# Primordial black hole formation with non-Gaussian curvature perturbations

Vicente Atal  Jaume Garriga  and Airam Marcos-Caballero

###### Abstract

In the context of transient constant-roll inflation near a local maximum, we
derive the non-perturbative field redefinition that relates a Gaussian random
field with the true non-Gaussian curvature perturbation. Our analysis shows
the emergence of a new critical amplitude $\zeta_{*}$, corresponding to
perturbations that prevent the inflaton from overshooting the local maximum,
thus becoming trapped in the false minimum of the potential. For potentials
with a mild curvature at the local maximum (and thus small non-Gaussianity),
we recover the known perturbative field redefinition. We apply these results
to the formation of primordial black holes, and discuss the cases for which
$\zeta_{*}$ is smaller or of the same order than the critical value for
collapse of spherically symmetric overdensities. In the latter case, we
present a simple potential for which the power spectrum needs an amplitude 10
times smaller that in the Gaussian case for producing a sizeable amount of
primordial black holes.

##  1 Introduction

It is certainly an appealing possibility that black holes could have been
formed primordially in an early period of evolution of our Universe [1]. These
primordial black holes (PBHs) could account for the dark matter in the
Universe [2] and be the seeds of the supermassive black holes present in the
center of most galaxies (for a recent review on the subject, see e.g. [3]). If
present, they could not only give an explanation for these intriguing
observations, but would shed light on the mechanism creating the primordial
inhomogeneities. For example, if inflation is the theory accounting for those
initial perturbations, then the creation of PBH would require a far from
canonical realization of inflation, possibly pointing towards its
microphysical origin.

In order to confront the hypothesis of their existence with observations, we
need to understand their formation process and statistical properties. If PBHs
come from large fluctuations of the primordial density perturbations, and if
those obey Gaussian statistics, then their description is well known [4].
Gaussianity of the primordial perturbations might seem to be supported by the
fact that during canonical and slow-roll single-field inflation departures
from Gaussianities are known to be small [5]. However, most models of PBH
creation during inflation require a large peak in the power spectrum, and this
amplification necessarily demands for departures from slow-roll [6]111A
notable exception, where no amplification of the power spectrum is needed, are
models where PBH are formed from the collapse of domain walls created during
inflation [7, 8].. Most calculations so far on the impact on the non-
Gaussianity in the computation of abundances has been performed assuming slow-
roll [9, 10], or considering arbitrary templates for the shape of the non-
Gaussianity [11, 12, 13, 14, 15]. When calculations have been done in the
transient constant-roll (CR) background, usually a perturbative approach has
been followed, focusing on the impact of the three-point function of the
curvature perturbation222The amplitude of non-Gaussianities in transient
periods of CR is different from their amplitude in a pure CR [17, 16, 18]. The
difference between both scenarios is that in the former the curvature
perturbations eventually freeze.[19, 22, 20, 21]. In this case it has been
shown that the curvature mode follows the so-called local template of non-
Gaussianities, which mean that the non-Gaussian field $\zeta$ can be written
in terms of a Gaussian field $\zeta_{g}$ through the following simple
transformation law

$\zeta_{g}\rightarrow\zeta=\zeta_{g}+\frac{3}{5}{f_{\rm NL}}\zeta_{\rm g}^{2}\
.$ (1.1)

In single-field models of inflation the parameter ${f_{\rm NL}}$ can be
calculated, and it is given by [22, 20]

$\displaystyle{f_{\rm NL}}$
$\displaystyle=\frac{5}{12}\left(-3+\sqrt{9-12\eta}\right)$ (1.2)
$\displaystyle=\frac{5}{12}\left(-6+\epsilon_{2}^{cr}\right)$ (1.3)

where $\eta\equiv V^{\prime\prime}/V<0$ is the curvature of the inflaton
potential at the top of the local maximum, and $\epsilon_{2}^{cr}$ is the
second slow roll parameter evaluated in the constant-roll period333We call
constant-roll whenever $\epsilon_{2}\leq-6$, thus incorporating ultra slow-
roll ($\epsilon_{2}=-6$).. In this scenario the cosmological background
interpolates between the constant-roll background and its dual graceful exit
(’slow-roll’) background, described by
$\epsilon_{2}^{ge}=-6+\epsilon_{2}^{cr}$ [23, 24, 25, 26, 20]. The amplitude
of ${f_{\rm NL}}$ can then be written as

$\displaystyle{f_{\rm NL}}$ $\displaystyle=\frac{5}{12}\epsilon_{2}^{ge}\ .$
(1.4)

Because after overshooting a local maximum the field evolves, at least during
a small period of time, in a large curvature potential, _à la_ hill-top,
$\epsilon_{2}^{ge}$ can naturally be $\mathcal{O}(1)$. This bump could
possibly set the end of inflation (because the initial velocity is
exponentially small after climbing the potential, it is still possible to get
a reasonable number of e-foldings), or it could be matched to a potential with
a smaller curvature, delaying the end of inflation.

The above relation shows that ${f_{\rm NL}}$ can be $\mathcal{O}(1)$, and this
turns out to be large enough to invalidate the Gaussian assumption for all
single-field models of inflation claiming to produce PBHs. This has been shown
in [20] by applying the criteria proposed in [27]. Now, the expression (1.1)
suggests that this is just the first order term of a general field
redefinition, only valid when ${f_{\rm NL}}\zeta\ll 1$. The large values of
${f_{\rm NL}}$ and $\zeta$ that describe most models of PBH formation suggest
that this truncation is not justified. A full non perturbative description of
the non Gaussianities is then important. As we will show in the section 2 of
this paper, the complete field redefinition is

$\zeta=-\frac{1}{\beta}\ln\left(1-\beta\zeta_{g}\right)$ (1.5)

with $\beta=6/5{f_{\rm NL}}$. For small values of $\beta$, the local
transformation law (1.1) is recovered.

Aside from this intrinsic non-Gaussianity, resulting from the microphysics of
inflation, large curvature perturbations are non-linearly related to the
density fluctuations. This effect can be taken into account by noting that
$\zeta$ is simply related in superhorizon scales to the so-called compaction
function ${\mathcal{C}}(r)\equiv\delta M/R$ with $\delta M$ the mass excess
over the FRW background, $R$ the areal radius and spherical symmetry has been
assumed. At superhorizon scales this function is time independent and takes
the following form444At the radius where this function is maximal, $r=r_{\rm
m}$, this function is related to the average density perturbation $\delta_{\rm
m}\equiv\delta(r=r_{\rm m})$ [28].

${\mathcal{C}}(r)=\frac{1}{3}\left(1-\left(1+r\partial_{r}\zeta\right)^{2}\right)\
.$ (1.6)

If at early times this function has an initial amplitude above a certain
threshold, then a black hole is formed [29]. This means that it is possible to
find the associated threshold for the curvature mode bypassing the need of
calculating the density perturbation at all scales (for which a linearised
relation with the curvature mode is usually assumed). This criteria for PBH
formation, and its impact in the calculation of abundances, was followed in
[30]. Quantitatively, larger amplitudes of the power spectrum, by a factor
$\sim 2$, are needed for reaching a desired abundances of PBH in comparison
with the linear theory [31, 32, 33].

In this work we will show, on the one hand, that when ${f_{\rm NL}}\sim 1$ the
amplitude of the power spectrum must be smaller by a similar factor compared
to the case with ${f_{\rm NL}}=0$, in order to attain the same abundace of
PBH. On the other hand, and because of the non-pertubative law (1.1), we will
show that for large values of ${f_{\rm NL}}$ perturbations will rather jump
into the false minimum of the potential than create spiky peaks. Black holes
will still be created, but this time through the collapse of bubbles of false
vacuum [7, 8].

