# Topological gravity in 3+1D and a possible origin of dark matter

Tianyao Fang  Department of Physics, The Chinese University of Hong Kong,
Shatin, New Territories, Hong Kong, China  Zheng-Cheng Gu
<EMAIL_ADDRESS>Department of Physics, The Chinese University of Hong
Kong, Shatin, New Territories, Hong Kong, China

###### Abstract

Dark matter is one of the deepest mystery of the universe. So far there is no
natural explanation why the dark matter should exist and even dominate the
universe. In this paper, we begin with a 3+1D topological gravity theory which
is super renormalizable with vanishing beta functions, then we argue that
Einstein gravity can emerge by condensing loop-like excitation from the
underlying topological gravity theory. In the meanwhile, the uncondensed loop-
like excitations serves as a natural candidate of dark matter and a
generalized Einstein equation can be derived in the presence of loop-
source(dark matter) background. Surprisingly, we find that such kind of dark
matter will not contribute to scalar curvature, however, it will become a
source of torsion. Finally, we derive the generalized Einstein equation in the
presence of Dirac field. Very different from the usual Einstein-Carton theory,
our theory further predicts that any type of normal matter, including Dirac
field will not produce torsion. All these unique predictions can be tested by
future experiments. Our framework suggests that topological invariant
principle might play a more profound role than the well-known general
covariance principle, especially towards understanding the nature of dark
matter and quantum gravity in 3+1D.

##  I Introduction

###  I.1 The dark matter puzzle

It is well known that dark matter makes up about 27% of the universe, which is
much bigger than all normal matter - adds up to less than 5% of the universe.
Nevertheless, it is rather hard to observe dark matter directly due to their
extremely weak interaction with normal matter. Actually it is long believed
that dark matter only interacts with normal matter via gravity. On the other
hand, Einstein’s general relativity severs as the most profound framework to
understand the basic structure of the universe, and the recent observation of
gravitational wave by LIGOLIGO  further verifies the correctness of such an
elegant theory. Therefore, it is very important to understand how dark matter
can be included into Einstein equation in a natural and covariant way, and it
is also very important to understand what makes dark matter so different from
normal matter. Although many phenomenological models are proposed to describe
dark matterDark1 ; Dark2 ; Dark3 , none of them can be derived naturally from
a general principle.

###  I.2 General covariance principle vs. topological invariance principle

The general covariance principle, which demands a geometric, coordinate-
independent formulation of physics, was long believed to be the mathematical
foundation of Einstein’s physical idea for general relativity. In particular,
the Hilbert-Einstein action $S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}R$ was
first proposed by Hilbert based on such a novel principle to derive the well
known (vacuum) Einstein equation. However, if we think about this principle
more carefully, the covariance principle actually can not lead to Einstein
equation in a unique way. For example, the higher order term $\sqrt{-g}R^{2}$
is also allowed by general covariance principle and such a term will induce
higher order corrections to the Einstein equation. In usual quantum field
theory such as gauge theory, one may argue that higher order terms should be
suppressed by the renormalization group flow. Unfortunately, as a non-
renormalizable theory, there is no natural reason to explain why higher order
terms such as $\sqrt{-g}R^{2}$ should not appear in the semi-classical limit.
Moreover, the general covariance principle is intrinsically inconsistent with
quantum mechanics since the Schrodinger equation requires an absolute time.

The topological aspects of Einstein gravity can be traced back to a century
ago. The so-called Einstein-Carton action $S=\frac{1}{32\pi
G}\int\varepsilon_{abcd}R^{ab}(\omega)\wedge e^{c}\wedge e^{d}$ was first
proposed by Elie Cartan to describe spinors in curved spacetime where torsion
was introduced. In such a first order formalism, Einstein gravity was
reformulated as a topological field theory(TFT). Actually, the TFT framework
not only provides us an alternative way to re-derive Einstein equation, but
also has the unique advantage to eliminate higher order terms such as $R^{2}$.
In fact, the topological invariance principle might play a more profound role
than the general covariance principle, and it might even help us to reveal the
deep mystery of dark matter.

In 2+1D, the Einstein-Carton action turns out to be exactly the same as the
Chern-Simons action of Pioncare gauge group.CS1  Three decades ago, Edward
Witten proposed to use such a new framework to resovle the long standing
renormalizabilty puzzle of 2+1D quantum gravity(at least perturbatively).
Although $2+1$D gravity is somewhat trivial due to the absence of propagating
gravitational wave and vanishing of space-time curvature, it still provides us
a concrete example of understanding quantum gravity via topological quantum
field theory(TQFT). Moreover, according to the correspondence between Chern-
Simons theory and CFT, the ADS3/CFT2 correspondence conjecture can also be
understood in a very natural wayADSCFT3 .

Nevertheless, the TQFT approach can not be easily generalized into $3+1$D due
to the following difficulties. (a) Einstein’s gravity in $3+1$D contains
propagating mode – the gravitational wave, therefore it is obviously not a
TQFT in the usual sense. (b) Our knowledge of higher dimensional TQFT is very
limited and there is no Chern-Simons like action in $3+1$D. Thanks to the
recent development of the classification of topological phases of quantum
matter in higher dimensionsChenlong ; Wenscience ; cobordism ; Wencoho , new
types of TQFT have been discovered in $3+1$D to describe the so-called three-
loop-braiding statistics. Here we argue that such types of TQFT are closely
related to Einstein gravity. In particular, we conjecture that gravitational
wave will disappear at an extremely high energy scale and $3+1$D quantum
gravity might be controlled by a TQFT renormalization group fixed point.

###  I.3 Main results and outline

In this paper, we first review a topological quantum gravity theory in 3+1D,
which can be regarded as a higher dimensional generalization of Chern-Simons
action of Pioncare gauge group. In the semi-classical limit, we argue that the
condensation of loop-like excitations in 3+1D topological quantum gravity will
lead to Einstein equation in a natural way. We conjecture that uncondensed
loop-like exciations can be a very promising candidate of dark matter. By
adding the loop source term, we can derive a generalized Einstein equation
which naturally includes dark matter sector. Surprisingly, we find that such
kind of dark matter will not contribute to scalar curvature and this unique
property makes the dark matter very different from the (massive) normal
matter. Finally, we derive the generalized Einstein equation in the presence
of Dirac field. Very different from the usual Einstein-Carton theory, we
predict that loop current is the only source of torsion while any type of
normal matter, including Dirac field will not contribute to torsion. All these
unique predictions can be tested in future experimental measurement.

The rest of the paper is organized as follows. In section II, we will review
the topological gravity theory in 3+1D and show that such a theory is actually
super renormalizable with vanishing beta functions. Then we will study the
canonical quantization of this theory in details. In section III, we will show
how Einstein gravity might emerge from the underlying topological gravity
theory in 3+1D by adding a topological mass term of the 2-form gauge field.
Physically, we conjecture that such a phenomenological theory might describe a
”loop condensing” phase. Remarkably, we find that the whole theory is still
super renormalizable with vanishing beta functions. In section IV, we propose
the loop-like extensive object as a natural candidate of dark matter and
derive a generalized Einstein equation in the presence of loop source term. In
section V, we also study the generalized Einstein equation in the presence of
Dirac fields. In section VI, we study the most general form of topological
action and discuss the uniqueness of the generalized Einstein equation.
Finally, there is a conclusion and discussion section.

##  II Topological gravity in 3+1D

Let us begin with the topological gravity theory in 3+1DTopgravity . Consider
the following topological invariant action:

$\displaystyle S_{top}$ $\displaystyle=$
$\displaystyle\frac{k_{1}}{4\pi}\int\varepsilon_{abcd}R^{ab}\wedge e^{c}\wedge
e^{d}+\frac{k_{2}}{2\pi}\int B_{ab}\wedge R^{ab}$ (1)
$\displaystyle+\frac{k_{3}}{2\pi}\int\widetilde{B}_{a}\wedge T^{a},$

where $e$ is the tetrad field, $R$ is the curvature tensor, $T$ is the torsion
tensor and $B,\widetilde{B}$ are 2-form gauge fields. It is straightforward to
verify that the above action is invariant under the following (twisted) 1-form
and 2-form gauge transformations, respectively:

$\displaystyle e^{a}$ $\displaystyle\rightarrow$ $\displaystyle e^{a}+Df^{a}$
$\displaystyle B_{ab}$ $\displaystyle\rightarrow$ $\displaystyle
B_{ab}-\frac{k_{3}}{2k_{2}}\left(\widetilde{B}_{a}f_{b}-\widetilde{B}_{b}f_{a}\right)$
$\displaystyle\widetilde{B}_{a}$ $\displaystyle\rightarrow$
$\displaystyle\widetilde{B}_{a}-\frac{k_{1}}{k_{3}}\varepsilon_{abcd}f^{b}R^{cd},$
(2)

and

$\displaystyle B_{ab}$ $\displaystyle\rightarrow$ $\displaystyle
B_{ab}+D\xi_{ab},$ (3) $\displaystyle\widetilde{B}_{a}$
$\displaystyle\rightarrow$ $\displaystyle\widetilde{B}_{a}+D\tilde{\xi}_{a}$
$\displaystyle B_{ab}$ $\displaystyle\rightarrow$ $\displaystyle
B_{ab}-\frac{k_{3}}{2k_{2}}\left(\tilde{\xi}_{a}\wedge
e_{b}-\tilde{\xi}_{b}\wedge e_{a}\right).$ (4)

Such an action can be regarded as the non-Abelian generalization of $AAdA+BF$
type TQFTaada1 ; aada2 ; aada3  of Poincare gauge group. Physically, it has
been shown that such kind of TQFT describes the so-called three-loop-braiding
statisticsloop1 ; loop2 . Apparently, as a TQFT, the action Eq. (1) is a super
renormalizable theory. Below we will discuss the coefficient quantization and
canonical quantization of such a theory.

###  II.1 Coefficient quantization and super renormalizability

Since the action Eq. (1) contains a cubic term, it is not easy to perform the
usual momentum shell path integral scheme to understand its renormalizability.
Similar to the Chern-Siomns theory in 2+1D, below we will show that all the
coefficients $k_{1},k_{2}$ and $k_{3}$ are quantized, which implies the
vanishing of beta functions and super renormalizability.

We first consider the coefficient quantization of the $BR$ term. We note that
the 2-form gauge transformation Eq. (3) will induce a boundary term for
$S_{top}$.

$\displaystyle\delta S_{top}$ $\displaystyle=$
$\displaystyle\frac{k_{2}}{2\pi}\int D\xi_{ab}\wedge R^{ab}$ $\displaystyle=$
$\displaystyle\frac{k_{2}}{2\pi}\int\left[D\left(\xi_{ab}\wedge
R^{ab}\right)+\xi_{ab}\wedge DR^{ab}\right]$ (5) $\displaystyle=$
$\displaystyle\frac{k_{2}}{2\pi}\int D\left(\xi_{ab}\wedge R^{ab}\right)$
$\displaystyle=$ $\displaystyle\frac{k_{2}}{2\pi}\int d\left(\xi_{ab}\wedge
R^{ab}\right),$

where we use the identity $DR=0$ in the second line(see Appendix A), and in
the last equation we use the fact that the covariant differentiation on a
scalar is just the same as usual partial differentiation. Considering a
manifold with a boundary, such a term becomes:

$\delta S_{top}=\frac{k_{2}}{4\pi}\int\xi_{ab}\wedge R^{ab}.$ (6)

To understand the quantization of $k_{2}$, we just need to consider the
maximum compact subgroup of $SO(3,1)$, which is $SO(3)$. If we choose
$\xi_{ab}=\varepsilon_{abc}e^{c}$ where $e$ is a tetrad field of 3D Euclidean
space, the boundary term can be simplified as:

$\delta S_{top}^{\prime}=\frac{k_{2}}{4\pi}\int\varepsilon_{abc}e^{a}\wedge
R^{bc},$ (7)

which is exactly the Chern-Simons action of gauge field $ISO(3)$ with
quantized coefficient $k_{2}=2\mathbb{Z}$. (See Appendix B for full details.)

Next we consider the coefficient quantization of $Ree$ term. Under the gauge
transformation Eq. (2), the torsion tensor $T$ will be transformed as:

$T^{a}\rightarrow T^{a}+R^{ab}f_{b}.$ (8)

Up to linear order, we obtain the variation of the action $S_{top}$ as:

$\displaystyle\delta S_{top}$ $\displaystyle=$
$\displaystyle\frac{k_{1}}{2\pi}\int\varepsilon_{abcd}R^{ab}\wedge
Df^{c}\wedge
e^{d}-\frac{k_{2}}{2\pi}\frac{k_{3}}{k_{2}}\int\widetilde{B}_{a}f_{b}\wedge
R^{ab}$ (9) $\displaystyle+\frac{k_{3}}{2\pi}\int\widetilde{B}_{a}\wedge
R^{ab}f_{b}-\frac{k_{3}}{2\pi}\frac{k_{1}}{k_{3}}\int\varepsilon_{abcd}T^{a}f^{b}\wedge
R^{cd}$ $\displaystyle=$
$\displaystyle\frac{k_{1}}{2\pi}\int\varepsilon_{abcd}\left(Df^{a}\wedge
e^{b}-T^{a}f^{b}\right)\wedge R^{cd}$ $\displaystyle=$
$\displaystyle\frac{k_{1}}{2\pi}\int\varepsilon_{abcd}\left(Df^{a}\wedge
e^{b}+f^{a}De^{b}\right)\wedge R^{cd}$ $\displaystyle=$
$\displaystyle\frac{k_{1}}{2\pi}\int\varepsilon_{abcd}d\left(f^{a}e^{b}\wedge
R^{cd}\right),$

where we derive the last line by using $DR=0$. It is obvious that if we choose
$f^{0}=1$ and $f^{i}=0$, Eq. (9) will have exactly the same boundary term as
$\delta S_{top}^{\prime}$. Therefore, we conclude that $k_{1}$ is also
quantized as $k_{1}=2\mathbb{Z}$.

Finally we consider the quantization of $k_{3}$. Under the second 2-form gauge
transformation in Eq. (4), the variation of total action reads:

$\displaystyle\delta S_{top}$ $\displaystyle=$
$\displaystyle-\frac{k_{2}}{2\pi}\frac{k_{3}}{k_{2}}\int\tilde{\xi}_{a}\wedge
e_{b}\wedge R^{ab}+\frac{k_{3}}{2\pi}\int T^{a}\wedge D\tilde{\xi}_{a}$ (10)
$\displaystyle=$ $\displaystyle-\frac{k_{3}}{2\pi}\int\tilde{\xi}_{a}\wedge
e_{b}\wedge R^{ab}+\frac{k_{3}}{2\pi}\int D\left(e^{a}\wedge
D\tilde{\xi}_{a}\right)$ $\displaystyle+\frac{k_{3}}{2\pi}\int e^{a}\wedge
R_{a}^{\ b}\wedge\tilde{\xi}_{b}$ $\displaystyle=$
$\displaystyle\frac{k_{3}}{2\pi}\int D\left(e^{a}\wedge
D\tilde{\xi}_{a}\right).$

If we choose $\tilde{\xi}^{a}=e^{a}$, and consider it on a flat space-time,
that is, $\omega_{\ b}^{a}=R_{\ b}^{a}=0,$ the equation above becomes:

$\delta S_{top}=\frac{k_{3}}{2\pi}\int de^{a}\wedge
de_{a}=\frac{k_{3}}{2\pi}\int F^{a}\wedge F_{a},$ (11)

where the field strength of $e^{a}$ is defined as
$F^{a}=\frac{1}{2}\left(\partial_{\mu}e_{\nu}^{a}-\partial_{\nu}e_{\mu}^{a}\right)dx^{\mu}\wedge
dx^{\nu}$. We note that the first Chern class
$\frac{1}{\left(2\pi\right)^{2}}\int F^{a}\wedge F_{a}=c_{1}$ should be an
integer111As a TQFT, here we define the tetrad field $e$ as a compact $U(1)$
gauge field, thus we finally have:

$\delta S_{top}=2\pi k_{3}c_{1}.$ (12)

Again, as the variation of the total action should be some integer multiplied
by $2\pi$, we conclude that $k_{3}=\mathbb{Z}$.

Apparently, as a TQFT, the action Eq. (2) is a super renormalizable quantum
theory with vanishing beta functions due to the coefficient qauntizations. It
is easy to see that $e$ and $\omega$ are of dimension 1, $B$ and
$\widetilde{B}$ are of dimension 2. Thus, when integrating out the fast mode
$\Lambda/s<p<\Lambda$ and recovering the integral region by doing the
rescaling transformation $p^{\prime}=sp$, we will have the following rescaling
law of the fields:

$\displaystyle e^{\prime}\left(p^{\prime}\right)$ $\displaystyle=$
$\displaystyle s^{-3}e_{<}\left(p^{\prime}/s\right)$
$\displaystyle\omega^{\prime}\left(p^{\prime}\right)$ $\displaystyle=$
$\displaystyle s^{-3}\omega_{<}\left(p^{\prime}/s\right)$ $\displaystyle
B^{\prime}\left(p^{\prime}\right)$ $\displaystyle=$ $\displaystyle
s^{-2}B_{<}\left(p^{\prime}/s\right)$
$\displaystyle\widetilde{B}^{\prime}\left(p^{\prime}\right)$ $\displaystyle=$
$\displaystyle s^{-2}\widetilde{B}_{<}\left(p^{\prime}/s\right).$ (13)

###  II.2 Canonical quantization

Now we consider the conanical quantization of the 3+1D topological gravity
theory. Without loss of generality, here and below we will choose
$k_{1}=k_{2}=2$ and $k_{3}=1$ for convenient. The canonical momentums of
$\omega_{i}^{ab}$ and $e_{i}^{a}$ can be defined as
$\Pi_{ab}^{i}=\frac{1}{2\pi}\varepsilon^{ijk}\varepsilon_{abcd}e_{j}^{c}e_{k}^{d}+\frac{1}{2\pi}\varepsilon^{ijk}B_{abjk}$
and $\pi_{a}^{i}=\frac{1}{4\pi}\varepsilon^{ijk}\widetilde{B}_{ajk}$. The
Lagrangian density of Eq. (2) can be rewritten as:

$\displaystyle\mathcal{L}_{Top}$ $\displaystyle=$
$\displaystyle\Pi_{ab}^{i}\partial_{0}\omega_{i}^{ab}+\pi_{a}^{i}\partial_{0}e_{i}^{a}+\frac{1}{2\pi}\varepsilon^{ijk}B_{ab0i}R_{jk}^{ab}+\frac{1}{4\pi}\varepsilon^{ijk}\widetilde{B}_{a0i}T_{jk}^{a}+\frac{1}{\pi}e_{0}^{a}(D_{i}\pi_{a}^{i}+\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abcd}R_{ij}^{bc}e_{k}^{d})$
(14)
$\displaystyle+\omega_{0}^{ab}(D_{i}\Pi_{ab}^{i}+\frac{1}{2}(\pi_{a}^{i}e_{bi}-\pi_{b}^{i}e_{ai})).$

Canonical quantization requires:

