$Q=(1,-1)$. The numerator factor in the modular ansatz is

$\displaystyle\mathcal{N}_{(1,-1)}$
$\displaystyle=\frac{E_{4}E_{6}^{3}\varphi_{2}(\phi)^{6}}{1492992}+\frac{E_{4}^{5}\varphi_{2}(\phi)^{5}\varphi_{0}(\phi)}{1990656}-\frac{E_{4}^{2}E_{6}^{2}\varphi_{2}(\phi)^{5}\varphi_{0}(\phi)}{221184}+\frac{5E_{4}^{3}E_{6}\varphi_{2}(\phi)^{4}\varphi_{0}(\phi)^{2}}{746496}$
$\displaystyle\
+\frac{5E_{6}^{3}\varphi_{2}(\phi)^{4}\varphi_{0}(\phi)^{2}}{1492992}-\frac{5E_{4}^{4}\varphi_{2}(\phi)^{3}\varphi_{0}(\phi)^{3}}{995328}-\frac{25E_{4}E_{6}^{2}\varphi_{2}(\phi)^{3}\varphi_{0}(\phi)^{3}}{2985984}$
(370) $\displaystyle\
+\frac{5E_{4}^{2}E_{6}\varphi_{2}(\phi)^{2}\varphi_{0}(\phi)^{4}}{497664}-\frac{13E_{4}^{3}\varphi_{2}(\phi)\varphi_{0}(\phi)^{5}}{5971968}-\frac{11E_{6}^{2}\varphi_{2}(\phi)\varphi_{0}(\phi)^{5}}{5971968}+\frac{E_{4}E_{6}\varphi_{0}(\phi)^{6}}{1492992}\,.$

The result for $Q=(1,0)$ string is given in Appendix B.2.

####  3.3.5 $B=\mathbb{F}_{1}$ with $SU(2)\times SU(2)$ gauge symmetry

Another interesting example is the supergravity with $T=1$ and $SU(2)^{2}$
gauge symmetry arising from F-theory compactified on an elliptic 3-fold over a
base $B=\mathbb{F}_{1}$ with two $SU(2)$ gauge algebras living on $e$ and $f$
curves in the base, respectively. This theory is anomaly free with 8
fundamentals for the first $SU(2)$, 14 fundamentals for the second $SU(2)$,
one bi-fundamental, and 202 neutral hypermultiplets, together with

$\displaystyle a=(-3,1)\,,\ b_{1}=(0,1)\,,\ b_{2}=(1,-1)\,,\
\Omega=\operatorname{diag}(1,-1)\,.$ (371)

The perturbative partition function of this theory is given by

$\displaystyle Z_{\mathrm{pert}}$
$\displaystyle=\operatorname{PE}\bigg{[}\frac{4y(y+y^{-1})}{(1-y)^{2}}\frac{1}{1-q}+\frac{2y}{(1-y)^{2}}\Big{(}e^{2\pi
i(2\phi_{1}+2\phi_{2})}+2+qe^{-2\pi
i(2\phi_{1}+2\phi_{2})}\Big{)}\frac{1}{1-q}$
$\displaystyle\qquad-\frac{y}{(1-y)^{2}}\Big{(}e^{2\pi
i(\pm\phi_{1}\pm\phi_{2})}+8e^{\pm 2\pi i\phi_{1}}+14e^{\pm 2\pi
i\phi_{2}}+202\Big{)}\frac{1+q}{1-q}\bigg{]}\,,$ (372)

where $\phi_{1}$ and $\phi_{2}$ are two gauge holonomies for $SU(2)\times
SU(2)$ gauge algebras. The first line contains the gravity, tensor and vector
multiplet contributions, and the second line is from the hypermultiplet
contributions. The Calabi-Yau hypersurface can be constructed from the
following toric data:

$\displaystyle\begin{array}[]{rrrr|rrrrr}\lx@intercol\hfil\nabla\hfil\lx@intercol\vrule\lx@intercol&l^{1}&l^{2}&l^{3}&l^{4}&l^{5}\\\
\hline\cr 0&0&0&-1&0&1&0&0&1\\\ 0&0&-1&0&0&0&0&0&1\\\ 1&0&2&3&0&0&1&0&0\\\
-1&0&2&3&-1&-2&1&1&0\\\ 0&-1&2&3&1&0&0&-1&1\\\ -1&1&2&3&1&0&0&0&0\\\
-1&0&1&2&0&2&0&-1&0\\\ 0&-1&1&2&0&0&0&1&-1\\\ 0&0&2&3&-1&1&-2&0&0\end{array}$
(383)

The Hodge numbers and Euler characteristics of the hypersurface are
$(h^{1,1},h^{2,1},\chi)=(5,201,-392)$. The hypersurface equation (50) from the
toric data can be brought into Weierstrass form with singular elliptic fibers
of Kodaira type $I_{2}$ living on $e$ and $f$ curves in the $\mathbb{F}_{1}$
base. The non-zero triple intersection numbers of divisors dual to Mori cone
generators are

$\displaystyle\begin{gathered}\kappa_{122}=2\,,\ \kappa_{123}=4\,,\
\kappa_{124}=4\,,\ \kappa_{125}=1\,,\ \kappa_{133}=6\,,\ \kappa_{134}=6\,,\
\kappa_{135}=2\,,\\\ \kappa_{144}=6\,,\ \kappa_{145}=2\,,\ \kappa_{222}=8\,,\
\kappa_{223}=16\,,\ \kappa_{224}=16\,,\ \kappa_{225}=3\,,\
\kappa_{233}=30\,,\\\ \kappa_{234}=30\,,\ \kappa_{235}=6\,,\
\kappa_{244}=26\,,\ \kappa_{245}=6\,,\ \kappa_{255}=1\,,\ \kappa_{333}=54\,,\
\kappa_{334}=54\,,\\\ \kappa_{335}=12\,,\ \kappa_{344}=48\,,\
\kappa_{345}=12\,,\ \kappa_{355}=2\,,\ \kappa_{444}=42\,,\ \kappa_{445}=10\,,\
\kappa_{455}=2\,.\end{gathered}$ (388)

We can compute the genus zero GV-invariants from the toric data, and identify
2-cycles in $\mathbb{F}_{1}$ and the Kähler parameters associated to
$l^{\alpha}$ as follows: $l^{1}$ and $l^{3}$ are identified with $e$ and $f$
curves, and $\\{l^{2},l^{4},l^{5}\\}$ are related to the 2-cycles with volumes
$\\{\tau-2\phi_{1},\phi_{1}-\phi_{2},\phi_{2}\\}$.

Let us now compute the elliptic genera of BPS strings using the modular ansatz
and the mirror results. The BPS string with tensor charge $Q=(k_{1},k_{2})$
has the following central charges:

$\displaystyle\begin{aligned}
&c_{L}=k_{1}^{2}-k_{2}^{2}+9(3k_{1}+k_{2})+6\,,\quad&&c_{R}=k_{1}^{2}-k_{2}^{2}+3(3k_{1}+k_{2})+6\,,\\\
&k_{l}=\frac{1}{2}(k_{1}^{2}+k_{2}^{2}-3k_{1}-k_{2})\,,\quad&&k_{SU(2)_{1}}=-k_{2}\,,\qquad
k_{SU(2)_{2}}=k_{1}+k_{2}\,,\end{aligned}$ (389)

with $k_{1}\geq 0$ and $k_{2}\geq-k_{1}$. We can write the modular ansatz for
the elliptic genus of this string as

$\displaystyle\begin{aligned}
Z_{(k_{1},k_{2})}&=\frac{1}{\eta(\tau)^{12(3k_{1}+k_{2})}}\frac{\mathcal{N}_{(k_{1},k_{2})}(\tau,\epsilon,\phi)}{\prod_{s_{1}=1}^{k_{1}}\varphi_{2}(\tau,s_{1}\epsilon)\cdot\prod_{s_{2}=1}^{k_{1}+k_{2}}\varphi_{2}(\tau,s_{2}\epsilon)}\\\
&\qquad\cdot\frac{1}{\prod_{i=1}^{2}\prod_{s_{i}=1}^{\kappa_{i}}\prod_{l=0}^{s_{i}-1}\varphi_{2}(\tau,(s_{i}-1-2l)\epsilon+2\phi_{i})}\,,\end{aligned}$
(390)

with $\kappa_{1}=k_{1}+k_{2}$ and $\kappa_{2}=k_{1}$.

For the string with charge $Q=(0,1)$, its elliptic genus has indices
$\\{-1,-1,1\\}$ for the $SU(2)$ symmetries with holonomies
$\\{\epsilon,\phi_{1},\phi_{2}\\}$, respectively. Using this together with the
genus zero GV-invariants computed from the toric data, we can fix the
numerator $\mathcal{N}_{(0,1)}$ in the modular ansatz as

$\displaystyle\mathcal{N}_{(0,1)}$
$\displaystyle=-\frac{E_{4}E_{6}\varphi_{2}(\phi_{1})^{3}\varphi_{2}(\phi_{2})}{216}+\frac{E_{4}^{2}\varphi_{2}(\phi_{1})^{2}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})}{432}+\frac{E_{6}\varphi_{2}(\phi_{1})\varphi_{0}(\phi_{1})^{2}\varphi_{2}(\phi_{2})}{432}$
$\displaystyle+\frac{E_{4}^{2}\varphi_{2}(\phi_{1})^{3}\varphi_{0}(\phi_{2})}{216}-\frac{E_{6}\varphi_{2}(\phi_{1})^{2}\varphi_{0}(\phi_{1})\varphi_{0}(\phi_{2})}{432}-\frac{E_{4}\varphi_{2}(\phi_{1})\varphi_{0}(\phi_{1})^{2}\varphi_{0}(\phi_{2})}{432}.$
(391)

For the string with $Q=(1,-1)$, its elliptic genus has indices $\\{-1,1,0\\}$
for $\\{\epsilon,\phi_{1},\phi_{2}\\}$. We again fix the numerator in the
modular ansatz using the genus zero GV-invariants. The result is given by

$\displaystyle\begin{aligned}
\mathcal{N}_{(1,-1)}&=-\frac{13E_{4}^{3}E_{6}\varphi_{2}(\phi_{1})\varphi_{2}(\phi_{2})^{4}}{31104}-\frac{11E_{6}^{3}\varphi_{2}(\phi_{1})\varphi_{2}(\phi_{2})^{4}}{31104}+\frac{E_{4}^{4}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})^{4}}{7776}\\\
&\quad+\frac{5E_{4}E_{6}^{2}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})^{4}}{7776}+\frac{31E_{4}^{4}\varphi_{2}(\phi_{1})\varphi_{2}(\phi_{2})^{3}\varphi_{0}(\phi_{2})}{62208}\\\
&\quad+\frac{41E_{4}E_{6}^{2}\varphi_{2}(\phi_{1})\varphi_{2}(\phi_{2})^{3}\varphi_{0}(\phi_{2})}{62208}-\frac{E_{4}^{2}E_{6}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})^{3}\varphi_{0}(\phi_{2})}{864}\\\
&\quad-\frac{E_{4}^{3}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})^{2}\varphi_{0}(\phi_{2})^{2}}{31104}+\frac{E_{6}^{2}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})^{2}\varphi_{0}(\phi_{2})^{2}}{31104}\\\
&\quad-\frac{11E_{4}^{3}\varphi_{2}(\phi_{1})\varphi_{2}(\phi_{2})\varphi_{0}(\phi_{2})^{3}}{62208}-\frac{13E_{6}^{2}\varphi_{2}(\phi_{1})\varphi_{2}(\phi_{2})\varphi_{0}(\phi_{2})^{3}}{62208}\\\
&\quad+\frac{E_{4}E_{6}\varphi_{0}(\phi_{1})\varphi_{2}(\phi_{2})\varphi_{0}(\phi_{2})^{3}}{2592}\,.\end{aligned}$
(392)

We note that the elliptic genera of strings with charge $Q=(k,-k)$ are not
generated by the Hecke transformation of the elliptic genus of a single string
coming from a D3 wrapped on a fiber curve $f$. Thus, this is not a heterotic
string in this case, which is expected since there is an $SU(2)$ gauge
symmetry supported on the fiber curve. Some higher string elliptic genera are
summarized in Appendix B.3.