##  2 The non-perturbative field redefinition

Single-field and canonical models of inflation can experience an enhancement
of the power spectrum at small scales if the background trajectory passes
through a phase of constant-roll. Many single-field models accomplishing this
with different microscopic motivations have been put forward in the literature
[34, 35, 36, 37, 38, 39, 40]. However they all belong to a very simple class
of models in which the background overshoots a local maximum. It is then
enough to consider the following expansion of the potential around this
maximum, i.e.

$V\left(\phi\right)=\frac{\eta}{2}(\phi-\phi_{c})^{2}$ (2.1)

where we consider $\eta<0$. In order to find the non perturbative relation
between the Gaussian and the non-Gaussian field we can make use of the $\delta
N$ formalism[41]. Here we closely follow the analysis of ref. [22]555For a
pure USR, i.e., without a smooth transition to a graceful exit phase, see [16,
27]. . The homogeneous solution in terms of the e-folding number $dN=Hdt$ is

$\phi(N)=\phi_{c}+c_{1}\,e^{\alpha N}+c_{2}\,e^{\beta N}$ (2.2)

with $\alpha=\frac{1}{2}\left(-3-\sqrt{9-12\eta}\right)$ and
$\beta=\frac{1}{2}\left(-3+\sqrt{9-12\eta}\right)$. Note that $\alpha<0$ and
$\beta\geq 0$, so at late times the solution proportional to $c_{2}$
dominates. The constants $c_{1}$ and $c_{2}$ are functions of the initial
velocity and position, given by

$\displaystyle c_{1}$
$\displaystyle=\frac{1}{\alpha-\beta}\left[\phi^{\prime}_{i}+\beta\left(\phi_{c}-\phi_{i}\right)\right]\
,$ (2.3) $\displaystyle c_{2}$
$\displaystyle=\frac{1}{\beta-\alpha}\left[\phi^{\prime}_{i}+\alpha\left(\phi_{c}-\phi_{i}\right)\right]\
.$ (2.4)

Without loss of generality, we can consider the inflaton field evolving from
negative to positive values, so that $\phi_{i}^{\prime}>0$ and
$\phi_{c}-\phi_{i}>0$. The field overshoots the local maximum of the potential
if at late times it goes towards $+\infty$, which means that $c_{2}$ must be
positive. On the other hand, if $c_{2}=0$ the field stops exactly at the local
maximum and if $c_{2}<0$ the field bounces back. We can now evaluate the
number of e-foldings from an initial time $N=0$ until the the beginning of the
slow-roll regime, which we call $N_{\rm SR}$. The slow-roll regime is the
regime in which the growing solution dominates, and we can always choose this
to happen at a value of $\phi=0$. In this case we find

$c_{2}\simeq-\phi_{c}\,e^{-\beta N_{\rm SR}}\ .$ (2.5)

We can now determine how a change in the initial conditions changes the number
of e-foldings to reach $\phi=0$. We find

$\frac{\delta c_{2}}{c_{2}}=-1+e^{-\beta\delta N|_{\phi=0}}\ .$ (2.6)

Note that there is no constraint on the amplitude of $\delta c_{2}$ with
respect to $c_{2}$. On a flat slicing at $N=N_{\rm SR}={\rm const}$,
$\delta\phi_{\rm SR}=\delta c_{2}e^{\beta N_{\rm SR}}$. From (2.2 we also have
that $c_{2}=\phi^{\prime}_{\rm SR}e^{-\beta N_{\rm SR}}/\beta$, so

$\frac{\delta c_{2}}{c_{2}}=\beta\frac{\delta\phi_{\rm SR}}{\phi^{\prime}_{\rm
SR}}\ .$ (2.7)

This mean that the Gaussian variable $\zeta_{g}\equiv-\delta\phi_{\rm
SR}/\phi^{\prime}_{\rm SR}$ is related to the real non-Gaussian perturbation
$\zeta\equiv\delta N$ as

$\zeta_{g}=\frac{1}{\beta}\left(1-e^{-\beta\zeta}\right)\ .$ (2.8)

By inverting this relation we find eq. (1.5). As we have already anticipated,
this identification has several interesting features. Curvature perturbations
are not well determined when the Gaussian variable has amplitudes larger than
$1/\beta$. Physically these are perturbations that prevent the system of
overshooting the local maximum. Given a background $c_{2}$, any perturbation
$\delta c_{2}$ such that $\delta c_{2}\leq-c_{2}$ will make $c_{2}\leq 0$,
preventing the field from overshooting the local maximum. From (2.7) this
corresponds to the case $\zeta_{g}\geq 1/\beta$. We depict these two regimes
in figure 1, where we show the fate of fluctuations larger than $1/\beta$,
that make the field become trapped in the false minimum, and fluctuations
smaller than $1/\beta$ for which the trajectory of the field towards to global
minimum is not altered. Regions that stay in the false minimum will form
bubbles of false vacuum, and the fate of those is eventually to create black
holes. Depending on their initial size, they might be standard black holes
coming from the ordinary collapse of the bubble at subhorizon scales, or
become black holes with an inflating internal region, that is, with a baby
universe in their interior [7, 8].

Figure 1: The potential introduced in eq. (4.1) (with shifted values for
$\phi$), where we describe the consequences of large fluctuations on top of
the background trajectory. The background solution overshoots the local
maximum. Fluctuations larger than $1/\beta$ at horizon crossing make the field
become trapped in the false minimum of the potential, while those smaller than
$1/\beta$ do not.

We end this section with two comments. First of all, the amplitude of $\beta$
can be written in terms of second slow-roll parameter $\epsilon_{2}$ at the
graceful exit phase. It holds that

$\displaystyle\beta$
$\displaystyle=\frac{1}{2}\left(-3+\sqrt{9-12\eta}\right)$ (2.9)
$\displaystyle=\frac{\epsilon_{2}^{ge}}{2}\ .$ (2.10)

Finally let us note that peaks of $\zeta_{g}$ are transformed into peaks of
$\zeta$. Therefore we can perfectly count the number of peaks in the Gaussian
variable to count the peaks in the non Gaussian one.

##  3 The typical high peak profile

The computation of the threshold for collapse as well as the abundances of
peaks are usually calculated using the mean profile of a peak as a
representative realisation of their full distribution. The mean profile of a
peak with amplitude $\mu$, averaged over all the possible values for
curvatures at the peak, is given by a integral of the form

$\mkern 1.5mu\overline{\mkern-1.5muF(r)\mkern-1.5mu}\mkern 1.5mu=\int\zeta
P(\zeta|\rm{peak})d\zeta$ (3.1)

where $P(\zeta|\rm{peak})$ is the conditional probability of $\zeta$ given
that the field has certain amplitude $\mu$ at the position of the peak $r=0$.
If the probability distribution function is Gaussian and if the peak is large,
in the sense that the amplitude $\mu$ is much larger than the variance
$\sigma_{0}$, i.e, $\nu\equiv\mu/\sigma_{0}\gg 1$, then the mean profile of
the peak is simply given by

$\mkern 1.5mu\overline{\mkern-1.5muF(r)\mkern-1.5mu}\mkern 1.5mu=\mu\psi(r)\
,$ (3.2)

where $\psi(r)$ is the correlation function $\psi(r)\equiv\langle
F(r)F(0)\rangle$, given by

$\psi(r)=\frac{1}{\sigma_{0}^{2}}\int\frac{dk}{k}\frac{\sin(kr)}{kr}P_{k}$
(3.3)

and $P_{k}$ is the power spectrum of the fluctuations. The variance of the
shape is given by

