dvips

arXiv:2309.xxxxx

Generalization of Instanton-Induced Inflation
and Dynamical Compactification

Jeongwon Ho<EMAIL_ADDRESS>Kyung Kiu Kim
<EMAIL_ADDRESS>Seoktae Koh<EMAIL_ADDRESS>and
Hyun Seok Yang<EMAIL_ADDRESS>authors contributed equally to
this work.

a Center for Quantum Spacetime, Sogang University, Seoul 121-741, Korea

b College of General Education, Kookmin University, Seoul 02707, Korea

c Department of Science Education, Jeju National University, Jeju, 63243,
Korea

d Institute for Gravitation and the Cosmos, The Pennsylvania State University,
University Park, PA 16802, USA

e Department of Physics and Photon Science, Gwangju Institute of Science and
Technology, Gwangju 61005, Korea

ABSTRACT

It was shown that Yang-Mills instantons on an internal space can trigger the
expansion of our four-dimensional universe as well as the dynamical
compactification of the internal space. We generalize the instanton-induced
inflation and dynamical compactification to general Einstein manifolds with
positive curvature and also to the FLRW metric with spatial curvature. We
explicitly construct Yang-Mills instantons on all Einstein manifolds under
consideration and find that the homogeneous and isotropic universe is allowed
only if the internal space is homogeneous. We then consider the FLRW metric
with spatial curvature as a solution of the eight-dimensional Einstein-Yang-
Mills theory. We find that open universe $(k=-1)$ admits bouncing solutions
unlike the other cases $(k=0,+1)$.

Keywords: Cosmic Inflation, Dynamical Compactification, Yang-Mills Instanton

##  1 Introduction

To understand the origin of the universe, we must combine the general
relativity with quantum theory, dubbed as quantum gravity. String theory is a
leading candidate for the fundamental theory of quantum gravity which requires
a six-dimensional internal space in addition to our four-dimensional spacetime
[1, 2]. However, most quantum gravity theories also require unification of
forces in which gravity plays an equivalent role with gauge theories. If so,
both gravity and gauge theory should play an important role in the early
evolution of the universe such as the cosmic inflation [3, 4, 5, 6] (see also
reviews [7] and [8]). In particular, this implies that nonperturbative objects
in gravity and gauge theories may play an important role for the inflationary
epoch of the early universe. These are gravitational instantons in gravity
such as Einstein manifolds [9] and Yang-Mills instantons in gauge theories
[10]. See also reviews [11], [12] and [13].

Instantons are defined as topologically nontrivial and nonsingular solutions
of classical equations such as Yang-Mills equations or Einstein equations that
minimize the action functional within their topological type [12].
Mathematically, an instanton is a self-dual or anti-self-dual curvature in a
vector bundle over a four-dimensional Riemannian manifold [14]. A
nonperturbative effect will be important in the strong coupling regime.
Because gravity is also a gauge theory like Yang-Mills theory, gravitational
instantons also satisfy the same kind of self-duality equations as Yang-Mills
instantons. Indeed it was shown [15, 16, 17, 18, 19] that Einstein manifolds
can be understood as Yang-Mills instantons for the local Lorentz group $SO(4)$
or $Spin(4)$. An important point is that Yang-Mills instantons formed in extra
dimensions provide a quantized energy density in our four-dimensional
spacetime [20]. Therefore, in order to test the role of instantons in the
inflationary epoch of the early universe, one may consider a four-dimensional
internal space on which Yang-Mills instantons are supported. Following the
motivation in [20], we will study the eight-dimensional Einstein-Yang-Mills
theory in a more general setup by considering the FLRW universe with spatial
curvature and general compact Einstein manifolds. We explicitly construct
Yang-Mills instantons on the internal space that generate the energy-momentum
in our four-dimensional spacetime.

In order for both string theory and inflation to be correct, the internal
space has to be compactified with a microscopic size and only four-dimensional
spacetime remains in a macroscopic scale after the end of cosmic inflation.
This picture suggests that the cosmic inflation of our four-dimensional
universe would be closely related to the dynamical compactification of extra
dimensions. A recent study has found a model realizing such behavior by
showing that the cosmic expansion of our four-dimensional spacetime causes a
dynamical compactification of extra dimensions simultaneously [20]. In this
paper we will clarify why such behavior must be generic for the eight-
dimensional Einstein-Yang-Mills theory.

An accelerating expansion from extra dimensions and a dynamical
compactification of extra dimensions are an old idea explored in many
literatures [21]-[38] (see also chapter 11.4 in [7]) and similar ideas using
monopoles and instantons in extra dimensions have appeared in [39]-[45].
However, their interconnections have not been significantly explored.

This paper is organized as follows. In section 2, we recapitulate the results
in [20]. We explain why Yang-Mills instantons formed in a four-dimensional
internal space generate the cosmic inflation of our four-dimensional spacetime
as well as the dynamical compactification of the internal space by analyzing
the Hamiltonian constraint in the Einstein-Yang-Mills theory. The Hamiltonian
constraint can be mapped to the graph of a quadratic equation describing the
size of the internal space and the graph is described by a concave parabola
with a symmetric axis located at a microscopic radius that characterizes a
fundamental scale. See Fig. 1. By carefully analyzing the behavior of the
graph, we argue based on the validity of classical description that the
internal space must have a minimum size at the position of the symmetric axis.
We also discuss how our instanton-induced inflation and dynamical
compactification can evade no-go theorems in [46, 47]. In section 3, we
generalize the instanton-induced inflation and dynamical compactification to
general Einstein manifolds with positive curvature. We explicitly construct
Yang-Mills instantons on all Einstein manifolds under consideration. We find
that the homogeneous and isotropic universe is possible only when the internal
space is homogeneous, so a nonhomogeneous Einstein manifold must be excluded.
In section 4, we then consider the FLRW metric with spatial curvature as a
solution of the eight-dimensional Einstein-Yang-Mills theory. We find that
open universe $(k=-1)$ admits bouncing solutions unlike the other cases
$(k=0,+1)$. In section 5, we discuss some more generalizations and possible
implications of the instanton-induced inflation, in particular, the dynamical
compactification of extra dimensions and the reheating mechanism in
inflationary cosmology. Appendix A provides useful algebraic relations for the
’t Hooft symbols which are crucially used for the explicit construction of
Yang-Mills instantons on a compact Einstein manifold. In appendix B, we
present the explicit solutions of Yang-Mills instantons on all Einstein
manifolds under consideration and confirm that a nonhomogeneous Einstein
manifold does not satisfy the cosmological principle.

##  2 Instanton-induced inflation and dynamical compactification

Let us recapitulate the basic idea in [20] how to use Yang-Mills instantons
for expanding Universe and dynamical compactification in a higher-dimensional
Einstein-Yang-Mills theory. Consider an eight-dimensional spacetime
$\mathcal{M}_{8}$ whose metric is given by

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
G_{MN}dX^{M}dX^{N}=e^{A}\otimes e^{A}$ (2.1) $\displaystyle=$ $\displaystyle
g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+e^{2f(x)}h_{\alpha\beta}(y)dy^{\alpha}dy^{\beta}=e^{m}\otimes
e^{m}+e^{a}\otimes e^{a},$

where
$X^{M}=(x^{\mu},y^{\alpha}),\;M,N=0,1,\cdots,7;\mu,\nu=0,1,2,3;\alpha,\beta=4,5,6,7$,
are local coordinates on $\mathcal{M}_{8}$ and
$e^{A}=(e^{m},e^{a})\;A,B=0,1,\cdots,7;m,n=0,1,2,3;a,b=4,5,6,7$, are
orthonormal vielbeins in $T^{*}\mathcal{M}_{8}$. The warped product metric
(2.1) is necessary in order to incorporate the back-reaction from Yang-Mills
instantons on a four-dimensional internal space. Consider an $SU(2)$-bundle
$\pi:E\to\mathcal{M}_{8}$ over $\mathcal{M}_{8}$ whose curvature is given by

$\displaystyle F$ $\displaystyle=$ $\displaystyle dA+A\wedge
A=\frac{1}{2}F_{MN}(X)dX^{M}\wedge dX^{N}$ (2.2) $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\partial_{M}A_{N}-\partial_{N}A_{M}+[A_{M},A_{N}]\Big{)}dX^{M}\wedge
dX^{N}$ (2.3)

where
$A=A^{i}_{M}(X)\tau^{i}dX^{M}=\big{(}A_{\mu}(x,y)dx^{\mu},A_{\alpha}(x,y)dy^{\alpha}\big{)}$
is a connection one-form of the $SU(2)$-bundle $E$ and
$\tau^{i}\;\big{(}i=1,2,3\big{)}$ are Lie algebra generators with a
normalization $\mathrm{Tr}\tau^{i}\tau^{j}=-\delta^{ij}$ and obey the
commutation relation

$[\tau^{i},\tau^{j}]=-\varepsilon^{ijk}\tau^{k}.$ (2.4)

In order to investigate the dynamical behavior of the eight-dimensional
spacetime $\mathcal{M}_{8}$ induced by Yang-Mills instantons on the internal
space $X_{4}$, let us consider the eight-dimensional Einstein-Yang-Mills
theory with the action

$S_{EYM}=\frac{1}{16\pi
G_{8}}\int_{\mathcal{M}_{8}}d^{8}X\sqrt{-G}R+\frac{1}{4g_{YM}^{2}}\int_{\mathcal{M}_{8}}d^{8}X\sqrt{-G}G^{MP}G^{NQ}\mathrm{Tr}F_{MN}F_{PQ}.$
(2.5)

The gravitational field equations read

$R_{MN}-\frac{1}{2}G_{MN}R=8\pi G_{8}T_{MN}$ (2.6)

with the energy-momentum tensor given by

$T_{MN}=-\frac{1}{g_{YM}^{2}}\mathrm{Tr}\Big{(}G^{PQ}F_{MP}F_{NQ}-\frac{1}{4}G_{MN}F_{PQ}F^{PQ}\Big{)}.$
(2.7)

The action (2.5) also leads to the equations of motion for Yang-Mills gauge
fields

$G^{MN}D_{M}F_{NP}=0.$ (2.8)

If $X_{4}$ is an Einstein manifold with a positive cosmological constant, that
is, $R^{(0)}_{\alpha\beta}=\lambda h_{\alpha\beta}$ with $\lambda>0$, then
$X_{4}$ is a closed compact manifold with
$vol(X_{4})=\int_{X_{4}}d^{4}y\sqrt{h}$ finite [9]. We are interested in the
gauge field configuration on the Einstein manifold $X_{4}$ described by

$A_{\mu}(x,y)=0,\qquad A_{\alpha}(x,y)=A_{\alpha}(y),$ (2.9)

for which the Yang-Mills action reduces to

$S_{YM}=\frac{1}{4g_{YM}^{2}}\int_{\mathcal{M}_{3,1}}d^{4}x\sqrt{-g}\int_{X_{4}}d^{4}y\sqrt{h}h^{\alpha\gamma}h^{\beta\delta}\mathrm{Tr}F_{\alpha\beta}F_{\gamma\delta}.$
(2.10)

For the gauge field configuration (2.9), the equations of motion (2.8) thus
take the simple form

$h^{\alpha\beta}D_{\alpha}F_{\beta\gamma}=0.$ (2.11)

Suppose that the gauge fields (2.9) are Yang-Mills instantons on $X_{4}$
satisfying the self-duality equation [12, 13]

$F_{\alpha\beta}=\pm\frac{1}{2}\frac{\varepsilon^{\xi\eta\gamma\delta}}{\sqrt{h}}h_{\alpha\xi}h_{\beta\eta}F_{\gamma\delta}.$
(2.12)

Then, Eq. (2.11) is automatically satisfied and the Yang-Mills instanton
solves the equations of motion (2.8) even in a warped spacetime with the
metric (2.1).