##  4 Emergent Strings and Swampland Conjectures

We shall now discuss the Swampland Conjectures associated with BPS strings. In
previous studies Lee:2018urn ; Lee:2019xtm , the authors performed a
comprehensive analysis of weak coupling limits in the moduli space of F-theory
compactifications to six dimensions. This analysis gave rise to the Emergent
String Conjecture, which states that every infinite-distance limit in the
moduli space is either a decompactification limit, where a tower of Kaluza-
Klein excitations become light, or an emergent string limit, characterized by
the emergence of an asymptotically tensionless critical string. In the limit,
the effective field theory breaks down due to a tower of infinitely many light
states, which is in accordance with the Swampland Distance Conjecture proposed
in Ooguri:2006in .

The Emergent String Conjecture has been rigorously tested for F-theory models
also in Lee:2018urn ; Lee:2019xtm ; Lee:2019wij . Specifically, when
considering the weak coupling limits of 2-form gauge fields while keeping the
gravitational coupling finite, these limits lie at infinite distance in the
tensor moduli space. In such a limit, there is always a shrinking rational
cycle $C_{0}$ with zero self-intersection in the Kähler base of the elliptic
fibration. Moreover, when a D3-brane wraps around this curve, it provides a
weakly-coupled tensionless string. By F-theory-heterotic duality, this string
is identified with the critical heterotic string compactified on a certain K3
surface.

In the presence of an Abelian gauge symmetry, the limit under consideration
corresponds to a weak gauge coupling limit. The emergent critical string in
the limit is always charged under the gauge symmetry. The study of the
elliptic genus of this heterotic string at weak gauge coupling limit plays a
crucial role in proving the sublattice Weak Gravity Conjecture (sWGC),
proposed in Heidenreich:2015nta ; Heidenreich:2016aqi , with respect to the
gauge symmetry. More precisely, as we discussed above, the elliptic genera in
the worldsheet CFTs encode the BPS protected states of strings interacting
with the bulk supergravity theory in six-dimensions. These elliptic genera
take the form of meromorphic Jacobi forms, and their charge lattice structure
is significantly constrained by modular invariance. For the emergent heterotic
string, its elliptic genus captures (non-BPS) charged particle excitations in
the perturbative heterotic string regime, and the masses of these particle
states become exponentially light at infinite distance in moduli space. By
utilizing the theory of weak Jacobi forms applied to the elliptic genus of the
heterotic string, we can explicitly determine both the charges and the masses
of these particles. Consequently, this enables us to concretely identify
superextremal particle states that populate a sublattice of the charge
lattice. This study, in turn, provides a mathematical proof of the sublattice
Weak Gravity Conjecture for 6d F-theory on elliptically fibered Calabi-Yau
3-folds with an Abelian gauge symmetry Lee:2018urn .

In this section, we will investigate strings and their BPS spectra at infinite
distance in the moduli space of specific 6-dimensional supergravity theories
that do not have an F-theory realization, and provide evidences to show that
these theories indeed satisfy the Swampland Conjectures. Additionally, we will
discuss an interesting infinite-distance limit in the moduli space where the
asymptotically tensionless string carrying a charge under the gauge symmetry
is not a critical string but rather a non-critical little string. Such little
strings are characteristic objects of 6d little string theories that arise in
the gravity decoupling limit.

###  4.1 Swampland conjectures in non-geometric theories

Let us consider the theory with $T=1$ and $SU(2)$ gauge group coupled to one
triple-symmetric and 84 fundamental hypermultiplets, i.e. $1\times{\bf
4}+84\times{\bf 2}$ hypermultiplets. This theory satisfies the anomaly
cancellation conditions in (2.1) with

$\displaystyle a=(-3,1)\ ,\quad b=(5,0)\ ,\quad\Omega={\rm diag}(1,-1)\ ,$
(393)

and $75$ neutral hypermultiplets, where $b$ is the charge vector supporting
the $SU(2)$ gauge algebra.

We will argue that a geometric realization of this theory in F-theory is not
possible. First, suppose that the theory has a geometric construction. In such
a case, it can be represented by an elliptic CY 3-fold over a Hirzebruch
surface $\mathbb{F}_{1}$ with $SU(2)$ gauge symmetry on a curve $b=5h$ with
$h\subset\mathbb{F}_{1}$ and $h^{2}=1$. It is important to note that a
geometric transition always exists from $\mathbb{F}_{1}$ to $\mathbb{P}^{2}$
by blowing down the exceptional curve in $\mathbb{F}_{1}$. This transition
results in a change in the field theory description from one tensor to 29
neutral hypermultiplets without affecting the gauge group and the charged
hypermultiplets. After this transition, we arrive at an F-theory model on a
base $\mathbb{P}^{2}$ with an $SU(2)$ gauge symmetry supported on a degree 5
curve within $\mathbb{P}^{2}$, i.e. $b=5\ell$ with $\ell\subset\mathbb{P}^{2}$
and $\ell^{2}=1$. This F-theory model is coupled to $1\times{\bf
4}+84\times{\bf 2}$ hypermultiplets.

However, it was shown in Klevers:2017aku  that a half hypermultiplet in the
triple-symmetric representation of $SU(2)$ can be realized in F-theory only
when the gauge group lives on a divisor with a triple point singularity, and a
degree 5 curve in $\mathbb{P}^{2}$ can have at most one triple point
singularity. This implies that the theory under consideration cannot
accommodate one full hypermultiplet in the triple-symmetric representation
which requires two triple point singularities.444A potential loophole in this
argument was presented in Raghuram:2020vxm . The quintic curve on
$\mathbb{P}^{2}$ that supports the triple point singularities actually reduces
to the union of a line and a quartic curve. When this happens, the $I_{2}$
singularity along the curve leads to an enhanced gauge algebra
$\mathfrak{su}(2)\times\mathfrak{su}(2)$. Therefore, the geometric model that
realizes the $\mathfrak{su}(2)$ gauge algebra on $b=5\ell$ with a full triple
symmetric hypermultiplet exhibits an automatic symmetry enhancement to the
$\mathfrak{su}(2)\times\mathfrak{su}(2)$ gauge algebra. Therefore, we can
conclude that the original theory with $T=1$ and one full hypermultiplet in
the triple-symmetric representation of $SU(2)$ gauge group cannot have a
geometric realization in F-theory, and thus it is a non-geometric theory.

Now, we will examine several Swampland Conjectures in this non-geometric
theory. To do so, we will investigate infinite distance limits in the moduli
space of the tensor branch. We first note that this theory can be Higgsed to a
geometric theory realized in F-theory on $\mathbb{F}_{1}$ without gauge
symmetry and charged matters. The tensor moduli space in the Higgsed theory is
described by the geometry of the Hirzebruch surface $\mathbb{F}_{1}$. The BPS
strings in the 6d theory come from D3-branes wrapped around holomorphic curves
in the base $\mathbb{F}_{1}$. The charge lattice of such BPS strings are
generated by two basis curves $e=(0,1)$ with $e^{2}=-1$ and $f=(1,-1)$ with
$f^{2}=0$. The only infinite distance limit in the moduli space of
$\mathbb{F}_{1}$, while keeping the gravity dynamical with the condition
$M^{4}_{\rm Pl}=\frac{1}{2}J\cdot J=1$, is

$\displaystyle J\cdot h=J\cdot(1,0)\ \sim\ t\ ,\qquad J\cdot f=J\cdot(1,-1)\
\sim\ t^{-1}\ ,\qquad t\ \rightarrow\ \infty,$ (394)

where $h=(1,0)$ denotes the base curve with $h^{2}=1$. In this limit, the
volume of a fiber curve vanishes, i.e. ${\rm vol}(f)\sim J\cdot f\rightarrow
0$, and thus the D3-brane wrapping this fiber curve becomes tensionless and
weakly coupled, which agrees with the Emergent String Conjecture. This
asymptotically tensionless string turns out to be the critical heterotic
string compactified on a K3 surface in a duality frame, as investigated in
Lee:2018urn .

As discussed in Tarazi:2021duw , the Higgsing process does not affect the
tensor multiplets and the string charge lattice $\Gamma$ because it involves
only the vector multiplet and the hypermultiplets. This implies that the
moduli space of the tensor branch and the charge lattice of BPS strings remain
unchanged both before and after the Higgsing procedure. This also means that
the charge lattice of BPS strings in the non-geometric theory, before
Higgsing, is spanned by the charges $(0,1)$ and $(1,-1)$. Thus, the original
non-geometric theory exhibits the same infinite distance limit described in
(394). Notably, this limit corresponds to a weak gauge coupling limit

$\displaystyle\frac{1}{g_{{\rm YM}}^{2}}=J\cdot b=J\cdot(5,0)\ \rightarrow\
\infty\ .$ (395)

The Emergent String Conjecture predicts an asymptotically tensionless string
in this weak coupling limit Lee:2019wij . Indeed, the string with tensor
charge $Q=(1,-1)$, with its tension expressed as $T_{(1,-1)}=J\cdot Q\sim
t^{-1}$, becomes tensionless in the limit with (394). This tensionless string
is evidently present in the string charge lattice $\Gamma$ in the non-
geometric theory, because it exists in the Higgsed theory and the process of
Higgsing does not affect the charge lattice.