$\frac{\left(\Delta
F\right)^{2}}{\sigma_{0}^{2}}=1-\frac{\psi^{2}}{1-\gamma^{2}}-\frac{1}{\gamma^{2}\left(1-\gamma^{2}\right)}\left(2\gamma^{2}\psi+\frac{R_{s}^{2}\nabla^{2}\psi}{3}\right)\frac{R_{s}^{2}\nabla^{2}\psi}{3}-\frac{5}{\gamma^{2}}\left(\frac{\psi^{\prime}}{r}-\frac{R_{s}^{2}\nabla^{2}\psi}{3}\right)^{2}-\frac{\psi^{\prime
2}}{\gamma^{2}}\ ,$ (3.4)

where $\gamma\equiv\sigma_{1}^{2}/(\sigma_{2}\sigma_{0})$,
$R_{s}\equiv\sqrt{3}\sigma_{1}/\sigma_{2}$, with $\sigma_{n}$ the $n$-moments
of the distribution [4]. The mean profile coming from the non-Gaussian
distribution found in the previous section is not well defined, since the
integral (3.1) diverges when $\zeta\rightarrow 1/\beta$. This is not an
impediment for analyzing how do most profiles look like. Most of the
realisations will have a Gaussian shape in between $\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu\pm\Delta F$, and so we
only need to see how these shapes are deformed by the presence of non-
Gaussianity. The deformation of the Gaussian shapes will depend on the
amplitude of the profile, but we only want to focus on profiles that could
possibly collapse to form BH. Here the identification between $\zeta(r)$ and
the compaction function $C(r)$ is helpful since the initial values of $C(r)$
leading to gravitational collapse are very well constrained. It holds that
[42, 43]666The dependence of the threshold on the shape of the initial density
contrast has been extensively studied (see e.g. [45]), as well as its impact
on the abundances of PBH [46]. For an exploration of these dependences on the
local shape of non-gaussianity, see [47].

$0.21<C_{\rm th}(r_{\rm m})<1/3\ ,$ (3.5)

where $r_{\rm m}$ is the radius at which $C(r)$ is maximum. The bound above
can then be easily translated into a bound on the amplitudes of $\zeta$ at the
peak, once a profile for $\zeta$ is given. This allows one to make reasonable
predictions for the abundance of PBH even in the case in which there is no
known exact value for the thresholds to collapse777By reasonable we mean that
it only translates into a small uncertainty in the amplitude of the primordial
power spectrum in order to reach a certain abundance. Of course, a small
variation of the threshold value keeping the amplitude of the power spectrum
fixed results in exponentially large uncertainties., as was for example done
in [20]. On the other hand, a comparison of the possible values for collapse
evaluated at the center of the profile leads to unbounded uncertainties.

For the power spectrum of the curvature perturbations, that determines the
correlation function $\psi$, we consider the following: from CMB scales up to
scales where the power spectrum grows we take a constant with amplitude
$A_{0}\sim 10^{-9}$. We model the growth of the power spectrum with a scale
dependence $k^{4}$. As argued in [44], this is the maximum possible growth of
the power spectrum. Most models actually saturate this bound during large part
of their growth. The decay of the power spectrum after the peak on the other
hand can be related to the amplitude of the non-Gaussianity [20]. The power
spectrum can then be modelled as

${\mathcal{P}}(k)=\begin{cases}A_{0}&\text{for }k<k_{0}\\\
{\mathcal{P}}_{0}\left(\frac{k}{k_{\rm p}}\right)^{4}&\text{for }k_{0}\leq
k\leq k_{\rm p}\\\ {\mathcal{P}}_{0}\left(\frac{k}{k_{\rm
p}}\right)^{-n}&\text{for }k\geq k_{\rm p}\ ,\end{cases}$ (3.6)

where ${\mathcal{P}}_{0}$ is the amplitude at the peak of the power spectrum.
The spectral index at scales smaller than $k_{\rm p}$, $n$, is given by

$n\simeq-3+\sqrt{9-12\eta_{V}}.$ (3.7)

By comparing (3.7) with the expression for $\beta$, we see that both are
related through

$\beta\simeq\frac{n}{2}\ .$ (3.8)

The abundance of peaks diverges when the power spectra decays slower than
$k^{-3}$, since they depend on the second momenta $\sigma_{2}$ [4]. Thus, in
order to make finite predictions we need to use a window function, which can
be done by considering the coarse grained power spectrum
${\mathcal{P}}(k,k_{\rm cut})$ given by

${\mathcal{P}}(k,r_{\rm f})\rightarrow W(k,r_{*}){\mathcal{P}}(k)\ .$ (3.9)

The choice of the window function might lead to uncertainties [48], although
small [49]. For simplicity, and because we concentrate on the dependence of
the non-Gaussianity on the abundances, we choose the top-hat window function
in momentum space. We typically need $A_{0}\ll{\mathcal{P}}_{0}$ in order to
have a significant abundance of PBH, so we can neglect the power at scales
larger than $k_{0}$. That is, we finally consider a power spectrum given by

${\mathcal{P}}(k,k_{\rm cut})=\begin{cases}0&\text{for }k<k_{0}\\\
{\mathcal{P}}_{0}\left(\frac{k}{k_{\rm p}}\right)^{4}\ ,&\text{for }k_{0}\leq
k\leq k_{\rm p}\\\ 0\ ,&\text{for }k>k_{\rm p}\ .\end{cases}$ (3.10)

where we choose the coarse graining scale $k_{\rm cut}$ to be the scale of the
peak of the power spectrum $k_{\rm p}$, since at this scale the abundace of
PBH will be larger. The correlation function determining the shape of the peak
is then simply given by

$\psi_{g}(r)\simeq\frac{4}{k_{\rm p}^{4}r^{4}}\left[-2+\left(2-k_{\rm
p}^{2}r^{2}\right)\cos\left(k_{\rm p}r\right)+2k_{\rm p}r\sin\left(k_{\rm
p}r\right)\right]$ (3.11)

where we have further assumed that $k_{0}\ll k_{\rm p}$.

In figure 2 we show how the Gaussian set of profiles is transformed by the
presence of non-Gaussianity. The non-Gaussian set of profiles is found by
applying the transformation law (1.5) to all Gaussian profiles ranging in the
interval $\mkern 1.5mu\overline{\mkern-1.5muF_{g}\mkern-1.5mu}\mkern
1.5mu\pm\Delta F_{g}$, with $\mkern
1.5mu\overline{\mkern-1.5muF_{g}\mkern-1.5mu}\mkern 1.5mu$ and $\Delta F_{g}$
given by (3.2) and (3.4) respectively. We take a fiducial value of
$\beta=1.35$ (corresponding to ${f_{\rm NL}}=1.25$), and vary the amplitude of
the profile such that the compaction function lies in the critical range
(3.5). We also take888Larger values of $\nu$ will have smaller dispersion in
their profiles and so in this sense we are consider the largest possible
dispersion. $\nu=5$. Interestingly, the dispersion of the profiles that could
possibly collapse is systematically smaller for the non-Gaussian case. The
reason is twofold. First, the radius $r_{\rm m}$ is smaller for the non-
Gaussian case (and close to the center of the profile the dispersion is
smaller). Secondly and most importantly, large value of the compaction
function are attained for smaller values of $\mu$, and thus, if we keep
$\nu\equiv\mu/\sigma$ fixed (since it is the most important parameter
controlling the abundances), the dispersion $\sigma$ will be smaller. From
here we deduce that while it is not possible to define a mean profile, it is
certainly possible to define a typical profile, given in this case by the
median profile of the distribution, which will be simply

$F_{\rm ng}=\frac{1}{\beta}\ln\left(1-\beta\mkern
1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu_{\rm g}\right)\ .$
(3.12)

In the next section we anayse the implications of such profile in the
formation and abundances of PBHs.