$\displaystyle[\omega_{i}^{ab}(x),\Pi_{cd}^{j}(y)]=i\delta_{i}^{j}\delta_{cd}^{ab}\delta(x-y),$
$\displaystyle[e_{i}^{a}(x),\pi_{b}^{j}(y)]=i\delta_{i}^{j}\delta_{b}^{a}\delta(x-y),$
$\displaystyle\ all\ other\ commutators=0,$ (15)

which are equivalent to:

$\displaystyle[\omega_{i}^{ab}(x),B_{cdjk}(y)]$ $\displaystyle=$
$\displaystyle i\pi\varepsilon_{ijk}\delta_{cd}^{ab}\delta(x-y),$
$\displaystyle[e_{i}^{a}(x),\widetilde{B}_{bjk}(y)]$ $\displaystyle=$
$\displaystyle i2\pi\varepsilon_{ijk}\delta_{b}^{a}\delta(x-y),$
$\displaystyle[B_{abij}(x),\pi_{c}^{k}(y)]$ $\displaystyle=$ $\displaystyle
i\varepsilon_{abcd}(e_{i}^{d}(x)\delta_{j}^{k}-e_{j}^{d}(x)\delta_{i}^{k})\delta(x-y),$

with the following flat-connection constraints:

$\displaystyle\mathcal{A}^{abi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon^{ijk}R_{jk}^{ab}=0,$
$\displaystyle\mathcal{B}^{ai}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon^{ijk}T_{jk}^{a}=0,$
$\displaystyle\mathcal{C}_{a}$ $\displaystyle=$ $\displaystyle
D_{i}\pi_{a}^{i}+\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abcd}R_{ij}^{bc}e_{k}^{d}=0,$
$\displaystyle\mathcal{D}_{ab}$ $\displaystyle=$ $\displaystyle
D_{i}\Pi_{ab}^{i}+\frac{1}{2}(\pi_{a}^{i}e_{bi}-\pi_{b}^{i}e_{ai})=0.$ (17)

It can be shown that these constraints are closed and satisfy the following
commutation relations:

$\displaystyle[\mathcal{A}^{abi}(x),\mathcal{D}_{cd}(y)]$ $\displaystyle=$
$\displaystyle-2i\delta_{[c}^{[a}\mathcal{A}_{\ d]}^{b]\ i}(x)\delta(x-y),$
$\displaystyle[\mathcal{B}^{ai}(x),\mathcal{C}_{b}(y)]$ $\displaystyle=$
$\displaystyle-i\mathcal{A}_{\ b}^{a\ i}(x)\delta(x-y),$
$\displaystyle[\mathcal{B}^{ai}(x),\mathcal{D}_{cd}(y)]$ $\displaystyle=$
$\displaystyle i\delta_{[c}^{a}\mathcal{B}_{d]}^{i}(x)\delta(x-y),$
$\displaystyle[\mathcal{C}^{a}(x),\mathcal{D}_{cd}(y)]$ $\displaystyle=$
$\displaystyle i\delta_{[c}^{a}\mathcal{C}_{d]}(x)\delta(x-y)$
$\displaystyle[\mathcal{D}^{ab}(x),\mathcal{D}_{cd}(y)]$ $\displaystyle=$
$\displaystyle 4i\delta_{[c}^{[a}\mathcal{D}_{\ d]}^{b]}(x)\delta(x-y),$
$\displaystyle\rm{others}$ $\displaystyle=$ $\displaystyle 0$ (18)

The proof of the above relations involves very tedious calculations and we
will leave the full details in Appendix C.

Similar to the 2 + 1D topological gravity, the phasespace to be quantized is
exactly the solutions of the above constraints divided by the group of gauge
transformations generated by the constraints. The quantum Hilbert space is the
flat connections of Poincare group modulo gauge transformations.(If we regard
$e$ and $\omega$ as coordinates while $\Pi$ and $\pi$ as momentums.)

Finally, we would like to stress that just like we use tensor category theory
to describe Chern-Simons theory, the algebraic tensor 2-category theoryTCAT1 ;
TCAT1  might also provide us an equivalent UV-complete description for such a
topological quantum gravity theory in $3+1$D.

##  III Loop condensation and the emergence of vacuum Einstein equation

To this point, one may wonder why we are interested in the $3+1$D topological
gravity theory which is somewhat trivial. Here we conjecture that quantum
gravity is actually controlled by a topological gravity fixed point and the
classical space-time vanishes at extremely high energy scale. Therefore it is
quite natural to expect the vanishing of curvature and torsion at that scale.
A possible way to generate interesting dynamics is condensing loop-like
excitations(i.e., flux lines in the context of gauge theory) of a $3+1$D TQFT.
If we further assume that the condensed loops carry a nontrivial linking Berry
phaseBB1 ; BB2 ; BB3 , a $\int Tr(B\wedge B)$ type topological mass term can
be generated. Let us consider the following action:

$\displaystyle S_{\theta}=-\frac{\theta}{2\pi}\int B_{ab}\wedge B^{ab}.$ (19)

Apparently, such a term breaks the 2-form gauge symmetry as well as the
translational gauge symmetry explicitly. A microscopic derivation of the above
term from loop condensation is beyond the scope of this paper. Here we just
introduce such a term phenomenologically to describe low energy dynamics and
ignore all the microscopic details of loop dynamics, which is the analog of
using massive gauge boson to describe Abelian Higgs phase and considering the
infinite massive limit for Higgs boson.

The classical equation of motion(EOM) for the total action $S$ reads:

$\displaystyle B^{ab}=\frac{1}{\theta}R^{ab},\quad
T^{a}=0,\quad\varepsilon_{abcd}e^{b}\wedge R^{cd}=-\nabla\tilde{B}_{a},$
$\displaystyle\varepsilon_{abcd}T^{c}\wedge
e^{d}+\frac{1}{2}(\tilde{B}_{a}\wedge e_{b}-\tilde{B}_{b}\wedge e_{a})=-\nabla
B_{ab}.$ (20)

Insert the first two equations into the last equation, we have:

$\displaystyle\frac{1}{2}(\tilde{B}_{a}\wedge e_{b}-\tilde{B}_{b}\wedge
e_{a})=-\frac{1}{\theta}\nabla R_{ab}=0.$ (21)

The equation above can be rewritten in a compact form as
$\varepsilon^{abcd}\tilde{B}_{a}\wedge e_{b}=0$, which further implies
$\tilde{B}^{a}=0$. Thus, we eventually derive the vacuum Einstein-Cartan
equation:

$\displaystyle\varepsilon_{abcd}e^{b}\wedge R^{cd}=0.$ (22)

Remarkably, for small $\theta$, the total action $S=S_{\rm{Top}}+S_{\theta}$
is still power-counting renormalizable since $S_{\theta}$ only contains
dimension 4 operator. Below we will show that the beta function still vanishes
even in the presence of $S_{\theta}$ term.

Since $S_{\theta}$ is a quadratic term, there is no mixed mode for this term.
Thus, we can separate the total action as:

$S=S_{0<}+S_{0>}+S_{0<,>}+S_{\theta<}+S_{\theta>},$ (23)

where $S_{0}\equiv S_{top}$ is the unperturbed action. To simplify the
notation, here $<$ represents slow mode, $>$ represents fast mode and $<,>$
represents mixed mode.

Thus, the partition function can be rewritten as:

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int
D_{<}e^{i(S_{0<}+S_{\theta<})}\int D_{>}e^{i(S_{0>}+S_{0<,>}+S_{\theta>})}$
(24) $\displaystyle=$ $\displaystyle\int
D_{<}e^{i(S_{0<}+S_{\theta<})}\frac{\int
D_{>}e^{iS_{\theta>}}e^{i(S_{0>}+S_{0<,>})}}{\int
D_{>}e^{i(S_{0>}+S_{0<,>})}}\int D_{>}e^{i(S_{0>}+S_{0<,>})}$ $\displaystyle=$
$\displaystyle\int D_{<}e^{i(S_{0<}+S_{\theta<})}\langle
e^{iS_{\theta>}}\rangle_{0},$

where $D_{<}(D_{>})$ is the integral over the slow (fast) modes of $e$,
$\omega$, $B$ and $\widetilde{B}$. Here $\langle\rangle_{0}$ denotes averaging
with respect to $S_{0>}+S_{0<,>}$. Since the beta function of $S_{0}$
vanishes, $\int D_{>}e^{i(S_{0>}+S_{0<,>})}$ must be a constant which is
independent of slow modes, thus we can drop it in the last line.

Let us denote $e^{i\delta S}=\langle e^{iS_{\theta>}}\rangle_{0}$ and we will
show such a term actually vanishes up to any order. To illustrate the basic
idea for the proof, below we just expand $\delta S$ up to the second order by
using cumulant expansion:

$i\delta S\simeq i\langle S_{\theta>}\rangle_{0}-\frac{1}{2}(\langle
S_{\theta>}^{2}\rangle_{0}-\langle S_{\theta>}\rangle_{0}^{2}).$ (25)

We first consider the term $i\langle S_{\theta>}\rangle_{0}$, which can be
computed from the following generating functional:

$\displaystyle Z[J]=\int D_{>}\exp\left[i(S_{0>}+S_{0<,>})+\frac{i}{\pi}\int
dpJ^{ab}(p)\wedge B_{>ab}(-p)\right].$

The expectation values can be obtained by differentiation of the generating
functional with respect to the source $J$:

$\langle S_{\theta>}\rangle_{0}=\frac{\pi\theta}{32}Z[0]^{-1}\int
dp\varepsilon_{\mu\nu\rho\sigma}(-i\frac{\delta}{\delta
J(p)_{ab\mu\nu}})(-i\frac{\delta}{\delta
J(-p)^{ab}_{\rho\sigma}})Z[J]\mid_{J=0},$ (27)

As $S_{0>}+S_{0<,>}$ depends on $B_{>}$ linearly, performing the integration
$\int DB_{>}$ will lead to delta functions. All terms containing $B_{>}$ can
be expressed as:

$\displaystyle S_{B_{>}}$ $\displaystyle=$ $\displaystyle\frac{1}{\pi}\int
dp(J^{ab}(p)+d\widetilde{\omega}^{ab}(p))\wedge B_{>ab}(-p)+\frac{1}{\pi}\int
dp_{1}dp_{2}dp_{3}[\Omega^{ac}(p_{1})\wedge\Omega^{\ b}_{c}(p_{2})$ (28)
$\displaystyle+2\widetilde{\omega}^{ac}(p_{1})\wedge\Omega^{\
b}_{c}(p_{2})+\widetilde{\omega}^{ac}(p_{1})\wedge\widetilde{\omega}^{\
b}_{c}(p_{2})]\wedge B_{>ab}(p_{3})\delta(p_{1}+p_{2}+p_{3}).$

Here we denote $\omega_{>}^{ab}\equiv\widetilde{\omega}^{ab}$ and
$\omega_{<}^{ab}\equiv\Omega^{ab}$ for convenience, and we also use $\int dp$
as a simplified notation of $\int\frac{d^{4}p}{(2\pi)^{4}}$. When we carry out
functional integral of the fast mode, we regard all slow modes as fixed
parameters. After integrating over $B_{>}$, the generating functional becomes:

$Z[J]=\int
De_{>}D\widetilde{B}_{>}\int\prod_{p}D\widetilde{\omega}(p)\delta(f_{ab\mu\nu}(p))e^{iS^{\prime}},$
(29)

where $S^{\prime}$ is the action that does not contain $B_{>}$ and $f$ is a
function of $\widetilde{\omega},\Omega,J$ in the form:

$\displaystyle f_{ab\mu\nu}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}[\frac{1}{2}J_{ab\mu\nu}(p)-ip_{\mu}\widetilde{\omega}_{ab\nu}(p)+\int
dp^{\prime}[\Omega_{ac\mu}(p^{\prime})\Omega^{c}_{\
b\nu}(p-p^{\prime})+\widetilde{\omega}_{ac\mu}(p^{\prime})\Omega^{c}_{\
b\nu}(p-p^{\prime})-\widetilde{\omega}_{bc\mu}(p^{\prime})\Omega^{c}_{\
a\nu}(p-p^{\prime})$ (30)
$\displaystyle+\widetilde{\omega}_{ac\mu}(p^{\prime})\widetilde{\omega}^{c}_{\
b\nu}(p-p^{\prime})]]-\mu\leftrightarrow\nu.$

Now we need to solve $f(p)_{ab\mu\nu}=0$ for $\widetilde{\omega}_{ab\mu}(p)$
at each momentum. However, for a fixed momentum $p$, there are 36 equations
but only 24 unknown $\widetilde{\omega}_{ab\mu}(p)$. In general, such kind of
overdetermined system has no solution and $Z[J]$ must be zero. Below we give a
more rigorous proof for $Z[J]=0$ for small but finite $J$.

Since $Z[0]$ is nonzero due to the renormalizable property of $S_{top}$, Eq.
(30) must have a solution for arbitrary $\Omega_{ab\mu}(p)$ at each $p$ in
momentum shell when $J_{ab\mu\nu}=0$. For small but finite $J$, we assume the
solutions of $\widetilde{\omega}$ can be expanded around the solutions for
$J=0$ with
$\widetilde{\omega}=\widetilde{\omega}_{0}+\delta\widetilde{\omega}$ and
$\widetilde{\omega}_{0}$ satisfies:

$\displaystyle-ip_{\mu}\widetilde{\omega}_{0ab\nu}(p)+\int
dp^{\prime}[\Omega_{ac\mu}(p^{\prime})\Omega^{c}_{\
b\nu}(p-p^{\prime})+\widetilde{\omega}_{0ac\mu}(p^{\prime})\Omega^{c}_{\
b\nu}(p-p^{\prime})-\widetilde{\omega}_{0bc\mu}(p^{\prime})\Omega^{c}_{\
a\nu}(p-p^{\prime})$
$\displaystyle+\widetilde{\omega}_{0ac\mu}(p^{\prime})\widetilde{\omega}^{\
c}_{0\ b\nu}(p-p^{\prime})]-\mu\leftrightarrow\nu=0.$ (31)

We begin with a simple case with $\Omega=0$. Eq. (31) becomes:

$-ip_{\mu}\widetilde{\omega}_{0ab\nu}(p)+\int
dp^{\prime}\widetilde{\omega}_{0ac\mu}(p^{\prime})\widetilde{\omega}^{\ c}_{0\
b\nu}(p-p^{\prime})-\mu\leftrightarrow\nu=0.$ (32)

Obviously $\widetilde{\omega}_{0}=0$ is a solution of the above equation. Up
to first order of $\delta\widetilde{\omega}$, Eq. (30) can be linearized as:

$ip_{\mu}\delta\widetilde{\omega}_{ab\nu}(p)-ip_{\nu}\delta\widetilde{\omega}_{ab\mu}(p)=J_{ab\mu\nu}(p).$
(33)

By multiplying $\varepsilon^{\mu\nu\rho\sigma}p_{\rho}$ on both sides, we
finally obtain:

$\varepsilon^{\mu\nu\rho\sigma}J_{ab\mu\nu}(p)p_{\rho}=0,$ (34)

which has no solutions for general source term $J_{ab\mu\nu}(p)$.

Now we consider general $\Omega$. For small but finite $J$, we can again only
keep up to first of $\delta\widetilde{\omega}$ for Eq. (30). Again, by
multiplying $\varepsilon^{\mu\nu\rho\sigma}$ on both sides, we obtain:

$\varepsilon^{\mu\nu\rho\sigma}\\{-ip_{\mu}\delta\widetilde{\omega}_{ab\nu}(p)+\int
dp^{\prime}[\omega^{\prime}_{ac\mu}(p^{\prime})\delta\widetilde{\omega}^{c}_{\
b\nu}(p-p^{\prime})-\omega^{\prime}_{bc\mu}(p^{\prime})\delta\widetilde{\omega}^{c}_{\
a\nu}(p-p^{\prime})]\\}=-\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}J_{ab\mu\nu}(p),$
(35)

where $\omega^{\prime}_{ab\mu}=\Omega_{ab\mu}+\widetilde{\omega}_{0ab\mu}$. In
general, such an overdetermined nonhomogeneous linear equation system also has
no solution for general source term $J$.

Thus, we conclude that the generating function $Z[J]$ must be zero for at
least small but finite source term $J$, and all the following expectation
values must vanish.

$\langle S_{\theta>}\rangle_{0}=\langle S^{2}_{\theta>}\rangle_{0}=...=0.$
(36)

This is simply because all the higher order terms can also be derived by
applying derivatives with respect to the generating functional $Z[J]$.
Therefore, we have $\delta S=0$ or $\langle e^{iS_{\theta>}}\rangle_{0}=1$. On
the other hand, the slow mode part reads:

$S_{\theta<}=-\frac{\theta}{8\pi}\int
dp_{1}\varepsilon^{\mu\nu\rho\sigma}B_{<\mu\nu}^{\
ab}\left(p_{1}\right)B_{<ab\rho\sigma}\left(-p_{1}\right).$ (37)

After recovering the integral region and rescaling of fields, $S_{\theta}$
becomes:

$\displaystyle S_{\theta}^{\prime}$ $\displaystyle=$
$\displaystyle-s^{-4+2+2}\frac{\theta}{8\pi}\int
dp^{\prime}_{1}\varepsilon^{\mu\nu\rho\sigma}B^{\prime
ab}_{\mu\nu}\left(p_{1}^{\prime}\right)B^{\prime}_{ab\rho\sigma}\left(-p_{1}^{\prime}\right)$
(38) $\displaystyle=$ $\displaystyle\frac{\theta}{8\pi}\int
dp^{\prime}_{1}\varepsilon^{\mu\nu\rho\sigma}B^{\prime
ab}_{\mu\nu}\left(p_{1}^{\prime}\right)B^{\prime}_{ab\rho\sigma}\left(-p_{1}^{\prime}\right),$

which has exactly the same form as $S_{\theta}$. Thus the beta function of
$S_{\theta}$ term vanishes as well, and the whole theory is still super
renormalizable.

Finally, we would like to stress that although the total action $S$ is super
renormalizable, it does not imply a UV-complete quantum gravity theory due to
the explicit broken of 2-form gauge symmetries by the $S_{\theta}$ term. How
to develop a theoretical framework, e.g. topological string field theory to
describe the loop condensation phase without 2-form gauge symmetry breaking
could be an essential step towards establishing a UV-complete theory of
quantum gravity, and we will leave this difficulty problem in our future
study. Below we will discuss the semi-classical consequence of loop
condensation and discuss it’s potential connection with the origin of dark
matter.

##  IV Generalized Einstein equation and a potential origin of dark matter

In the previous section, we argue that the condensation of loop-like
excitations coupled with the 2-form gauge field $B$(more precisely, that is
the flux line of gauge field $\omega$, and we will call it $\omega$-loop
below) will naturally lead to the emergence of vacuum Einstein equation.
However, those loop-like excitations coupled with 2-form gauge field
$\widetilde{B}$(more precisely, that is the flux line of $e$ gauge field, and
we will call it $e$-loop below) could not condense simultaneously due to the
presence of $Ree$ term which describes non-trivial three-loop braiding process
between $\omega$-loop and $e$-loop with $e$-loop as the base loop. Therefore,
linked $e$-loops must be confined and they can not be observed at low energy.
Nevertheless, those unlinked $e$-loop still can survive at low energy and
these extensive objects could be a natural candidate of dark matter and the
framework developed here will allow us to derive a generalized Einstein
equation with loop-like extensive matter source.

