KOBE-TH-23-02

KU-PH-034

EPHOU-23-007

# Three-generation solutions of equations of motion
in heterotic supergravity

Maki<EMAIL_ADDRESS>∗Department of Physics, Kobe
University, Kobe 657-8501, Japan Takanao<EMAIL_ADDRESS>∗Department of Physics, Kobe University, Kobe 657-8501, Japan Hikaru
<EMAIL_ADDRESS>∗Department of Physics, Kobe
University, Kobe 657-8501, Japan

###### Abstract

We study the generation number of massless fermions in compactifications
recently found in heterotic supergravity. The internal spaces are products of
two-dimensional spaces of constant curvatures, and the standard embedding is
not assumed. The generation number is constrained by the equations of motion
and the Bianchi identity. In the case that the Euler characteristics of the
three internal submanifolds are $(2,-2,-2)$, the generation number is three or
less. We also show that three-generation solutions exist for arbitrary Euler
characteristics, and we present some explicit solutions.

##  1 Introduction

The origin of three-generation quarks and leptons is one of the mysteries of
the standard model. There may be an answer in compactifications of the extra
dimensions in the superstring theories. Since the mid-’80s, supersymmetric
compactifications, like Calabi-Yau manifolds [1], are mainly considered in
this context. In particular, toroidal orbifold compactifications, which are
singular limits of certain Calabi-Yau compactifications, have been well
studied to derive three-generation models, e.g. Refs. [2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15].

However, supersymmetry has not been discovered by collider experiments, so
low-energy supersymmetry may not be a necessary condition. If supersymmetry is
not assumed, we have to solve the equations of motion to find compact spaces
(for discussions on the relation between the supersymmetry conditions and the
equations of motion, see Refs. [16, 17]). It is difficult in general since
they are second-order non-linear partial differential equations.

Recently, non-supersymmetric exact solutions of equations of motion in the
heterotic supergravity are found [18]. They are direct products of Minkowski
spacetime and 2-manifolds: $S^{2}\times T^{2}\times H^{2}/\Gamma$,
$S^{2}\times S^{2}\times H^{2}/\Gamma$, $S^{2}\times H^{2}/\Gamma\times
H^{2}/\Gamma$, where $S^{2}$, $T^{2}$, $H^{2}/\Gamma$ denote spaces of
positive, zero, and negative constant curvatures (a sphere, a torus, and a
compact hyperbolic manifold [19, 20]). There are many advantages of
considering such manifolds: the equations of motion are simplified to
algebraic equations, the curvatures of compact dimensions are discretely
fixed, the scale of compact space volumes is determined by $\alpha^{\prime}$
(except $T^{2}$), the standard embedding needs not to be assumed, the 3-form
$H$-field is naturally zero, and the Yukawa couplings may be calculated by
applying previous researches on two-dimensional compactifications ($S^{2}$ was
studied in Refs. [21, 22] and $H^{2}/\Gamma$ in Ref. [23]). Product spaces of
2-manifolds can satisfy the Bianchi identity for $H$ without the standard
embedding [24]. In this case, the generation number of fermions $N_{gen}$ is
determined by gauge fluxes [25] and not by the Euler number $\chi$ only; the
well-known relation $N_{gen}=|\chi/2|$ does not hold.

In this paper, we study the possible $N_{gen}$ in these compactifications by
considering both the equations of motion and the Bianchi identity. In Sec. 2,
we summarize these equations under the assumption that the internal space is a
product of two-dimensional spaces of constant curvatures. In Sec. 3, the
explicit relation between $N_{gen}$ and fluxes is given. By the relation and
an equation of motion, we obtain $N_{gen}=0$ for $S^{2}\times T^{2}\times
H^{2}/\Gamma$. In Sec. 4, we consider the simplest case that the Euler
characteristics of $H^{2}/\Gamma$ are $-2$. In this case, we show $N_{gen}\leq
3$ for the $S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$. On the other hand,
larger $N_{gen}$ (in our scope, 9) is possible for $S^{2}\times S^{2}\times
H^{2}/\Gamma$. Finally, we prove that the $N_{gen}=3$ solutions always exist
for both $S^{2}\times S^{2}\times H^{2}/\Gamma$ and $S^{2}\times
H^{2}/\Gamma\times H^{2}/\Gamma$ with arbitrary Euler characteristics in Sec.
5.