Figure 2: _top)_ The mean and 1-$\sigma$ contours of the Gaussian profiles and
how they transform under non Gaussianity ($\beta=1.35$). For the non-Gaussian
profiles these corresponds to the median and the 16% and 84% quartiles. We
vary the amplitude $\mu$ so that profiles span all possible values for the
compaction function that could in principle form PBHs, limits given by eq.
(3.5). We keep $\nu=5$ fixed. _bottom)_ Compaction functions in the critical
range for the Gaussian and non Gaussian case (corresponding to the profiles
shown above).

##  4 Implications for PBH

The implications of the non-Gaussianity in the abundance of PBH will depend on
whether the critical value for spherical collapse ($\zeta_{c}$) is smaller or
of the same order than the critical value for staying in the false minimum of
the potential $(\zeta_{*}$). In the following we will separately discuss both
cases.

###  4.1 Perturbative limit $\zeta_{c}<\zeta_{*}$

We first consider the perturbative limit, for which the critical value for
collapse ($\zeta_{c}$) is smaller than amplitude to end in the false minimum
($\zeta_{*}$). We take $\beta=1.25$, and the same critical value for the
compaction function for both the Gaussian and non-Gaussian profiles, since the
normalized profiles do not vary substantially with respect to each other for
such values of $\beta$. We take the threshold to be $C_{\rm th}=0.255$ since
the profiles are similar to those already studied in the literature, and for
which the threshold has been determined [46]. For the Gaussian case this
amplitude of the compaction function is attained for $\mu=0.54$. In the non-
Gaussian case, this threshold is attained for $\mu=0.44$. In order to keep the
value of $\nu$ constant, the power spectrum needs an amplitude $\sim 1.5$
smaller in the non-Gaussian case. For this amplitude of $\beta$, this is an
effect of a similar order than the one coming from the non-linear relation
between $\zeta$ and $\delta_{m}$, but acts in the opposite direction. In the
next section, we will see that this effect can be largely enhanced when the
theory enters in the non-perturbative regime.

###  4.2 Non-perturbative limit $\zeta_{c}\simeq\zeta_{*}$

Now we consider the non perturbative limit for which
$\zeta_{c}\simeq\zeta_{*}$. For this, larger values of $\beta$ should be
attained, larger than in current models in the literature. We propose a simple
model where this is realised. We consider the Starobinsky potential [50] with
a small bump given by a Gaussian function, i.e.

$V(\phi)=V_{0}^{4}\left(1-e^{-\sqrt{2/3}\phi}\right)^{2}\left[1+Ae^{-\left(\phi-\phi_{0}\right)^{2}/\left(2\sigma^{2}\right)}\right]\
.$ (4.1)

In figure 3 we show the power spectrum as well as the evolution of the second
slow-roll parameter $\epsilon_{2}$, which determines the amplitude of $\beta$.

Figure 3: _left)_ Power spectrum for the model given by the potential (4.1),
together with the following parameters: $\\{A=1.80325\times
10^{-3},\phi_{0}=3.7,\sigma^{2}=9\times 10^{-5}\\}$. _right)_ Evolution of the
second slow roll parameter $\epsilon_{2}$ as a function of time (measured in
e-foldings $N$). The dashed line corresponds to $\epsilon_{2}$ coming from the
solution (2.2) [20]. We see that it describes well the transition from the CR
to the graceful exit phase.

The predicted value for $\epsilon_{2}$ at the graceful exit period is
$\epsilon_{2}^{ge}=12$, and so999As can be seen in the figure, the actual
value is slightly larger. This means that $f_{\rm NL}$ will correspondingly be
slightly larger. Note that the fact that there is a large evolution of
$\epsilon_{2}$ after the transition does not affect the amplitude of ${f_{\rm
NL}}$ at scales corresponding to the transition, where the peak of the power
spectrum is, since perturbations freeze when $\epsilon_{2}=\epsilon_{2}^{ge}$
[20]. $\beta=6$. In figure 4 we show the profile as well as the compaction
function as a function of the amplitude $\mu$. We see that for values of $\mu$
very close to the non perturbative critical value $1/\beta$, in this
particular case $\mu=0.999/\beta$, the compaction function is still below the
lowest of the standard critical threshold for the compaction function, namely
$C_{\rm th}=0.21$. Note that if we choose amplitudes $\mu$ even closer to the
threshold $1/\beta$, lets say $\mu=c/\beta$ with $c>0.999$, the compaction
function will eventually rise above its threshold for collapse. However, the
contribution of these fluctuations to the total abundance of PBH will
correspond to the integration of the probability density function in a very
narrow range of the gaussian variable, that is from $c/\beta$ to $1/\beta$,
where in this case $0.999<c<1$. On the other hand, the contributions from
fluctuations that jump into the false minimum comes from fluctuations in the
range $1/\beta$ to $\infty$, which will then dominate the production of PBH.

Finally let us note that because the inflaton field traverses the bump in
$\sim 2$ e-foldings, the distance of the CMB scales to the end of inflation is
only shifted by 2-efoldings with respect to the original model without the
bump. This means that this modification of the original potential will not
alter its consistency with CMB data.

Figure 4: The variation of the median profile and median compaction function
in the non-Gaussian case, with $\beta=6$. While the amplitude is very close to
the non-perturbative bound $1/\beta$ ($\mu=0.999/\beta$), the maximum of the
compaction function is still lower than the lowest possible threshold for
collapse, $C_{\rm th}=0.21$.

We conclude that now the theory will produce BHs when the amplitude of
$\zeta\sim 1/\beta\sim 0.17$, as opposed to the Gaussian case in which
$\zeta_{c}\sim 0.54$. This difference corresponds to a factor 10 in the
amplitude of the power spectrum101010Note however that both scenarios are
phenomenologically different and might lead to a different spectrum of masses
for the resulting PBH.. Larger factors might be attained with a larger
curvature at the local maxima, but the fine tuning in the potential keeps
increasing.

##  5 Conclusions

In this paper we have calculated the amplitude and shape of non-Gaussianities
in models of inflation where the field overshoots a local maximum. These
models have gained much attention since they might create enough PBH to
account for the dark matter in the Universe. The impact of non-Gaussianities
on the abundance of PBH has so far only been estimated, in transient constant-
roll backgrounds, through a perturbative determination of its amplitude and
shape, more precisely, via the computation of the three-point function. Here
we have shown that via the $\delta N$ formalism is possible to find a non-
linear relation between a Gaussian random field $\zeta_{g}$ with the true
curvature mode $\zeta$, given by eq. (1.5). This relation incorporates the
fact that large perturbations will be trapped in the false minimum, signaling
the presence of a second critical amplitude $\zeta_{*}=1/\beta$. It also
reduces to the standard local template of non-Gaussianities in the small
perturbation regime. We have computed the effect of such law in the case in
which $\zeta_{*}$ is smaller or of the same order than the usual critical
value for collapse of a large inhomogeneity. In the latter case, black holes
will be created through the collapse of bubbles of false vacuum, rather than
the usually considered collapse of a large overdensity. We have constructed a
simple model in which this is the case, and for which amplitudes of the power
spectrum 10 times smaller are needed in comparison with the Gaussian estimate
in order to have a significant PBH abundance.

## Acknowledgments

We thank Jose J. Blanco-Pillado, Cristiano Germani, Ilia Musco and Sam Young
for useful discussions. VA and JG are supported by FPA2016-76005 -C2-2-P,
MDM-2014-0369 of ICCUB (Unidad de Excelencia Maria de Maeztu),
AGAUR2014-SGR-1474 and AGAUR2017-SGR-754. AM-C acknowledges a postdoctoral
contract from the Universidad de Cantabria postdoctoral program, the Spanish
research project, ref. ESP2017-83921-C2-1-R (AEI/FEDER, EU), the Spanish
Ministry MINECO grant (FPA2015-64041-C2-1P) and the Basque Government grant
(IT-979-16).

