bbb\right]=
\frac{1}{2}{\beta}^{~a_1}_{\mu_3~a_2}\delBRS\left[g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right].\end{aligned}$$ All fields in the BRS-transformation, $g$, $\partial\chi$, $\omega$, and $\SSS$, are nilpotent, therefore $\widehat{\QQQ}_\beta$ is BRS-null due to the remark in section \[ap1\]. Next, the anti-ghost form is taken as the vector field $\xi^\mu$. As a consequence, the conserved charge can be obtained as; $$\begin{aligned}
\QQQ_{\tilde\ccc}&:=&\frac{1}{2}\left(\iota_{\tilde\ccc}\www^\cdott\right)\SSS_\cdott=
\frac{1}{2}g^{\mu_1\mu_2}{\tilde\chi}^{~a_1}_{\mu_3~a_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}.\end{aligned}$$ The BRS-transformation of this charge is obtained as; $$\begin{aligned}
\widehat{\QQQ}_{\tilde\ccc}&:=&\frac{i}{2}
\delBRS\left[\QQQ_{\tilde\ccc}\right],\\&=&
\frac{i}{4}{\tilde\chi}^{~a_1}_{\mu_1~a_2}\delBRS\left[g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right]\nonumber\\
&~&-\frac{1}{4}{\beta}^{~a_1}_{\mu_1~a_2}\left(g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right),\end{aligned}$$ where a factor $i/2$ is just a convention. Then again by taking the BRS-transformation of this, one can get $$\begin{aligned}
\delBRS\left[\widehat{\QQQ}_{\tilde\ccc}\right]&=&-\frac{1}{2}
{\beta}^{~a_1}_{\mu_1~a_2}\delBRS\left[g^{\mu_1\mu_2}\partial_{\mu_1}\chi^{\mu_3}\omega_{\mu_2}^{~a_2a_3}~\SSS_{a_3a_1}\right]=
-\widehat{\QQQ}_\bbb,\end{aligned}$$ where nilpotent of $(g\omega\SSS)$ is used. Therefore, the charge $\QQQ_{\tilde\ccc}$ is not nilpotent, but $\widehat{\QQQ}_{\tilde\ccc}$ is.

Since charges $\QQQ_\xi$ is conserved[@Kurihara2018] and external derivative and the BRS-transformation are commute each other as (\[dBRS\]), both charges are conserved as $$\begin{aligned}
d\widehat{\QQQ}_\xi&=&d\left(\delBRS\left[\QQQ_\xi\right]\right)=~\delBRS\left[d\QQQ_\xi\right]=~0,\end{aligned}$$ where $\xi=\tilde{\ccc},\bbb$. In conclusion, the two conserved charges $\widehat{\QQQ}_\bbb$ and $\widehat{\QQQ}_{\tilde\ccc}$ are obtained as $$\begin{aligned}
\widehat{\QQQ}_\bbb&=&\frac{1}{2}\beta^{~\bcdot}_{\mu~\ast}~\delBRS\left[\widehat{\QQQ}_{~\bcdot}^{\mu~\ast}\right],\\
\widehat{\QQQ}_{\tilde\ccc}&=&\frac{i}{4}\tilde\chi^{~\bcdot}_{\mu~\ast}~\delBRS\left[\widehat{\QQQ}_{~\bcdot}^{\mu~\ast}\right]-\frac{1}{4}\beta^{~\bcdot}_{\mu~\ast}~\widehat{\QQQ}_{~\bcdot}^{\mu~\ast},\\\end{aligned}$$ where $$\begin{aligned}
\widehat{\QQQ}_{~~b}^{\mu a}&:=&g^{\nu_1\nu_2}\partial_{\nu_1}\chi^{\mu}\omega_{\nu_2}^{~ac}~\SSS_{cb},\end{aligned}$$ which satisfy $$\begin{aligned}
\left\{
\begin{array}{cll}
d\widehat{\QQQ}_\bbb&=&d\widehat{\QQQ}_{\tilde\ccc}~=~0,\\
\delBRS\left[\widehat{\QQQ}_{\tilde\ccc}\right]&=&-\widehat{\QQQ}_\bbb,\\
\delBRS\left[\widehat{\QQQ}_{\bbb}\right]&=&0.
\end{array}
\right.\end{aligned}$$ Moreover, the charge $\widehat{\QQQ}_{\bbb}$ is a generator of the BRS-transformation for operators, such as $$\begin{aligned}
i\lambda~\delBRS\left[\widehat\Omega\right]&=&\left[\widehat{\Omega},\lambda~\widehat{\QQQ}_\bbb\right],\label{BRSgen0}\end{aligned}$$ where $\widehat\Omega$ is any operator and $\lambda=i(-1)^{\pm\widehat{\QQQ}_{\tilde\ccc}}$[@Nakanishi:1977gt]. Especially by taking $\widehat\Omega=\widehat{\QQQ}_{\tilde\bbb}$ or $\widehat{\QQQ}_{\tilde\ccc}$, one can obtain $$\begin{aligned}
\widehat{\QQQ}_{\bbb}^2&=&0,\label{BRSgen1}\\
\left[\widehat{\QQQ}_{\tilde\ccc},\widehat{\QQQ}_\bbb\right]&=&-i\widehat{\QQQ}_\bbb.\label{BRSgen2}\end{aligned}$$ A relation $(\ref{BRSgen1})$ immediately follows from nilpotent of $\widehat{\QQQ}_{\bbb}$ and $\lambda=-1$ in (\[BRSgen0\]). A proof of the relation (\[BRSgen2\]) is given in Appendix \[appC\].

Discussions
===========

In a previous section, we performed quantization of our theory according to the canonical method in details. In this section, we discuss physics based on quantum general relativity developed in a previous section.