An important point is that Yang-Mills instantons on the internal space $X_{4}$
generate a nontrivial energy-momentum tensor given by

$\displaystyle
T_{\mu\nu}=\frac{1}{4g_{YM}^{2}}\widetilde{g}_{\mu\nu}\mathrm{Tr}F_{\alpha\beta}F^{\alpha\beta},$
$\displaystyle
T_{\alpha\beta}=-\frac{e^{-2f(x)}}{g_{YM}^{2}}\mathrm{Tr}\Big{(}h^{\gamma\delta}F_{\alpha\gamma}F_{\beta\delta}-\frac{1}{4}h_{\alpha\beta}F_{\gamma\delta}F^{\gamma\delta}\Big{)}=0,$
(2.13) $\displaystyle T_{\mu\alpha}=0,$

where $\widetilde{g}_{\mu\nu}(x)=e^{-4f(x)}g_{\mu\nu}(x)$ and, for
convenience, all indices are raised and lowered with the product metric (2.1)
with $f(x)=0$. Let us denote the energy-momentum tensor $T_{\mu\nu}$ as the
form

$T_{\mu\nu}=-\frac{1}{g_{YM}^{2}}\widetilde{g}_{\mu\nu}(x)\rho_{n}(y),$ (2.14)

where $\rho_{n}(y)=-\frac{1}{4}\mathrm{Tr}F_{\alpha\beta}F^{\alpha\beta}$ is
the instanton density on $X_{4}$ which is uniform along the four-dimensional
spacetime $\mathcal{M}_{3,1}$. The coupling in Eq. (2.14) implies an
interesting behavior of four-dimensional spacetime ${\cal M}_{3,1}$. The
energy-momentum tensor $T_{\mu\nu}$ depends on the size of internal space
$X_{4}$ characterized by the warp factor $e^{2f(x)}$. For a fixed instanton
density, to be specific, if the internal space becomes small, i.e., $f(x)$
decreasing, then the energy-momentum tensor $T_{\mu\nu}$ becomes large, i.e.,
${\cal M}_{3,1}$ more expanding, or vice versa. This behavior leads to a
dynamical mechanism for the compactification of an internal space through the
inflation of our four-dimensional spacetime. Recently, Ref. [48] generalized
the eight-dimensional Einstein-Yang-Mills theory by including coupling with a
perfect-fluid matter that admits several interesting solutions such as
bouncing universes and oscillatory solutions converging to a de Sitter
spacetime.

Suppose that the solution of an $SU(2)$ Yang-Mills instanton on an Einstein
manifold $X_{4}$ is known. In section 3, we will recapitulate how to
explicitly find a Yang-Mills instanton on $X_{4}$. Then, it is enough to solve
the Einstein equations (2.6) for the warped product geometry (2.1) which take
the form

$\displaystyle
R_{\mu\nu}^{(0)}-\frac{1}{2}g_{\mu\nu}R_{(g)}=4\big{(}\nabla^{(g)}_{\mu}\partial_{\nu}f+\partial_{\mu}f\partial_{\nu}f\big{)}-\big{(}4\nabla^{2}_{(g)}f+10g^{\rho\sigma}\partial_{\rho}f\partial_{\sigma}f-2e^{-2f}\lambda\big{)}g_{\mu\nu}$
$\displaystyle\hskip 93.89418pt-\frac{8\pi
G_{8}}{g_{YM}^{2}}e^{-4f(x)}\rho_{n}(y)g_{\mu\nu},$ (2.15) $\displaystyle
R_{\alpha\beta}^{(0)}-2\lambda
h_{\alpha\beta}=3e^{2f(x)}\Big{(}\frac{1}{6}R_{(g)}-\nabla^{2}_{(g)}f-2g^{\mu\nu}\partial_{\mu}f\partial_{\nu}f\Big{)}h_{\alpha\beta},$
(2.16)

where $R_{\mu\nu}^{(0)}$ and $R_{\alpha\beta}^{(0)}$ are the Ricci tensors
when $f=0$ and $\nabla^{(g)}_{\mu}$ is a covariant derivative with respect to
the metric $g_{\mu\nu}(x)$. A close inspection on the dependence of
$\mathcal{M}_{3,1}$ and $X_{4}$ in Eq. (2.16) shows that a solution exists
only if the internal space must be a space with a constant Ricci scalar, i.e.,
$R_{(h)}=4\lambda$ = constant. With this condition, Eq. (2) further requires
that its consistent solution exists only if the instanton density
$\rho_{n}(y)$ is constant. This means that the instantons are uniformly
smeared out on the internal space $X_{4}$. Note that the internal space
$X_{4}$ in our case is a compact Einstein manifold. It is known [9] that an
Einstein manifold with a positive cosmological constant must be compact and
four examples are known:
$\mathbb{S}^{4},\,\mathbb{C}P^{2},\,\mathbb{S}^{2}\times\mathbb{S}^{2}$ and
Page where the Page [49] is an inhomogeneous Einstein manifold on the
nontrivial product as a twisted $\mathbb{S}^{2}$ bundle over $\mathbb{S}^{2}$.
We will show that the Page space does not admit a constant instanton density,
so it cannot be a solution of the above equations. It is interesting to note
that a homogeneous and isotropic universe cannot be obtained from an
inhomogeneous internal space such as the Page space. This result suggests that
the uniformity of the internal space is closely related to the cosmological
principle of our universe.

To solve the above equations, let us try the specific ansatz [7]

$ds^{2}=-dt^{2}+e^{2h(t)}d\mathbf{x}\cdot
d\mathbf{x}+e^{2f(t)}h_{\alpha\beta}(y)dy^{\alpha}dy^{\beta},$ (2.17)

where we have assumed the flat FLRW metric for our four-dimensional
spacetime.111In our metric ansatz (2.17), $h(t)=\ln a(t)$ where $a(t)$ is a
usual scale factor of the four-dimensional FLRW universe, so
$\dot{h}=\frac{\dot{a}}{a}$ corresponds to the Hubble parameter. Accordingly,
$\ddot{h}=\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}$. The Einstein
equations (2) and (2.16) for the metric (2.17) leads to the following
differential equations [20]

$\displaystyle
E_{1}=2\dot{f}^{2}+4\dot{f}\dot{h}+\dot{h}^{2}+\frac{8}{\zeta_{c}^{2}}e^{-2f}-\frac{1}{3\zeta_{I}^{2}}e^{-4f}=0,$
(2.18) $\displaystyle
E_{2}=2\ddot{h}+3\dot{h}^{2}+4\ddot{f}+8\dot{f}\dot{h}+10\dot{f}^{2}+\frac{24}{\zeta_{c}^{2}}e^{-2f}-\frac{1}{\zeta_{I}^{2}}e^{-4f}=0,$
(2.19) $\displaystyle
E_{3}=\ddot{h}+2\dot{h}^{2}+\ddot{f}+3\dot{f}\dot{h}+2\dot{f}^{2}+\frac{4}{\zeta_{c}^{2}}e^{-2f}=0,$
(2.20)

where $\zeta_{c}^{2}\equiv\frac{12}{\lambda}$ and
$\zeta_{I}^{2}\equiv\frac{g_{YM}^{2}}{8\pi G_{8}\rho_{n}}$. In general
relativity, the metric component $g_{00}$ is not a dynamical variable but a
Lagrange multiplier. Thus $E_{1}=0$ derived from the variation $\delta g_{00}$
is not an evolution equation but a (Hamiltonian) constraint that initial
conditions for the evolution equations, $E_{2}=0$ and $E_{3}=0$, must satisfy.
Indeed, one can show that
$\dot{E}_{1}+\big{(}4\dot{f}(t)+3\dot{h}(t)\big{)}E_{1}=\dot{h}(t)E_{2}+4\dot{f}(t)E_{3}$.
Therefore, if the evolution equations are solved with initial conditions
chosen to be consistent with $E_{1}=0$, the constraint equation will always be
satisfied. Therefore the constraint equation will give us an important
information for the time evolution of an eight-dimensional spacetime
$\mathcal{M}_{8}$.

The solution space is most easily understood by studying the constraint
equation (2.18). Let us define a graph for the constraint
$\mathcal{Y}:=E_{1}=0$ as

$\mathcal{Y}=a\mathcal{X}^{2}+b\mathcal{X}+c=a\Big{(}\mathcal{X}+\frac{b}{2a}\Big{)}^{2}+\Big{(}c-\frac{b^{2}}{4a}\Big{)},$
(2.21)

where
$\mathcal{X}=e^{-2f},\;a=-\frac{1}{3\zeta_{I}^{2}}<0,\;b=\frac{2\lambda}{3}\geq
0$ and $c=\dot{h}^{2}+2\dot{f}^{2}+4\dot{f}\dot{h}$. The graph (2.21) is thus
described by a concave parabola with the symmetric axis
$\mathcal{X}_{s}=-\frac{b}{2a}=\lambda\zeta_{I}^{2}>0$. Then the constraint
equation can be understood as finding $\mathcal{X}$-intercepts for given
velocity data $(\dot{h},\dot{f})$ which determines the $y$-intercept $c$. One
can easily infer the behavior of a solution from Fig. 1. The quadratic
equation $\mathcal{Y}(\mathcal{X})=0$ has two (in general, complex) roots
$\mathcal{X}_{L}$ and $\mathcal{X}_{R}$ given by

$\mathcal{X}_{L}=-\frac{b}{2a}\left(1-\sqrt{1-\frac{4ac}{b^{2}}}\right),\qquad\mathcal{X}_{R}=-\frac{b}{2a}\left(1+\sqrt{1-\frac{4ac}{b^{2}}}\right).$
(2.22)

From our definition $\mathcal{X}=e^{-2f}>0$, these $\mathcal{X}$-intercepts
must be positive. The symmetric axis $\mathcal{X}_{s}>0$ is positive and
solely determined by the ratio $12\frac{\zeta_{I}^{2}}{\zeta_{c}^{2}}$. Hence
it does not depend on velocities $(\dot{h},\dot{f})$. It means that the
parabola in Fig. 1 simply moves up or down along the symmetric axis
$\mathcal{X}=\mathcal{X}_{s}$ depending on velocities $(\dot{h},\dot{f})$.
Note that the $\mathcal{X}$-intercept $\mathcal{X}_{L}$ becomes positive only
when the $y$-intercept $c$ is negative. From the definition of
$c=\dot{h}^{2}+2\dot{f}^{2}+4\dot{f}\dot{h}$, this requires $\dot{f}\dot{h}<0$
for any solution of the $\mathcal{X}_{L}$-branch. Thus there are two
possibilities for the solution:

$\dot{f}\dot{h}<0\quad\Longleftrightarrow\quad\left\\{\begin{array}[]{ll}\dot{f}<0,\quad\dot{h}>0,&\hbox{$\mathcal{X}_{L}$
moving to the right;}\\\ \dot{f}>0,\quad\dot{h}<0,&\hbox{$\mathcal{X}_{R}$
moving to the left.}\end{array}\right.$ (2.23)

The right-moving solution describes an eight-dimensional spacetime where our
four-dimensional spacetime is expanding and the internal space is contracting
while the left-moving solution does the opposite.

