CERN-TH-2022-032

LMU-ASC 10/22
UPR-1316-T

One Loop to Rule Them All:

Eight and Nine Dimensional String Vacua from Junctions

Mirjam Cvetič1,2,3, Markus Dierigl4, Ling Lin5, Hao Y. Zhang1

1 Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, PA 19104, USA
2 Department of Mathematics,
University of Pennsylvania, Philadelphia, PA 19104, USA
3 Center for Applied Mathematics and Theoretical Physics,
University of Maribor, SI20000 Maribor, Slovenia
4 Arnold-Sommerfeld-Center for Theoretical Physics,
Ludwig-Maximilians-Universität, 80333 München, Germany
5CERN Theory Department, CH-1211 Geneva, Switzerland

String and 5-brane junctions are shown to succinctly classify all known 8d
${\cal N}=1$ string vacua. This requires an extension of the description for
ordinary $[p,q]$-7-branes to consistently include O7+-planes, which then
naturally encodes the dynamics of $\mathfrak{sp}_{n}$ gauge algebras,
including their $p$-form center symmetries. Central to this analysis are loop
junctions, i.e., strings/5-branes which encircle stacks of 7-branes and O7+’s.
Loop junctions further signal the appearance of affine symmetries of emergent
9d descriptions at the 8d moduli space’s boundaries. Such limits reproduce all
9d string vacua, including the two disconnected rank $(1,1)$ moduli
components.

###### Contents

  1. 1 Introduction
  2. 2 String and 5-brane junctions
    1. 2.1 Basics of $[p,q]$-7-branes and string junctions
    2. 2.2 The junction lattice
    3. 2.3 Extended (co-)weights and higher-form center symmetries
  3. 3 Junctions on O7+ and center symmetries of $\mathfrak{sp}$ dynamics
  4. 4 8d string vacua and their global structure from junctions
    1. 4.1 Gauge group topology from global null junctions
    2. 4.2 Duality to heterotic and CHL descriptions
    3. 4.3 A rank $(2,2)$ momentum lattice
    4. 4.4 Examples
  5. 5 9d vacua via affine 7-brane stacks
    1. 5.1 Global structure of 9d vacua of rank 17
    2. 5.2 9d CHL vacua via junctions
    3. 5.3 Disconnected moduli branches for 9d rank $(1,1)$ vacua
  6. 6 Conclusions and outlook
  7. A Deriving the evenness condition on O7+ via 8d CHL strings
  8. B Embedding O7+ into $\widehat{\text{E}}_{8}$
  9. C All 8d supergravity vacua via $[p,q]$-7-branes
    1. C.1 Non-maximally-enhanced cases
    2. C.2 No O7+
    3. C.3 One O7+
    4. C.4 Two O7+’s

##  1 Introduction

Supergravity theories in a large number of dimensions form an ideal laboratory
to investigate the manifestation of quantum gravitational consistency
conditions in the low-energy limit of string theory. In particular, they
provide a concrete class of models which corroborates the conjecture of
“string universality”, or “string lamppost principle”, stating that all
consistent (super)gravity theories arise from string theory.

Formulating and sharpening the relevant conditions on consistent effective
theories of quantum gravity are at the heart of the Swampland Program [1, 2].
Arguably, among the best motivated of these constraints is that quantum
gravity theories should have no exact global symmetries. This statement can
also be applied to higher-form global symmetries, such as 1-form center
symmetries of non-Abelian gauge sectors. The condition demands that these
symmetries are either gauged or broken. However, if they are to be gauged, one
needs to demand the absence of obstructions/anomalies to turning on the gauge
fields of these generalized symmetries. This absence of anomalies of center
1-form symmetries can lead to severe restrictions on the global topology of
the allowed gauge groups in supersymmetric theories [3, 4]. Similarly, the
absence of global symmetries requires certain topological invariants called
bordism groups to be trivial [5], once more leading to powerful constraints,
in particular on the total rank of the gauge symmetry, of consistent
supergravity theories in more than six dimensions [6, 7].

Being confronted with the set of supergravity models that pass the above
consistency tests, the remaining question is whether all of these can be
realized in string theory. To answer this we therefore need good control of
the realization of the global form of the gauge groups, i.e., the fate of the
center 1-form symmetries in string theory constructions. In the present work
we focus on compactifications to eight and nine dimensions (8d and 9d) with 16
supercharges (i.e., ${\cal N}=1$).

A powerful approach that successfully utilizes the machinery of geometry is
F-theory [8], which ties the algebraic and arithmetic properties of elliptic
K3-surfaces to 8d gauge theories with ADE gauge algebras of total rank
$(2,18)$.111Throughout this work, we collect the number $a$ of independent
gravi-photons, and the maximal non-Abelian gauge rank $r$ into a pair $(a,r)$,
which we often refer to as the (total) gauge rank. At generic values of
moduli, the gauge algebra is hence $\mathfrak{u}(1)^{a+r}$. In this context,
the global gauge group structure is encoded in the Mordell–Weil group of
rational sections of the elliptic fibration [9, 10, 11]. Under M-/F-theory
duality, this can be phrased in terms of gauging and breaking higher-form
symmetries [12], that is reflected geometrically in the gluing of torsional
homology cycles in local patches containing the non-Abelian gauge dynamics
[13].222The investigation of generalized symmetries within the geometric
engineering framework has received broad attention in recent literature [14,
15, 16, 17, 3, 4, 18, 19, 20, 21, 13, 22, 23, 24, 25, 26]. Moreover, through
suitable deformations that correspond to infinite distance points in the 8d
moduli space, the F-theory geometry also classifies 9d ${\cal N}=1$ string
vacua with gauge rank $(1,17)$ [27, 28]. This is consistent with the dual
heterotic description, where the 9d moduli space — described via the rank
$(1,17)$ Narain lattice — is contained in that of the 8d moduli space, with a
rank $(2,18)$ Narain lattice description (see [29] for a recent comprehensive
study).

However, the dictionary between geometry and physics is less understood in the
presence of so-called frozen singularities [30, 31, 32, 33]. While these are
known to be the necessary ingredient for an F-theory description of
$\mathfrak{sp}$ gauge algebras on the 8d ${\cal N}=1$ moduli branches of gauge
ranks $(2,10)$ and $(2,2)$, the characterization of, e.g., the gauge group
topology is no longer purely geometric (i.e., given by the Mordell–Weil group)
[34]. Likewise, it is not immediately clear how to identify decompactification
limits on these moduli spaces. On the other hand, advances in the Swampland
program [35] strongly suggest, that all 8d ${\cal N}=1$ vacua should have a
characterization in terms of an elliptically-fibered K3.

As we will demonstrate in this work, string junctions provide a _unified
framework_ that encompasses all these features. In this description, the
underlying elliptic K3 is encoded in the configuration of $[p,q]$-7-branes of
type IIB string theory, whose $[p,q]$-type are in one-to-one correspondence to
elliptic singularities characterized by an $SL(2,\mathbb{Z})$-monodromy
$M_{[p,q]}$. The junctions are then $(p,q)$-strings or -5-branes stretched
between the 7-branes. In their original formulation [36, 37, 38] that is
equivalent to F-theory without frozen singularities, junctions describe the 8d
gauge dynamics with ADE gauge algebras,333String junctions have been also used
to construct lower-dimensional theories, see [39, 40, 41, 42, 43, 44, 45, 46,
47]. as well as their higher-form symmetries [13]. To account for a junction
description of all 8d ${\cal N}=1$ vacua, we extend the discussion to include
O7+-planes, which are the IIB avatars of frozen singularities, and have the
same monodromy as an elliptic $\text{D}_{8}$ singularity [30, 31, 32].

A concept that will be key to this work are so-called fractional null
junctions, which are certain fractional (and hence, unphysical)
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges encircling
all 7-branes, i.e., _loop junctions_. In the absence of O7+-planes, these are
known to be equivalent to Mordell–Weil torsion of the underlying elliptic K3
[48, 49]. To correctly account for the electric and magnetic center symmetries
and the gauge group topology for the $\mathfrak{sp}$ gauge symmetries that
arise in the presence of O7+, it turns out to be instrumental to study
separately $(p,q)$-strings and -5-branes. The key difference is, while any
integer number of 5-branes can end on an O7+, the number of _string_ -prongs
there must be even. Indeed, with this modification, we find that the junction
description of 8d vacua with one O7+ is equivalent to so-called CHL vacua [50,
51] of rank $(2,10)$, including the characterization of the global gauge group
structure [34, 52]. Moreover, it is straightforward to include two O7+-planes,
thereby giving a junction-esque classification of 8d string vacua with gauge
rank $(2,2)$ including their gauge group topology, for which there is no known
heterotic or CHL string description.