##  2 Compactification

###  2.1 Equations of motion

We review the compactification found in Ref. [18] and summarize the conditions
to be solved in the following sections. We consider the bosonic part of the
Lagrangian for heterotic supergravity [26, 17]:

$\displaystyle
L=\sqrt{-g}e^{-2\phi}\left[R+4(\nabla\phi)^{2}-\frac{1}{12}H_{MNP}H^{MNP}+\frac{\alpha^{\prime}}{8}R_{MNPQ}R^{MNPQ}-\frac{\alpha^{\prime}}{8}\textrm{tr}(F_{MN}F^{MN})\right],$
(1)

where $g$ is the determinant of the metric $g_{MN}$ and the indices run
$M,N,P,Q=0,\ldots,9$. $R_{MNPQ}$ is the Riemann tensor, and the Ricci scalar
$R$ and the Ricci tensor $R_{MN}$ are defined as
$R=g^{NQ}R_{NQ}=g^{MP}g^{NQ}R_{MNPQ}$, and $F_{MN}$ is the gauge field
strength of the gauge group $E_{8}\times E_{8}$.

The equation of motion for the three-form field $H$ is

$\displaystyle\partial^{M}(e^{-2\phi}H_{MNP})=0.$ (2)

We take dilaton $\phi$ constant and $H$ to vanish:

$\displaystyle\partial_{M}\phi=0,\quad H_{MNP}=0,$ (3)

and then the equation of motion for $H$ (2) is satisfied. Such a configuration
with unbroken supersymmetry leads to Ricci-flat compactification [1]. In this
paper, we do not assume low-energy supersymmetry. The other equations of
motion [27, 17] become

$\displaystyle
R+\frac{\alpha^{\prime}}{8}R_{MNPQ}R^{MNPQ}-\frac{\alpha^{\prime}}{8}\textrm{tr}(F_{MN}F^{MN})$
$\displaystyle=0,$ (4) $\displaystyle
R_{MN}+\frac{\alpha^{\prime}}{4}R_{MPQR}{R_{N}}^{PQR}-\frac{\alpha^{\prime}}{4}\textrm{tr}(F_{MP}{F_{N}}^{P})$
$\displaystyle=0,$ (5) $\displaystyle\nabla_{M}F^{MN}+[A_{M},F^{MN}]$
$\displaystyle=0.$ (6)

By multiplying Eq. (5) with $g^{MN}$ and comparing with Eq. (4), the dilaton
equation becomes much simpler:

$\displaystyle R=0.$ (7)

In addition to the above equations of motion, the curvature 2-form
$\mathcal{R}$ and the gauge field 2-form $F$ have to satisfy the Bianchi
identity

$\displaystyle
0=dH=\frac{\alpha^{\prime}}{4}(\textrm{tr}\mathcal{R}\wedge\mathcal{R}-\textrm{tr}F\wedge
F)$ (8)

for the Green-Schwarz anomaly cancellation mechanism [28].

###  2.2 Curvatures and gauge field configurations

We are going to solve Eqs. (5), (6), (7), and (8). We assume that the ten-
dimensional spacetime $M^{10}$ is a product of four manifolds:

$\displaystyle M^{10}=M_{0}\times M_{1}\times M_{2}\times M_{3},$ (9)

where $M_{0}$ is the four-dimensional Minkowski spacetime and $M_{i}\
(i=1,2,3)$ are two-dimensional spaces of constant curvature. Note that the
antisymmetric tensor field $H_{MNP}$ is naturally zero for two-dimensional
$M_{i}$. The metric of $M^{10}$ is block diagonal and depends only on the
coordinates of corresponding submanifolds:
$g_{mn}^{(i)}=g_{mn}^{(i)}(x^{(i)})$, where the indices $m$ and $n$ are
tangent to $M_{i}$. The nonzero Riemann tensor components are then

$\displaystyle R_{mnpq}^{(i)}$
$\displaystyle=\lambda_{i}(g_{mp}^{(i)}g_{nq}^{(i)}-g_{mq}^{(i)}g_{np}^{(i)}),$
(10)

where $\lambda_{i}$ is a constant sectional curvature. The manifold with
$\lambda_{i}>0$ is a sphere $S^{2}$, $\lambda_{i}=0$ is a torus $T^{2}$, and
$\lambda_{i}<0$ is a compact hyperbolic manifold $H^{2}/\Gamma$ [19, 20].

For the gauge field strength $F_{MN}=F_{AMN}T_{A}$ ($T_{A}$ are generators of
$E_{8}\times E_{8}$), we assume that it is also block-diagonal for $M$ and
$N$, and nonzero only for $U(1)_{A}$ components $(A=1,2,\ldots,11)$. The range
of $A$ is set to leave the gauge group $SO(10)$ in $E^{8}\times E^{8}\supset
SO(10)\times U(1)_{1}\times\dots\times U(1)_{11}$. For each block of
$F_{AMN}$, we assume the Freund-Rubin configuration [29]

$\displaystyle F_{Ai,mn}$
$\displaystyle=\sqrt{g_{i}}f_{Ai}\epsilon_{mn}^{(i)},$ (11)

where $g_{i}=\textrm{det}(g_{mn}^{(i)})$, $f_{Ai}$ is a constant (sometimes
called a flux density [30]), and $\epsilon_{mn}^{(i)}$ is a Levi-Civita symbol
for $M_{i}$ (for example, $\epsilon_{45}^{(1)}=-\epsilon_{54}^{(1)}=1$).