$\mathcal{X}=e^{-2f}$$\mathcal{X}_{L}$$\mathcal{X}_{R}$$\mathcal{X}_{s}=\lambda\zeta_{I}^{2}$$\mathcal{Y}$$c$
Figure 1:  The graph of $\mathcal{Y}=a\mathcal{X}^{2}+b\mathcal{X}+c$.

A simple argument shows that $f(t)$ in the $\mathcal{X}_{L}$-branch is a
(strictly) monotonic function of $t$.222The figures in Ref. [20] clearly have
shown this behaviour. Suppose that the function $f(t)$ in the
$\mathcal{X}_{L}$-branch is not monotonic. It means that the function $f(t)$
can pass through the region where $\dot{f}=0$ at some time. Since the
$\mathcal{X}_{L}$-branch does not allow $c>0$, $\dot{h}=0$ there and so $c=0$
(i.e., $\mathcal{X}_{L}=0$). Therefore $\dot{f}=0$ in the
$\mathcal{X}_{L}$-branch also enforces $\dot{h}=0$, in which the eight-
dimensional spacetime becomes $\mathcal{M}_{8}=\mathbb{R}^{3,1}\times X_{4}$
with $vol(X_{4})\to\infty$. The evolution equations, $E_{2}=E_{3}=0$, then
imply that this spacetime must continue thereafter. This behavior is an
exceptional case possible only in a particular contracting solution but cannot
appear in generic cases, especially in expanding solutions. Therefore, the
warp factor $f(t)$ in the $\mathcal{X}_{L}$-branch must be a (strictly)
monotonic function of $t$ for generic solutions. It leads to an interesting
picture [22] that the cosmic expansion of our four-dimensional spacetime
necessarily accompanies a dynamical compactification of extra dimensions.
However, the $\mathcal{X}_{R}$-branch admits a solution satisfying
$\dot{f}\dot{h}>0$, so it does not necessarily require a similar behavior as
the $\mathcal{X}_{L}$-branch. Now we will argue that the
$\mathcal{X}_{R}$-branch corresponds to a quantum branch so that it should be
discarded within a classical approximation.

A notable point is that the size of internal space depends on the warp factor
$e^{2f(t)}=\mathcal{X}^{-1}$ whose value is determined by solving the
constraint equation $\mathcal{Y}=E_{1}=0$. First, note that the graph in Fig.
1 is a concave parabola, so the parabola cannot move down below the
$\mathcal{X}$-axis since the parabola below the $\mathcal{X}$-axis means that
there is no real solution of Eq. (2.21). This implies that the internal space
has a minimum size for an $\mathcal{X}_{L}$-branch solution while a maximum
size for an $\mathcal{X}_{R}$-branch solution, whose size is set by the value
$\mathcal{X}_{s}$ of the symmetric axis of the parabola. In order to estimate
the explicit value of $\mathcal{X}_{s}$, let us use the relation,
$G_{8}=g_{YM}^{2}l_{s}^{2}$, well-known in string theory (see §13.1.3 in [1]
and §13.3 in [2]) where $l_{s}$ characterizes a fundamental scale as the size
of string. Using the explicit result $\rho_{1}=\frac{3}{4R^{4}}$ for the
single instanton on $\mathbb{S}^{4}$, we get333There was a minor error in Eq.
(3.19) in [20]. Considering the difference between $SU(2)$ generators in
gravity and gauge theory, it should read as
$\rho_{1}(y)=\frac{12}{\zeta_{c}^{4}}=\frac{3}{4R^{4}}$. See the footnote 2 in
[19] for the explanation of this difference.

$\mathcal{X}_{s}=\frac{R^{2}}{2\pi l_{s}^{2}},$ (2.24)

where $R=\frac{\zeta_{c}}{2}$ is the radius of the internal space
$\mathbb{S}^{4}$. Since the size of internal space is given by $Re^{f(x)}$, we
see that $(Re^{f})\geq\sqrt{2\pi}l_{s}\approx 2.5l_{s}$ for the
$\mathcal{X}_{L}$-branch whereas $0\leq(Re^{f})\leq\sqrt{2\pi}l_{s}\approx
2.5l_{s}$ for the $\mathcal{X}_{R}$-branch. In particular, the
$\mathcal{X}_{R}$-branch travels at most $2.5l_{s}$ distances. Moreover, when
$\mathcal{X}\to\infty$, i.e. $e^{f}\to 0$, the $\mathcal{X}_{R}$-branch hits
the spacetime (internal) singularity. If the fundamental scale $l_{s}$ is
comparable to the Planck length $l_{P}=10^{-35}$ m, the Planck distance travel
cannot be described by a classical geometry. Hence it is reasonable that the
$\mathcal{X}_{R}$-branch can be ignored within the classical approximation.

We emphasize that the existence of the minimum size of the internal space is
due to the Yang-Mills instanton smeared over the compact space $X_{4}$.
Without the instanton, the graph in Eq. (2.21) is replaced by a linear
equation, i.e., $\mathcal{Y}=b\mathcal{X}+c$ with the same
$c=\dot{h}^{2}+2\dot{f}^{2}+4\dot{f}\dot{h}$ [30]. In this case, there is only
a single branch which can travel the entire positive range, $\mathcal{X}>0$,
but still requires the condition (2.23) to have a real root. Even in this
case, $f(t)$ must be a monotonic function of $t$ since $\dot{f}=0$ enforces
$c>0$ for generic cases, where $\mathcal{X}=e^{-2f}$ becomes negative,
physically not allowed. The monotonicity of the function $f(t)$ implies that
there is no bounce. A similar behavior is also expected for the case with
Yang-Mills instantons. Actually the same reasoning implies that the bounce
$\mathcal{X}_{L}(\dot{f}<0)\rightleftarrows\mathcal{X}_{L}(\dot{f}>0)$ is not
allowed either for any solution of the differential equations (2.18)-(2.20).
However, this situation will change if the four-dimensional metric is FLRW
with spatial curvature, as will be shown in section 4, or considering a
coupling with matters [48].

Some comments are in order. There are no-go theorems stating that there are no
nonsingular de Sitter compactifications for large class of gravity theories
[46, 47]. We will discuss how our instanton-induced inflation and dynamical
compactification can evade such no-go theorems. Maldacena and Nuñez consider a
$D$-dimensional warped metric [46]

$ds^{2}=\Omega^{2}(y)\left(g^{(d)}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+h^{(D-d)}_{\alpha\beta}(y)dy^{\alpha}dy^{\beta}\right).$
(2.25)

Using the $D$-dimensional Einstein equations, they showed that the warping
factor should satisfy the equation

$\frac{d}{(D-2)\Omega^{D-2}}\nabla_{(h)}^{2}\Omega^{D-2}=R(g)-\Omega^{2}\left(T^{\mu}_{\;\;\mu}-\frac{d}{D-2}T^{M}_{\;\;\;M}\right),$
(2.26)

where $R(g)$ is the scalar curvature of the $d$-dimensional metric $g$. A
simple manipulation with Eq. (2.26) shows that the de Sitter spacetime with
$R(g)>0$ is not allowed without a bare positive cosmological constant.

A main reason for our model to evade the Maldacena-Nuñez argument is that the
warp factor in Eq. (2.17) is time-dependent whereas that in (2.25) depends
only on the coordinates of the internal space.444Indeed, the authors in [46]
pointed out (see the footnote $i$) that there can be expanding (or
contracting) universe solutions when some scalar fields are time-dependent.
Let us clarify this aspect for the eight-dimensional Einstein-Yang-Mills
theory (2.5). The Einstein equations (2.6) can be written as the form

$\displaystyle R_{MN}=8\pi
G_{8}\left(T_{MN}-\frac{1}{6}G_{MN}T^{L}_{\;\;L}\right).$ (2.27)

Contracting $(\mu,\nu)$-components leads to

$\displaystyle
R(g)=-6\biggl{(}\dot{h}^{2}+4\dot{f}\dot{h}+\frac{1}{3\zeta_{I}^{2}}\mathcal{X}^{2}\biggr{)}.$
(2.28)

It can also be obtained by contracting Eq. (2) with $g^{\mu\nu}$ and using
Eqs. (2.19) and (2.20). If the metric were time-independent,
$R(g)=-\frac{2}{\zeta_{I}^{2}}\mathcal{X}^{2}<0$, so we could not obtain the
de Sitter space. However, if the metric is time-dependent, we can obtain
$R(g)>0$ as long as the term $\dot{f}\dot{h}$ is negative and large enough.
Therefore, in order to have a de Sitter space including expanding universes,
the time-dependence is crucial so that $\dot{f}\dot{h}$ has a negative value.
Thus the dynamical compactification of an internal space is an inevitable
consequence of an inflating universe in our toy model.

The no-go theorem in [46] has been generalized in [47] to $D$-dimensional
time-dependent metrics

$ds^{2}=g^{(d)}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+e^{2\rho(x)}h^{(D-d)}_{\alpha\beta}(y)dy^{\alpha}dy^{\beta}.$
(2.29)

Starting with the eleven-dimensional supergravity $(D=11)$, the effective
potential in $d$ dimensions has been obtained by the compactification to
$(11-d)$-dimensional curved internal manifolds. After the Weyl scaling

$g^{(d)}_{\mu\nu}(x)=e^{2\omega(x)}q^{(d)}_{\mu\nu}(x)$ (2.30)

with

$\omega(x)=-\frac{D-d}{d-2}\rho(x),$ (2.31)

the $d$-dimensional effective theory is defined in the Einstein frame with the
potential

$V(\chi)=V_{R}e^{-\lambda_{1}\chi}+V_{G}e^{-\lambda_{2}\chi}$ (2.32)

where
$\lambda_{1}=\frac{6}{\sqrt{(d-2)(11-d)}},\;\lambda_{2}=\frac{2(d+1)}{\sqrt{(d-2)(11-d)}}$,
and $\chi=3\sqrt{\frac{(11-d)}{d-2}}\rho$ is scaled to have a canonically
normalized kinetic term of scalar field, $\frac{1}{2}(\nabla\chi)^{2}$. In Eq.
(2.32), $V_{R}=-R(h)$ is the scalar curvature of the internal manifold and
$V_{G}$ (which is always positive) is the average of the four-form $G$-flux in
the eleven-dimensional supergravity. Note that $\lambda_{1}\leq\lambda_{2}$
for $d\geq 2$. Then one can show that the effective potential (2.32) does not
admit a de Sitter vacuum $\chi_{0}$ where $V^{\prime}(\chi_{0})=0$ and
$V(\chi_{0})>0$. In particular, for the region where $V>0$,
$\frac{|\partial_{\chi}V|}{V}$ is a monotonically decreasing function for both
$V_{R}>0$ and $V_{R}<0$ and satisfies the swampland bound [47].