In addition, we also propose a junction description for decompactification
limits to 9d including O7+-planes. Parallel to the 9d uplifts of the rank
$(2,18)$ setting [27, 28] (see also [53] for a related discussion of 10d
uplifts of 9d heterotic vacua), we identify the corresponding infinite
distance limits with O7+-planes by the emergence of loop junctions that
affinize the 8d gauge algebra. Again, the subtle differences from having
modified boundary conditions for strings and 5-branes can be cross-checked
with the momentum lattice description of the CHL string for uplifting 8d rank
$(2,10)$ theories to 9d rank $(1,9)$ theories. Like in 8d, the junction
description naturally encodes the gauge group topology of 9d vacua. For the
rank $(2,2)$ theories without a momentum lattice analog, the 9d theories with
rank $(1,1)$ that result from the junction description live on two
disconnected moduli branches that are only connected through an
$S^{1}$-reduction to 8d, which matches other stringy constructions [54]. This
further establishes junctions as a complimentary framework to sharpen aspects
of the Swampland Distance conjecture [2] in string compactifications.

The rest of the paper is organized as follows. After reviewing the junction
framework with ordinary $[p,q]$-7-branes in Section 2, we discuss, in Section
3, the modified boundary conditions for strings and 5-branes on an O7+-plane
that give rise to the correct higher-form symmetries of $\mathfrak{sp}$ gauge
algebras in 8d. In Section 4, we then describe global 8d models by “gluing”
together local patches with 7-brane stacks involving O7+-planes. A particular
focus will be on the gauge group topology that is encoded in the fractional
null junctions. We then examine, in Section 5, the infinite distance limits
described via 7-branes and junctions that correspond to 9d ${\cal N}=1$ vacua,
for which we will also determine the global gauge group structure. The
appendices contain some technical aspects, as well as the full list of gauge
group topologies for all 8d vacua with maximally-enhanced non-Abelian
symmetries in Appendix C.

##  2 String and 5-brane junctions

String junctions provide an efficient way to classify electrically charged
states with respect to gauge symmetries localized on 7-brane stacks in type
IIB string theory. Therefore, they also contain information about the electric
1-form center symmetries and the global realization of the 8d gauge group [48,
49, 13]. The magnetically dual perspective is provided by 5-brane webs, which
can also be described by junctions [55, 56, 57, 58].

In this section we recall some properties of string and 5-brane junctions in
the presence of a general $[p,q]$-7-brane stack. This provides a local
construction of the charged states. Importantly, the charge under the center
symmetry is related to the appearance of certain _fractional junctions_ , the
extended weight junctions, that determine the global properties of the model
[13]. We then generalize the discussion of string and 5-brane junctions to
backgrounds containing O7+-planes, whose geometric interpretation in F-theory
is more challenging [32, 33]. With the help of string junctions we can
successfully extract the correct properties of these configurations and
identify the electric center symmetries also for symplectic gauge groups. This
analysis is repeated with 5-brane junctions, which, as opposed to the ADE
algebras realized without O7+’s, have a subtle distinction from string
junctions that is precisely needed to correctly account for the magnetic
center symmetry.

###  2.1 Basics of $[p,q]$-7-branes and junctions

In this section we will recall the basics of the junction description for 8d
${\cal N}=1$ dynamics from type IIB compactifications [36, 38]. The key
players are spacetime filling $[p,q]$-7-branes $\mathbf{X}_{[p,q]}$, which we
will denote with square brackets. A single 7-brane must have coprime $p$ and
$q$. In the plane perpendicular to its worldvolume, $\mathbf{X}_{[p,q]}$
induces a singularity in the axio-dilaton profile $\tau=C_{0}+ie^{-\phi}$,
composed of the RR 0-form $C_{0}$ and the dilaton field $\phi$, which is
characterized by an $SL(2,\mathbb{Z})$ monodromy

$\displaystyle M_{[p,q]}=\begin{pmatrix}1+pq&-p^{2}\\\
q^{2}&1-pq\end{pmatrix}\in SL(2,\mathbb{Z})\,,$ (2.1)

which acts on $\tau$ by a Möbius transformation, and in the doublet
representation,

$\displaystyle\left(\begin{smallmatrix}B_{2}\\\
C_{2}\end{smallmatrix}\right)\mapsto
M_{[p,q]}\left(\begin{smallmatrix}B_{2}\\\ C_{2}\end{smallmatrix}\right)\,,$
(2.2)

on the NSNS- and RR-2-form fields $(B_{2},C_{2})$. This monodromy can be
captured by a branch cut in the perpendicular plane that emanates form the
7-brane. In the following it will prove useful to introduce conventions for
some special 7-branes, that will later appear in the construction of non-
Abelian gauge algebras

$\begin{split}&\mathbf{A}=\mathbf{X}_{[1,0]}:\quad
M_{[1,0]}=\begin{pmatrix}1&-1\\\ 0&1\end{pmatrix}\,,\\\
&\mathbf{B}=\mathbf{X}_{[1,-1]}:\quad M_{[1,-1]}=\begin{pmatrix}0&-1\\\
1&2\end{pmatrix}\,,\\\ &\mathbf{C}=\mathbf{X}_{[1,1]}:\quad
M_{[1,1]}=\begin{pmatrix}2&-1\\\ 1&0\end{pmatrix}\,,\\\
&\mathbf{N}=\mathbf{X}_{[0,1]}:\quad M_{[0,1]}=\begin{pmatrix}1&0\\\
1&1\end{pmatrix}\,.\end{split}$ (2.3)

In a local model, where the perpendicular plane is non-compact (i.e., is
$\mathbb{R}^{2}=\mathbb{C}\ni z$), it is customary to extend the branch cuts
all downwards (meeting at $z=-i\infty$, without crossing each other before).
Starting from a configuration describing a certain 8d vacuum, we can obtain
another one on the same 8d ${\cal N}=1$ moduli space by moving the 7-branes.
When $\mathbf{X}_{[p_{1},q_{1}]}$ crosses the branch cut of
$\mathbf{X}_{[p_{2},q_{2}]}$ from the left to right, the $[p,q]$-type changes
according to:

$\displaystyle\begin{split}\underrightarrow{{\bf X}_{[p_{1},q_{1}]}}{\bf
X}_{[p_{2},q_{2}]}\rightarrow{\bf X}_{[p_{2},q_{2}]}{\bf X}_{[p_{1}+D\cdot
p_{2},\,q_{1}+D\cdot q_{2}]}\,,\end{split}$ (2.4)

where $D\equiv\det\left(\begin{smallmatrix}p_{1}&p_{2}\\\
q_{1}&q_{2}\end{smallmatrix}\right)$, and the arrow indicating the branch-cut-
crossing 7-brane. Likewise, when it crosses from right to left, one has

$\displaystyle{\bf X}_{[p_{2},q_{2}]}\underleftarrow{{\bf
X}_{[p_{1},q_{1}]}}\rightarrow$ $\displaystyle{\bf X}_{[p_{1}+D\cdot
p_{2},\,q_{1}+D\cdot q_{2}]}{\bf X}_{[p_{2},q_{2}]}\,.$ (2.5)

For any concrete configuration, we can arrange the 7-branes along a horizontal
axis, and denote them as
$\mathbf{X}_{[p_{1},q_{1}]}\mathbf{X}_{[p_{2},q_{2}]}\,...$ by labelling from
left to right. The $SL(2,\mathbb{Z})$ monodromy around any (connected) part of
this chain is the product of the individual monodromies of the encircled
branes _from right to left_. To obtain a valid global configuration (i.e.,
where the perpendicular plane is $\mathbb{P}^{1}$) describing an 8d ${\cal
N}=1$ supergravity model, tadpole cancellation requires exactly 24
$[p,q]$-7-branes, whose overall monodromy must be the identity. Note that,
while their _relative_ $[p,q]$-types are pivotal for distinguishing different
physical configurations, an _overall_ $SL(2,\mathbb{Z})\ni g$ transformation,

$\displaystyle\left[\begin{smallmatrix}p_{i}\\\
q_{i}\end{smallmatrix}\right]\mapsto g\left[\begin{smallmatrix}p_{i}\\\
q_{i}\end{smallmatrix}\right]\,,\quad M_{[p_{i},q_{i}]}\mapsto
gM_{[p_{i},q_{i}]}g^{-1}\quad\text{for all }\,i\,,$ (2.6)

does not matter physically for a model (local or global) described by a
collection of 7-branes $\mathbf{X}_{[p_{i},q_{i}]}$.

BPS-particles in 8d arise from $(p,q)$-strings — a bound state of $p$
fundamental and $q$ D-strings with electric charge
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$ under
$\left(\begin{smallmatrix}B_{2}\\\ C_{2}\end{smallmatrix}\right)$ — anchored
on the 7-branes of the same $[p,q]$-type, and extending as a directed line
into the perpendicular plane. Their magnetically dual objects, which are four-
dimensional in 8d, are given by $(p,q)$-5-branes — a bound state of $p$ NS5-
and $q$ D5-branes with magnetic charge $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$ under $\left(\begin{smallmatrix}B_{2}\\\
C_{2}\end{smallmatrix}\right)$ — that share four common spatial directions as
the 7-brane and also project to lines in the perpendicular plane. Since
strings and 5-branes can fuse and split, so long as the overall
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge is conserved
at every vertex, they form junctions, see left of Figure 1. Note that, by
flipping the direction on any prong, its $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge acquires a minus sign.444To make contact
with the F-theory description of type IIB, note that (directed) junctions can
be interpreted as (oriented) 2-cycles in an elliptic K3, on which M2- and
M5-branes can be wrapped, which are the objects dual to strings and 5-branes
under M-/F-theory duality. See, e.g., [13] for details of this correspondence.