The configuration of Eq. (11) satisfies the equation of motion for $F_{MN}$
(6). The other equations of motion (5), (7) and the Bianchi identity (8) are
much simplified (we take $\alpha^{\prime}=2$ for simplicity) [18]:

$\displaystyle\lambda_{1}+\lambda_{2}+\lambda_{3}$ $\displaystyle=0,$ (12)
$\displaystyle\lambda_{i}+\lambda_{i}^{2}-\sum_{A=1}^{11}f_{Ai}^{2}$
$\displaystyle=0,$ (13) $\displaystyle\sum_{A=1}^{11}f_{Ai}f_{Aj}$
$\displaystyle=0\quad(i\neq j).$ (14)

The first equation of motion (12) does not allow curvatures $\lambda_{i}$ to
be all positive or all negative. If $\lambda_{i}=0$, the second equation of
motion (13) necessitates $f_{Ai}=0$ for all $A$. In this case, the generation
number is zero as we will see in the following section. We consider the cases
with $\lambda_{i}\neq 0$. We set $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$
without loss of generality, then $M_{1}=S^{2}$ and $M_{3}=H^{2}/\Gamma$. For
the second submanifold $M_{2}$, both $S^{2}$ and $H^{2}/\Gamma$ are possible.

###  2.3 Euler characteristics and flux quantization

The equations (12), (13) and (14) are actually integer equations, as we will
see below. Firstly, the Ricci scalar is related to an integer by the Gauss-
Bonnet theorem:

$\displaystyle\int_{M_{i}}\frac{R^{(i)}}{2}\sqrt{g_{i}}d^{2}x=\textrm{vol}(M_{i})\lambda_{i}=2\pi\chi_{i},$
(15)

where $R^{(i)}$ is the Ricci scalar of $M_{i}$ and $\chi_{i}$ is the Euler
characteristic ($\chi_{i}=2$ for $S_{2}$ and a negative even number for
$H_{2}/\Gamma$) of $M_{i}$. In addition, the gauge field strength satisfies
the flux quantization condition [24]

$\displaystyle\int_{M_{i}}F_{Ai}=\textrm{vol}(M_{i})f_{Ai}=2\pi n_{Ai},$ (16)

where $n_{Ai}$ is an integer. For $\chi_{i}\neq 0$, from Eqs. (15) and (16),
we find

$\displaystyle f_{Ai}=\frac{n_{Ai}}{\chi_{i}}\lambda_{i}.$ (17)

By substituting Eqs. (16) and (17) into Eqs. (13) and (14), the equations of
motion and the Bianchi identity become integer equations.

In summary of this section, the equations to be solved in this paper are:

$\displaystyle\lambda_{1}+\lambda_{2}+\lambda_{3}$ $\displaystyle=0,$ (18)
$\displaystyle\lambda_{i}$
$\displaystyle=\frac{\chi_{i}^{2}}{\sum_{A=1}^{11}n_{Ai}^{2}-\chi_{i}^{2}}\quad\left(\sum_{A=1}^{11}n_{Ai}^{2}\neq\chi_{i}^{2}\right),$
(19) $\displaystyle\sum_{A=1}^{11}n_{Ai}n_{Aj}$ $\displaystyle=0\quad(i\neq
j).$ (20)

##  3 Number of generations

In this section, we obtain an expression of the generation number $N_{gen}$ by
the fluxes $n_{Ai}$ by applying the general formula in Ref. [25].

We consider the decomposition of $E_{8}$ and a gaugino in 248 representation
[31]:

$\displaystyle E_{8}$ $\displaystyle\supset SO(10)\times SU(4),$ (21)
$\displaystyle 248$
$\displaystyle=(45,1)+(1,15)+(10,6)+(16,4)+(\overline{16},\overline{4})$ (22)

One generation of the standard model fermions can be in a 16 representation.
The number of the generation is [25]:

$\displaystyle N_{gen}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\frac{1}{6}\left|\int\text{tr}(F^{3})\right|.$
(23)

As stated in the previous section, we consider $U(1)$ fluxes in $SU(4)$. To be
concrete, we take the basis of the Cartan subalgebra of the $\mathfrak{su}(4)$
as