Hilbert space and physical states {#discs}
---------------------------------

A tangent representation of the spin and surface forms is considered to introduce a Hilbert space. A set of square integrable real-functions $L^2(\MM)$ is introduced as the Hilbert space $\HH$. Spin forms obtained as a solution of the Einstein equation are assumed to be square integrable, and thus the space of coefficient functions of spin forms $\varpi$ are a subset of the Hilbert space as $\varpi\subset\HH\subset L^2$ The bar-dual space of $\varpi$, which is denoted $\overline{\varpi}$, is simply referred to as the dual space of $\varpi$. The dual space $\overline{\varpi}$ is also square integrable, and thus the Gel’fand triple becomes $\varpi\subseteq\HH\subseteq\overline{\varpi}$. For the spin form $\www^{ab}$ and its dual form $\overline\www^{ab}$, their coefficient functions are written as $\omega^{ab}_{~~\mu}\in\varpi$ and $\frac{1}{2}\epsilon_{ab\cdott}\hspace{.2em}\omega^\cdott_{~\mu}\in\overline\varpi$, respectively. A standard bilinear from $\langle \overline{\www}|\www\rangle$ is defined as; $$\begin{aligned}
\langle \overline\www|\www\rangle&:=&\frac{1}{2}\overline\www_\cdott\otimes\www^\cdott=
\frac{1}{4}\epsilon_{\bcdott\bcdott}\hspace{.2em}
\omega^{~~\cdott}_{\mu_1}\hspace{.2em}\omega^{~~\cdott}_{\mu_2}\hspace{.2em}
dx^{\mu_1}\otimes dx^{\mu_2}.\end{aligned}$$ Under this bilinear from, the functional space $\varpi$ can be regarded as the pseudo-Riemannian manifold with a metric tensor of; $$\begin{aligned}
g_{\mu\nu}&:=&\left[{\bm g}\left(\overline{\www},\www\right)\right]_{\mu\nu}=
\frac{1}{4}\epsilon_{\bcdott\bcdott}\hspace{.2em}
\omega^{~~\cdott}_{\mu}\hspace{.2em}\omega^{~~\cdott}_{\nu}.\end{aligned}$$ A norm of the spin form $\|\www\|$ can be defined using the bilinear form as; $$\begin{aligned}
\|\www\|^2:=\int_{\Sigma_2}\langle\overline\www|\www\rangle\in\R,\end{aligned}$$ where ${\Sigma_2}$ is an appropriate $2$-dimensional sub-manifold ${\Sigma_2}\subset\M$. A form $d\www^{ab}$ is considered as a $1$-from defined on $\varpi$ whose coefficient functions are $d\omega^{~ab}_\mu=(\partial_\bcdot\omega^{~ab}_\mu)\eee^\bcdot\in\Omega^1(\varpi)$. An external derivative of tensor components can be transformed to that of the corresponding forms as shown in (\[diag\]). A element of the dual space $\overline{\varpi}$ is a linear combination of spin forms $\omega^{ab}_{~~\mu}\in\varpi$, and thus factional spaces $\varpi$ and $\overline{\varpi}$ are equivalent. Therefore, the Hilbert space can be taken as $\HH=\varpi$, because the Gel’fand triple is now $\varpi\subseteq\HH\subseteq \varpi$.

The commutation relations corresponding to the Poisson brackets (\[pb1\]) and (\[pb2\]) are $$\begin{aligned}
\left[\widehat{\vomega}^{ab},\widehat{\vomega}^{cd}\right]&=&
\left[\widehat{\SSS}_{ab},\widehat{\SSS}_{cd}\right]~=~0\label{cr00}\end{aligned}$$ and (\[CR1\]). The above expressions of the commutation relations are formal ones. More precisely, the representation of (\[CR1\]) must be understood as $$\begin{aligned}
\left[\widehat{\omega}_{\mu}^{~~ab}(x),\widehat{\Varepsilon_{\nu}^{c}\Varepsilon_{\rho}^{d}}(y)\right]
&=&-i
\epsilon^{abcd}
\epsilon_{\mu\nu\rho\sigma}
\delta^{(3)}(x^{\mu,\nu,\rho}-y^{\mu,\nu,\rho})
\int\delta^{(1)}(x^{\sigma}-y^{\sigma})dx^{\sigma}.\end{aligned}$$ Fixing $\sigma=0$ ([*time coordinate*]{}) instead taking a sum and performs the integration with respect to $dx^0$, one can obtain the standard [*equal-time*]{} commutation relation.

The operator can be taken as $$\begin{aligned}
\widehat{\vomega}^{ab}=\vomega^{ab},&~&
\widehat{\SSS}_{ab}=i\frac{\delta~}{\delta\vomega^{ab}},\end{aligned}$$ reproducing the commutation relations (\[CR1\]) and (\[cr00\]). A Schr[" o]{}dinger equation becomes $$\begin{aligned}
\widehat{\HHH}_G|\Psi(\omega)\rangle&=&E_G|\Psi(\omega)\rangle,\end{aligned}$$ where $\widehat{\HHH}_G$ is the Hamiltonian operator corresponding to (\[HG\]), $E_G$ is the energy eigenvalue and $|\Psi(\omega)\rangle\in\cal{H}$. While the existence of the norm is ensured, the negative norm states $\langle\vomega | \vomega \rangle<0$ are still included in $\cal{H}$, that corresponds to a negative energy state.

For an explicit discussion, a following simple $(1\hspace{-.1em}+\hspace{-.1em}3)$ coordinate decomposition is considered: Assuming that the global space-time manifold is filled with congruence of geodesics whose tangent vector is time-like at any point on the line. A coordinate $x^0$ is taken along the time-like vector on those geodesics, and other three coordinates are taken as the orthonormal base. By using that coordinate system, the three-dimensional boundary is simply taken as a manifold at $x^0=\tau$ (an [*equal-time*]{} boundary) and the state is written as $|\Psi_\tau\rangle$. The expectation value of the operator $\widehat{\cal{O}}$ can be represented as followsWe note that the operator $\widehat{{\cal O}}(\widehat{\vomega},\widehat{\SSS})$ cannot be determined uniquely from the classical function ${\cal O}\left(\vomega,\SSS\right)$ due to the non-commutativity of quantum operators. This problem is known as the operator ordering problem in general. The operator ordering problem in quantum general relativity is discussed in ref.[@2017arXiv171207964K].: $$\begin{aligned}
\overline{{\cal O}}&=&
\langle\Psi_\tau|\widehat{{\cal O}}\left(\widehat{\vomega},\widehat{\SSS}\right)|\Psi_\tau\rangle=
\langle\Psi_\tau|\int{\cal O}\left(\vomega,\SSS\right)|\Psi_\tau\rangle=\int{\cal O}\left(\vomega,\SSS\right).\end{aligned}$$ For instance, the Hamiltonian operator has an expected value given as $E_G=\int_{\partial\Sigma} \iota_{0}\HHH_G$, which is the total energy in the universe on a three-dimensional space at given time defined above. A trivial ground-state of $\www^{ab}=0$ exists with the eigenvalue of $E_G=0$. By also using the torsion less condition, one can obtain $d\eee^a=0$, which means that the ground state corresponds to the flat Lorentz spacetime. This is one of general results for the Chern–Weil theory: the critical point of the Chern-Weil action gives a flat connection.