As charged objects of the 2-form fields $(B_{2},C_{2})$, strings and 5-branes
also experience $SL(2,\mathbb{Z})$ monodromies as they are transported around
7-branes. This action can be represented in the perpendicular plane, after
choosing the branch cuts, by an analogous transformation

$\displaystyle\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)\rightarrow\left(\begin{smallmatrix}r^{\prime}\\\
s^{\prime}\end{smallmatrix}\right)=M_{[p,q]}\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)=\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)+(qr-ps)\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)\,$ (2.7)

on the $\left(\begin{smallmatrix}r\\\ s\end{smallmatrix}\right)$-charges of a
junction-prong as it crosses the branch cut of a $[p,q]$-7-brane, see middle
of Figure 1. Finally, in analogy to Hanany–Witten transitions [59], the same
junction can be expressed, by moving the branch-cut-crossing prong across the
7-brane, as a junction with an additional prong on the 7-brane, see right of
Figure 1.

Figure 1: Strings and 5-branes, which are represented as lines in the
perpendicular plane, form junctions, where the $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge at each vertex is conserved (left). In the
presence of 7-branes, they undergo monodromy transformations (2.7) when they
cross a branch cut (middle). By a Hanany–Witten transition, the same junction
can be represented as having a prong on the 7-brane (right).

To have non-Abelian gauge dynamics in 8d, we have to collide 7-branes to form
stacks. Strings that stretch between different constituents of a stack then
become light and form massless W-bosons of the enhanced gauge symmetry. In
terms of the 7-brane types (2.3), ADE-gauge algebras are realized when the
following stacks form555We have chosen a particular $SL(2,\mathbb{Z})$-frame
that is common in the literature, but any $SL(2,\mathbb{Z})$-conjugated
configuration would obviously give the same gauge algebra.:

$\displaystyle\begin{array}[]{| c | c | c |}\hline\cr\text{Lie algebra}&\text{brane constituents}&\text{monodromy}\\\ \hline\cr\hline\cr\mathfrak{su}_{n}&\mathbf{A}^{n}&\begin{pmatrix}1&-n\\\ 0&1\end{pmatrix}\\\ \hline\cr\mathfrak{so}_{2n}&\mathbf{A}^{n}\mathbf{B}\mathbf{C}&\begin{pmatrix}-1&n-4\\\ 0&-1\end{pmatrix}\\\ \hline\cr\mathfrak{e}_{n\geq 1}&{\bf A}^{n-1}{\bf B}{\bf C}^{2}&\begin{pmatrix}-2&2n-9\\\ -1&n-5\end{pmatrix}\\\ \hline\cr\tilde{\mathfrak{e}}_{n\geq 0}&{\bf A}^{n}{\bf X}_{[2,-1]}{\bf C}&\begin{pmatrix}-3&3n-11\\\ -1&n-4\end{pmatrix}\\\ \hline\cr\end{array}$ (2.13)

where we have used exponents to group the same type of branes that are appear
consecutively. The overall monodromy of a 7-brane stack is the product of the
individual branes from right to left; e.g.,
$M_{\mathfrak{so}_{2n}}=M_{[1,1]}M_{[1,-1]}M_{[1,0]}^{n}$. The realizations of
the exceptional algebras are physically equivalent, i.e., equal up to 7-brane
moves inside the stack and $SL(2,\mathbb{Z})$ conjugations, for $n\geq
2$,666We use the standard identifications
$\mathfrak{e}_{2}\cong\mathfrak{su}_{2}\oplus\mathfrak{u}(1)$,
$\mathfrak{e}_{3}\cong\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}$,
$\mathfrak{e}_{4}\cong\mathfrak{su}_{5}$,
$\mathfrak{e}_{5}\cong\mathfrak{so}_{10}$. while
$\mathfrak{e}_{1}\cong\mathfrak{su}_{2}$ and
$\tilde{\mathfrak{e}}_{1}\cong\mathfrak{u}(1)$; finally, the
$\tilde{\mathfrak{e}}_{0}$ configuration corresponds to a trivial gauge
algebra. There are additional strongly coupled versions of the Lie algebra
$\mathfrak{su}_{n}$ with $n\in\\{2,3\\}$ of the form
$\mathbf{A}^{n+1}\mathbf{C}$. In the remaining part of this section, we will
focus mainly on the “standard” cases $\mathfrak{su}_{n}$, $\mathfrak{so}_{2n}$
and $\mathfrak{e}_{n\geq 6}$, while $\tilde{\mathfrak{e}}_{n}$ will be
relevant in Section 5. Of course there is a beautiful relation between the
7-branes stacks above with their induced $SL(2,\mathbb{Z})$ monodromies, and
the classification of singularities in elliptic fibrations by Kodaira, which
is central in F-theory (see [60, 61] for recent reviews and additional
references). In the following, we will focus solely on the junction
perspective.

###  2.2 The junction lattice

In the following, we give an abstract definition of junctions as lines in the
plane perpendicular to the 7-branes satisfying the axioms above. In principle,
one has to specify if they represent $(p,q)$-strings or 5-brane webs to attach
physical meaning to them.

Consider the junctions formed by a single prong extending from one 7-brane
$\mathbf{X}_{[p,q]}$, which we denote with a lower case letter as
$\mathbf{x}_{[p,q]}$, and sometimes call a unit junction. In analogy to the
different types defined in (2.3), there are then also junctions

$\displaystyle\mathbf{a}\,,\enspace\mathbf{b}\,,\enspace\mathbf{c}\,,\enspace\mathbf{n}\,.$
(2.14)

Since a general string or 5-brane junction takes the form of a linear
combination of the individual prongs, the set of all physical junctions
(strings or 5-branes) on a 7-brane configuration,
$\mathbf{X}_{[p_{1},q_{1}]}\mathbf{X}_{[p_{2},q_{2}]}\cdots\mathbf{X}_{[p_{i},q_{i}]}\cdots$,
form a $\mathbb{Z}$-module,

$\displaystyle J_{\text{phys}}=\big{\\{}{\bf
j}=\sum_{i}a^{i}\mathbf{x}_{[p_{i},q_{i}]}\,|\,a^{i}\in\mathbb{Z}\big{\\}}\,.$
(2.15)

One important physical invariant is the net, or _asymptotic_
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge of a junction
${\bf j}$, given by $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)_{\text{asymp}}=\sum_{i}a^{i}\left(\begin{smallmatrix}p_{i}\\\
q_{i}\end{smallmatrix}\right)$.

One further defines a symmetric bi-linear pairing $(.,.)$ on this module as
follows. For the basis junction $\mathbf{x}_{[p_{i},q_{i}]}$ (note that the
ordering of the 7-branes is important), one defines777Here, we simply present
the rules as stated in [38]. It can be shown that they agree with the
geometric intersection pairing for the elliptic K3 of the dual F-theory
description.

$\displaystyle\begin{split}&\big{(}\mathbf{x}_{[p_{i},q_{i}]},\mathbf{x}_{[p_{j},q_{j}]}\big{)}=\big{(}\mathbf{x}_{[p_{j},q_{j}]},\mathbf{x}_{[p_{i},q_{i}]}\big{)}=\begin{cases}-1\,,&\text{if
}\,i=j\\\ \tfrac{1}{2}\det\left(\begin{smallmatrix}p_{i}&p_{j}\\\
q_{i}&q_{j}\end{smallmatrix}\right)\,,&\text{if
}\,\mathbf{X}_{[p_{i},q_{i}]}\,\text{ is on the left of
}\,\mathbf{X}_{[p_{j},q_{j}]}\,.\end{cases}\end{split}$ (2.16)

By linearly extending to the module $J_{\text{phys}}$, we endow it with a
lattice structure, which will be called the (physical) junction lattice. For
example, consider an arrangement of only $\mathbf{A}$, $\mathbf{B}$ and
$\mathbf{C}$ branes which are ordered “alphabetically”,

$\displaystyle\mathbf{A}_{1}\,\cdots\,\mathbf{A}_{\alpha}\,\cdots\,\mathbf{B}_{1}\,\cdots\,\mathbf{B}_{\beta}\,\cdots\,\mathbf{C}_{1}\,\cdots\,\mathbf{C}_{\gamma}\,.$
(2.17)