Since the $B\wedge B$ term also breaks the 1-form gauge transformation Eq.
(2), the expectation value of $e$ could also be non-zero in the loop
condensation phase:

$\left\langle e_{\mu}^{a}\right\rangle=\frac{1}{l_{p}}\delta_{\mu}^{a},$ (39)

where $l_{p}$ is a fundamental length scale induced by loop condensation, and
we can naturally interpret such a length scale as the Planck length, which
uniquely determines the Newton’s constant. Moreover, the non-zero expectation
value of $e$ also naturally explains the origin of a flat background metric of
the universe. In our usual convention, the flat background metric is defined
as $e_{\mu}^{a}=\delta_{\mu}^{a}$, thus it is more convenient to redefine
$\widetilde{e}=l_{p}e$ such that
$\left\langle\widetilde{e}_{\mu}^{a}\right\rangle=\delta_{\mu}^{a}$. The
$S_{top}$ can be rewritten as:(Without confusing, we just replace all
$\widetilde{e}$ by $e$)

$\displaystyle S_{top}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
l_{p}^{2}}\int R^{ab}\wedge e^{c}\wedge
e^{d}\varepsilon_{abcd}+\frac{1}{\pi}\int R^{ab}\wedge B_{ab}$ (40)
$\displaystyle+\frac{1}{2\pi l_{p}}\int T^{a}\wedge\widetilde{B}_{a}.$

To simplify the coefficients in our future calculation, we further rescale
$B\rightarrow\pi B$, $\widetilde{B}\rightarrow 2\pi\widetilde{B}$, and define
$\pi l_{p}^{2}=\kappa=16\pi G$, where $G$ is the Newton constant. Thus,
$S_{top}$ can be rewritten as:

$\displaystyle S_{top}$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa}\int
R^{ab}\wedge e^{c}\wedge e^{d}\varepsilon_{abcd}+\int R^{ab}\wedge B_{ab}$
(41) $\displaystyle+\frac{1}{l_{p}}\int T^{a}\wedge\widetilde{B}_{a}.$

Now, we consider the total action $S$ with both $S_{\theta}$ and loop-source
term coupled with the 2-form gauge field $\widetilde{B}$:

$S=S_{top}-\frac{\theta\pi}{2}\int B^{ab}\wedge
B_{ab}-\int\sigma^{a}\wedge\widetilde{B}_{a},$ (42)

where $\sigma^{a}$ is the $e$-loop current and
$\sigma^{a}\wedge\widetilde{B}_{a}$ is a topological invariant (minimal)
coupling term.

Below we will derive the generalized Einstein equation from the above total
action $S$.

(1)Variation with respect to $B$ gives:

$\displaystyle R^{ab}=\theta\pi B^{ab},\ \text{in terms of components}\
R_{\mu\nu}^{ab}=\theta\pi B_{\mu\nu}^{ab}.$ (43)

(2)Variation with respect to $\widetilde{B}$ gives:

$T^{a}=l_{p}\sigma^{a},\ \text{in terms of components}\
T_{\mu\nu}^{a}=l_{p}\sigma_{\mu\nu}^{a}.$ (44)

(3)Variation with respect to $\omega$ gives:

$\frac{1}{2\kappa}\int D\delta\omega^{ab}\wedge e^{c}\wedge
e^{d}\varepsilon_{abcd}+\int D\delta\omega^{ab}\wedge
B_{ab}+\frac{1}{l_{p}}\int\widetilde{B}_{a}\wedge\delta\omega^{ab}\wedge
e_{b}=0.$ (45)

Integral by parts, we have:

$\frac{1}{\kappa}\int\delta\omega^{ab}\wedge De^{c}\wedge
e^{d}\varepsilon_{abcd}+\int\delta\omega^{ab}\wedge
DB_{ab}+\frac{1}{2l_{p}}\int\delta\omega^{ab}\wedge\left(\widetilde{B}_{a}\wedge
e_{b}-\widetilde{B}_{b}\wedge e_{a}\right)=0.$ (46)

From Eq. (43), we further have $DB_{ab}=\frac{1}{\theta\pi}DR_{ab}=0$. Using
the definition $De^{c}=T^{c}$, we finally obtain:

$\frac{1}{\kappa}T^{c}\wedge
e^{d}\varepsilon_{abcd}+\frac{1}{2l_{p}}\left(\widetilde{B}_{a}\wedge
e_{b}-\widetilde{B}_{b}\wedge e_{a}\right)=0.$ (47)

To derive the components of the above equation, we wedge $e^{f}$ on both
sides:

$\displaystyle\frac{1}{4l_{p}}\left(\widetilde{B}_{aij}e^{i}\wedge e^{j}\wedge
e_{b}-\widetilde{B}_{bij}e^{i}\wedge e^{j}\wedge e_{a}\right)\wedge e^{f}$
(48) $\displaystyle=$ $\displaystyle-\frac{1}{2\kappa}T_{\ ij}^{c}e^{i}\wedge
e^{j}\wedge e^{d}\wedge e^{f}\varepsilon_{abcd},$

Comparing with both sides, we obtain:

$\displaystyle\frac{1}{2l_{p}}\left(\widetilde{B}_{aij}\eta_{bd}\varepsilon^{ijdf}-\widetilde{B}_{bij}\eta_{ad}\varepsilon^{ijdf}\right)$
(49) $\displaystyle=$ $\displaystyle-\frac{1}{\kappa}T_{\
ij}^{c}\delta_{abc}^{ijf}$ $\displaystyle=$
$\displaystyle-\frac{2}{\kappa}\left(T_{\ ab}^{f}+T_{\
ca}^{c}\delta_{b}^{f}-T_{\
cb}^{c}\delta_{a}^{f}\right)\equiv-\frac{2}{\kappa}Q_{\ ab}^{f},$

where the coefficient 2 comes from $T_{\ ij}^{c}=-T_{\ ji}^{c}$. Note that
with the Lorentz signature, contracting of $\varepsilon$ will lead to an extra
minus sign, i.e., $\varepsilon^{ijfd}\varepsilon_{abcd}=-\delta_{abc}^{ijf}$.
According to the above definition, $Q_{\ ab}^{f}$ is also antisymmetric for
the two lower indices. Now we lower down the index $f$ and multiply both sides
by $\varepsilon^{abcd}$, the above equation becomes:

$\frac{1}{l_{p}}\varepsilon^{abcd}\widetilde{B}_{a}^{\
ij}\varepsilon_{ijbf}=-\varepsilon^{abcd}\frac{2}{\kappa}Q_{fab}.$ (50)

Expand the contraction of the total antisymmteric tensor
$\varepsilon^{abcd}\varepsilon_{ijbf}=\delta_{ijf}^{acd}$, we obtain:

$\frac{1}{l_{p}}\left(\widetilde{B}_{f}^{\ cd}+\widetilde{B}_{a}^{\
ac}\delta_{f}^{d}-\widetilde{B}_{a}^{\
ad}\delta_{f}^{c}\right)=-\varepsilon^{abcd}\frac{1}{\kappa}Q_{fab}.$ (51)

By contracting $f$ and $c$, we further have:

$-\frac{2}{l_{p}}\widetilde{B}_{a}^{\
ad}=-\varepsilon^{abcd}\frac{1}{\kappa}Q_{abc}.$ (52)

Since $\varepsilon^{abcd}\eta_{ac}T_{\ fb}^{f}=0$, we can simplify the above
equation as:

$\frac{1}{l_{p}}\widetilde{B}_{a}^{\
ad}=\varepsilon^{abcd}\frac{1}{2\kappa}T_{abc}.$ (53)

Put this expression into Eq. (51), we finally get the components of
$\widetilde{B}_{f}$:

$\frac{1}{l_{p}}\widetilde{B}_{f}^{\
cd}=-\frac{1}{\kappa}\varepsilon^{abcd}Q_{fab}+\frac{1}{2\kappa}\left(\varepsilon^{abgd}\delta_{f}^{c}-\varepsilon^{abgc}\delta_{f}^{d}\right)T_{abg}.$
(54)

Thus, we end up with:

$\displaystyle\frac{1}{l_{p}}\widetilde{B}_{a}^{\ ij}\varepsilon_{ijkl}$
$\displaystyle=$
$\displaystyle\frac{2}{\kappa}\delta_{kl}^{ij}Q_{aij}-\frac{1}{\kappa}\delta_{akl}^{bcd}T_{bcd}$
(55) $\displaystyle=$
$\displaystyle\frac{4}{\kappa}Q_{akl}-\frac{6}{\kappa}T_{\left[akl\right]}.$

Apply the relation between the torsion and contorsion ($T^{a}=K^{a}_{\
b}\wedge e^{b}$, and the components
$K_{abc}=-\frac{1}{2}(T_{abc}+T_{bca}-T_{cab})$. As seen in Appendix A), we
can further simplify the above expression as:

$\displaystyle\frac{1}{l_{p}}\widetilde{B}_{a}^{\ ij}\varepsilon_{ijkl}$
$\displaystyle=$
$\displaystyle\frac{2}{\kappa}\left(T_{akl}-T_{kla}-T_{lak}\right)+\frac{4}{\kappa}\left(T^{b}_{\
bk}\eta_{al}-T^{b}_{\ bl}\eta_{ak}\right)$ (56) $\displaystyle=$
$\displaystyle\frac{4}{\kappa}K_{kla}+\frac{4}{\kappa}\left(T^{b}_{\
bk}\eta_{al}-T^{b}_{\ bl}\eta_{ak}\right).$

Actually $\widetilde{B}_{f}$ can be written in a more compact form:

$\widetilde{B}_{a}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge K^{cd},$
(57)

The full details of the above derivation can be found in Appendix D.

(4)Variation with respect to the last variable $e$ gives:

$\displaystyle\frac{1}{\kappa}\int R^{ab}\wedge e^{c}\wedge\delta
e^{d}\varepsilon_{abcd}+\frac{1}{l_{p}}\int\widetilde{B}_{d}\wedge D\delta
e^{d}=0,$ (58)

which leads to:

$\frac{1}{\kappa}R^{ab}\wedge
e^{c}\varepsilon_{abcd}-\frac{1}{l_{p}}D\widetilde{B}_{d}=0.$ (59)

Decompose $R$ as $R=\widetilde{R}+\widetilde{D}K+K\wedge K$. (See more details
in Appendix A, where $\widetilde{D}$ is covariant derivative with Christofell
connection $\Gamma$ and $\widetilde{R}$ is the torsion-free curvature tensor
written in terms of $\Gamma$.) In such a way, we can decompose the above
equation as:

$\frac{1}{\kappa}\left(\widetilde{R}^{ab}+K_{\ g}^{a}\wedge
K^{gb}\right)\wedge e^{c}\varepsilon_{abcd}+\frac{1}{l_{p}}K^{a}_{\
d}\wedge\widetilde{B}_{a}+\frac{1}{\kappa}\widetilde{D}K^{ab}\wedge
e^{c}\varepsilon_{abcd}-\frac{1}{l_{p}}\widetilde{D}\widetilde{B}_{d}=0,$ (60)

where we also separate the covariant derivative of $\widetilde{B}_{d}$ into
two parts $D\widetilde{B}_{d}=\widetilde{D}\widetilde{B}_{d}-K^{a}_{\
d}\wedge\widetilde{B}_{a}$. By substituting Eq. (57), we can finally simplify
the above equation as:

$\frac{1}{\kappa}\left(\widetilde{R}^{ab}+K_{\ g}^{a}\wedge
K^{gb}\right)\wedge
e^{c}\varepsilon_{abcd}+\frac{1}{\kappa}\varepsilon_{abmn}e^{b}\wedge
K^{mn}\wedge K^{a}_{\ d}=0.$ (61)

We note that the last two terms of Eq. (60) cancel out and
$e^{c}$,$\varepsilon_{abcd}$ can be taken out from $\widetilde{D}$ since
$\widetilde{D}e=\widetilde{D}\varepsilon_{abcd}=0$. To derive the components
of the above equation, just wedge $e^{f}$ on both sides:

$-\frac{1}{2\kappa}\widetilde{R}_{\ \
ij}^{ab}\delta_{abd}^{ijf}-\frac{1}{\kappa}K_{\ ci}^{a}K_{\ \
j}^{cb}\delta_{abd}^{ijf}+\frac{1}{\kappa}\delta_{amn}^{ijf}K^{mn}_{\ \ \
i}K_{\ dj}^{a}=0,$ (62)

where we replace the index from $g$ to $c$ in the second term. The first term
in left hand side is the usual Einstein tensor:

$\displaystyle\widetilde{R}_{\ \ ij}^{ab}\delta_{abd}^{ijf}$ $\displaystyle=$
$\displaystyle 2\widetilde{R}_{\ \ ab}^{ab}\delta_{d}^{f}+2\widetilde{R}_{\ \
bd}^{fb}+2\widetilde{R}_{\ \ da}^{af}=2\widetilde{R}\delta_{\
d}^{f}-4\widetilde{R}_{d}^{f}\equiv-4\widetilde{G}_{\ d}^{f}.$ (63)

The second term can be expressed as:

$\displaystyle K_{\ ci}^{a}K_{\ \ j}^{cb}\delta_{abd}^{ijf}$ $\displaystyle=$
$\displaystyle K_{\ ca}^{a}K_{\ \ b}^{cb}\delta_{d}^{f}+K_{\ cb}^{f}K_{\ \
d}^{cb}+K_{\ cd}^{a}K_{\ \ a}^{cf}-K_{\ cb}^{a}K_{\ \
a}^{cb}\delta_{d}^{f}-K_{\ ca}^{a}K_{\ \ d}^{cf}-K_{\ cd}^{f}K_{\ \ a}^{ca}$
$\displaystyle=$ $\displaystyle K_{\ ca}^{a}K_{\ \ b}^{cb}\delta_{d}^{f}+2K_{\
cb}^{f}K_{\ \ d}^{cb}-K_{\ cb}^{a}K_{\ \ a}^{cb}\delta_{d}^{f}-2K_{\
ca}^{a}K_{\ \ d}^{cf}.$

Recall the definition of contorsion, we have $K_{\
ca}^{a}=-\frac{1}{2}\left(T_{\ ca}^{a}+T_{ca}^{\ \ a}-T_{a\ c}^{\
a}\right)=-T_{\ ca}^{a}$. Without causing confusion, we define $T_{\
ac}^{a}\equiv T_{c}$ here to simplify the above expression as:

$K_{\ ci}^{a}K_{\ \
j}^{cb}\delta_{abd}^{ijf}=-\left(T_{a}T^{a}+K_{abc}K^{bca}\right)\delta_{d}^{f}+2K_{\
cb}^{f}K_{\ \ d}^{cb}-2T_{c}K_{\ \ d}^{cf}.$ (65)

The last term can be expressed as:

$\delta_{amn}^{ijf}K^{mn}_{\ \ \ i}K_{\ dj}^{a}=2\left(K^{jf}_{\ \ \ i}K_{\
dj}^{i}+K^{ij}_{\ \ \ i}K_{\ dj}^{f}+K^{fi}_{\ \ \ i}K_{\
dj}^{j}\right)=2(K^{fab}K_{dba}+T^{a}K^{f}_{\ da}-T^{f}T_{d}).$ (66)

By substituting all these expressions into Eq. (62), we finally obtain:

$\displaystyle\widetilde{G}_{fd}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(T_{a}T^{a}+K_{abc}K^{bca}\right)\eta_{fd}+K_{fcb}K_{\
\ d}^{cb}-T_{c}K_{\ fd}^{c}-K_{fab}K_{d}^{\ ba}-T_{a}K_{fd}^{\ \
a}+T_{f}T_{d}$ (67) $\displaystyle=$
$\displaystyle-\frac{l_{p}^{2}}{2}\left(\sigma_{a}\sigma^{a}+\Sigma_{abc}\Sigma^{bca}\right)\eta_{fd}+l_{p}^{2}\left(\Sigma_{fcb}\Sigma_{\
\ d}^{cb}-\sigma_{c}\Sigma_{\ fd}^{c}-\Sigma_{fab}\Sigma_{d}^{\
ba}-\sigma_{a}\Sigma_{fd}^{\ \ a}+\sigma_{f}\sigma_{d}\right),$

where $\sigma_{a}=\sigma_{\ ba}^{b},$ and
$\Sigma_{aij}=-\frac{1}{2}\left(\sigma_{aij}+\sigma_{ija}-\sigma_{jai}\right)$.
Symmetrize the index $f$ and $d$, we can derive a generalized Einstein
equation in the presence of loop-like extensive matter field:

$\displaystyle\frac{1}{\kappa}\widetilde{G}_{fd}=-\frac{1}{2\pi}\left(\sigma_{a}\sigma^{a}+\Sigma_{abc}\Sigma^{bca}\right)\eta_{fd}+\frac{1}{2\pi}\left(\Sigma_{fab}\Sigma_{\
\ d}^{ab}+\Sigma_{dab}\Sigma_{\ \
f}^{ab}\right)+\frac{1}{2\pi}\sigma_{a}\left(\sigma_{fd}^{\ \
a}+\sigma_{df}^{\ \ a}\right)-\frac{1}{\pi}\Sigma_{fab}\Sigma_{d}^{\
ba}+\frac{1}{\pi}\sigma_{f}\sigma_{d}.$ (68)

Surprisingly, if we interpret the right hand side of the above equation as the
energy-momentum tensor of loop-like extensive matter, its trace vanishes,
indicating the vanishing of scalar curvature in the above equation.

$\displaystyle\eta^{fd}\widetilde{G}_{fd}=-R$ $\displaystyle=$
$\displaystyle-2\left(T_{a}T^{a}+K_{abc}K^{bca}\right)+\left(K_{dab}K_{\ \
d}^{ab}\right)+T_{a}T^{a}-K_{fab}K^{fba}+T_{f}T^{f}$ (69) $\displaystyle=$
$\displaystyle-2\left(T_{a}T^{a}+K_{abc}K^{bca}\right)+2T_{a}T^{a}+K_{abc}K^{bca}-K_{abc}K^{acb}=0.$

Finally, the anti-symmetric part of Eq. (67) further requires:

$\frac{1}{2}\left(\sigma_{fab}\sigma_{\ \ d}^{ab}-\sigma_{dab}\sigma_{\ \
f}^{ab}\right)+\sigma_{\ ab}^{a}\left(\sigma_{fd}^{\ \ b}-\sigma_{df}^{\ \
b}\right)=0.$ (70)

Unlike the usual Einstein-Carton theory, it does not vanish automatically.
However, it can be satisfied if we assume the internal loop current
$\sigma_{abc}$ is totally antisymmetric. Of course, the most general solution
of the above equation is still unclear and we will leave this problem in our
future work.