Due to an existence of BRS generators of (\[BRSgen1\]), (\[BRSgen2\]), and nilpotent of the BRS operator on forms, the physical state can be defined according to the Nakanishi–Kugo–Ojima formalism as $$\begin{aligned}
\widehat{\QQQ}_\bbb|\mathrm{phys}\rangle&=&\widehat{\QQQ}_{\tilde\ccc}|\mathrm{phys}\rangle=~0,\end{aligned}$$ where $|\mathrm{phys}\rangle\in{\cal H}_{\mathrm{phys}}\subset{\cal H}$. As a consequence, $\langle\mathrm{phys}|\mathrm{phys}\rangle\geq0$ follows due to the Kugo–Ojima theorem. A transition operator $\widehat{{\mathcal S}}$ is introduced such that the matrix element $$\begin{aligned}
p\left(\tau_2:\tau_1\right)&=&\langle\Psi_{\tau_2}|\widehat{{\mathcal S}}|\Psi_{\tau_1}\rangle\end{aligned}$$ gives the transition probability from the state $|\Psi_{\tau_1}\rangle$ at $x^0=\tau_1$ to another state $|\Psi_{\tau_2}\rangle$ at $x^0=\tau_2$, where $|\Psi_{\tau_{1,2}}\rangle\in{\cal H}_{\mathrm{phys}}$ and $\widehat{{\mathcal S}}$ is a functional constructed by the spin, surface, auxiliary, and Faddeev-Popov forms. Again the Kugo–Ojima theorem ensures the unitarity of this transition matrix[@nakanishi1990covariant]. In conclusion, we can state from above discussion that a time evolution of the universe should keep the unitarity of the state vector of quantum gravity.

Renormalizability {#renorm}
-----------------

Renormalizability has not been discussed so far in this study because our quantization is carried out non-perturbatively. However, as is well-known, a simple power-counting shows that quantum general relativity is non-renormalizable. In reality, for the pure gravitational theory including neither matter fields nor a cosmological constant, the following facts are known by direct calculations: ’t Hooft and Veltman confirmed that the on-shell S-matrix element of quantum general relativity is finite at one-loop[@'tHooft:1974bx], however the theory is non-renormalizable, if an interaction with scalar particles is included. On the other hand, Berends and Gastmans reported that the amplitude of the graviton-graviton scattering is not unitary even at that tree-level[@BERENDS197599]. Beyond the one-loop level, Goroff and Sagnotti reported that the theory is non-renormalizable at a $2$-loop[@Goroff198581; @GOROFF1986709].

Even though at first glance, the unitarity violation reported in [@BERENDS197599] appears to contradict our results, this is not the case. First, the metric tensor or vierbein used to be a standard variable of the perturbation expansion. However, the appropriate phase-space variables as the Chern–Weil action in $4$-dimensions are the spin and surface forms. The standard perturbation gravity has been using an improper expansion variable. Moreover, the covariant perturbation gravity, developed by DeWitt[@DeWitt:1967ub; @DeWitt:1967uc] and used in studies to show the non-renormalizability[@'tHooft:1974bx; @BERENDS197599; @Goroff198581; @GOROFF1986709], uses the background field method, in which the metric tensor is separated into the (classical) background metric $g_{\mu\nu}$ and the quantum field $h_{\mu\nu}$ such as $
g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}.
$ On the other hand, the ground state of the quantum Hamiltonian must have a zero-point energy due to the quantum effect. Even though quantum general relativity in the covariant perturbation method is non-renormalizable in previous studies, we cannot conclude that quantum general relativity itself is non-renormalizable[^2]. Llewellyn Smith suggested that the renormalizability and the tree-level unitarity of the theory are equivalent[@LLEWELLYNSMITH1973233], as has been confirmed for some concrete examples, and with no exceptions reported to date. Since our formalism does not employ an interaction picture, but a Heisenberg picture instead, it is nothing to do with perturbative expansion with respect to a coupling constant. A method to solve quantum general relativity in a Heisenberg picture is discussed by Nakanishi and Abe, and summarized in Ref.[@doi:10.1143/PTP.111.301]. There are still possibilities that a renormalizable perturbation of quantum general relativity exists.

Summary
=======

In this study, we have re-formulated general relativity as a Chern–Weil theory on a $4$-dimensional space-time manifold. In a standard formulation, there is no topological invariance of a Chern-Simons type in $4$-dimensional space time because a translation invariant action cannot be constructed. A key idea to construct a topological invariant in an even-dimensional space is using a co-translation instead of a simple translation. The Einstein–Hilbert action can be identified a second Chern class and is invariant under the co-translation operation up to total derivative, when a cosmological constant does not exist.

The theory with topological invariants introduced in this report is different from (so-called) the topological quantum field theory (QTFT)[@witten1988; @Atiyah1988], that is, simply put, the theory whose $2$-point correlation function is given as to independent from the metric. As a consequence, there exhibit no dynamics. Quantum general relativity proposed in this report has the $4$-dimensional Lagrangian given as the second Chern class in a five-dimensional manifold. As a consequence, the theory has topological invariants as the de Rham classes due to the Chern–Weil theory[@zbMATH03077491], For instance, Zanelli shows a natural quantization of the gravitational constant on odd-dimensional spacetime using a Euler classes[@Zanelli:1994ti]. A similar analysis can be performed in a $4$-dimension space time[@Kurihara2018].

This construction of the action integral suggests that a proper set of phase-space variables are the spin and surface forms. A canonical quantization based on those canonical variables is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism. Since general relativity is a constraint system with global $GL(1,3)$ and local $SO(1,3)$ symmetries, we have to fix unphysical degree of freedom by implementing a gauge fixing and a ghost Lagrangian. After the gauge fixing, still the BRS symmetry is remaining. We gave a complete set of the quantum Lagrangian and BRS transformation operator with auxiliary and ghost fields in a self consistent manner. Next, an appropriate Hilbert space and physical states are introduced into the theory. As a consequence of the Kugo–Ojima theorem, the positivity of the physical states and the unitarity of the transition matrix were shown in this study. These results suggest that the renormalizability of quantum general relativity is worth reconsidering.

I appreciate the kind hospitality of all members of the theory group of Nikhef, particularly Prof. J. Vermaseren and Prof. E. Laenen. A major part of this study has been conducted during my stay at Nikhef in 2017. In addition, I would like to thank Dr. Y. Sugiyama and Prof. J. Fujimoto for their continuous encouragement and fruitful discussions.