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Eight-dimensional non-geometric heterotic strings
and enhanced gauge groups

Yusuke Kimura1
1Tokyo Institute of Technology, International Research Frontiers Initiative,
Nagatsuta-cho, Midori-ku, Yokohama 226-8503, Japan

(Dated: May 30, 2022)

###### Contents

  1. 1 Introduction
  2. 2 Review of 8D non-geometric heterotic constructions with unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra
    1. 2.1 Review of non-geometric heterotic constructions by Malmendier and Morrison
    2. 2.2 5-branes and enhancements in non-Abelian gauge algebra
  3. 3 Maximally enhanced non-Abelian gauge groups in 8D non-geometric heterotic strings with unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$
    1. 3.1 Points of most enhanced non-Abelian gauge algebras in the 8D non-geometric moduli space and F-theory duals
    2. 3.2 Sample model
    3. 3.3 Possible extensions
  4. 4 Concluding remarks

##  1 Introduction

In this study, we discuss the gauge group in elementary particle physics.
Elementary particles are fundamental subatomic particles that cannot be
subdivided into smaller particles. The elementary particle theory is employed
to investigate the interaction among such fundamental particles. The notion of
the gauge group arises in the theoretical framework of elementary particles to
describe the intersections among the particles. Studies on elementary particle
physics show that gauge groups are crucial in describing the forces or
interactions among elementary particles.

First, we introduce the notion of a group in mathematics; it refers to a set
of elements such that one can “multiply” any two elements. This operation is
associative, implying that $(g\cdot h)\cdot k=g\cdot(h\cdot k)$. A
distinguished element exists in the group and is referred to as an identity,
$e$, such that for any element $g$ in a group, the multiplication of $g$ by
identity $e$, $g\cdot e$, is $g$: $g\cdot e=e\cdot g=g$. Furthermore, every
element $g$ in the group has an inverse $g^{-1}$ such that the multiplication
of $g$ with its inverse $g^{-1}$ yields identity $e$: $g\cdot
g^{-1}=g^{-1}\cdot g=e$. These rules define a group. Although the description
of the rules may appear slightly abstract at first, the rules are very
general. Owing to this generality, many objects in mathematics are groups.
Several representative examples of groups are found to be familiar notions.
For example, a set of integers, $\mathbb{Z}$, with addition + regarded as
“multiplication,” is a group. All possible permutations of $n$ objects is
another example of a group, which is referred to as the symmetric group,
$S_{n}$. Some geometric symmetries also form a group.

Groups can be categorized into two classes: commutative groups and non-
commutative ones. For group $G$, if the multiplication of any two elements
commutes, that is, $a\cdot b=b\cdot a$ for any two elements $a$ and $b$ in
$G$, then group $G$ is said to be commutative. A commutative group is also
referred to as the Abelian group. $\mathbb{Z}$ (with addition) is an example
of the Abelian group. When there is a pair of elements, $a$ and $b$, that does
not commute, $a\cdot b\neq b\cdot a$, group $G$ is said to be non-commutative.
In elementary particle physics, when gauge group $G$ is non-commutative, $G$
is referred to as a non-Abelian gauge group.

Some recent developments in elementary particle physics can be attributed to
string theory. This theory is considered as a candidate for the consistent
quantum gravitational theory. The quantum field theoretical formulation of
gravity originally had a divergence problem. To overcome this problem, the
notion of point particles was replaced with that of particles with a one-
dimensional structure, or “strings,” leading to the formulation and
construction of string theory. However, string theory research has revealed
that such “string” particles can only consistently exist in high-dimensional
space-time, whose dimensions are larger than four.

Because the real world is four dimensional, comprising three spatial
dimensions and one time dimension, many researchers in string theory consider
that the extra-dimensional space, referred to as a compactification space, is
tiny and escapes experimental observations. Studies on string theory show that
the geometry of the compactification space is closely related to the structure
of the gauge group.

In this study, we aim to introduce and review some recent developments on this
aspect of string theory, including our own results.

String theory has several different versions of formulations. These are
considered equivalent, at least under certain circumstances, and the different
versions of string theories are closely related. The several formulations of
string theory are not completely independent theories; instead, they can be
viewed as different viewpoints of an identical theory under some conditions.
The formulations of sting theory include heterotic strings [1, 2, 3] and
F-theory [4, 5, 6].

How does the geometry of the compactification space determine the gauge group
structure? Based on F-theory, this point can be explained as follows: in type
IIB superstring theory, Ramond-Ramond 0-form $C_{0}$ and dilation $\phi$
arise. Type IIB superstring theory is considered to possess
$SL_{2}(\mathbb{Z})$ symmetry, wherein the $SL_{2}(\mathbb{Z})$ group acts on
bosons, including these particles. In the formulation of F-theory [4, 5, 6],
Ramond-Ramond scalar $C_{0}$ and dilaton $\phi$ are treated in a unified
manner as the axio-dilaton, $C_{0}+ie^{-\phi}$, which is identified with the
modular parameter, $\tau$, of an elliptic curve. Because the modular parameter
of an elliptic curve has $SL_{2}(\mathbb{Z})$ symmetry, this identification is
useful for technical reasons. Here, an elliptic curve is in real dimensions a
two-dimensional surface with one hole. Elliptic curves are frequently
considered in algebraic geometry, wherein the geometries are considered over
complex numbers in many situations. Elliptic curves have a complex dimension
of 1, and owing to this property, they are referred to as curves, although
they are surfaces in real dimensions.

Given these backgrounds, compactification spaces in F-theory are “elliptic
fibrations,” which mean spaces consisting of elliptic curves that lie over the
base spaces.

Physical objects known as “branes” appear in string theory. String particles
can be of two types: an open string and a closed string. Open strings mediate
the forces of the gauge groups. An open string has two ends on a brane. In
F-theory, there are 7-branes and they are essentially related to the non-
Abelian gauge group formed in the theory.

7-branes are wrapped on a subspace in the base space. Elliptic curves are
generically smooth over the base space, but over a complex codimension one
subspace they degenerate to singular fibers. 7-branes are wrapped on the
components in this subspace of the base space.

The different ways in which the elliptic fibers degenerate to the singular
fibers were classified for elliptically fibered complex surfaces [7, 8]. A
correspondence [6, 9] exists between the ways in which the elliptic fibers
degenerate and the non-Abelian gauge groups formed on the 7-branes in
F-theory. The non-Abelian gauge groups are given as Lie groups and, hence,
have $ADE$ types. Kodaira’s classification [7, 8] of the singular fibers of an
elliptic fibration also has $ADE$ types. Non-Abelian gauge group arising on
7-branes that are wrapped on a component in the base space, and the
degeneration of an elliptic fiber lying over that component, have identical
$ADE$ types.

