CERN-TH-2021-214
CTPU-PTC-21-41
ZMP-HH/21-24

Physics of Infinite Complex Structure Limits in eight Dimensions

Seung-Joo Lee1, Wolfgang Lerche2, and Timo Weigand3

1Center for Theoretical Physics of the Universe,
Institute for Basic Science, Daejeon 34126, South Korea
2CERN, Theory Department,
1 Esplande des Particules, Geneva 23, CH-1211, Switzerland
3II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22607 Hamburg, Germany
3Zentrum für Mathematische Physik, Universität Hamburg,
Bundesstrasse 55, 20146 Hamburg, Germany

††Email: seungjoolee at ibs.re.kr, wolfgang.lerche at cern.ch, timo.weigand at
desy.de

###### Abstract

We investigate infinite distance limits in the complex structure moduli space
of F-theory compactified on K3 to eight dimensions. While this is among the
simplest possible arenas to test ideas about the Swampland Distance
Conjecture, it is nevertheless non-trivial enough to improve our understanding
of the physics for these limiting geometries, including phenomena of
emergence. It also provides a perspective on infinite distance limits from the
viewpoint of open strings. The paper has two quite independent themes. In the
main part we show that all degenerations of elliptic K3 surfaces at infinite
distance as analysed in the companion paper [1] can be interpreted as
(partial) decompactification or emergent string limits in F-theory, in
agreement with the Emergent String Conjecture. We present a unified geometric
picture of the possible towers of states that can become light and illustrate
our general claims via the connection between Kulikov models of degenerating
K3 surfaces and the dual heterotic string. As an application we classify the
possible maximal non-abelian Lie algebras and their Kac-Moody and loop
extensions that can arise in the infinite distance limits. In the second part
we discuss the infinite distance behaviour of certain exact quartic gauge
couplings. We encounter a tension with the hypothesis that effective couplings
should be fully generated by integrating out massive states. We show that by
appropriately renormalizing the string coupling, at least partial emergence
can be achieved.

###### Contents

  1. 1 Introduction
  2. 2 Review: Affine Algebras and infinite Distance Limits
  3. 3 Kulikov Models as Emergent String Limits or Decompactifications
    1. 3.1 Refined Kulikov models for elliptic K3 surfaces
    2. 3.2 Type II.b Kulikov models as emergent string limits
    3. 3.3 Type II.a Kulikov models as decompactification limits
    4. 3.4 Type III.a Kulikov models as partial decompactification limits
    5. 3.5 Type III.b Kulikov models as weak coupling plus decompactification limits
  4. 4 Infinite Distance Limits in the $E_{7}\times E_{8}$-Weierstrass Model
    1. 4.1 The $E_{7}\times E_{8}$-Weierstrass model and its heterotic dual
    2. 4.2 Systematics of affine enhancements in the $E_{7}\times E_{8}$-Weierstrass model
    3. 4.3 Type II degenerations
    4. 4.4 Type III degenerations
  5. 5 Maximal Non-abelian Gauge Algebras in 9d from Type III.a Kulikov Models
    1. 5.1 A-series and D-series
    2. 5.2 Maximal $D$-series: construction
  6. 6 Emergence of Quartic Gauge Couplings?
  7. 7 Discussion
  8. A Siegel Modular Forms: Definitions and Properties

##  1 Introduction

Understanding the structure of the landscape of consistent quantum gravity
theories, as opposed to the swampland of effective theories without a UV
completion involving gravity, is one of the most ambitious goals in modern
theoretical physics. Numerous criteria defining the boundary between both
types of theories have been proposed within the Swampland Program initiated in
[2]. Among these, the Swampland Distance Conjecture [3] arguably plays a
central role: It underlies the de Sitter Conjecture in asymptotic regions of
moduli space [4], serves as inspiration for the Anti-de Sitter Conjecture [5],
the Spin-2 Conjecture [6] and the Gravitino Mass Conjecture [7, 8], predicts a
holographically dual CFT Distance Conjecture [9, 10] and, quite generally,
illustrates the tension between parametrically large field traversions in
cosmology and validity of the effective field theory, to name but a few
aspects.111For further details and a collection of related works we refer to
the reviews [11, 12, 13, 14]. The conjecture asserts that at infinite distance
in moduli space, a tower of states becomes light exponentially fast in any
quantum gravity theory. From a conceptual point of view, the fascination
behind this claim lies in the fact that every quantum gravity theory must
enter a new phase at infinite distance in its moduli space where the original
effective description breaks down.

Given its special status within the Swampland Program, the Distance Conjecture
clearly begs for a more fundamental explanation. In [15] it was proposed that
the relevant towers of states, which become light at the parametrically
fastest rate on the infinite boundaries of moduli space, are either a (dual)
Kaluza-Klein tower or the tower of excitations of an emergent asymptotically
weakly-coupled fundamental string. If generally true this would drastically
simplify our global picture of the quantum gravity landscape, to the extent
that it would reduce the boundaries of moduli space in the asymptotic regime
to well controlled theories.222A related, but independent idea stressing the
role of strings for the asymptotics of moduli space is the Distant Axionic
String Conjecture of [16, 17]. This would demystify the breakdown of the
effective theory at infinite distance and also offer a new perspective on the
Emergence Proposal of [18, 19, 20].333An explanation of the Swampland Distance
Conjecture based on entropy arguments was recently proposed in [21].

This Emergent String Conjecture has been successfully tested for a number of
different corners of string and M-theory in various dimensions and in
situations with as little as four supercharges. Apart from higher dimensional
compactifications [22], this includes $N=1$ supersymmetric compactifications
of F-theory to six [23] and four [24, 25] dimensions and of M-theory to five
[15] and four [26] dimensions, as well as 4d $N=2$ supersymmetric
compactifications of Type IIA [15] and Type IIB [27]. In all these cases, the
moduli space whose infinite distance limits were investigated corresponds to
the Kähler moduli space of the underlying Calabi-Yau variety. Intricate
features of Kähler geometry guarantee that whenever the parametrically
lightest tower of states does not correspond to a Kaluza-Klein tower, a unique
(oftentimes solitonic) string emerges. The geometry works in such a way that
its excitations lie parametrically at the same scale as a Kaluza-Klein tower
and there exists a duality frame in which the emergent string is weakly
coupled. Depending on the nature of the moduli space under investigation [27,
25], this rests on a remarkable conspiracy of classical and quantum effects.

By contrast, an explicit confirmation or falsification of the Emergent String
Conjecture in the complex structure moduli space of string compactifications
has so far proven to be very difficult. In [20, 28, 29, 30, 31] the Swampland
Distance Conjecture as such has been quantitatively confirmed for Type IIB
compactifications near the asymptotic boundary of complex structure moduli
space, by arguing for the appearance of a tower of BPS states from D3-branes
wrapping asymptotically vanishing 3-cycles. The mathematical structure of the
boundaries of complex structure moduli spaces has been scrutinised from the
point of view of asymptotic Hodge theory in a series of further works [32, 33,
34]. This program in particular has lead to important insights into the
possibilities of moduli stabilisation in flux compactifications [35, 36, 37],
a topic of central relevance in string theory. Irrespective of this progress,
it is fair to say that a clear physics interpretation of the infinite towers
of states, or even an identification of the parametrically leading towers, is
yet to be obtained. By mirror symmetry, Type IIB string theory on Calabi-Yau
threefolds is equivalent to corresponding Type IIA compactifications, for
which the asymptotically massless states at infinite distance in Kähler moduli
space have a clear interpretation as Kaluza-Klein states or emergent string
excitations [15]. It is therefore desirable to obtain a comparable
understanding of the asymptotic physics directly from the point of view of the
complex structure moduli space.

In this paper we embark on this program in the arguably simplest non-trivial
context, namely of F-theory compactified to eight dimensions on elliptically
fibered K3 surfaces. While at first sight eight-dimensional theories might
look straightforward to analyse, especially when viewed as compactifications
of the heterotic string on $T^{2}_{\rm het}$, the structure of infinite
distance limits turns out to be surprisingly intricate from the geometric
point of view. The focus of our work is therefore on understanding the
geometry of complex structure degenerations and its match to physics at
infinite distance. Our analysis builds on the refinement of the so-called Type
III Kulikov models for elliptic K3 surfaces that was elaborated on in the
companion paper [1], as well as in the independent series of works of [38,
39]. We will obtain perfect agreement between the refined Kulikov models and
the possible decompactification or emergent string limits as suggested by the
Emergent String Conjecture. For an illustration, see Figure -1199. We will
give a more detailed summary of the main points of this analysis, which are
the subject of Sections 2 \- 5, in the second part of this introduction.

Figure -1199: Infinite distance complex structure degenerations for F-theory
on elliptic K3 surface, and the physics in the various asymptotic regimes.

Moreover, in the largely independent Section 6 of this paper, we comment on
some other aspects of emergence, profiting from the canonical, highly
constrained structure of the theory in eight dimensions. Specifically, we
address the interesting proposal [18, 40, 19, 20, 41, 12] that gauge couplings
(and by courageous extrapolation, all couplings) emerge in the IR just from
integrating out loops of massive states. In particular, their divergence in
the decompactification limit arises from integrating out an asymptotically
massless tower of KK states or string excitations. In geometric duality
frames, such as F-theory, singularities of gauge couplings generally arise
from degenerating geometries, for example, from D-branes tied to collapsing or
coincident cycles; the prime example being the conifold singularity of Calabi-
Yau manifolds. One customarily says that the geometry automatically integrates
out the asymptotically massless states.

However, these singularities arise at tree-level of the geometric
compactifications, and the question arises whether there exist other duality
frames where the singularity, or rather the RG flow for that matter, literally
arises from integrating out of loops of massive states. For the conifold
singularity in Type II strings compactified on Calabi-Yau threefolds, it is
known that the relevant dual perturbative frame is the heterotic string, for
which the singularity in the coupling arises at one-loop order due to a
massless state circulating in the loop [42].

While this picture is very appealing, it is not a priori guaranteed that for
any kind of large distance singularity in the moduli space there exists a
duality frame for which the singularity arises from perturbative loops of
asymptotically massless states. In fact, already in the early literature [3]
it was pointed out that this picture may fail for the special case of Kaluza-
Klein gauge symmetries, for which the large distance singularity arises from
to the diverging volume of the compactification space. For such a case one
expects at most partial emergence, in the sense that only part of the
divergence may arise at the quantum level [40, 19, 12].

We will address this issue for eight-dimensional theories, which are very
restricted in that the perturbative dual frame is unambiguously the heterotic
string compactified on $T_{\rm het}^{2}$, while non-perturbative effects are
absent. An extra benefit is that it is the quartic, and not the quadratic
gauge couplings that are dimensionless, and this disentangles the issues of
gauge fields becoming dynamical at low energies, metric on moduli space, and
the running of one-loop diagrams. That is, the quadratic gauge couplings have
positive mass dimension and so vanish in the UV; in this sense, gauge dynamics
emerges in the IR. But these couplings are not renormalized at one loop order,
so one cannot really say that it is the integrating-out of massive states that
makes these fields dynamical.

As mentioned, the relevant one-loop exact, BPS saturated gauge couplings in
eight dimensions are quartic, and these share a lot (but not all) of the
properties of the well-investigated quadratic couplings of $N=2$
supersymmetric strings in four dimensions. We will focus only on such
couplings of a certain kind, namely on those of the $U(1)$ gauge fields that
arise via KK reduction from ten dimensions. Our point will be to show that
certain 1-loop amplitudes, in fact, do not exhibit any singularity in the
decompactification limit, but rather all the divergence arises at tree level.
This seems to be in tension even with just partial emergence.444 As we will
mention later, this phenomenon appears also in four dimensions and is
therefore not an artifact of eight dimensions.

Subsequently we refine the discussion and argue how this situation can be
ameliorated, by sweeping part of the problem into a renormalization of the
string coupling. In this way, at least partial emergence can be recovered.

The lesson is that the notion of emergence as the phenomenon of literally
integrating out states at the quantum level seems to be overly restrictive, at
least in the current framework.555A way out might be that the quantum
emergence of KK gauge fields is inherited from an emergence of gravity itself
in the higher dimensional theory [12]. We thank Irene Valenzuela for a
discussion of this possibility. Because of dualities, there is no absolute
distinction between what may be called tree-level or one-loop effect anyway.
Indeed the key features of the Swampland Distance Conjecture, such as the
appearance of massless towers of states at infinite distances in moduli space,
can be well captured in a purely geometric formulation at tree level,
irrespective of whether or not there exists a duality frame for which the
singularity arises at the quantum level.

### Physical interpretation of Kulikov models

After this general overview, we now summarize in more detail the results of
Sections 2 \- 5, which concern the infinite distance limits in the complex
structure moduli space of F-Theory on K3, and in particular their physical
interpretation.

According to the classic theory of semi-stable degenerations [43], a K3
surface generally splits into a union of several surface components at
infinite distance in complex structure moduli space. The possible
degenerations go by the name of Kulikov Type II and Type III models [44, 45,
46, 47].666Type I models appear at finite distance in moduli space and are not
of interest to us here. The theory of asymptotic Hodge structures guarantees
the existence of one (Type III) or two (Type II) transcendental elliptic
curves whose calibrated volumes vanish at infinite distance. From the point of
view of M-theory on the degenerating space, we therefore expect one or two
towers of asymptotically massless states from M2-branes wrapped an arbitrary
number of times along these vanishing cycles. This parallels the situation for
Type IIB compactifications on Calabi-Yau threefolds, where towers of states
appear from wrapped D3-branes along asymptotically vanishing 3-cycles, as in
[20, 28, 29, 30, 31]. However, it turns out that these states form in general
only part of the of the asymptotically light towers. This makes it crucial to
understand the nature of the asymptotically vanishing cycles and to interpret
the physics of the associated states.

It is at this stage that a refined geometric picture of the degeneration comes
into play. As is well-known to string theorists already from the works [48,
49, 50], the elliptic Type II Kulikov degenerations fall into two
qualitatively different sectors, called Type II.a and II.b in [51].

Type II.a models describe the famous stable degeneration limits where the
elliptic K3 breaks up into the union of two rational elliptic surfaces
intersecting over an elliptic curve. Such limits are dual to the $T^{2}$
compactification of the heterotic string in the limit of large volume, $T_{\rm
het}\to i\infty$. Indeed, we will argue that the two towers of wrapped
M2-branes have a clear interpretation in F-theory in terms of string
junctions: these form the two imaginary roots that enhance the non-abelian
gauge algebra to the double-loop algebra

$G_{\infty}=(\hat{E}_{9}\oplus\hat{E}_{9})/\sim\qquad\quad\text{(Type
II.a)}\,,$ (1.1)

where the quotient indicates that the two imaginary roots of the first double-
loop algebra $\hat{E}_{9}$ and those of the second $\hat{E}_{9}$ are
identified. These string junctions are the F-theoretic incarnation of two
towers of Kaluza-Klein states in the dual heterotic frame, as expected in the
stable degeneration limit. This establishes the Type II.a limits as complete
decompactification limits from 8d to 10d.

By contrast, the Type II.b Kulikov models represent limits in which the 10d
axio-dilaton diverges as $\tau_{\rm IIB}\to i\infty$, corresponding to weakly
coupled Type IIB orientifold limits. While it is true that the two towers of
states from wrapped M2-branes along the two vanishing cycles become
asymptotically light, they do not furnish the only parametrically leading
light states. Rather, an M2-brane wrapped on a degenerating 1-cycle in the
fiber gives rise to an asymptotically tensionless string, which is identified
with the perturbative Type IIB string. Its (denser) tower of excitations sits
at the same scale as the M2-brane states, and the limit is to be interpreted
as an emergent string limit [15] in 8d, rather than a decompactification
limit.

While the Type II limits have been known from the classic works [49, 50, 51],
the remaining Type III limits at infinite distance have only recently been
understood systematically for elliptic K3 surfaces [1, 38, 39], by analysing
their associated Weierstrass models. In particular, the refined classification
in terms of Type III.a and III.b limits has been established in our companion
paper [1] in this way. In both cases, the Weierstrass model of the degenerate
K3 surface breaks up into a chain of possibly degenerate elliptic fibrations,
where the generic elliptic fibers of the $i^{\rm th}$ components are of Type
I${}_{n_{i}}$ in the language of Kodaira’s classification of elliptic
fibrations.777Recall that a smooth elliptic fiber is of Type I0, while a fiber
of Type Ik with $k>1$ is a nodal curve and the result of shrinking the $(k,0)$
cycle. It is guaranteed that $n_{i}>0$ for the middle components and hence
their fibrations degenerate, while three possibilities arise for the end
components:

  * •

Both end components are rational elliptic surfaces (Type III.a of first kind);

  * •

one end coomponent is a rational elliptic surface and the other one is a
surface with generic In>0 fibers (Type III.a of second kind);

  * •

both end components are surfaces with generic In>0 fibers (Type III.b).