To this end, we see that the loop-like extensive matter has several intriguing
features which make it to be a very promising candidate of dark matter. (a)
First of all, there would be no other interactions between loop-like extensive
matter and the normal matter except gravity. This is simply because loop-like
extensive objects will not couple to the 1-form gauge fields and they will not
participate all the interactions in Standard Model. (b) The energy-momentum
tensor of loop-like extensive matter does not contribute to scalar curvature
and such a unique property makes it very hard to be observed in a local way.
Nevertheless, the existences of loop-like extensive matter will still greatly
influence the large scale structure of galaxies in the universe. (c) Finally,
the loop-like extensive objects are the most natural topological excitations
in $3+1$D. Actually all point like objects should be regulated as loop-like
objects at Planck scale, and this is one of the most important motivation in
super-string theory. However, the loop-like extensive objects we proposed in
this paper are very different from the critical string in super-string theory
and they can even exist in $3+1$D without super symmetry.(This is because
these topological excitations only rely on the topological sectors of the
background space-time manifold, which is very different from the critical
string that couples to the metric of the background space-time manifold.)
Thus, in our framework, dark matter is more fundamental than the normal
matter, and it is not a surprise why dark matter should dominate the universe.
Below, we will discuss the origin of fermionic normal matter and derive the
generalized Einstein equation in the presence Dirac field.

##  V Generalized Einstein equation with Dirac field

In this section, we will discuss the most general action including fermionic
normal matter field, i.e., the Dirac field. Below we will consider the
following total action:

$S^{\prime}=S_{top}+S_{Dirac}-\frac{\theta\pi}{2}\int B^{ab}\wedge
B_{ab}-\int\sigma^{a}\wedge\widetilde{B}_{a},$ (71)

where $S_{top}$ defined in Eq. (41), and $S_{Dirac}$ reads:

$\displaystyle S_{Dirac}$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\int\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge\left(\frac{1}{2}\overline{\psi}\gamma^{d}\nabla\psi+\left(h.c.\right)\right)$
(72) $\displaystyle+\frac{1}{4!}\int\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge e^{d}m\overline{\psi}\psi.$

Here we use the following gamma matrix representation:

$\gamma^{0}=\left(\begin{array}[]{cc}0&-i\\\ -i&0\end{array}\right),\
\gamma^{j}=\left(\begin{array}[]{cc}0&-i\sigma^{j}\\\
i\sigma^{j}&0\end{array}\right),$ (73)

and $\overline{\psi}$ is defined as
$\overline{\psi}=i\psi^{\dagger}\gamma^{0}$. $\nabla$ is the usual covariant
derivative for spinor field:

$\nabla=d+\Omega=d+\Pi_{ab}\omega^{ab}.$ (74)

with $\Pi_{ab}=\frac{1}{8}\left[\gamma_{a},\gamma_{b}\right]$.

Here we express the Dirac action in a topological invariant way, which
indicates the topological origination of fermions. In a recent work, it was
proposed that all elementary fermions can be regarded as topological Majorana
zero modes at Planck scale. However, since topological Majorana zero modes
must attach onto linked closed-loops in $3+1$D, it is very natural to assume
all these topological zero modes arise from linked $e$-loops. After the
condensation of $\omega$-loop, the linked $e$-loops will be confined at Planck
scale and they can naturally be regarded as point-like particles. Of course,
the action of bosonic matter fields, i.e.. gauge fields and Higgs field can
not be written in such a topological invariant way, which suggests that the
bosonic matter are not fundamental and they should be regarded as the
collective motion of fermionic Dirac fields. How to derive the whole Standard
Model from an underlying topological action is a very important question and
we will leave this problem to our future work.

It is easy to verify that we can convert Eq. (72) to the conventional Dirac
action by applying the following identities:

$\displaystyle\varepsilon_{abcd}e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}$
$\displaystyle=$
$\displaystyle\det(e)\varepsilon_{\mu\nu\rho\sigma}e^{\sigma}_{d},$
$\displaystyle\varepsilon_{abcd}e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}e^{d}_{\sigma}$
$\displaystyle=$ $\displaystyle\det(e)\varepsilon_{\mu\nu\rho\sigma},$ (75)

which lead to:

$\displaystyle S_{Dirac}=-\int
d^{4}xe\left[\frac{1}{2}\overline{\psi}\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi+\left(h.c.\right)+m\overline{\psi}\psi\right],$

where $e\equiv\det(e^{a}_{\mu})$.

Below we will perform the variational calculations for total action
$S^{\prime}$.

(1)Variation with respect to $B$ gives:

$R^{ab}=\theta\pi B^{ab},\ \text{in terms of components}\
R_{\mu\nu}^{ab}=\theta\pi B_{\mu\nu}^{ab}.$ (77)

(2)Variation with respect to $\widetilde{B}$ gives:

$T^{a}=l_{p}\sigma^{a},\ \text{in terms of components}\ \
T_{\mu\nu}^{a}=l_{p}\sigma_{\mu\nu}^{a}.$ (78)

(3)Variation with respect to $\overline{\psi}$ gives:

$\displaystyle\frac{1}{3!}\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge\gamma^{d}\nabla\psi+\frac{1}{4!}\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge e^{c}\wedge e^{d}m\psi$
$\displaystyle-\frac{1}{4}\varepsilon_{abcd}T^{a}\wedge e^{b}\wedge
e^{c}\gamma^{d}\psi=0.$ (79)

The detailed derivation of above equation can be found in Appendix E. To
obtain the components of the above equation, we can apply the identity Eq.
(75) and finally have:

$\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi+m\psi-\frac{1}{2}T_{\
ab}^{a}\gamma^{b}\psi=0.$ (80)

We note that the components of $-\frac{1}{4}\varepsilon_{abcd}T^{a}\wedge
e^{b}\wedge e^{c}\gamma^{d}\psi$ can be expressed as:

$-\frac{1}{8}\varepsilon_{abcd}\varepsilon^{\mu\nu\rho\sigma}T^{a}_{\mu\nu}e^{b}_{\rho}e^{c}_{\sigma}\gamma^{d}\psi=\frac{1}{2}ee^{\mu}_{a}e^{\nu}_{d}T^{a}_{\mu\nu}\gamma^{d}\psi=\frac{1}{2}eT^{a}_{\
ab}\gamma^{b}\psi.$ (81)

The components of other terms can also be derived in a similarly way. We see
that this EOM is very different from the usual Einstein-Cartan theory with
Dirac field. Because in our case torsion is proportional to the loop source
and has nothing to do with fermion current, there would be no fermion coupling
term in Eq. (80). If we separate the spin connection $\omega^{ab}$ into
$\Gamma^{ab}$ and $K^{ab}$, the EOM of the Dirac field can further be
simplified as:

$\displaystyle\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Pi^{cd}(\Gamma_{cd\mu}+K_{cd\mu})\right)\psi+m\psi-\frac{1}{2}T_{\
ab}^{a}\gamma^{b}\psi$ (82) $\displaystyle=$
$\displaystyle\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Pi^{cd}\Gamma_{cd\mu}\right)\psi+m\psi+\frac{1}{4}K_{acd}\gamma^{acd}\psi$
$\displaystyle+\frac{1}{2}K_{\ da}^{a}\gamma^{d}\psi-\frac{1}{2}T_{\
ab}^{a}\gamma^{b}\psi$ $\displaystyle=$
$\displaystyle\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Pi^{cd}\Gamma_{cd\mu}\right)\psi+m\psi-\frac{l_{p}}{8}\sigma_{abc}\gamma^{abc}\psi=0$

where we use the identity:

$\displaystyle\gamma^{a}\Pi^{cd}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\gamma^{a}\Pi^{cd}+\Pi^{cd}\gamma^{a}\right)+\frac{1}{4}(\eta^{ac}\gamma^{d}-\eta^{ad}\gamma^{c})$
(83) $\displaystyle=$
$\displaystyle\frac{1}{4}\gamma^{acd}+\frac{1}{4}(\eta^{ac}\gamma^{d}-\eta^{ad}\gamma^{c}),$

with $\gamma^{abc}=\gamma^{[a}\gamma^{b}\gamma^{c]}$ and $K_{\ ba}^{a}=T_{\
ab}^{a}$.

Interestingly, in the presence of dark matter(loop current) background, the
extra term $-\frac{l_{p}}{8}\sigma_{abc}\gamma^{abc}$ can be regarded as a
small correction to the fermion mass. In general, the loop current background
could have a non-uniform distribution, which implies a small fluctuation of
fermion mass with dark matter background. Of course, as the correction is
proportional to the Planck length $l_{p}$, it is not clear if such a tiny
change of fermion mass can be measured by current experiment or not.

Next we can further compute the variation of $S^{\prime}$ with respect to $e$
and $\omega$. We first consider the variation of Dirac field with respect to
$e$ and compute energy momentum tensor $t_{d}$ for Dirac field.

$\displaystyle\delta_{e}S_{Dirac}$ $\displaystyle=$
$\displaystyle\frac{1}{2!}\int\varepsilon_{abcd}\delta e^{a}\wedge e^{b}\wedge
e^{c}\wedge\left(\frac{1}{2}\overline{\psi}\gamma^{d}\nabla\psi+\left(h.c.\right)\right)+\frac{1}{3!}\int\varepsilon_{abcd}\delta
e^{a}\wedge e^{b}\wedge e^{c}\wedge e^{d}m\overline{\psi}\psi$ (84)
$\displaystyle=$
$\displaystyle\frac{1}{2!}\int\varepsilon_{abcd}\left(\frac{1}{2}\overline{\psi}\gamma^{a}\nabla\psi+\left(h.c.\right)\right)\wedge
e^{b}\wedge e^{c}\wedge\delta
e^{d}+\frac{1}{3!}\int\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge\delta e^{d}m\overline{\psi}\psi$ $\displaystyle\equiv$
$\displaystyle-\int t_{d}\wedge\delta e^{d},$

where

$t_{d}=-\frac{1}{2!}\varepsilon_{abcd}\left(\frac{1}{2}\overline{\psi}\gamma^{a}\nabla\psi+\left(h.c.\right)\right)\wedge
e^{b}\wedge e^{c}-\frac{1}{3!}\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}m\overline{\psi}\psi.$ (85)

To simplify the energy momentum tensor, we can make use of the EOM for Dirac
field. We wedge $e^{f}$ on the right to make it a 4-form and derive the
components of the above equation as:

$\displaystyle-\frac{1}{2!}\varepsilon_{abcd}\varepsilon^{ibcf}\left(\frac{1}{2}\overline{\psi}\gamma^{a}\nabla_{i}\psi+\left(h.c.\right)\right)-\frac{1}{3!}\varepsilon_{abcd}\varepsilon^{abcf}m\overline{\psi}\psi$
(86) $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\overline{\psi}\gamma^{a}\nabla_{a}\psi+\left(h.c.\right)+m\overline{\psi}\psi\right)\delta^{f}_{d}-\left(\frac{1}{2}\overline{\psi}\gamma^{f}\nabla_{d}\psi+\left(h.c.\right)\right)$
$\displaystyle=$
$\displaystyle-\left(\frac{1}{2}\overline{\psi}\gamma^{f}\nabla_{d}\psi+\left(h.c.\right)\right).$

The first term of the second line vanishes on shell. On the other hand, we
find the component of
$-\frac{1}{3!}\varepsilon_{gabc}\left(\frac{1}{2}\overline{\psi}\gamma^{g}\nabla_{d}\psi+\left(h.c.\right)\right)e^{a}\wedge
e^{b}\wedge e^{c}$ can also be written as:

$-\frac{1}{3!}\varepsilon_{gabc}\varepsilon^{abcf}\left(\frac{1}{2}\overline{\psi}\gamma^{g}\nabla_{d}\psi+\left(h.c.\right)\right)=-\left(\frac{1}{2}\overline{\psi}\gamma^{f}\nabla_{d}\psi+\left(h.c.\right)\right),$
(87)

which is exactly the same as Eq. (86). Thus we can simplify the energy
momentum tensor as:

$\displaystyle t_{d}$ $\displaystyle=$
$\displaystyle-\frac{1}{3!}\varepsilon_{fabc}\left(\frac{1}{2}\overline{\psi}\gamma^{f}\nabla_{d}\psi+\left(h.c.\right)\right)e^{a}\wedge
e^{b}\wedge e^{c}\equiv\frac{1}{3!}\varepsilon_{fabc}t_{\ d}^{f}e^{a}\wedge
e^{b}\wedge e^{c},$ (88)

where $t_{\
d}^{f}=-\frac{1}{2}\overline{\psi}\gamma^{f}e_{d}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi+\left(h.c.\right).$

Similarly, we can also consider the variation of Dirac field with respect to
$\omega$:

$\displaystyle\delta_{\omega}S_{Dirac}$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\frac{1}{2}\int\varepsilon_{cdfg}e^{c}\wedge
e^{d}\wedge
e^{f}\wedge\delta\omega^{ab}\overline{\psi}\left(\gamma^{g}\Pi_{ab}+\Pi_{ab}\gamma^{g}\right)\psi$
$\displaystyle=$
$\displaystyle\frac{1}{3!}\frac{1}{2}\int\varepsilon_{gcdf}\delta\omega^{ab}\wedge
e^{c}\wedge e^{d}\wedge
e^{f}\overline{\psi}\left(\gamma^{g}\Pi_{ab}+\Pi_{ab}\gamma^{g}\right)\psi$
$\displaystyle\equiv$
$\displaystyle-\int\delta\omega^{ab}\wedge\widetilde{T}_{ab},$ (90)

where

$\widetilde{T}_{ab}=-\frac{1}{3!}\frac{1}{2}\varepsilon_{gcdf}\overline{\psi}\left(\gamma^{g}\Pi_{ab}+\Pi_{ab}\gamma^{g}\right)\psi
e^{c}\wedge e^{d}\wedge e^{f}=\frac{1}{3!}\widetilde{T}_{\
ab}^{g}\varepsilon_{gcdf}e^{c}\wedge e^{d}\wedge e^{f},$ (91)

and $\widetilde{T}_{\
ab}^{g}=-\frac{1}{2}\overline{\psi}\left(\gamma^{g}\Pi_{ab}+\Pi_{ab}\gamma^{g}\right)\psi=-\frac{1}{4}\overline{\psi}\gamma^{[a}\gamma^{b}\gamma^{c]}\psi$,
which is totally antisymmetric with respect to all indices.

Now we are ready to compute the variation with respect to $\omega$ and $e$ for
the total action $S^{\prime}$.

(4)Variation of the total action $S^{\prime}$ with respect to $\omega$ yields:

$\delta_{\omega}S^{\prime}=\frac{1}{2\kappa}\int D\delta\omega^{ab}\wedge
e^{c}\wedge e^{d}\varepsilon_{abcd}+\int D\delta\omega^{ab}\wedge
B_{ab}+\frac{1}{l_{p}}\int\widetilde{B}_{a}\wedge\delta\omega^{ab}\wedge
e_{b}-\int\delta\omega^{ab}\wedge\widetilde{T}_{ab}=0.$ (92)

Similar as the case without Dirac filed, integral by parts leads to:

$\frac{1}{\kappa}T^{c}\wedge
e^{d}\varepsilon_{abcd}+\frac{1}{2l_{p}}\left(\widetilde{B}_{a}\wedge
e_{b}-\widetilde{B}_{b}\wedge e_{a}\right)=\widetilde{T}_{ab}.$ (93)

The solution of $\widetilde{B}_{a}$ to the above equation is:

$\widetilde{B}_{d}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge
K^{cd}+\frac{l_{p}}{2}\varepsilon_{abcd}e^{b}\wedge e^{f}\widetilde{T}_{f}^{\
cd},$ (94)

See in Appendix D for full details.

(5)Finally we consider the variation of $S^{\prime}$ with respect to the last
variable $e$:

$\delta_{e}S^{\prime}=\frac{1}{\kappa}\int R^{ab}\wedge e^{c}\wedge\delta
e^{d}\varepsilon_{abcd}+\frac{1}{l_{p}}\int\widetilde{B}_{d}\wedge D\delta
e^{d}-\int t_{d}\wedge\delta e^{d}=0,$ (95)

which gives:

$\frac{1}{\kappa}R^{ab}\wedge
e^{c}\varepsilon_{abcd}-\frac{1}{l_{p}}D\widetilde{B}_{d}=t_{d}.$ (96)

Again, decompose $R$ and the covariant derivative in to torsion-free part and
torsion part, we obtain:

$\frac{1}{\kappa}\left(\widetilde{R}^{ab}+K_{\ g}^{a}\wedge
K^{gb}\right)\wedge
e^{c}\varepsilon_{abcd}+\frac{1}{l_{p}}\widetilde{B}_{a}\wedge K_{\
d}^{a}-t_{d}+\frac{1}{\kappa}\widetilde{D}K^{ab}\wedge
e^{c}\varepsilon_{abcd}-\frac{1}{l_{p}}\widetilde{D}\widetilde{B}_{d}=0.$ (97)

By substituting the solution of $\widetilde{B}_{a}$, we can simplify the above
equation as:

$\frac{1}{\kappa}\left(\widetilde{R}^{ab}+K_{\ g}^{a}\wedge
K^{gb}\right)\wedge
e^{c}\varepsilon_{abcd}+\frac{1}{\kappa}\varepsilon_{abmn}e^{b}\wedge
K^{mn}\wedge K_{\ d}^{a}=t_{d}-\frac{1}{2}\varepsilon_{abmn}e^{b}\wedge
e^{g}\wedge K_{\ d}^{a}\widetilde{T}_{g}^{\
mn}+\frac{1}{2}\varepsilon_{dbmn}e^{b}\wedge
e^{g}\widetilde{D}\widetilde{T}_{g}^{\ mn}.$ (98)

The components of the left hand side are the same as the case without Dirac
field. The components of the right hand side can be obtained by wedge $e^{f}$
on the right side:

$\displaystyle\frac{1}{3!}\varepsilon_{gabc}\varepsilon^{abcf}t^{g}_{\
d}-\frac{1}{2}\varepsilon_{abmn}\varepsilon^{bgif}K_{\
di}^{a}\widetilde{T}_{g}^{\
mn}+\frac{1}{2}\varepsilon_{dbmn}\varepsilon^{bgif}\widetilde{D}_{i}\widetilde{T}_{g}^{\
mn}$ (99) $\displaystyle=$ $\displaystyle t^{f}_{\ d}-\left(K_{\
di}^{a}\widetilde{T}_{a}^{\ if}+K_{\ da}^{a}\widetilde{T}_{g}^{\ fg}+K_{\
di}^{f}\widetilde{T}_{g}^{\
gi}\right)+\left(\widetilde{D}_{i}\widetilde{T}_{d}^{\
if}+\widetilde{D}_{i}\widetilde{T}_{g}^{\
gi}\delta^{f}_{d}+\widetilde{D}_{d}\widetilde{T}_{g}^{\ fg}\right)$
$\displaystyle=$ $\displaystyle t^{f}_{\
d}+\widetilde{T}^{abf}K_{dab}+\widetilde{D}_{i}\widetilde{T}_{\ \ d}^{if},$

where we use the total antisymmetric property of $\widetilde{T}$.