Heterotic strings and F-theory are closely related through the duality
relation [4, 5, 6, 10, 11]. Heterotic strings on the 2-torus, $T^{2}$, are
dual to F-theory on elliptically fibered K3 surfaces. In [12], a class of
heterotic strings was constructed on $T^{2}$ that did not have a geometric
interpretation 111Discussion of connections $O^{+}(\Lambda^{2,2})$-modular
forms and non-geometric heterotic strings can be found in [13]. Non-geometric
type II theories were studied in [14, 15].. The authors in [12] constructed
“non-geometric” heterotic strings with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra and F-theory duals. The
main subjects of this study are non-geometric heterotic strings [12] and the
structure of heterotic/F-theory duality [4, 5, 6, 10, 11]. In this study, we
review the construction of non-geometric heterotic strings in [12] and present
some developments on this topic. The points in the eight-dimensional (8D) non-
geometric heterotic moduli with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra, at which the non-
Abelian gauge groups are maximally enhanced, were determined in [16]. From the
heterotic viewpoint, the gauge groups formed at these point do not allow for a
perturbative interpretation [16]. The K3 surfaces at these points on the dual
F-theory are deformations of stable degenerations 222An elliptic K3 surface
degenerates into two 1/2 K3 surfaces intersecting along an elliptic curve in
the stable degeneration limit. The duality relation of heterotic strings and
F-theory becomes rigorous when the stable degeneration limit is considered on
the F-theory side. [11, 17], which arise owing to the coincidence of 7-branes,
as demonstrated in [16]. This effect can be viewed on the dual heterotic side
owing to the insertion of 5-branes [16]. We also discuss these issues. Recent
progress in non-geometric heterotic strings can also be found, for example, in
[18, 19, 20, 21, 22, 23, 24, 25]. Related discussions on the lattice
structures in K3 surfaces and the duality of heterotic strings and F-theory
can be found in recent studies, e.g. in [26, 27]. Some elliptically fibered K3
surfaces with extended gauge groups also appear in the Swampland Program [28].

In section 2, we briefly review the construction of 8D non-geometric heterotic
strings given in [12]. In sections 3.1 and 3.2, we then review maximally
enhanced non-Abelian gauge algebras in the 8D non-geometric heterotic strings
with unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra deduced
in [16]. We mention the discussion of nonperturbative effect of inserting
5-branes considered in [16]. Application of the Shioda–Tate formula [29, 30,
31] is mentioned in section 3.1. Details of this argument in the context of
F-theory can be found, e.g. in [32]. The global structure of the gauge groups
formed in F-theory [33, 34, 35] are also mentioned in section 3.2. Recent
studies on this aspect with non-trivial Mordell–Weil torsions can be found,
e.g. in [36, 37, 38, 39]. Two potential directions for extending [16] are
proposed in section 3.3. For the first direction of extension, a related
method was discussed in [40]. The second direction involves lattice theoretic
techniques. Discussions of some lattice theoretic techniques in the context of
F-theory/heterotic duality can be found in [41, 42]. We state our concluding
remarks in section 4.

##  2 Review of 8D non-geometric heterotic constructions with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra

###  2.1 Review of non-geometric heterotic constructions by Malmendier and
Morrison

There is a duality relation between F-theory and heterotic strings [4, 5, 6,
10, 11], which states that F-theory on a K3 fibered Calabi–Yau (CY) $n+1$-fold
and heterotic theory on an elliptically fibered CY $n$-fold are physically
equivalent.

Eight-dimensional heterotic string on a 2-torus, $T^{2}$, and F-theory on an
elliptically fibered K3 surface describe physically equivalent theories. A
comparison of the moduli spaces of the two 8D theories reveals some aspects of
the duality relation.

The moduli space of the 8D heterotic strings is the Narain space [43], which
is quotient

$D_{2,18}/O(\Lambda^{2,18}).$ (1)

Here, $D_{2,18}$ represents the symmetric space of $O(2,18)$, namely
$D_{2,18}=O(2)\times O(18)\backslash O(2,18)$. The 8D F-theory moduli on
elliptic K3 surfaces with a global section is given as

$D_{2,18}/O^{+}(\Lambda^{2,18}),$ (2)

where $O^{+}(\Lambda^{2,18})=O(\Lambda^{2,18})\cap O^{+}(2,18)$. The double
cover of the heterotic moduli space (1) matches with the F-theory moduli space
(2), and this relation defines the duality of 8D heterotic strings and
F-theory.

The Narain space (1) decomposes into the product of the complex structure
moduli, the moduli of the Wilson line expectation values, and Kähler moduli in
a suitable limit [44]. The action of group $O^{+}(\Lambda^{2,18})$ mixes the
three moduli spaces in this decomposition. As a result, heterotic strings
possessing the symmetry of $O^{+}(\Lambda^{2,18})$ do not allow for a
geometric interpretation. Such theories are “non-geometric.” Malmendier and
Morrison [12] built such 8D non-geometric heterotic strings on the two-torus
with the symmetry group that is a subgroup in $O^{+}(\Lambda^{2,18})$.

Utilizing the F-theory/heterotic duality, they constructed the 8D non-
geometric heterotic strings as duals of F-theory with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra. 8D F-theory with
unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra is realized on
elliptic K3 surfaces with $E_{8}E_{7}$ singularity. The moduli space of
F-theory on elliptic K3 surfaces with $E_{8}E_{7}$ singularity with a global
section is given as [12]

$D_{2,3}/O^{+}(L^{2,3}).$ (3)

$L^{2,3}$ denotes the orthogonal complement of lattice $E_{8}\oplus E_{7}$
inside $\Lambda^{2,18}$. There is an isomorphism between $D_{2,3}$ and the
Siegel upper half-space of genus two, $\mathbb{H}_{2}$, i.e.
$D_{2,3}\cong\mathbb{H}_{2}$. This isomorphism induces the correspondence of
the ring of $O^{+}(L^{2,3})$-modular forms and the Siegel modular forms with
genus two of even weight [45].

An elliptically fibered K3 surface with a global section can be described by
using the Weierstrass equation, and those with the $E_{8}E_{7}$ singularity
are given by the following Weierstrass form:

$y^{2}=x^{3}+(\alpha t^{4}+\gamma t^{3})x+t^{7}+\beta t^{6}+\delta t^{5}.$ (4)

The specified values for $\alpha$, $\beta$, $\gamma$, and $\delta$ for the K3
surface determine a point in $D_{2,3}$, which via the isomorphism
$D_{2,3}\cong\mathbb{H}_{2}$ specifies a point in $\mathbb{H}_{2}$. We denote
the specified point in $\mathbb{H}_{2}$ by $\underline{\tau}$ to align it with
the notation used in [12]. For this, $\alpha$, $\beta$, $\gamma$, and $\delta$
are given in terms of Igusa’s generators [46] as follows [47, 48, 49] up to
scale factors:

$\alpha=\frac{-\psi_{4}(\underline{\tau})}{48},\hskip
13.65732pt\beta=\frac{-\psi_{6}(\underline{\tau})}{864},\hskip
13.65732pt\gamma=-4\chi_{10}(\underline{\tau}),\hskip
13.65732pt\delta=\chi_{12}(\underline{\tau}).$ (5)

One then considers a line bundle, $\mathcal{L}$, on a two-torus $T^{2}$, and
select sections, $\alpha,\beta,\gamma,\delta$, of $\mathcal{L}^{\otimes
4},\mathcal{L}^{\otimes 6},\mathcal{L}^{\otimes 10},\mathcal{L}^{\otimes 12}$,
respectively. Compactification on $T^{2}$ with the identification of the
sections (5) yields the construction of non-geometric heterotic strings with
$O^{+}(L^{2,3})$-symmetry given in [12].

###  2.2 5-branes and enhancements in non-Abelian gauge algebra

When the defining equation of an elliptic K3 surface is given, by computing
the discriminant of the equation one can determine the singularity type of the
surface. When the Weierstrass equation of the surface, $y^{2}=x^{3}+fx+g$, is
given, then the discriminant is described by

$\Delta=4f^{3}+27g^{2}.$ (6)

This particularly means that the discriminant of the K3 surface (4) is given
by

$\Delta=t^{9}\big{(}4(\alpha t+\gamma)^{3}+27t(t^{2}+\beta
t+\delta)^{2}\big{)}.$ (7)

For generic values of $\alpha,\beta,\gamma,\delta$, i.e. for generic points in
the complex structure moduli space, the singularity type is $E_{8}E_{7}$. For
special values of $\alpha,\beta,\gamma,\delta$ (equivalently, at the
corresponding specific points in the moduli,) the singularity types are
enhanced. This physically means that the gauge algebras are enhanced at those
points.