Type III.a models are similar to the Type II.a models described above, in the
following sense: At the intersection of the rational elliptic end component(s)
with the adjacent surface component with generic fibers of Type In>0, one
vanishing transcendental torus is localised. M2-branes wrapped thereon can be
interpreted as string junctions responsible for the enhancement of an
$E_{9-n}$ algebra to the affine algebra $\hat{E}_{9-n}$. For Type III.a limits
with two rational elliptic end components, the symmetry algebra hence contains
an affine algebra

$G_{\infty}\supset(\hat{E}_{9-n}\oplus\hat{E}_{9-m})/\sim\,,\quad 1\leq
n,m\leq 9\,,\qquad\quad\text{(Type III.a of first kind)}\,,$ (1.2)

while in models with only a single rational elliptic component only one affine
algebra factor can arise. The imaginary root responsible for the affinisation
of the symmetry algebra gives rise to a single tower of Kaluza-Klein modes,
and the limit is therefore a partial decompactification limit from 8d to 9d.

Type III.b limits, on the other hand, are weak coupling limits in which
$\tau_{\rm IIB}\to i\infty$, while at the same time the complex structure of
the Type IIB torus diverges. Such limits must therefore be full
decompactifications from 8d to 10d. This might seem, at first sight, to be in
tension with the fact that one finds only a single, rather than two
independent light towers from wrapped M2-branes. This tower can be interpreted
as the winding tower along the small 1-cycle on $T^{2}_{\rm IIB}$. The missing
second tower must correspond to a proper supergravity Kaluza-Klein tower that
cannot be detected as easily from geometry, even though the physical
interpretation of the limit is unambiguous.

As explained before, the limits of Type II.a and III.a correspond to
decompactification limits in the heterotic duality frame, where the infinite
distance limit is to be understood as the limit

  * •

$T_{\rm het}\to i\infty$ with $U_{\rm het}$ finite (Type II.a);

  * •

$T_{\rm het}\to i\infty$ with $T_{\rm het}/U_{\rm het}$ finite (Type III.a).

In the original F-theory duality frame, by contrast, the infinite directions
in moduli space can be interpreted as non-compact directions in the moduli
space of 7-branes. While naively one might think that the open string or brane
moduli space, on a compactification space of finite volume and at finite
values of the complex structure, should be compact, this is in general not
true if one includes mutually non-local 7-branes. The non-compact directions
appear when certain such $(p,q)$ branes approach each other. More precisely,
as explained in [52], the affinisation of an $E_{n}$ algebra to $\hat{E}_{n}$
appears, in the brane picture, when a single 7-brane of certain $(p,q)$-type
approaches a stack of branes with some gauge algebra $E_{n}$. In such a
situation the elliptic fiber degenerates to a non-minimal type in the sense of
Kodaira’s classification, i.e., the vanishing orders of the Weierstrass
sections $f$ and $g$ both exceed the critical values of $4$ and $6$. The Type
II and Type III Kulikov models are obtained by removing these singularities by
a chain of blowups in the base. This establishes the following generic
correspondence:
Non-minimal Kodaira fibers in codimension one $\Longleftrightarrow$ Affine or
loop extensions of Lie algebras $\Longleftrightarrow$ Decompactification
limits in a dual frame at infinite distance

In Section 2 we begin by reviewing the affinisation of Lie algebras in
F-theory in the language of $(p,q)$ 7-branes, as established in [52]. In
Section 3 we elaborate on the physical interpretation of the possible infinite
distance limits of elliptic K3 surfaces, as probed by F-theory. An important
test of the resulting picture is provided in Section 4, where we
systematically construct infinite distance limits for a Weierstrass model with
minimal non-abelian gauge algebra $E_{7}\oplus E_{8}$; the geometric results
there can be compared quantitatively with the dual heterotic description via
the mirror map worked out in [53]. In Section 5 we classify the possible
maximal enhancements of non-abelian gauge algebras in 9d, as determined by the
Type III.a Kulikov models, and find complete agreement with the previous
results of [54, 55, 56].

##  2 Review: Affine Algebras and infinite Distance Limits

Infinite distance limits in the complex structure moduli space of F-theory on
elliptic K3 surfaces can be approached from several different angles. The most
systematic one is via the theory of semi-stable degenerations of K3 surfaces,
and we will follow this route beginning with Section 3. To facilitate the
physical interpretation of the geometric results, it is beneficial to first
sharpen our intuition on the possible phenomena that we expect to encounter at
infinite distance in complex structure moduli space.

An obvious non-compact direction is the 10d string coupling, $g_{s}$, i.e. the
imaginary part of the axio-dilaton, $\tau=C_{0}+\frac{i}{g_{s}}$. It is
geometrised in F-theory as the complex structure modulus of the elliptic
fiber. In absence of other competing effects, infinite distance limits along
the direction $\tau\to i\infty$ in the moduli space will describe weak
coupling limits. These can be superimposed with infinite distance limits in
the complex structure of the torus $T^{2}_{\rm IIB}$ of the associated Type
IIB orientifold.

A less evident type of non-compact directions occurs in genuinely non-
perturbative configurations with mutually non-local 7-branes, and this
corresponds to the formation of certain affine, or more generally loop
algebras. Before developing a clear geometric description for such limits in
the subsequent sections, we will first take a complementary viewpoint as
provided via the formalism of string junctions, which we now briefly review
following the discussion in [52].

When we compactify F-theory to eight dimensions, the physical compactification
space, as seen from the Type IIB string perspective, is a rational curve which
forms the base $B$ of an elliptic K3 surface $X$. The 24 singular fibers of
the K3 surface correspond to the locations of 7-branes of general
$[p,q]$-type. Upon encircling such a location counter-clockwise, as depicted
in Figure -1198, a general $(r,s)$ string undergoes an $SL(2,\mathbb{Z})$
monodromy of the form

$\displaystyle\left(\begin{matrix}r\\\ s\end{matrix}\right)\rightarrow
M_{[p,q]}\left(\begin{matrix}r\\\ s\end{matrix}\right)\,,\qquad
M_{[p,q]}=\left(\begin{matrix}1+pq&-p^{2}\\\ q^{2}&1-pq\end{matrix}\right)\,.$
(2.1)

Figure -1198: Monodromy around a $[p,q]$ 7-brane in F-theory on K3.

Suitable combinations of such $[p,q]$ type 7-branes are well known to realize
Lie algebras of ADE type [57]. Adopting the conventions of [58], we consider
the following set of 7-branes $X_{[p,q]}$ to serve as building blocks:

$\displaystyle A=X_{[1,0]}\,,\qquad B=X_{[1,-1]}\,,\qquad C=X_{[1,1]}\,.$
(2.2)

Then gauge theories of ADE type are constructed as configurations of branes
including the ones in Table 2.1.888Recall that as for characterizing the
monodromies, one considers a brane combination $X_{[p_{1},q_{1}]}\ldots
X_{[p_{n},q_{n}]}$ as a chain and takes the branch cut induced by the
$SL(2,\mathbb{Z})$ backreaction of the collection of $[p,q]$ 7-branes as a
straight line flowing down from $X_{[p,q]}$. The monodromy associated with a
brane configuration written as such a chain $X_{[p_{1},q_{1}]}\ldots
X_{[p_{n},q_{n}]}$ is then computed as the matrix product
$M_{[p_{n},q_{n}]}\cdot\ldots\cdot M_{[p_{1},q_{1}]}$.

The singularity in moduli space that leads to an enhancement to some finite
Lie algebra of ADE type corresponds to a finite distance motion in the 7-brane
moduli space. This can be formulated as a finite distance deformation in the
complex structure moduli space of the associated elliptic K3 surface $X$. The
corresponding singularities in the elliptic fiber were famously classified by
Kodaira and Néron and can be realized by tuning the vanishing orders of the
characteristic functions $f$ and $g$ of the Weierstrass model for $X$, as
recalled in Table 2.2. This raises the natural question whether also infinite
distance deformations in this brane moduli space are possible. As we will
argue, such deformations correspond to the formation of infinite dimensional
loop algebras, rather than ordinary ADE Lie algebras.

In [58] it was explained how an exceptional Lie algebra999Here $E_{1}=A_{1}$,
$E_{2}=A_{1}\oplus u(1)$, $E_{3}=A_{2}\oplus A_{1}$, $E_{4}=A_{4}$,
$E_{5}=D_{5}$. $E_{N}$ can be enhanced to a loop algebra, more precisely an
affine algebra of Kac-Moody type, by addition of a single extra brane.
Specifically, for $N=1\ldots,8$, the finite exceptional Lie algebra $E_{N}$
can be enhanced into the affine algebra $\hat{E}_{N}$ by adding to the brane
stack $A^{N-1}BCC$ an extra 7-brane $X_{[3,1]}$:

$\displaystyle
E_{N}\stackrel{{\scriptstyle+X_{[3,1]}}}{{\longrightarrow}}\hat{E}_{N}\,,\qquad
N=1,\ldots,8\,,$ (2.3)
$\displaystyle\hat{E}_{N}=E_{N}X_{[3,1]}=A^{N-1}BCCX_{[3,1]}=A^{N-1}BCBC\,.$
(2.4)

A second series of affine enhancements can be constructed as

$\displaystyle\hat{\tilde{E}}_{N}\stackrel{{\scriptstyle+X_{[4,1]}}}{{\longrightarrow}}\hat{\tilde{E}}_{N}\,,\qquad
N=0,\ldots,8\,,$ (2.5)
$\displaystyle\hat{\tilde{E}}_{N}=(A^{N}X_{[2,-1]}C)X_{[4,1]}\,,$ (2.6)

which for $2\leq N\leq 8$ turns out to be equivalent to the series (2.3).
However for $N=1$ and $N=0$ this yields independent enhancements, with
$\tilde{E}_{1}=u(1)$ and $\tilde{E}_{0}=\emptyset$.

$G$ | branes | Monodromy $M_{G}$
---|---|---
$A_{N}$ | $A^{N+1}$ | $\left(\begin{matrix}1&-N-1\\\ 0&1\end{matrix}\right)$
$D_{N}$ | $A^{N}BC$ | $\left(\begin{matrix}-1&N-4\\\ 0&-1\end{matrix}\right)$
$E_{N}$ | $A^{N-1}BCC$ | $\left(\begin{matrix}-2&2N-9\\\ -1&N-5\end{matrix}\right)$
Table 2.1: Brane content and monodromies for ADE groups on elliptic K3
surfaces.

The resulting brane configuration induces an $SL(2,\mathbb{Z})$ monodromy of
the form

$\qquad M_{\hat{E}_{N}}=M_{\hat{\tilde{E}}_{N}}=\left(\begin{matrix}1&9-N\\\
0&1\end{matrix}\right)\,.$ (2.7)

In particular, it leaves invariant the charges of a $(1,0)$ string encircling
the configuration because101010For simplicity of notation, here and in the
sequel we will not explicity refer to $\hat{\tilde{E}}_{N}$ as opposed to
$\hat{E}_{N}$ unless the difference is important.

$M_{\hat{E}_{N}}\delta_{1}=\delta_{1}\,,\qquad\delta_{1}=\left(\begin{matrix}-1\\\
0\end{matrix}\right)\,.$ (2.8)

Such a $(1,0)$ string winding around a non-trivial path encircling the branes
gives rise to a particle in the eight-dimensional field theory, which becomes
massless in the limit for which all branes in the $\hat{E}_{N}$ configuration
coincide, i.e. precisely when $E_{N}$ enhances to $\hat{E}_{N}$. See Figure
-1197. The string junctions associated with this state satisfy the relations
[58]

$\displaystyle\delta_{1}\cdot\delta_{1}=0\,,\qquad\delta_{1}\cdot\alpha^{E_{N}}_{j}=0\
$ (2.9)

within the string junction lattice, where $\alpha^{E_{N}}_{j}$ is a string
junction associated to one of the simple roots of the Lie algebra $E_{N}$.
Group theoretically both conditions identify $\delta_{1}$ as the imaginary
root within the root lattice of the affine Lie algebra $\hat{E}_{N}$ [58]. The
possible rank-one affine enhancements which can be obtained in this manner are
[58]

$\displaystyle\hat{E}_{8},\quad\hat{E}_{7},\quad\hat{E}_{6},\qquad\hat{E}_{5}=\hat{D}_{5},$
(2.10)
$\displaystyle\hat{E}_{4}=\hat{A}_{4},\quad\hat{E}_{3}=\widehat{A_{2}\oplus
A_{1}},\quad\hat{E}_{2}=\widehat{A_{1}\oplus
u(1)},\quad\hat{E}_{1}=\hat{A}_{1},\quad\hat{\tilde{E}}_{1}=\widehat{u(1)},\quad\hat{\tilde{E}}_{0}=\hat{\emptyset}\,.$

In this list we include for $\hat{E}_{3}$ and $\hat{E}_{2}$ the imaginary
roots for both involved Lie algebras, but identify them such that the total
rank increases just by one.

Since the condition for a string junction $J$ to give rise to a BPS state is
that $J\cdot J\geq-2$, any multiple

$n\,\delta_{1}\,,\qquad n\in\mathbb{Z}$ (2.11)

gives rise to a BPS state. As a result, we obtain an infinite tower of
massless BPS states from the collision of an $E_{N}$ brane stack and an
$X_{[3,1]}$ brane, or equivalently in the limit of enhancing $E_{N}$ to its
affine algebra $\hat{E}_{N}$.

Based on a monodromy analysis, ref. [58] furthermore identifies a number of
rank-two enhancements. The only one which turns out to make an appearance in
our context is the enhancement from $E_{8}$ to the double-loop algebra
$\hat{E}_{9}$, which is realised as the brane configuration

$\displaystyle\hat{E}_{9}=A\hat{E}_{8}=AE_{8}X_{[3,1]}=A^{8}BCCX_{[3,1]}=A^{8}BCBC\,.$
(2.12)

Its monodromy matrix

$M_{\hat{E}_{9}}=\left(\begin{matrix}1&0\\\ 0&1\end{matrix}\right)$ (2.13)

leaves invariant a 2-dimensional lattice of string junctions spanned by

$\displaystyle\delta_{1}=\left(\begin{matrix}-1\\\
0\end{matrix}\right)\,,\qquad\delta_{2}=\left(\begin{matrix}0\\\
-1\end{matrix}\right)\,.$ (2.14)

These satisfy

$\displaystyle\delta_{i}\cdot\delta_{j}=0\,,\qquad\delta_{1}\cdot\alpha^{E_{8}}_{j}=0\,$
(2.15)

and therefore act as the two imaginary roots in the enhancement of $E_{8}$ to
the double loop algebra $\hat{E}_{9}$ (which in fact is not an affine
algebra). In the limit of enhancement, one finds two towers of massless BPS
states from the junctions

$\displaystyle n_{1}\delta_{1}\,,\qquad n_{2}\delta_{2}\,,\qquad
n_{i}\in\mathbb{Z}\,.$ (2.16)

The appearance of one or two infinite towers of massless states suggests that
the affine or double loop enhancements occur at infinite distance in the
moduli space. We will show that this intuition is indeed correct, and that the
deformations leading to loop algebras are in fact the only possible infinite
distance degenerations in the complex structure moduli space of F-theory on K3
except for the weak coupling limits (possibly followed by an infinite distance
degeneration in the complex structure of the Type IIB torus $T^{2}_{\rm
IIB}$). In the Weierstrass models, loop enhancements will be identified as
deformations giving rise to certain non-minimal Kodaira singularities over
codimension-one loci on the base $B$. To remove the non-minimal singularities,
a sequence of blowups leads to a degeneration of the K3 into multiple
components, which we identify with certain Kulikov models [1]. From a physics
perspective, the tower of states associated with the imaginary roots $\delta$
play the role of a Kaluza-Klein tower in the dual heterotic string and signals
decompactification from 8d, either to 9d for affine enhancements to
$\hat{E}_{N}$ with $N\leq 8$, or to 10d for the loop enhancement to
$\hat{E}_{9}$. In general, these states may form only a subset of the
parametrically leading towers of states which become light, and one of our
tasks will be to identify the full set of leading towers in order to establish
the correct physical interpretation of the infinite distance limits.

Branes | Algebra | Kodaira Type | ${\rm ord}(f)$ | ${\rm ord}(g)$ | ${\rm ord}(\Delta)$
---|---|---|---|---|---
$A^{n+1}$ | $A_{n}$ | In+1 | 0 | 0 | $n+1$
$A^{n}BC$ | $D_{n}$ | I${}^{\ast}_{n-4}$ | 2 | 3 | $n+2$
$A^{5}BC^{2}$ | $E_{6}$ | IV∗ | $\geq 3$ | 4 | 8
$A^{6}BC^{2}$ | $E_{7}$ | III∗ | 3 | $\geq 5$ | $9$
$A^{7}BC^{2}$ | $E_{8}$ | II∗ |  $\geq$ 4 | $5$ | $10$
Table 2.2: Vanishing orders for a Weierstrass model
$y^{2}=x^{3}+fxz^{4}+gz^{6}$ with discriminant $\Delta=4f^{3}+27g^{2}$
realising the ADE brane configurations of Table 2.1 as singularities in the
fiber according to the Kodaira classification.

##  3 Kulikov Models as Emergent String Limits or Decompactifications

We now present the physical interpretation of the infinite distance limits in
the complex structure moduli space of elliptic K3 surfaces. We begin with a
quick review of the geometric results from our companion paper [1] which
refines the classification of infinite distance limits via Kulikov models in a
way suitable for our purposes. The four canonical types of such geometric
infinite distance, Type II.a and II.b [51] and Type III.a and III.b [1] (see
also [39]) are then subsequently analysed from the point of view of F-theory.

###  3.1 Refined Kulikov models for elliptic K3 surfaces

Infinite distance limits in the complex structure moduli space of an elliptic
K3 surface can be studied within the framework of semi-stable degenerations
[43]. For simplicity we will restrict ourselves to one-parameter degenerations
as this turns out to be sufficient for understanding the asymptotic physics.