Topological theories {#apBF}
====================

In this report, theories which have topological (characteristic) classes are referred to as the topological theory. We note that the [topological theory]{} is not the same as, so called the “topological field theory”, which has the metric independent correlation functions and does not have any dynamics. Three examples of the [topological theory]{}s are summarized under following setups in this Appendix.

On a $n$-dimensional manifold $\M$, a principal bundle $P(\M,G)$ is defined, where $\M$ is the base manifold and $G$ is the structural Lie-group. The connection one-form $\AAA$ and the corresponding curvature $2$-form $\FFF=d\AAA+\AAA\wedge\AAA$ are equipped on the base manifold $\M$. Here $\AAA$ is a Lie-algebra valued one-form $\AAA\in\ggg$, where $\ggg$ is a Lie-algebra of the structural group $G$.

Chern–Simons topological theory
-------------------------------

In 1988, Witten shows that general relativity in the $(1+2)$-dimensional space can be considered as a Chern–Simons [topological theory]{}[@witten1988]. The base manifold $\M$ is a Riemannian manifold with $(1+2)$ space-time dimension and the structural group is $ISO(1,2)$. The connection one-form and curvature $2$-form can be respectively taken as $\AAA_0$ and $\FFF_0$ as (\[PCoF\]) and (\[PCaF\]) in the three-dimension. The action is given as; $$\begin{aligned}
\I_{CS}&=&\frac{1}{4}\int_{\Sigma_4}\FFF_0\wedge\FFF_0
=\frac{1}{2}\int_{\Sigma_4}Tr\left[
\AAA_0\wedge d\AAA_0+\frac{2}{3}\AAA_0\wedge\AAA_0\wedge\AAA_0
\right],\nonumber\\
&=&\frac{1}{2}\int_{\Sigma_4}d\left(
\epsilon_{\cdott\bcdot}~\eee^\bcdot\wedge\RRR^\cdott
\right)=\frac{1}{2}\int_{\partial\Sigma_4={\Sigma}_3}\left(
\epsilon_{\cdott\bcdot}~\eee^\bcdot\wedge\RRR^\cdott
\right),\end{aligned}$$ where ${\Sigma}_4$ is an appropriate simply connected and orientable $4$-dimensional manifold. The relation between ${\Sigma}_4$ and ${\Sigma}_3$ is the same as that in (\[csaction0\]). This is nothing other than the Einstein–Hilbert action[^3] in three space-time dimension, and thus it is shown that general relativity can be constructed as the Chern-Simons [topological theory]{}. This coincidence is rather accidental only in the three dimensional space[@0264-9381-29-13-133001; @doi:10.1063/1.4990708]. While the Chern–Simins [topological theory]{} in three-dimension does not have any dynamical degree, this is not due to a topological aspect of the theory. From a simple counting of number of [d.o.f$.$]{}, one can understand that the dynamical [d.o.f$.$]{} in three-dimensional general relativity is zero at the classical level.

BF topological theory
---------------------

In 1977, the BF [topological theory]{} is introduced at first by Plebański[@doi:10.1063/1.523215], while the term “BF theory” did not exist yet at that time. In 1989, Horowitz first treated general relativity of the BF theory as the topological theory in general $n$ space-time dimension[@Horowitz1989]. Review articles of the BF [topological theory]{} can be found in [@1991PhR...209..129B; @Krasnov2011].

Some additional constraints are necessary to apply on the original BF [topological theory]{} to treat general relativity[@doi:10.1063/1.523215]. This constraint, called the simplicity condition, in four space time dimension is discussed by Gielen and Oritti[@Gielen:2010cu] (linear constraints) and Celada, González and Montesinos[@0264-9381-33-21-213001] (constraints on $\CC$ formalism). A reason why the BF [topological theory]{} requires the constraints is as follows: The base manifold $\M$ is taken as four space time dimensional Riemannian manifold and the structural group is taken as $SO(1,3)$ (or $SO(4)$). The connection one-form and curvature $2$-form can be respectively taken as the spin-form $\www^{ab}$ and curvature $2$-form $\RRR^{ab}$ as introduced in section \[prep\]. In addition to them, new Lie-algebra valued $2$-form $\BBB^{ab}$ is introduced, and then the BF topological action is introduced as; $$\begin{aligned}
\tilde{\I}_{BF}&=&\frac{1}{2}\int_{{\Sigma}_4}\epsilon_{\cdott\cdott}~\BBB^\cdott\wedge\RRR^\cdott,\end{aligned}$$ where ${\Sigma}_4$ is an appropriate simply connected and orientable $4$-dimensional manifold. It cannot be simply recognized as the topological action of the structural group $SO(1,3)$ (or $SO(4)$), because the $2$-form $\BBB^{ab}$ does not belong to the principal bundle in general. Instead, the form $\BBB^{ab}$ can be understood as a connection form on a 2-bundle, and it forms principal 2-bundle in the higher-gauge theory[@doi:10.1063/1.1790048]. While the action $\tilde{\I}_{BF}$ can be topological by means of the 2-gauge theory, it is not coincide with the Einstein–Hilbert action. To convert the BF [topological theory]{} to a gravitational theory, additional constraints must be implement as a Lagrange multiplier term[@0264-9381-16-7-303] such as; $$\begin{aligned}
{\I}_{BF}&=&\frac{1}{2}\int_{{\Sigma}_4}\left(
\epsilon_{\cdott\cdott}~\BBB^\cdott\wedge\RRR^\cdott
-\frac{1}{2}\phi_{\cdott\cdott}\BBB^\cdott\wedge\BBB^\cdott
\right),\end{aligned}$$ where $\phi_{abcd}$ is a scalar symmetric traceless matrix. The simplicity condition appears as the equation of motion obtained by taking variation with respect to $\phi$. After implementing the constraint term, the action $\I_{BF}$ does not have a topological characteristic class any more. As one of solutions of the simplicity condition, the surface form $\eee^a\wedge\eee^b$ can be taken as the $2$-form $\BBB^{ab}$, and thus a shape of the BF action coincides with the Einstein–Hilbert action. This coincidence of the shape of the action is true only for solutions of the equation of motion (on-shell condition) in the classical level, and the on-shell condition cannot be simply true after quantization of the BF theory. This is the reason why quantization of the BF gravitational theory is complicated[@0264-9381-33-21-213001]. In addition, the spin form (spin connection) is not treated as a fundamental form as a canonical conjugate of $\BBB^{ab}$ in quantization of the BF theory. As a consequence, a quantum BF theory is different from the quantum theory proposed in this article.