From the dual heterotic perspective, the enhanced gauge algebra can be
attributed to the presence of 5-branes [12]. From the discriminant (7), the
authors of [12] deduced the 5-brane solutions in the moduli space. The two
types of 5-brane solutions deduced in [12] are as follows:

$\displaystyle\gamma$ $\displaystyle=0$ (8) $\displaystyle q$
$\displaystyle=0.$

Here, $q$ denotes a polynomial in coefficients $\alpha,\beta,\gamma$, and
$\delta$ in the Weierstrass equation (4), given as follows:

$\displaystyle q=$ $\displaystyle
11664\,\delta^{5}-5832\,\beta^{2}\delta^{4}+864\,\alpha^{3}\delta^{4}+16\,\alpha^{6}\delta^{3}$
(9)
$\displaystyle+216\,\alpha^{3}\beta^{2}\delta^{3}+16200\,\alpha\gamma^{2}\delta^{3}-2592\,\alpha^{2}\beta\gamma\delta^{3}+729\,\beta^{4}\delta^{3}$
$\displaystyle-5670\,\alpha\beta^{2}\gamma^{2}\delta^{2}-13500\,\beta\gamma^{3}\delta^{2}+216\,\alpha^{4}\beta^{2}\gamma^{2}\delta+2700\,\alpha^{2}\beta^{2}\gamma^{4}-3420\,\alpha^{3}\beta\gamma^{3}\delta$
$\displaystyle+4125\,\alpha^{2}\gamma^{4}\delta+729\,\alpha\beta^{4}\gamma^{2}\delta+888\,\alpha^{4}\gamma^{2}\delta^{2}+6075\,\beta^{3}\gamma^{3}\delta-16\,\alpha^{6}\beta\gamma^{3}+16\,\alpha^{5}\gamma^{4}$
$\displaystyle-216\,\alpha^{3}\beta^{3}\gamma^{3}+16\,\alpha^{7}\gamma^{2}\delta-5625\,\alpha\beta\gamma^{5}-729\,\beta^{5}\gamma^{3}+3125\,\gamma^{6}.$

##  3 Maximally enhanced non-Abelian gauge groups in 8D non-geometric
heterotic strings with unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$

###  3.1 Points of most enhanced non-Abelian gauge algebras in the 8D non-
geometric moduli space and F-theory duals

Herein, we discuss the points in the 8D F-theory moduli with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra constructed in [12],
wherein the non-Abelian gauge algebras are maximally enhanced. These points in
the moduli are special, that is, the gauge groups for the dual heterotic
theories at the points do not allow for the perturbative interpretations [16].
We briefly discuss this point.

First, we explain a method [16] for determining the points in the moduli at
which the non-Abelian gauge algebras are maximally enhanced. The maximally
enhanced non-Abelian gauge algebras are also deduced by applying the method
mentioned in [16]. To achieve this goal, a modern technique in algebraic
geometry was used. As reviewed in section 2.1, Siegel modular forms played a
role in the construction in [12] to investigate the aspects of the
heterotic/F-theory duality. The classification of the elliptic fibrations of
(some of) K3 surfaces with Picard number 20 is another useful mathematical
technique that clarifies the duality relation of F-theory and heterotic
strings.

Every K3 surface, $S$, has a Picard number, $\rho(S)$, which varies over the
complex structure moduli. The set of K3 surfaces with a higher Picard number
has a lower dimension in the complex structure moduli. Quantitatively, the
dimension of the complex structure moduli of the complex K3 surfaces with
Picard number $\rho$ is $20-\rho$. Intuitively, a K3 surface with a higher
Picard number is rarer in terms of the whole complex structure moduli.

The highest Picard number for complex K3 surfaces is 20, and they constitute a
discrete set of points in the complex structure moduli.

Physically, the Picard number of an elliptic K3 surface is directly related to
the gauge group formed in F-theory on that K3 surface. There is an equality
known as the Shioda–Tate formula [29, 30, 31] for an elliptic K3 surface with
a global section, which holds between the Picard number, rank of lattice
generated by the components in the singular fibers not meeting the zero
section, and the rank of the Mordell–Weil group. A global section of an
elliptic K3 surface is intuitively a copy of the base $\mathbb{P}^{1}$, and
when an elliptic fibration admits a global section, one can define an
“addition” of any two global sections. The set of global sections under the
“addition” form a group, referred to as the Mordell–Weil group. The rank of
the Mordell–Weil group of an elliptic K3 surface yields the number of U(1)
factors formed in F-theory on that K3 [6]. Physically, the Shioda–Tate formula
implies that for F-theory on an elliptic K3 surface $S$, the sum of the rank
of the non-Abelian gauge group and the number of U(1)s coming from the
Mordell–Weil group is equal to the Picard number $\rho(S)$ minus 2.

This indicates that the sum of the ranks of the non-Abelian gauge groups and
U(1)s increases as the Picard number of K3 surface increases, with the maximum
of the sum being $20-2=18$.

The result in mathematics [50, 51] showed that the complex K3 surfaces of the
highest Picard number 20 are labelled by triplets of integers (modulo an
appropriate group). This comes from the fact that such K3 surfaces are
specified by the “transcendental lattices” of their own [50, 51], the
intersection forms of which are $2\times 2$ integer matrices. In brief, the
complex K3 surfaces of the highest Picard number 20 are labelled by the
intersection forms [50, 51]:

$\begin{pmatrix}2a&b\\\ b&2c\end{pmatrix},$ (10)

where $a,b,c$ are integers. $a\geq c\geq b\geq 0$ can be assumed by using the
$GL_{2}(\mathbb{Z})$ action.

A complex elliptic K3 surface $f:S_{\rm K3}\rightarrow\mathbb{P}^{1}$ of
Picard number 20 with a section is referred to as extremal when the
Mordell–Weil group of the fibration $f$, $MW(S_{\rm K3},f)$, is finite.
F-theory on an extremal fibration has non-Abelian gauge group of rank 18, that
is the highest for 8D F-theory on an elliptic K3 surface.

The complex structures of the K3 surfaces that admit extremal fibrations are
completely classified in [52]. This result can be used to determine the points
in the 8D F-theory moduli with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra, wherein the non-
Abelian gauge algebras are maximally enhanced. In [16], the following were
deduced:

$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}\oplus\mathfrak{su}(3)\oplus\mathfrak{su}(2),\hskip
14.22636pt\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}\oplus\mathfrak{su}(4),\hskip
14.22636pt\mathfrak{e}_{8}^{2}\oplus\mathfrak{su}(3),\hskip
14.22636pt\mathfrak{e}_{8}^{2}\oplus\mathfrak{su}(2)^{2}.$ (11)

Considering the rank of the gauge groups at these points, we can say that the
gauge groups on the dual heterotic strings do not allow for a perturbative
interpretation [16]. As demonstrated in [16], the K3 surfaces at these points
on the dual F-theory are deformations of the stable degenerations that arise
owing to the coincidence of the 7-branes. This can be attributed to the non-
perturbative effects of the insertion of 5-branes from the dual heterotic
perspective [16].