A (one-parameter) degeneration ${\cal X}$ of K3 surfaces is a family of K3
surfaces $X_{u}$, where $u\in D=\\{u\in\mathbb{C}:|u|<1\\}$. The central fiber
of the degeneration, $X_{u=0}$, is the degenerate surface which we aim to
study. The degeneration is semi-stable if $X_{0}$ is a union of surfaces each
appearing with multiplicity one such that the total family ${\cal X}$ is
smooth as a threefold and the singularities of $X_{0}$, if there are any, are
locally of normal crossing form. By the important results of [59, 44, 45, 46],
every degeneration of K3 surfaces can be brought into the form of a Kulikov
model: This means that the degeneration is semi-stable and in addition the
family ${\cal X}$ is Calabi-Yau. For general K3 surfaces, the Kulikov models
enjoy a classification into models of Type I (at finite distance), Type II and
Type III (both at infinite distance).

It is known from the general theory of degenerations of K3 surfaces [43] that
the central fiber $X_{0}$ of a Type II Kulikov model exhibits two
transcendental elliptic curves, $\gamma_{1}$ and $\gamma_{2}$, whose
calibrated volumes vanish,

$\displaystyle{\rm vol}(\gamma_{i})=\int_{\gamma_{i}}\Omega_{0}=0\,.$ (3.1)

Here $\Omega_{0}$ denotes the (2,0) form on the degenerate surface $X_{0}$.
For models of Kulikov Type III there exists only a single such transcendental
elliptic curve of asymptotically vanishing volume. If we compactify M-theory
on such K3 surfaces, we therefore obtain one or two towers of asymptotically
massless BPS particles from M2-branes that wrap the vanishing tori arbitrarily
many times. However, as we will see, in general these particles form only a
subset of the asymptotically massless towers, and sometimes they do not form
the parametrically leading towers. A full understanding of the asymptotic
physics therefore requires a refined geometric picture of the degeneration and
an analysis of the nature of states which become massless in compactifications
of M- or F-theory.

Figure -1197: Monodromy and imaginary junctions for affine Lie algebras
$\hat{E}_{N}$ (left) and for the loop algebra $\hat{E}_{9}$ (right).

Fortunately, if we restrict ourselves to degenerations of elliptically fibered
K3 surfaces, such a geometric picture has been obtained for Type II
degenerations in [51] and recently for Type III degenerations in our companion
paper [1], as well as in [39]. By blowing down all exceptional curves in the
fiber, one obtains a family $\cal Y$ of Weierstrass models ${Y}_{u}$. In
particular, to the degenerate K3 surface $X_{0}$ one associates a Weierstrass
model $Y_{0}$.

Kulikov models of Type I correspond to the degenerations of the elliptic fiber
which lie at finite distance in the complex structure moduli. The degeneration
$Y_{0}$ is a Weierstrass model over a single rational curve $B_{0}$ and has
singularities in the elliptic fiber of the familiar Kodaira-Néron types as
listed in Table 2.2.

For the infinite distance degenerations of elliptic Kulikov Types II and III,
on the other hand, the base $B_{0}=\cup_{i=0}^{P}B^{i}$ is a chain of rational
curves $B^{i}$. The surface $Y_{0}$ correspondingly decomposes into a chain of
surfaces

$Y_{0}=\cup_{i=0}^{P}Y^{i}\,,$ (3.2)

where each surface component $Y^{i}$ is a possibly degenerate Weierstrass
model over $B^{i}$. By a combination of birational transformations and base
changes, elliptic Type II and Type III Kulikov models can be brought into
certain canonical forms, which are distinguished as follows:

  1. 1.

Elliptic Type II models [51]: The degeneration $\cal X$ can be brought into
the canonical form where $X_{0}=X^{1}\cup X^{2}$ such that $E:=X^{1}\cap
X^{2}$ is an elliptic curve.

     * •

Type II.a (see also [48, 49]): $X^{1}$ and $X^{2}$ are two rational elliptic
surfaces, i.e. elliptic fibrations with 12 singular fibers. The intersection
curve $E$ is the smooth elliptic fiber over the intersection point $B^{1}\cap
B^{2}$ in the base.

     * •

Type II.b (see also [50]): The base $B_{0}$ is a non-degenerate rational
curve, and $X^{1}$ and $X^{2}$ are both rationally fibered over $B_{0}$. The
intersection $X^{1}\cap X^{2}$ is a ramified double cover of $B_{0}$, which
indeed describes an elliptic curve $E$.

  2. 2.

Elliptic Type III models [1, 39]: The Weierstrass degeneration $\cal Y$
associated with $\cal X$ can be brought into a canonical form such that each
intermediate surface $Y^{i}$, $i=1,\ldots,{P-1}$ appearing in (3.2) is a
degenerate Weierstrass model over a rational base component $B^{i}$ with
generic fibers of Kodaira Type I${}_{n_{i}}$, $n_{i}>0$. Over special points
of $B^{i}$, the singularity can enhance to an $A$-type singularity (in the
sense of the Kodaira classification of Table 2.2).111111Note that our
characterisation of the singularity enhancements is in the context of the
degenerate K3 surface, while the stated feature of generic I${}_{n_{i}}$
fibers is observed in the context of the 3-fold $\mathcal{Y}$. Hereafter we
will not explicitly distinguish the contexts unless potential confusions
arise. Without loss of generality one can assume that at least one
intermediate surfaces are present (i.e. $P>1$) and that no special fibers are
located in the intersection loci of the components.

     * •

Type III.a: One or both of the end components $Y^{0}$ and $Y^{P}$ are rational
elliptic surfaces. If the end component is not rational elliptic, it is a
degenerate Weierstrass model with In>0 fibers over generic points and
precisely two singularities of $D$-type (along with possibly extra $A$-type
singularities).

     * •

Type III.b: Each of the two end components is a degenerate Weierstrass model
with In>0 fibers over generic points and precisely two singularities of
$D$-type (along, possibly, with extra $A$-type singularities).

The family $\cal Y$ of Weierstrass model can be parametrized as

$\displaystyle Y_{u}:\qquad y^{2}=x^{3}+f_{u}(s,t)xz^{4}+g_{u}(s,t)z^{6}\,,$
(3.3)

with discriminant

$\Delta_{u}=4f_{u}^{3}+27g_{u}^{2}\,.$ (3.4)

For $u\neq 0$, $f_{u}(s,t)$ and $g_{u}(s,t)$ are homogeneous polynomials of
degree $8$ and $12$, respectively, in homogenous coordinates $[s:t]$ of the
base $B_{u}=\mathbb{P}^{1}_{[s:t]}$. The degeneration occurs in the limit
$u\to 0$. Note that, for notational simplicity, we will hereafter omit the
subscript $u$ in the Weierstrass sections $f$ and $g$, as well as in the
discriminant $\Delta$, of which $u$-dependence will be assumed in the context.
Then, Kulikov models of Type II.b have the property that the Weierstrass
sections in the limit $u\to 0$ take the special form

$f|_{u=0}=-3h^{2}\,,\qquad g|_{u=0}=2h^{3}\,,$ (3.5)

while otherwise all vanishing orders respect the minimality bound according to
Kodaira’s classification.121212The latter condition is satisfied if and only
if the degree-$4$ polynomial $h$ has four distinct roots. As detailed in [1],
models of Type II.a, III.a and III.b, on the other hand, are in one-to-one
correspondence with Weierstrass models which, for $u=0$, acquire one or
several non-minimal fibers. If we choose one of the non-minimal fibers to lie
at $s=0$, this implies for the vanishing orders that131313The classification
of vanishing orders is understood modulo base change. This means that one has
to consider the maximal vanishing orders which can be obtained by transforming
$u\to u^{k}$ for some integer $k\geq 1$.

$\displaystyle{\rm ord}_{\cal Y}(f,g,\Delta)|_{u=s=0}=(4+m,6+n,12+k)\,,\qquad
m\geq 0\,,n\geq 0\,,k\geq 0\,,$ (3.6)

with the following correspondence:

  * •

If $k=0$, blowups and base changes can bring the model into Kulikov form of
Type II.a.

  * •

If $m=0$, $n=0$ and $k>0$, the model can be brought into Kulikov form of Type
III.a or III.b.

  * •

If $m>0$ and $n>0$ (and hence $k>0$), the model can be of Kulikov form of Type
I (finite distance) or Type II/III.

Models of Type III.a with only a single rational elliptic end component
require, in addition to the above non-minimal singularity, precisely one of
the two types of tunings that we will describe below, while Models of Type
III.b require the both. The relevant tunings are as follows:

  1. 1.

The Weierstrass sections restricted to the curve $u=0$ take the form

$\displaystyle f|_{u=0}=-3s^{4}\hat{h}^{2}\,,\qquad
g|_{u=0}=2s^{6}\hat{h}^{3}\,,$ (3.7)

where $\hat{h}$ is a polynomial of degree $2$ in $[s:t]$.

  2. 2.

If we denote by $f_{-}$ and $g_{-}$ the parts of $f$ and $g$ consisting only
of the terms of degrees $d_{f}\leq 4$ and, respectively, $d_{g}\leq 6$ in $s$,
whose vanishing orders in $u$ are precisely $(4-d_{f})P$ and $(6-d_{g})P$,
then these parts take the form [1]

$f_{-}=-3t^{4}(L_{0}u^{2P}t^{2}+L_{1}u^{P}st+L_{2}s^{2})^{2}\,,\quad
g_{-}=2t^{6}(L_{0}u^{2P}t^{2}+L_{1}u^{P}st+L_{2}s^{2})^{3}\,.$ (3.8)

Here $L_{i}\in\mathbb{C}$ and the parameter $P$ coincides with the required
number of blowups (i.e. the number of components in (3.2)). For the precise
statement see [1].

With this preparation, we are now in a position to analyse the towers of
states that become asymptotically massless in both types of limits. This can
be approached either in the language of M-theory as before, or via duality
directly in F-theory, by analysing the spectrum of branes that wrap the
degenerate K3 surface.

###  3.2 Type II.b Kulikov models as emergent string limits

Let us begin with the Kulikov Type II.b limits, which were first described in
this language in the F-theory literature in [50]. The physics of such limits
is a perturbative weak coupling limit of Sen type [60, 61].

By definition [51], a Type II.b Kulikov model is constructed as a blowup of a
singular elliptic fibration over the base $\mathbb{P}^{1}_{[s:t]}$ for which
the generic elliptic fiber has degenerated to a nodal curve. This degeneration
is the result of the collapse of the $(1,0)$ cycle in the elliptic fiber,
which we denote by $S_{A}$, while the dual $(0,1)$ cycle $S_{B}$ of the
elliptic fiber survives in the nodal fiber. Blowing up the node replaces the
nodal fiber by a Kodaira Type I2 fiber consisting of a pair of rational curves
intersecting, generically, in two points $p_{1}$ and $p_{2}$. See Figure -1196
for an illustration. Over four points, $p_{1}$ and $p_{2}$ come together to
form a double point. The two intersection points $p_{1}$ and $p_{2}$ define a
bi-section, i.e. a double cover of the base branched over the four points
where $p_{1}$ and $p_{2}$ coincide. The 2-cycle defined by the bi-section is
identified with the elliptic curve $E=X^{1}\cap X^{2}$.

To understand the origin of the transcendental 2-cycles $\gamma_{i}$, $i=1,2$,
as advertised in (3.1), note that the group $H_{1}(E,\mathbb{Z})$ of the
intersection curve $E=X^{1}\cap X^{2}$ is spanned by two 1-cycles, which we
denote by $\sigma_{1}$ and $\sigma_{2}$. By suitably combining the collapsed
one-cycle $S_{A}$ with each $\sigma_{i}$ one constructs two homologically non-
trivial transcendental 2-cycles with the topology of a torus, and these two
2-cycles are the objects $\gamma_{1}$ and $\gamma_{2}$ whose calibrated volume
(3.1) vanishes in the degenerate situation.

Figure -1196:  Kulikov Type II.b degeneration. Note that the nodes exchange
under mondromy around the singular configuration shown in the center.

If we compactify M-theory on $X_{0}=X^{1}\cup X^{2}$, we obtain the following
asymptotically massless objects in seven dimensions: First, an M2-brane
wrapping the vanishing cycle $S_{A}$ gives rise to an asymptotically
tensionless, fundamental Type II string. This string is non-BPS because
$S_{A}$ is actually homologically trivial. Second, wrapping an M2-brane
arbitrarily many times along either of the two calibrated 2-cycles,
$\gamma_{1}$ and $\gamma_{2}$, yields a tower of BPS particles. Note that the
existence of these BPS particles is guaranteed because the 2-cycles
$\gamma_{i}$ have the topology of a torus and can therefore be wrapped
arbitrarily often. The BPS particles are to be identified with the winding
states from wrapping the Type II string along the two 1-cycles $\sigma_{1}$
and $\sigma_{2}$ of $E$. The latter sit parametrically at the same scale as
the string excitation tower. The winding tower is of course dual to a
corresponding Kaluza-Klein tower of supergravity modes of the same parametric
mass scale. All in all, the asymptotic spectrum of the Type II.b Kulikov
degeneration carries the signature of an Emergent String Limit as defined in
[15].

Note that if it were not for the asymptotically tensionless string obtained by
wrapping an M2-brane along $S_{A}$, we would have incorrectly characterised
the infinite distance limit as a KK decompactification. However the KK modes
form only part of the spectrum, and due to the higher density of states it is
actually the tower of string excitations which is defining the asymptotic
physics as an equi-dimensional weak coupling limit.

From the perspective of F-theory in 8d, this behaviour is of course no
surprise. To realise an I2 fiber over generic points of the base, the
vanishing orders of $f$, $g$ and $\Delta$ of the Weierstrass model (3.3) must
become

$\displaystyle{\rm ord}_{\cal Y}(f,g,\Delta))|_{u=0}=(0,0,2)\quad\text{at
generic points of }\,\,\mathbb{P}^{1}_{[s:t]}\,.$ (3.9)

The behaviour of the 10d Type IIB axio-dilaton, $\tau=C_{0}+\frac{i}{g_{s}}$,
at a generic point of the base can then be read off from the $j$-function:

$\displaystyle
j(\tau)\sim\frac{f^{3}}{\Delta}\sim\frac{1}{u^{2}}\quad\Longrightarrow\quad\tau\to
i\infty\qquad{\rm for}\,\,\,u\to 0\,.$ (3.10)

A convenient way to realise the vanishing orders is to consider the
degeneration [60]

$\displaystyle f_{u}=-3h^{2}+u\,\eta\,,\qquad
g_{u}=-2h^{3}+u\,h\,\eta-\frac{u^{2}}{12}\chi\,,\qquad\Delta=-9\,u^{2}\,h^{2}\,(\eta^{2}-h\chi)+{\cal
O}(u^{3})\,,$ (3.11)

where $h$, $\eta$ and $\chi$ are polynomials of degree $4$, $8$ and $12$. In
the Type IIB orientifold interpretation, the limit $u\to 0$ gives rise to an
orientifold compactification on the torus $E$ and the four zeroes of $h=0$
correspond to the location of the orientifold planes. If we disregard the
overall factor of $u^{2}$ in $\Delta$ as

$\Delta=u^{2}\Delta^{\prime}\,,$ (3.12)

the vanishing orders at the orientifold become

${\rm ord}(f_{0},g_{0},\Delta^{\prime}_{0})|_{h=0}=(2,3,2)\,,$ (3.13)

where the subscripts indicate that $u=0$ has been taken to begin with. Away
from these localised regions, the theory reduces to a weakly coupled,
perturbative Type IIB string, which is non-BPS in 8d since the Kalb-Ramond
field $B_{2}$ is projected out by the orientifold projection. The string tower
appears parametrically at the same scale as the two KK towers associated with
each one-cycle of $E$, and both scales vanish in units of the 8d Planck scale.

Finally, as remarked in [1], a more general parametrisation of weak coupling
limits is to simply take

$f_{u}=-3h^{2}+u^{a}\eta\,,\qquad g_{u}=-2h^{3}+u^{b}\rho\,,$ (3.14)

for integral $a\geq 1$, $b\geq 1$ and generic sections $\eta$ and $\rho$ of
suitable degree. In this case, the general theorems of [51] guarantee the
existence of a birational transformation which, when followed by a suitable
base change, brings the fibration into Kulikov form of Type II.b.

###  3.3 Type II.a Kulikov models as decompactification limits

The spectrum of asymptotically massless states in limits of Kulikov Type II.a
is very different. The two del Pezzo (dP9) surfaces $X^{1}$ and $X^{2}$ into
which the K3 surface degenerates are each elliptically fibered over a rational
curve, $B^{1}$ and $B^{2}$, respectively. In other words, the base
$\mathbb{P}^{1}_{[s:t]}$ of the K3 surface $X_{u}$ splits for $u=0$ into the
union of $B^{1}$ and $B^{2}$ intersecting at a single point ${\cal
P}=B^{1}\cap B^{2}$, and the elliptic double curve $E=X^{1}\cap X^{2}$
represents the common elliptic fiber over ${\cal P}$. This geometry was
described in the F-theory literature early on in [48, 49, 50]. The two
2-cycles $\gamma_{i}$ can therefore be constructed by fibering the two one-
cycles $\sigma_{i}$ in $H_{1}(E,\mathbb{Z})$ over a 1-cycle $\Sigma$ that
encircles the intersection point ${\cal P}$ on either $B^{1}$ or $B^{2}$. The
geometry is sketched in Figure -1195.