For this 7-brane configuration, we have

$\displaystyle\begin{split}&(\mathbf{a}_{\alpha},\mathbf{a}_{\alpha^{\prime}})=-\delta_{\alpha,\alpha^{\prime}}\,,\quad(\mathbf{b}_{\beta},\mathbf{b}_{\beta^{\prime}})=-\delta_{\beta,\beta^{\prime}}\,,\quad(\mathbf{c}_{\gamma},\mathbf{c}_{\gamma^{\prime}})=-\delta_{\gamma,\gamma^{\prime}}\,,\\\
&(\mathbf{a}_{\alpha},\mathbf{b}_{\beta})=-\tfrac{1}{2}\,,\quad(\mathbf{a}_{\alpha},\mathbf{c}_{\gamma})=\tfrac{1}{2}\,,\quad(\mathbf{b}_{\beta},\mathbf{c}_{\gamma})=1\,.\end{split}$
(2.18)

An important property of the pairing (2.16) is that it is invariant under
7-brane motions. That is, given a fixed set of 7-branes
$\mathbf{X}_{[p_{i},q_{i}]}$, the lattice $(J_{\text{phys}},(\cdot,\cdot))$
changes only up to a unimodular transformation (i.e., change of basis) when we
move the 7-branes. To see this it suffices to consider a two-branes
configuration $\mathbf{X}_{[p_{1},q_{1}]}\mathbf{X}_{[p_{2},q_{2}]}$ with
$J_{\text{phys}}=\\{a^{1}\mathbf{x}_{[p_{1},q_{1}]}+a^{2}\mathbf{x}_{[p_{2},q_{2}]}\\}$,
for which the pairing matrix is $\left(\begin{smallmatrix}-1&D/2\\\
D/2&-1\end{smallmatrix}\right)$, with
$D=\det\left(\begin{smallmatrix}p_{1}&p_{2}\\\
q_{1}&q_{2}\end{smallmatrix}\right)$. After moving
$\mathbf{X}_{[p_{1},q_{1}]}$ across the branch cut to the right, as in (2.4)
(the other direction, (2.5), works analogously), the configuration
$\mathbf{X}_{[p_{2},q_{2}]}\mathbf{X}_{[p_{1}+Dp_{2},q_{1}+Dq_{2}]}\equiv\mathbf{X}_{l}\mathbf{X}_{r}$
has the lattice

$\displaystyle
J_{\text{phys}}=\left\\{a_{l}\mathbf{x}_{l}+a_{r}\mathbf{x}_{r}\right\\}\,,\
\text{with}\
(\mathbf{x}_{i},\mathbf{x}_{j})=\left(\begin{smallmatrix}-1&-\tfrac{D}{2}\\\
-\tfrac{D}{2}&-1\end{smallmatrix}\right)=\left(\left(\begin{smallmatrix}-D&1\\\
1&0\end{smallmatrix}\right)^{-1}\right)^{T}\left(\begin{smallmatrix}-1&\tfrac{D}{2}\\\
\tfrac{D}{2}&-1\end{smallmatrix}\right)\left(\begin{smallmatrix}-D&1\\\
1&0\end{smallmatrix}\right)^{-1}.$ (2.19)

The unimodular transformation $\left(\begin{smallmatrix}-D&1\\\
1&0\end{smallmatrix}\right)$ precisely traces how the original unit prongs
$\\{\mathbf{x}_{[p_{i},q_{i}]}\\}$ are expressed in terms of the new basis
$\\{\mathbf{x}_{l},\mathbf{x}_{r}\\}$ after the 7-brane transition (2.4),

$\displaystyle\mathbf{x}_{[p_{1},q_{1}]}\rightarrow-D\mathbf{x}_{l}+\mathbf{x}_{r}\,,\quad\mathbf{x}_{[p_{2},q_{2}]}\rightarrow\mathbf{x}_{l}\,.$
(2.20)

#### Loop junctions and their self-pairings

A junction type that will be particularly important to our discussions are
_loop junctions_. These are formed by encircling a collection of 7-branes with
an $\left(\begin{smallmatrix}r\\\ s\end{smallmatrix}\right)$-charge, that
undergoes $SL(2,\mathbb{Z})$ transformations as it crosses their branch cuts.
As a convention for nomenclature, we use the $\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)$-charge it starts out with to label the loop
junction $\bm{\ell}_{(r,s)}$, even if its $(p,q)$-type changes after it comes
back, see Figure 2. If the overall monodromy of the encircled stack is $M$,
then such a loop has asymptotic charge $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)=(M-\mathbb{1})\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)$. For two loops, $\bm{\ell}_{(r,s)}$ and
$\bm{\ell}_{(u,v)}$, encircling the same 7-branes, one clearly has
$\bm{\ell}_{(r,s)}+\bm{\ell}_{(u,v)}=\bm{\ell}_{(r+u,s+v)}$. In principle, any
such loop can be turned into the standard basis 2.15 with prongs on 7-branes
by pulling the loop across the encircled 7-branes via a Hanany–Witten
transition, which allows to compute pairings involving loop junctions.
However, since the loop does not touch the encircled 7-branes, but only sees
their overall monodromy, the self-pairing of a loop should be computable just
with this data.

Figure 2: A loop junction $\bm{\ell}_{(r,s)}$ around a collection of 7-branes
with overall monodromy $M$. The asymptotic charge
$\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)=\left(\begin{smallmatrix}r^{\prime}\\\
s^{\prime}\end{smallmatrix}\right)-\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)=(M-\mathbb{1})\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)$ is in general non-zero.

To do so, first consider the junction ${\bf
j}=\mathbf{x}_{[p,q]}+\mathbf{x}_{[r,s]}$ as depicted on the left of Figure 3.
According to (2.16), we have

$\displaystyle({\bf j},{\bf j})=({\bf x}_{[p,q]},{\bf x}_{[p,q]})+({\bf
x}_{[r,s]},{\bf x}_{[r,s]})+2({\bf x}_{[p,q]},{\bf
x}_{[r,s]})=-2+\det\begin{pmatrix}p&r\\\ q&s\end{pmatrix}\,.$ (2.21)

As pointed out in [38], this result can also be interpreted as the sum of the
contributions from the two end points of the 7-branes (each contribution
$-1$), and the contribution of the 3-pronged vertex. The latter must therefore
be

$\displaystyle\det\begin{pmatrix}p&r\\\ q&s\end{pmatrix}=ps-
rq=\det\begin{pmatrix}r&-(p+r)\\\
s&-(q+s)\end{pmatrix}=\det\begin{pmatrix}-(p+r)&p\\\ -(q+s)&q\end{pmatrix}\,,$
(2.22)

i.e., the determinant of two of the three $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge vectors, arranged in their counter-clockwise
ordering (and all prongs either ingoing or outgoing).

Figure 3: The self-pairing of a 3-pronged junction (left) can be separated
into contributions from the ends on 7-branes and the vertex, see (2.21). When
there are no prongs ending on 7-branes, such as for loop juncions (right), the
only contribution is that of the vertex.

This logic can now be easily applied to compute self-pairings of loop
junctions. Since such a junction has no endpoints on 7-branes, the only
contribution to the self-pairing must come from the 3-pronged vertex. For the
junction $\bm{\ell}_{(r,s)}$ in Figure 2, this contribution evaluates to
(after accounting for the signs necessary to have all prongs in- or outgoing)

$\displaystyle(\bm{\ell}_{(r,s)},\bm{\ell}_{(r,s)})=\det\begin{pmatrix}p&r\\\
q&s\end{pmatrix}=-\det\begin{pmatrix}r&r^{\prime}\\\
s&s^{\prime}\end{pmatrix}\,.$ (2.23)

As a consistency check, consider a loop junction $\bm{\ell}_{(r,s)}$ around a
single $[p,q]$-7-brane such that the asymptotic charge is
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$ (see right of Figure
3), i.e.,

$\displaystyle(M_{[p,q]}-\mathbb{1})\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)=(qr-ps)\left(\begin{smallmatrix}p\\\
s\end{smallmatrix}\right)\stackrel{{\scriptstyle!}}{{=}}\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)\quad\Leftrightarrow\quad(qr-
ps)\stackrel{{\scriptstyle!}}{{=}}1\,,$ (2.24)

which always has a solution for $(r,s)$ since the labels of a single
$\mathbf{X}_{[p,q]}$ must be coprime. Then, the self-pairing is
$(\bm{\ell}_{(r,s)},\bm{\ell}_{(r,s)})=\det\left(\begin{smallmatrix}p&r\\\
q&s\end{smallmatrix}\right)=ps-qr=-1=(\mathbf{x}_{[p,q]},\mathbf{x}_{[p,q]})$.
This was expected, since by construction, this loop is equivalent, by a
Hanany–Witten transition, to the unit junction $\mathbf{x}_{[p,q]}$.