Thus, the components of Eq. (98) read:

$\displaystyle\widetilde{G}_{fd}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(T_{a}T^{a}+K_{abc}K^{bca}\right)\eta_{fd}+K_{fcb}K_{\
\ d}^{cb}-T_{c}K_{\ fd}^{c}-K_{fab}K_{d}^{\ ba}-T_{a}K_{fd}^{\ \
a}+T_{f}T_{d}+\frac{\kappa}{2}\left(t_{fd}+\widetilde{T}_{abf}K_{d}^{\
ab}+\widetilde{D}_{i}\widetilde{T}_{\ fd}^{i}\right)$ (100) $\displaystyle=$
$\displaystyle-\frac{l_{p}^{2}}{2}\left(\sigma_{a}\sigma^{a}+\Sigma_{abc}\Sigma^{bca}\right)\eta_{fd}+l_{p}^{2}(\Sigma_{fcb}\Sigma_{\
\ d}^{cb}-\sigma_{c}\Sigma_{\ fd}^{c}-\Sigma_{fab}\Sigma_{d}^{\
ba}-\sigma_{a}\Sigma_{fd}^{\ \ a}+\sigma_{f}\sigma_{d})$
$\displaystyle+\frac{\kappa l_{p}}{2}\left(\widetilde{T}_{abf}\Sigma_{d}^{\
ab}+\widetilde{T}_{abf}\Sigma_{\ \
d}^{ab}\right)+\frac{\kappa}{2}\left(-\frac{1}{2}\overline{\psi}\gamma_{f}e_{d}^{\mu}\left(\partial_{\mu}+\Gamma_{ab\mu}\Pi^{ab}\right)\psi+\left(h.c.\right)+\widetilde{D}_{i}\widetilde{T}_{\
fd}^{i}\right),$

where in the second step, we separate $t_{fd}$ into torsion free part and
torsion part:

$\displaystyle t_{fd}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\overline{\psi}\gamma_{f}e_{d}^{\mu}\left(\partial_{\mu}+\Gamma_{ab\mu}\Pi^{ab}\right)\psi-\frac{1}{2}K_{\
\
d}^{ab}\overline{\psi}\gamma_{f}\Pi_{ab}\psi-\frac{1}{2}\left(-\partial_{\mu}\overline{\psi}+\overline{\psi}\Gamma_{ab\mu}\Pi^{ab}\right)\gamma_{f}e_{d}^{\mu}\psi-\frac{1}{2}K_{\
\ d}^{ab}\overline{\psi}\Pi_{ab}\gamma_{f}\psi$ (101) $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}\overline{\psi}\gamma_{f}e_{d}^{\mu}\left(\partial_{\mu}+\Gamma_{ab\mu}\Pi^{ab}\right)\psi+\left(h.c.\right)\right)+\widetilde{T}_{abf}K_{\
\ d}^{ab}.$

Finally, after symmetrizing the index $f$ and $d$, we end up with the
generalized Einstein equation with both dark matter and Dirac field:

$\displaystyle\frac{1}{\kappa}\widetilde{G}_{fd}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\left(\sigma_{a}\sigma^{a}+\Sigma_{abc}\Sigma^{bca}\right)\eta_{fd}+\frac{1}{2\pi}\left(\Sigma_{fab}\Sigma_{\
\ d}^{ab}+\Sigma_{dab}\Sigma_{\ \
f}^{ab}\right)+\frac{1}{2\pi}\sigma_{a}\left(\sigma_{fd}^{\ \
a}+\sigma_{df}^{\ \ a}\right)-\frac{1}{\pi}\Sigma_{fab}\Sigma_{d}^{\
ba}+\frac{1}{\pi}\sigma_{f}\sigma_{d}$ (102)
$\displaystyle+\frac{l_{p}}{4}\left[\widetilde{T}_{abf}(\Sigma_{d}^{\
ab}+\Sigma_{\ \ d}^{ab})+\widetilde{T}_{abd}\left(\Sigma_{f}^{\ ab}+\Sigma_{\
\
f}^{ab}\right)\right]-\frac{1}{8}\left[\overline{\psi}\left(\gamma_{f}e_{d}^{\mu}+\gamma_{d}e_{f}^{\mu}\right)\left(\partial_{\mu}+\Gamma_{ab\mu}\Sigma^{ab}\right)\psi+(h.c.)\right].$

The first line is the same as Eq. (68), which is the energy momentum tensor of
the dark matter. The first term in the second line can be regarded as the
coupling between the dark matter and Dirac field, and the second term in the
second line is the usual energy momentum tensor of Dirac field.

In addition, the antisymmetric part of Eq. (100) can be simplified as:

$\frac{1}{2}\left(\sigma_{fcb}\sigma_{\ \ d}^{cb}-\sigma_{dcb}\sigma_{\ \
f}^{cb}\right)+\sigma_{a}\left(\sigma_{fd}^{\ \ a}-\sigma_{df}^{\ \
a}\right)=\frac{\pi}{4}l_{p}(\sigma_{f}^{\
ab}\widetilde{T}_{dab}-\sigma_{d}^{\ ab}\widetilde{T}_{fab}).$ (103)

The detailed derivation can be found in Appendix F. Again, this equation
doesn’t have a obvious solution. Even with totally antisymmetric internal loop
current $\sigma_{abc}$, this equation are not satisfied automatically due to
the additional terms on the right-hand side. However, since the right-hand
side is proportional to $l_{p}$, the totally antisymmetric internal loop
current $\sigma_{abc}$ is still an approximate solution in the limit
$l_{p}\rightarrow 0$.

##  VI Other possible topological terms and the uniqueness of generalized
Einstein equation

In the loop condensation phase, due to the breaking of 2-form and 1-form gauge
transformations, we can in principle consider more general topological terms
which are invariant under local Lorentz transformation. It is very important
to understand the effect of these terms to justify the uniqueness of the
generalized Einstein equation. Below we will consider the following most
general topological invariant action(in the absence of Dirac field):

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int\varepsilon_{abcd}R^{ab}\wedge e^{c}\wedge
e^{d}+\int R^{ab}\wedge B_{ab}+\frac{1}{l_{p}}\int
T^{a}\wedge\widetilde{B}_{a}-\frac{\theta\pi}{2}\int B^{ab}\wedge
B_{ab}-\int\sigma^{a}\wedge\widetilde{B}_{a}$ (104)
$\displaystyle-\frac{\lambda}{2}\int T^{a}\wedge T_{a}+\frac{\psi}{2}\int
R^{ab}\wedge e_{a}\wedge e_{b}.$

The above action includes all possible topological invariant terms except
$\widetilde{B}^{a}\wedge\widetilde{B}_{a}$. This is because the loops coupled
with $B$ and $\widetilde{B}$ can not be condensed simultaneously due to the
presence of $Ree$ term, as discussed in section IV. Similar as before,
variation with respect to $B$ and $\widetilde{B}$ gives:

$R^{ab}=\theta\pi B^{ab},\ \ \ T^{a}=l_{p}\sigma^{a}.$ (105)

Variation with respect to $\omega$ gives:

$\frac{1}{\kappa}T^{c}\wedge
e^{d}\varepsilon_{abcd}+(\frac{1}{l_{p}}\widetilde{B}+(\psi-\lambda)T)_{[a}\wedge
e_{b]}=0.$ (106)

As seen in Appendix D, the general solution of $\widetilde{B}_{a}$ reads:

$\widetilde{B}_{a}=-l_{p}(\psi-\lambda)T_{a}+\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge
K^{cd},$ (107)

Finally, variation with respect to $e^{d}$ gives:

$\displaystyle\frac{1}{\kappa}\varepsilon_{abcd}R^{ab}\wedge
e^{c}-\frac{1}{l_{p}}D\widetilde{B}_{d}+\lambda DT_{d}+\psi R_{ad}\wedge
e^{a}$ (108) $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\varepsilon_{abcd}\left(\widetilde{R}^{ab}+\widetilde{D}K^{ab}+K_{\
g}^{a}\wedge K^{gb}\right)\wedge
e^{c}-\frac{1}{l_{p}}\left(\widetilde{D}\widetilde{B}_{d}-\widetilde{B}_{a}\wedge
K^{a}_{\ d}\right)+(\psi-\lambda)R_{ad}\wedge e^{a},$

where in the second step we expand $R$ and the covariant derivative. We also
apply the structure equation $DT_{d}=R^{a}_{\ b}\wedge e^{b}$ (See Appendix
A). By substituting Eq. (107) in, we finally have:

$\varepsilon_{abcd}\widetilde{R}^{ab}\wedge e^{c}+\varepsilon_{abcd}K_{\
f}^{a}\wedge K^{fb}\wedge e^{c}+\varepsilon_{abmn}e^{b}\wedge K^{mn}\wedge
K_{\ d}^{a}=0,$ (109)

which is exactly the same as Eq. (61). By adding the Dirac field into the
action Eq. (104), we will still get the same field equation as Eq. (98).

Thus we conclude that the generalized Einstein equation we derive here is very
unique and additional topological invariant term will not change the EOM. Of
course, the cosmology constant term can always be include in our framework
naturally since it can also be rewritten as a topological invariant form with
$S_{\Lambda}=\frac{\Lambda}{4!}\int\epsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge e^{d}$. Moreover, all the topological invariant terms are
dimension 4 operators and the whole theory is still power counting
renormalizable. All these nice properties suggest that topological invariant
principle is more profound than general covariance principle.

##  VII Conclusion and discussions

In this paper, we begin with a 3+1D topological gravity theory which is super
renormalizable, then we argue that Einstein gravity will naturally emerge from
such a topological gravity theory via $\omega$-loop condensation. Although a
rigorous way of treating loop condensation requires a new framework of string
field theory222As loop like excitations arise from the underlying 3+1D TQFT
can be regarded as a non-smooth string, therefore, we believe such a string
field theory is very different from the usual super string theory which can
only be defined in 10 or 11 dimensional spacetime to describe loop like
extensive excitations, the semiclassicial limit can still be achieved by
simply adding a topological mass term for the 2-form gauge field $B$ which
couples to $\omega$-loop. Remarkably, such a semi-classical theory is still
super renormalizable.

Moreover, our approach indicates that loop-like extensive objects could be the
most general form of matter source(including both normal matter and dark
matter) in 3+1D space time. In particular, those uncondensed (deconfined)
loop-like excitations can be naturally interpreted as dark matter. By
introducing a loop source term of (unlinked) $e$-loop(that is, the topological
invariant coupling between $e$-loop current and the corresponding 2-form gauge
field $\tilde{B}$), we obtain a generalized Einstein equation which naturally
includes the dark matter sector. Surprisingly, we find that the trace of the
energy-momentum tensor vanishes for dark matter. As a result, such kind of
”conformal” dark matter will not contribute to scalar curvature. On the other
hand, we predict that the current of such kind of dark matter will naturally
contribute to torsion.

In addition, we also investigate the essential difference between our theory
and the usual Einstein-Carton theory by introducing Dirac fields.
Interestingly, we find that the fermion current will not contribute to
torsion. From the renormalization group perspective, the traditional Einstein-
Carton theory is inconsistent with the relativistic quantum field theory. This
is because the torsion produced by Dirac spinor will lead to four fermion
interactions, however, relativistic quantum field theory tells us that four
fermion interactions are irrelevant in $3+1$D and they should not appear in
the semiclassical limit(for any fixed background metric). Thus, we claim that
all normal matters actually will not produce torsion while only dark matter
can be the possible source of torsion. Such a strong prediction can be tested
in future experiments. Finally, we consider all the allowed topological
invaraint terms and we find all these terms will not change the generalized
Einstein equation.

On the other hand, although all normal matter looks like point like particle,
it has been argued for decades that there should be no point-like particle at
Planck scale due to the UV divergence in quantum field theory. In usual super
string theory, it was conjectured that all elementary particles are actually
different vibration modes of critical strings at Planck scale. However, super
string theory requires both super symmetry and extra space-time dimensions,
but none of them are observed experimentally so far. Very recently, a
topological scenario suggests that all elementary particles are actually
topological zero modes of linked loop-like excitations. Such an approach
naturally explains the origin of three generations of elementary particles and
gives rise to the correct neutrino mass mixing angles. In our theory, when
$\omega$-loop condenses, the linked $e$-loops will be confined and they can be
regarded as fermionic particles if topological Majroana modes can be further
attached on to these linked $e$-loops. Apparently, understanding the dynamics
and quantum effect of loop-like extensive matter will become very important
but challenging direction in the future. In particular, the notation of loop
condensation might unify all fundamental forces and tell us the origin of four
dimensional space-time!

##  VIII Acknowledgement

We would like to thank Yongshi Wu and Ming-Chung Chu for helpful discussions.
This work is supported by a grant from the Research Grants Council of the Hong
Kong Special Administrative Region, China (Project No. AoE/P-404/18).

##  Appendix A Structure equations

Below we will list several useful identities and related definitions.

For $R_{\ b}^{a}\in\Lambda_{\left(1,1\right)}^{2}$, it is straightforward to
verify that:

$\displaystyle DR$ $\displaystyle=$ $\displaystyle
D\left(d\omega+\omega\wedge\omega\right)=d\left(d\omega+\omega\wedge\omega\right)+\omega\wedge
R-R\wedge\omega$ (110) $\displaystyle=$ $\displaystyle
d\omega\wedge\omega-\omega\wedge d\omega+\omega\wedge
d\omega+\omega\wedge\omega-d\omega\wedge\omega-\omega\wedge\omega$
$\displaystyle=$ $\displaystyle 0.$

By replacing $\omega$ with the torsion free connection $\Gamma$ in the above
equation, we have:

$\widetilde{D}\widetilde{R}=0,$ (111)

where $\widetilde{D}$ is the covariant derivative with respect to Christofell
connection $\Gamma$, and $\widetilde{R}$ is written in terms of $\Gamma$ , and
the explict form of $\Gamma$ reads:

$\displaystyle\Gamma_{\mu\nu}^{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{\rho\lambda}\left(\partial_{\mu}g_{\nu\lambda}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu}\right)$
(112) $\displaystyle=$
$\displaystyle\frac{1}{2}e^{a\rho}e^{\lambda}_{b}(\partial_{\mu}e^{a}_{\nu}e_{a\lambda}+e^{a}_{\nu}\partial_{\mu}e_{a\lambda}+\partial_{\nu}e^{a}_{\mu}e_{a\lambda}$
$\displaystyle+e^{a}_{\mu}\partial_{\nu}e_{a\lambda}-\partial_{\lambda}e^{a}_{\mu}e_{a\nu}-\partial_{\lambda}e^{a}_{\nu}e_{a\mu}).$

Through a coordinate transformation, we can also write it with internal index:

$\displaystyle\Gamma_{\ b\mu}^{a}$ $\displaystyle=$ $\displaystyle
e^{a}_{\rho}\partial_{\mu}e^{\rho}_{b}+e^{a}_{\rho}\Gamma_{\nu\mu}^{\rho}e^{\nu}_{b}$
(113) $\displaystyle=$ $\displaystyle
e^{a}_{\rho}\partial_{\mu}e^{\rho}_{b}+\frac{1}{2}e^{a\lambda}e^{\nu}_{b}(\partial_{\mu}e^{c}_{\nu}e_{c\lambda}+e^{c}_{\nu}\partial_{\mu}e_{c\lambda}+\partial_{\nu}e^{c}_{\mu}e_{c\lambda}$
$\displaystyle+e^{c}_{\mu}\partial_{\nu}e_{c\lambda}-\partial_{\lambda}e^{c}_{\mu}e_{c\nu}-\partial_{\lambda}e^{c}_{\nu}e_{c\mu})$
$\displaystyle=$
$\displaystyle\frac{1}{2}[e^{\rho}_{b}(\partial_{\rho}e_{\mu}^{a}-\partial_{\mu}e_{\rho}^{a})+e^{a\rho}(\partial_{\mu}e_{b\rho}-\partial_{\rho}e_{b\mu})$
$\displaystyle+e^{a\lambda}e^{\nu}_{b}e^{c}_{\mu}(\partial_{\nu}e_{c\lambda}-\partial_{\lambda}e_{c\nu})].$

For $T^{a}\in\Lambda_{\left(1,0\right)}^{2}$, we have:

$\displaystyle DT^{a}$ $\displaystyle=$ $\displaystyle d\left(de^{a}+\omega_{\
b}^{a}\wedge e^{b}\right)+\omega_{\ b}^{a}\wedge\left(de^{b}+\omega_{\
c}^{b}\wedge e^{c}\right)$ $\displaystyle=$ $\displaystyle d\omega_{\
b}^{a}\wedge e^{b}-\omega_{\ b}^{a}\wedge de^{b}+\omega_{\ b}^{a}\wedge
de^{b}+\omega_{\ b}^{a}\wedge\omega_{\ c}^{b}\wedge e^{c}$ $\displaystyle=$
$\displaystyle R_{\ b}^{a}\wedge e^{b}.$

Again, if we take $\widetilde{D}$ on $e^{a}$:

$\displaystyle\widetilde{D}e^{a}$ $\displaystyle=$ $\displaystyle
de^{a}+\Gamma_{\ b}^{a}\wedge e^{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}e_{\nu}^{a}-\partial_{\nu}e_{\mu}^{a}+\Gamma_{\
b\mu}^{a}e_{\nu}^{b}-\Gamma_{\ b\nu}^{a}e_{\mu}^{b}\right)dx^{\mu}\wedge
dx^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e_{\rho}^{a}e_{c}^{\rho}\left(\partial_{\mu}e_{\nu}^{c}-\partial_{\nu}e_{\mu}^{c}+\Gamma_{\
b\mu}^{c}e_{\nu}^{b}-\Gamma_{\ b\nu}^{c}e_{\mu}^{b}\right)dx^{\mu}\wedge
dx^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e_{\rho}^{a}\left(\Gamma_{\nu\mu}^{\rho}-\Gamma_{\mu\nu}^{\rho}\right)dx^{\mu}\wedge
dx^{\nu}=0.$

Thus, it is convenient to separate the spin connection into two pieces:

$\omega_{\ b}^{a}=\Gamma_{\ b}^{a}+K_{\ b}^{a},$ (116)

where $K_{\ b}^{a}$ is called contorsion. We can get the relation between $T$
and $K$ from Eq. (A), which is:

$T^{a}=De^{a}=\widetilde{D}e^{a}+K_{\ b}^{a}\wedge e^{b}=K_{\ b}^{a}\wedge
e^{b},$ (117)

and the corresponding component form reads:

$K_{ab\mu}=-\frac{1}{2}(e^{\rho}_{b}T_{a\rho\mu}+e^{\rho}_{a}T_{b\mu\rho}-e^{c}_{\mu}e^{\rho}_{a}e^{\sigma}_{b}T_{c\rho\sigma}).$
(118)

By using the decomposition (116), we can rewrite the curvature tensor into:

$\displaystyle R$ $\displaystyle=$ $\displaystyle
d(\Gamma+K)+\left(\Gamma+K\right)\wedge\left(\Gamma+K\right)$ (119)
$\displaystyle=$ $\displaystyle d\Gamma+dK+\Gamma\wedge\Gamma+\Gamma\wedge
K+K\wedge\Gamma+K\wedge K$ $\displaystyle=$
$\displaystyle\widetilde{R}+\widetilde{D}K+K\wedge K.$

Finally, it is easy to show that $\varepsilon_{abcd}$ is a covariant constant:

$\displaystyle D\varepsilon_{abcd}$ $\displaystyle=$ $\displaystyle
d\varepsilon_{abcd}-\omega_{\ a}^{f}\varepsilon_{fbcd}-\omega_{\
b}^{f}\varepsilon_{afcd}-\omega_{\ c}^{f}\varepsilon_{abfd}-\omega_{\
d}^{f}\varepsilon_{abcf}$ (120) $\displaystyle=$ $\displaystyle 0.$

The first term $d\varepsilon_{abcd}$ is zero. For the left four terms, we can
consider their components. Without loss of generality, we set
$a=1,b=1,c=2,d=3$ (other cases like $a\neq b\neq c\neq d$ or $a=b=c\neq d$ are
very esay to check), then the relevant nonzero terms are $\omega_{\
a}^{f}\varepsilon_{fbcd}=\omega_{\ 1}^{0}$ and $\omega_{\
b}^{f}\varepsilon_{afcd}=-\omega_{\ 1}^{0}$. These two terms cancel each other
out, so $D\varepsilon_{abcd}=0$.