Figure -1195:  Kulikov Type II.a degeneration. The base degenerates into two
$\mathbb{P}^{1}$s that form the bases of two elliptic del Pezzo surfaces dP9.

Viewed as a geometric 1-cycle on $B^{1}$ or $B^{2}$, $\Sigma$ can be slipped
off the base sphere by deforming it to the anti-podal pole and is therefore
trivial, but fibering both $\sigma_{i}$ over $\Sigma$ near ${\cal P}$ indeed
gives rise to two non-trivial 2-cycles $\gamma_{i}$. The asymptotic vanishing
of their calibrated volume (3.1) for degeneration parameter $u=0$ is an
intuitive consequence of the fact that $\Sigma$ can be contracted towards the
intersection point ${\cal P}$.

The seven-dimensional compactification of M-theory on K3 is therefore
characterised by the appearance of two towers of asymptotically massless
particles which arise from M2-branes wrapped any number of times on the two
2-cycles $\gamma_{1}$ and $\gamma_{2}$. Unlike for limits of Type II.b, these
towers of BPS particles are not accompanied by a tower of excitations from an
asymptotically tensionless string. The reason is that the M2 brane cannot be
wrapped along the 1-cycle $\Sigma$ on either of the base components, because
viewed in isolation $\Sigma$ can simply be slipped off $B^{1}$ or $B^{2}$.

From the point of the view of the dual F-theory compactification on K3, these
two towers of BPS particles correspond to two towers of asymptotically
massless particles in 8d: These arise from string junctions of the form
$n_{1}\,\delta_{1}$ and $n_{2}\,\delta_{2}$, where $n_{i}\in\mathbb{Z}$
represents the winding numbers. If we denote by $\sigma_{1}$ and $\sigma_{2}$
the $(1,0)$ and $(0,1)$ cycles on $E$, then an M2-brane on $\gamma_{1}$
uplifts to a $(1,0)$ string and the cycle on $\gamma_{2}$ uplifts to a $(0,1)$
string, each encircling the intersection point ${\cal P}$ along $\Sigma$. The
important difference to the M-theory viewpoint is that in F-theory, it is not
possible to freely slip the $(1,0)$ and $(0,1)$ strings on $\Sigma$ to the
other pole of the curves $B^{1}$ or $B^{2}$, because the strings would have to
cross the 12 7-branes located on either of the two curves.

Nonetheless we can interpret the $(1,0)$ and $(0,1)$ strings on $\Sigma$ as
string junctions encircling the entire configuration of 12 branes on $B^{1}$
or $B^{2}$. This makes contact with the discussion in Section 2. The combined
monodromy associated with the 12 singular fibers on a dP9 is unity, and the
associated brane configuration would enhance to the double loop algebra
$\hat{E}_{9}$ if all 12 branes coincide on the dP9. The wrapped strings
therefore correspond precisely to the two junctions $\delta_{1}$ and
$\delta_{2}$ of Section 2. In fact, as we will discuss in more detail in
Section 4.3, the Type II.a Kulikov model is the result of blowing up a
singular Weierstrass model over $\mathbb{P}^{1}_{[s:t]}$, for which a complex
structure degeneration enforces the collision of the 12 constituent branes of
an $\hat{E}_{9}$ configuration at up to two points. At these points, the
singularity in the elliptic fiber is non-minimal in the sense of the Kodaira
classification, i.e. the Weierstrass vanishing orders become

${\rm ord}_{\cal Y}(f,g,\Delta)|_{u=0}=(\geq 4,6,12)\quad{\rm or}\quad(4,\geq
6,12)\,.$ (3.15)

Prior to blowing up the non-minimal singularity in the base, one obtains two
towers of asymptotically massless states of the form $n_{1}\delta_{1}$ and
$n_{2}\delta_{2}$ localised at the singularity. The blowup gives a
regularisation of the non-minimal singularity by separating the 12 branes in
such a way that the resulting fiber types are of minimal form. This comes at
the cost of degenerating the base into several components. In this regularised
geometry, the massless towers of states are still visible as the same types of
string junctions, now encircling the intersection point ${\cal P}$, as
described above, or equivalently in M-theory, from the towers of M2-branes
along $\gamma_{1}$ and $\gamma_{2}$. Note that the situation is symmetric
between both components of the Type II.a Kulikov model and in particular the
string junctions on $B^{1}$ and $B^{2}$ are homologous in the junction lattice
[62]. The total symmetry algebra of the system in the infinite distance limit
is therefore

$\displaystyle G_{\infty}=(\hat{E}_{9}\oplus\hat{E}_{9})/\sim\,,$ (3.16)

where the quotient identifies the two imaginary roots $\delta_{i}$ from each
of the first and from the second factor.

These two towers of states are interpreted as two Kaluza-Klein towers in the
dual heterotic string, signalling the decompactification of the two circles of
the heterotic torus $T^{2}_{\rm het}$. We will elaborate on this
interpretation in detail in Section 4.3. This identifies the Type II.a limits
as decompactification limits from 8d to 10d. This is of course nothing but the
well-known statement that stable degenerations (of Type II.a) realise large
volume limits of the dual heterotic string [48, 49, 50]. Clearly, the gauge
algebra in the decompactified theory is simply

$G_{\rm 10d}=E_{8}\oplus E_{8}\,,$ (3.17)

the maximal finite Lie subalgebra of (3.16).

###  3.4 Type III.a Kulikov models as partial decompactification limits

We now characterise the physics of F-theory on a K3 surface in an infinite
distance limit of Kulikov Type III.a. In such a limit, the theory undergoes a
partial decompactification to nine dimensions. This can be traced back to the
appearance of one or two factors of an $\hat{E}_{9-n}$ affine algebra for some
values $1\leq n\leq 9$, as we now explain from F-theory.

Figure -1194:  Rational elliptic end component of a Kulikov Type III.a
degeneration intersecting a neighbouring positive $I_{n}$ component from the
right.

In models of Type III.a, at least one of the two end components of the
degenerate Weierstrass model $Y_{0}$ given in (3.2) is a rational elliptic
surface, which we will often refer to as a dP9 surface. As reviewed in Section
3.1 such configurations are obtained by suitable blowups of non-minimal
singularities of the type (3.6) with $m=n=0$ and $k>0$.141414Recall that the
configuration could be of other Kulikov types if $k>0$ was achieved by $m>0$
and $n>0$. For details see [1]. Generically, this gives rise to a chain of
surfaces with rational elliptic surfaces on both ends, $Y^{0}$ and $Y^{P}$. An
additional tuning of the form (3.7) or (3.8) results in a configuration where
only one of the end components is rational elliptic.

Let us first assume that there is only one rational elliptic end component,
which we identify w.l.o.g. with the surface $Y^{0}$. Its neighbouring surface
component $Y^{1}$ is a degenerate fibration over $B^{1}$ with fibers of Type
I${}_{n_{1}>0}$ in codimension zero. In the sequel we will set $n_{1}=:n$. The
two curves $B^{0}$ and $B^{1}$ intersect at a point $\cal P$. The fiber over
$\cal P$ is the intersection curve $C^{0,1}=Y^{0}\cap Y^{1}$. This fiber
$C^{0,1}$ is a nodal curve, obtained by collapsing the $(1,0)$ cycle $S_{A}$
of the elliptic fiber. See Figure -1194 for an illustration. If $n>1$, then
after blowing up the nodal point, the curve $C^{0,1}$ is replaced within
$X_{0}$ by a cycle of $n$ intersecting rational curves intersecting like the
nodes of the affine Dynkin diagram of $SU(n-1)$

What is important for us is that the $SL(2,\mathbb{Z})$ monodromy picked up
upon encircling the intersection point $\cal P$ along a 1-cycle $\Sigma$ on
$B^{0}$ is represented by the matrix

$\displaystyle M_{{\rm I}_{n}}=\left(\begin{matrix}1&-n-1\\\
0&1\end{matrix}\right)\,.$ (3.18)

This implies that the dP9 surface $Y^{0}$ must generically have $(12-n)$
singular fibers away from $\cal P$. The monodromy upon encircling the location
of the associated $(12-n)$ 7-branes is

$\displaystyle M^{-1}_{{\rm I}_{n}}=M_{\hat{E}_{9-{n}}}\,,$ (3.19)

which identifies the 7-branes on $Y^{0}$ away from the intersection point
$\cal P$ as the constituents of an $\hat{E}_{9-n}$ configuration. The
enhancement to $\hat{E}_{9-n}$ occurs in the original Weierstrass model prior
to blowup as a consequence of the non-minimal singularity (3.6), and the
blowup partially separates the branes. We therefore claim that there is the
following dictionary between affine $\hat{E}_{9-n}$ algebras in F-theory and
the geometry of the blowup of a Weierstrass model with non-minimal
singularities:

$\hat{E}_{9-n}$ enhancements

in F-theory

$\Longleftrightarrow$

Rational elliptic end components of a Type III.a

Kulikov Weierstrass model with

$({\rm deg}(f),{\rm deg}(g),{\rm deg}(\Delta))=(4,6,12-n)$

away from the other surface components

where ${\rm deg}$ denotes the sum of vanishing orders. By analogy with our
discussion of the Type II.a models in Section 3.3, it is now clear that we can
construct one independent elliptic transcendental 2-cycle $\gamma$ by
transporting the (1,0) cycle $S_{A}$ in the elliptic fiber along the pinching
1-cycle $\Sigma$ around the intersection point $\cal P$. This is possible
because the $(1,0)$ cycle in the elliptic fiber is left invariant by the
monodromy $M_{{\rm I}_{n}}$ upon encircling $\cal P$. The calibrated volume of
$\gamma$ in the degenerate K3 surface $X_{0}$ vanishes. Unlike for a Type II.a
degeneration, the $(0,1)$ cycle in the elliptic fiber is not left invariant by
this monodromy. This explains why there is only a single, rather than two,
homologically independent transcendental 2-torus of vanishing volume.

M2-branes wrapping $\gamma$ give rise to a tower of BPS states in the seven-
dimensional M-theory compactification, which uplifts to a tower of BPS states
in the 8d F-theory. In the F-theory language this tower is identified with the
tower of string junctions $k\,\delta_{1}$, $k\in\mathbb{Z}$, where
$\delta_{1}$ is a $(1,0)$ string that encircles the point $\cal P$.
Equivalently, it can be viewed as a string junction encircling the $(12-n)$
7-branes located on the base $B^{0}$ away from the intersection point $\cal
P$.

Compared to the Type II.a limit, we therefore encounter only a single tower of
asymptotically massless BPS particles, corresponding to the single loop
enhancement to build an $\hat{E}_{9-n}$ algebra for $n\geq 1$. Such loop
algebras are affine Lie algebras of Kac-Moody type. Furthermore, this tower of
BPS states can be interpreted as Kaluza-Klein tower signalling a
decompactification to 9d in the language of the dual heterotic string. We will
confirm this interpretation by a careful match with the dual heterotic side in
Section 4.4.

If both end components $Y^{0}$ and $Y^{P}$ are dP9 surfaces, the total non-
abelian part of the symmetry algebra of the configuration is

$\displaystyle
G_{\infty}=H\oplus(\hat{E}_{n_{0}}\oplus\hat{E}_{n_{P}})/\sim\,,$ (3.20)

where the quotient indicates that the two string junctions associated with the
imaginary roots of the two affine algebras are not independent but rather are
identified. If only one end component is a dP9 surface, the second affine
factor does not appear. The factor $H$ refers to the non-abelian symmetry
algebra localised on all components with generic fibers of Type Ik>0. The non-
abelian part of the gauge algebra in the 9d decompactification limit is the
maximal finite Lie subalgebra of (3.20), which is given by

$\displaystyle G_{\rm 9d}=H\oplus E_{n_{0}}\oplus E_{n_{P}}\,.$ (3.21)

We will give further evidence for this picture in Section 4.4.

There is a second difference as compared to the Kulikov Type II.a limit: In a
Type III.a model, the intermediate surface components $Y^{i}$,
$i=1,\ldots,P-1$ (and possibly one of the two end components as well), are
I${}_{n_{i}}$ surfaces with $n_{i}>0$. Since the enhancements at the
intersection of the components are all of Kodaira Type Ik, the generic
I${}_{n_{i}}$ fibers over the different components are mutually local with
respect to one another. Therefore, the 10d axio-dilaton coupling vanishes
along these components. Moreover there is a $(1,0)$ string which is obtained
from an M2-brane that wraps the vanishing $(1,0)$ cycle in the fiber of the
components $Y^{i}$, with $n_{i}>0$. It becomes asymptotically tensionless in
units of the 8d Planck scale, but only locally in the region of the degenerate
compactification space.

Nonetheless, the limit is not an 8d weak coupling limit. First, as just
mentioned, the $(1,0)$ string becomes tensionless only away from the rational
elliptic end component(s), where the generic fiber is non-degenerate and hence
the $(1,0)$ cycle is not contracted. This already shows that there does not
exist a globally weakly coupled frame in which the $(1,0)$ string plays the
role of the fundamental string. Second, the mass scale of the tower of states
arising from the string junctions $k\,\delta_{1}$ vanishes faster than the
vanishing rate of the excitations of the (local) $(1,0)$ string. This is
because the tower of string junctions is composed of asymptotically
tensionless $(1,0)$ strings winding, in addition, around a vanishing cycle
encircling $\cal P$. This subtle point justifies the interpretation as a
decompactification, rather than as equidimensional weak coupling limit.

We note that the local weak coupling nature along the surface components away
from the rational elliptic end components manifests itself also in the
structure of Kodaira fibers over special points of the base. As reviewed in
Section 3.1, it was found in [1] that the only possible fiber enhancements on
the surface components with generic In>0 fibers are to singularities of
$A$-type or $D$-type. These are the singularities which are compatible, at
least locally, with weak coupling. On an end component with In>0 fibers, the
$D_{k}$ enhancements in Table 2.2 can furthermore occur also for values $0\leq
k<4$: Such enhancements are known to be absent, by Kodaira’s classification,
on non-degenerate elliptic surfaces, corresponding to a non-perturbative
obstruction to forming bound states of branes of the form $A^{k}BC$ for $0\leq
k<4$. In locally weakly coupled regions, on the other hand, no such
obstruction should occur, and this is in perfect agreement with the existence
of the corresponding $D_{k}$-type fibers on the weakly coupled surface
components. Examples of this phenomenon can be found in [1].

Finally, in the next subsection we will provide an interpretation of the extra
constraint (3.7) which distinguishes the Type III.a limits with one or two
rational elliptic end surfaces.

###  3.5 Type III.b Kulikov models as weak coupling plus decompactification
limits

As we now show, the physics of a Kulikov Type III.b limit is that of a weak
coupling limit dual to a perturbative Type IIB orientifold, in combination
with a large complex structure limit for the torus $T^{2}_{\rm IIB}$ on which
the Type IIB theory is compactified. This leads to an asymptotic
decompactification from 8d to 10d.

First, the codimension-zero I${}_{n_{i}>0}$ fibers in all surface components
$Y^{i}$ indicate the existence of a weakly coupled duality frame that is
globally defined on the base $B_{0}$ in the infinite distance limit. The
I${}_{n_{i}}$ singularities in the fibers $Y^{i}$ are induced by the shrinking
of one-cycles, and since the I${}_{n_{i}}$ singularities of all components are
mutually local with respect to each other (otherwise the components $Y^{i}$
would intersect each other in fibers different from Ik type), they are in fact
copies of the same one-cycle, $S_{A}$, that shrinks in all components. An
M2-brane wrapped along $S_{A}$ gives rise to a critical string in the
uncompactified dimensions which is asymptotically tensionless as measured in
7d Planck units. This string uplifts to the weakly coupled Type IIB string in
F-theory, whose tower of excitations is asymptotically tensionless compared to
the 8d Planck scale. In addition, one finds a tower of massless states from
the winding modes of this string along the 1-cycle $\Sigma$ on each base
component $B^{i}$ that shrinks at the intersection of two components. The
fibration of $S_{A}$ over $\Sigma$ defines one transcendental elliptic curve
$\gamma$ of asymptotically vanishing volume. The same tower of asymptotically
massless states can hence be equivalently understood in M-theory as the tower
of M2-branes wrapping this elliptic curve $\gamma$ multiple times. It is clear
that the mass scale associated with this latter tower of particles asymptotes
to zero faster than the excitations of the weakly coupled fundamental string
from the M2-brane wrapped on $S_{A}$ alone. This is because

$\displaystyle\frac{M^{2}_{\rm str}}{M^{2}_{11}}\sim{\cal V}_{11}(S_{A})\to
0\,,\qquad\quad\frac{M_{{\rm w}}}{M_{11}}\sim{\cal V}_{11}(S_{A})\times{\cal
V}_{11}(\Sigma)\sim{\cal V}_{11}(\gamma)\to 0\,,$ (3.22)

such that

$\frac{M_{{\rm w}}}{M_{\rm str}}\sim M_{\rm str}{\rm Vol}(\Sigma)\to 0\,,$
(3.23)

where ${\cal V}_{11}(\ldots)$ measures volumes in units of the 11d Planck
scale, $M_{11}$.