Chern–Weil topological theory
-----------------------------

This is the theory intrduced in this report. The base manifold $\M$ is taken as Riemannian manifold with $(1\hspace{-.2em}+\hspace{-.2em}3)$ space time dimension and the structural group is $(1\hspace{-.2em}+\hspace{-.2em}3)$-dimensional co-Poincaré group introduced in section \[newsym\]. The connection $1$-form and curvature $2$-form can be respectively taken as $\AAA$ given in (\[c2f\]) and $\FFF$ given in (\[Pcon\]), introduced also in section \[newsym\]. The action integral of the Cern–Weil [topological theory]{} can be written as; $$\begin{aligned}
\I_{CW}&=&\frac{1}{4}\int_{\widetilde{\M}_5}\mathrm{Tr}\left[
\FFF\wedge\FFF
\right]=\frac{1}{2}\int_{\M_4}\SSS_\cdott\wedge\RRR^\cdott,\end{aligned}$$ as given by equations (\[csaction1\]) and (\[csaction0\]). In contrast with the BF gravitational theory, both forms $\SSS_{ab}$ and $\RRR^{ab}(\www)$ are directly obtained from the principal bundle, and thus the Einstein–Hilbert action itself keeps the characteristic class (second Chern-class), which is ensured by the Chern–Weil theory[@zbMATH03077491; @doi:10.1063/1.4990708]. The simplicity condition of the BF gravitational theory corresponds to the definition of the surface form in the Chern–Weil [topological theory]{}, and thus it is exact after quantization. Quantization of the Chern–Weil [topological theory]{} can be performed with a gauge fixing term with respect to the global $GL(1,3)$ and local $ISO(1,3)$ as shown in section \[canonical\].

In contrast with the Chern–Simons theory in a three-dimensional space time, the Chern–Weil theory in a $4$-dimensional space time has dynamical [d.o.f$.$]{} after quantization. Among ten [d.o.f$.$]{} on the symplectic fields of the vierbein and spin connection, two physical degrees are remaining for dynamical [d.o.f$.$]{}, corresponding to two spin states of graviton. Quantization of constrained system is performed using the Kugo–Ojima formalism in section \[6-1\]. At first, the auxiliary, ghost and anti-ghost fields are introduced to fix the gauge and unphysical [d.o.f$.$]{} in the system. Then the BRS transformations are required on all of physical and unphysical fields. At this stage, number of constraints due to the BRS transformations is the same as a total [d.o.f$.$]{} in the system, and thus there is no dynamical degree if all constraints are independent each other. In reality, as shown in section \[BRSCharge\], all constraints are not independent and there are two conserved BRS charges, $\widehat{\QQQ}_\bbb$ and $\widehat{\QQQ}_{\tilde\ccc}$, in the system. Therefore, the system still has two dynamical [d.o.f$.$]{} after quantization.

Proof of nilpotency {#ap1}
===================

\
The coordinate vectors are one of the most fundamental vectors on $T\MM$. The nilpotent can be confirmed as $$\begin{aligned}
&~&
\delBRS\left[\delBRS\left[x^{\mu}\right]\right]
~=~\delBRS\left[\chi^\mu\right]
~=~\delBRS\left[g^{\mu\nu}\chi_\nu\right],\\
&~&~=~
g^{\mu\rho}\left(\partial_\rho\chi^\nu\right)\chi_\nu+
g^{\rho\nu}\left(\partial_\rho\chi^\mu\right)\chi_\nu-g^{\mu\nu}g_{\nu\rho}\left(
\partial^\rho\chi^\sigma+
\partial^\sigma\chi^\rho
\right)\chi_\sigma,\\
&~&~=~
\left(\partial^\mu\chi^\nu\right)\chi_\nu+
\left(\partial^\nu\chi^\mu\right)\chi_\nu-\left(\partial^\mu\chi^\sigma\right)\chi_\sigma-
\left(\partial^\sigma\chi^\mu\right)\chi_\sigma=0.\end{aligned}$$\
Starting from the BRS-transformation of the metric tensor (\[BRSg\]), $$\begin{aligned}
\delBRS\left[\delBRS\left[g_{\mu\nu}\right]\right]&=&
\delBRS\left[
-g_{\mu\rho}\partial_\nu\chi^\rho
\right]
+\delBRS\left[\mu\leftrightarrow\nu\right],\nonumber\\
&=&\left\{-\delBRS\left[g_{\mu\rho}\right]\partial_\nu\chi^\rho
+g_{\mu\rho}\left(\partial_\nu\delBRS\left[x^\sigma\right]\right)\partial_\sigma\chi^\rho\right\}
+\left\{\mu\leftrightarrow\nu\right\},\nonumber\\
&=&\left\{
g_{\mu\sigma}\left(\partial_\rho\chi^\sigma\right)\partial_\nu\chi^\rho+
g_{\mu\rho}\left(\partial_\nu\chi^\sigma\right)\partial_\sigma\chi^\rho\right\}
+\left\{\mu\leftrightarrow\nu\right\}=0,\end{aligned}$$ where anti-commutativity of the ghost filed is used.\
\
Since the ghost field has two parts, nilpotent is checked separately. First for the $\chi_\mu$, $$\begin{aligned}
\delBRS\left[\delBRS\left[\chi_\mu\right]\right]
&=&
\delBRS\left[\delBRS\left[g_{\mu\nu}\chi^\nu\right]\right]
~=~\delBRS\left[\delBRS\left[g_{\mu\nu}\right]\right]\chi^\nu=0,\end{aligned}$$ where $\delBRS[\chi^\mu]=0$ and nilpotent of the metric tensor are used. Direct calculation from (\[BRSchi\]) gives the same result, too. The second part becomes $$\begin{aligned}
\delBRS\left[\delBRS\left[\chi^a_{~b}\right]\right]&=&
\delBRS\left[\chi^a_{~c}\chi^{c}_{~b}\right]
~=~
\chi^a_{~c_2}\chi^{c_2}_{~~c_1}\chi^{c_1}_{~~b}-\chi^a_{~c_1}\chi^{c_1}_{~~c_2}\chi^{c_2}_{~~b}=0,\end{aligned}$$ due to anti-commutativity of the ghost field.