The metric ansatz (2.29) has the same form as ours in (2.1). We should
emphasize that the no-go theorem in [47] is a result derived from the
effective potential (2.32), which is defined in $d$-dimensional Einstein frame
with the metric $q_{\mu\nu}$. However, we did not consider any
compactification (or dimensional reduction) to four-dimensional spacetime.
Instead, we have considered the dynamical evolution of the entire eight-
dimensional spacetime, rather than addressing the dynamics in a lower-
dimensional effective field theory. Furthermore, we describe the evolution in
the Einstein frame of the eight-dimensional spacetime rather than the four-
dimensional spacetime. As we have noticed, the evolution of the four-
dimensional universe and the internal space is deeply interconnected, so in
our case, it is necessary to consider the entire evolution of the eight-
dimensional spacetime. The description based on the four-dimensional effective
field theory is not sufficient to capture the dynamical response of the
internal space. As a result, the no-go theorem derived from the lower-
dimensional effective potential (2.32) cannot be applied to the eight-
dimensional spacetime, where the internal space undergoes dynamical
compactification.

It was shown in [47] that the swampland inequality, $\frac{|\nabla
V|}{V}\geq\lambda_{1}$, related to the no-go theorem discussed above, is a
consequence of the fact that the $D$-dimensional theory satisfies the strong
energy condition. It should also hold true in our case. The strong energy
condition states that for any timelike vector $V^{M}$, $R_{MN}V^{M}V^{N}\geq
0$, or in any local coordinate systems, $R_{00}\geq 0$. Since the
$(\mu,\nu)$-components of the eight-dimensional Ricci tensor are given by

$\displaystyle
R_{\mu\nu}=R_{\mu\nu}^{(0)}-4\left(\nabla_{\mu}^{(g)}\partial_{\nu}f+\partial_{\mu}f\partial_{\nu}f\right),$
(2.33)

a simple calculation with the metric (2.17) shows that

$\displaystyle
R_{00}=\frac{1}{2}\biggl{(}\frac{40}{\zeta_{c}^{2}}\mathcal{X}-\frac{1}{\zeta_{I}^{2}}\mathcal{X}^{2}+10\dot{f}^{2}+20\dot{f}\dot{h}+5\dot{h}^{2}\biggr{)},$
(2.34)

where Eqs. (2.19) and (2.20) are used again. Substituting (2.18) into (2.34)
leads to

$\displaystyle R_{00}=\frac{1}{3\zeta_{I}^{2}}\mathcal{X}^{2}>0.$ (2.35)

Therefore, we find that the strong energy condition is satisfied in our model.
However, it does not imply that a $D$-dimensional gravity theory does not
admit a de Sitter-like solution which describes an expanding or contracting
universe, as we discussed above.

##  3 Generalization to other closed Einstein manifolds

In the previous section, we observed that Yang-Mills instantons on the
internal space can trigger the expansion of our Universe and the dynamical
compactification of extra dimensions. However, in order to realize a
homogeneous and isotropic universe, it is necessary to have an instanton
solution uniformly smeared over a compact Einstein manifold, ensuring a
constant instanton density. In this section, we will construct an instanton
solution on a general Einstein manifold with positive curvature, which
includes
$\mathbb{S}^{4},\,\mathbb{C}P^{2},\,\mathbb{S}^{2}\times\mathbb{S}^{2}$ and
Page $=\mathbb{C}P^{2}\sharp\overline{\mathbb{C}P^{2}}$ [49]. This list
encompasses most, if not all, of the simple types of compact Einstein
manifolds with positive curvature [9]. See [50] for a classification of
compact 4-dimensional Einstein manifolds.

We aim to solve the instanton equation (2.12) on a compact Einstein manifold,
denoted as $X_{4}$. Although the equations of motion (2.8) have been reduced
to the problem solving the instanton equation (2.12) on a compact Einstein
manifold $X_{4}$, it is, in general, difficult to find an instanton solution
on the Einstein manifold. Fortunately there exists a general method [15, 16]
to find an $SU(2)$ Yang-Mills instanton on $X_{4}$ if the metric on $X_{4}$ is
known. Let us briefly recapitulate the method and apply it to find the
instanton solution on $X_{4}$.

Consider an Einstein manifold $X_{4}$ which may be one of
$\mathbb{S}^{4},\,\mathbb{C}P^{2},\,\mathbb{S}^{2}\times\mathbb{S}^{2}$ and
Page. The metric on $X_{4}$ takes the form

$ds_{4}^{2}=h_{\alpha\beta}(y)dy^{\alpha}dy^{\beta}=e^{a}\otimes e^{a}.$ (3.1)

Using the metric, one can calculate the spin connection
$\omega_{ab}=\omega_{ab\alpha}dy^{\alpha}$ and curvature tensor
$R_{ab}=\frac{1}{2}R_{ab\alpha\beta}dy^{\alpha}\wedge dy^{\beta}$ by solving
the structure equations [11, 12]

$\displaystyle T^{a}=de^{a}+\omega_{ab}\wedge e^{b}=0,$ (3.2) $\displaystyle
R_{ab}=d\omega_{ab}+\omega_{ac}\wedge\omega_{cb}.$ (3.3)

The underlying idea is that Einstein gravity can be formulated as a gauge
theory of the Lorentz group, where spin connections play a role of gauge
fields and Riemann curvature tensors correspond to their field strengths.
Another important point is that Riemann curvature tensors, denoted as
$R_{ab}$, are $so(4)$-valued two-forms in
$\Omega^{2}(X_{4})=\Lambda^{2}T^{*}X_{4}$. These facts are combined with the
well-known theorems [14]:

Self-duality. On an orientable Riemannian four-manifold, the 2-forms decompose
into the vector spaces of self-dual and anti-self-dual 2-forms,

$\Omega^{2}=\Omega_{+}^{2}\oplus\Omega_{-}^{2}$ (3.4)

defined by the $\pm 1$ eigenspaces of the Hodge star operator
$*:\Omega^{2}\to\Omega^{2}$.

Lie group isomorphism. There is a global isomorphism for the four-dimensional
Lorentz group, i.e., $SO(4)=SU(2)_{+}\otimes SU(2)_{-}/\mathbb{Z}_{2}$. It
also leads to the splitting of the Lie algebra

$so(4)=su(2)_{+}\oplus su(2)_{-}.$ (3.5)

Indeed, these two decompositions are deeply related to each other due to the
canonical vector space isomorphism between the Clifford algebra
$\mathbb{C}l(4)$ in four dimensions and the exterior algebra
$\Omega^{*}M=\bigoplus_{k=0}^{4}\Lambda^{k}T^{*}M$ over a four-dimensional
Riemannian manifold $M$.

We can apply these two decompositions to the spin connection and curvature
tensor. First let us apply the decomposition (3.5) to spin connections:

$\omega_{ab}=A^{(+)i}\eta^{i}_{ab}+A^{(-)i}\overline{\eta}^{i}_{ab}$ (3.6)

where $\eta^{i}_{ab}$ and $\overline{\eta}^{i}_{ab}$ are the ’t Hooft symbols
satisfying the self-duality relation

$\eta^{i}_{ab}=\frac{1}{2}\varepsilon_{abcd}\eta^{i}_{cd},\qquad\overline{\eta}^{i}_{ab}=-\frac{1}{2}\varepsilon_{abcd}\overline{\eta}^{i}_{cd}.$
(3.7)

See appendix A for the algebraic identities for the ’t Hooft symbols. Thus the
spin connections are split into a pair of $SU(2)_{+}$ and $SU(2)_{-}$ gauge
fields. Note that the index $i=(1,2,3)$ refers to the $su(2)_{\pm}$ Lie
algebra index. Accordingly the Riemann curvature tensors $R_{ab}$ are also
decomposed into a pair of $SU(2)_{+}$ and $SU(2)_{-}$ field strengths:

$R_{ab}=F^{(+)i}\eta^{i}_{ab}+F^{(-)i}\overline{\eta}^{i}_{ab},$ (3.8)

where $F^{(\pm)i}=\frac{1}{2}F^{(\pm)i}_{ab}e^{a}\wedge
e^{b}=dA^{(\pm)i}-\varepsilon^{ijk}A^{(\pm)j}\wedge A^{(\pm)k}$. The second
decomposition (3.4) is that the six-dimensional vector space of two-forms
splits canonically into the sum of three-dimensional vector spaces of self-
dual and anti-self-dual two forms:

$F^{(+)i}_{ab}=f_{(++)}^{ij}\eta^{j}_{ab}+f_{(+-)}^{ij}\overline{\eta}^{j}_{ab},\qquad
F^{(-)i}_{ab}=f_{(-+)}^{ij}\eta^{j}_{ab}+f_{(--)}^{ij}\overline{\eta}^{j}_{ab},$
(3.9)

where each component satisfies the self-duality equation according to Eq.
(3.7). Recently, the decomposition (3.9) has been used significantly in [51]
to study the scalar invariants of the curvature tensor. One can show [16]
that, if $X_{4}$ is an Einstein manifold obeying the equation
$R_{\alpha\beta}=\lambda h_{\alpha\beta}$ with $\lambda$ a cosmological
constant,

$f^{ij}_{(+-)}=0=f^{ij}_{(-+)}.$ (3.10)

That is, $SU(2)_{+}$ and $SU(2)_{-}$ field strengths in the Riemann curvature
tensor (3.9) satisfy the self-duality equations

$F^{(\pm)i}_{ab}=\pm\frac{1}{2}\varepsilon_{abcd}F^{(\pm)i}_{cd}.$ (3.11)

Using the Riemannian metric over $X_{4}$ in Eq. (3.1), the self-duality
equations (3.11) can be expressed in a curved frame as

$F^{(\pm)i}_{\alpha\beta}=\pm\frac{1}{2}\frac{\varepsilon^{\xi\eta\gamma\delta}}{\sqrt{h}}h_{\alpha\xi}h_{\beta\eta}F^{(\pm)i}_{\gamma\delta},$
(3.12)

which is exactly the same as the instanton equation (2.12).

Therefore, one can construct an $SU(2)$ Yang-Mills instanton on a general
Einstein manifold $X_{4}$ that obeys Eq. (3.12) whenever the metric on $X_{4}$
is known. We will illustrate this prescription with an explicit example,
taking $X_{4}$ to be $\mathbb{S}^{4}$. Before delving into details, let us
summarize the results in advance. In order to realize a homogeneous and
isotropic universe, it is necessary to construct Yang-Mills instantons
uniformly smeared out over a compact Einstein manifold $X_{4}$. We find that
such instantons are allowed for all compact Einstein manifolds except the Page
space. Table 1 provides a summary of the geometric properties of these compact
Einstein manifolds, including the Kähler, spin, and homogeneous structures [9,
11]. The emerging conclusion is that the uniformity of internal space is
essential for realizing a homogeneous and isotropic universe.

| $\mathbb{S}^{4}$ | $\mathbb{C}P^{2}$ | $\mathbb{S}^{2}\times\mathbb{S}^{2}$ | Page
---|---|---|---|---
Kähler? | No | Yes | Yes | No
Spin? | Yes | No | Yes | No
Homogeneous? | Yes | Yes | Yes | No
Table 1: Compact Einstein manifolds

The de Sitter metric on $X_{4}=\mathbb{S}^{4}$ with radius
$R=\frac{\zeta_{c}}{2}$ is

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
h_{\alpha\beta}dy^{\alpha}dy^{\beta}=\frac{\zeta_{c}^{4}}{(r^{2}+\zeta_{c}^{2})^{2}}dy^{\alpha}dy^{\beta}$
(3.13) $\displaystyle=$
$\displaystyle\frac{1}{(1+(\frac{r}{2R})^{2})^{2}}\left(dr^{2}+r^{2}\sigma^{2}_{i}\right)=e^{a}\otimes
e^{a}.$