##  Appendix B Coefficient quantization of ISO(3) Chern-Simons action

To construct a Euclidean 3 dimensional gravity theory with the form $e\wedge
R$, we need to use the gauge group $ISO(3)$ which is semidirect product of
translation group $R(3)$ and rotation group $SO(3)$. Fortunately, the non-
compact part $R(3)$ can be contracted to a point, therefore the topological
properties of $ISO(3)$ and $SO(3)$ are the same. This can be proved by long
exact sequence:

$\displaystyle...\rightarrow\pi_{n}(R3)\rightarrow\pi_{n}(ISO(3))\rightarrow\pi_{n}(SO(3))\rightarrow$
$\displaystyle\pi_{n-1}(R3)\rightarrow...\rightarrow\pi_{0}(R3)\rightarrow\pi_{0}(ISO(3)).$
(121)

$\pi_{n}(R3)=0$ and $\pi_{n-1}(R3)=0$ lead to
$\pi_{n}(ISO(3))=\pi_{n}(SO(3))$. Thus we can use $ISO(3)$ to construct gauge
theory and use $SO(3)$ to compute its topological properties. We define the
rotational generators as $J^{ab}$ and the translational generators as $P^{a}$.
The non-degenerate invariant bilinear form can be written as
$W=\varepsilon_{abc}P^{a}J^{bc}$. So the invariant quadratic form reads:

$\left\langle P_{a},J_{bc}\right\rangle=\varepsilon_{abc},\ \ \left\langle
J_{a},J_{b}\right\rangle=\left\langle P_{a},P_{b}\right\rangle=0.$ (122)

To make $W$ an invariant form ($\left[P_{a},W\right]=\left[J_{ab},W\right]=0$,
the commutative relations of $ISO(3)$ generator must satisfy:

$\left[J_{ab},J_{cd}\right]=\delta_{bc}J_{ad}-\delta_{bd}J_{ac}-\delta_{ac}J_{bd}+\delta_{ad}J_{bc},$
(123)

and

$\left[P_{a},J_{bc}\right]=\delta_{ab}P_{c}-\delta_{ac}P_{b},\ \
\left[P_{a},P_{b}\right]=0.$ (124)

Let us use these definitions to construct gauge theory for the group $ISO(3)$.
The gauge field is a Lie-algebra-valued one form

$A_{i}=e_{i}^{a}P_{a}+\frac{1}{2}\omega_{i}^{ab}J_{ab},$ (125)

Thus, $\delta S_{BR}$ can be written as the usual three-form Chern-Simons
action of gauge group $ISO(3)$, where:

$\displaystyle\int Tr\left(A\wedge dA\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\varepsilon_{abc}\left(e^{a}\wedge
d\omega^{bc}+\omega^{ab}\wedge de^{c}\right)$ (126) $\displaystyle=$
$\displaystyle\int\varepsilon_{abc}e^{a}\wedge d\omega^{bc}.$

and

$\displaystyle\frac{2}{3}\int Tr\left(A\wedge A\wedge A\right)$
$\displaystyle=$ $\displaystyle\frac{1}{3}\int
Tr\left([A_{i},A_{j}]dx^{i}\wedge dx^{j}\wedge A\right)$ (127)
$\displaystyle=$ $\displaystyle\frac{1}{3}\int
Tr\left[\left(e^{a}\wedge\omega^{bc}\left[P_{a},J_{bc}\right]+\frac{1}{4}\omega^{ab}\wedge\omega^{cd}\left[J_{ab},J_{bd}\right]\right)\wedge
A\right]$ $\displaystyle=$ $\displaystyle\frac{1}{3}\int
Tr\left[\left(2e^{a}\wedge\omega_{a}^{\ c}+\omega_{\
b}^{a}\wedge\omega^{bc}\right)\wedge A\right]$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int\left(\varepsilon_{cdf}e^{a}\wedge\omega_{a}^{\
c}\wedge\omega^{df}+\varepsilon_{acd}e^{d}\wedge\omega_{\
b}^{a}\wedge\omega^{bc}\right).$

To simplify this result, we compute the covariant derivative on the scalar:

$\displaystyle D\left(\varepsilon_{abc}e^{a}\wedge\omega^{bc}\right)$
$\displaystyle=$
$\displaystyle\varepsilon_{abc}\left(De^{a}\wedge\omega^{bc}-e^{a}\wedge
D\omega^{bc}\right)$ (128) $\displaystyle=$
$\displaystyle\varepsilon_{abc}\left(de^{a}\wedge\omega^{bc}-e^{a}\wedge
d\omega^{bc}\right),$

which gives:

$\displaystyle\varepsilon_{abc}\omega_{\ d}^{a}\wedge
e^{d}\wedge\omega^{bc}=2\varepsilon_{abc}e^{a}\wedge\omega_{\
d}^{b}\wedge\omega^{dc}.$ (129)

Thus we have:

$\displaystyle\frac{2}{3}\int Tr\left(A\wedge A\wedge
A\right)=\int\varepsilon_{abc}e^{a}\wedge\omega_{\ d}^{b}\wedge\omega^{dc}.$
(130)

Combining these two equations, we obtain:

$\displaystyle\int Tr\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$
(131) $\displaystyle=$
$\displaystyle\int\varepsilon_{abc}e^{a}\wedge\left(d\omega^{bc}+\omega_{\
d}^{b}\wedge\omega^{dc}\right)$ $\displaystyle=$
$\displaystyle\int\varepsilon_{abc}e^{a}\wedge R^{bc}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{k_{2}}\delta S^{\prime}_{top}.$

Now we have proven that $\delta S^{\prime}_{top}$ can be written as the usual
three-form Chern-Simons action on a three dimensional manifold $M$.
Considering that this three dimensional manifold is the boundary of the four
dimensional manifold $W$, it’s easy to prove that:

$\frac{k_{2}}{2\pi}Tr\int_{W}F\wedge
F=\frac{k_{2}}{2\pi}Tr\int_{M}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge
A\right).$ (132)

Defining:

$\displaystyle I_{W}=\frac{k_{2}}{2\pi}Tr\int_{W}F\wedge F,$ (133)

where $F$ is field strength defined as:

$\displaystyle
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\left[A_{\mu},A_{\nu}\right].$
(134)

Now we can choose another different four dimensional space $W^{\prime}$ whose
boundary is $M$ too. To require these two integral to be equivalent, they must
differ by $2n\pi$:

$\displaystyle I_{W}-I_{W^{\prime}}=I_{X}=2n\pi.$ (135)

We can see that in the above equation, the integral region glue together with
no boundary. Now on a closed four-manifold $X,$ for $SU(2)$, the quantity
$Tr\int_{X}F\wedge F/8\pi^{2}\in\mathbb{Z}$. When $SU(2)$ is replaced by
$SO(3)$, which is 2-fold covered by $SU(2)$, we have $Tr\int_{X}F\wedge
F/8\pi^{2}\in\mathbb{Z}/4$. Thus:

$I_{X}=\frac{k_{2}}{2\pi}Tr\int_{X}F\wedge F=m\pi k_{2}=2n\pi.$ (136)

with $m$ an arbitrary integer. Finally, we can conclude that
$k_{2}=2\mathbb{Z}$.

##  Appendix C Commutation relations among the constraints

When we compute the commutation relations, it is convenient to consider it in
the momentum space. Let us define Fourier transformation as:

$A(p)=\int d^{4}xe^{ipx}A(x),\ \
A(x)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ipx}A(p).$ (137)

For convenience, we denote $\int\frac{d^{4}p}{(2\pi)^{4}}$ as $\int dp$
without causing confuse. The commutation relation in momentum space reads:

$\displaystyle[e_{i}^{a}(p),\pi_{b}^{j}(q)]$ $\displaystyle=$
$\displaystyle\int d^{4}xd^{4}ye^{i(px+qy)}[e_{i}^{a}(x),\pi_{b}^{j}(y)]$
(138) $\displaystyle=$ $\displaystyle
i(2\pi)^{4}\delta_{i}^{j}\delta_{b}^{a}\delta(p+q).$

(1)For $[\mathcal{B}^{ai}(x),\mathcal{C}_{b}(y)]$, we can first calculate the
commutator containing one derivative term in momentum space:

$\displaystyle\varepsilon^{ijk}[\partial_{j}e_{k}^{a}(x),\omega_{b\ l}^{\
f}(y)\pi_{f}^{l}(y)]$ (139) $\displaystyle=$ $\displaystyle(-i)\int
e^{-i(px+qy+q^{\prime}y)}dpdqdq^{\prime}\varepsilon^{ijk}p_{j}[e_{k}^{\
a}(p),\omega_{b\ l}^{\ f}(q)\pi_{f}^{l}(q^{\prime})]$ $\displaystyle=$
$\displaystyle\int dpdqdq^{\prime}e^{-i(px+qy-
py)}\varepsilon^{ijk}p_{j}\omega_{b\ l}^{\
f}(q)(2\pi)^{4}\delta_{k}^{l}\delta_{f}^{a}\delta(p+q^{\prime})$
$\displaystyle=$ $\displaystyle\int dpdqe^{-i(px+qy-
py)}\varepsilon^{ijk}p_{j}\omega_{b\ k}^{\ a}(q)$ $\displaystyle=$
$\displaystyle-\int dpdqe^{-i(px+qy)}\varepsilon^{ijk}p_{j}\omega_{\
bk}^{a}(p+q),$

and

$\displaystyle\varepsilon^{ijk}[\omega_{\
fj}^{a}(x)e_{k}^{f}(x),\partial_{l}\pi_{b}^{l}(y)]$ (140) $\displaystyle=$
$\displaystyle-(-i)\int
dpdp^{\prime}dqe^{-i(px+p^{\prime}x+qy)}\varepsilon^{ijk}q_{l}[\omega_{\
fj}^{a}(p)e_{k}^{f}(p^{\prime}),\pi_{b}^{l}(q)]$ $\displaystyle=$
$\displaystyle\int
dpdp^{\prime}dqe^{-i(px+p^{\prime}x+qy)}\varepsilon^{ijk}q_{l}\omega_{\
fj}^{a}(p)(2\pi)^{4}\delta_{k}^{l}\delta_{b}^{f}\delta(p^{\prime}+q)$
$\displaystyle=$ $\displaystyle-\int dpdqe^{-i(px-
qx+qy)}\varepsilon^{ijk}q_{j}\omega_{\ bk}^{a}(p)$ $\displaystyle=$
$\displaystyle-\int dpdqe^{-i(px+qy)}\varepsilon^{ijk}q_{j}\omega_{\
bk}^{a}(p+q).$

The commutator containing two derivative terms is
$\varepsilon^{ijk}[\partial_{j}e_{k}^{a}(x),\partial_{l}\pi_{b}^{l}(y)]$. It
is vanishing due to $\varepsilon^{ijk}p_{j}p_{k}=0$. Together with these two
terms we have:

$\displaystyle\varepsilon^{ijk}([\partial_{j}e_{k}^{a}(x),\omega_{b\ j}^{\
f}(y)\pi_{f}^{l}(y)]+[\omega_{\
fj}^{a}(x)e_{k}^{f}(x),\partial_{l}\pi_{b}^{l}(y)])$ (141) $\displaystyle=$
$\displaystyle-\int dpdqe^{-i(px+qy)}\varepsilon^{ijk}(p_{j}+q_{j})\omega_{\
bk}^{a}(p+q)$ $\displaystyle=$ $\displaystyle-\varepsilon^{ijk}\int
dqe^{-iq(y-x)}\int dke^{-ikx}k_{j}\omega_{\ bk}^{a}(k)$ $\displaystyle=$
$\displaystyle-i\varepsilon^{ijk}\partial_{j}\omega_{\ bk}^{a}(x)\delta(y-x).$

The commutator without derivative is trivial:

$\varepsilon^{ijk}[\omega_{\ fj}^{a}(x)e_{k}^{f}(x),\omega_{b\ l}^{\
f}(y)\pi_{f}^{l}(y)]=-i\varepsilon^{ijk}\omega_{\ fj}^{a}\omega_{\
bk}^{f}(x)\delta(x-y).$ (142)

Thus we finally have
$\varepsilon^{ijk}[D_{j}e_{k}^{a}(x),D_{l}\pi_{b}^{l}(y)]=-i\frac{1}{2}\varepsilon^{ijk}R_{\
bjk}^{a}(x)\delta(x-y)$, or
$[\mathcal{B}^{ai}(x),\mathcal{C}_{b}(y)]=-i\mathcal{A}_{\ b}^{a\
i}(x)\delta(x-y)$.

(2)For $[\mathcal{C}_{a}(x),\mathcal{C}_{b}(y)],$ the commutator containing
one derivative reads:

$\displaystyle[\partial_{i}\pi_{a}^{i}(x),\frac{1}{2}\varepsilon^{ijk}\varepsilon_{bcdf}R_{ij}^{cd}(y)e_{k}^{f}(y)]+[\frac{1}{2}\varepsilon^{ijk}\varepsilon_{acdf}R_{ij}^{cd}(x)e_{k}^{f}(x),\partial_{i}\pi_{b}^{i(y)}]$
(143) $\displaystyle=$ $\displaystyle\int
dpdqe^{-i(px+qy)}\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abcd}(p_{k}+q_{k})R_{ij}^{cd}(p+q)$
$\displaystyle=$
$\displaystyle\frac{1}{2}i\varepsilon^{ijk}\varepsilon_{abcd}\partial_{i}R_{jk}^{cd}(x)\delta(y-x),$

and the commutator without derivative term yields:

$\displaystyle\frac{1}{2}i\varepsilon^{ijk}\varepsilon_{abcd}\left(\omega_{\
fk}^{c}(x)R_{ij}^{fd}(x)+\omega_{\
fk}^{d}(x)R_{ij}^{cf}(x)\right)\delta(x-y).$ (144)

Thus
$[\mathcal{C}_{a}(x),\mathcal{C}_{b}(y)]=\frac{1}{2}i\varepsilon^{ijk}\varepsilon_{abcd}D_{k}R_{ij}^{cd}(x)\delta(x-y)=0$.
The commutator $[\mathcal{D}_{ab}(x),\mathcal{D}_{cd}(y)]$ is similar to this
one and we don’t show the calculation details again.

(3)For $[\mathcal{A}^{abi}(x),\mathcal{D}_{cd}(y)],$ the commutator containing
one derivative reads:

$\displaystyle\varepsilon^{ijk}[\partial_{j}\omega_{k}^{ab}(x),\omega_{c\
i}^{\ f}(y)\Pi_{fd}^{i}(y)+\omega_{d\ i}^{\
f}(y)\Pi_{cf}^{i}(y)]+\varepsilon^{ijk}[\omega_{\
fj}^{a}(x)\omega_{k}^{fb}(x),\partial_{i}\Pi_{cd}^{i}(y)]$ (145)
$\displaystyle=$ $\displaystyle-\int
dpdqe^{-i(px+qy)}\frac{1}{2}(p_{j}+q_{j})\varepsilon^{ijk}(\omega_{\
ck}^{a}(p+q)\delta_{d}^{b}-\omega_{\
ck}^{b}(p+q)\delta_{d}^{a}-c\leftrightarrow d)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}i\varepsilon^{ijk}(\partial_{j}\omega_{\
ck}^{a}(x)\delta_{d}^{b}-\partial_{j}\omega_{\
ck}^{b}(x)\delta_{d}^{a}+\partial_{j}\omega_{\
dk}^{b}(x)\delta_{c}^{a}-\partial_{j}\omega_{\
dk}^{a}(x)\delta_{c}^{b})\delta(x-y)$ $\displaystyle=$
$\displaystyle-2i\varepsilon^{ijk}\partial_{j}\omega_{\
[dk}^{[b}(x)\delta_{c]}^{a]}\delta(x-y),$

where braket $[]$ means applying the antisymmetrization only with respect to
the internal index, leaving the spacetime index unchanged (always with the
total weight 1, e.g., $\omega_{[ab]}=\frac{1}{2}(\omega_{ab}-\omega_{ba})$).
The commutator without derivative term gives:

$\displaystyle\varepsilon^{ijk}[\omega_{\
fj}^{a}(x)\omega_{k}^{fb}(x),\omega_{c\ i}^{\ f}(y)\Pi_{fd}^{i}(y)+\omega_{d\
i}^{\ f}(y)\Pi_{cf}^{i}(y)]$ (146) $\displaystyle=$ $\displaystyle
i\varepsilon^{ijk}(\omega_{\ fj}^{a}(x)\omega_{c\ k}^{\
g}(x)\delta_{gd}^{fb}+\omega_{\ fj}^{b}(x)\omega_{c\ k}^{\
g}(x)\delta_{gd}^{af}+\omega_{\ fj}^{a}(x)\omega_{d\ k}^{\
g}(x)\delta_{cg}^{fb}+\omega_{\ fj}^{a}(x)\omega_{d\ k}^{\
g}(x)\delta_{cg}^{af})\delta(x-y)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}i\varepsilon^{ijk}(\omega_{\ fj}^{b}(x)\omega_{\
di}^{f}(x)\delta_{c}^{a}-\omega_{\ fj}^{a}(x)\omega_{\
di}^{f}(x)\delta_{c}^{b}+\omega_{\ fj}^{a}(x)\omega_{\
ci}^{f}(x)\delta_{d}^{b}-\omega_{\ fj}^{b}\omega_{\
ci}^{f}(x)\delta_{d}^{a}(x))\delta(x-y)$ $\displaystyle=$
$\displaystyle-2i\varepsilon^{ijk}\omega_{\ fj}^{[b}(x)\omega_{\
[di}^{|f|}(x)\delta_{c]}^{a]}\delta(x-y).$

Together with the above two terms, we have
$[\mathcal{A}^{abi}(x),\mathcal{D}_{cd}(y)]=-i\varepsilon^{ijk}\delta_{[c}^{[a}R_{\
d]jk}^{b]}(x)\delta(x-y)=-2i\delta_{[c}^{[a}\mathcal{A}_{\ d]}^{b]\
i}(x)\delta(x-y)$.