A tensor $\partial_{\mu}\chi^{\nu}$ is also nilpotent as $$\begin{aligned}
\delBRS\left[\delBRS\left[\partial_{\mu}\chi^{\nu}\right]\right]&=&
-\delBRS\left[\partial_{\mu}\chi^{\rho}\partial_{\rho}\chi^{\nu}\right]=~
-\partial_{\mu}\chi^{\rho_1}\partial_{\rho_1}\chi^{\rho_2}\partial_{\rho_2}\chi^{\nu}
+\partial_{\mu}\chi^{\rho_1}\partial_{\rho_1}\chi^{\rho_2}\partial_{\rho_2}\chi^{\nu}=~0.\end{aligned}$$ $$\begin{aligned}
\delBRS\left[\delBRS\left[\eee^a\right]\right]&=&
\delBRS\left[\eee^b\chi^a_{~b}\right]=~
\eee^{b_1}\chi^{b_2}_{~~b_1}\chi^a_{~b_2}+\eee^{b_2}\chi^a_{~b_1}\chi^{b_1}_{~~b_2}=0,\end{aligned}$$ due to anti-commutativity of the ghost field.\
\
One can trace the same calculation as a case of the vierbein form due to $\delBRS\left[d\chi^{ab}\right]=0$. Detailed calculations are omitted here.\
\
The volume form is global scalar and their BRS-transformation is expected to vanish, which can be confirmed as $$\begin{aligned}
\delBRS\left[\vvv\right]&=&\frac{1}{4!}\epsilon_{\cdott\cdott}
\delBRS\left[\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\right]~=~
\frac{1}{3!}\epsilon_{a_1\bcdot\cdott}
\delBRS\left[\chi^{a_1}_{~~a_2}\eee^{a_2}\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\right]~=~0,\end{aligned}$$ due to $\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\wedge\eee^\bcdot\propto\epsilon^{\cdott\cdott}$ and $\chi^{a_1}_{~~a_2}=0$ when $a_1=a_2.$\
\
The BRS-transformation of the surface form is given by $$\begin{aligned}
\delBRS\left[\SSS_{ab}\right]&=&\frac{1}{2}\epsilon_{abc_1c_2}\delBRS\left[\eee^{c_1}\wedge\eee^{c_2}\right]
=~\epsilon_{abc_1c_2}\chi^{c_1}_{~c_3}\eee^{c_3}\wedge\eee^{c_2}~
\left(=~\ccc^\bcdot\wedge\overline{\eee}_{ab\bcdot}\right).\end{aligned}$$ Applying the BRS-transformation on it again, one can get $$\begin{aligned}
&~&\delBRS\left[\delBRS\left[\SSS_{ab}\right]\right]~=~
\epsilon_{abc_1c_2}\delBRS\left[\chi^{c_1}_{~c_3}\eee^{c_3}\wedge\eee^{c_2}\right],\\
&~&~=~\epsilon_{abc_1c_2}\Bigl\{
\chi^{c_1}_{~c_4}\chi^{c_4}_{~c_3}\eee^{c_3}\wedge\eee^{c_2}-
\chi^{c_1}_{~c_3}\chi^{c_3}_{~c_4}\eee^{c_4}\wedge\eee^{c_2}-
\chi^{c_1}_{~c_3}\chi^{c_2}_{~c_4}\eee^{c_3}\wedge\eee^{c_4}
\Bigr\}=0,\end{aligned}$$ because first term is the same as the second term and the third term is symmetric with $c_1$ and $c_2$ exchange.\
\
The BRS-transformation of $\ccc^a$ is given by $$\begin{aligned}
\delBRS\left[\ccc^a\right]&=&
\delBRS\left[\chi^a_{~b}~\Varepsilon^b_\mu~dx^\mu\right],\\
&=&\chi^a_{~b_1}\chi^{b_1}_{~b_2}\Varepsilon^{b_2}_\mu dx^\mu-
\chi^a_{~b_1}~\Varepsilon^{b_2}_\mu \chi^{b_2}_{~b_1}dx^\mu
+\chi^a_{~b}~\left(\partial_\mu\chi^\nu\right)\Varepsilon^b_\nu~dx^\mu
-\chi^a_{~b}~\Varepsilon^b_\mu d\chi^\mu=0,\end{aligned}$$

\
Nilpotent of other forms are trivial and the proof is omitted here.\
\
The quantum Lagrangian must be the BRS-null. The gauge-fixing and Fadeef-Popov terms, $\LLL_{GF}+\LLL_{FP}$, will be constructed to satisfy the BRS-null condition. Therefore, the proof for the gravitational Lagrangian is given here by $$\begin{aligned}
\delBRS\left[\LLL_G\right]&=&
\frac{1}{2}\delBRS\left[\left(
d\vomega^\cdott
+\vomega^\bcdot_{~\ast}\wedge\vomega^{\ast\bcdot}\right)\wedge{\SSS}_\cdott
-\frac{\Lambda}{3!}\vvv\right].\end{aligned}$$ The BRS-transformation for the volume form is vanished by itself. For the derivative term, $$\begin{aligned}
\delBRS\left[
d\vomega^\cdott\wedge{\SSS}_\cdott
\right]&=&
\epsilon_{abc_2c_3}\chi_{~c_1}^{b}d\vomega^{ac_1}\wedge\eee^{c_2}\wedge\eee^{c_3}+
\epsilon_{abc_2c_3}\vomega^{ac_1}\wedge d\chi_{~c_1}^{b}\wedge\eee^{c_2}\wedge\eee^{c_3}\\&~&+
\epsilon_{abc_1c_2}\chi^{c_1}_{~c_3}~d\vomega^{ab}\wedge\eee^{c_3}\wedge\eee^{c_2}\\
&=&2~\vomega^{ac_1}\wedge d\chi^{b}_{~c_1}\wedge\SSS_{ab},\end{aligned}$$ where first- and third-terms are cancelled each other. Remnant term is transformed as $$\begin{aligned}
\delBRS\left[\vomega^\bcdot_{~\ast}\wedge\vomega^{\ast\bcdot}\wedge{\SSS}_\cdott\right]&=&
\epsilon_{abc_2c_3}\chi^{c_2}_{~c_4}\vomega^{ac_1}\wedge\vomega_{c_1}^{~~b}\wedge\eee^{c_4}\wedge\eee^{c_3}+
\epsilon_{abc_3c_4}\chi^{c_1}_{~c_2}\vomega^{ac_1}\wedge\vomega_{c_2}^{~~b}\wedge\eee^{c_3}\wedge\eee^{c_4}\\&~&+
\epsilon_{abc_3c_4}\chi^{b}_{~c_2}\vomega^{ac_1}\wedge\vomega_{c_1}^{~~c2}\wedge\eee^{c_3}\wedge\eee^{c_4}-
2~\vomega^{ac_1}\wedge d\chi^b_{~c_1}\wedge\SSS_{ab},\\
&=&-2~\vomega^{ac_1}\wedge d\chi^b_{~c_1}\wedge\SSS_{ab}.\end{aligned}$$ In the r.h.s of the first line, the second term is zero as itself, and first- and third-terms are cancelled each other. Therefore one can confirmed $\delBRS\left[\LLL_G\right]=0$ after summing up all terms. If we use a following remake, we can give simpler proofs for above forms.\
\
If both of two fields, $\alpha$ and $\beta$, are nilpotent, $\alpha\beta$ is also nilpotent.\
[*Proof:*]{}\
If a field $X$ is nilpotent, signatures of the Leibniz rule satisfy $\epsilon_{X}=-\epsilon_{\delta X}$ due to $\delBRS[\delBRS[X]]=0$ and (\[Leib\]), where $\epsilon_{X}$ ($\epsilon_{\delta X}$) is a signature of $X$ ($\delBRS[X]$), respectively. Therefore $$\begin{aligned}
\delBRS\left[\delBRS\left[\alpha\beta\right]\right]&=&
\epsilon_{\alpha}\delBRS\left[\alpha\right]\delBRS\left[\beta\right]+
\epsilon_{\delta\alpha}\delBRS\left[a\right]\delBRS\left[\beta\right]
~=~0.\end{aligned}$$