The orthonormal vierbeins $e^{a}\;(a=1,2,3,4)$ in the metric (3.13) are
defined by

$e^{a}=\left(1+\Big{(}\frac{r}{2R}\Big{)}^{2}\right)^{-1}\left\\{r\sigma_{x},r\sigma_{y},r\sigma_{z},dr\right\\},$
(3.14)

where $\sigma^{i}\;(i=1,2,3)$ are left-invariant 1-forms on the group manifold
$SU(2)\cong\mathbb{S}^{3}$ and satisfy the exterior algebra

$d\sigma^{i}+\varepsilon^{ijk}\sigma^{j}\wedge\sigma^{k}=0.$ (3.15)

It is straightforward to calculate the spin connection by solving the torsion-
free condition (3.2). They read

$\omega_{ij}=-\varepsilon_{ijk}\sigma^{k},\qquad\omega_{i4}=\left(1-\frac{rf^{\prime}}{f}\right)\sigma^{i}$
(3.16)

where $f(r)=1+\Big{(}\frac{r}{2R}\Big{)}^{2}$ and $f^{\prime}=\frac{df}{dr}$.
Using the definition (3.6), the corresponding $SU(2)$ gauge fields can easily
be read off from the spin connections and they are given by

$\displaystyle A^{(+)i}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\eta^{i}_{ab}\omega_{ab}\hskip
113.81102ptA^{(-)i}=\frac{1}{4}\overline{\eta}^{i}_{ab}\omega_{ab}$ (3.17)
$\displaystyle=$
$\displaystyle-\frac{rf^{\prime}}{2f}\sigma^{i}=\left(-1+\frac{1}{f}\right)\sigma^{i},\hskip
54.06006pt=-\left(1-\frac{rf^{\prime}}{2f}\right)\sigma^{i}=-\frac{1}{f}\sigma^{i},$

where we have used the explicit representation of the ’t Hooft symbols. (See
appendix A.) The $SU(2)$ field strengths are determined by the gauge fields
(3.17) as

$\displaystyle F^{(+)i}$ $\displaystyle=$
$\displaystyle-\frac{1}{2R^{2}}e^{4}\wedge
e^{i}+\frac{1}{4R^{2}}\varepsilon^{ijk}e^{j}\wedge e^{k},$ (3.18)
$\displaystyle F^{(-)i}$ $\displaystyle=$
$\displaystyle\frac{1}{2R^{2}}e^{4}\wedge
e^{i}+\frac{1}{4R^{2}}\varepsilon^{ijk}e^{j}\wedge e^{k}.$ (3.19)

It is straightforward to check the self-duality (3.11).

It may be instructive to point out that the $SU(2)$ gauge fields in (3.17)
coincide with the instanton solution on $\mathbb{R}^{4}$. (See Remark 2, p.
296, in [12].) Using the coordinates on $\mathbb{R}^{4}$, the left-invariant
1-forms $\sigma^{i}$ on $\mathbb{S}^{3}$ are represented by [52]

$\sigma^{i}=-\frac{1}{r^{2}}\eta^{i}_{\alpha\beta}y^{\alpha}dy^{\beta}$

and then the gauge fields $A^{(+)i}$ in (3.17), for example, are equal to

$A^{(+)i}=\frac{1}{r^{2}+\zeta_{c}^{2}}\eta^{i}_{\alpha\beta}y^{\alpha}dy^{\beta}$
(3.20)

where the diameter $2R$ of $\mathbb{S}^{4}$ is identified with the instanton
size $\zeta_{c}$. These gauge fields precisely correspond to the gauge fields
of a self-dual $SU(2)$ instanton on $\mathbb{R}^{4}$ [10, 13]. It is not
merely a coincidence. The self-duality equation (2.12) in four dimensions is
conformally invariant, extending to the conformal compactification
$\mathbb{S}^{4}=\mathbb{R}^{4}\cup\\{\infty\\}$. Consequently, the solution on
$\mathbb{S}^{4}$ is the same as the finite-action solution on
$\mathbb{R}^{4}$. This is also related to the fact that the single instanton
bundle is the Hopf fibration of $\mathbb{S}^{7}$ [12].

Now let us check whether the instanton density
$\rho_{n}=-\frac{1}{4}\mathrm{Tr}F_{\alpha\beta}F^{\alpha\beta}$ is constant.
The following formula makes the calculation easier;

$F^{(\pm)i}\wedge F^{(\pm)i}=\pm 2\rho_{n}\sqrt{h}d^{4}y$ (3.21)

where $\sqrt{h}d^{4}y=e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4}$ is the volume
form on $X_{4}$. If $\rho_{n}$ is constant, the left-hand side in Eq. (3.21)
must be a constant multiple of the volume form. For the instanton field
strengths in Eqs. (3.18) and (3.19), we get

$F^{(\pm)i}\wedge F^{(\pm)i}=\pm\frac{3}{2R^{4}}\sqrt{h}d^{4}y.$ (3.22)

This result exactly coincides with Eq. (3.19) in [20] (see footnote 3). So we
confirm that the Yang-Mills instanton obtained from the spin connection of
$\mathbb{S}^{4}$ admits a constant density.

The same calculation can also be done for other compact Einstein manifolds;
$X_{4}=\mathbb{C}P^{2},\,\mathbb{S}^{2}\times\mathbb{S}^{2}$ and Page
$=\mathbb{C}P^{2}\sharp\overline{\mathbb{C}P^{2}}$. The Einstein equations (2)
depend only on the instanton density which must be constant for the existence
of a solution. Therefore, it is sufficient to show that the instanton density
on $X_{4}$ is constant, as the equations of motion (2.8) are automatically
solved by Yang-Mills instantons obeying the self-duality (2.12). Further
details are provided in appendix B.

##  4 Generalization to FLRW with spatial curvature

In this section, we will consider general homogeneous and isotropic spacetimes
$\mathcal{M}^{3,1}$ with spatial curvature. For this purpose, we take a
general FLRW metric

$\displaystyle
ds_{4}^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}=-dt^{2}+e^{2h}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(4.1)

where $k$ is normalized as $-1,\,\,0$, or 1 and $d\Omega^{2}$ is the metric of
unit sphere $\mathbb{S}^{2}$. The four-dimensional spacetime metric in Eq.
(2.17) is now replaced by the general FLRW metric (4.1). Then the Einstein
equations (2) and (2.16) lead to

$\displaystyle E_{1}=$
$\displaystyle\,\,\dot{h}^{2}+4\dot{f}\dot{h}+2\dot{f}^{2}+\frac{8}{\zeta_{c}^{2}}e^{-2f}-\frac{1}{3\zeta_{I}^{2}}e^{-4f}+ke^{-2h}=0,$
(4.2) $\displaystyle E_{2}=$
$\displaystyle\,\,2\ddot{h}+4\ddot{f}+3\dot{h}^{2}+8\dot{f}\dot{h}+10\dot{f}^{2}+\frac{24}{\zeta_{c}^{2}}e^{-2f}-\frac{1}{\zeta_{I}^{2}}e^{-4f}+ke^{-2h}=0,$
(4.3) $\displaystyle E_{3}=$
$\displaystyle\,\,\ddot{h}+2\dot{h}^{2}+\ddot{f}+3\dot{f}\dot{h}+2\dot{f}^{2}+\frac{4}{\zeta_{c}^{2}}e^{-2f}+ke^{-2h}=0.$
(4.4)

Note that new terms proportional to $k$ have been appeared. But the constraint
$E_{1}=0$ is still stable under time evolution, i.e.,
$\dot{E}_{1}+\big{(}4\dot{f}(t)+3\dot{h}(t)\big{)}E_{1}=\dot{h}(t)E_{2}+4\dot{f}(t)E_{3}$.

First, instead of solving the equations of motion, we will analyze the
constraint equation (4.2) qualitatively, using the logic similar to that for
the $k=0$ case. It is convenient to rescale the variables by absorbing the
model parameters as

$\displaystyle t\rightarrow\frac{\zeta_{c}^{2}}{4\sqrt{3}\zeta_{I}}t,\qquad
f\rightarrow f+\log\left(\frac{\zeta_{c}}{2\sqrt{3}\zeta_{I}}\right),\qquad
h\rightarrow
h+\frac{1}{2}\log\left(\frac{\zeta_{c}^{4}}{48\zeta_{I}^{2}}\right),$ (4.5)

The constraint equation (4.2) can then be expressed with the rescaled
parameters as

$\displaystyle\mathcal{Y}(\mathcal{X},\mathcal{Z})\equiv
4\dot{f}\dot{h}+2\dot{f}^{2}+\dot{h}^{2}-\mathcal{X}^{2}+2\mathcal{X}+k\mathcal{Z}=0,$
(4.6)

where $\mathcal{X}\equiv e^{-2f}>0$ and $\mathcal{Z}\equiv e^{-2h}>0$. The
contour plot in the first quadrant of $(\mathcal{X},\mathcal{Z})$-plane is
shown in Fig. 2 and each contour line represents $c\equiv
4\dot{f}\dot{h}+2\dot{f}^{2}+\dot{h}^{2}$. If we solve Eq. (4.6) in terms of
$\mathcal{X}$, the roots are given by

$\displaystyle\mathcal{X}_{\pm}=1\mp\sqrt{1+4\dot{f}\dot{h}+2\dot{f}^{2}+\dot{h}^{2}+k\mathcal{Z}}.$
(4.7)

The effect of introducing a new term, $k\mathcal{Z}$, is only to shift the
$\mathcal{Y}$-intercept in Fig. 1 from $c$ to $c_{k}\equiv c+k\mathcal{Z}$, as
shown in Fig. 2, but not to change the position of the symmetry axis
$\mathcal{X}_{s}=1$ (or $\lambda\zeta_{I}^{2}$). Therefore $\mathcal{X}_{-}$
corresponds to a quantum branch for the same reason as the $k=0$ case, so we
discard the $\mathcal{X}_{-}$-branch. Solutions in the
$\mathcal{X}_{+}$-branch are constrained by $0\leq\mathcal{X}_{+}\leq 1$ as
seen from Eq. (4.7).

Figure 2:  Contour plot of the constraint equation (4.6) for $k=-1$ (left) and
$k=1$ (right) in $(\mathcal{X}=e^{-2f},\,\mathcal{Z}=e^{-2h})$-plane. Each
contour represents $c=4\dot{f}\dot{h}+2\dot{f}^{2}+\dot{h}^{2}$.