$\displaystyle\begin{split}T_{1}&=\text{diag}[-1,0,0,1],\\\
T_{2}&=\text{diag}[0,-1,0,1],\\\ T_{3}&=\text{diag}[0,0,-1,1].\end{split}$
(24)

The $U(1)_{A}$ fluxes of $A=4,\dots,11$ reside in another $E_{8}$ and are not
relevant for $N_{gen}$. Then the generation number is, by using
$F=\sum_{A=1}^{3}F_{A}T_{A}$,

$\displaystyle N_{gen}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\frac{1}{6}\left|\int\text{tr}\\{(\text{diag}[-F_{1},-F_{2},-F_{3},F_{1}+F_{2}+F_{3}])^{3}\\}\right|$
$\displaystyle=\frac{1}{(2\pi)^{3}}\frac{1}{6}\left|\int\left(\left(\sum_{A=1}^{3}F_{A}\right)^{3}-\sum_{A=1}^{3}F_{A}^{3}\right)\right|.$
(25)

As an example, we calculate one of the second terms in Eq. (25) explicitly:

$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{6}\int F_{A}^{3}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int_{M_{1}}F_{A1}\int_{M_{2}}F_{A2}\int_{M_{3}}F_{A3}$
$\displaystyle=\prod_{i=1}^{3}n_{Ai}.$ (26)

We have used the flux quantization condition (16). The other terms can be
obtained similarly. The result is:

$\displaystyle N_{gen}=$
$\displaystyle\left|\prod_{i=1}^{3}\sum_{A=1}^{3}n_{Ai}-\sum_{A=1}^{3}\prod_{i=1}^{3}n_{Ai}\right|.$
(27)

We calculate $N_{gen}$ by this formula in this paper.

If one of the submanifolds $M_{i}$ is $T^{2}$, $\lambda_{i}=0$ and Eq. (13)
implies $n_{Ai}=0$ for all $A$, then $N_{gen}=0$. Thus, non-zero generation
numbers are possible only for the compactifications with $S^{2}\times
H^{2}/\Gamma\times H^{2}/\Gamma$ or $S^{2}\times S^{2}\times H^{2}/\Gamma$,
and we consider these manifolds below.

##  4 Ranges of generation numbers

###  4.1 $S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$ case

Since $\lambda_{1}>0$ and $\chi_{1}=2$ in Eq. (19), $n_{A1}$ have to satisfy

$\displaystyle\sum_{A=1}^{11}n_{A1}^{2}\geq 5.$ (28)

To obtain nonzero $N_{gen}$, at least one of $\\{n_{Ai}\\}_{A=1,2,3}$ for
every $i$ have to be nonzero. This condition and Eq. (19) give
$\lambda_{2}<-1$, $\lambda_{3}<-1$. By Eq. (18), we obtain $\lambda_{1}>2$ and
it means

$\displaystyle\sum_{A=1}^{11}n_{A1}^{2}<6.$ (29)

Thus, there is only one possibility in this case:

$\displaystyle\sum_{A=1}^{11}n_{A1}^{2}=5,$ (30)

and it corresponds to $\lambda_{1}=4$. On the other hand, the conditions
$\lambda_{2}<0$, $\lambda_{3}<0$ means

$\displaystyle\sum_{A=1}^{11}n_{Ai}^{2}<\chi_{i}^{2}\quad(i=2,3).$ (31)

In general, $|\chi_{i}|\ (i=2,3)$ and the fluxes can be arbitrarily large. The
generation number can be (almost) any integer in our compactifications.
However, it would be natural that the Euler characteristics and fluxes are not
so large. To be concrete, we consider the simplest case $\chi_{2}=\chi_{3}=-2$
in this section below. We can show $N_{gen}\leq 3$ in this case.

From Eq. (31), there are three possibilities: $\sum_{A}n_{Ai}^{2}=1,2,3$
($i=2,3$). These correspond to $\lambda_{2},\lambda_{3}=-4/3,-2,-4$. Only one
pattern of curvatures can satisfy the equation of motion (18):

$\displaystyle\lambda_{1}=4,\quad\lambda_{2}=\lambda_{3}=-2.$ (32)

Equivalently, the fluxes need to satisfy

$\displaystyle\sum_{A=1}^{11}n_{A1}^{2}=5,\
\sum_{A=1}^{11}n_{A2}^{2}=\sum_{A=1}^{11}n_{A3}^{2}=2.$ (33)

Then $n_{A1}=(2,1,\vec{0})$ or $(1,1,1,1,1,\vec{0})$, $n_{A2}=(1,1,\vec{0})$,
$n_{A3}=(1,1,\vec{0})$ or their permutations and signs of each component can
be changed. The fluxes that contribute to $N_{gen}$ are $A=1,2,3$, and these
fluxes satisfy

$\displaystyle\left|\sum_{A=1}^{3}n_{A1}\right|\leq 3,\
\left|\sum_{A=1}^{3}n_{Ai}\right|\leq 2\ (i=2,3).$ (34)

The Bianchi identity (20) restricts $N_{gen}$ further. It requires each column
of $n_{Ai}$ to be orthogonal to the others. Since only two components of
columns $i=2,3$ are non-zero, then $n_{A1}n_{A2}n_{A3}=0$ for every $A$. The
second term of $N_{gen}$ in Eq. (27) thus vanishes.