In fact, we will identify the tower associated with $\gamma$ as the winding
modes of the weakly coupled fundamental Type IIB string wrapped on an
asymptotically vanishing one-cycle on $T^{2}_{\rm IIB}$. The vanishing of the
one-cycle indicates that the complex structure of $T^{2}_{\rm IIB}$ undergoes
an infinite distance limit, in addition to the weak coupling limit $\tau\to
i\infty$. Since the Kähler volume of the compactification space remains
constant in units of the Type IIB string scale, such a large complex structure
limit indicates a complete decompactification to 10d. In addition to the tower
of winding states described above, there must thus arise a tower of
supergravity Kaluza-Klein modes associated with the dual 1-cycle of
$T^{2}_{\rm IIB}$ whose volume becomes inversely proportional to the one of
the shrinking cycle. This tower of states is not visible as wrapped M2-branes
in M-theory and needs to be inferred in a more indirect manner.

We can understand these statements more clearly in the language of the
degenerating Weierstrass model as follows. Since all components of $Y_{0}$
have generic fibers of Kodaira Type I${}_{n_{i}>0}$, we can blow down all but
one components to arrive at a (highly singular) family of Weierstrass models
${\widehat{\cal Y}}$, which feature a degenerate central element
$\widehat{Y}_{0}$ whose codimension-zero fibers are guaranteed to be of type
In>0. Therefore its Weierstrass model can be written in the general form
(3.14) for certain $\eta$ and $\rho$. At the same time, the fact that the
original degeneration leads to a multi-component central fiber $Y_{0}$ implies
the existence of non-minimal Kodaira fibers in $\widehat{Y}_{0}$. They must
then occur at the zeroes of $h$. This follows from the fact that at each of
the four zeroes of $h$, the degree $8$ function $f$ and the degree $12$
function $g$ vanish to order $2$ and $3$, respectively, which leaves no room
for additional zeroes of $f$ and $g$ at different points. A non-minimal
Kodaira singularity at one of the zeroes of $h$ in turn is only possible if at
least two zeroes of $h$ coincide on ${\widehat{Y}_{0}}$. Since from a Type IIB
perspective the zeroes of $h$ are the locations of the O7-planes, this means
that at least one pair of O7-planes coalesces in the limit. When this happens,
one must now distinguish two qualitatively different situations:

  * •

Generically, i.e., without further tuning of the parameters beyond the
described collision of O7-planes via that of zeros of $h$, we will be taken
away from a uniform weak coupling limit by introducing strongly coupled
localised codimension-one objects on the base. This manifests itself in such a
way that after performing the necessary blowups, which bring us back to the
Weierstrass family ${\cal Y}$, at least one of the end components of $Y_{0}$
is not of In type with positive $n$, but rather is a rational elliptic
surface. That is, the degeneration is of Type III.a. The rational elliptic
component is the one which contains the location of the colliding O7-planes
after the blowup and the axio-dilaton is generically of ${\cal O}(1)$. The
situation is thus best described as a 9d decompactification limit of the dual
heterotic string, as explained in Section 3.4.

  * •

In contrast, by a special tuning of the parameters of the Weierstrass model
underlying the family ${\widehat{\cal Y}}$, we can remain at weak coupling
despite the collision of the O7-planes, namely if the weak coupling limit is
taken at a faster rate than the limit leading to the collision. This way, the
blowup indeed results in a Type III.b degeneration, in which all surface
components of $Y_{0}$ have generic In>0 fibers. The required tuning is
precisely the additional condition (3.8).

The interpretation of such infinite-distance complex structure limits in terms
of weakly coupled Type IIB strings is suggested by noting that the distance of
the O7-planes measures the length of the 1-cycles of the Type IIB
compactification space $T^{2}_{\rm IIB}$. More precisely, the non-trivial
1-cycles on $T^{2}_{\rm IIB}$ correspond to the one-cycles encircling pairs of
zeroes of the section $h$ on the base $B_{0}$ of $\widehat{Y_{0}}$ introduced
above. Colliding two of these zeroes is therefore equivalent to a complex
structure degeneration of $T^{2}_{\rm IIB}$ for which a one-cycle shrinks. To
the extent that the Kähler volume in Type IIB string units is unaffected by
the complex structure degenerations, this leads to a decompactification to
10d, rather than to 9d.

##  4 Infinite Distance Limits in the $E_{7}\times E_{8}$-Weierstrass Model

In this section, we systematically analyse a representative class of infinite
distance limits in the complex structure moduli space of an elliptic K3
surface from the point of view F-theory/ heterotic duality. We will obtain a
clear physical interpretation of the infinite distance limits in the moduli
space of F-theory via the explicitly known mirror map to the heterotic moduli
in terms of Siegel modular forms. This illustrates and lends additional
support to our claims of the previous sections concerning the asymptotic
physics of the infinite distance limits.

For concreteness, we consider a one-parameter family (3.3) of K3 surfaces
described by a Weierstrass model over base $\mathbb{P}^{1}_{[s:t]}$ which at a
generic point $u\ \neq 0$ in moduli space gives rise to a non-abelian gauge
algebra $E_{7}\times E_{8}$. This model is sufficiently simple to allow for a
systematic treatment, and at the same time exhibits all qualitative properties
of the infinite distance limits of Kulikov Type II.a and III.a as described in
the previous section.151515The $E_{7}\oplus E_{8}$ algebra is inconsistent
with limits of Type II.b or III.b. We will construct these infinite distance
limits by enhancing the exceptional gauge algebras to loop algebras
$\hat{E}_{9-n}$. This will be achieved by constructing non-minimal
singularities in the Weierstrass model $\cal Y$ corresponding to the vanishing
orders (3.6) with $m=0$ or $n=0$, i.e.,

$\displaystyle{\rm ord}_{\cal Y}(f,g,\Delta)|_{u=s=0}=(4+m,6,12+k)\quad{\rm
or}\quad(4,6+n,12+k)\,,$ (4.1)

at a point on the base $\mathbb{P}^{1}_{[s:t]}$, which, here, is taken to be
$s=0$. We will see explicitly how degenerations with $k=0$ are of Type II.a
and give rise to complete decompactifications of the 8d theory to 10d, as can
be read off from the appearance of two loop algebras of Type $\hat{E}_{9}$.
For $k>0$, on the other hand, the degenerations will be of Type III.a and we
will realise affine Lie algebras of type $(\hat{E}_{m}\times\hat{E}_{8})/\sim$
with $m=7,8$. The effective theories will in turn be interpreted as partial
decompactifications to 9d by carefully translating the model into the dual
heterotic frame via the formalism developed in [53] and studied further in
[63, 64]. Moreover, depending on the parametrisation of the degeneration, we
will find an intricate structure of various branches leading to different
gauge algebras in the effective 9d theory.

After introducing the $E_{7}\times E_{8}$ Weierstrass model in Section 4.1, we
will explain how to systematically construct non-minimal singularities of the
form (4.1) in Section 4.2 and outline the structure of the resulting
degenerate Kulikov Type II or Type III models. In Sections 4.3 and 4.4 we will
illustrate the general picture for a particular parametrisation of the
degeneration. This will lead to different branches of the blowup theory, and
we will analyse the gauge symmetry in the effective 9d theory both within
F-theory and from the perspective of the dual heterotic theory.

###  4.1 The $E_{7}\times E_{8}$-Weierstrass model and its heterotic dual

Our starting point is the family of Weierstrass models
$y^{2}=x^{3}+fxz^{4}+gz^{6}$ over $\mathbb{P}^{1}_{[s:t]}$ defined by the
degree 8 and 12 functions161616The parameters $a$, $b$, $c$, $d$, $e$ depend
on $u$, which we suppress, and for notational simplicity we also drop the
subscript of $f_{u}$ and $g_{u}$, as already mentioned.

$\displaystyle f=t^{3}s^{4}(a\,t+c\,s)\,,\qquad
g=t^{5}s^{5}(d\,s^{2}+b\,s\,t+e\,t^{2})\,.$ (4.2)

For generic values of the complex parameters $a$, $b$, $c$, $d$, $e$, the
vanishing orders of $f$, $g$ and their discriminant $\Delta=4f^{3}+27g^{2}$
are given by

${\rm ord}(f,g,\Delta)|_{t=0}=(3,5,9)\,,\qquad{\rm
ord}(f,g,\Delta)|_{s=0}=(4,5,10)\,.$ (4.3)

This identifies the non-abelian part of the gauge algebra in the 8d
compactification of F-theory on this family of K3 surfaces as $E_{7}\oplus
E_{8}$, for generic values of the parameters. For special choices of
parameters, the non-abelian gauge algebra can enhance further.

The complex parameters $a$, $b$, $c$, $d$, $e$ are subject to the 2-parameter
family of rescalings

$\displaystyle(x,y,z,t,s,a,b,c,d,e)\sim(\lambda^{2\tau}x,\lambda^{3\tau}y,z,\lambda^{\sigma}t,s,\lambda^{4\tau-4\sigma}a,\lambda^{6\tau-6\sigma}b,\lambda^{4\tau-3\sigma}c,\lambda^{6\tau-5\sigma}d,\lambda^{6\tau-7\sigma}e)$

for arbitrary $\tau,\sigma\in\mathbb{C}^{\ast}$. All-in-all we are therefore
left with a moduli space of complex dimension three.

This moduli space can be most conveniently characterized in terms of the dual
heterotic string compactification on $T^{2}_{\rm het}$.171717We will drop the
subscripts from now on when referring to the quantities in the heterotic
duality frame. The duality map, which is essentially the mirror map, has been
worked out in detail in [53] and was studied further in [63, 64]. In the patch
$e=1$ of moduli space, the remaining four parameters of the K3 Weierstrass
model can be identified as

$\displaystyle a=-\frac{\psi_{4}({\Omega})}{48}\,,\quad
b=-\frac{\psi_{6}({\Omega})}{864}\,,\quad c=-4\chi_{10}({\Omega})\,,\quad
d=\chi_{12}({\Omega})\,,\quad e=1\,.$ (4.5)

Here $\psi_{k}$ and $\chi_{m}$ are Siegel modular forms181818Definitions and
some of the most relevant properties are reviewed in Appendix A. of genus
$g=2$ of the indicated weight. They depend via

$\Omega:=\left(\begin{matrix}T&V\\\ V&U\end{matrix}\right)$ (4.6)

on the Kähler modulus $T$, complex structure modulus $U$ and Wilson line
modulus $V$ of the dual heterotic string compactified on $T^{2}$. For later
purposes, we also introduce the variables

$q_{T}\equiv e^{2\pi iT}\,,\qquad q_{U}\equiv e^{2\pi iU}\,,\qquad q_{V}\equiv
e^{2\pi iV}.$ (4.7)

Another quantity that will be relevant for us is the discriminant factor

$Q(a,b,c,d)\sim\frac{\chi^{2}_{35}}{\chi_{10}}\,,$ (4.8)

where the cusp form $\chi_{35}$ is the single odd generator of the ring
$M_{*}(\Gamma_{2})$ of genus $g=2$ Siegel modular forms (A.3). The polynomial
$Q$ is defined as follows [53]: For generic values of the parameters $a$, $b$,
$c$, $d$, the discriminant of the Weierstrass model takes the form

$\displaystyle\Delta=t^{9}s^{10}\Delta^{\prime}\,,$ (4.9)

where $\Delta^{\prime}$ is a polynomial of degree five in $[s:t]$. The
discriminant of this polynomial factors further into two polynomials

$\displaystyle{\rm Discr}(\Delta^{\prime})=P(a,b,c,d)\times Q(a,b,c,d)\,.$
(4.10)

If $P(a,b,c,d)=0$, the model acquires an additional Type II Kodaira fiber over
some point on the base, while otherwise whenever $Q(a,b,c,d)=0$ has a single
zero, there occurs an I2 singularity, which is characteristic of an $SU(2)$
enhancement.191919Such codimension one singularities are called Humbert
surfaces, and if multiple roots of $Q(a,b,c,d)$ coincide, higher codimension
intersections of Humbert surfaces occur that are called Shimura curves or
complex multiplication points. The nomenclature is not important for our
purposes. Explicitly, one finds that [53]

$\begin{split}Q(a,b,c,d)&=-16\,a^{6}b\,c^{3}-216\,a^{3}\,b^{3}\,c^{3}-729\,b^{5}\,c^{3}+16\,a^{5}\,c^{4}+2700\,a^{2}\,b^{2}\,c^{4}-5625\,a\,b\,c^{5}\cr&+3125\,c^{6}+16\,a^{7}\,c^{2}\,d+216\,a^{4}\,b^{2}\,c^{2}\,d+729\,a\,b^{4}\,c^{2}\,d-3420\,a^{3}\,b\,c^{3}\,d+6075\,b^{3}\,c^{3}\,d\cr&+4125\,a^{2}\,c^{4}\,d+888\,a^{4}\,c^{2}\,d^{2}-5670\,a\,b^{2}\,c^{2}\,d^{2}-13500\,b\,c^{3}\,d^{2}+16\,a^{6}\,d^{3}+216\,a^{3}\,b^{2}\,d^{3}\cr&+729\,b^{4}\,d^{3}-2592\,a^{2}\,b\,c\,d^{3}+16200\,a\,c^{2}\,d^{3}+864\,a^{3}\,d^{4}-5832\,b^{2}\,d^{4}+11664\,d^{5}\,.\end{split}$
(4.11)

###  4.2 Systematics of affine enhancements in the $E_{7}\times
E_{8}$-Weierstrass model

Our goal is to construct the possible infinite distance limits in the three-
dimensional complex structure moduli space. With the help of two scaling
relations it can always be arranged that the parameters $(a,b,c,d,e)$ take
finite values. In this scheme, the infinite distance limits that are not of
weak coupling type202020Such limits occur by degenerating the elliptic fiber
over generic points of the base, as systematised in [1]. can be obtained by
arranging for non-minimal Kodaira singularities, i.e. vanishing orders for
$(f,g,\Delta)$ of order $(4,6,12)$ or beyond, at points in codimension one on
the base. With all parameters kept finite, there are a priori three different
ways of achieving this in the family (4.2) of Weierstrass models:

  1. 1.

Non-minimality only at $t=0$ requires taking $c\to 0$ and $d\to 0$, while
keeping $e\neq 0$.

  2. 2.

Non-minimality only at $s=0$ requires taking $e\to 0$ while keeping
$(c,d)\neq(0,0)$.

  3. 3.

Non-minimality both at $s=0$ and $t=0$ requires $c\to 0$, $d\to 0$ and $e\to
0$.

However, it is easy to see that due to the rescaling symmetry (4.1) these
three types of limits can be partially mapped into one another, at least in
certain regimes of moduli space. Note first that since we are interested in
identifying relations between asymptotically massless towers, it is convenient
to express the infinite distance limits as one-parameter limits. Consider then
a general limit in which

$\displaystyle c\sim u^{\gamma}\,,\quad d\sim u^{\delta}\,,\quad e\sim
u^{\epsilon}\,\quad{\rm with}\,\quad u\to 0\,,\qquad\gamma,\delta,\epsilon\geq
0\,.$ (4.12)

The two-parameter family (4.1) of rescalings contains the one-parameter family
(for $\lambda^{\tau}=\lambda^{\sigma}=:\lambda$)

$\displaystyle(a,b,c,d,e)\sim(a,b,\lambda c,\lambda d,\lambda^{-1}e)\,.$
(4.13)

Then taking $\lambda=u^{\epsilon}$, as long as

$\displaystyle\gamma>\epsilon\,,\qquad\delta>\epsilon\,,$ (4.14)

we can go to a patch in which $c\to 0$ and $d\to 0$ while $e$ stays finite and
of order one in the limit $u\to 0$. Importantly, we still have the freedom to
perform a rescaling which identifies

$\displaystyle(a,b,c,d)\sim(\lambda^{4}a,\lambda^{6}b,\lambda^{10}c,\lambda^{12}d)$
(4.15)

without leaving the patch $e=1$, in agreement with the modular behaviour of
the Siegel modular forms (4.5).

For concreteness, we will be considering regimes in moduli space which fall
into this class. In the regime of small $c$ and $d$, the combinations of
Siegel modular forms that appear in (4.5) and (4.11) admit expansions in
$q_{T}$, $q_{U}$ and $q_{V}$ of the form (c.f. Appendix A)

$\begin{split}c&\sim\chi_{10}\sim
q_{T}q_{U}(-2+q_{V}+\frac{1}{q_{V}})+\ldots\,,\cr d&\sim\chi_{12}\sim
q_{T}q_{U}(10+q_{V}+\frac{1}{q_{V}})+\ldots\,,\cr
4a^{3}+27b^{2}&\sim(\psi_{4}^{3}-\psi_{6}^{2})\sim q_{T}+q_{U}+\ldots\,,\cr
Q&\sim\chi^{2}_{35}/\chi_{10}\sim
q_{T}^{3}q_{U}^{3}(q_{T}-q_{U})^{2}q_{V}(q_{V}+\frac{1}{q_{V}})^{2}+\ldots\,.\end{split}$
(4.16)

The infinite series of subleading terms in $\chi_{10}$, $\chi_{12}$ as well as
the terms in $\psi_{4}$ and $\psi_{6}$ underlying the quoted expression for
$\psi_{4}^{3}-\psi_{6}^{2}$ can be ignored as long as

$\displaystyle\frac{q_{T}}{q_{V}}\ll 1\,,\qquad\frac{q_{U}}{q_{V}}\ll 1\,.$
(4.17)

This condition is consistent with the condition for the modulus of Wilson line
to lie within the fundamental domain given by

$\displaystyle{\rm Im}(U){\rm Im}(T)\geq({\rm Im}(V))^{2}\,.$ (4.18)

Similarly, the subleading terms for $\chi^{2}_{35}/\chi_{10}$ can be ignored
as long as

$\displaystyle\frac{q_{T}}{q_{V}^{2}}\ll 1\,,\qquad\frac{q_{U}}{q_{V}^{2}}\ll
1\,.$ (4.19)

In the regime of small $q_{V}$, the condition (4.19) is stronger than (4.17)
and can therefore be violated within the fundamental domain. The expansion
(4.16) of $Q$ can only be applied if (4.19) holds.