It may be instructive to analyze the behavior of solutions in the
$\mathcal{X}_{+}$-branch in more detail. As was pointed out above, the
$\mathcal{Y}$-intercept in the graph defined by Eq. (4.6) for a given
$\mathcal{Z}$ is given by $c_{k}=c+k\mathcal{Z}$, which must be negative to
satisfy $\mathcal{X}_{+}>0$. Therefore, for $k=1$, it is necessary to require
$\dot{f}\dot{h}<0$ in order to obey $c_{+}=c+\mathcal{Z}<0$. Hence the
behavior of solutions for $k=1$ is similar to that for the $k=0$ case. The
three-dimensional solution flows for $k=+1$ are depicted in Fig. 3. Because
the behavior of solutions with $k=+1$ is similar to the case with $k=0$, no
bounce appears in the $k=+1$ case for the same reason as the $k=0$ case.
However, the $k=-1$ case is very different from the other cases. Since the
$\mathcal{Y}$-intercept in this case is given by $c_{-}=c-\mathcal{Z}$, the
condition $c_{-}<0$ does not necessarily require the condition
$\dot{f}\dot{h}<0$. So the function $f(t)$ needs not be monotonic. In other
words, a bounce solution is allowed for the $k=-1$ case, where a solution
makes a smooth transition from a region where $\dot{f}>0$ to a region where
$\dot{f}<0$ or vice virsa. Fig. 4 clearly shows this behavior.555This behavior
is somewhat counterintuitive. One might naively expect that a bouncing
solution would appear in the $k=1$ case where the four-dimensional spacetime
$\mathcal{M}_{3,1}$ contains a compact space $\mathbb{S}^{3}$, so back-
reactions from the compact space $\mathbb{S}^{3}$ would cause a bouncing
behavior. But it is not the case. Rather, the bounce solution appears in the
open universe $(k=-1)$. Thus it will be interesting to clarify a detailed
mechanism of the bouncing behavior for the $k=-1$ case.

Figure 3:  Solution flows with $k=1$. The middle and right figures show the
projection onto constants $X=0$ and $X=0.5$, respectively, where $H=\dot{h}$.
The size of arrow head in the left figure denotes the rapidity of time
evolution.

Figure 4:  Solution flows with $k=-1$. The middle and right figures show the
projection onto constants $X=0$ and $X=0.5$, respectively, where $H=\dot{h}$.
These flows contain bouncing solutions.

Figures 3 and 4 have been obtained by solving numerically the flow equations
which rewrite Eqs. (4.2)-(4.4) as the form

$\displaystyle\ddot{f}=-3\dot{f}H-4\dot{f}^{2}+2\mathcal{X}^{2}-3\mathcal{X}\,$
(4.8)
$\displaystyle\dot{H}=4\dot{f}H+4\dot{f}^{2}-H^{2}-3\mathcal{X}^{2}+4\mathcal{X}\,$
(4.9) $\displaystyle\dot{\mathcal{X}}=-2\dot{f}\mathcal{X},$ (4.10)

where $H\equiv\dot{h}$ is the Hubble parameter and $\mathcal{Z}$ is eliminated
using the constraint equation (4.6),

$\displaystyle
k\mathcal{Z}=\mathcal{X}^{2}-2\mathcal{X}-4\dot{f}H-2\dot{f}^{2}-H^{2}.$
(4.11)

This set of equations generates three-dimensional flows depending on the data
$(\dot{f},H,\mathcal{X})$. Given the initial condition
$\left(\dot{f},H,\mathcal{X}\right)\big{|}_{t=t_{i}}$, these flows show the
full-time evolution of metric functions. The numerical analysis is separately
done for $k=+1$ and $k=-1$ cases by solving the constraint (4.11) for each
case.

Finally, we briefly comment on the four-dimensional Ricci scalar $R(g)$ for
the FLRW metric (4.1). The Ricci scalar (2.28) is now replaced by

$\displaystyle\tilde{R}(g)$ $\displaystyle\equiv$
$\displaystyle\frac{\zeta_{c}^{4}}{2^{5}3^{2}\zeta_{I}^{2}}R(g)=-4\dot{f}\dot{h}-k\mathcal{Z}-\dot{h}^{2}-\mathcal{X}^{2}$
(4.12) $\displaystyle=$ $\displaystyle-c_{k}+2\dot{f}^{2}-\mathcal{X}^{2},$

after taking into account the rescaling (4.5). In order to obtain a de Sitter-
like solution, it is necessary to have $\tilde{R}(g)>0$. For $k=1$,
$c_{+}=c+\mathcal{Z}<0$ requires $\dot{f}\dot{h}<0$ and the condition
$c_{+}<0$ is stronger than that for the $k=0$ case, $c<0$. Therefore, in order
for the four-dimensional FLRW spacetime to have a positive Ricci scalar, the
condition $\dot{f}\dot{h}<0$ is more stringent than the flat case. However,
the stringent condition is not necessary for the $k=-1$ case since, even if
$\dot{f}\dot{h}>0$, $\tilde{R}(g)$ could be positive while obeying
$c_{-}=c-\mathcal{Z}<0$. As a result, bouncing solutions are also allowed in
this case as was illustrated by Fig. 4.

##  5 Discussion

Although we have generalized the instanton-induced inflation and dynamical
compactification in Ref. [20] to general Einstein manifolds with positive
curvature and also to the FLRW metric with spatial curvature, we have omitted
(or postponed) an important class of generalization: Turning on Yang-Mills
gauge fields on $\mathcal{M}_{3,1}$ as well as those on $X_{4}$. We can
consider a more general gauge field configuration

$A_{\mu}(x,y)=A_{\mu}(x),\qquad A_{\alpha}(x,y)=A_{\alpha}(y),$ (5.1)

instead of Eq. (2.9). However, a time-dependent as well as static finite-
energy solution does not exist in pure $SU(2)$ Yang-Mills theory on
$\mathbb{R}^{3,1}$ [53, 54]. Recently it was shown in [55, 56] that nontrivial
solutions with finite energy and finite action can be constructed in the pure
$SU(2)$ Yang–Mills theory if the theory is defined on four-dimensional de
Sitter $dS_{4}$ and anti-de Sitter $AdS_{4}$ spaces. Indeed, if we take
$h(t)=\ln\cosh(t)$ with $k=1$ and $h(t)=\ln|\sin(t)|$ with $k=-1$ in Eq.
(4.1), the FLRW metric describes the de Sitter space $dS_{4}$ and the anti-de
Sitter space $AdS_{4}$, respectively. Therefore it will be interesting to
embed the nontrivial solutions in [55, 56] into the eight-dimensional
Einstein-Yang-Mills theory (2.5) with the ansatz (5.1). We hope to address
this issue in the near future.

Our toy model can be embedded into string theory by introducing a two-
dimensional surface $\Sigma$ such that ten-dimensional spacetime becomes
$\mathcal{M}_{8}\times\Sigma$ (see [20] for a brief discussion of such an
embedding). Then it is natural to expect that the inflation of our four-
dimensional spacetime has to simultaneously occur with the dynamical
compactification of extra dimensions because only the four-dimensional
spacetime in $\mathcal{M}_{8}\times\Sigma$ remains in a macroscopic scale
after the end of cosmic inflation. The internal space must be stabilized in
some way with a microscopic size after the inflation. We have observed that
the internal space in $\mathcal{M}_{8}$ has a minimum size. Our model is
controlled by two parameters denoted by
$\zeta_{c}^{2}\equiv\frac{12}{\lambda}$ and
$\zeta_{I}^{2}\equiv\frac{g_{YM}^{2}}{8\pi G_{8}\rho_{n}}$. These are
basically determined by the curvature of an internal Einstein manifold $X_{4}$
and the instanton density on $X_{4}$, respectively. Their ratio
$\mathcal{X}_{s}=\frac{12\zeta_{I}^{2}}{\zeta_{c}^{2}}$ turns out to be a
fundamental scale which is precisely the minimum size of the internal space.
Therefore an interesting question is: When the extra dimensions approach to
this scale, what happens?

For a solution in the $\mathcal{X}_{L}$-branch moving to the right, the
(3+1)-dimensional spacetime is expanding whereas the four-dimensional internal
space is contracting. Since the internal space $X_{4}$ has the minimum size at
$\mathcal{X}=\mathcal{X}_{s}$ which is roughly the size of Planck scale and
the warp factor $f(t)$ must be a (strictly) monotonic function for $k=0,1$,
the inflation should end near the minimum size while the internal space would
be stabilized there. A crude speculation is that there will be a huge quantum
back-reaction from the instanton over the tiny internal space when the extra
dimensions approach to the minimum size. Then the quantum back-reaction would
excite Yang-Mills gauge fields as well as metric fluctuations in the eight-
dimensional Einstein-Yang-Mills theory. It is expected that the inflation
energy will be transferred to ubiquitous fluctuations and rapidly decreased by
producing a large amount of radiations and matters. Ultimately, the cosmic
inflation will end and the extra dimensions would be stabilized around the
critical size $\mathcal{X}_{s}$. If so, the dynamical stabilization of the
internal space may be closely related to the matter generation in our four-
dimensional spacetime. This phenomenon would correspond to a reheating
mechanism in inflationary cosmology [7, 8].

We showed that the open universe with $k=-1$ (negatively curved) allows a
bounce solution. In four-dimensional Einstein gravity coupled with a scalar
field, it is known [57] that the existence of bouncing solution depends on the
spatial curvature. For a flat or open universe, the possibility of a bounce is
precluded by the null energy condition. For a closed universe, on the other
hand, the bounce can take place when the curvature term balances with the
total energy of the universe. But the bounce solution appeared in the open
universe in our case. There is no violation of the null energy condition. This
might be possible due to an energy exchange between the four-dimensional
spacetime and the internal space and the open universe might be more favorable
for the energy exchange. It should be interesting to clarify a detailed
mechanism of the bouncing behavior for the $k=-1$ case.

Another interesting feature in our model is that the homogeneity of internal
space is crucial to realize a homogeneous and isotropic universe. We found
that the Page space [49] does not admit such universe that satisfies the
cosmological principle. In order to incorporate the Page space within our
framework, it would be necessary to consider a generalized Bianchi-type metric
with a warp factor depending on the extra dimensions too, which may be
responsible for an inhomogeneous and anisotropic universe [58]. However, it is
not clear whether such a solution could be useful to explain our universe.

## Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF)
with grant numbers NRF-2019R1A2C1007396 (KK), NRF-2021R1A2C1005748 (SK) and
NRF-2018R1D1A1B0705011314 (HSY). JH was supported by the Ministry of Education
through the Center for Quantum Spacetime (CQUeST) of Sogang University
(NRF-2020R1A6A1A03047877) and NRF-2020R1A2C1014371. We acknowledge the
hospitality at APCTP where part of this work was done.

##  Appendix A ’t Hooft symbols

The explicit components of the ’t Hooft symbols $\eta^{i}_{ab}$ and
${\overline{\eta}}^{i}_{ab}$ for $i=1,2,3$ are given by

$\displaystyle\begin{array}[]{l}{\eta}^{i}_{ab}={\varepsilon}^{i4ab}+\delta^{ia}\delta^{4b}-\delta^{ib}\delta^{4a},\\\
{\overline{\eta}}^{i}_{ab}={\varepsilon}^{i4ab}-\delta^{ia}\delta^{4b}+\delta^{ib}\delta^{4a}\end{array}$
(A.3)

with ${\varepsilon}^{1234}=1$. They satisfy the following relations [51]

$\displaystyle\eta^{(\pm)i}_{ab}=\pm\frac{1}{2}{\varepsilon_{ab}}^{cd}\eta^{(\pm)i}_{cd},$
(A.4)
$\displaystyle\eta^{(\pm)i}_{ab}\eta^{(\pm)i}_{cd}=\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}\pm\varepsilon_{abcd},$
(A.5)
$\displaystyle\varepsilon_{abcd}\eta^{(\pm)i}_{de}=\mp(\delta_{ec}\eta^{(\pm)i}_{ab}+\delta_{ea}\eta^{(\pm)i}_{bc}-\delta_{eb}\eta^{(\pm)i}_{ac}),$
(A.6) $\displaystyle\eta^{(\pm)i}_{ab}\eta^{(\mp)j}_{ab}=0,$ (A.7)
$\displaystyle\eta^{(\pm)i}_{ac}\eta^{(\pm)j}_{bc}=\delta^{ij}\delta_{ab}+\varepsilon^{ijk}\eta^{(\pm)k}_{ab},$
(A.8)
$\displaystyle\eta^{(\pm)i}_{ac}\eta^{(\mp)j}_{bc}=\eta^{(\pm)i}_{bc}\eta^{(\mp)j}_{ac},$
(A.9)
$\displaystyle\varepsilon^{ijk}\eta^{(\pm)j}_{ab}\eta^{(\pm)k}_{cd}=\delta_{ac}\eta^{(\pm)i}_{bd}-\delta_{ad}\eta^{(\pm)i}_{bc}-\delta_{bc}\eta^{(\pm)i}_{ad}+\delta_{bd}\eta^{(\pm)i}_{ac},$
(A.10)

where $\eta^{(+)i}_{ab}\equiv\eta^{i}_{ab}$ and
$\eta^{(-)i}_{ab}\equiv{\overline{\eta}}^{i}_{ab}$.