The latter inequality of (34) is satisfied in three cases:
$\left|\sum_{A=1}^{3}n_{Ai}\right|=0\text{ or }2$ for one of $i\in\
\\{2,3\\}$, or
$\left|\sum_{A=1}^{3}n_{A2}\right|=\left|\sum_{A=1}^{3}n_{A3}\right|=1$. In
the first case, $N_{gen}=0$. In the second case,
$\left|\sum_{A=1}^{3}n_{Ai}\right|$ cannot be 2 for both $i=2,3$ by the
Bianchi identity. For example, if $(n_{A3})=(1,1,0,\dots)$, the Bianchi
identity necessitate $(n_{A2})=(0,0,1,\dots)$ and $(n_{A1})=(0,0,1,\dots)$ or
$(\pm 1,\mp 1,1,\dots)$, and then $N_{gen}\leq 2$. In the last case,

$\displaystyle N_{gen}=\left|\sum_{A=1}^{3}n_{A1}\right|\cdot 1\cdot 1\leq 3.$
(35)

The equality $N_{gen}=3$ can hold and an example is shown in solution 1 of
Tab. 1. Thus, the maximal generation number is three when
$\chi_{2}=\chi_{3}=-2$.

###  4.2 $S^{2}\times S^{2}\times H^{2}/\Gamma$ case

Next, in the $M_{2}=S_{2}$ case, the sectional curvatures are not uniquely
determined, and larger $N_{gen}$ are possible. The curvatures satisfying the
equations of motion (18), (19) are:

$\displaystyle(\lambda_{1},\lambda_{2},\lambda_{3})=(2,2,-4),(1,1,-2),\left(\frac{4}{3},\frac{2}{3},-2\right),\left(1,\frac{1}{3},-\frac{4}{3}\right),\left(\frac{2}{3},\frac{2}{3},-\frac{4}{3}\right).$
(36)

For example, let us consider the case
$(\lambda_{1},\lambda_{2},\lambda_{3})=(\frac{2}{3},\frac{2}{3},-\frac{4}{3})$
and $\chi_{3}=-2$. In this case, the fluxes are

$\displaystyle\sum_{A=1}^{11}n_{A1}^{2}=\sum_{A=1}^{11}n_{A2}^{2}=10,\
\sum_{A=1}^{11}n_{A3}^{2}=1.$ (37)

These equations constrain the fluxes $n_{A1}$ and $n_{A2}$ to be permutations
of $(3,1,\vec{0})$, $(2,2,1,1,\vec{0})$, $(2,1,1,1,1,1,1,\vec{0})$ or
$(1,1,1,1,1,1,1,1,1,1,0)$ and $n_{A3}$ to be a permutation of $(1,\vec{0})$
(signs of each component can be changed). We have checked all the cases that
also satisfy the Bianchi identity (20). We have found $N_{gen}\leq 9$ and
$N_{gen}\neq 5,7$. The fluxes that give $N_{gen}=9$ are, for example,

$\displaystyle(n_{Ai})=\begin{pmatrix}3&0&0\\\ 0&3&0\\\ 0&0&1\\\ 1&0&0\\\
0&1&0\\\ \vec{0}&\vec{0}&\vec{0}\end{pmatrix},$ (38)

where $\vec{0}$ is a 6-component zero vector. The fluxes that give $N_{gen}=3$
are shown in solution 4 of Tab. 1.

##  5 Three generation solutions

In this section, we show that three-generation solutions exist for arbitrary
Euler characteristics of $H_{2}/\Gamma$, in both $M_{2}=S_{2}$ and
$H_{2}/\Gamma$ cases. Samples of the solutions are summarized in Tab. 1.