We now assert that the tensionless string arising in the weak coupling limit
(394) of the non-geometric theory is, in fact, the heterotic string on a K3
surface. The evidences for this claim are as follows. Firstly, the central
charges of the tensionless string match those of the heterotic string.
Specifically, the string with charge $Q=(1,-1)$ has

$\displaystyle Q\cdot Q=0\ ,\quad c_{L}=24\ ,\quad c_{R}=12\ ,\quad k_{l}=-1\
,$ (396)

which precisely agree with the properties of the heterotic string. Secondly,
this string after the Higgsing reduces to the string coming from a D3-brane
wrapping a fiber curve $f$ in $\mathbb{F}_{1}$. This is indeed a weakly
coupled, perturbative heterotic string, as discussed above. Lastly, as we will
show shortly, the elliptic genera for $n$ strings with charge $nQ=(n,-n)$, can
be derived from the Hecke transformation of the elliptic genus for a single
string with $n=1$. This is a distinguished feature of the critical heterotic
strings, reflecting the fact that they do not form bound states
Dijkgraaf:1996xw . Altogether, these evidences strongly support our claim that
the string with charge $(1,-1)$ corresponds to the heterotic string in 6d.

The last evidence can be easily proven as follows. We first prove that the
solution to the modular ansatz of the form (31) for the elliptic genus of $n$
strings with charge $nQ=(n,-n)$ is unique by employing the argument given in
Sec. 5.1.1 in Gu:2017ccq . Let us define $\mathcal{Z}_{kQ}$ as

$\displaystyle
1+\sum_{k=1}^{\infty}w^{k}Z_{kQ}=\operatorname{PE}\left[\sum_{k=1}^{\infty}w^{k}\mathcal{Z}_{kQ}\right]\,.$
(397)

Suppose that the elliptic genera of charges $kQ=(k,-k)$ with $k<n$ are fully
fixed, and that there are two distinct sets of coefficients $\\{c_{i}\\}$ and
$\\{\tilde{c}_{i}\\}$ in the modular ansatz (33) of $Z_{nQ}$ which satisfy the
GV-invariant condition (42). The difference between two modular ansätze

$\displaystyle
Z_{nQ}(c_{i})-Z_{nQ}(\tilde{c}_{i})=\mathcal{Z}_{nQ}(c_{i})-\mathcal{Z}_{nQ}(\tilde{c}_{i})$
(398)

is of the form (31) that has the same modular properties with $Z_{nQ}$, and
can be expressed as the form within the plethystic exponential in (42). The
‘lemma’ in Gu:2017ccq  is that Jacobi forms which have such properties can
have at most a single $\varphi_{-2,1}(\tau,\epsilon)$ factor in the
denominator. Thus, for a weak Jacobi form $\mathcal{N}_{nQ}^{\mathrm{diff}}$,
the difference of two ansätze is given by

$\displaystyle
Z_{nQ}(c_{i})-Z_{nQ}(\tilde{c}_{i})=\frac{1}{\eta(\tau)^{24n}}\frac{\mathcal{N}_{nQ}^{\mathrm{diff}}(\tau,\epsilon,\phi)}{\varphi_{2}(\tau,\epsilon)}\,.$
(399)

Now note that the elliptic genus $Z_{nQ}$ has an index $-n$ with respect to
$\epsilon$, so $\mathcal{N}_{nQ}^{\mathrm{diff}}$ has an index $-n+1$ with
respect to $\epsilon$. However, since there is no weak Jacobi form of a
negative index, $\mathcal{N}_{nQ}^{\mathrm{diff}}$ always vanishes if $n>1$.
This implies that GV-invariant condition uniquely fixes all the unknown
coefficients in the modular ansatz of $Z_{nQ}$ when $n>1$. Using this
uniqueness, we conclude that the elliptic genus for the strings with charge
$nQ$ is given by the Hecke transformation of $Z_{Q}$ as

$\displaystyle
1+\sum_{n=1}^{\infty}w^{n}Z_{nQ}(\tau,\epsilon,\phi)=\exp(\sum_{n=1}^{\infty}\frac{w^{n}}{n}\sum_{ad=n}\sum_{b(\operatorname{mod}d)}Z_{Q}\Big{(}\frac{a\tau+b}{d},a\epsilon,a\phi\Big{)})\,,$
(400)

where $Q=(1,-1)$. The reasons for this are as follows. First, $Z_{nQ}$
obtained using this equation exhibits correct modular properties: it has
weight zero, index $-n$ for $\epsilon$ and $5n$ for $\phi$. This is a property
of the Hecke transformation given in Appendix A. More detailed study for the
Hecke transformation is given in Eichler:1995 . Second, it is known that the
partition function of the heterotic string is generated by the Hecke
transformation Dijkgraaf:1996xw , so $Z_{nQ}$ from (400) is an elliptic genus
of a heterotic string. In other words, it has the same denominator structure
as (31) and satisfy the GV-invariant condition in (42). Thus, $Z_{nQ}$ from
(400) is a candidate for the elliptic genus we are interested in. Due to the
uniqueness of the modular ansatz, it must be the correct answer for the
elliptic genus of the strings with charge $nQ=(n,-n)$. This suggests that
strings with charge $Q=(1,-1)$ in our non-geometric theory is a heterotic
string.

Let us turn to the computation of the elliptic genus of the string with charge
$Q=(1,-1)$. This string has the central charges given in (396) with
$k_{SU(2)}=5$. Thus we can take the modular ansatz for the elliptic genus as

$\displaystyle
Z_{(1,-1)}^{b=(5,0)}(\tau,\epsilon,\phi)=\frac{\mathcal{N}_{(10,5)}(\tau,\phi)}{\eta(\tau)^{24}\varphi_{2}(\tau,\epsilon)}\
.$ (401)

The numerator $\mathcal{N}_{(10,5)}$ is a weak Jacobi form of weight 10 and
index 5 for the $SU(2)$ gauge group. It can be expressed as a polynomial in
the modular forms $E_{4}(\tau),E_{6}(\tau),\varphi_{2}(\tau,\phi)$, and
$\varphi_{0}(\tau,\phi)$ with 10 unknown coefficients. To narrow down these
coefficients, we can impose the following two conditions. First, the ground
state in the elliptic genus is an $SU(2)$ singlet. Second, this elliptic genus
should reduce to the elliptic genus of the heterotic string on K3 without any
gauge symmetry after the Higgsing. By enforcing these conditions, we can
determine 6 coefficients. In order to determine the remaining 4 coefficients
in the modular ansatz, we will use the claim discussed above, which asserts
that this string in the weak gauge coupling is a perturbative heterotic
string.

The elliptic genus counts left-moving excitations of the strings, and the
excitation level $\ell$ is captured by the power of $q$ in the $q$-expansion.
In the case of the heterotic string, the left-moving string excitations, along
with right-moving pairs by level-matching, provide particle states in the bulk
6d theory. In particular, the states at $\ell=0$, corresponding to the
$q^{0}$-th order, generate the massless sector in the 6d theory. It then
follows that the elliptic genus for the string of $Q=(1,-1)$ at $q^{0}$-th
order, denoted as $n_{0}$, must capture the chiral index of the massless
matter content in the 6d theory Lee:2018urn  as

$\displaystyle n_{0}|_{\mathfrak{q}}=2(V+3+T-H)|_{\mathfrak{q}}\ ,$ (402)

where the subscript $\mathfrak{q}$ indicates the restriction to the states
with a specific charge $\mathfrak{q}$. This condition imposes significant
constraints on the elliptic genus for the heterotic string. All the remaining
coefficients in the modular ansatz are determined by this condition.
Remarkably, the conditions for $\mathfrak{q}\geq 2$ are sufficient to fix all
the unknown coefficients, and as a result, the conditions at $\mathfrak{q}=0$
and $\mathfrak{q}=1$ are automatically satisfied.

The result after fixing the coefficients is

$\displaystyle\mathcal{N}_{(10,5)}(\tau,\phi)=$
$\displaystyle-\frac{7E_{4}^{5}\varphi_{2}(\phi)^{5}+E_{4}^{2}E_{6}^{2}\varphi_{2}(\phi)^{4}}{995328}-\frac{19E_{4}^{4}\varphi_{2}(\phi)^{3}\varphi_{0}(\phi)^{2}}{497664}+\frac{E_{4}^{3}E_{6}\varphi_{2}(\phi)^{4}\varphi_{0}(\phi)}{23328}$
$\displaystyle-\frac{(65E_{4}^{3}+55E_{6}^{2})\varphi_{2}(\phi)\varphi_{0}(\phi)^{4}}{E_{4}^{4}\varphi_{2}(\phi)^{3}\varphi_{0}(\phi)^{2}}+\frac{5E_{4}^{2}E_{6}\varphi_{2}(\phi)^{2}\varphi_{0}(\phi)^{3}}{62208}$
$\displaystyle-\frac{7E_{4}E_{6}^{2}\varphi_{2}(\phi)^{3}\varphi_{0}(\phi)^{2}}{62208}+\frac{E_{4}E_{6}\varphi_{0}(\phi)^{5}}{124416}-\frac{E_{6}^{3}\varphi_{2}(\phi)^{4}\varphi_{0}(\phi)}{373248}\
.$ (403)

Plugging this into the modular ansatz, we compute the elliptic genus which can
be expanded as

$\displaystyle-$ $\displaystyle
Z_{(1,-1)}^{b=(5,0)}(\tau,\epsilon,\phi)\varphi_{-2,1}(\tau,\epsilon)=-2q^{-1}+(140+170\xi^{\pm
1}-2\xi^{\pm 2}+2\xi^{\pm 3})q^{0}$ (404)
$\displaystyle\qquad\qquad+\left(77336+62752\xi^{\pm 1}+30604\xi^{\pm
2}+8456\xi^{\pm 3}+908\xi^{\pm 4}+56\xi^{\pm 5}\right)q+\mathcal{O}(q^{2})\ ,$

where $\xi=e^{2\pi i\phi}$ and $\xi^{\pm n}\equiv\xi^{n}+\xi^{-n}$. So we can
uniquely determine the elliptic genus for the string with charge $(1,-1)$,
which is consistent with the structure of the critical heterotic string on K3.

The left-moving excitations captured by the elliptic genus, when properly
paired with right-moving excitations, realize actual charged (but non-BPS)
particles states in the 6d supergravity. In the tensionless limit, these
states provide a tower of asymptotically massless charged states. This results
in the break down of the low-energy effective theory, which is expected since
the tensionless limit lies at infinite distance in the moduli space.