After these preliminaries we now consider infinite distance limits in the
patch where $e=1$ by taking $c\to 0$ and $d\to 0$. As it turns out, the
physically different infinite distance limits are distinguished by the
relative vanishing order of $c$ and $d$ with respect to the vanishing of the
combination $4a^{3}+27b^{2}$. This motivates considering the vanishing orders

$\displaystyle 4a^{3}+27b^{2}\sim u^{k}\,,\qquad c\sim u^{n}\,,\quad d\sim
u^{m}\,\qquad n\geq 1,m\geq 1\,,k\geq 0\,,$ (4.20)

where the value of $k$ coincides with that in (4.1). Models with such
vanishing orders require

$l={\rm min}(n,m)$ (4.21)

blowups to resolve the non-minimal singularity at $t=0$.

As we will see, the physics at infinite distance can be classified as follows:

  1. 1.

Limits with $k=0$ give rise to a Kulikov Type II degeneration of the elliptic
K3 surface. This follows from the fact that in such cases after the blowup
both end components contain 12 branes each, as expected for a $\hat{E}_{9}$
configuration. Correspondingly, the expansion of modular forms implies that in
this case, $q_{T}\to 0,q_{U}={\cal O}(1)$ or vice versa, which amounts to
decompactification to 10 dimensions.

  2. 2.

Limits with $k\geq 1$ are of Kulikov Type III, corresponding to a
$\hat{E}_{8-N}$ configuration for $N=0$ or $N=1$ on the two end components.
The expansion of the modular forms is consistent with the fact that $q_{T}\to
0$ and $q_{U}\to 0$ at a finite ratio, signalling a partial decompactification
to 9 dimensions.

For otherwise generic coefficients, we will find Type III limits with

$\displaystyle{\rm Im}(T)=(l-1){\rm Im}(U)\,.$ (4.22)

Here we made use of the duality symmetry to make sure that, without loss of
generality, ${\rm Im}(T)\geq{\rm Im}(U)$. For each choice of $l$, different
branches are obtained by special tunings of the remaining coefficients, with
the following properties: If $n>m$, the non-abelian part of the gauge algebra
in 9 dimensions always includes an $E_{8}\oplus E_{8}$ factor, which can be
enhanced to $E_{8}\oplus E_{8}\oplus SU(2)$. If $n\leq m$, the non-abelian
part of the gauge group in 9 dimensions can be $E_{8}\oplus E_{8}$ or
$E_{8}\oplus E_{8}\oplus SU(2)$, or $E_{7}\oplus E_{8}$ with possible
enhancement to $E_{7}\oplus E_{8}\oplus SU(2)$.212121As will be explained at
the end of Section 4.4, the algebra of maximal enhancement $G_{\rm
9d}=E_{7}\oplus E_{8}\oplus SU(3)$ occurs outside the patch of moduli space
characterised by (4.14).

###  4.3 Type II degenerations

To illustrate the general picture, we consider a class of limits with $n=m=4$
for the parameters defined in (4.20). It turns out convenient to parametrise
the coefficients appearing in (4.2) as

$\displaystyle\begin{split}&&a=a_{0}\,,\qquad
b&=\frac{2ia_{0}^{3/2}}{3\sqrt{3}}(1+y+b_{1}u)+b_{2}u^{2}+{\cal O}(u^{3})\\\
&&c=-a_{0}(i\sqrt{3}+x)u^{4}+a_{0}pu^{5}+{\cal O}(u^{6})\,,\qquad
d&=a_{0}^{3/2}u^{4}+d_{1}u^{5}+{\cal O}(u^{6})\,,\end{split}$ (4.23)

where the infinite distance limit is enforced by taking $u\to 0$.

As a warmup, we take all parameters $b_{1}$, $a_{0}$, $p$, $x$ and $y$ to be
generic and of order one. At $u=0$, the discriminant factorises as

$\displaystyle\Delta|_{u=0}=t^{12}s^{10}\Delta^{\prime}\,.$ (4.24)

In particular, the Weierstrass vanishing orders $({\rm ord}(f),{\rm
ord}(g),{\rm ord}(\Delta))$ at $t=0$ are of type $(4,6,12)$, indicating a non-
minimal singularity in the fiber.

A Weierstrass model with only minimal singularities is obtained via a sequence
of four blowups,

$\displaystyle(t,e_{i},x,y)$ $\displaystyle\to$
$\displaystyle(te_{i+1},e_{i}e_{i+1},xe_{i+1}^{2},ye_{i+1}^{3})\,\qquad
i=0,1,2,3\,,$ (4.25)

where we have set $e_{0}=u$. After each step the Weierstrass equation exhibits
an overall factor of $e_{i+1}^{6}$, and to obtain the proper transform of the
Weierstrass equation we divide by this factor and consider the resulting
equation. This amounts to a rescaling
$(x,y,z,f,g)\sim(\lambda^{2}x,\lambda^{3}y,z,\lambda^{4}f,\lambda^{6}g)$ for
$\lambda=e_{i+1}$, which does not affect the Calabi-Yau condition. The process
terminates after four steps. The resulting fibration is a Weierstrass model
$Y_{0}$ which degenerates into a chain of five components,

$\displaystyle Y_{0}=\cup_{i=0}^{4}Y^{i}\,.$ (4.26)

Each component $Y^{i}$ is an in general degenerate elliptic fibration over a
rational base curve

$B^{i}=\\{e_{i}=0\\}\,,$ (4.27)

where we have defined $e_{0}=u$. Here “degenerate” refers to the fact that the
Weierstrass model may have Kodaira fibers of Type I${}_{n_{i}}$ over generic
points of $B^{i}$, as explained generally in [1], but apart from this it has
only minimal Kodaira fibers over special points of $B^{i}$. These codimension-
one singularities can be resolved, but we keep the description as a
Weierstrass model.

After the blowup, the Stanley-Reisner ideal of coordinates that are not
allowed to vanish simultaneously contains in particular the elements

$\begin{split}\\{(e_{4},e_{2}),(e_{4},e_{1}),(e_{4},u),(e_{4},s),(e_{3},t),(e_{3},e_{1}),(e_{3},u),(e_{3},s),(e_{2},u),(e_{2},t),(e_{2},s),\cr(e_{1},s),(e_{1},t)(e_{1},s)(u,t)\\}\in{\rm
SRI}\,.\end{split}$ (4.28)

To analyse the geometry of the Weierstrass models $Y^{i}$, we restrict $f$,
$g$ and $\Delta$ to the individual components by setting $e_{i}=0$, and using
the Stanley-Reisner ideal (4.28) to set all coordinates that are not allowed
to vanish on this locus equal to one. In particular, for the discriminant
factor $\Delta^{\prime}$ in (4.24) this gives

$\begin{split}\Delta^{\prime}|_{e_{4}=0}&=e_{3}P_{2}(e_{3},t;x,a_{0})+yQ_{3}(e_{3},t;y,a_{0})\cr\Delta^{\prime}|_{u=0}&=e_{1}P_{1}(e_{1},s;x,a_{0})+yQ_{2}(e_{1},s;y,a_{0})\cr\Delta^{\prime}|_{e_{i}=0}&=4a_{0}^{3}y(2+y)\,,\qquad\qquad
i=1,2,3\,,\end{split}$ (4.29)

where $P_{i}$ and $Q_{j}$ are polynomials of indicated degrees in the first
variables and we also indicate their dependence on the parameters of the
model. From this we draw the following conclusions: The fiber over generic
points of each base component $B^{i}$ is of Kodaira Type I0. The two end
components $Y^{0}$ and $Y^{4}$ have 12 singular fibers away from the
intersection with the other components and hence represent rational elliptic,
or dP9, surfaces. The component $Y^{4}$ has a Type III∗ singularity at $t=0$,
corresponding to an algebra $E_{7}$, and three more I1 singularities at
generic points on $B^{4}$. Moreover, $Y^{0}$ exhibits a Type II∗ singularity
at $s=0$, corresponding to an algebra $E_{8}$, together with 2 more I1
singularities over generic points on $B^{0}$. The intermediate components
$Y^{i}$, $i=1,2,3$, are trivial fibrations over the respective $B^{i}$, and
all adjacent components intersect in an elliptic curve.

The degeneration is therefore an example of a Kulikov Type II model222222It is
guaranteed that the model can be brought into the stable form of a Type II.a
model [51], but we do not make this further step explicit here. with symmetry
group

$\displaystyle G_{\rm\infty}=\hat{E}_{9}\oplus\hat{E}_{9}/\sim\,.$ (4.30)

As reviewed in Section 2, the string junctions associated with the imaginary
roots $\delta_{1}$ and $\delta_{2}$ within $\hat{E}_{9}$ can be interpreted as
the Kaluza-Klein towers from the decompactification to 10 dimensions in the
heterotic duality picture. Indeed, this can be seen explicitly by applying the
duality map (4.5) in its expanded version (4.16). Up to order one
coefficients, one finds

$\begin{split}\chi_{10}\sim\chi_{12}\sim
u^{4}\,,\qquad\psi_{4}^{3}-\psi_{6}^{2}\sim
1\,,\qquad\chi^{2}_{35}/\chi_{10}\sim u^{12}\,,\end{split}$ (4.31)

which indicates a scaling

$q_{T}\sim u\,,\qquad q_{U}\sim 1\,,\qquad q_{V}\sim 1\,.$ (4.32)

This is of course precisely the behaviour $T\to i\infty$ with $U$ (and $V$) of
order one, for the dual heterotic moduli. It goes of course without saying
that the gauge algebra in the 10d limit of the theory is $G_{\rm
10d}=E_{8}\oplus E_{8}$, the gauge algebra of the 10d heterotic string, which
corresponds to the maximal finite Lie algebra within (4.30).

###  4.4 Type III degenerations

To enhance the non-minimal singularity at $t=0$ further, we must arrange for
$4a^{3}+27b^{2}$ to vanish as $u\to 0$, which amounts to setting the parameter
$y=0$ in (4.23). Now the Weierstrass vanishing orders at $t=0$ become
$(4,6,13)$ in the limit $u\to 0$, and a rich structure of physically
inequivalent branches opens up. The non-minimal singularity can still be
removed by the same blowup procedure as shown in (4.25). The crucial
difference is that now the generic fiber of the intermediate surface
components $Y^{i}$, $i=1,2,3$, is of Type I${}_{n_{i}>0}$. This can be
inferred from the third line in (4.29), which shows that the discriminant
along the intermediate $B^{i}$ vanishes if $y=0$. This is the hallmark of a
Type III.a elliptic Kulikov model [1].

Let us now discuss a number of specializations:

####  $b_{1}$, $x$ generic of ${\cal O}(1)$

For generic values of the parameters $b_{1}$, $a_{0}$, $p$, $x$ after the four
blowups (4.25), the discriminant of the properly transformed Weierstrass model
takes the form

$\displaystyle\Delta=e_{1}e_{2}e_{3}t^{9}s^{10}\Delta^{\prime}\,,$ (4.33)

with

$\begin{split}\Delta^{\prime}|_{e_{4}=0}&=a_{0}^{3}(-12t^{2}x+3e_{3}t(-3+8i\sqrt{3}x+4x^{2})+4e_{3}^{2}(3i\sqrt{3}+9x-3i\sqrt{3}x^{2}-x^{3}))\cr\Delta^{\prime}|_{e_{3}=0}&=-4a_{0}^{3}(2b_{1}e_{4}+3e_{2}x)\cr\Delta^{\prime}|_{e_{2}=0}&=-8a_{0}^{3}b_{1}\cr\Delta^{\prime}|_{e_{1}=0}&=-4a_{0}^{3/2}(2a_{0}^{3/2}b_{1}u-3i\sqrt{3}e_{2})\,\cr\Delta^{\prime}|_{e_{0}=0}&=3(9e_{1}+4i\sqrt{3}a_{0}^{3/2}s)\,.\end{split}$
(4.34)

The geometry is depicted in Figure -1193. From (4.33) we infer that the
intermediate surface components $Y^{i}$, $i=1,2,3$, have I1 singular fibers in
codimension zero, i.e. over generic points of their base $B^{i}$. The end
components $Y^{0}$ and $Y^{4}$ are dP9 surfaces intersecting the adjacent
surfaces $Y^{1}$ and $Y^{3}$ in such an I1 fiber. Away from the respective
intersection points, there are 11 singular fibers, distributed as a III∗
Kodaira fiber ($E_{7}$) at $t=0$ and 2 I1 fibers over generic points of
$B^{4}$ and, respectively, as a II∗ Kodaira fiber ($E_{8}$) at $s=0$ along
with one extra I1 fiber on $B^{0}$. One of the originally twelve 7-branes on
each of the end components has moved to the adjacent component, that is to
$Y^{1}$ or $Y^{3}$, respectively.

Figure -1193:  Kulikov Type III degeneration of $E_{7}\times E_{8}$
Weierstrass model (4.34) for generic values of the parameters $b_{1}$ and $x$.

Upon encircling the intersection point $B^{0}\cap B^{1}$ clockwise on $B^{0}$,
and likewise the intersection $B^{3}\cap B^{4}$ on $B^{4}$, one picks up a
monodromy

$\displaystyle M^{-1}_{I_{1}}=M_{\hat{E}_{8}}=\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right)\,.$ (4.35)

This transformation leaves the string junction $\delta_{1}=(1,0)$ invariant,
which can be identified with the imaginary root of the affine Lie algebra
$\hat{E}_{8}$. This affine algebra is obtained upon colliding the 11 branes
located on $B^{0}$ and $B^{4}$ away from the intersection points. The blowup
has separated the 11 branes in such a way as to give rise only to minimal
Kodaira singularities in the fiber. The non-abelian part of the symmetry
algebra of the theory in the limit is identified as

$\displaystyle G_{\infty}=\hat{E}_{8}\oplus\hat{E}_{8}/\sim\,.$ (4.36)

As explained in Section 3.4, $n$ copies of the string junction $\delta_{1}$,
for $n\in\mathbb{Z}$, give rise to a tower of massless BPS particles in the
effective theory. These play the role of a tower of Kaluza-Klein modes
signalling the decompactification of the dual heterotic theory from 8d to 9d.
Furthermore we claim that the non-abelian part of the gauge algebra in the
effectively 9d theory attained in the limit is $G_{\rm 9d}=E_{8}\oplus E_{8}$,
the maximal non-abelian Lie algebra within (4.36).

These two statements can be checked explicitly with the help of the dictionary
(4.5) and (4.16), which now gives

$\begin{split}\chi_{10}\sim\chi_{12}\sim
u^{4}\,,\qquad\psi_{4}^{3}-\psi_{6}^{2}\sim
u\,,\qquad\chi^{2}_{35}/\chi_{10}\sim u^{14}\,.\end{split}$ (4.37)

From the behaviour of $\psi_{4}^{3}-\psi_{6}^{2}$ in the limit under
consideration, we infer that either $q_{T}\sim u$ or $q_{U}\sim u$. W.l.o.g we
take $q_{U}\sim u$. Then $\frac{q_{T}}{q_{V}}\sim u^{3}$ by inspection of
$\chi_{10}$ and $\chi_{12}$. With these two scaling relations, the behaviour
of $\chi^{2}_{35}/\chi_{10}$ shows that altogether

$\displaystyle q_{U}\sim u\,,\qquad q_{T}\sim u^{3}\,,\qquad q_{V}={\cal
O}(1)\,.$ (4.38)

The limit hence corresponds to a partial decompactification to 9d, for which

$\displaystyle{\rm Im}(T)\sim 3\,{\rm Im}(U)\,,\qquad{\rm Im}(T)\to\infty\,.$
(4.39)

The vanishing of the heterotic Wilson line in 9 dimensions is reflected by
$V={\cal O}(1)$, which is realised in the present model. On this branch in
moduli space, it is not possible to produce an additional $SU(2)$ factor
because this would require that either $U=T$ or $2V=T$ (or, after exchange of
$T$ and $U$, $2V=U$) in the dual heterotic model, which is incompatible with
(4.38).

Correspondingly in F-theory, generating an additional $SU(2)$ factor would
require the collision of two single branes: This is not possible in the
Kulikov model realising the branch under consideration with generic values for
the parameters $b_{1}$ and $x$, because the two branes are localised on
different components.

####  $x=0$, $b_{1}$ generic of ${\cal O}(1)$

Figure -1192:  Kulikov Type III degeneration of $E_{7}\times E_{8}$
Weierstrass model (4.34) for generic value of the parameter $b_{1}$ and $x=0$.
The left shows how for $x=0$ one 7-brane from $B^{4}$ and one from $B^{3}$
coalesce at the intersection. The right picture shows the result of a base
change and blowups. The intermediate components over $B^{3}$ up to $B^{6}$
have generic I2 fibers and are not explicitly drawn.