If we introduce two families of $4\times 4$ matrices defined by

$[\tau^{i}_{+}]_{ab}\equiv\frac{1}{2}\eta^{i}_{ab},\qquad[\tau^{i}_{-}]_{ab}\equiv\frac{1}{2}{\overline{\eta}}^{i}_{ab},$
(A.11)

the matrices in (A.11) provide two independent spin $s=\frac{3}{2}$
representations of $su(2)$ Lie algebra. According to the definition (A.3),
they are explicitly given by

$\displaystyle\tau^{1}_{+}=\frac{1}{2}\begin{pmatrix}0&0&0&1\\\ 0&0&1&0\\\
0&-1&0&0\\\ -1&0&0&0\\\
\end{pmatrix},\;\;\tau^{2}_{+}=\frac{1}{2}\begin{pmatrix}0&0&-1&0\\\
0&0&0&1\\\ 1&0&0&0\\\ 0&-1&0&0\\\
\end{pmatrix},\;\;\tau^{3}_{+}=\frac{1}{2}\begin{pmatrix}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\ \end{pmatrix},$
$\displaystyle\tau^{1}_{-}=\frac{1}{2}\begin{pmatrix}0&0&0&-1\\\ 0&0&1&0\\\
0&-1&0&0\\\ 1&0&0&0\\\
\end{pmatrix},\;\;\tau^{2}_{-}=\frac{1}{2}\begin{pmatrix}0&0&-1&0\\\
0&0&0&-1\\\ 1&0&0&0\\\ 0&1&0&0\\\
\end{pmatrix},\;\;\tau^{3}_{-}=\frac{1}{2}\begin{pmatrix}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\\\ \end{pmatrix}.$

The relations in (A.8) and (A.9) immediately show that $\tau^{i}_{\pm}$
satisfy $su(2)$ Lie algebras, i.e.,

$[\tau^{i}_{\pm},\tau^{j}_{\pm}]=-\varepsilon^{ijk}\tau^{k}_{\pm},\qquad[\tau^{i}_{\pm},\tau^{j}_{\mp}]=0.$
(A.12)

##  Appendix B Einstein manifolds as Yang-Mills instantons

In this appendix, we show in detail how to construct the explicit solution of
Yang-Mills instantons on another Einstein manifolds, namely
$X_{4}=\mathbb{C}P^{2},\,\mathbb{S}^{2}\times\mathbb{S}^{2}$ and Page. We also
investigate whether these manifolds admit a constant instanton density.

###  B.1 $\mathbb{C}P^{2}$

The metric on $\mathbb{C}P^{2}$ takes the form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
h_{\alpha\beta}dy^{\alpha}dy^{\beta}=\frac{1}{f}(r^{2}\sigma_{1}^{2}+r^{2}\sigma_{2}^{2})+\frac{1}{f^{2}}(r^{2}\sigma^{2}_{3}+dr^{2})=e^{a}\otimes
e^{a}$ (B.1)

where

$f(r)=1+\frac{\lambda r^{2}}{6}$

and the orthonormal vierbeins are defined by

$e^{a}=\left\\{\frac{r\sigma_{x}}{\sqrt{f}},\frac{r\sigma_{y}}{\sqrt{f}},\frac{r\sigma_{z}}{f},\frac{dr}{f}\right\\}.$

The spin connections read as

$\displaystyle\omega_{12}=(f^{-1}-2)\sigma^{3}=-\frac{1+\frac{\lambda
r^{2}}{3}}{1+\frac{\lambda
r^{2}}{6}}\sigma^{3},\quad\omega_{13}=\frac{1}{\sqrt{f}}\sigma^{2},\quad\omega_{23}=-\frac{1}{\sqrt{f}}\sigma^{1},$
$\displaystyle\omega_{14}=\frac{1}{\sqrt{f}}\sigma^{1},\quad\omega_{24}=\frac{1}{\sqrt{f}}\sigma^{2},\quad\omega_{34}=\left(1-\frac{rf^{\prime}}{f}\right)\sigma^{3}=\frac{1-\frac{\lambda
r^{2}}{6}}{1+\frac{\lambda r^{2}}{6}}\sigma^{3}.$

The corresponding $SU(2)$ gauge fields can be read off from the spin
connections as

$\displaystyle A^{(+)1}=A^{(+)2}=0,\qquad A^{(+)3}=-\frac{1}{4}\lambda
re^{3},$ $\displaystyle A^{(-)1}=-\frac{1}{r}e^{1},\quad
A^{(-)2}=-\frac{1}{r}e^{2},\quad A^{(-)3}=-\frac{1+f}{2r}e^{3}.$ (B.2)

The field strengths are determined from the above gauge fields as

$\displaystyle F^{(+)1}=F^{(+)2}=0,\quad
F^{(+)3}=dA^{(+)3}=\frac{\lambda}{2}(e^{1}\wedge e^{2}+e^{3}\wedge e^{4}),$
(B.3) $\displaystyle
F^{(-)i}=\frac{\lambda}{6}\left(\frac{1}{2}\varepsilon^{ijk}e^{j}\wedge
e^{k}-e^{i}\wedge e^{4}\right).$ (B.4)

In particular, the self-dual sector is an Abelian gauge field. The self-
duality (3.11) is manifest.

As in section 3, in order to check whether the instanton density,
$\rho_{n}=-\frac{1}{4}\mathrm{Tr}F_{\alpha\beta}F^{\alpha\beta}$, is constant,
let us calculate the quantity

$F^{(\pm)i}\wedge F^{(\pm)i}=\pm 2\rho_{n}\sqrt{h}d^{4}y$ (B.5)

where $\sqrt{h}d^{4}y=e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4}$ is the volume
form on $\mathbb{C}P^{2}$. For the instanton solution in Eqs. (B.3) and (B.4),
we get

$F^{(+)i}\wedge F^{(+)i}=\frac{\lambda^{2}}{2}\sqrt{h}d^{4}y,\qquad
F^{(-)i}\wedge F^{(-)i}=-\frac{\lambda^{2}}{6}\sqrt{h}d^{4}y.$ (B.6)

So we confirm that the instanton density on $\mathbb{C}P^{2}$ is also
constant.

###  B.2 $\mathbb{S}^{2}\times\mathbb{S}^{2}$

Let us take the metric on $X_{4}=\mathbb{S}^{2}\times\mathbb{S}^{2}$ as the
form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
h_{\alpha\beta}dy^{\alpha}dy^{\beta}=\sum_{\flat=1}^{2}R_{\flat}^{2}(d\theta_{\flat}^{2}+\sin^{2}\theta_{\flat}d\phi_{\flat}^{2})=e^{a}\otimes
e^{a}$ (B.7)

where the orthonormal vierbeins are given by

$e^{a}=\left\\{R_{1}d\theta_{1},R_{1}\sin\theta_{1}d\phi_{1},R_{2}d\theta_{2},R_{2}\sin\theta_{2}d\phi_{2}\right\\}.$

The nonzero spin connections are

$\omega_{12}=-\frac{\cot\theta_{1}}{R_{1}}e^{2},\qquad\omega_{34}=-\frac{\cot\theta_{2}}{R_{2}}e^{4}.$

The corresponding gauge fields consist of two $U(1)$ gauge fields

$\displaystyle
A^{(+)3}=-\frac{\cot\theta_{1}}{2R_{1}}e^{2}-\frac{\cot\theta_{2}}{R_{2}}e^{4},$
$\displaystyle
A^{(-)3}=-\frac{\cot\theta_{1}}{2R_{1}}e^{2}+\frac{\cot\theta_{2}}{R_{2}}e^{4}.$
(B.8)

The field strengths are determined by the above gauge fields as

$\displaystyle F^{(+)3}=dA^{(+)3}=\frac{1}{2}\left(\frac{e^{1}\wedge
e^{2}}{R_{1}^{2}}+\frac{e^{3}\wedge e^{4}}{R_{2}^{2}}\right),$ (B.9)
$\displaystyle F^{(-)3}=dA^{(-)3}=\frac{1}{2}\left(\frac{e^{1}\wedge
e^{2}}{R_{1}^{2}}-\frac{e^{3}\wedge e^{4}}{R_{2}^{2}}\right).$ (B.10)

The self-duality (3.11) requires that $R_{1}=R_{2}$. For the above field
strengths, we obtain

$F^{(\pm)3}\wedge F^{(\pm)3}=\pm\frac{1}{2R_{1}^{2}R_{2}^{2}}\sqrt{h}d^{4}y.$
(B.11)

Thus we get a constant instanton density even if $R_{1}\neq R_{2}$. It is
possible because $X_{4}=\mathbb{S}^{2}\times\mathbb{S}^{2}$ is a product
space.