In Lee:2018urn ; Lee:2018spm , the Sublattice Weak Gravity Conjecture (sLWGC)
has been rigorously verified in F-theory on elliptically fibered Calabi-Yau
3-folds by examining the spectrum of charged states originating from the
emergent heterotic string at the weak gauge coupling limit. The sLWGC
postulates that any $U(1)$ gauge theory coupled to gravity must have a
sublattice in the charge lattice such that for any site in the sublattice,
there exists a superextremal particle-like state satisfying the condition that
its charge-to-mass ratio surpasses that of a extremal Reissner-Nordström black
hole Heidenreich:2016aqi . This is a refined version of the Weak Gravity
Conjecture first proposed in Arkani-Hamed:2006emk . The WGC for 6d
$\mathcal{N}=(1,0)$ supergravity, accounting for contributions from massless
scalar fields, leads to the extremality bound Lee:2018urn ; Lee:2018spm ,

$\displaystyle\mathfrak{q}^{2}\,g_{\rm YM}^{2}\geq\frac{M^{2}}{M_{\rm
pl}^{4}}\ .$ (405)

Using the modular properties of the elliptic genus and the theory of Jacobi
forms, we can prove sLWGC for 6d F-theory models with Abelian gauge group
factors. For this, we first note the fact that any weak Jacobi form of weight
$w$ and index $m$ can be expanded as Eichler:1995

$\displaystyle\varphi_{w,m}(\tau,\phi)=\sum_{\ell\in\mathbb{Z}\,{\rm
mod}\,2m}h_{\ell}(\tau)\theta_{m,\ell}(\tau,\phi)\ ,$ (406)

where $h_{\ell}(\tau)$ represents a vector-valued weakly-holomorphic modular
form of weight $w-1/2$ and

$\displaystyle\theta_{m,\ell}(\tau,\phi)=\sum_{k\in\mathbb{Z}}q^{(\ell+2mk)^{2}/4m}\xi^{\ell+2mk}\
$ (407)

is the theta function of index $m$ and characteristic $\ell$. The numerator of
the elliptic genus of the heterotic string, as demonstrated in equation (401)
for our specific case, takes the form of a weak Jacobi form, and therefore it
admits an expansion in terms of the theta functions. The sector at $\ell=0$ is
of particular interest. This specific sector is particularly intriguing as it
pertains to the sublattice associated with the left-moving ground state, which
corresponds to the leading coefficient in the expansion
$h_{0}(\tau)=2+\mathcal{O}(q)$. So it always has non-trivial contribution to
the elliptic genus and thus to the particle spectrum in the 6d supergravity.
The excitation level and the charge for the states in this sublattice can be
easily extracted from the theta function at $\ell=0$, which are

$\displaystyle n(k)=mk^{2}\ ,\quad\mathfrak{q}_{k}=2mk\ ,\quad k\in\mathbb{Z}\
.$ (408)

The charged states arising from the string states populating in this
sublattice turn out to be the superextremal particles satisfying the
extremality bound as Lee:2018urn

$\displaystyle\mathfrak{q}_{k}^{2}=4mn(k)=4m\left(\frac{M_{k}^{2}}{4mg_{\rm
YM}^{2}}+1\right)>\frac{M_{k}^{2}}{g_{\rm YM}^{2}}\ ,$ (409)

with the normalization $M_{\rm Pl}=1$ at a weak gauge coupling limit, e.g.
$t\rightarrow\infty$. Here, we used the mass formula for the excitations of
the heterotic string, $M_{k}^{2}=4T(n(k)-1)$. This provides a proof for the
Sublattice Weak Gravity Conjecture in the context of F-theory compactification
on elliptic Calabi-Yau 3-folds with a single $U(1)$ factor. Also, the
analogous proof for F-theory constructions involving multiple Abelian factors
is outlined in Lee:2018spm .

The extension of this proof to non-Abelian gauge theories can be carried out
by employing the same idea. The sLWGC for non-Abelian gauge fields is a
stronger conjecture compared to the conventional Weak Gravity Conjecture
mentioned earlier. It asserts that any gauge theory in dimensions $d\geq 5$
with a non-Abelian gauge group $G$ coupled to gravity must have a finite-index
Weyl-invariant sublattice $\Gamma_{0}$ of the weight lattice $\Gamma_{G}$ such
that for every dominant weight ${\bf\lambda}_{\bf R}\in\Gamma_{0}$, there
exists a superextremal particle-like state transforming in the $G$ irrep ${\bf
R}$ with highest weight ${\bf\lambda}_{\bf R}$ Heidenreich:2017sim . The proof
of this conjecture for F-theory compactified on elliptic or genus one fibered
Calabi-Yau 3-folds when it has a tensionless charged heterotic string at weak
gauge coupling limit was presented in Cota:2020zse , which was achieved once
again by utilizing the modular properties of Jacobi forms appearing in the
elliptic genus of the heterotic string.

These proofs for the sLWGC can also be applied to the non-geometric theory,
specifically in the case where $T=1$ and $SU(2)$ gauge group on $b=(5,0)$ is
coupled to $1\times{\bf 4}+84\times{\bf 2}$ hypermultiplets. The elliptic
genus of the string with charge $(1,-1)$ in this theory can be expressed as
(401) and the numerator given in (4.1) is a weak Jacobi form. Therefore, by
relying on our claim that this string becomes a heterotic string at weak gauge
coupling limit and using the property of weak Jacobi forms in (406), we can
demonstrate that this supergravity theory has the superextremal particles,
which arise from the tensionless heterotic string, satisfying the extremality
condition555The translation from the bound in (405) to the one in (409) was
performed for F-theory models in Lee:2018urn ; Lee:2018spm . The same bound
can be obtained for the theories in this subsection by substituting the Kähler
form $J$ and the curve classes $C,C_{0}$ in their F-theory models by those in
our theories as $J_{\rm theirs}=J_{\rm ours}$, $C=b$, and $C_{0}=(1,-1)$.
(409), and thus satisfies the (non-Abelian) Sublattice Weak Gravity
Conjecture. As this theory also satisfies the other Swampland Conjectures like
the Distance Conjecture and the Emergent String Conjecture, it seems plausible
that this field theory can be consistently coupled to quantum gravity, and
thus it resides outside the Swampland.

Another interesting supergravity theory is the $SU(2)$ gauge theory with $T=1$
and $\frac{3}{2}\times{\bf 4}+1\times{\bf 3}+93\times{\bf 2}+52\times{\bf 1}$
hypermultiplets. This is an anomaly free theory with the anomaly vectors and
the intersection form

$\displaystyle a=(-3,1)\ ,\quad b=(6,0)\ ,\quad\Omega={\rm diag}(1,-1)\ .$
(410)

In contrast to the previous example involving $SU(2)$ gauge group on
$b=(5,0)$, we cannot rule out the possibility of this theory being realized in
F-theory. The realization of this theory demands three triple point
singularities on the curve $b=(6,0)$ to accommodate three triple symmetric
half-hypermultiplets. This configuration can be achieved in the F-theory
framework Klevers:2017aku . However, the explicit construction of the
corresponding elliptic Calabi-Yau 3-fold is unknown at present. Thus the
geometric method, such as the mirror technique, to investigate this theory is
not currently feasible. Nonetheless, the technique developed in this
subsection can still be employed to carry out concrete tests for various
Swampland Conjectures in this theory.

The moduli space of this theory coincides with that of the previous theory
described in (393), and the charge lattice of BPS strings is also spanned by
the basis vectors $(0,1)$ and $(1,-1)$. The weak gauge coupling limit, while
gravity is kept dynamical, is located at infinite distance in the moduli space

$\displaystyle\frac{1}{g^{2}_{\rm YM}}=J\cdot b=J\cdot(6,0)\ \rightarrow\
\infty\ .$ (411)

In this limit, the string with the tensor charge $(1,-1)$ again becomes
tensionless. The existence of this string is ensured by its presence in the
Higgsed theory, which corresponds to the F-theory model on an elliptic 3-fold
over a base $\mathbb{F}_{1}$. Based on the same reasons accounted for proving
the emergence of the heterotic string in the previous example, which relies on
its central charges, Higgsing, and special properties of its elliptic genus,
we claim that this string is also the heterotic string on a K3. This assertion
enables us to calculate the elliptic genus of this tensionless string.

The central charges of the string with charge $(1,-1)$ are given by

$\displaystyle Q\cdot Q=0\ ,\quad c_{L}=24\ ,\quad c_{R}=12\ ,\quad k_{l}=-1\
,\quad k_{SU(2)}=6\ .$ (412)

From this, we can take the following ansatz for the elliptic genus of this
string:

$\displaystyle
Z_{(1,-1)}^{b=(6,0)}(\tau,\epsilon,\phi)=\frac{\mathcal{N}_{(10,6)}(\tau,\phi)}{\eta(\tau)^{24}\varphi_{-2,1}(\tau,\epsilon)}\
.$ (413)

The numerator $\mathcal{N}_{(10,6)}$ is now a weak Jacobi form of weight $10$
and index $5$ for the $SU(2)$ gauge group. It contains 12 unknown coefficients
when expressed in terms of the basis modular forms. By using the properties of
the heterotic string at $q^{-1}$ and $q^{0}$ orders in the $q$-expansion, we
can uniquely fix the numerator,

$\displaystyle\mathcal{N}_{(10,6)}(\tau,\phi)=$
$\displaystyle-\frac{(5E_{4}^{4}+3E_{6}^{2})E_{4}^{2}\varphi_{2}(\phi)^{5}\varphi_{0}(\phi)}{1990656}+\frac{(6E_{4}^{4}E_{6}-2E_{4}E_{6}^{3})\varphi_{2}(\phi)^{6}}{5971968}$
(414)
$\displaystyle-\frac{(13E_{4}^{3}+11E_{6}^{2})\varphi_{2}(\phi)\varphi_{0}(\phi)^{5}}{5971968}-\frac{(9E_{4}^{3}+11E_{6}^{2})E_{4}\varphi_{2}(\phi)^{3}\varphi_{0}(\phi)^{3}}{1492992}$
$\displaystyle+\frac{(29E_{4}^{3}+E_{6}^{2})E_{6}\varphi_{2}(\phi)^{4}\varphi_{0}(\phi)^{2}}{2985984}+\frac{5E_{4}^{2}E_{6}\varphi_{2}(\phi)^{2}\varphi_{0}(\phi)^{4}}{497664}+\frac{E_{4}E_{6}\varphi_{0}(\phi)^{6}}{1492992}\
.$

Substituting this numerator into the modular ansatz, the elliptic genus is
expanded as

$\displaystyle-$ $\displaystyle
Z_{(1,-1)}^{b=(6,0)}(\tau,\epsilon,\phi)\varphi_{-2,1}(\tau,\epsilon)=-2q^{-1}+(96+189\xi^{\pm
1}+3\xi^{\pm 3})q^{0}$ (415) $\displaystyle\quad+\left(69156+60858\xi^{\pm
1}+31752\xi^{\pm 2}+12169\xi^{\pm 3}+1890\xi^{\pm 4}+189\xi^{\pm 5}+8\xi^{\pm
6}\right)q+\mathcal{O}(q^{2})\,.$

The result is consistent with the structure of the elliptic genus of the
heterotic string on K3: all coefficients present in the $q$-expansion are
integers, the chiral index for the massless states are correctly captured, and
when $\xi$ is set to $1$, this is Higgsed to the elliptic genus of the
heterotic string associated to a rational fiber curve in a base
$\mathbb{F}_{1}$ in an F-theory.