A qualitative change occurs if we set the parameter $x=0$: One 7-brane on the
base component $B^{4}$ and the 7-brane on the component $B^{3}$ both wander
towards the intersection point $B^{3}\cap B^{4}$. See Figure -1192. This has
two related effects: First, at the collision point, it seems at first sight
that an additional $SU(2)$ gauge symmetry might arise. Second, according to
our general logic the singularity at $t=0$ in the limit $u=0$ is of type
$\hat{E}_{7}$, since the degrees of $(f,g,\Delta)$ decrease to $(4,6,10)$ away
from the intersection point.

Concerning the potential $SU(2)$ enhancement, however, we note that the
intersection locus $B^{3}\cap B^{4}$ is a nodal singularity of the base and
hence the physics interpretation of the 7-branes thereon is a priori obscure.
To obtain a better understanding we go back to the degenerate Weierstrass
model prior to the blowups and perform a base change $u\to u^{2}$, as
explained generally in [1]. Such a rescaling of the parameter $u$ cannot
change the interpretation of the physics. Removing the non-minimal
singularities in the fiber of the model after base change now requires a total
of 8 blowups with blowup coordinates $e_{i}$, $i=1,\ldots,8$. One obtains a
chain of 9 degenerate Weierstrass models, $Y_{0}=\cup_{i=0}^{8}Y^{i}$, with
I${}_{n_{i}}$ fibers over generic points of the following form:

$\displaystyle{\rm I}_{0}-{\rm I}_{2}-{\rm I}_{2}-{\rm I}_{2}-{\rm I}_{2}-{\rm
I}_{2}-{\rm I}_{2}-{\rm I}_{1}-{\rm I}_{0}\,.$ (4.40)

Here the left end component $Y^{8}$ (with base $B^{8}:\\{e_{8}=0\\}$) is
intersected by $t=0$ and the right end component $Y^{0}$ by $s=0$. The base
change resolves the relative position of two 7-branes which seem to coalesce
for $x=0$: In the new model, both branes are located at two generic points
within the base component $B^{7}$ given by the zeroes of

$\displaystyle 9e_{6}^{2}+8b_{1}e_{8}^{2}=0\,.$ (4.41)

This shows that as long as $b_{1}\neq 0$, the two branes cannot coincide and
therefore $x=0$ with $b_{1}\neq 0$ does not lead to an additional $SU(2)$
factor.

We conclude that for $x=0$ the symmetry group in the Kulikov model under
consideration is

$\displaystyle G_{\infty}=(\hat{E}_{7}\oplus\hat{E}_{8})/\sim\,$ (4.42)

and the non-abelian part of the gauge group in the partial decompactification
limit to 9d is $G_{\rm 9d}=E_{7}\times E_{8}$.

This behaviour is consistent with the heterotic dual model: The scaling (4.37)
of the parameters of the Weierstrass model remains unchanged by setting $x=0$
except for

$\displaystyle Q\sim\chi^{2}_{35}/\chi_{10}$ $\displaystyle\sim
q_{T}^{3}q_{U}^{3}(q_{T}-q_{U})^{2}q_{V}(q_{V}+\frac{1}{q_{V}})^{2}\sim
u^{15}\,.$ (4.43)

Naively, this is compatible with the scaling

$\displaystyle q_{U}\sim u\,,\qquad\frac{q_{T}}{q_{V}}\sim u^{3}\,,\qquad
q_{V}\sim u^{1/2}\,,$ (4.44)

from which one would conclude that $2V=U$ (corresponding to an $SU(2)$
enhancement), but this is too quick: For this solution the convergence
condition (4.19) is badly violated and the expansion (4.43) breaks down. Hence
there is no indication of an $SU(2)$ enhancement on the heterotic side either
and we are left with the non-abelian part of the 9d gauge algebra given by

$\displaystyle G_{\rm 9d}=E_{7}\oplus E_{8}\,.$ (4.45)

####  $b_{1}=0$, $x={\cal O}(1)$ and generic

If, by contrast, we set $b_{1}=0$, while keeping $x={\cal O}(1)$ and non-zero,
we enter a new branch because the vanishing order of the discriminant (4.33)
along the component $e_{2}=0$ increases to

$\displaystyle\Delta=e_{1}e_{2}^{2}e_{3}t^{9}s^{10}\Delta^{\prime}$ (4.46)

with

$\begin{split}\Delta^{\prime}|_{e_{4}=0}&=a_{0}^{3}(-12t^{2}x+3e_{3}t(-3+8i\sqrt{3}x+4x^{2})+4e_{3}^{2}(3i\sqrt{3}+9x-3i\sqrt{3}x^{2}-x^{3}))\,,\cr\Delta^{\prime}|_{e_{3}=0}&=-12a_{0}^{3}x\,,\cr\Delta^{\prime}|_{e_{2}=0}&=12ia_{0}^{3/2}(\sqrt{3}b_{2}e_{1}e_{3}+ia_{0}^{3/2}e_{1}^{2}x+\sqrt{3}e_{3}^{2})\,,\cr\Delta^{\prime}|_{e_{1}=0}&=12i\sqrt{3}a_{0}^{3/2}\,.\cr\end{split}$
(4.47)

Figure -1191:  Kulikov Type III degeneration of $E_{7}\times E_{8}$
Weierstrass model (4.34) for generic value of the parameter $x$ and $b_{1}=0$.

The 7-brane from the component $e_{3}=0$ has moved onto the component
$e_{2}=0$, see Figure -1191. This is reflected in a change of vanishing orders
of the modular forms to

$\chi_{10}\sim\chi_{12}\sim u^{4}\,,\qquad\psi_{4}^{3}-\psi_{6}^{2}\sim
u^{2}\,,\qquad\chi^{2}_{35}/\chi_{10}\sim u^{16}\,.$ (4.48)

Such behaviour requires that the heterotic variables scale as

$\displaystyle q_{T}\sim u^{2}\,,\qquad q_{U}\sim u^{2}\,,\qquad q_{V}={\cal
O}(1)\,.$ (4.49)

Again, $V={\cal O}(1)$ implies that the Wilson line in 9 dimensions is
ineffective and the gauge algebra is enhanced to $G_{\rm 9d}=E_{8}\oplus
E_{8}$ in the 9-dimensional limit.

The only way to achieve an additional gauge enhancement along this branch is
if the two roots of $\Delta^{\prime}|_{e_{2}=0}$ coincide, which occurs for

$\displaystyle x=-\frac{i\sqrt{3}b_{2}^{2}}{4a_{0}^{3/2}}\,,$ (4.50)

leading to the (non-abelian) symmetry group

$\displaystyle
G_{\infty}=SU(2)\oplus(\hat{E}_{8}\oplus\hat{E}_{8})/\sim\,\qquad\Longrightarrow\,\,G_{\rm
9d}=SU(2)\oplus E_{8}\oplus E_{8}\,.$ (4.51)

Since the two coinciding 7-branes belong to the two different groups of 12
branes on K3, each associated with a full $\hat{E}_{9}$ loop algebra, we
expect such enhancement to correspond to the locus where, in heterotic
coordinates, $T=U$. This can be achieved on the branch under consideration,
where $q_{T}\sim q_{U}\sim u^{2}$. By contrast, an enhancement by taking
$2V=T$ cannot be obtained on this branch, where $V$ and $T$ are separated by
two orders in $u$.

The vanishing order of the discriminant polynomial on the branch with
$b_{1}=0$ is

$\displaystyle Q\sim\frac{\chi^{2}_{35}}{\chi_{10}}\sim
x^{2}u^{16}\left(9a_{0}^{3/2}b_{2}^{2}-12i\sqrt{3}a_{0}^{3}x-2i\sqrt{3}a_{0}^{3/2}b_{2}^{2}x-8a_{0}^{3}x^{2}+{\cal
O}(u)\right)\,.$ (4.52)

The polynomial in brackets vanishes for

$\displaystyle x=-\frac{i\sqrt{3}b_{2}^{2}}{4a_{0}^{3/2}}\qquad{\rm or}\qquad
x=-\frac{i3\sqrt{3}}{2}\,.$ (4.53)

The first solution indeed corresponds to the $SU(2)$ enhancement locus found
above. We interpret the vanishing orders as indicative of the asymptotic
scaling

$\displaystyle q_{T}\sim u^{2}\,,\quad q_{U}=q_{T}+{\cal O}(u^{5/2})\,,\quad
q_{V}={\cal O}(1)\,.$ (4.54)

#### Branch $b_{1}=0$, $x=0$

Finally, a novel branch opens up for $b_{1}=0$ and $x=0$. The discriminant
becomes

$\displaystyle\Delta=e_{1}e_{2}^{2}e_{3}^{2}t^{9}s^{10}\Delta^{\prime}$ (4.55)

with

$\begin{split}\Delta^{\prime}|_{e_{4}=0}&=3ia_{0}^{3}(4\sqrt{3}e_{3}+3it)\,,\cr\Delta^{\prime}|_{e_{3}=0}&=-3a_{0}^{3/2}(3a_{0}^{3/2}e_{2}^{2}-(4i\sqrt{3}d_{1}+4a_{0}^{3/2}p)e_{2}e_{4}-4i\sqrt{3}b_{2}e_{4}^{2})\,,\cr\Delta^{\prime}|_{e_{2}=0}&=12i\sqrt{3}a_{0}^{3/2}(b_{2}e_{1}+e_{3})\,,\cr\Delta^{\prime}|_{e_{1}=0}&=12i\sqrt{3}a_{0}^{3/2}\,,\cr\Delta^{\prime}|_{u=0}&=3(4i\sqrt{3}a_{0}^{3/2}+9e_{1})\,.\end{split}$
(4.56)

Figure -1190:  Kulikov Type III degeneration of $E_{7}\times E_{8}$
Weierstrass model (4.34) for $b_{1}=0$ and $x=0$.

The geometry is depicted in Figure -1190. Consider first the surface component
$Y^{4}$. Apart from the $E_{7}$ singularity at $t=0$, an additional 7-brane is
located at $4\sqrt{3}e_{3}+3it=0$. The intersection curve $Y^{4}\cap Y^{3}$ is
an I2 fiber, as required for consistency of the rational surface $Y^{4}$,
which exhibits a brane content of total monodromy $M^{-1}_{{\rm
I}_{2}}=M_{\hat{E}_{7}}$ away from the intersection locus. The two additional
7-branes (which would enhance $\hat{E}_{7}$ further to $\hat{E}_{9}$) are
located at two points of $B^{3}$, given by the vanishing locus of the
polynomial $\Delta^{\prime}|_{e_{3}=0}$.

Similarly, on $Y^{0}$ we recognise an $E_{8}$ singularity at $s=0$ along with
one more I1 fiber at a generic position given by
$4i\sqrt{3}a_{0}^{3/2}s+9e_{1}=0$. Consistently with the total monodromy, the
intersection $Y^{0}\cap Y^{1}$ is an I1 fiber. The 7-brane which would enhance
$\hat{E}_{8}$ to $\hat{E}_{9}$ is located on $B^{2}$ at $b_{2}e_{1}+e_{3}=0$.

Let us now analyse the potential $SU(2)$ enhancements from the point of view
of the Weierstrass model. An obvious possibility occurs when the two 7-branes
on the base component $B^{3}$ coalesce, i.e. when the roots of the polynomial
$\Delta^{\prime}|_{e_{3}=0}$ coincide. By inspection of
$\Delta^{\prime}|_{e_{3}=0}$, this happens for

$\displaystyle
3i\sqrt{3}a_{0}^{3/2}b_{2}-3d_{1}^{2}+2i\sqrt{3}a_{0}^{3/2}d_{1}p+a_{0}^{3}p^{2}=0\,.$
(4.57)

From the previous discussion it is clear that the resulting $SU(2)$ factor
embeds into one of the two perturbative $E_{8}$ factors, i.e. the $SU(2)$
enhancement is the result of a Wilson line breaking $E_{8}\to E_{7}\oplus
SU(2)$, leading altogether to

$G_{\rm 9d}=E_{7}\oplus E_{8}\oplus SU(2)\,.$ (4.58)

This locus must correspond to the solution $2V=T$ (or, after exchange
symmetry, $2V=U$) on the heterotic side. Note that the scaling of the modular
forms

$\begin{split}&\chi_{10}\sim\chi_{12}\sim
u^{4}\,,\qquad\psi_{4}^{3}-\psi_{6}^{2}\sim u^{2}(b_{2}+b_{3}u+{\cal
O}(u^{2}))\,,\cr&\chi^{2}_{35}/\chi_{10}\sim
u^{18}b_{2}^{2}(3i\sqrt{3}a_{0}^{3/2}b_{2}-3d_{1}^{2}+2i\sqrt{3}a_{0}^{3/2}d_{1}p+a_{0}^{3}p^{2}+{\cal
O}(u))\,.\end{split}$ (4.59)

takes us out of the domain of convergence (4.19) of the expansions. This
impedes a simple comparison with the heterotic dual theory.

The second possible $SU(2)$ enhancement naively occurs if we set $b_{2}=0$. On
this locus, one brane on the base component $B^{3}$ and the brane on $B^{2}$
each move towards the intersection point of both base components. However, we
must again be careful in interpreting this configuration. A base change $u\to
u^{2}$ followed by eight blowups, similar to the one underlying the
configuration (4.40), show that for $b_{2}=0$ no actual gauge enhancement
occurs.

The same conclusion holds also for the putative intersection of both special
$SU(2)$ loci, i.e. for

$\displaystyle
b_{2}=0\,,\qquad-3d_{1}^{2}+2i\sqrt{3}a_{0}^{3/2}d_{1}p+a_{0}^{3}p^{2}=0\,.$
(4.60)

While naively one would conclude an enhancement to $E_{8}\oplus E_{7}\oplus
SU(3)$, this is an artefact of the collision of the three branes at the
intersection point of two base components. A base change $u\to u^{6}$,
followed by 24 blowups, reveals that for this special choice of parameters the
gauge algebra remains $G_{\rm 9d}=E_{8}\oplus E_{7}$, with no additional
enhancement.

#### Maximal enhancement $G_{\rm 9d}=E_{7}\oplus E_{8}\oplus SU(3)$

The picture just described and the pattern of non-abelian enhancements do not
change qualitatively by further tuning the vanishing orders of the coefficient
$b$ in (4.23) to higher order in $u$. This raises the question if the maximal
enhancement $G_{\rm 9d}=E_{7}\oplus E_{8}\oplus SU(3)$ can be achieved as an
infinite distance limit of F-theory. This is possible, but in a regime of
moduli space outside the realm of (4.14). To find a parameterisation of the
limit leading to $G_{\rm 9d}=E_{7}\oplus E_{8}\oplus SU(3)$, we start from the
Weierstrass model (4.2) and first tune the coefficients such as to realise an
I3 singularity at a generic point ${\cal P}$ away from $s=0$ or $t=0$. To be
concrete, let us take ${\cal P}=\\{s-t=0\\}$. Then the I3 singularity is
obtained if we take for instance

$\begin{split}&a=\left(\frac{-3}{16}\right)^{1/3}(1-4u)\,,\qquad
c=\left(\frac{-3}{16}\right)^{1/3}4u\,,\cr&d=u+u^{2}\,,\qquad
b=\frac{1}{6}-u-2u^{2}\,,\qquad e=u^{2}\,.\end{split}$ (4.61)

In the limit $u\to 0$ a non-minimal singularity with ${\rm
ord}(f,g,\Delta)|_{s=u=0}=(4,6,13)$ arises. As it turns out, to achieve a
configuration with no further non-minimalities, one must first perform a base
change $u\to u^{2}$, followed by 4 blowups beginning at $s=u=0$ and 2 more
blowups beginning at $t=u=0$. This leads to a chain of 7 surface components
with generic fibers of the form

$\displaystyle{\rm I}_{0}-{\rm I}_{1}-{\rm I}_{2}-{\rm I}_{3}-{\rm I}_{4}-{\rm
I}_{2}-{\rm I}_{0}\,,$ (4.62)

where $t=0$ lies on the right end component and $s=0$ on the left. The ${\rm
I}_{4}$-surface contains the I3 singularity in the fiber over a special point
of its base. Altogether the geometry realises the enhancement
$G_{\infty}=SU(3)\oplus(\hat{E}_{7}\oplus\hat{E}_{8})/\sim$, which is
interpreted as a gauge $G_{\rm 9d}=E_{7}\oplus E_{8}\oplus SU(3)$ in the 9d
decompactification limit.

##  5 Maximal Non-abelian Gauge Algebras in 9d from Type III.a Kulikov Models

As an immediate application of the classification of Type III.a Kulikov models
in [1] and their physical interpretation given in Section 3.4, we can
determine the maximal non-abelian gauge algebras in the nine-dimensional
theories that are obtained as decompactification limits at infinite distance
in the complex structure moduli space. As a consistency check, we will find a
perfect match with the results of [55, 56] (and the previous analysis in
[54]), which analyse the maximal non-abelian enhancements in heterotic string
theory compactified on $S^{1}$. Note that throughout this work, we are
considering F-theory on K3 surfaces without any non-trivial gauge background
turned on, which would reduce the rank of the theory.232323Such rank-reduced
setups have been considered recently from the viewpoint of the Swampland
program in [65, 66, 67, 68]. Hence the total rank in the original eight-
dimensional setting is $r=18$. It decreases to $r=17$ in the nine-dimensional
decompactification limit since one of the abelian gauge groups in eight
dimensions corresponds to the KK $U(1)$.