(4) For $[\mathcal{B}^{ai}(x),\mathcal{D}_{cd}(y)],$ the commutator containing
one derivative reads:

$\displaystyle[\varepsilon^{ijk}\omega_{\
fj}^{a}(x)e_{k}^{f}(x),\partial_{i}\Pi_{cd}^{i}(y)]+[\varepsilon^{ijk}\partial_{j}e_{k}^{a}(x),\frac{1}{2}(\pi_{c}^{i}(y)e_{di}(y)-\pi_{d}^{i}(y)e_{ci}(y))]$
(147) $\displaystyle=$ $\displaystyle\int
dpdqe^{-i(px+qy)}\varepsilon^{ijk}[q_{j}e_{fk}\delta_{cd}^{af}+\frac{1}{2}p_{j}(e_{dk}\delta_{c}^{a}-e_{ck}\delta_{d}^{a})]$
$\displaystyle=$ $\displaystyle\int
dpdqe^{-i(px+qy)}\frac{1}{2}\varepsilon^{ijk}(p_{j}+q_{j})(e_{dk}\delta_{c}^{a}-e_{ck}\delta_{d}^{a})$
$\displaystyle=$ $\displaystyle
i\varepsilon^{ijk}\partial_{j}e_{[dk}(x)\delta_{c]}^{a}\delta(x-y).$

and the commutator without derivative term gives:

$\displaystyle[\varepsilon^{ijk}\omega_{j}^{af}(x)e_{fk}(x),\omega_{c\ i}^{\
g}(y)\Pi_{gd}^{i}(y)+\omega_{d\ i}^{\
g}(y)\Pi_{cg}^{i}(y)]+[\varepsilon^{ijk}\omega_{j}^{af}(x)e_{fk}(x),\frac{1}{2}(\pi_{c}^{i}(y)e_{di}(y)-\pi_{d}^{i}(y)e_{ci}(y))]$
(148) $\displaystyle=$ $\displaystyle i\varepsilon^{ijk}[\omega_{c\ j}^{\
g}(x)e_{fk}(x)\delta_{gd}^{af}+\omega_{d\ j}^{\
g}(x)e_{fk}(x)\delta_{cg}^{af}+\frac{1}{2}(\omega_{\
cj}^{a}(x)e_{dk}(x)-\omega_{\ dj}^{a}(x)e_{ck}(x))]\delta(x-y)$
$\displaystyle=$ $\displaystyle\frac{1}{2}i\varepsilon^{ijk}(\omega_{d\ j}^{\
f}(x)e_{fk}(x)\delta_{c}^{a}-\omega_{c\ j}^{\
f}(x)e_{fk}(x)\delta_{d}^{a})\delta(x-y).$

Together with these two terms, we have
$[\mathcal{B}^{ai}(x),\mathcal{D}_{cd}(y)]=i\varepsilon^{ijk}D_{j}e_{[dk}(x)\delta^{a}_{c]}\delta(x-y)=i\delta_{[c}^{a}\mathcal{B}_{d]}^{i}(x)\delta(x-y).$

(5)For $[\mathcal{C}^{a}(x),\mathcal{D}_{cd}(y)],$ the commutator
$[D_{i}\pi_{a}^{i}(x),\mathcal{D}_{cd}(y)]$ is similar to
$[\mathcal{B}^{ai}(x),\mathcal{D}_{cd}(y)]$, so we have
$[D_{i}\pi_{a}^{i}(x),\mathcal{D}_{cd}(y)]=iD_{i}\pi_{[d}^{i}(x)\delta_{c]}^{a}\delta(x-y).$
Next we need to compute the commutator
$[\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abcd}R_{ij}^{bc}(x)e_{k}^{d}(x),D_{i}\Pi_{cd}^{i}(y)]$
(for the sake of indices, we rewrite it as
$[\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)R_{jk}^{mn}(x),D_{i}\Pi_{cd}^{i}(y)]$).
This term is similar to $[\mathcal{A}^{abi}(x),\mathcal{D}_{cd}(y)]$ except
for an extra variable $e_{i}^{b}$. The calculation of commutators without
derivative is trivial. We focus on the commutator containing one derivative,
which gives rise to:

$\displaystyle[\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)\partial_{j}\omega_{k}^{mn}(x),\omega_{c\
i}^{\ g}(y)\Pi_{gd}^{i}(y)+\omega_{d\ i}^{\
g}(y)\Pi_{cg}^{i}(y)]+[\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)\omega_{\
fj}^{m}(x)\omega_{k}^{fn}(x),\partial_{i}\Pi_{cd}^{i}(y)]$ (149)
$\displaystyle=$ $\displaystyle(-i)\int
dpdqe^{-i(px+qy)}\varepsilon^{ijk}\varepsilon_{abmn}[\int
dp_{1}e_{i}^{b}(p-p_{1})p_{1j}\omega_{k}^{mn}(p_{1}),\int dq_{1}(\omega_{c\
i}^{\ g}(q-q_{1})\Pi_{gd}^{i}(q_{1})+\omega_{d\ i}^{\
g}(q-q_{1})\Pi_{cd}^{i}(q_{1})]$ $\displaystyle+(-i)\int
dpdqe^{-i(px+qy)}\varepsilon^{ijk}\varepsilon_{abmn}[\int
dp_{1}dp_{2}e_{i}^{b}(p-p_{1}-p_{2})\omega_{\
fj}^{m}(p_{1})\omega_{k}^{fn}(p_{2}),q_{i}\Pi_{cd}^{i}(q)]$ $\displaystyle=$
$\displaystyle\int dpdqe^{-i(px+qy)}\varepsilon^{ijk}\varepsilon_{abmn}\\{\int
dp_{1}e_{i}^{b}(p-p_{1})p_{1j}[\omega_{c\ k}^{\
g}(q+p_{1})\delta_{gd}^{mn}+\omega_{d\ k}^{\ g}(q+p_{1})\delta_{cg}^{mn}]$
$\displaystyle+\int dp_{2}e_{i}^{b}(p+q-p_{2})q_{j}[\omega_{f\ k}^{\
n}(p_{2})\delta_{cd}^{mf}+\omega_{f\ k}^{\ m}(p_{2})\delta_{cd}^{fn}]\\}$
$\displaystyle=$ $\displaystyle\int
dpdqe^{-i(px+qy)}\varepsilon^{ijk}\varepsilon_{abmn}\int
dp_{1}e_{i}^{b}(p-p_{1})\times(p_{1j}+q_{j})[\omega_{c\ k}^{\
m}(q+p_{1})\delta_{d}^{n}+\omega_{d\ k}^{\ n}(q+p_{1})\delta_{c}^{m}]$
$\displaystyle=$ $\displaystyle
i\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)\left(\partial_{j}\omega_{c\
k}^{\ m}(x)\delta_{d}^{n}+\partial_{j}\omega_{d\ k}^{\
n}(x)\delta_{c}^{m}\right)\delta(x-y),$

Thus we have:

$\displaystyle[\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abcd}e_{i}^{b}(x)R_{jk}^{mn}(x),D_{i}\Pi_{cd}^{i}(y)]=\frac{1}{2}i\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)\left(R_{c\
jk}^{\ m}(x)\delta_{d}^{n}+R_{d\ jk}^{\
n}(x)\delta_{c}^{m}\right)\delta(x-y).$ (150)

Together with:

$\displaystyle[\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abcd}e_{i}^{b}(x)R_{jk}^{cd}(x),\frac{1}{2}(\pi_{c}^{i}(y)e_{di}(y)-\pi_{d}^{i}(y)e_{ci}(y))]=\frac{1}{4}i\varepsilon^{ijk}R_{jk}^{mn}(x)(e_{di}(x)\varepsilon_{acmn}-e_{ci}(x)\varepsilon_{admn})\delta(x-y),$
(151)

we obtain:

$\displaystyle[\frac{1}{2}\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)R_{jk}^{mn}(x),\mathcal{D}_{cd}(y)]$
$\displaystyle=$
$\displaystyle\frac{1}{2}i\varepsilon^{ijk}\varepsilon_{abmn}e_{i}^{b}(x)(R_{c\
jk}^{\ m}(x)\delta_{d}^{n}+R_{d\ jk}^{\ n}(x)\delta_{c}^{m})\delta(x-y)$ (152)
$\displaystyle+\frac{1}{4}i\varepsilon^{ijk}R_{jk}^{mn}(x)(e_{di}(x)\varepsilon_{acmn}-e_{ci}(x)\varepsilon_{admn})\delta(x-y)$
$\displaystyle=$
$\displaystyle\frac{1}{4}i\varepsilon^{ijk}e_{i}^{b}(x)R_{jk}^{mn}(x)(\eta_{ac}\varepsilon_{dbmn}-\eta_{ad}\varepsilon_{cbmn})\delta(x-y).$

The second equation can be proven by checking their components in two cases:
$a=c\neq d$ and $a\neq c\neq d$. We ignore the spacetime index $i,j,k$ here.
Without loss of generality, we consider:

(1) $a=c=0,$ $d=1.$ The nonzero component of the first line is
$\frac{i}{2}(e^{2}R^{\ 3}_{0}-e^{3}R^{\ 2}_{0})-\frac{i}{2}e_{0}R^{23},$ and
the nonzero component of the second line is
$\frac{i}{2}(e^{2}R^{30}+e^{3}R^{02}+e^{0}R^{23})$. They are the same if we
raise up the index 0.

(2) $a=2,$ $c=0,$ $d=1.$ The nonzero component of the first line is
$\frac{i}{2}(-R_{1}^{\ 3}e^{1}-R_{0}^{\ 3}e^{0})+\frac{i}{2}(R_{1}^{\
3}e^{1}+R_{0}^{\ 3}e^{0})=0;$. The second line is zero apparently. Thus we
finally prove:

$\displaystyle[\mathcal{C}^{a}(x),\mathcal{D}_{cd}(y)]$ $\displaystyle=$
$\displaystyle
i\left(D_{i}\pi_{[d}^{i}(x)\delta_{c]}^{a}+\frac{1}{2}i\delta_{[c}^{[a}\varepsilon_{d]bmn}\varepsilon^{ijk}e_{i}^{b}(x)R_{jk}^{mn}(x)\right)\delta(x-y)$
(153) $\displaystyle=$ $\displaystyle
i\left(D_{i}\pi_{[d}^{i}(x)\delta_{c]}^{a}+\frac{1}{2}i\delta_{[c}^{[a}\varepsilon_{d]mnb}\varepsilon^{ijk}R_{ij}^{mn}(x)e_{k}^{b}(x)\right)\delta(x-y)$
$\displaystyle=$ $\displaystyle
i\delta_{[c}^{a}\mathcal{C}_{d]}(x)\delta(x-y).$

##  Appendix D The explicit solution of $\widetilde{B}$

Assuming $\widetilde{B}_{d}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge
K^{cd}+\frac{l_{p}}{2}\varepsilon_{abcd}e^{b}\wedge e^{f}\widetilde{T}_{f}^{\
cd}$. We can prove that this solution satisfy Eq. (93), which is:

$\frac{1}{\kappa}T^{c}\wedge
e^{d}\varepsilon_{abcd}+\frac{1}{2l_{p}}\left(\widetilde{B}_{a}\wedge
e_{b}-\widetilde{B}_{b}\wedge e_{a}\right)=\widetilde{T}_{ab}.$ (154)

First we prove that:

$\frac{1}{2}\left(\varepsilon_{acmn}e^{c}\wedge K^{mn}\wedge
e_{b}-\varepsilon_{bcmn}e^{c}\wedge K^{mn}\wedge e_{a}\right)=-T^{c}\wedge
e^{d}\varepsilon_{abcd},$ (155)

or equivalently:

$\frac{1}{2}\left(\varepsilon_{acmn}e_{b}-\varepsilon_{bcmn}e_{a}\right)\wedge
K^{mn}\wedge e^{c}=K_{\ f}^{c}\wedge e^{f}\wedge e^{d}\varepsilon_{abcd}.$
(156)

this can be verified by considering its components. Without loss of generality
we set $a=0,$ $b=1$. The nonzero component of left hand side is

$e_{1}\wedge\left(K^{12}\wedge e^{3}+K^{31}\wedge
e^{2}\right)+e_{0}\wedge\left(K^{02}\wedge e^{3}+K^{30}\wedge e^{2}\right),$
(157)

where we have used $e_{i}\wedge e_{i}=0$. The nonzero component of right hand
side is

$\displaystyle\left(K_{\ 0}^{2}\wedge e^{0}\wedge e^{3}+K_{\ 1}^{2}\wedge
e^{1}\wedge e^{3}\right)-\left(K_{\ 0}^{3}\wedge e^{0}\wedge e^{2}+K_{\
1}^{3}\wedge e^{1}\wedge e^{2}\right)$ (158) $\displaystyle=$ $\displaystyle
e_{1}\wedge\left(K^{12}\wedge e^{3}+K^{31}\wedge
e^{2}\right)+e_{0}\wedge\left(K^{02}\wedge e^{3}+K^{30}\wedge e^{2}\right),$

so the left hand side is equal to right hand side.

Then we prove:

$\frac{1}{4}\widetilde{T}_{f}^{\ mn}\left(\varepsilon_{acmn}e^{c}\wedge
e^{f}\wedge e_{b}-\varepsilon_{bcmn}e^{c}\wedge e^{f}\wedge
e_{a}\right)=\widetilde{T}_{ab},$ (159)

or equivalently:

$-\frac{1}{4}\widetilde{T}_{f}^{\ mn}e^{f}\wedge
e^{c}\wedge\left(\varepsilon_{acmn}e_{b}-\varepsilon_{bcmn}e_{a}\right)=\frac{1}{6}\widetilde{T}_{\
ab}^{g}\varepsilon_{gcdf}e^{c}\wedge e^{d}\wedge e^{f}.$ (160)

We can also verify it by considering the components. Setting $a=0,$ $b=1$, the
nonzero component left hand side is:

$\displaystyle-\frac{1}{2}\left(\widetilde{T}_{0}^{\ 12}e^{0}\wedge
e^{3}+\widetilde{T}_{0}^{\ 31}e^{0}\wedge e^{2}\right)\wedge
e_{1}-\frac{1}{2}\left(\widetilde{T}_{1}^{\ 02}e^{1}\wedge
e^{3}+\widetilde{T}_{1}^{\ 30}e^{1}\wedge e^{2}\right)\wedge e_{0}$ (161)
$\displaystyle=$ $\displaystyle\widetilde{T}^{012}e_{0}\wedge e_{1}\wedge
e^{3}-\widetilde{T}^{013}e_{0}\wedge e_{1}\wedge e^{2},$

and the nonzero component of right hand side is:

$\displaystyle-\widetilde{T}_{\ 01}^{3}e^{0}\wedge e^{1}\wedge
e^{2}+\widetilde{T}_{\ 01}^{2}e^{0}\wedge e^{1}\wedge e^{3}$ (162)
$\displaystyle=$ $\displaystyle\widetilde{T}^{012}e_{0}\wedge e_{1}\wedge
e^{3}-\widetilde{T}^{013}e_{0}\wedge e_{1}\wedge e^{2},$

Therefore, the left hand side is equal to right hand side. Combining the above
two parts, we finaly prove:

$\widetilde{B}_{d}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge
K^{cd}+\frac{l_{p}}{2}\varepsilon_{abcd}e^{b}\wedge e^{f}\widetilde{T}_{f}^{\
cd}.$ (163)

Without Dirac field, $\widetilde{B}_{d}$ has a very simple solution:

$\widetilde{B}_{d}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge K^{cd}.$
(164)

In order to solve the $\widetilde{B}_{d}$ for the most general TQFT
action(without Dirac field):

$\frac{1}{\kappa}T^{c}\wedge
e^{d}\varepsilon_{abcd}+(\frac{1}{l_{p}}\widetilde{B}+(\psi-\lambda)T)_{[a}\wedge
e_{b]}=0,$ (165)

we can just regard
$\widetilde{B}^{\prime}_{d}=\widetilde{B}_{d}+l_{p}(\psi-\lambda)T_{a}$ as the
new variable. Then it becomes the same equation as the previous case without
Dirac field, and we have:

$\widetilde{B}^{\prime}_{d}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge
K^{cd}\Rightarrow\widetilde{B}_{d}=\frac{l_{p}}{\kappa}\varepsilon_{abcd}e^{b}\wedge
K^{cd}-l_{p}(\psi-\lambda)T_{a}.$ (166)

##  Appendix E EOM and energy-momentum tensor of the Dirac field

To compute the variation with respect to $\overline{\psi}$, which gives the
EOM of the Dirac field, we foucs on the hermitian conjugate term in Eq.(72):

$\displaystyle\frac{1}{3!}\int\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge\left(\frac{1}{2}\overline{\psi}\gamma^{d}\nabla\psi\right)^{\dagger}$
(167) $\displaystyle=$
$\displaystyle\frac{1}{3!}\frac{1}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge
e^{c}\wedge\left(-d\overline{\psi}\gamma^{d}\psi+\omega_{fg}\overline{\psi}\Pi^{fg}\gamma^{d}\psi\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2!}\frac{1}{2}\int\varepsilon_{abcd}\
de^{a}\wedge e^{b}\wedge
e^{c}\overline{\psi}\gamma^{d}\psi+\frac{1}{3!}\frac{1}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge e^{c}\wedge\overline{\psi}\gamma^{d}d\psi$
$\displaystyle+\frac{1}{3!}\frac{i}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge
e^{c}\wedge\omega_{fg}\overline{\psi}\left(\gamma^{d}\Pi^{fg}+[\Pi^{fg},\gamma^{d}]\right)\psi$
$\displaystyle=$
$\displaystyle\frac{1}{3!}\frac{1}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge
e^{c}\wedge\overline{\psi}\gamma^{d}\nabla\psi-\frac{1}{2!}\frac{1}{2}\int\varepsilon_{abcd}\
de^{a}\wedge e^{b}\wedge
e^{c}\overline{\psi}\gamma^{d}\psi-\frac{1}{3!}\frac{1}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge e^{c}\wedge\omega_{\ f}^{d}\overline{\psi}\gamma^{f}\psi$
$\displaystyle=$
$\displaystyle\frac{1}{3!}\frac{1}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge
e^{c}\wedge\overline{\psi}\gamma^{d}\nabla\psi-\frac{1}{2!}\frac{1}{2}\int\varepsilon_{abcd}\
de^{a}\wedge e^{b}\wedge
e^{c}\overline{\psi}\gamma^{d}\psi-\frac{1}{2!}\frac{1}{2}\int\varepsilon_{abcd}\omega_{\
f}^{a}\wedge e^{f}\wedge e^{b}\wedge e^{c}\overline{\psi}\gamma^{d}\psi$
$\displaystyle=$
$\displaystyle\frac{1}{3!}\frac{1}{2}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge
e^{c}\wedge\overline{\psi}\gamma^{d}\nabla\psi-\frac{1}{2!}\frac{1}{2}\int\varepsilon_{abcd}T^{a}\wedge
e^{b}\wedge e^{c}\overline{\psi}\gamma^{d}\psi.$

In the third step we use the following identity:

$\displaystyle\omega_{fg}\left[\Pi^{fg},\gamma^{d}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{8}\omega_{fg}\left(\gamma^{f}\gamma^{g}\gamma^{d}-\gamma^{g}\gamma^{f}\gamma^{d}-\gamma^{d}\gamma^{f}\gamma^{g}+\gamma^{d}\gamma^{g}\gamma^{f}\right)$
(168) $\displaystyle=$
$\displaystyle\frac{1}{8}\omega_{fg}\left(2\eta^{gd}\gamma^{f}-2\eta^{fd}\gamma^{g}+2\eta^{gd}\gamma^{f}-2\eta^{fd}\gamma^{g}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\omega_{fg}\left(\eta^{gd}\gamma^{f}-\eta^{fd}\gamma^{g}\right)$
$\displaystyle=$ $\displaystyle-\omega_{\ f}^{d}\gamma^{f},$

where in the last step we use the identity:

$\varepsilon_{abcd}\omega_{\ f}^{d}=-\varepsilon_{dbcf}\omega_{\
a}^{d}-\varepsilon_{adcf}\omega_{\ b}^{d}-\varepsilon_{abdf}\omega_{\ c}^{d}.$
(169)

Thus the action can be rewritten as:

$S_{Dirac}=\frac{1}{3!}\int\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge\overline{\psi}\gamma^{d}\nabla\psi+\frac{1}{4!}\int\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge e^{c}\wedge
e^{d}m\overline{\psi}\psi-\frac{1}{2!}\frac{1}{2}\int\varepsilon_{abcd}T^{a}\wedge
e^{b}\wedge e^{c}\overline{\psi}\gamma^{d}\psi.$ (170)

Variation with respect to $\overline{\psi}$ gives:

$\frac{1}{3!}\varepsilon_{abcd}e^{a}\wedge e^{b}\wedge
e^{c}\wedge\gamma^{d}\nabla\psi+\frac{1}{4!}\varepsilon_{abcd}e^{a}\wedge
e^{b}\wedge e^{c}\wedge e^{d}m\psi-\frac{1}{4}\varepsilon_{abcd}T^{a}\wedge
e^{b}\wedge e^{c}\gamma^{d}\psi=0.$ (171)

##  Appendix F The antisymmetric part of generalized Einstein equaiton with
Dirac field

The antisymmetric part of Eq. (100) reads:

$\displaystyle\frac{1}{2\pi}\left(\Sigma_{fcb}\Sigma_{\ \
d}^{cb}-\Sigma_{dcb}\Sigma_{\ \ f}^{cb}\right)$
$\displaystyle-\frac{1}{2\pi}\sigma_{c}\left(\Sigma_{\ fd}^{c}-\Sigma_{\
df}^{c}\right)+\frac{1}{4}l_{p}\left(\widetilde{T}_{fab}\Sigma_{d}^{\
ab}-\widetilde{T}_{dab}\Sigma_{f}^{\ ab}\right)$ (172)
$\displaystyle-\frac{1}{\pi}\sigma_{a}\Sigma_{fd}^{\ \
a}+\frac{1}{2}\widetilde{D}_{i}\widetilde{T}_{\
fd}^{i}+\frac{1}{2}t_{[fd]}=0,$

or equivalently

$\frac{1}{4}\left(\sigma_{fcb}\sigma_{\ \ d}^{cb}-\sigma_{dcb}\sigma_{\ \
f}^{cb}\right)+\frac{1}{2}\sigma_{a}\left(\sigma_{fd}^{\ \ a}-\sigma_{df}^{\ \
a}\right)=-\frac{\pi}{2}\widetilde{D}_{i}\widetilde{T}_{\
fd}^{i}-\frac{\pi}{2}t_{[fd]}-\frac{\pi}{4}l_{p}\left(\widetilde{T}_{fab}\Sigma_{d}^{\
ab}-\widetilde{T}_{dab}\Sigma_{f}^{\ ab}\right),$ (173)

where
$\Sigma_{aij}=-\frac{1}{2}\left(\sigma_{aij}+\sigma_{ija}-\sigma_{jai}\right)$
is the contorsion of loop source, $t_{\
d}^{f}=-\frac{1}{2}\overline{\psi}\gamma^{f}e_{d}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi+\left(h.c.\right)$
is the energy momentum tensor of Dirac field, and $\Omega_{\mu}=\omega_{\ \
\mu}^{ab}\Pi_{ab}$. The Dirac current is defined as
$\widetilde{T}^{abc}=-\frac{1}{2}\overline{\psi}\left(\gamma^{a}\Pi^{bc}+\Pi^{bc}\gamma^{a}\right)\psi$.
In the following we will prove that by using the EOM of the Dirac field:

$\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi+m\psi-\frac{1}{2}T_{\
ab}^{a}\gamma^{b}\psi=0,$ (174)

we can simplify the expression $\widetilde{D}_{i}\widetilde{T}_{\
fd}^{i}+t_{[fd]}=-\frac{1}{2}(T_{\ ab}^{f}\widetilde{T}^{abd}-T_{\
ab}^{d}\widetilde{T}^{abf})$. For convenience, we define $T_{\ ab}^{a}\equiv
T_{b}$.

First we try to express $\widetilde{D}_{i}\widetilde{T}^{ifd}$ as:

$\widetilde{D}_{i}\widetilde{T}^{ifd}=-\frac{1}{2}\partial_{i}[\overline{\psi}\left(\gamma^{i}\Pi^{fd}+\Pi^{fd}\gamma^{i}\right)\psi]+\Gamma_{\
bi}^{i}\widetilde{T}^{bfd}+\Gamma_{\ bi}^{f}\widetilde{T}^{ibd}+\Gamma_{\
bi}^{d}\widetilde{T}^{ifb}.$ (175)

Applying the commutation relation between gamma matrices, we obtain:

$\gamma^{i}\Pi^{fd}=\Pi^{fd}\gamma^{i}+\frac{1}{2}(\eta^{fi}\gamma^{d}-\eta^{di}\gamma^{f}),$
(176)

such that Eq. (175) becomes:

$\displaystyle\widetilde{D}_{i}\widetilde{T}^{ifd}$ $\displaystyle=$
$\displaystyle-(\partial_{i}\overline{\psi})\gamma^{i}\Pi^{fd}\psi-\overline{\psi}\Pi^{fd}\gamma^{i}(\partial_{i}\psi)+\Gamma_{\
bi}^{i}\widetilde{T}^{bfd}+\Gamma_{\ bi}^{f}\widetilde{T}^{ibd}+\Gamma_{\
bi}^{d}\widetilde{T}^{ifb}$ (177)
$\displaystyle-\frac{1}{4}(\partial^{d}\overline{\psi}\gamma^{f}\psi-\partial^{f}\overline{\psi}\gamma^{d}\psi+\overline{\psi}\gamma^{d}\partial^{f}\psi-\overline{\psi}\gamma^{f}\partial^{d}\psi).$

Actually the second line of Eq. (177) will cancel out the partial derivative
terms of $t_{[fd]}$. Therefore, we can just focus on the first line. Applying
the EOM of Dirac field, we get:

$(\partial_{i}\overline{\psi})\gamma^{i}\Pi^{fd}\psi=\overline{\psi}(\Omega_{i}\gamma^{i}+m+\frac{1}{2}T_{i}\gamma^{i})\Pi^{fd}\psi,$
(178)

and

$\overline{\psi}\Pi^{fd}\gamma^{i}(\partial_{i}\psi)=\overline{\psi}\Pi^{fd}(-\gamma^{i}\Omega_{i}-m+\frac{1}{2}T_{i}\gamma^{i})\psi.$
(179)

Combining them and using the definition
$\widetilde{T}^{abc}=-\frac{1}{2}\overline{\psi}\left(\gamma^{a}\Pi^{bc}+\Pi^{bc}\gamma^{a}\right)\psi,$
we obtain:

$\displaystyle\widetilde{D}_{i}\widetilde{T}^{ifd}$ $\displaystyle=$
$\displaystyle-\overline{\psi}(\Omega_{i}\gamma^{i}\Pi^{fd}-\Pi^{fd}\gamma^{i}\Omega_{i})\psi+\omega_{\
bi}^{i}\widetilde{T}^{bfd}+\Gamma_{\ bi}^{f}\widetilde{T}^{ibd}+\Gamma_{\
bi}^{d}\widetilde{T}^{ifb}$
$\displaystyle-\frac{1}{4}(\partial^{d}\overline{\psi}\gamma^{f}\psi-\partial^{f}\overline{\psi}\gamma^{d}\psi+\overline{\psi}\gamma^{d}\partial^{f}\psi-\overline{\psi}\gamma^{f}\partial^{d}\psi)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\overline{\psi}(\Omega^{f}\gamma^{d}-\Omega^{d}\gamma^{f})\psi-\overline{\psi}(\omega_{\
ai}^{f}\Pi^{da}\gamma^{i}-\omega_{\
ai}^{d}\Pi^{fa}\gamma^{i}+\Pi^{fd}\gamma^{a}\omega_{ab}^{\ \ b})\psi+\omega_{\
bi}^{i}\widetilde{T}^{bfd}+\Gamma_{\ bi}^{f}\widetilde{T}^{ibd}+\Gamma_{\
bi}^{d}\widetilde{T}^{ifb}$
$\displaystyle-\frac{1}{4}(\partial^{d}\overline{\psi}\gamma^{f}\psi-\partial^{f}\overline{\psi}\gamma^{d}\psi+\overline{\psi}\gamma^{d}\partial^{f}\psi-\overline{\psi}\gamma^{f}\partial^{d}\psi),$

where we have applied the relation $T_{b}+\Gamma_{\ bi}^{i}=\omega_{\ bi}^{i}$
and the identity:

$\displaystyle\Omega_{i}\gamma^{i}\Pi^{fd}$ $\displaystyle=$
$\displaystyle\Pi^{fd}\gamma^{i}\Omega_{i}+\frac{1}{2}\Omega_{i}(\eta^{fi}\gamma^{d}-\eta^{di}\gamma^{f})+\frac{1}{2}\omega_{abi}(\eta^{bd}\Pi^{fa}+\eta^{bf}\Pi^{ad}+\eta^{ad}\Pi^{bf}+\eta^{af}\Pi^{db})\gamma^{i}+\frac{1}{2}\Pi^{fd}(\eta^{bi}\gamma^{a}-\eta^{ai}\gamma^{b})\omega_{abi}$
(181) $\displaystyle=$
$\displaystyle\Pi^{fd}\gamma^{i}\Omega_{i}+\frac{1}{2}(\Omega^{f}\gamma^{d}-\Omega^{d}\gamma^{f})+\omega_{\
ai}^{f}\Pi^{da}\gamma^{i}-\omega_{\
ai}^{d}\Pi^{fa}\gamma^{i}+\Pi^{fd}\gamma^{a}\omega_{ab}^{\ \ b}.$

Next we consider $t_{[fd]}$. The explicit form of $t_{[fd]}$ is:

$\displaystyle t_{[fd]}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\partial_{d}\overline{\psi}\gamma_{f}\psi-\partial_{f}\overline{\psi}\gamma_{d}\psi+\overline{\psi}\gamma_{d}\partial_{f}\psi-\overline{\psi}\gamma_{f}\partial_{d}\psi)-\frac{1}{4}\overline{\psi}(\gamma_{f}\Omega_{d}+\Omega_{d}\gamma_{f}-\gamma_{d}\Omega_{f}-\Omega_{f}\gamma_{d})\psi.$
(182)

Adding them together, we obtain:

$\displaystyle\widetilde{D}_{i}\widetilde{T}^{ifd}+t^{[fd]}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\overline{\psi}(\Omega^{f}\gamma^{d}-\Omega^{d}\gamma^{f}+\gamma^{f}\Omega^{d}-\gamma^{d}\Omega^{f})\psi-\overline{\psi}(\omega_{\
ai}^{f}\Pi^{da}\gamma^{i}-\omega_{\
ai}^{d}\Pi^{fa}\gamma^{i}+\Pi^{fd}\gamma^{a}\omega_{ab}^{\ \ b})\psi$ (183)
$\displaystyle+\omega_{\ bi}^{i}\widetilde{T}^{bfd}+\Gamma_{\
bi}^{f}\widetilde{T}^{ibd}+\Gamma_{\ bi}^{d}\widetilde{T}^{ifb}.$

According to commutation relation in Eq. (176), the first term in Eq. (183)
can be expressed as:

$\displaystyle-\frac{1}{4}\overline{\psi}(\Omega^{f}\gamma^{d}-\Omega^{d}\gamma^{f}+\gamma^{f}\Omega^{d}-\gamma^{d}\Omega^{f})\psi$
$\displaystyle=$
$\displaystyle-\frac{1}{8}\overline{\psi}(\eta^{af}\gamma^{b}-\eta^{bf}\gamma^{a})\omega_{ab}^{\
\
d}\psi+\frac{1}{8}\overline{\psi}(\eta^{ad}\gamma^{b}-\eta^{bd}\gamma^{a})\omega_{ab}^{\
\ f}\psi$ (184) $\displaystyle=$
$\displaystyle\frac{1}{4}\overline{\psi}(\omega_{i}^{\
fd}\gamma^{i}-\omega_{i}^{\ df}\gamma^{i})\psi,$

and the second term in Eq. (183) can be expressed as:

$-\overline{\psi}(\omega_{\ ai}^{f}\Pi^{da}\gamma^{i}-\omega_{\
ai}^{d}\Pi^{fa}\gamma^{i}+\Pi^{fd}\gamma^{a}\omega_{ab}^{\ \ b})\psi=\omega_{\
ab}^{f}\widetilde{T}^{abd}-\omega_{\ ab}^{d}\widetilde{T}^{abf}-\omega_{\
ab}^{b}\widetilde{T}^{afd}-\frac{1}{4}\overline{\psi}(\omega_{i}^{\
fd}\gamma^{i}-\omega_{i}^{\ df}\gamma^{i})\psi.$ (185)

Combining these two terms, Eq. (183) can be simplified as:

$\displaystyle\widetilde{D}_{i}\widetilde{T}^{ifd}+t^{[fd]}$ $\displaystyle=$
$\displaystyle\omega_{\ ab}^{f}\widetilde{T}^{abd}-\omega_{\
ab}^{d}\widetilde{T}^{abf}-\omega_{\ ab}^{b}\widetilde{T}^{afd}+\omega_{\
bi}^{i}\widetilde{T}^{bfd}+\Gamma_{\ bi}^{f}\widetilde{T}^{ibd}+\Gamma_{\
bi}^{d}\widetilde{T}^{ifb}$ (186) $\displaystyle=$ $\displaystyle K_{\
ab}^{f}\widetilde{T}^{abd}-K_{\ ab}^{d}\widetilde{T}^{abf}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(T_{\ ab}^{f}\widetilde{T}^{abd}-T_{\
ab}^{d}\widetilde{T}^{abf}\right).$

By substituting it into Eq. (173), the antisymmetric part of EOM can be
simplified as:

$\displaystyle\frac{1}{2}\left(\sigma_{fcb}\sigma_{\ \
d}^{cb}-\sigma_{dcb}\sigma_{\ \ f}^{cb}\right)+\sigma_{a}\left(\sigma_{fd}^{\
\ a}-\sigma_{df}^{\ \ a}\right)$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}l_{p}\left(\sigma_{fab}\widetilde{T}^{ab}_{\ \
d}-\sigma_{dab}\widetilde{T}^{ab}_{\ \
f}\right)-\frac{\pi}{2}l_{p}\left(\widetilde{T}_{fab}\Sigma_{d}^{\
ab}-\widetilde{T}_{dab}\Sigma_{f}^{\ ab}\right)$ (187) $\displaystyle=$
$\displaystyle\frac{\pi}{2}l_{p}\left(\sigma_{fab}\widetilde{T}^{ab}_{\ \
d}-\sigma_{dab}\widetilde{T}^{ab}_{\ \
f}\right)+\frac{\pi}{4}l_{p}\left(\widetilde{T}_{fab}\sigma_{d}^{\
ab}-\widetilde{T}_{dab}\sigma_{f}^{\ ab}\right)$ $\displaystyle=$
$\displaystyle\frac{\pi}{4}l_{p}\left(\sigma_{fab}\widetilde{T}^{ab}_{\ \
d}-\sigma_{dab}\widetilde{T}^{ab}_{\ \ f}\right).$

## References

  * (1) BP Abbott, et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  * (2) Gerard Jungmana, Marc Kamionkowskibc, and Kim Griestd, Supersymmetric dark matter, Physics Reports 267, 195 (1996).
  * (3) Gianfranco Bertonea, Dan Hooperb, and Joseph Silkb, Particle dark matter: evidence, candidates and constraints, Physics Reports 405, 279 (2005).
  * (4) Nima Arkani-Hamed, Douglas P. Finkbeiner, Tracy R. Slatyer, and Neal Weiner, A theory of dark matter, Physical Review D 79, 015014 (2009).
  * (5) J Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  * (6) E Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  * (7) C Rovelli, and L Smolin, Knot theory and quantum gravity, Physical Review Letters 61, 1155 (1988).
  * (8) C Rovelli, and L Smolin, Loop space representation of quantum general relativity, Nuclear Physics B 331, 80 (1990).
  * (9) C Rovelli, and L Smolin, Spin networks and quantum gravity, Physical Review D 52, 5743 (1995).
  * (10) L Smolin, and A Starodubtsev, General relativity with a topological phase: an action principle, arXiv:0311163
  * (11) L Freidel, and A Starodubtsev, Quantum gravity in terms of topological observables, arXiv:0501191
  * (12) E Witten, 2+ 1 dimensional gravity as an exactly soluble system, Nuclear Physics B 311, 46 (1988).
  * (13) Edward Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359
  * (14) Daniel Freedman, Peter van Nieuwenhuizen, and Sergio Ferrara, Progress toward a theory of supergravity, Phys. Rev. D 13, 3214 (1976).
  * (15) E Cremmer, B Julia, and J Scherk, Supergravity in theory in 11 dimensions, Physics Letters B 76, 409 (1978).
  * (16) X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  * (17) X. Chen, Z.-C. Gu, Z.-X. Liu, X.-G. Wen, Symmetry-Protected Topological Orders in Interacting Bosonic Systems, Science 338, 1604 (2012).
  * (18) A. Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology, arXiv:1403.1467; A. Kapustin, Bosonic Topological Insulators and Paramagnets: a view from cobordisms, arXiv:1404.6659.
  * (19) X.-G. Wen, Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty)$ non-linear $\sigma$-models, Phys. Rev. B 91, 205101 (2015).
  * (20) Zheng-Cheng Gu, The emergence of 3+ 1D Einstein gravity from topological gravity, arXiv:1709.09806
  * (21) Peng Ye and Zheng-Cheng Gu, Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases, Phys. Rev. B 93, 205157 (2016).
  * (22) Juven Wang, Xiao-Gang Wen, and Shing-Tung Yau, Quantum Statistics and Spacetime Surgery, arXiv:1602.05951
  * (23) Pavel Putrov, Juven Wang, and Shing-Tung Yau, Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions, Annals of Physics 384, 254 (2017).
  * (24) Chenjie Wang and Michael Levin, Braiding statistics of loop excitations in three dimensions Phys. Rev. Lett. 113, 080403 (2014).
  * (25) Shenghan Jiang, Andrej Mesaros, and Ying Ran, Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding, Phys. Rev. X 4, 031048 (2014).
  * (26) Liang Kong and Xiao-Gang Wen, Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions, arXiv:1405.5858
  * (27) Tian Lan, Liang Kong, and Xiao-Gang Wen, A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons, arXiv:1704.04221
  * (28) Kapustin A. and Seiberg N. Coupling a QFT to a TQFT and duality, J. High Energ. Phys. (2014) 2014: 1
  * (29) Peng Ye and Zheng-Cheng Gu, Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulatorsPhys. Rev. X 5, 021029 (2015).
  * (30) Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett , Generalized Global Symmetries , J. High Energ. Phys. (2015) 2015: 172