###  3.2 Sample model

We describe the point in the moduli space with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra, where the 8D F-theory
has
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}\oplus\mathfrak{su}(3)\oplus\mathfrak{su}(2)$
gauge algebra as a sample. This model was studied in [16]. Based on the
condition that an elliptic K3 surface with a section yielding this model has
the corresponding singularity type $E_{8}E_{7}A_{2}A_{1}$, the Weierstrass
equation of the K3 surface was deduced in [16] as follows:

$y^{2}=x^{3}+\sqrt[3]{4}t^{3}(\frac{-15t}{4}+3)x+t^{5}(t^{2}+\frac{-5t}{2}+10)$
(12)

with discriminant

$\Delta\sim t^{9}(t-4)^{2}(t+1)^{3}.$ (13)

In [16], the transcendental lattice of the extremal K3 fibration (12) was
deduced to have the following intersection form: $\begin{pmatrix}6&0\\\
0&2\end{pmatrix}$. Equations (12) and (13) imply [16] that the 7-brane stack
supporting the $\mathfrak{e}_{8}$ algebra is located at $t=\infty$ in base
$\mathbb{P}^{1}$; the 7-brane stack supporting the $\mathfrak{e}_{7}$ algebra
is at $t=0$; the 7-brane stack supporting the $\mathfrak{su}(3)$ algebra is at
$t=-1$; the 7-brane stack supporting the $\mathfrak{su}(2)$ algebra is at
$t=4$. In [16], the Weierstrass equation (12) was confirmed to satisfy a
5-brane solution [12] $q=0$ in (8).

To determine the global structure [34] of the gauge group formed in F-theory,
one needs to specify the torsion part of the Mordell–Weil group of an elliptic
fibration on which the theory is compactified. In fact, for the situation
where the BPS solitons form a faithful representation of the gauge group $G$,
then for F-theory on an elliptic fibration the fundamental group,
$\pi_{1}(G)$, of the gauge group is isomorphic to the Mordell–Weil torsion of
that elliptic fibration, as discussed in [34]. The Mordell–Weil torsion of the
K3 surface with the singularity type $E_{8}E_{7}A_{2}A_{1}$ is trivial [52].
Therefore, the global structure of the gauge group formed in F-theory on the
K3 surface (12) is [16] $E_{8}\times E_{7}\times SU(3)\times SU(2)$.

As demonstrated in [16], the elliptic K3 surface (12) arises as a deformation
of sum $X_{[II^{*},1,1]}\cup X_{[III^{*},2,1]}$, wherein a 7-brane over which
type $I_{1}$ fiber lies and 7-brane over which type $I_{2}$ fiber lies
coincide, and two 7-branes over which type $I_{1}$ fibers lie become
coincident. Here, $X_{[II^{*},1,1]}$ and $X_{[III^{*},2,1]}$ represent a
rational elliptic surface with one $II^{*}$ type fiber and two $I_{1}$ type
fibers [53], and a rational elliptic surface with one $III^{*}$ type fiber,
one $I_{2}$ type fiber, and one $I_{1}$ type fiber, respectively. This
indicates that the K3 surface (12) arises as a deformation of stable
degeneration as a result of coincident 7-branes. The effect of the coincident
7-branes and gauge enhancements on the heterotic side corresponds to the
insertion of 5-branes.

###  3.3 Possible extensions

We discuss a few possible directions for extending the study on the
enhancements in gauge groups in the 8D moduli space in [16].

Investigating the maximal enhancements of gauge algebras in the 8D F-theory
moduli with unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra,
where the maximally enhanced gauge algebras are not purely non-Abelian, but
also include U(1) factors may be interesting.

Such points in the moduli space can be found by constructing a moduli space of
elliptically fibered K3 surfaces with a section, whose singularity types
include $E_{8}E_{7}$ with rank 17. A K3 surface at a generic point of this
moduli space has Picard number 19, and the Mordell–Weil rank is zero.
Subsequently, one searches for special points in the moduli space, where the
corresponding K3 surfaces acquire additional independent global sections. The
Mordell–Weil ranks of the K3 surface increase to one at these special points
in the moduli space, and U(1) forms in F-theory on those points. This yields
the construction of F-theory models with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra, wherein the gauge
groups are maximally enhanced with one U(1) factor.

For example, one can construct the moduli space of elliptic K3 surfaces with a
global section with singularity type $E_{8}E_{7}A_{2}$, and one determines
points in this moduli where the Mordell–Weil rank rises to one.
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}\oplus\mathfrak{su}(3)\oplus\mathfrak{u}(1)$
gauge algebra forms at those points in the moduli.

Another potential direction is to analyze points in the moduli where the gauge
algebras are enhanced to rank 17. The lattice-theoretical approach can be used
in this direction to help study models with gauge groups of rank 17.

The lattice called the Néron–Severi lattice plays a role in this approach. The
Néron–Severi lattice is a sublattice of the K3 lattice $\Lambda_{\rm K3}$,
where the K3 lattice $\Lambda_{\rm K3}$ is the second integral cohomology
group $H^{2}(K3,\mathbb{Z})$ with the structure [54]

$\Lambda_{\rm K3}\cong E_{8}(-1)^{2}\oplus\begin{pmatrix}0&1\\\
1&0\end{pmatrix}^{3}.$ (14)

The Néron–Severi lattice, $NS(S)$, contains a considerable amount of geometric
information of a K3 surface, $S$. When there is a lattice embedding of
$H\oplus G$ inside the Néron–Severi lattice, $NS(S)$, where $H$ is the
hyperbolic plane and the image of $H$ inside $NS(S)$ contains a pseudo-ample
class [12], the lattice embedding ensures [50] that the K3 surface has an
elliptic fibration with a section of singularity type $G$.

Therefore, one can determine whether the $\mathfrak{g}$ gauge algebra forms in
F-theory on an elliptic K3 surface with a section by analyzing whether the
Néron–Severi lattice of that K3 surface contains $H\oplus G$ ($H$ in the
Néron–Severi lattice must include a pseudo-ample class), where $G$ is the
corresponding singularity type of the $\mathfrak{g}$ gauge algebra.

According to the classification result of the singularity types of the
elliptic K3 surfaces in [55], the K3 surfaces in the dual 8D F-theory moduli
with unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra with
enhanced non-Abelian gauge groups of rank 17 must have one of the following
three singularity types: $E_{8}^{2}A_{1}$, $E_{8}E_{7}A_{1}^{2}$, and
$E_{8}E_{7}A_{2}$. The heterotic duals of F-theory on K3 surfaces with a
section possessing these singularity types should satisfy at least one of the
two 5-brane solutions (8).

Elliptic K3 surfaces with a section with each of these singularity types
constitute a one-dimensional (1D) complex structure moduli. At special points
in these 1D moduli spaces, either the singularity type is enhanced, resulting
in an enhancement in gauge algebra, or the Mordell–Weil rank increases to one
to form U(1).

##  4 Concluding remarks

In this study, we reviewed the construction of 8D non-geometric heterotic
strings and dual F-theory with unbroken
$\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra and certain relevant
studies. We also discussed related concepts and potential directions for
future extensions. We believe that this article clarifies that previous
studies have made progress in elucidating some aspects of heterotic/F-theory
duality.

The 8D non-geometric heterotic strings and dual F-theory with unbroken
$\mathfrak{e}_{7}\oplus\mathfrak{e}_{7}$ algebra was constructed by the
authors in [23].

Constructions of six-dimensional (6D) non-geometric heterotic strings with
unbroken $\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}$ gauge algebra and dual
F-theory were also discussed in [12]. Related to the extremal fibrations
discussed in section 3.1, 6D F-theory models on elliptically fibered CY
3-folds, constructed by fibering extremal K3 fibrations over $\mathbb{P}^{1}$,
were analyzed in [16, 24].

## Acknowledgments

We would like to thank Shigeru Mukai for discussions.

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