#### (Co-)weight lattices from junctions

For a single brane stack of ADE type (2.13), the physical junctions _without_
asymptotic charges are generated by

$\begin{split}\mathfrak{su}_{n}:&\quad\bm{\alpha}_{i}=\mathbf{a}_{i}-\mathbf{a}_{i+1}\,,\enspace
i\in\\{1,\dots,n-1\\}\,,\\\
\mathfrak{so}_{2n}:&\quad\bm{\alpha}_{i}=\mathbf{a}_{i}-\mathbf{a}_{i+1}\,,\enspace
i\in\\{1,\dots,n-1\\}\,,\enspace\bm{\alpha}_{n}=\mathbf{a}_{n-1}+\mathbf{a}_{n}-\mathbf{b}-\mathbf{c}\,,\\\
\mathfrak{e}_{n}:&\quad\bm{\alpha}_{i}=\mathbf{a}_{i}-\mathbf{a}_{i+1}\,,\enspace
i\in\\{1,\dots,n-2\\}\,,\enspace\bm{\alpha}_{n-1}=\mathbf{a}_{n-2}+\mathbf{a}_{n-1}-\mathbf{b}-\mathbf{c}_{1}\,,\enspace\bm{\alpha}_{n}=\mathbf{c}_{1}-\mathbf{c}_{2}\,,\end{split}$
(2.25)

where we have indexed 7-branes and their associated unit junctions of the same
$[p,q]$-type. Computing their mutual bi-linear pairing of $\bm{\alpha}_{i}$
one finds

$\displaystyle(\bm{\alpha}_{i},\bm{\alpha}_{j})=A_{ij}\,,$ (2.26)

with $A_{ij}$ the negative Cartan matrix. Indeed, strings represented by the
junctions above are associated to the W-bosons which lead to the enhanced
gauge symmetry on the 7-brane stack. We will call them _root junctions_ for
obvious reasons, and they span the root junction lattice of the ADE algebra
$\Lambda_{\text{r}}\subset J_{\text{phys}}$.

In complete analogy to representation theory (save for a minus sign for the
pairing), the bi-linear pairing allows the definition of the _coroot
junctions_ , whose span is the coroot junction lattice $\Lambda_{\text{cr}}$,
as follows

$\displaystyle\bm{\alpha}^{\vee}_{i}=\frac{2}{-(\bm{\alpha}_{i},\bm{\alpha}_{i})}\bm{\alpha}_{i}\,.$
(2.27)

Since for ADE algebras all roots have length-square $2$, these coincide with
the root junctions. However, physically, these should be thought of as the
magnetically dual states, and hence arise from 5-brane webs represented by the
junctions. We can therefore also identify the pairing between two junctions,
where one represents a string and the other a 5-brane, as the Dirac-pairing
between electric and magnetic operators of the 8d gauge theory.

One further defines the _weight junctions_ ${\bf w}_{i}$, which are dual to
the coroot junctions with respect to $(.,.)$ (or, more precisely, its
$\mathbb{Q}$-linear extension),

$\displaystyle({\bf w}_{i},\bm{\alpha}^{\vee}_{j})=-\delta_{ij}\,.$ (2.28)

They span the weight junction lattice $\Lambda_{\text{w}}$, and correspond to
the electric states of the gauge symmetry if they represent a string.
Similarly, one defines the _coweight junctions_ ${\bf w}_{i}^{\vee}$ and their
lattice $\Lambda_{\text{cw}}$ via

$\displaystyle({\bf w}_{i}^{\vee},\bm{\alpha}_{j})=-\delta_{ij}\,,$ (2.29)

which, when representing a 5-brane, is a magnetic state.

Note that the (co-)weights and (co-)roots are in a very real sense localized
degrees of freedom. For any additional 7-brane $\mathbf{X}_{[r,s]}$ that is
added to the system, we can explicitly compute from (2.25) that
$(\mathbf{x}_{[r,s]},\bm{\alpha}^{(\vee)}_{j})=0$. Therefore, any junction
that has no prong on the 7-brane stack represents an uncharged state under the
gauge symmetry on that stack.

For ADE algebras the coweights and weights again agree, and there the
distinction between string and 5-brane junctions is only of formal nature.
However, it will become important once we include O7+-planes. Before that, we
have to introduce the concept of so-called extended (co-)weight junctions
[38].

###  2.3 Extended (co-)weights and higher-form center symmetries

In general, the (co-)weight junctions,

$\displaystyle{\bf
w}^{\vee}_{i}=\sum_{j}(-A^{-1})_{ij}\bm{\alpha}_{j}\,,\quad{\bf
w}_{i}=\sum_{j}(-\tilde{A}^{-1})_{ij}\bm{\alpha}^{\vee}_{j}\,,$ (2.30)

with $\widetilde{A}_{ij}=(\bm{\alpha}_{i}^{\vee},\bm{\alpha}_{j}^{\vee})$,
will have fractional coefficients in front of the unit prongs
$\mathbf{x}_{[p_{i},q_{i}]}$. This implies that they are not physical
junctions on their own. However, they can be made physical by adding certain
other fractional junctions with _non-zero_ asymptotic
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges, resulting
in an integer (i.e., physical) junction with a prong that extends away from
the 7-brane stack. Equivalently, it formalizes the intuition that non-adjoint
matter states (carrying weights that are not roots) on a 7-brane stack arise
from open strings that have ends on other 7-branes (possibly at infinity).

As a simple example, consider $\mathfrak{g}=\mathfrak{su}_{2}$, realized on an
$\mathbf{A}_{1}\mathbf{A}_{2}$-stack. While the (co-)weight junction ${\bf
w}^{(\vee)}=\tfrac{1}{2}(\mathbf{a}_{1}-\mathbf{a}_{2})$ without any
asymptotic $(p,q)$-charge is non-physical, we can consider the unit string
junctions $\mathbf{a}_{1}$ or $\mathbf{a}_{2}$, each of which carries an
asymptotic $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$ charge. From
$(\mathbf{a}_{1},\bm{\alpha}^{\vee})=(\mathbf{a}_{1},\mathbf{a}_{1}-\mathbf{a}_{2})=-(\mathbf{a}_{2},\mathbf{a}_{1}-\mathbf{a}_{2})=-1$,
we expect these (string) junctions to be fundamental matter of the
$\mathfrak{su}_{2}$. Note that we can formally write

$\displaystyle\mathbf{a}_{1}=\tfrac{1}{2}(\mathbf{a}_{1}+\mathbf{a}_{2})+{\bf
w}\,,\quad\mathbf{a}_{2}=\tfrac{1}{2}(\mathbf{a}_{1}+\mathbf{a}_{2})-{\bf
w}\,.$ (2.31)

Because $\tfrac{1}{2}(\mathbf{a}_{1}+\mathbf{a}_{2})\equiv\bm{\omega}$ has
asymptotic charge $\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$,
and satisfies $(\bm{\omega},\bm{\alpha})=0$, we can interpret the above
rewriting as separating the $\mathfrak{su}_{2}$ gauge charges of the unit
junctions, captured by the summand proportional to $\bf w$, from the
asymptotic $SL(2,\mathbb{Z})$-charges, captured by $\bm{\omega}$. By
linearity, this separation can be done for any physical junction ${\bf
j}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}$. For $\mathfrak{su}_{2}$, the
state corresponding to ${\bf j}=s{\bf w}+k\bm{\omega}\in J_{\text{phys}}$ is a
weight of an spin-$s/2$ representation, which has charge $s\mod 2$ under the
$\mathbb{Z}_{2}$-center. It is easy to see in this case, the physicality
condition, i.e., for ${\bf j}$ to have integer number of prongs on the
7-branes, relates $s\equiv k\mod 2$. Therefore, the coefficient of any
physical junction in front of $\bm{\omega}$ provides an equivalent way to
encode the center charge of that corresponding state. This line of argument
can be generalized to any ADE-stack [38].

Figure 4: Construction of extended weight junctions. Since the
$\left(\begin{smallmatrix}r\\\ s\end{smallmatrix}\right)$-charges that appear
in the loop are in general fractional, the prongs ending on the 7-branes after
pulling the loop across also have fractional coefficients.