In this context, even without an explicit F-theory construction for this
theory at present, it is still possible to examine certain Swampland
Conjectures. The characteristics of the string with charge $(1,-1)$ including
the consistent elliptic genus computed above offer compelling evidence that
supports the emergence of a tensionless heterotic string at weak gauge
coupling. By noting that the elliptic genus of the heterotic string encodes
the particle states in the 6d supergravity as well as their charge-to-mass
ratios, which can be derived from the special feature of weak Jacobi forms
discussed in (406), we can indeed confirm that this theory satisfies the non-
Abelian Sublattice Weak Gravity Conjecture.

###  4.2 Emergent little strings

In certain supergravity theories, infinite distance limits in the moduli space
correspond to strong gauge coupling limits. In this subsection, we will
highlight one such instance. As we will see, the theory also has
asymptotically tensionless strings in a neighborhood of the infinite distance
limit, consistent with the Emergent String Conjecture. However, it is
important to note that the tensionless strings, which are charged under the
gauge group, are not the heterotic string in this case. Instead, they are non-
critical strings at strong coupling in a 6d little string theory (LST) that
interacts with gravity.

Consider the 6d supergravity with $T=1$ and $SU(2)$ gauge group coupled to
$16\times{\bf 2}+215\times{\bf 1}$ hypermultiplets. This theory is anomaly
free with the charge vectors and the intersection form,

$\displaystyle a=(-3,1)\ ,\quad b=(1,-1)\ ,\quad\Omega={\rm diag}(1,-1)\ .$
(416)

This theory can be engineered in F-theory compactification on an elliptically
fibered 3-fold with $(h^{1,1},h^{2,1})=(4,214)$. The base surface of the CY3
is a Hirzebruch surface $\mathbb{F}_{1}$ and the fiber curve
$f\subset\mathbb{F}_{1}$ with $f^{2}=0$ supports an $SU(2)$ gauge algebra.
This model is a Calabi-Yau hypersurface and its toric data is given as
follows:

$\displaystyle\begin{array}[]{rrrr|rrrr}\lx@intercol\hfil\nabla\hfil\lx@intercol\vrule\lx@intercol&l^{1}&l^{2}&l^{3}&l^{4}\\\
\hline\cr 0&0&0&-1&0&1&0&1\\\ 0&0&-1&0&0&0&0&1\\\ -1&-1&2&3&0&0&1&0\\\
0&1&2&3&1&0&0&0\\\ 0&-1&2&3&1&0&-1&0\\\ 1&0&1&2&0&2&0&-1\\\
1&0&2&3&0&-2&1&1\\\ 0&0&2&3&-2&1&-1&0\end{array}$ (426)

The Mori cone curve associated with $l^{1}$ and $l^{3}$ are fiber $f$ and base
$e$ classes of the Hirzebruch surface $\mathbb{F}_{1}$, respectively, and
$l^{2}+2l^{4}$ is related to the elliptic fiber.

The only infinite distance limit in the moduli space, while the volume of
$\mathbb{F}_{1}$ stays finite, is the infinite volume limit of the base curve
$h$, i.e. ${\rm Vol}(h)\rightarrow\infty$. In the limit, the fiber curve $f$
shrinks to zero size, i.e. ${\rm Vol}(f)\rightarrow 0$. This implies that the
$SU(2)$ gauge coupling diverges because its inverse is controlled by the
volume of the fiber curve. Thus, this limit represents the strong coupling
regime in the moduli space for the $SU(2)$ gauge group.

Let us consider D3-branes wrapped around the fiber curve $f$ with zero self-
intersection. Since the volume of the wrapped curve approaches zero in the
infinite distance limit, these wrapped D3-branes yield tensionless strings in
the supergravity theory. It turns out that there are two distinct strings
which become tensionless as the gauge coupling diverges.

The first one is the critical heterotic string, which coincides with the
emergent strings previously highlighted in Lee:2018urn . The oscillations of
this tensionless heterotic string produces a tower of massless particle
states, leading to the breakdown of the effective field theory. This is
consistent with expectations from the Distance Conjecture at infinite distance
in the moduli space. We note, however, that this string is not charged under
the $SU(2)$ gauge field since the wrapped fiber curve does not intersect with
the fiber curve that supports the gauge algebra. So, this means that the
previous proof of the Weak Gravity Conjecture for F-theory models hosting only
Abelian gauge factors in Lee:2018urn ; Lee:2018spm , which heavily leans on
the distinct features of the charged heterotic strings appearing at infinite
distance limits, does not apply in this case, even though the theory has an
F-theory realization.

The second string, which becomes tensionless in the infinite distance limit,
is the instantonic string for the $SU(2)$ gauge group. The worldvolume theory
on the instantonic strings contains bosonic zero modes parametrizing the
instanton moduli space, and thus they are distinguished from the heterotic
string which solely involves fermionic zero modes apart from the center of
mass modes. It is clear that the instantonic strings carry non-trivial charges
for the $SU(2)$ gauge symmetry. These strings are instanton solitons in the 6d
LST with $SU(2)$ gauge group and 16 fundamental hypermultiplets, which can be
realized on a D5-brane in the Type I string theory Witten:1995gx ;
Gimon:1996rq .

We will now demonstrate that the elliptic genus of the second tensionless
string matches exactly with that of the non-critical string in the 6d LST. To
support this claim, we will directly calculate the elliptic genus of the
string arising from a wrapped D3-brane around the fiber curve $f$, which hosts
the $SU(2)$ singularity. This string is an instantonic string with

$\displaystyle c_{L}=24\ ,\quad c_{R}=12\ ,\quad k_{l}=-1\ ,\quad k_{SU(2)}=0\
.$ (427)

Hence, we can take the modular ansatz for the elliptic genus as

$\displaystyle
Z_{(1,-1)}^{b=(1,-1)}(\tau,\phi)=\frac{\mathcal{N}_{(8,4)}(\tau,\phi)}{\eta(\tau)^{24}\varphi_{2}(\tau,\epsilon)\varphi_{2}(\tau,2\phi)}\
.$ (428)

Here, the denominator factor depending on the $SU(2)$ gauge chemical potential
is chosen for the bosonic zero mode of a single $SU(2)$ instanton moduli
space. The numerator is a weak Jacobi form of weight 8 and $SU(2)$ index $4$.
By comparing it to the genus zero GV-invariant obtained from the mirror
analysis, we can uniquely determine the numerator,

$\displaystyle\mathcal{N}_{(8,4)}(\tau,\phi)=$
$\displaystyle-\frac{(E_{4}^{4}+5E_{4}E_{6}^{2})\varphi_{2}(\phi)^{4}}{648}+\frac{(E_{4}^{3}-E_{6}^{2})\varphi_{2}(\phi)^{2}\varphi_{0}(\phi)^{2}}{2592}$
$\displaystyle-\frac{E_{4}E_{6}(\varphi_{0}(\phi)^{2}-3E_{4}\varphi_{2}(\phi)^{2})\varphi_{2}(\phi)\varphi_{0}(\phi)}{216}\
.$ (429)

Given this result, the elliptic genus (428) perfectly matches the elliptic
genus of a single $SU(2)$ little string which was computed in Kim:2018gak
using the ADHM construction for this little string in Johnson:1998yw  and also
in Kim:2023glm  using the blowup equation.666More precisely, only the states
charged under the $SU(2)$ gauge symmetry match, after switching off all the
$SO(32)$ flavor fugacities, which we checked numerically and also up to
$q^{10}$ order in $q$-expansion. The charge-neutral sector can differ from the
LST results, as the partition functions of local theories are defined up to
certain factors independent of dynamical parameters like $\phi$. One can also
compute the elliptic genera of strings from multi-wrapped D3-branes on $f$,
and check that they agree with those of the strings at higher instanton
numbers in the LST. This verifies that the strings corresponding to D3-branes
wrapped around a fiber curve $f$ with an $SU(2)$ singularity are little
strings in the 6d LST.

It is also worth noting that when we turn off the $SU(2)$ gauge fugacity (by
setting $\phi\rightarrow 0$), this elliptic genus matches that of the
heterotic string on K3. This reflects the fact that when we Higgs the $SU(2)$
gauge symmetry, the theory reduces to the 6d F-theory corresponding to a
Hirzebruch base $\mathbb{F}_{1}$ without gauge singularities, and the string
in the Higgsed theory is dual to a six-dimensional heterotic string in the
emergent string limit. In fact, the string arising from a D3-brane wrapped
over a fiber curve away from the $SU(2)$ singularity before the Higgsing
coincides with the heterotic string after the Higgsing. This is because the
string remains unaffected during the Higgsing process as it does not interact
with the $SU(2)$ gauge field. From this perspective, we can say that the
elliptic genus in the emergent string limit encodes two spectra: the BPS
states of the little strings carrying $SU(2)$ gauge charges and those of the
charge-neutral string dual to a 6d heterotic string. In this case, the
elliptic genus of the tensionless string cannot capture the $SU(2)$ charged
particle states in the bulk 6d supergravity since the emergent heterotic
string does not carry any gauge charge. Thus, as mentioned earlier, we cannot
confirm the Weak Gravity Conjecture for theories of this kind using the
argument from references Lee:2018urn ; Lee:2018spm  for Abelian gauge groups
(also Cota:2020zse  for non-Abelian groups) which is based on the spectrum of
a charged heterotic string.

##  5 Conclusions

In this work, we have explored BPS strings and their behavior at infinite-
distance limits in the moduli space of 6d $\mathcal{N}=(1,0)$ supergravity
theories. Our main tool for this exploration is the exact spectrum of BPS
states in 2d worldsheet CFTs living on these strings. In the first part of
this work, we outlined a computational strategy that uses a modular bootstrap
approach for calculating the elliptic genera of BPS strings in 6d
supergravity. We introduced a generalized modular ansatz that potentially
covers all string configurations, including instantonic strings for 6d gauge
groups. We then confirmed the validity of this approach through explicit
calculations, using topological strings and mirror symmetry in the context of
F-theory compactifications on elliptically fibered CY 3-folds.

In the second part, we examined various Swampland conjectures in 6d
supergravity theories, which are not easily described by geometric
constructions in F-theory compactifications. These theories have exotic matter
representations of the gauge group, specifically the triple symmetric
representation of the $SU(2)$ group. Such representations are difficult or
even impossible to explicitly realize using Weierstrass models in F-theory.
Our investigation strongly suggests that these models give rise to emergent
heterotic strings at weak gauge coupling limit, which is located at infinite
distance in the moduli space, and the effective field theory breaks down due
to a tower of asymptotically massless charged states. This is consistent with
both the Swampland Distance and Emergent String Conjectures. Furthermore, the
excitations of these strings provide states occupying a sublattice in the
charge lattice that meets the criteria of a sublattice version of the Weak
Gravity Conjecture. Our results demonstrate that it is possible to rigorously
test the Swampland conjectures in certain EFTs without the need for geometric
realizations or specific string theory embeddings.