Equations of motion {#app2}
===================

From the classical Lagrangian form, the torsion-less condition and the Einstein equation are obtained as the equations of motion by requiring a stationary condition for a variation of the action. The same procedure can extract Euler-Lagrange equations from the quantum Lagrangian. Here the quantum Lagrangian is summarized as $$\begin{aligned}
&~&\LLL_{QG}=\LLL_G+\LLL_{GF}+\LLL_{FP},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(\ref{QLGF})\\
&~&
\left\{
\begin{array}{llc}
\LLL_G&=&~~~~~~\frac{1}{2}\left(
\RRR^\cdott
-\frac{\Lambda}{3!}\overline{\SSS}^\cdott\right)\wedge{\SSS}_\cdott,~~~~~~~~~~~~~~~~~~~~~~(\ref{Lagrangian44})\\
\LLL_{GF}&=&
-\frac{1}{2}\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}
\right)\wedge\SSS_\cdott,~~~~~~~~~~~~~~~~~~~~(\ref{LGF})\\
\LLL_{FP}&=&
-\frac{i}{2}\left(
d\tilde{\ccc}^\cdott+
\alpha
\tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right)
\wedge\ccc^\star\wedge\overline{\eee}_{\star\cdott},~~~~~~~~~~~~~~~(\ref{LFP})
\end{array}
\right.\end{aligned}$$ Euler-Lagrange equations are obtained as follows:

$$\begin{aligned}
\TTT^a&=&d\eee^a+\www^a_{~\bcdot}\wedge\eee^\bcdot~=~0.\end{aligned}$$ This is the torsion-less condition as the the same as a case of the classical Lagrangian.\
$$\begin{aligned}
\frac{1}{2}\overline{\left(\RRR^\cdott\wedge\eee^\bcdot\right)}_a-\Lambda\VVV_a
-\frac{1}{2}\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\eee}_{a\cdott}
-\frac{i}{2}\left(d\tilde{\ccc}^\cdott+\alpha\tilde{\ccc}^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\ccc}_{a\cdott}
&=&0,\label{ELE2}\end{aligned}$$ where $\VVV_a=\epsilon_{ab_1b_2b_3}\eee^{b_1}\wedge\eee^{b_2}\wedge\eee^{b_3}/3!$. The first two terms give the Einstein equation without matter fields. Third- and fourth-terms newly appeared from the gauge-fixing and Feddeev–Popov Lagrangian forms.\
$$\begin{aligned}
d\SSS_{ab}-\alpha\left(
\bbb^\bcdot_a\wedge\SSS_{\bcdot b}-i\tilde{\ccc}^\bcdot_a\wedge\ccc^\bcdot\wedge\overline{\eee}_{\cdott b}
\right)&=&0.\end{aligned}$$ We note that $\bbb$ and $\delta_\bbb$ are anti-commute each other and the variation operator is applied from the left. When the Landau-gauge is used, the de Donder gauge-fixing condition $d\SSS_{ab}=0$ is obtained.\
$$\begin{aligned}
\left(d\tilde{\ccc}^\cdott+
\alpha
\tilde{\ccc}^{\bcdot}_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\eee}_{a\bcdot\bcdot}&=&0,\label{ELE3}\end{aligned}$$ where the anti-commutation among $\ccc$, $\delta\ccc$ and $\bbb$ is used.\
$$\begin{aligned}
\epsilon_{ab\cdott}\left(
d\left(\ccc^\bcdot\wedge\eee^\bcdot\right)-\alpha
\bbb^\bcdot_{~\ast}\wedge\ccc^\ast\wedge\eee^\bcdot
\right)\nonumber&=&
\epsilon_{ab\cdott}\left(
d\ccc^\bcdot-\alpha
\bbb^\bcdot_{~\ast}\wedge\ccc^\ast
\right)\wedge\eee^\bcdot
~=~0,\end{aligned}$$ where the de Donder condition is used.