###  5.1 $S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$ case

In this case, we search for the solution with $\lambda_{1}=4$,
$\lambda_{2}=\lambda_{3}=-2$. In a special case with $\chi_{2}=\chi_{3}=-2$,
we can find a solution as shown in Tab. 1 (solution 1). In the other cases, we
can set $\chi_{3}\leq-4$ without loss of generality, and we can choose

$\displaystyle(n_{Ai})=\begin{pmatrix}2&0&-1\\\ 1&0&2\\\ 0&1&0\\\
\vec{0}&\vec{n}_{2}^{\prime}&\vec{0}\\\
\vec{0}&\vec{0}&\vec{n}_{3}^{\prime}\end{pmatrix},$ (39)

where $\vec{n}_{2}^{\prime}=(n_{42},\dots,n_{72})$ and
$\vec{n}_{3}^{\prime}=(n_{83},\dots,n_{11,3})$ are 4-component vectors. Such
fluxes satisfy the Bianchi identity (20) and the three-generation condition.
From Eq. (19), the curvatures $\lambda_{2}=\lambda_{3}=-2$ can be obtained by
the fluxes satisfying

$\displaystyle\sum_{A=4}^{7}n_{A2}^{2}$
$\displaystyle=\frac{\chi_{2}^{2}}{2}-1,$ (40)
$\displaystyle\sum_{A=8}^{11}n_{A3}^{2}$
$\displaystyle=\frac{\chi_{3}^{2}}{2}-5.$ (41)

Since $\chi_{2}\leq-2$, $\chi_{3}\leq-4$ and both are even numbers, the right-
hand sides are positive integers. According to Lagrange’s four-square theorem
(see Ref. [32] for a recent discussion), every positive integer can be written
as the sum of at most four squares. Interestingly, we have four integer
parameters on the left-hand sides, so the solutions of
$\lambda_{2}=\lambda_{3}=-2$ always exist. For example, if $\chi_{3}=-4$,
$\vec{n}_{3}^{\prime}=(1,1,1,0)$. In Tab. 1, we show three explicit solutions
with $(\chi_{2},\chi_{3})=(-2,-2),\ (-2,-4),\ (-4,-4)$.

###  5.2 $S^{2}\times S^{2}\times H^{2}/\Gamma$ case

To satisfy the equation of motion (12), we search for the solution with
$\lambda_{1}=\lambda_{2}=2/3$, $\lambda_{3}=-4/3$. The curvatures
$\lambda_{1}=\lambda_{2}=2/3$ can be obtained by the fluxes satisfying

$\displaystyle\sum_{A=1}^{11}n_{A1}^{2}=\sum_{A=1}^{11}n_{A2}^{2}=10.$ (42)

Then we choose

$\displaystyle(n_{Ai})=\begin{pmatrix}3&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&0&0\\\
0&3&0\\\ \vec{0}&\vec{0}&\vec{n}_{3}^{\prime\prime}\end{pmatrix}.$ (43)

where $\vec{n}_{3}^{\prime\prime}=(n_{63},\dots,n_{11,3})$ is a 6-component
vector. The Bianchi identity (20) and the three-generation condition are
satisfied. From Eq. (19), the condition $\lambda_{3}=-\frac{4}{3}$ becomes

$\displaystyle\sum_{A=6}^{11}n_{A3}^{2}$
$\displaystyle=\frac{\chi_{3}^{2}}{4}-1.$ (44)

If $\chi_{3}=-2$, the right-hand side is 0, then
$\vec{n}_{3}^{\prime\prime}=\vec{0}$. If $\chi_{3}\leq-4$, the right-hand side
is a positive integer, so there is a $\vec{n}_{3}^{\prime\prime}$ that
satisfies this equation by Lagrange’s four-square theorem. Thus, the solutions
of $N_{gen}=3$ always exist.

For example, if $\chi_{3}=-6$, $\vec{n}_{3}^{\prime\prime}=(2,2,0,0,0,0)$.
Three sample solutions with $\chi_{3}=-2,-4,-6$ are shown in Tab. 1.

| solution 1 | solution 2 | solution 3 | solution 4 | solution 5 | solution 6
---|---|---|---|---|---|---
$\chi_{2}$ | -2 | -2 | -4 | 2 | 2 | 2
$\chi_{3}$ | -2 | -4 | -4 | -2 | -4 | -6
$(n_{Ai})$ | $\begin{pmatrix}2&0&0\\\ 1&0&0\\\ 0&1&1\\\ 0&1&-1\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}2&0&-1\\\ 1&0&2\\\ 0&1&0\\\ 0&1&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&1\\\ 0&0&1\\\ 0&0&1\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}2&0&-1\\\ 1&0&2\\\ 0&1&0\\\ 0&2&0\\\ 0&1&0\\\ 0&1&0\\\ 0&1&0\\\ 0&0&1\\\ 0&0&1\\\ 0&0&1\\\ 0&0&0\end{pmatrix}$ | $\begin{pmatrix}3&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&0&0\\\ 0&3&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}3&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&0&0\\\ 0&3&0\\\ 0&0&1\\\ 0&0&1\\\ 0&0&1\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}$ | $\begin{pmatrix}3&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&0&0\\\ 0&3&0\\\ 0&0&2\\\ 0&0&2\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix}$
Table 1: Samples of three-generation solutions. These Euler characteristics
$\chi_{i}$ and fluxes $(n_{Ai})$ satisfy the equations of motion (18), (19),
the Bianchi identity (20), and $N_{gen}=3$ [$N_{gen}$ is given in Eq. (27)].
Solutions 1-3 are for $S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$ cases, and
solutions 4-6 are for $S^{2}\times S^{2}\times H^{2}/\Gamma$ cases.