Expanding our approach to explore other non-geometric supergravity theories
could be quite interesting. Specifically, if the theory includes a BPS string
whose central charges match those given in (396), we can argue that this
string becomes a critical heterotic string in a duality frame. The elliptic
genus of this string is strongly constrained by the properties of the
heterotic string. So, by combining this knowledge with the modular bootstrap
technique, we might be able to exactly calculate the elliptic genus. If we
find an elliptic genus that is consistent with both the features of the
heterotic string and the modular ansatz, as in the example we discussed in
Section 4, then we could use this to solidly confirm certain Swampland
conjectures like the Weak Gravity Conjecture. This would offer strong evidence
that these non-geometric theories have consistent UV-completions and thus
being in the landscape. In these cases, it would also be fascinating to
further examine whether these non-geometric theories cannot be realized in any
string framework, thereby potentially serving as counterexamples to the String
Lamppost Principle (SLP). Conversely, if we cannot find a consistent elliptic
genus for the string, it would strongly suggest that the supergravity theory
in question is inconsistent when coupled to gravity and hence belongs to the
Swampland. Therefore, this analysis could serve as an additional Swampland
constraint for 6d supergravity theories.

###### Acknowledgements.

We would like to thank Yuta Hamada, Seung-Joo Lee, Sungwoo Nam, Shing Yan Li
and Xin Wang for useful discussion and conversation. HH thanks the hospitality
of POSTECH where part of this work was done. The work of HH is supported in
part by JSPS KAKENHI Grant Number JP18K13543 and JP23K03396. HK and MK are
supported by Samsung Science and Technology Foundation under Project Number
SSTF-BA2002-05 and by the National Research Foundation of Korea (NRF) grant
funded by the Korea government (MSIT) (2023R1A2C1006542).

##  Appendix A Modular forms and Jacobi forms

In this appendix, we review basic definitions and properties of the modular
forms and Jacobi forms.

###  A.1 Modular forms

Let $\mathcal{H}=\\{z\in\mathbb{C}\mid\Im z>0\\}$ be the upper half plane of
the complex plane. _Modular form_ is a function $f:\mathcal{H}\to\mathbb{C}$
satisfying

for an integer $k$, which is called the _weight_ of the modular form. The
modular forms are periodic under $\tau\to\tau+1$, and they have Fourier
expansion of the form

$\displaystyle f(\tau)=\sum_{n\in\mathbb{Z}}c(n)q^{n}\qquad(q=e^{2\pi
i\tau})\,.$ (430)

$f$ is called a holomorphic (resp. cusp or weakly holomorphic) modular form if
$c(n)=0$ unless $n\geq 0$ (resp. $n>0$ or $n\geq-N$ for some $N\geq 0$).

An important example of the modular forms is the Eisenstein series

$\displaystyle
E_{2k}(\tau)=\frac{1}{2\zeta(2k)}\sum_{(m,n)\neq(0,0)}\frac{1}{(m+n\tau)^{2k}}=1+\frac{(2\pi
i)^{2k}}{\zeta(2k)(2k-1)!}\sum_{n=1}^{\infty}\sigma_{2k-1}(n)q^{n}\,,$ (431)

of weight $2k$, where $\zeta(s)$ is the Riemann zeta function and
$\sigma_{k}(n)=\sum_{d|n}d^{k}$ is the divisor function. When $k>1$,
$E_{2k}(\tau)$ is a holomorphic modular form, while $E_{2}(\tau)$ transforms
as

$\displaystyle
E_{2}\quantity(\frac{a\tau+b}{c\tau+d})=(c\tau+d)^{2}E_{2}(\tau)-\frac{6i}{\pi}c(c\tau+d)\,.$
(432)

$E_{2}$ is an example of _quasi-modular form_ , meaning that it is a
holomorphic part of the non-holomorphic function
$\hat{E}_{2}=E_{2}-3/(\pi\Im\tau)$ which transforms like a modular form of
weight 2. The ring of holomorphic modular forms on $\mathrm{SL}(2,\mathbb{Z})$
is freely generated by two Eisenstein series $E_{4}(\tau)$ and $E_{6}(\tau)$,
i.e., any holomorphic modular form $f(\tau)$ of weight $k$ can be written as
$f=\sum_{4\alpha+6\beta=k}c_{\alpha\beta}E_{4}^{\alpha}E_{6}^{\beta}$ for some
constants $c_{\alpha\beta}\in\mathbb{C}$. For instance, the 24-th power of the
Dedekind eta function defined by

$\displaystyle\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})\,,$ (433)

is an example of weight 12 cusp modular form called the modular discriminant,
and it can be represented as
$\eta(\tau)^{24}=(E_{4}(\tau)^{3}-E_{6}(\tau)^{2})/1728$.

###  A.2 Jacobi forms

A function $\varphi_{k,m}:\mathcal{H}\times\mathbb{C}\to\mathbb{C}$ is called
a _Jacobi form_ if it satisfies two transformation properties

(434)

where $k\in\mathbb{Z}$ is called the _weight_ and $m\in\mathbb{Z}_{\geq 0}$ is
called the _index_ or _level_. It has a Fourier expansion of the form

$\displaystyle\varphi_{k,m}(\tau,z)=\sum_{n,r}c(n,r)q^{n}\xi^{r}\,,$ (435)

where $\xi=e^{2\pi iz}$. The Jacobi form $\varphi_{k,m}$ is called a
holomorphic (resp. cusp or weak) Jacobi form if $c(n,r)=0$ unless $4nm\geq
r^{2}$ (resp. $4nm>r^{2}$ or $n\geq 0$). The transformation property (434)
implies that the Fourier expansion can be written as

$\displaystyle\varphi_{k,m}(\tau,z)=\sum_{\ell\in\mathbb{Z}}h_{\ell}(\tau)q^{\ell^{2}/4m}\xi^{\ell}=\sum_{\ell\in\mathbb{Z}/2m\mathbb{Z}}h_{\ell}(\tau)\vartheta_{m,\ell}(\tau,z)\,,$
(436)

where $h_{\ell}(\tau)=h_{\ell+2m\lambda}(\tau)$ for $\lambda\in\mathbb{Z}$ and

$\displaystyle\vartheta_{m,\ell}(\tau,z)=\sum_{n\equiv\ell(\operatorname{mod}2m)}q^{n^{2}/4m}\xi^{n}=\sum_{k\in\mathbb{Z}}q^{(\ell+2mk)^{2}/4m}\xi^{\ell+2mk}$
(437)

is the theta function of index $m$ and characteristic $\ell$. If
$\varphi_{k,m}(\tau,z)$ is a holomorphic (resp. cusp of weak) Jacobi form,
then $h_{\ell}(\tau)$ is a weight $k-1/2$ holomorphic (resp. cusp or weakly
holomorphic) vector-valued modular form. The ring of weak Jacobi form is
freely generated over the ring of modular forms on
$\mathrm{SL}(2,\mathbb{Z})$. The generators of the ring are given by

$\displaystyle\varphi_{-2,1}(\tau,z)=-\frac{\theta_{1}(\tau,z)^{2}}{\eta(\tau)^{6}}\,,\quad\varphi_{0,1}(\tau,z)=4\sum_{i=2}^{4}\frac{\theta_{i}(\tau,z)^{2}}{\theta_{i}(\tau,0)^{2}}\,,$
(438)

where

$\displaystyle\begin{aligned}
&\theta_{1}(\tau,z)=-i\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\frac{1}{2}(n+1/2)^{2}}\xi^{n+1/2}\,,\quad&&\theta_{2}(\tau,z)=\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}(n+1/2)^{2}}\xi^{n+1/2}\,,\\\
&\theta_{3}(\tau,z)=\sum_{n\in\mathbb{Z}}q^{\frac{n^{2}}{2}}\xi^{n}\,,\quad&&\theta_{4}(\tau,z)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\frac{n^{2}}{2}}\xi^{n}\,,\end{aligned}$
(439)

are Jacobi theta functions. Hence, any weak Jacobi form $\varphi_{k,m}$ of
weight $k$, index $m$ can be uniquely represented as

$\displaystyle\varphi_{k,m}(\tau,z)=\underset{a_{3}+a_{4}=m,a_{i}\in\mathbb{Z}_{\geq
0}}{\sum_{4a_{1}+6a_{2}-2a_{3}=k}}c_{a_{i}}E_{4}(\tau)^{a_{1}}E_{6}(\tau)^{a_{2}}\varphi_{-2,1}(\tau,z)^{a_{3}}\varphi_{0,1}(\tau,z)^{a_{4}}\,,$
(440)

where $c_{a_{i}}\in\mathbb{C}$ are some constants. For a Jacobi form
$\varphi_{k,m}$ of weight $k$ and index $m$, one can introduce the _Hecke
operator_ defined as Eichler:1995

$\displaystyle
T_{n}\varphi_{k,m}(\tau,z)=n^{k-1}\sum_{ad=n}\sum_{b(\operatorname{mod}d)}d^{-k}\varphi_{k,m}\left(\frac{a\tau+b}{d},az\right)\,,$
(441)

which is a weight $k$, index $nm$ Jacobi form. This operator is useful for
considering the elliptic genera of heterotic strings.

The weak Jacobi forms are invariant under the Weyl group transformation of
$SU(2)$ acting on $z$. This can be generalized into other Lie algebras. Let
$\mathfrak{g}$ be a simple Lie algebra of rank $l$,
$\mathfrak{h}_{\mathbb{C}}\cong\mathbb{C}^{l}$ be the complexification of its
Cartan subalgebra, $W$ be its Weyl group, and $\langle\cdot,\cdot\rangle$ be
the Killing form on $\mathfrak{h}_{\mathbb{C}}$. A _Weyl invariant Jacobi
form_ is a function
$\varphi_{k,m}:\mathcal{H}\times\mathfrak{h}_{\mathbb{C}}\to\mathbb{C}$ which
has Weyl-invariance $\varphi_{k,m}(\tau,wz)=\varphi_{k,m}(\tau,z)$ and
satisfies two transformation properties

(442)

where $Q^{\vee}$ is the coroot lattice of $\mathfrak{g}$. The Fourier
expansion of the Weyl invariant Jacobi form which is a counterpart of (436) is

$\displaystyle\varphi_{k,m}(\tau,z)=\sum_{\lambda\in
P/mQ^{\vee}}h_{\lambda}(\tau)\vartheta_{m,\lambda}(\tau,z)$ (443)

where $P$ is the weight lattice of $\mathfrak{g}$ and the theta function
associated to the integral lattice $mQ^{\vee}$ is given by Kac:1984mq

$\displaystyle\vartheta_{m,\lambda}(\tau,z)=\underset{\mu\equiv\lambda(\operatorname{mod}mQ^{\vee})}{\sum_{\mu\in
P}}q^{\frac{1}{2m}\langle\mu,\mu\rangle}e^{2\pi i\langle\mu,z\rangle}\,.$
(444)

The ring of Weyl-invariant Jacobi form is freely generated over the ring of
$\mathrm{SL}(2,\mathbb{Z})$ modular forms except $\mathfrak{g}=E_{8}$ case
Wirthmuller . In this paper, we use $SU(3)$ Weyl invariant Jacobi forms in
subsection 3.3.3, which is generated by three fundamental Jacobi forms
$\varphi_{k}^{A_{2}}$ of weight $-k$ and index $1$ with $k=0,2,3$. They are
given by

$\displaystyle\varphi_{k}^{A_{2}}(\tau,\phi_{1},\phi_{2})=\mathcal{Z}^{3-k}\left.\prod_{j=1}^{3}\frac{i\theta_{1}(\tau,x_{j})}{\eta(\tau)^{3}}\right|_{\sum
x_{j}=0}\,,\quad(k=0,2,3)\,,$ (445)

where

$\displaystyle\mathcal{Z}=\frac{1}{2\pi
i}\left(\sum_{j=1}^{3}\frac{\partial}{\partial
x_{j}}+\frac{\pi^{2}}{3}E_{2}(\tau)\sum_{j=1}^{3}x_{j}\right)$ (446)

and $x_{1}=\phi_{1}$, $x_{2}=-\phi_{1}+\phi_{2}$, $x_{3}=-\phi_{2}$.