###  5.1 A-series and D-series

The decompactification limits of F-theory to nine dimensions correspond to two
kinds of Kulikov Type III.a degenerations: either both end surface components
$Y^{0}$ and $Y^{P}$ are dP9 surfaces with $(12-n_{1})$ and $(12-n_{P-1})$
7-branes, respectively, localised away from the intersection with the
neighbouring surface components; or $Y^{0}$ is a dP9 surface and $Y^{P}$ is an
In>0 surface with 2 $D$-type singularities and possibly additonal $A$-type
singularities.

In the first case, the non-abelian part of the symmetry algebra is

$G_{\infty}=H\oplus(\hat{E}_{n}\oplus\hat{E}_{m})/\sim\,,$ (5.1)

where $H$ refers to a direct sum of $A$-type Lie algebras and the ranks of the
involved E-type algebras are determined as $n=9-n_{1}$ and $m=9-n_{P-1}$.
Indeed, we know from the discussion of the Type III.a models that the
codimension-two singularities on the intermediate surface components $Y^{i}$
must all be of $A$ type. To determine the maximal possible non-abelian
enhancements in this class of models, we merely need to note that out of the
total of 24 7-branes (or singular fibers) of the original K3 surface, $(n+3)$
constitute the first $\hat{E}_{n}$ factor and $(m+3)$ the second
$\hat{E}_{m}$. This identifies

$\displaystyle G_{\infty}=A_{17-n-m}\oplus(\hat{E}_{n}\oplus\hat{E}_{m})/\sim$
(5.2)

as the maximal non-abelian symmetry algebra of a Type III.a model, for which
the remaining $18-n-m$ branes all coincide to form the $A_{17-n-m}$
configuration. Such configurations realise the maximal enhancement pattern

$\displaystyle G_{\rm 9d}^{\rm max}=A_{17-n-m}\oplus(E_{n}\oplus
E_{m})\,,\qquad 0\leq n,m\leq 8\,,\qquad n\neq 2,m\neq 2\,$ (5.3)

in the asymptotic nine dimensional theory. The reason why we have excluded the
values $n,m=2$ is because these do not yield a maximal enhancement. Indeed,
recall from (2.10) that $\hat{E}_{2}$ yields a gauge algebra $A_{1}+u(1)$ in
9d, whose non-abelian component agrees with the gauge algebra associated with
$\hat{E}_{1}$. Therefore the non-abelian gauge algebra of the configuration
with $n=2$ can be embedded into one with $n=1$.

It remains to analyse the second kind of Type III.a models, where the end
component $Y^{0}$ is a dP9 surface with symmetry algebra $\hat{E}_{k}$ with
$k=9-n_{1}$ and the end component $Y^{P}$ has generic I${}_{n_{P}>0}$ fibers.
The maximal non-abelian enhancements occur if both $D$-type algebras from
$Y^{P}$ and the extra $A$-type factors from the components different from
$Y^{0}$ combine into a single $D$-type algebra. Maximality in particular
requires that one of the $D$-type configuration consists only of a pair of
branes of type $B$ and $C$, in the notation of Section 2, hence realising the
trivial algebra $D_{0}$. Counting branes thus yields for the maximal non-
abelian symmetry algebras:

$\displaystyle G_{\infty}=D_{17-k}\oplus\hat{E}_{k}\,.$ (5.4)

This realises the second series for the maximal 9d gauge algebras that can be
realized via geometry:

$\displaystyle G_{\rm 9d}^{\rm max}=D_{17-k}\oplus E_{k}\,,\qquad 0\leq k\leq
8\,,k\neq 2\,.$ (5.5)

Recalling that $E_{1}=A_{1}$, $E_{2}=A_{1}\oplus u(1)$, $E_{3}=A_{2}\oplus
A_{1}$, $E_{4}=A_{4}$, $E_{5}=D_{5}$, one can check explicitly that the 44
inequivalent configurations of the $A$-series (5.3) and the $D$-series (5.5)
generate the complete list of maximal non-abelian gauge symmetries that can be
realized by the heterotic string compactified on $S^{1}$ [55].

We note in passing that the same conclusions can also be reached by taking the
realisation of the affine algebras $\hat{E}_{k}$ of Section 2 as the starting
point and explicitly constructing the possible maximal brane configurations
whose total $SL(2,\mathbb{Z})$ monodromy multiplies to unity.

It is also clear that the series of maximal $A$-type enhancements can be
constructed explicitly by a generalisation of the procedure that was
exemplified for the special case
$G_{\infty}=A_{2}\oplus(\hat{E}_{7}\oplus\hat{E}_{8})/\sim$ at the end of the
previous section. Instead of spelling this out for the other elements of the
$A$-series, let us now sketch the realisation of the maximal $D$-type
enhancements, which engineers the second type of Tye III.a Kulikov models with
only one rational elliptic end component.

###  5.2 Maximal $D$-series: construction

The second maximal non-abelian series (5.5) with a $D_{17-n}\oplus\hat{E}_{n}$
enhancement can be realised along the following lines:

  1. 1.

The starting point is a singularity with Weierstrass vanishing orders ${\rm
ord}(f,g,\Delta)|_{t=0}=(2,3,19-n)$, realising a $D_{17-n}$ enhancement at
$t=0$.

  2. 2.

In a second step, one engineers a singularity with non-minimal vanishing
orders

${\rm ord}(f,g,\Delta)|_{s=0}=(4,6,*)$ (5.6)

in an infinite distance limit parametrised as $u\to 0$ such that at the same
time the fiber over a generic point degenerates to an Ik singularity. This
requires a degeneration of the form (3.7). The infinite distance limit then
realises a (local, but not global) weak coupling limit in which the 10d string
coupling vanishes asymptotically at least away from a non-minimal singularity.

  3. 3.

The non-minimal singularity at $s=0$ in the limit $u\to 0$ is removed by one
or several blowups, starting at $s=u=0$.

After the blowups, the point $s=0$ lies on the base of a rational elliptic
component surface. The total degree of $(f,g,\Delta)$ on this component away
from the intersection point with the other surface components is $(4,6,3+n)$.
This indicates that the blowups have separated the branes associated with a
$\hat{E}_{n}$ singularity in the limit $u\to 0$. The $\hat{E}_{n}$
configuration contributes an asymptotically massless tower of BPS states which
we interpret as the Kaluza-Klein tower for the partial decompactification to
9d, as before.

  4. 4.

The component containing $t=0$, on the other hand, is not a rational elliptic
surface because the elliptic fiber degenerates over every point of the base
component. This behaviour is inherited from the Ik degeneration over generic
points of the K3 surface prior to the blowup. For a maximal symmetry algebra,
in addition to the overall degeneration, the component should exhibit the
$(2,3,19-n)$ singularity at $t=0$ as well as one more $D$-type singularity
with vanishing orders $(2,3,2)$ at a generic position away from $t=0$. As
explained at the end of Section 3.4, the latter singularity, which is a
$D_{0}$ fiber, cannot occur at finite distance for an elliptic K3, but is
realisable here due to the local weak coupling nature.

Let us spell this out in more detail for the example of an asymptotic
$G_{\infty}=D_{16}\oplus\hat{E}_{1}$ algebra, which realises the maximal non-
abelian gauge algebra $G_{\rm 9d}=D_{16}\oplus A_{1}$ in the 9d infinite
distance limit. Conveniently, the $D_{16}$ singularity at $t=0$ can be
engineered as a global Tate model [69], which is a parametrisation of the
functions $f$ and $g$ of the Weierstrass model as

$\displaystyle f=-\frac{1}{48}(\beta_{2}^{2}-24\beta_{4})\,,\qquad
g=-\frac{1}{864}(-\beta_{2}^{3}+36\beta_{2}\beta_{4}-216\beta_{6})$ (5.7)

for

$\beta_{2}=a_{1}^{2}+4a_{2}\,,\qquad\beta_{4}=a_{1}a_{3}+2a_{4}\,,\qquad\beta_{6}=a_{3}^{2}+4a_{6}\,.$
(5.8)

Each $a_{n}$ is a section of the line bundle $K^{-n}$, where by $K$ we denote
the canonical bundle of the base. In the present case, this means that $a_{n}$
is a polynomial of degree $2n$ in the homogeneous coordinates $[s:t]$ of the
base $\mathbb{P}^{1}$ of the elliptic K3, which therefore can be parametrised
as

$\displaystyle a_{n}=\sum_{i=0}^{2n}a_{n,i}\,t^{i}\,s^{2n-i}\,.$ (5.9)

A $D$-type singularity at $t=0$ arises if one specialises the $a_{n}$ such
that they vanish to a certain order at $t=0$, concretely [69]

$\displaystyle{\rm
ord}(a_{1},a_{2},a_{3},a_{4},a_{6})|_{t=0}=\begin{cases}(1,1,k+1,k+1,2k+1)&\text{for}\,D_{2k+2}\\\
(1,1,k,k+1,2k+1)&\text{for}\,D_{2k+1}\end{cases}\,.$ (5.10)

Applied to $D_{16}$, this means that we must set

$\displaystyle a_{1,0}=0\,,\quad a_{2,0}=0\,,\quad a_{3}=0\,,\quad
a_{4}=a_{4,8}t^{8}\,,\quad a_{6}=0\,,$ (5.11)

which indeed leads to vanishing orders ${\rm ord}(f,g,\Delta)|_{t=0}=(2,3,18)$
as required for $D_{16}$.

In the next step we engineer a non-minimal singularity with vanishing orders
${\rm ord}(f,g,\Delta)|_{s=0}=(4,6,*)$ in an infinite distance limit of the
form (3.7). To this end we first parametrise the remaining Tate monomials as

$\displaystyle a_{i,j}=\sum_{k\geq 0}a^{(k)}_{i,j}\,u^{k}$ (5.12)

and tune

$\displaystyle a^{(0)}_{2,3}=-\frac{1}{2}a^{(0)}_{1,1}\,a^{(0)}_{1,2}\,,\qquad
a^{(0)}_{2,4}=-\frac{1}{4}(a^{(0)}_{1,2})^{2}\,,\qquad a^{(0)}_{4,8}=0\,.$
(5.13)

With this prescription,

$\displaystyle f|_{u=0}=-\frac{1}{48}t^{2}s^{4}p_{1}(s,t)^{2}\,,\qquad
g|_{u=0}=\frac{1}{864}t^{3}s^{6}p_{1}(s,t)^{3}$ (5.14)

for a linear polynomial
$p_{1}(s,t)=4a_{2,1}s+((a^{(0)}_{1,1})^{2}+4a^{(0)}_{2,2})t$, while the
discriminant takes the form

$\displaystyle\Delta=-\frac{1}{16}(a^{(1)}_{4,8})^{2}\,t^{18}\,u^{2}\,\tilde{\Delta}^{\prime}\,.$
(5.15)

Note that in the limit $u\to 0$, the fiber over a generic point on the base
degenerates into an I2 fiber, as intended.

Figure -1189:  Kulikov Type III Weierstrass model with
$D_{16}\times\hat{E}_{1}$ enhancement.

The non-minimal singularity at $s=0$ in the limit $u\to 0$ can be removed by a
blowup. The blowup turns out to be simplified if one performs a base change

$u\to u^{4}\,,$ (5.16)

after which

$\displaystyle\Delta=-\frac{1}{16}{(a^{(1)}_{4,8})^{2}}\,t^{18}\,u^{8}\,\Delta^{\prime}\,.$
(5.17)

For this degeneration, the simple blowup

$\displaystyle(s,u)\to(s\,e_{1},u\,e_{1})$ (5.18)

gives rise to a Type III.a Kulikov model consisting of two intersecting
components. See Figure -1189 for an illustration; as before, we replace the
coordinate symbol $u$ by $e_{0}$ after the blowup. Over the component
$e_{0}=0$, the elliptic fiber degenerates as indicated by the power of
$e_{0}^{8}$ in the discriminant. The fibration over the component $e_{1}=0$,
on the other hand, is a rational elliptic surface with 12 singular fibers. Of
these, $8$ singular fibers are accounted for by the intersection with the
component $e_{0}=0$ carrying the I8 singularity. The remaining 4 singular
fibers are located away from intersection point $e_{1}=e_{0}=0$. Only the
latter correspond to physical 7-branes located at points of the base. More
precisely, over $e_{1}=0$ we can set $t=1$ and obtain

$\displaystyle f|_{e_{1}=0}=p_{4}(s,e_{0})\,,\qquad
g|_{e_{1}=0}=s^{2}p_{2}(s,e_{0})\,,\qquad\Delta|_{e_{1}=0}=q_{4}(s,e_{0})\,,$
(5.19)

for some polynomials of the indicated degrees. According to our general
prescription, the rational elliptic surface contributes a $\hat{E}_{1}$ factor
to the asymptotic symmetry algebra. This corresponds to a non-abelian gauge
algebra component $A_{1}$ in the 9d theory at infinite distance.242424As
discussed after (2.5), the $4$ singular fibers may in principle lead to an
extensioin to $\hat{\tilde{E}}_{1}$. We leave it to future work to clarify
this ambiguity.

Over the surface patch with base $e_{0}=0$ we have, on the other hand,

$\displaystyle f|_{e_{0}=0}=t^{2}\,p^{2}_{1}(e_{1},t)\,,\qquad
g|_{e_{0}=0}=t^{3}\,p^{3}_{1}(e_{1},t)\,,\qquad\Delta^{\prime}|_{e_{0}=0}=p^{2}_{1}(e_{1},t)\,,$
(5.20)

where $\Delta^{\prime}$ is the residual discriminant after factoring out
$-\frac{1}{16}(a^{(1)}_{4,8})^{2}\,t^{18}$. Apart from the $(2,3,18)$
singularity at $t=0$ we observe a non-minimal singularity with vanishing
orders $(2,3,2)$ at $p_{1}(e_{1},t)=0$. Altogether, this component carries 20
of the original 24 7-branes on K3, complementing the 4 7-branes which form the
$\hat{E}_{1}$ singularity on the dP9 surface over $e_{1}=0$.

This procedure can be generalised to construct also the remaining maximal
enhancements in the $D$-series, albeit not all of them as degenerations of
global Tate models.

##  6 Emergence of Quartic Gauge Couplings?

We now change tack and comment on the large distance asymptotics of the
quartic gauge couplings in eight dimensions, by mostly making use of known
results for the dual heterotic string. These couplings are the analogs of the
well-investigated quadratic gauge couplings in $N=2$ supersymmetric string
theories in four dimensions (see, for example, the early works [70, 71]).
Indeed, in eight dimensions it is not the gauge kinetic terms, but rather the
quartic couplings that exhibit logarithmic one-loop corrections: the quadratic
gauge kinetic terms do not receive any quantum corrections and become
irrelevant in the UV. Distinguishing quadratic and quartic terms in eight
dimensions may thus shed some light on the notion of emergence, from the point
of view that not only the quadratic terms but all couplings might be emergent
in string theory.

At one loop order, the relevant, bosonic structure of the quartic terms takes
the schematic form

$\mathcal{S}_{(4)}^{{\rm 8d,1-loop}}\ \sim\ \int
d^{8}x\sqrt{-G^{(8)}}\left(\frac{1}{g(t)^{2}}\,[F^{4}]+\theta(t)[F\wedge
F\wedge F\wedge F]\right)\,,$ (6.1)

where $[\ast]$ summarily denotes the possible gauge and Lorentz contractions
pertaining to a variety of superinvariants. The moduli dependent quantum
couplings $g(t)$, $\theta(t)$ are known to be BPS-saturated and one-loop exact
in the dual frame of the heterotic string. As such they can be computed [72,
73, 74, 75, 76] by generalizing the methods originally developed in [70, 71]
for four-dimensional theories, and the notions of Borcherds lifts and Kac-
Moody algebras apply here as well. Computations in F-theory geometry were
presented in [74, 75, 77], and for Type I strings e.g. in [78, 72, 73, 76, 79,
80].

We will consider here only abelian gauge fields, namely those which arise from
the Kaluza-Klein compactification and which are the superpartners of the
moduli $t$. They have quite different properties as compared to the gauge
fields that are already present in ten dimensions. As we will see, this is
true in particular from the point of view of emergence, which is why we will
focus on them.

We will consider the parity-even couplings that arise from KK reduction of the
heterotic string action in ten dimensions. The part relevant for us reads
schematically [81, 82]

$\displaystyle\mathcal{S}^{{\rm 10d}}\ \sim$ $\displaystyle\\!\\!\\!\\!\\!\int
d^{10}x\,\sqrt{-G^{(10)}}\left[e^{-2\phi^{(10)}}\left(R+H^{2}+R^{2}+R^{4}\dots\right)_{{\rm
tree}}\right.$ $\displaystyle\qquad\qquad\qquad+\left.\left(R^{4}+B\wedge
R\wedge R\wedge R\wedge R+\dots\right)_{{\rm 1-loop}}+\dots\right].$

Here $H=dB+\Omega_{L}+\Omega_{F}$ is the Chern-Simons corrected three-form
field strength, where $\Omega_{L}=\omega\wedge