##  6 Conclusion

In this paper, we have discussed how the generation number $N_{gen}$ is
restricted by both the equations of motion and the Bianchi identity in
$S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$ and $S^{2}\times S^{2}\times
H^{2}/\Gamma$. In Section 2, we have reviewed the compactification and
summarized the equation of motion (18), (19), and the Bianchi identity (20).
These equations constrain the Euler characteristics $\chi_{i}$ and the flux
quantization numbers $n_{Ai}$. We have seen in Section 3 that the generation
number $N_{gen}$ is calculated by the flux quantization numbers
$n_{Ai}\,\,(A=1,2,3)$ [Eq. (27)]. In Section 4, we have studied how the
generation numbers $N_{gen}$ satisfying the equations of motion (18), (19),
and Bianchi identity (20) can be obtained when the Euler characteristics
$\chi_{i}$ of $H^{2}/\Gamma$ are $-2$. We have obtained three-generation
solutions in both $S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$ and
$S^{2}\times S^{2}\times H^{2}/\Gamma$ cases. In particular, in the case of
$S^{2}\times H^{2}/\Gamma\times H^{2}/\Gamma$, the generation number $N_{gen}$
is at most 3. In Section 5, we have shown that three-generation solutions
exist for arbitrary Euler characteristics of $H^{2}/\Gamma$.

These results are interesting for two reasons. One is that, in general, flux
quantization numbers $n_{Ai}$ can take arbitrary integers, but they can be
restricted considerably by the equations of motion and the Bianchi identity.
The second is that we have obtained three-generation solutions in
compactifications without supersymmetry. To derive four-dimensional realistic
models, we will study fermion mass hierarchy and flavor mixings as well as CP
violation in these compactifications.

## Acknowledgments

The authors thank the Yukawa Institute for Theoretical Physics at Kyoto
University, where this work was initiated during the YITP-W-22-09 on ”Strings
and Fields 2022”. This work was supported by JSPS KAKENHI Grants No.JP
21J20739 (M. T.) and JP 20J20388 (H. U.).