First, notice that, by a Hanany–Witten transition,
$\bm{\omega}=\bm{\ell}_{(0,-\nicefrac{{1}}{{2}})}$ is a loop junction around
the $\mathbf{A}_{1}\mathbf{A}_{2}$ stack. For a general stack with monodromy
$M$, one defines $\bm{\omega}$, which are called extended weight junctions, as
the generators of all loop junctions $\bm{\ell}_{(r,s)}$ (with possibly
fractional $(r,s)$) encircling the stack that have integer asymptotic
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge, i.e.,

$\displaystyle(M-\mathbb{1})\left(\begin{smallmatrix}r\\\
s\end{smallmatrix}\right)\in\mathbb{Z}^{2}\,.$ (2.32)

For the ADE algebras realized via the stacks as given in (2.13), a standard
basis for these are denoted $\bm{\omega}_{p,q}$ with asymptotic charges

$\displaystyle\bm{\omega}_{p}:\quad\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)_{\text{asymp}}=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)\,,\quad\bm{\omega}_{q}:\quad\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)_{\text{asymp}}=\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)\,.$ (2.33)

For $\mathfrak{su}_{n}$ stacks, $(M-\mathbb{1})$ has only rank 1, so there is
only one generator, $\bm{\omega}_{p}$ with asymptotic
$\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$ charge. Due to the
generally fractional $(r,s)$-prong that crosses the branch cuts, the prongs
that end on the constituent branes of the stack are also fractional after a
Hanany–Witten transition, see Figure 4. Explicitly, the extended weight
junctions and their pairings are given by888Note that we can infer
$\big{(}\bm{\omega}_{p},\bm{\omega}_{q}\big{)}$ from the self-pairing of
$\bm{\omega}_{p}+\bm{\omega}_{q}=\bm{\ell}_{(r_{p},s_{p})}+\bm{\ell}_{(r_{q},s_{q})}=\bm{\ell}_{(r_{p}+r_{q},s_{p}+s_{q})}$,
which can be computed from the contribution of the single 3-pronged vertex, as
in Figure 3.

$\displaystyle\begin{split}\mathfrak{su}_{n}\quad(\mathbf{A}^{n}):&\quad\bm{\omega}_{p}=\bm{\ell}_{(0,-\nicefrac{{1}}{{n}})}=\tfrac{1}{n}\sum_{i=1}^{n}\mathbf{a}_{i}\,,\quad(\bm{\omega}_{p},\bm{\omega}_{p})=-\tfrac{1}{n}\,,\\\
\mathfrak{so}_{2n}\quad(\mathbf{A}^{n}\mathbf{B}\mathbf{C}):&\quad\begin{cases}\bm{\omega}_{p}=\bm{\ell}_{(-\nicefrac{{1}}{{2}},0)}=\tfrac{1}{2}(\mathbf{b}+\mathbf{c})\,,\\\
\bm{\omega}_{q}=\bm{\ell}_{(1-\nicefrac{{n}}{{4}},-\nicefrac{{1}}{{2}})}=\tfrac{1}{2}(\sum_{i}\mathbf{a}_{i}-\mathbf{b}+\mathbf{c}-n\bm{\omega}_{p})\,,\end{cases}\
(\bm{\omega}_{\alpha},\bm{\omega}_{\beta})=\begin{pmatrix}0&0\\\
0&\tfrac{n}{4}-1\end{pmatrix}_{\alpha\beta},\\\
\mathfrak{e}_{6}\quad(\mathbf{A}^{5}\mathbf{B}\mathbf{C}^{2}):&\quad\begin{cases}\bm{\omega}_{p}=\bm{\ell}_{(0,\nicefrac{{1}}{{3}})}=-\tfrac{1}{3}\sum_{i=1}^{5}\mathbf{a}_{i}+\tfrac{4}{3}\mathbf{b}+\tfrac{2}{3}\sum_{i=1}^{2}\mathbf{c}_{i}\,,\\\
\bm{\omega}_{q}=\bm{\ell}_{(-1,-1)}=\sum_{i=1}^{5}\mathbf{a}_{i}-3\mathbf{b}-\sum_{i=1}^{2}\mathbf{c}_{i}\,,\end{cases}\
(\bm{\omega}_{\alpha},\bm{\omega}_{\beta})=\begin{pmatrix}\tfrac{1}{3}&-\tfrac{1}{2}\\\
-\tfrac{1}{2}&1\end{pmatrix}_{\alpha\beta},\\\
\mathfrak{e}_{7}\quad(\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2}):&\quad\begin{cases}\bm{\omega}_{p}=\bm{\ell}_{(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}})}=-\tfrac{1}{2}\sum_{i=1}^{6}\mathbf{a}_{i}+2\mathbf{b}+\sum_{i=1}^{2}\mathbf{c}_{i}\,,\\\
\bm{\omega}_{q}=\bm{\ell}_{(-\nicefrac{{5}}{{2}},-\nicefrac{{3}}{{2}})}=\tfrac{3}{2}\sum_{i=1}^{6}\mathbf{a}_{i}-5\mathbf{b}-2\sum_{i=1}^{2}\mathbf{c}_{i}\,,\end{cases}\
(\bm{\omega}_{\alpha},\bm{\omega}_{\beta})=\begin{pmatrix}\tfrac{1}{2}&-1\\\
-1&\tfrac{5}{2}\end{pmatrix}_{\alpha\beta},\\\
\mathfrak{e}_{8}\quad(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2}):&\quad\begin{cases}\bm{\omega}_{p}=\bm{\ell}_{(2,1)}=-\sum_{i=1}^{7}\mathbf{a}_{i}+4\mathbf{b}+2\sum_{i=1}^{2}\mathbf{c}_{i}\,,\\\
\bm{\omega}_{q}=\bm{\ell}_{(-7,-3)}=3\sum_{i=1}^{7}\mathbf{a}_{i}-11\mathbf{b}-5\sum_{i=1}^{2}\mathbf{c}_{i}\,,\end{cases}\
(\bm{\omega}_{\alpha},\bm{\omega}_{\beta})=\begin{pmatrix}1&-\tfrac{5}{2}\\\
-\tfrac{5}{2}&7\end{pmatrix}_{\alpha\beta}.\end{split}$ (2.34)

All physical junctions associated to a 7-brane stack, i.e., junctions with
prongs of only integer $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge, can be written uniquely in terms of a
linear combination of weight and extended weight junctions

$\displaystyle\mathbf{j}=\sum_{i}a^{i}{\bf
w}_{i}+a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}\,,\quad
a^{i},a^{p},a^{q}\in\mathbb{Z}\,.$ (2.35)

Physically, this means that a physical $(p,q)$-string/-5-brane is fully
characterized by its asymptotic electric/magnetic
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge under
$(B_{2},C_{2})$, and the weight/coweight charges under the 7-brane gauge
algebras.

In turn, it can be verified that every possible weight junction ${\bf
w}=\sum_{i}a^{i}{\bf w}_{i}$ ($a^{i}\in\mathbb{Z}$) of the gauge algebra
$\mathfrak{g}$ can be completed into a physical junction by the addition of an
integer linear combination ${\bf
j}_{e}=a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}$ of extended weights [38].
Such integer linear combination is not unique and is determined only up to
multiples $n^{p}\bm{\omega}_{p}+n^{q}\bm{\omega}_{q}$ which have integer
charges for each prong. This non-uniqueness can be understood as the fact that
${\bf j}_{e}$ is determined by the charge of ${\bf w}$ under the center
$Z(\widetilde{G})$ of the simply-connected group $\widetilde{G}$ associated to
$\mathfrak{g}$. Intuitively, this is expected because the charge under the
center of a specific state ${\bf w}$ is encoded in its prefactors of the
weight basis ${\bf w}_{i}$, which in turn introduces fractional prongs that
can only be cancelled by the extended weights. Analogously to how weights can
be screened by the W-bosons for observers “at infinity”, there are multiples
of specific asymptotic $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charges that can be added and subtracted without
affecting the local gauge dynamics on the 7-branes.999Describing the gauge
dynamics by F-theory on a non-compact K3, this is reflected by the homology of
the asymptotic boundary exhibiting discrete torsion, associated to the fact
that $n^{p}\times\text{(A-cycle)}+n^{q}\times\text{(B-cycle)}$ on the generic
torus fiber shrinks at the singularity [13].

The precise connection between higher-form symmetries and extended weights
have been described in [13]. Formally, we can define the lattice,

$\displaystyle
J_{\text{ext}}=\\{\mathbf{j}_{e}=a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}\,|\,a^{p},a^{q}\in\mathbb{Z}\\}\,,$
(2.36)

whose elements are arbitrary integer linear combinations of extended weights
that may be fractional. Then, the screening arguments for the center
symmetries, together with the junction characterization of gauge degrees of
freedom, translates into:

$\displaystyle
Z(\widetilde{G}_{\text{ADE}})=\frac{\text{weights}}{\text{roots}}=\frac{\text{coweights}}{\text{coroots}}=\frac{J_{\text{ext}}}{J_{\text{phys}}\cap
J_{\text{ext}}}\,,$ (2.37)

where $(J_{\text{phys}}\cap J_{\text{ext}})$ denotes extended weight junctions
that are themselves physical, i.e., do not contain fractional prongs. Note
that, since $\bm{\omega}_{\circ}=\bm{\ell}_{(r_{\circ},s_{\circ})}$ are loop
junctions of the form depicted on the left of Figure 4, $(J_{\text{phys}}\cap
J_{\text{ext}})$ are precisely the loops
$\ell_{(r,s)}=n^{p}\ell_{(r_{p},s_{p})}+n^{q}\ell_{(r_{q},s_{q})}$ with
integer $(r,s)$. Concretely, in terms of the extended weights summarized in
(2.34), one finds