The BRS-transformation may give another set of equations, which must be consistent with above equations: $$\begin{aligned}
\delBRS\left[\TTT^a\right]&=&\chi^a_{~\bcdot}~d\eee^\bcdot+d\chi^a_{~\bcdot}\wedge\eee^\bcdot
+\chi^a_{~\bcdot}~\vomega^{~\bcdot}_{\ast}\wedge\eee^\ast
+\chi_{\ast}^{~\bcdot}~\vomega^{a}_{~\bcdot}\wedge\eee^\ast
-d\chi^a_{~\bcdot}\wedge\eee^\bcdot
+\chi^\bcdot_{~\ast}\vomega^a_{~\bcdot}\wedge\eee^\ast
\nonumber\\
&=&\chi^a_{~\bcdot}~\TTT^\bcdot~=~0.\end{aligned}$$ This is consistent with the torsion-less condition. The BRS-transformation for the volume form is vanished and last two terms are cancelled each other such as $$\begin{aligned}
\delBRS\left[i\left(d\tilde{\ccc}^\cdott+\alpha\tilde{\ccc}^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\ccc}_{a\cdott}\right]\nonumber&=&
-\left(d\bbb^\cdott+\alpha\bbb^\bcdot_{~\ast}\wedge\bbb^{\ast\bcdot}\right)\wedge\overline{\eee}_{a\cdott}\end{aligned}$$ Therefor, the BRS-transformation of (\[ELE2\]) is given by $$\begin{aligned}
0&=&
\epsilon_{abc\bullet}\delBRS\left[\left(d\www^{ab}+\www^a_{~\bcdot}\wedge\www^{\bcdot b}\right)\wedge\eee^c\right]\\
&=&\epsilon_{abc\bullet}\Bigl\{
\chi^b_{~\bcdot}\left(d\www^{a\bcdot}+\www^a_{~\ast}\wedge\www^{\ast\bcdot}\right)\wedge\eee^c
+
\left(d\www^{ab}+\www^a_{~\bcdot}\wedge\www^{\bcdot b}\right)\wedge\ccc^c
\Bigr\}.\end{aligned}$$ This is consistent with the Einstein equation. $$\begin{aligned}
\delBRS\left[d\eee^a-\alpha\left(
\bbb^a_{~\bcdot}\wedge\eee^\bcdot-i\tilde{\ccc}_\bcdot^a\wedge\ccc^\bcdot
\right)\right]&=&d\ccc^a=0,\end{aligned}$$ where $\alpha$-terms are cancelled each other. The BRS-transformation of (\[ELE3\]) gives an equation of motion for $\bbb$ in (\[ELE2\]). $$\begin{aligned}
\epsilon_{ab\cdott}\delBRS\left[
d\left(\ccc^\bcdot\wedge\eee^\bcdot\right)
-\alpha
\bbb^\bcdot_{~\ast}\wedge\ccc^\ast\wedge\eee^\bcdot
\right]&=&0,\end{aligned}$$ where $\epsilon_{ab\cdott}\ccc^\bcdot\wedge\ccc^\bcdot=0$ is used. This is not an equation, but an identity.

proof of (\[BRSgen2\]) {#appC}
======================

The proof of (\[BRSgen2\]) can be given as follows: At first, shorthand notations are introduced to omit indices in following calculations for a while such as, $$\begin{aligned}
\widehat{\QQQ}_\bbb=\frac{1}{2}\widehat{\bm\beta}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right],&~~&
\widehat{\QQQ}_{\tilde\ccc}=\frac{i}{4}\widehat{\bm\chi}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right]
-\frac{1}{4}\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ},\\\end{aligned}$$ because the correspondence of indices is clear in them. By using these notions, the commutation relation can be represented as $$\begin{aligned}
-8i\left[\widehat{\QQQ}_{\tilde\ccc},\widehat{\QQQ}_\bbb\right]&=&
\left[
\widehat{\bm\beta}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right],
\widehat{\bm\chi}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right]
\right]+i
\left[
\widehat{\bm\beta}\bcdot\delBRS\left[\widehat{\bm\QQQ}\right],
\widehat{\bm\beta}\bcdot \widehat{\bm\QQQ}
\right],\nonumber\\&=&
\widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right], \widehat{\bm\chi}\right]\delBRS\left[\widehat{\bm\QQQ}\right]
+\widehat{\bm\chi}
\left[\widehat{\bm\beta},\delBRS\left[\widehat{\bm\QQQ}\right]
\right]\delBRS\left[\widehat{\bm\QQQ}\right]\nonumber\\&~&
+i~
\widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ}\right]+i
\left[
\widehat{\bm\beta},
\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ}
\right]\delBRS\left[\widehat{\bm\QQQ}\right].\nonumber\label{DD}\\\end{aligned}$$ where $[\widehat{\bm\beta},\widehat{\bm\chi}]=0$ is used. The first term of (\[DD\]) becomes $$\begin{aligned}
\widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right], \widehat{\bm\chi}\right]\delBRS\left[\widehat{\bm\QQQ}\right]&=&\widehat{\bm\beta}
\left(
\delBRS\left[\widehat{\bm\QQQ}\right]\widehat{\bm\chi}-
\widehat{\bm\chi}~\delBRS\left[\widehat{\bm\QQQ}\right]
\right)\delBRS\left[\widehat{\bm\QQQ}\right],
\nonumber\\&=&
\widehat{\bm\beta}
\left(
\delBRS\left[
\left\{\widehat{\bm\QQQ},\widehat{\bm\chi}\right\}
\right]+i\left[\widehat{\bm\beta},\widehat{\bm\QQQ}\right]
\right)\delBRS\left[\widehat{\bm\QQQ}\right],\\&=&
i\widehat{\bm\beta}~
\left[\widehat{\bm\beta},\widehat{\bm\QQQ}\right]
\delBRS\left[\widehat{\bm\QQQ}\right]=~4\widehat{\QQQ}_\bbb,\end{aligned}$$ where $\delBRS\left[\left\{\widehat{\bm\QQQ},\widehat{\bm\chi}\right\}\right]=0$ due to (\[fCR3\]), and (\[fCR2\]) are used. Note that $\widehat{\bm\chi}$ and $\widehat{\bm\QQQ}$ have $\epsilon_X=-1$ in (\[Leib\]). The second term of (\[DD\]) is zero, because $$\begin{aligned}
\left[\widehat{\bm\beta},\delBRS\left[\widehat{\bm\QQQ}\right]
\right]&=&
\delBRS\left[
\left[
\widehat{\bm\beta},\widehat{\bm\QQQ}
\right]
\right]=~0,\end{aligned}$$ due to (\[fCR2\]). The third term is also zero as $$\begin{aligned}
i\widehat{\bm\beta}\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ}\right]&=&
i\widehat{\bm\beta}\widehat{\bm\beta}
\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\QQQ}\right]+
i\widehat{\bm\beta}~
\delBRS\left[\left[\widehat{\bm\QQQ},\widehat{\bm\beta}\right]\right]\widehat{\bm\QQQ}=~0,\end{aligned}$$ due to (\[fCR2\]). A relation $\left[\delBRS\left[\widehat{\bm\QQQ}\right],\widehat{\bm\QQQ}\right]=0$ can be confirmed by direct calculations. The last term of (\[DD\]) becomes $$\begin{aligned}
i\left[
\widehat{\bm\beta},
\widehat{\bm\beta}\bcdot\widehat{\bm\QQQ}
\right]\delBRS\left[\widehat{\bm\QQQ}\right]&=&
i\widehat{\bm\beta}~
\left[\widehat{\bm\beta},\widehat{\bm\QQQ}\right]
\delBRS\left[\widehat{\bm\QQQ}\right]=~4\widehat{\QQQ}_\bbb.\end{aligned}$$

[^1]: See, for instance, section 7 in Ref.[@fre2012gravity]

[^2]: See section 5 in Ref.[@nakanishi1990covariant]

[^3]: The cosmological term in the Einstein–Hilbert action is omitted in this Appendix.