## References

  * [1] P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, “Vacuum configurations for superstrings,” Nucl. Phys. B 258 (1985) 46–74.
  * [2] L. E. Ibanez, J. E. Kim, H. P. Nilles, and F. Quevedo, “Orbifold Compactifications with Three Families of SU(3) x SU(2) x U(1)**n,” Phys. Lett. B 191 (1987) 282–286.
  * [3] M. V. Libanov and S. V. Troitsky, “Three fermionic generations on a topological defect in extra dimensions,” Nucl. Phys. B 599 (2001) 319–333, arXiv:hep-ph/0011095.
  * [4] J. M. Frere, M. V. Libanov, and S. V. Troitsky, “Three generations on a local vortex in extra dimensions,” Phys. Lett. B 512 (2001) 169–173, arXiv:hep-ph/0012306.
  * [5] H. Abe, T. Kobayashi, and H. Ohki, “Magnetized orbifold models,” JHEP 09 (2008) 043, arXiv:0806.4748 [hep-th].
  * [6] H. Abe, K.-S. Choi, T. Kobayashi, and H. Ohki, “Three generation magnetized orbifold models,” Nucl. Phys. B 814 (2009) 265–292, arXiv:0812.3534 [hep-th].
  * [7] T.-H. Abe, Y. Fujimoto, T. Kobayashi, T. Miura, K. Nishiwaki, and M. Sakamoto, “$Z_{N}$ twisted orbifold models with magnetic flux,” JHEP 01 (2014) 065, arXiv:1309.4925 [hep-th].
  * [8] T.-h. Abe, Y. Fujimoto, T. Kobayashi, T. Miura, K. Nishiwaki, M. Sakamoto, and Y. Tatsuta, “Classification of three-generation models on magnetized orbifolds,” Nucl. Phys. B 894 (2015) 374–406, arXiv:1501.02787 [hep-ph].
  * [9] M. Sakamoto, M. Takeuchi, and Y. Tatsuta, “Zero-mode counting formula and zeros in orbifold compactifications,” Phys. Rev. D 102 no. 2, (2020) 025008, arXiv:2004.05570 [hep-th].
  * [10] T. Kobayashi, H. Otsuka, M. Sakamoto, M. Takeuchi, Y. Tatsuta, and H. Uchida, “Zero-mode wave functions by localized gauge fluxes,” arXiv:2211.04596 [hep-th].
  * [11] T. Kobayashi, H. Otsuka, M. Sakamoto, M. Takeuchi, Y. Tatsuta, and H. Uchida, “Index theorem on magnetized blow-up manifold of $T^{2}/\mathbb{Z}_{N}$,” arXiv:2211.04595 [hep-th].
  * [12] H. Imai, M. Sakamoto, M. Takeuchi, and Y. Tatsuta, “Index and winding numbers on $T^{2}/\mathbb{Z}_{N}$ orbifolds with magnetic flux,” arXiv:2211.15541 [hep-th].
  * [13] K. Hoshiya, S. Kikuchi, T. Kobayashi, Y. Ogawa, and H. Uchida, “Classification of three-generation models by orbifolding magnetized $T^{2}\times T^{2}$,” PTEP 2021 no. 3, (2021) 033B05, arXiv:2012.00751 [hep-th].
  * [14] S. Kikuchi, T. Kobayashi, K. Nasu, and H. Uchida, “Classifications of magnetized T4 and T4/Z2 orbifold models,” JHEP 08 (2022) 256, arXiv:2203.01649 [hep-th].
  * [15] S. Kikuchi, T. Kobayashi, K. Nasu, S. Takada, and H. Uchida, “Number of zero-modes on magnetized $T^{4}/Z_{N}$ orbifolds analyzed by modular transformation,” arXiv:2211.07813 [hep-th].
  * [16] S. Ivanov, “Heterotic supersymmetry, anomaly cancellation and equations of motion,” Phys. Lett. B 685 (2010) 190–196, arXiv:0908.2927 [hep-th].
  * [17] O. Lechtenfeld, C. Nolle, and A. D. Popov, “Heterotic compactifications on nearly Kahler manifolds,” JHEP 09 (2010) 074, arXiv:1007.0236 [hep-th].
  * [18] T. Tsuyuki, “Minkowski spacetime and non-Ricci-flat compactification in heterotic supergravity,” Phys. Rev. D 104 no. 6, (2021) 066009, arXiv:2106.03625 [hep-th].
  * [19] I. Y. Arefeva and I. V. Volovich, “Hyperbolic Manifolds as Vacuum Solutions in Kaluza-Klein Theories,” Nucl. Phys. B 274 (1986) 619–632.
  * [20] D. Orlando and S. C. Park, “Compact hyperbolic extra dimensions: a M-theory solution and its implications for the LHC,” JHEP 08 (2010) 006, arXiv:1006.1901 [hep-th].
  * [21] J. P. Conlon, A. Maharana, and F. Quevedo, “Wave Functions and Yukawa Couplings in Local String Compactifications,” JHEP 09 (2008) 104, arXiv:0807.0789 [hep-th].
  * [22] S. Imai and Y. Tatsuta, “Wavefunctions on S2 with flux and branes,” JHEP 07 (2019) 005, arXiv:1902.04307 [hep-th].
  * [23] M. Honda, “Magnetized Riemann Surface of Higher Genus and Eta Quotients of Semiprime Level,” arXiv:2008.11461 [hep-th].
  * [24] E. Witten, “Some Properties of O(32) Superstrings,” Phys. Lett. B 149 (1984) 351–356.
  * [25] I. Bars and M. Visser, “Number of Massless Fermion Families in Superstring Theory,” Phys. Lett. B 163 (1985) 118–122.
  * [26] E. A. Bergshoeff and M. de Roo, “The Quartic Effective Action of the Heterotic String and Supersymmetry,” Nucl. Phys. B 328 (1989) 439–468.
  * [27] K. Becker and S. Sethi, “Torsional Heterotic Geometries,” Nucl. Phys. B 820 (2009) 1–31, arXiv:0903.3769 [hep-th].
  * [28] M. B. Green and J. H. Schwarz, “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory,” Phys. Lett. B 149 (1984) 117–122.
  * [29] P. G. O. Freund and M. A. Rubin, “Dynamics of Dimensional Reduction,” Phys. Lett. B 97 (1980) 233–235.
  * [30] A. R. Brown and A. Dahlen, “Spectrum and stability of compactifications on product manifolds,” Phys. Rev. D 90 no. 4, (2014) 044047, arXiv:1310.6360 [hep-th].
  * [31] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory Vol. 2: 25th Anniversary Edition.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 11, 2012.
  * [32] M. Eastwood and B. Moore, “Modular forms, projective structures, and the four squares theorem,” arXiv:2108.06433 [math.NT].