$\displaystyle\begin{array}[]{c|c}\mathfrak{g}&J_{\text{ext}}/(J_{\text{phys}}\cap
J_{\text{ext}})\\\
\hline\cr\mathfrak{su}_{n}&\displaystyle\frac{\\{a^{p}\bm{\omega}_{p}\\}}{(n\bm{\omega}_{p})}\cong\big{\\{}a^{p}\
\text{mod }n\big{\\}}\cong\mathbb{Z}_{n}\\\
\mathfrak{so}_{4n}&\displaystyle\frac{\\{a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}\\}}{(2\bm{\omega}_{p},2\bm{\omega}_{q})}\cong\\{(a^{p}\
\text{mod }2,a^{q}\ \text{mod }2)\\}\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\mathfrak{so}_{4n+2}&\displaystyle\frac{\\{a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}\\}}{(2\bm{\omega}_{p},4\bm{\omega}_{q},\bm{\omega}_{p}+2\bm{\omega}_{q})}\cong\big{\\{}2a^{p}+a^{q}\mod
4\big{\\}}\cong\mathbb{Z}_{4}\\\
\mathfrak{e}_{6}&\displaystyle\frac{\\{a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}\\}}{(3\bm{\omega}_{p},\bm{\omega}_{q})}\cong\big{\\{}a^{p}\
\text{mod }3\\}\cong\mathbb{Z}_{3}\\\
\mathfrak{e}_{7}&\displaystyle\frac{\\{a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}\\}}{(2\bm{\omega}_{p},2\bm{\omega}_{q},\bm{\omega}_{p}+\bm{\omega}_{q})}\cong\big{\\{}a^{p}+a^{q}\mod
2\big{\\}}\cong\mathbb{Z}_{2}\end{array}$ (2.44)

In the language of higher-form symmetries (see also [13]), a physical
string/5-brane junction ${\bf j}=\sum_{i}a^{i}{\bf
w}_{i}+a^{p}\bm{\omega}_{p}+a^{q}\bm{\omega}_{q}$ carries an electric/magnetic
$Z(\widetilde{G})$ 1-form/5-form symmetry charge prescribed by (2.44).

##  3 Junctions on O7+ and center symmetries of $\mathfrak{sp}$ dynamcis

So far, we have reviewed the junction framework for ordinary $[p,q]$-7-branes,
which succinctly encode the 8d ${\cal N}=1$ gauge dynamics with simply-laced
gauge algebras. However, field theoretically, one can also have
$\mathfrak{sp}_{n}$ algebras. In the type IIB string constructions these are
linked to the presence of O7+-planes, which was not considered systematically
within the junction framework previously. Therefore, we need to generalize the
above analysis.

First, we note that the O7+-plane, unlike the the O7--plane, does not split,
at finite string coupling, into constituents represented by ordinary
$[p,q]$-7-branes. Therefore, we will represent it by a single, albeit special,
7-brane. The monodromy generated by one O7+-plane is in the same
$SL(2,\mathbb{Z})$ conjugacy class as a 7-brane stack with
$\mathfrak{g}=\mathfrak{so}_{16}$. In the following local analysis, we use the
same presentation as in (2.13),

$\displaystyle M_{\text{O7}^{+}}=\begin{pmatrix}-1&4\\\ 0&-1\end{pmatrix}\,.$
(3.1)

There are multiple ways of arguing for this physically. The prevalent
interpretation of an O7+ in recent literature [30, 31] is as the remnant of
“freezing” the $\mathfrak{so}_{16}$ gauge dynamics on an ordinary 7-brane
stack, see also [32, 33].

However, even after two decades, the freezing operation remains somewhat
mysterious. In particular, a geometric derivation of its effect on higher-form
symmetries in the M-theory frame [15, 16] appears to be challenging. However,
as we will argue now, one can obtain a complete picture, at least in the IIB
duality frame, of the mechanism using junctions. The key distinction to
$[p,q]$-branes is that the physicality condition for prongs that end on an O7+
_differ_ between strings and 5-branes.

From the perturbative IIB picture, only pairs of fundamental strings can end
on an O7+ [62], which one can see in a perturbative picture via Chan–Paton
factors. Via various dualities it can also be argued that only an even number
of D-strings can end on the O7+-plane, see [62, 63]. Thus, the physical string
junctions emanating from the O7+-plane have $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge restricted by $p,q\in 2\mathbb{Z}$. In
contrast, $(p,q)$-5-branes can end with any integer number on O7+. This is
relevant, e.g., in the construction of 5d SCFTs via 5-brane webs [64]. Hence,
a prong of a physical 5-brane junction can end with arbitrary integer
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge on an O7+. As
we will see momentarily, these conditions naturally give rise to a consistent
description of $\mathfrak{sp}_{n}$ gauge algebras, including their center
symmetries, from the junction lattice. Moreover, once we have set up the
notation in Section 4, we can derive these conditions independently in global
models with one O7+-plane that realize 8d rank $(2,10)$, by appealing to the
dual description of these models via the CHL-string (see Appendix A).

It is worthwhile to compare the boundary conditions for strings and 5-branes
on O7+ with the unfrozen $\mathfrak{so}_{16}$-stack. First, since both
generate the same monodromy, the loop junctions that generate all integer
asymptotic $\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges
in the presence of a single O7+ are the same as for an $\mathfrak{so}_{16}$
stack (2.34),

$\displaystyle\bm{\omega}_{p}^{\text{O7}^{+}}=\bm{\ell}_{(-\nicefrac{{1}}{{2}},0)}\,,\quad\bm{\omega}_{q}^{\text{O7}^{+}}=\bm{\ell}_{(-1,-\nicefrac{{1}}{{2}})}\,.$
(3.2)

We will call these the extended weight junctions of the O7+, even though a
single O7+ (unlike its unfrozen cousin) has no gauge dynamics, and hence no
root or weight lattice to begin with. By collapsing the loop or, equivalently,
performing a Hanany–Witten transition across the orientifold plane, these
extended weight junctions have a $\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$ and a $\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)$ prong, respectively, on the O7+. The set of
junctions ending/emanating from one O7+, which is entirely characterized by
its total $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)=\left(\begin{smallmatrix}a^{p}\\\
a^{q}\end{smallmatrix}\right)$-charge, can therefore be written as

$\displaystyle{\bf
j}=a^{p}\,\bm{\omega}_{p}^{\text{O7}^{+}}+a^{q}\,\bm{\omega}_{q}^{\text{O7}^{+}}\,,\quad
a^{p},a^{q}\in\mathbb{Z}\,.$ (3.3)

Physical string junctions must then have $(a^{p},a^{q})\equiv(0\ \text{mod
2},0\ \text{mod 2})$, which agrees with the physicality condition for
junctions on an $\mathfrak{so}_{16}$-stack that has no (unscreenable) gauge
charge, see (2.44). Equivalently, any string loop $\bm{\ell}_{(r,s)}$ that
encircles the O7+ is physical if and only if $r$ and $s$ are both integer. In
contrast, a physical 5-brane junction with odd $(a^{p},a^{q})$, which can end
on an O7+, would not be admissible on an $\mathfrak{so}_{16}$-stack without
picking up (unscreenable) gauge charge. In particular, this means that
_physical_ 5-brane loops $\bm{\ell}_{(r,s)}$ around the O7+ could have _half-
integer_ valued $r$ and $s$.

To fully incorporate O7+’s in the junction framework, we also need to define
the bi-linear pairing. It is hard to come up with a rule for prongs ending on
the O7+ by appealing to any geometric counterpart in a dual M-/F-theory
picture, because of the presence of frozen singularities there. However, since
the extended weights can be viewed as loops which is only sensitive to the
induced $SL(2,\mathbb{Z})$ monodromy $M$, but not the “microscopics” of a
7-brane stack, one would naturally expect that the pairing of such junctions
is insensitive to whether $M$ is sourced by an O7+, or an $\mathfrak{so}_{16}$
stack. Following the discussion around Figure 3, we can therefore directly
compute (rather than define, which would require further justifications) from
the loop junction representation:

$\displaystyle\big{(}\bm{\omega}_{p}^{\text{O7}^{+}},\bm{\omega}_{p}^{\text{O7}^{+}}\big{)}=0\,,\quad\big{(}\bm{\omega}_{q}^{\text{O7}^{+}},\bm{\omega}_{q}^{\text{O7}^{+}}\big{)}=1\,,\quad\big{(}\bm{\omega}_{p}^{\text{O7}^{+}},\bm{\omega}_{q}^{\text{O7}^{+}}\big{)}=0\,.$
(3.4)

Considering O7+-planes together with general $[r,s]$-7-branes (which we assume
to be on the left of the O7+), one further finds

$\displaystyle(\bm{\omega}_{p}^{\text{O7}^{+}},\mathbf{x}_{[r,s]})=-\tfrac{s}{2}\,,\quad(\bm{\omega}_{q}^{\text{O7}^{+}},\mathbf{x}_{[r,s]})=\tfrac{r}{2}\,.$
(3.5)