##  Appendix B Expressions of Modular ansatz at higher string orders

In this appendix, we report the results of the numerator of the ansatz for
higher string order.

###  B.1 $B=\mathbb{P}^{2}$ with $SU(2)$ gauge symmetry

#####  $d=2$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

The discriminants of Picard-Fuchs operators are $\Delta_{1}=(1-4z_{2})$ and

The numerator in the modular ansatz for the 3-string elliptic genus is

#####  $d=3$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

#####  $d=4$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

#####  $d=5$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

#####  $d=6$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

#####  $d=7$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

#####  $d=8$ case

The numerator in the modular ansatz for the 2-string elliptic genus is

###  B.2 $B=\mathbb{F}_{1}$ with $SU(2)$ gauge symmetry

We summarize the numerator in the modular ansatz for the elliptic genus of the
string with charge $Q=(1,0)$ considered in the subsection 3.3.4.

#####  $d=1$ case

#####  $d=2$ case

#####  $d=3$ case

#####  $d=4$ case

#####  $d=5$ case

#####  $d=6$ case

###  B.3 $B=\mathbb{F}_{1}$ with $SU(2)\times SU(2)$ gauge symmetry

We present the numerator in the modular ansatz for the elliptic genus of the
string with charge $Q=(1,0)$ and $Q=(0,2)$ considered in the subsection 3.3.5.