###  B.3 Page

The final example is the Page space
$X_{4}=\mathbb{C}P^{2}\sharp\overline{\mathbb{C}P^{2}}$ which is an
$\mathbb{S}^{2}$ bundle over $\mathbb{S}^{2}$ [49]. It is constructed by
attaching the complex projective space and its complex conjugate, removing a
4-ball from each manifold, and gluing them together along the resulting
$\mathbb{S}^{3}$ boundaries. The metric on the Page space takes on a
complicated form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{\lambda}\frac{1+\nu^{2}}{3+\nu^{2}}\left(U^{-1}d\chi^{2}+\frac{U}{4}\sin^{2}\chi(d\psi+\cos\theta
d\phi)^{2}+\frac{1-\nu^{2}\cos^{2}\chi}{4\nu}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right),$ (B.12) $\displaystyle=$ $\displaystyle e^{a}\otimes e^{a}$

where

$U(\chi)=1-\frac{2\nu^{2}\sin^{2}\chi}{(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)}$

and $\nu$ is the positive root of $\nu^{4}+4\nu^{3}-6\nu^{2}+12\nu-3=0$ which
works out to be $0.2817$. The orthonormal vierbeins are defined by

$\sqrt{\frac{\lambda}{3}\frac{3+\nu^{2}}{1+\nu^{2}}}e^{a}=\left\\{\sqrt{\frac{1-\nu^{2}\cos^{2}\chi}{4\nu}}d\theta,\sqrt{\frac{1-\nu^{2}\cos^{2}\chi}{4\nu}}\sin\theta
d\phi,\frac{\sqrt{U}}{2}\sin\chi(d\psi+\cos\theta
d\phi),\frac{d\chi}{\sqrt{U}}\right\\}.$

Here we choose $\lambda=3\frac{1+\nu^{2}}{3+\nu^{2}}$ for simplicity. After a
little algebra, we get the spin connections

$\displaystyle\omega_{12}=-\sqrt{\frac{4\nu}{1-\nu^{2}\cos^{2}\chi}}\cot\theta
e^{2}+fe^{3},\quad\omega_{13}=fe^{2},\quad\omega_{23}=-fe^{1},$
$\displaystyle\omega_{14}=ge^{1},\quad\omega_{24}=ge^{2},\quad\omega_{34}=\left(\sqrt{U}\cot\chi+\frac{2(\nu^{2}-1)}{U(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)}g\right)e^{3},$
(B.13)

where

$f=\frac{\sqrt{U}\nu}{1-\nu^{2}\cos^{2}\chi}\sin\chi,\qquad g=f\nu\cos\chi.$

The corresponding $SU(2)$ gauge fields are given by

$\displaystyle A^{(\pm)1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(-f\pm
g)e^{1},\qquad A^{(\pm)2}=\frac{1}{2}(-f\pm g)e^{2},$ $\displaystyle
A^{(\pm)3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\sqrt{\frac{4\nu}{1-\nu^{2}\cos^{2}\chi}}\cot\theta
e^{2}+fe^{3}\pm\Big{(}\sqrt{U}\cot\chi+\frac{2(\nu^{2}-1)}{U(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)}g\Big{)}e^{3}\right).$

It is straightforward (but a bit tedious) to calculate the $SU(2)$ field
strengths:

$\displaystyle F^{(\pm)1}$ $\displaystyle=$
$\displaystyle\frac{U}{2}\frac{\nu(1-\nu^{2})\cos\chi\mp\nu^{2}(\cos^{2}\chi-\sin^{2}\chi)\pm\nu^{4}\cos^{4}\chi}{(1-\nu^{2}\cos^{2}\chi)^{2}}(e^{1}\wedge
e^{4}\pm e^{2}\wedge e^{3}),$
$\displaystyle-\frac{\nu^{3}(1-\nu^{2})\cos\chi\sin^{2}\chi}{(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)^{3}}(1\mp\nu\cos\chi)(e^{1}\wedge
e^{4}\pm e^{2}\wedge e^{3}),$ $\displaystyle F^{(\pm)2}$ $\displaystyle=$
$\displaystyle\frac{U}{2}\frac{\nu(1-\nu^{2})\cos\chi\mp\nu^{2}(\cos^{2}\chi-\sin^{2}\chi)\pm\nu^{4}\cos^{4}\chi}{(1-\nu^{2}\cos^{2}\chi)^{2}}(e^{2}\wedge
e^{4}\pm e^{3}\wedge e^{1})$ (B.14)
$\displaystyle-\frac{\nu^{3}(1-\nu^{2})\cos\chi\sin^{2}\chi}{(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)^{3}}(1\mp\nu\cos\chi)(e^{2}\wedge
e^{4}\pm e^{3}\wedge e^{1}),$ $\displaystyle F^{(\pm)3}$ $\displaystyle=$
$\displaystyle-\frac{\nu}{2(1-\nu^{2}\cos^{2}\chi)}\left(U^{\prime}\sin\chi+\frac{2(1-\nu^{2})U\cos\chi}{1-\nu^{2}\cos^{2}\chi}\right)(e^{3}\wedge
e^{4}\pm e^{1}\wedge e^{2})$
$\displaystyle\pm\frac{\nu}{2(1-\nu^{2}\cos^{2}\chi)}\left(4-\frac{\nu
U\sin^{2}\chi(3+\nu^{2}\cos^{2}\chi)}{1-\nu^{2}\cos^{2}\chi}\right)(e^{3}\wedge
e^{4}\pm e^{1}\wedge e^{2}).$

One can show that the Ricci tensor is given by (after recovering the factor
$\frac{3}{\lambda}\frac{1+\nu^{2}}{3+\nu^{2}}$)

$\displaystyle R^{(0)}_{ab}=\lambda\,\text{diag}\left(c,c,1,1\right),$ (B.15)

where
$c=\frac{\nu^{4}-8\nu^{3}+9\nu^{2}-24\nu+3\left(\nu^{2}+1\right)\nu^{2}\cos
2\chi}{3\nu^{4}-3\nu^{2}-6+3\left(\nu^{2}+1\right)\nu^{2}\cos 2\chi}$. Using
the quartic equation $\nu^{4}+4\nu^{3}-6\nu^{2}+12\nu-3=0$, one can show that
$c=1$. Therefore the Page space is an Einstein manifold that satisfies the
Einstein equation $R^{(0)}_{\alpha\beta}=\lambda h_{\alpha\beta}$. Instead of
showing the full expressions, we present the plots of the instanton densities
in Fig. 5. The full expression is highly complex and lacks insight. The figure
clearly illustrates that Yang-Mills instantons on the Page space exhibit an
inhomogeneous density.

Figure 5:  The instanton densities $\rho^{(\pm)}$ for self-dual and anti-self-
dual instantons on $X_{4}=$ Page.

###  B.4 Squashed $\mathbb{S}^{4}$

In this example, we aim to understand the importance of the homogeneity of an
internal space $X_{4}$ in achieving a constant instanton density on $X_{4}$.
We examine another nonhomogeneous space obtained by deforming a round sphere
$\mathbb{S}^{4}$ whose deformation is parameterized by three axis-length
parameters $(r,l,\tilde{l})$. The metric of squashed 4-sphere is given by
$ds^{2}=e^{a}\otimes e^{a}$ with the vierbeins [59]

$\left\\{e^{1}=\sin\rho\tilde{\sigma}^{1},\;e^{2}=\sin\rho\tilde{\sigma}^{2},\;e^{3}=\sin\rho\tilde{\sigma}^{3}+hd\rho,\;e^{4}=gd\rho\right\\},$

where

$\displaystyle f$ $\displaystyle\equiv$
$\displaystyle\sqrt{l^{2}\sin^{2}\theta+\tilde{l}^{2}\cos^{2}\theta},$
$\displaystyle g$ $\displaystyle\equiv$
$\displaystyle\sqrt{r^{2}\sin^{2}\rho+l^{2}\tilde{l}^{2}f^{-2}\cos^{2}\rho},$
$\displaystyle h$ $\displaystyle\equiv$
$\displaystyle\frac{\tilde{l}^{2}-l^{2}}{f}\cos\rho\sin\theta\cos\theta,$

and $\tilde{\sigma}^{i}\,(i=1,2,3)$ are dreibeins of the three-dimensional
ellipsoid given in polar coordinates $(\varphi,\chi,\theta)$ by

$\tilde{\sigma}^{1}=l\cos\theta
d\varphi,\;\tilde{\sigma}^{2}=\tilde{l}\sin\theta
d\chi,\;\tilde{\sigma}^{3}=fd\theta.$

It is straightforward to calculate the $SU(2)$ gauge fields

$\displaystyle A^{(\pm)1}$ $\displaystyle=$
$\displaystyle\pm\frac{\tilde{l}^{2}}{2gf^{2}}\cot\rho
e^{1}+\frac{\cot\theta}{2f\sin\rho}e^{2},$ $\displaystyle A^{(\pm)2}$
$\displaystyle=$ $\displaystyle\pm\frac{l^{2}}{2gf^{2}}\cot\rho
e^{2}+\frac{\tan\theta}{2f\sin\rho}e^{1},$ (B.16) $\displaystyle A^{(\pm)3}$
$\displaystyle=$
$\displaystyle\pm\frac{l^{2}\tilde{l}^{2}}{2gf^{4}}\cot\rho\left(e^{3}-\frac{h}{g}e^{4}\right),$

and their field strengths

$\displaystyle F^{(\pm)1}$ $\displaystyle=$
$\displaystyle\pm\frac{\tilde{l}^{2}}{2\sin\rho}d\psi\wedge
e^{1}\pm\frac{\tilde{l}^{2}(\tilde{l}^{2}-l^{2})\psi\sin
2\theta}{4f^{3}\sin^{2}\rho}e^{1}\wedge(e^{3}-g^{-1}he^{4})$
$\displaystyle+\frac{l^{2}r^{2}}{2f^{4}g^{2}}e^{2}\wedge(e^{3}-g^{-1}he^{4}),$
$\displaystyle F^{(\pm)2}$ $\displaystyle=$
$\displaystyle\pm\frac{l^{2}}{2\sin\rho}d\psi\wedge
e^{2}\pm\frac{l^{2}(\tilde{l}^{2}-l^{2})\psi\sin
2\theta}{4f^{3}\sin^{2}\rho}e^{2}\wedge(e^{3}-g^{-1}he^{4})$
$\displaystyle-\frac{\tilde{l}^{2}r^{2}}{2f^{4}g^{2}}e^{1}\wedge(e^{3}-g^{-1}he^{4}),$
$\displaystyle F^{(\pm)3}$ $\displaystyle=$
$\displaystyle\pm\frac{l^{2}\tilde{l}^{2}}{2f^{2}\sin\rho}d\psi\wedge(e^{3}-g^{-1}he^{4})+\frac{r^{2}}{2f^{2}g^{2}}e^{1}\wedge
e^{2},$

where $\psi\equiv\frac{\cos\rho}{gf^{2}}$ and
$d\psi=-\frac{r^{2}\sin\rho}{g^{4}f^{2}}e^{4}+\frac{h}{g^{3}f^{6}\sin\rho}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)(e^{3}-g^{-1}he^{4})$.

The field strengths in Eq. (B.4) are neither self-dual nor anti-self-dual, so
the squashed four-sphere is not an Einstein manifold. According to the result
(3.10), $SU(2)$ gauge fields constructed from a four-manifold $X_{4}$ become
self-dual or anti-self-dual only if $X_{4}$ is Einstein, i.e., obeying the
vacuum Einstein equation, $R_{\alpha\beta}=\lambda h_{\alpha\beta}$ [16]. As a
result, $SU(2)$ field strengths in Eq. (B.4) do not satisfy the self-duality
(3.11). Therefore the squashed four-sphere would be relevant as an internal
manifold only when matters are coupled with gravity. It is straightforward to
see how the instanton density behaves for the squashed four-sphere. The
instanton density for the $SU(2)$ field strengths in (B.4) exhibits a
nontrivial distribution as expected:

$\displaystyle F^{(\pm)i}\wedge
F^{(\pm)i}=\pm\frac{3}{2}\frac{l^{2}\tilde{l}^{2}r^{4}}{f^{6}g^{6}}\sqrt{h}d^{4}y,$
(B.18)

where $e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4}=\sqrt{h}d^{4}y$ is the volume
form on the squashed four-sphere. When $l=\tilde{l}=r$, it becomes the round
four-sphere $\mathbb{S}^{4}$ which is an Einstein manifold. In this case, the
$SU(2)$ gauge fields in (B.4) become instanton connections and the instanton
density (B.18) becomes constant, i.e., $F^{(\pm)i}\wedge
F^{(\pm)i}=\pm\frac{3}{2r^{4}}\sqrt{h}d^{4}y$ which is consistent with (3.22)
in the unit $R=r$. But the squashed $\mathbb{S}^{4}$ has a non-uniform
distribution. This result supports our claim that the instanton density on a
nonhomogeneous manifold is not uniform in general.

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