## References

  * (1) C. Vafa, _The String landscape and the swampland_ , hep-th/0509212.
  * (2) H.-C. Kim, G. Shiu and C. Vafa, _Branes and the Swampland_ , _Phys. Rev. D_ 100 (2019) 066006 [1905.08261].
  * (3) S.-J. Lee and T. Weigand, _Swampland Bounds on the Abelian Gauge Sector_ , _Phys. Rev. D_ 100 (2019) 026015 [1905.13213].
  * (4) H.-C. Kim, H.-C. Tarazi and C. Vafa, _Four-dimensional $\mathbf{\mathcal{N}=4}$ SYM theory and the swampland_, _Phys. Rev. D_ 102 (2020) 026003 [1912.06144].
  * (5) S. Katz, H.-C. Kim, H.-C. Tarazi and C. Vafa, _Swampland Constraints on 5d $\mathcal{N}=1$ Supergravity_, _JHEP_ 07 (2020) 080 [2004.14401].
  * (6) C. Angelantonj, Q. Bonnefoy, C. Condeescu and E. Dudas, _String Defects, Supersymmetry and the Swampland_ , _JHEP_ 11 (2020) 125 [2007.12722].
  * (7) P. Cheng, R. Minasian and S. Theisen, _Anomalies as obstructions: from dimensional lifts to swampland_ , _JHEP_ 01 (2022) 068 [2106.14912].
  * (8) Y. Hamada and C. Vafa, _8d supergravity, reconstruction of internal geometry and the Swampland_ , _JHEP_ 06 (2021) 178 [2104.05724].
  * (9) H.-C. Tarazi and C. Vafa, _On The Finiteness of 6d Supergravity Landscape_ , 2106.10839.
  * (10) L. Martucci, N. Risso and T. Weigand, _Quantum gravity bounds on $\mathcal{N}$ = 1 effective theories in four dimensions_, _JHEP_ 03 (2023) 197 [2210.10797].
  * (11) S.-J. Lee, W. Lerche and T. Weigand, _A Stringy Test of the Scalar Weak Gravity Conjecture_ , _Nucl. Phys. B_ 938 (2019) 321 [1810.05169].
  * (12) S.-J. Lee, W. Lerche and T. Weigand, _Emergent strings, duality and weak coupling limits for two-form fields_ , _JHEP_ 02 (2022) 096 [1904.06344].
  * (13) S.-J. Lee, W. Lerche and T. Weigand, _Emergent strings from infinite distance limits_ , _JHEP_ 02 (2022) 190 [1910.01135].
  * (14) M.-x. Huang, S. Katz and A. Klemm, _Topological String on elliptic CY 3-folds and the ring of Jacobi forms_ , _JHEP_ 10 (2015) 125 [1501.04891].
  * (15) R. Gopakumar and C. Vafa, _On the gauge theory / geometry correspondence_ , _Adv. Theor. Math. Phys._ 3 (1999) 1415 [hep-th/9811131].
  * (16) R. Gopakumar and C. Vafa, _M theory and topological strings. 1._ , hep-th/9809187.
  * (17) R. Gopakumar and C. Vafa, _M theory and topological strings. 2._ , hep-th/9812127.
  * (18) S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, _Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces_ , _Commun. Math. Phys._ 167 (1995) 301 [hep-th/9308122].
  * (19) S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, _Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces_ , _Nucl. Phys. B_ 433 (1995) 501 [hep-th/9406055].
  * (20) S.-J. Lee, W. Lerche and T. Weigand, _Tensionless Strings and the Weak Gravity Conjecture_ , _JHEP_ 10 (2018) 164 [1808.05958].
  * (21) M. Montero, G. Shiu and P. Soler, _The Weak Gravity Conjecture in three dimensions_ , _JHEP_ 10 (2016) 159 [1606.08438].
  * (22) B. Heidenreich, M. Reece and T. Rudelius, _Sharpening the Weak Gravity Conjecture with Dimensional Reduction_ , _JHEP_ 02 (2016) 140 [1509.06374].
  * (23) B. Heidenreich, M. Reece and T. Rudelius, _Evidence for a sublattice weak gravity conjecture_ , _JHEP_ 08 (2017) 025 [1606.08437].
  * (24) D. Klevers, D. R. Morrison, N. Raghuram and W. Taylor, _Exotic matter on singular divisors in F-theory_ , _JHEP_ 11 (2017) 124 [1706.08194].
  * (25) A. Sagnotti, _A Note on the Green-Schwarz mechanism in open string theories_ , _Phys. Lett. B_ 294 (1992) 196 [hep-th/9210127].
  * (26) V. Kumar and W. Taylor, _A Bound on 6D N=1 supergravities_ , _JHEP_ 12 (2009) 050 [0910.1586].
  * (27) C. Cheung and G. N. Remmen, _Positivity of Curvature-Squared Corrections in Gravity_ , _Phys. Rev. Lett._ 118 (2017) 051601 [1608.02942].
  * (28) Y. Hamada, T. Noumi and G. Shiu, _Weak Gravity Conjecture from Unitarity and Causality_ , _Phys. Rev. Lett._ 123 (2019) 051601 [1810.03637].
  * (29) H.-C. Kim, S. Kim and J. Park, _6d strings from new chiral gauge theories_ , 1608.03919.
  * (30) H. Shimizu and Y. Tachikawa, _Anomaly of strings of 6d $\mathcal{N}=\left(1,0\right)$ theories_, _JHEP_ 11 (2016) 165 [1608.05894].
  * (31) J. Polchinski, _Monopoles, duality, and string theory_ , _Int. J. Mod. Phys. A_ 19S1 (2004) 145 [hep-th/0304042].
  * (32) V. Braun and D. R. Morrison, _F-theory on Genus-One Fibrations_ , _JHEP_ 08 (2014) 132 [1401.7844].
  * (33) C. Vafa, _Evidence for F theory_ , _Nucl. Phys. B_ 469 (1996) 403 [hep-th/9602022].
  * (34) D. R. Morrison and C. Vafa, _Compactifications of F theory on Calabi-Yau threefolds. 1_ , _Nucl. Phys. B_ 473 (1996) 74 [hep-th/9602114].
  * (35) D. R. Morrison and C. Vafa, _Compactifications of F theory on Calabi-Yau threefolds. 2._ , _Nucl. Phys. B_ 476 (1996) 437 [hep-th/9603161].
  * (36) M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov and C. Vafa, _Geometric singularities and enhanced gauge symmetries_ , _Nucl. Phys. B_ 481 (1996) 215 [hep-th/9605200].
  * (37) S. Katz, D. R. Morrison, S. Schafer-Nameki and J. Sully, _Tate’s algorithm and F-theory_ , _JHEP_ 08 (2011) 094 [1106.3854].
  * (38) P. S. Aspinwall and D. R. Morrison, _Nonsimply connected gauge groups and rational points on elliptic curves_ , _JHEP_ 07 (1998) 012 [hep-th/9805206].
  * (39) P. S. Aspinwall, S. H. Katz and D. R. Morrison, _Lie groups, Calabi-Yau threefolds, and F theory_ , _Adv. Theor. Math. Phys._ 4 (2000) 95 [hep-th/0002012].
  * (40) C. Mayrhofer, D. R. Morrison, O. Till and T. Weigand, _Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory_ , _JHEP_ 10 (2014) 016 [1405.3656].
  * (41) E. Witten, _Phase transitions in M theory and F theory_ , _Nucl. Phys. B_ 471 (1996) 195 [hep-th/9603150].
  * (42) A. Grassi and D. R. Morrison, _Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds_ , math/0005196.
  * (43) S. H. Katz and C. Vafa, _Matter from geometry_ , _Nucl. Phys. B_ 497 (1997) 146 [hep-th/9606086].
  * (44) A. Grassi and D. R. Morrison, _Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds_ , _Commun. Num. Theor. Phys._ 6 (2012) 51 [1109.0042].
  * (45) A. Grassi and T. Weigand, _On topological invariants of algebraic threefolds with ( $\mathbb{Q}$-factorial) singularities_, 1804.02424.
  * (46) V. Sadov, _Generalized Green-Schwarz mechanism in F theory_ , _Phys. Lett. B_ 388 (1996) 45 [hep-th/9606008].
  * (47) D. Klevers and W. Taylor, _Three-Index Symmetric Matter Representations of SU(2) in F-Theory from Non-Tate Form Weierstrass Models_ , _JHEP_ 06 (2016) 171 [1604.01030].
  * (48) M. Del Zotto, J. J. Heckman, A. Tomasiello and C. Vafa, _6d Conformal Matter_ , _JHEP_ 02 (2015) 054 [1407.6359].
  * (49) P. Arras, A. Grassi and T. Weigand, _Terminal Singularities, Milnor Numbers, and Matter in F-theory_ , _J. Geom. Phys._ 123 (2018) 71 [1612.05646].
  * (50) V. Kumar, D. R. Morrison and W. Taylor, _Mapping 6D N = 1 supergravities to F-theory_ , _JHEP_ 02 (2010) 099 [0911.3393].
  * (51) V. Kumar, D. R. Morrison and W. Taylor, _Global aspects of the space of 6D N = 1 supergravities_ , _JHEP_ 11 (2010) 118 [1008.1062].
  * (52) F. Benini, R. Eager, K. Hori and Y. Tachikawa, _Elliptic Genera of 2d ${\mathcal{N}}$ = 2 Gauge Theories_, _Commun. Math. Phys._ 333 (2015) 1241 [1308.4896].
  * (53) M. Del Zotto and G. Lockhart, _On Exceptional Instanton Strings_ , _JHEP_ 09 (2017) 081 [1609.00310].
  * (54) J. Gu, M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, _Refined BPS invariants of 6d SCFTs from anomalies and modularity_ , _JHEP_ 05 (2017) 130 [1701.00764].
  * (55) N. Bobev, M. Bullimore and H.-C. Kim, _Supersymmetric Casimir Energy and the Anomaly Polynomial_ , _JHEP_ 09 (2015) 142 [1507.08553].
  * (56) B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, _F-Theory, Spinning Black Holes and Multi-string Branches_ , _JHEP_ 01 (2016) 009 [1509.00455].
  * (57) C. F. Cota, A. Klemm and T. Schimannek, _State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture_ , _JHEP_ 05 (2021) 030 [2012.09836].
  * (58) J. Kim, K. Lee and J. Park, _On elliptic genera of 6d string theories_ , _JHEP_ 10 (2018) 100 [1801.01631].
  * (59) M. Del Zotto and G. Lockhart, _Universal Features of BPS Strings in Six-dimensional SCFTs_ , _JHEP_ 08 (2018) 173 [1804.09694].
  * (60) M. Eichler and D. Zagier, _The Theory of Jacobi Forms_ , no. 55 in Progress in Mathematics. Birkhäuser, 1985, https://doi.org/10.1007/978-1-4684-9162-3.
  * (61) M. Del Zotto, J. Gu, M.-X. Huang, A.-K. Kashani-Poor, A. Klemm and G. Lockhart, _Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs_ , _JHEP_ 03 (2018) 156 [1712.07017].
  * (62) H.-C. Kim, M. Kim and Y. Sugimoto, _Blowup equations for little strings_ , _JHEP_ 05 (2023) 029 [2301.04151].
  * (63) V. V. Batyrev, _Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties_ , _J. Alg. Geom._ 3 (1994) 493 [alg-geom/9310003].
  * (64) R. Altman, J. Gray, Y.-H. He, V. Jejjala and B. D. Nelson, _A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List_ , _JHEP_ 02 (2015) 158 [1411.1418].
  * (65) K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa et al., _Mirror symmetry_ , vol. 1 of _Clay mathematics monographs_. AMS, Providence, USA, 2003.
  * (66) M. Kreuzer and H. Skarke, _Complete classification of reflexive polyhedra in four-dimensions_ , _Adv. Theor. Math. Phys._ 4 (2000) 1209 [hep-th/0002240].
  * (67) Y.-C. Huang and W. Taylor, _Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers_ , _JHEP_ 02 (2019) 087 [1805.05907].
  * (68) Y.-C. Huang and W. Taylor, _On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds_ , _JHEP_ 03 (2019) 014 [1809.05160].
  * (69) Y.-C. Huang and W. Taylor, _Fibration structure in toric hypersurface Calabi-Yau threefolds_ , _JHEP_ 03 (2020) 172 [1907.09482].
  * (70) M. Kreuzer and H. Skarke, _Calabi-Yau four folds and toric fibrations_ , _J. Geom. Phys._ 26 (1998) 272 [hep-th/9701175].
  * (71) V. Braun, _Toric Elliptic Fibrations and F-Theory Compactifications_ , _JHEP_ 01 (2013) 016 [1110.4883].
  * (72) T. Oda and H. S. Park, _Linear Gale transforms and Gel’fand-Kapranov-Zelevinskij decompositions_ , _Tohoku Mathematical Journal_ 43 (1991) 375 .
  * (73) P. Berglund, S. H. Katz and A. Klemm, _Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties_ , _Nucl. Phys. B_ 456 (1995) 153 [hep-th/9506091].
  * (74) T. W. Grimm and H. Hayashi, _F-theory fluxes, Chirality and Chern-Simons theories_ , _JHEP_ 03 (2012) 027 [1111.1232].
  * (75) M. Kreuzer and H. Skarke, _PALP: A Package for analyzing lattice polytopes with applications to toric geometry_ , _Comput. Phys. Commun._ 157 (2004) 87 [math/0204356].
  * (76) J. Rambau, _Topcom: Triangulations of point configurations and oriented matroids_ , in _Proceedings of the International Congress of Mathematical Software_ , 2002, https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/684.
  * (77) W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, “Singular 4-3-0 — A computer algebra system for polynomial computations.” http://www.singular.uni-kl.de, 2022.
  * (78) The Sage Developers, _SageMath, the Sage Mathematics Software System (Version 10.0)_ , 2023.
  * (79) M. Demirtas, A. Rios-Tascon and L. McAllister, _CYTools: A Software Package for Analyzing Calabi-Yau Manifolds_ , 2211.03823.
  * (80) M. Kreuzer and H. Skarke, “Calabi-Yau data.” http://hep.itp.tuwien.ac.at/~kreuzer/CY/.
  * (81) A. Strominger, _SPECIAL GEOMETRY_ , _Commun. Math. Phys._ 133 (1990) 163.
  * (82) I. M. Gel’fand, A. V. Zelevinsky and M. M. Kapranov, _Hypergeometric functions and toral manifolds_ , _Funct Anal Its Appl_ 23 (1989) 94.
  * (83) I. M. Gel’fand, M. M. Kapranov and A. V. Zelevinsky, _Generalized Euler integrals and A-hypergeometric functions_ , _Advances in Mathematics_ 84 (1990) 255.
  * (84) M. Demirtas, M. Kim, L. McAllister, J. Moritz and A. Rios-Tascon, _Computational Mirror Symmetry_ , 2303.00757.
  * (85) M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, _Holomorphic anomalies in topological field theories_ , _Nucl. Phys. B_ 405 (1993) 279 [hep-th/9302103].
  * (86) M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, _Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes_ , _Commun. Math. Phys._ 165 (1994) 311 [hep-th/9309140].
  * (87) P. Candelas, A. Font, S. H. Katz and D. R. Morrison, _Mirror symmetry for two parameter models. 2._ , _Nucl. Phys. B_ 429 (1994) 626 [hep-th/9403187].
  * (88) T. M. Chiang, A. Klemm, S.-T. Yau and E. Zaslow, _Local mirror symmetry: Calculations and interpretations_ , _Adv. Theor. Math. Phys._ 3 (1999) 495 [hep-th/9903053].
  * (89) A. Klemm and E. Zaslow, _Local mirror symmetry at higher genus_ , _AMS/IP Stud. Adv. Math._ 23 (2001) 183 [hep-th/9906046].
  * (90) M. Aganagic, A. Klemm, M. Marino and C. Vafa, _The Topological vertex_ , _Commun. Math. Phys._ 254 (2005) 425 [hep-th/0305132].
  * (91) A. Iqbal and C. Kozcaz, _Refined topological strings on local ${\mathrm{\mathbb{P}}}^{2}$_, _JHEP_ 03 (2017) 069 [1210.3016].
  * (92) M.-x. Huang, K. Sun and X. Wang, _Blowup Equations for Refined Topological Strings_ , _JHEP_ 10 (2018) 196 [1711.09884].
  * (93) H.-C. Kim, M. Kim, S.-S. Kim and K.-H. Lee, _Bootstrapping BPS spectra of 5d/6d field theories_ , _JHEP_ 04 (2021) 161 [2101.00023].
  * (94) V. Kumar, D. S. Park and W. Taylor, _6D supergravity without tensor multiplets_ , _JHEP_ 04 (2011) 080 [1011.0726].
  * (95) Y.-C. Huang and W. Taylor, _Mirror symmetry and elliptic Calabi-Yau manifolds_ , _JHEP_ 04 (2019) 083 [1811.04947].
  * (96) C. T. C. Wall, _Classification Problems in Differential Topology. V_ , _Invent Math_ 1 (1966) 355.
  * (97) A. Klemm, J. Manschot and T. Wotschke, _Quantum geometry of elliptic Calabi-Yau manifolds_ , 1205.1795.
  * (98) H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , _Nucl. Phys. B_ 766 (2007) 21 [hep-th/0605264].
  * (99) N. Raghuram, W. Taylor and A. P. Turner, _Automatic enhancement in 6D supergravity and F-theory models_ , _JHEP_ 07 (2021) 048 [2012.01437].
  * (100) R. Dijkgraaf, G. W. Moore, E. P. Verlinde and H. L. Verlinde, _Elliptic genera of symmetric products and second quantized strings_ , _Commun. Math. Phys._ 185 (1997) 197 [hep-th/9608096].
  * (101) N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, _The String landscape, black holes and gravity as the weakest force_ , _JHEP_ 06 (2007) 060 [hep-th/0601001].
  * (102) B. Heidenreich, M. Reece and T. Rudelius, _The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff_ , _Eur. Phys. J. C_ 78 (2018) 337 [1712.01868].
  * (103) E. Witten, _Small instantons in string theory_ , _Nucl. Phys. B_ 460 (1996) 541 [hep-th/9511030].
  * (104) E. G. Gimon and J. Polchinski, _Consistency conditions for orientifolds and D-manifolds_ , _Phys. Rev. D_ 54 (1996) 1667 [hep-th/9601038].
  * (105) C. V. Johnson, _On the (0,4) conformal field theory of the throat_ , _Mod. Phys. Lett. A_ 13 (1998) 2463 [hep-th/9804201].
  * (106) V. G. Kac and D. H. Peterson, _Infinite dimensional Lie algebras, theta functions and modular forms_ , _Adv. Math._ 53 (1984) 125.
  * (107) K. Wirthmüller, _Root systems and Jacobi forms_ , _Compositio Mathematica_ 82 (1992) 293.