####  $\mathfrak{sp}$ gauge algebras and their higher-form symmetries from
junctions

From the perturbative IIB picture, it is well-known that we can generate 8d
$\mathfrak{sp}_{n}$ gauge dynamics on a 7-brane stack formed by $n$
$\mathbf{A}$-branes on top of one O7+. This 7-brane stack, of the form
$\mathbf{A}^{n}{\bf O7}^{+}$, has the same monodromy as an
$\mathfrak{so}_{16+2n}$ stack:

$\displaystyle M_{\mathfrak{sp}_{n}}=M_{\mathbf{A}^{n}{\bf
O7}^{+}}=\begin{pmatrix}-1&4+n\\\ 0&-1\end{pmatrix}\,.$ (3.6)

This allows us to straightforwardly define the extended weight junctions, as
loops $\ell_{(r,s)}$ around the entire stack (including the O7+) such that an
asymptotic $\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$\- (for
$\bm{\omega}_{p}$) or $\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)$-charge (for $\bm{\omega}_{q}$) remains. Then, after
performing the suitable Hanany–Witten transitions, we find

$\displaystyle\begin{split}&\bm{\omega}_{p}^{\mathfrak{sp}_{n}}=\bm{\ell}_{(-\nicefrac{{1}}{{2}},0)}=\bm{\omega}_{p}^{\text{O7}^{+}}\,,\quad\bm{\omega}_{q}^{\mathfrak{sp}_{n}}=\bm{\ell}_{(-1-\nicefrac{{n}}{{4}},-\nicefrac{{1}}{{2}})}=\tfrac{1}{2}\sum_{i=1}^{n}\mathbf{a}_{i}-\tfrac{n}{2}\bm{\omega}_{p}^{\text{O7}^{+}}+\bm{\omega}_{q}^{\text{O7}^{+}}\,.\end{split}$
(3.7)

Since $M_{\mathfrak{sp}_{n}}=M_{\mathfrak{so}_{16+2n}}$, the loop junctions
look identical to those of $\mathfrak{so}_{16+2n}$. Hence, with
$J_{\text{ext}}=\\{a^{p}\bm{\omega}_{p}^{\mathfrak{sp}_{n}}+a^{q}\bm{\omega}_{q}^{\mathfrak{sp}_{n}}\
|\ a^{p},a^{q}\in\mathbb{Z}\\}$, the physicality condition on linear
combinations of extended junctions is captured by

$\displaystyle J_{\text{ext}}/\big{(}J_{\text{ext}}\cap J_{\text{phys,
strings}})=\begin{cases}\big{\\{}(a^{p}\ \text{mod 2},a^{q}\ \text{mod
2})\\}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}\,,&n\ \text{even}\,,\\\
\big{\\{}2a^{p}+a^{q}\ \text{mod 4}\\}\cong\mathbb{Z}_{4}\,,&n\
\text{odd}\,.\end{cases}$ (3.8)

On the other hand, for 5-brane junctions, we have

$\displaystyle J_{\text{ext}}/\big{(}J_{\text{ext}}\cap J_{\text{phys,
5-branes}})=\big{\\{}a^{q}\ \text{mod 2}\\}\cong\mathbb{Z}_{2}\,.$ (3.9)

Comparing to the discussion around (2.44), one might be tempted to identify
$Z(Sp(n))_{\text{electric}}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ or
$\mathbb{Z}_{4}$, and $Z(Sp(n))_{\text{magnetic}}=\mathbb{Z}_{2}$, which is
clearly not correct.

To rectify this, we must instead consider in detail the role of the extended
weights as completing weights and coweights into physical strings and
5-branes, respectively. In analogy to the ADE-stacks, we first construct the
$\mathfrak{sp}_{n}$ roots $\bm{\alpha}_{i}$ as string junctions with no
asymptotic $\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge,
that stretch between the constituents of this stack. A basis for such
junctions are

$\displaystyle\bm{\alpha}_{i}=\mathbf{a}_{i}-\mathbf{a}_{i+1}\,,\enspace
i\in\\{1,\dots,n-1\\}\,,\enspace\bm{\alpha}_{n}=2\mathbf{a}_{n}-2\bm{\omega}_{p}^{\text{O7}^{+}}\,,$
(3.10)

see also Figure 5. With this and the bi-linear pairings in (3.4), one
straightforwardly verifies

$\displaystyle(\bm{\alpha}_{n},\bm{\alpha}_{n})=-4\,,\quad(\bm{\alpha}_{i},\bm{\alpha}_{n})=2\,\delta_{i,n-1}\,,\quad(\bm{\alpha_{i}},\bm{\alpha}_{j})=\begin{cases}-2\,,&i=j\,,\\\
1\,,&|i-j|=1\,,\\\ 0\,,&\text{else}\,,\end{cases}\quad 1\leq i,j\leq n-1\,,$
(3.11)

which precisely reproduces the negative of the Cartan matrix of an
$\mathfrak{sp}_{n}$ algebra. In particular, we see that, while the short roots
(those with length squared 2) arise from single-pronged strings between the
$\mathbf{A}$-branes, just as for ADE-algebras, the long root $\bm{\alpha}_{n}$
is only a generator due to the evenness condition for strings ending on the
O7+.

Figure 5: Root junctions of $\mathfrak{sp}$ algebra (the double arrow denotes
the factor of 2 required by evenness on O7+).

As a non-simply-laced algebra, $\mathfrak{sp}$ has different coroots than
roots:

$\displaystyle\bm{\alpha}_{i}^{\vee}=\frac{2\bm{\alpha}_{i}}{-(\bm{\alpha}_{i},\bm{\alpha}_{i})}=\begin{cases}\bm{\alpha}_{i}\,,&1\leq
i\leq n-1\,,\\\ \frac{1}{2}\bm{\alpha}_{n}={\bf
a}_{n}-\bm{\omega}^{O7{{}^{+}}}_{p}\,,&i=n\,.\end{cases}$ (3.12)

Since these are the magnetic objects in a gauge theory context, they arise
from 5-brane junctions, which, consistent with the boundary conditions, can
have a single prong on the O7+ that is needed to form the short coroot
$\bm{\alpha}^{\vee}_{n}$ with length squared 1.

The distinction between strings and 5-branes are of course also important for
the weights and coweights. The weight junctions are obtained as dual to the
coroot junctions. Defining the matrix
$\widetilde{A}_{ij}=\big{(}\bm{\alpha}^{\vee}_{i},\bm{\alpha}_{j}^{\vee}\big{)}$,
these can be written as

$\displaystyle\mathbf{w}_{i}=\big{(}-\widetilde{A}^{-1}\big{)}_{ij}\,\bm{\alpha}^{\vee}_{j}=\sum_{j}\text{min}\\{i,j\\}\,\bm{\alpha}_{j}^{\vee}=\sum_{j=1}^{n-1}\text{min}\\{i,j\\}\,\bm{\alpha}_{j}+\tfrac{1}{2}\text{min}\\{i,n\\}\,\bm{\alpha}_{n}\,.$
(3.13)

Similarly, one obtains the coweight junctions as duals of the root junctions.
With the negative Cartan matrix $A=(\alpha_{i},\alpha_{j})$ one has

$\displaystyle\mathbf{w}_{i}^{\vee}=\big{(}-A^{-1}\big{)}_{ij}\,\bm{\alpha}_{j}=\begin{cases}\sum_{j=1}^{n-1}\text{min}\\{i,j\\}\,\bm{\alpha}^{\vee}_{j}+i\,\bm{\alpha}^{\vee}_{n}\,,&i<n\,,\\\
\sum_{j=1}^{n}\frac{j}{2}\,\bm{\alpha}_{j}^{\vee}\,,&i=n\,,\end{cases}$ (3.14)

expressed as fractional linear combinations of coroots.

In terms of the junctions, the electric center symmetry,

$\displaystyle
Z\big{(}Sp(n)\big{)}=\text{weights}/\text{roots}\cong\frac{\\{\tfrac{k}{2}\bm{\alpha}_{n}\\}}{(\bm{\alpha}_{n})}\cong\mathbb{Z}_{2},$
(3.15)

is precisely generated by multiples of
$\tfrac{1}{2}\bm{\alpha}_{n}=\mathbf{a}_{n}-\bm{\omega}_{p}^{\text{O7}^{+}}$,
which is unphysical as a string junction because of the odd prong on the O7+.
To obtain a physical junction with the same $\mathfrak{sp}_{n}$ gauge charge,
we must therefore add linear combinations of the extended weights (3.7) with
integer prongs on $\mathbf{A}$, and odd numbers of prong of
$\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)=\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$-charge on the O7+. For even $n$, this requirement
is met by
$a^{p}\bm{\omega}_{p}^{\mathfrak{sp}_{n}}+a^{q}\bm{\omega}_{q}^{\mathfrak{sp}_{n}}$
with $a^{p}\equiv 1\ \text{mod }2$ and $a^{q}\equiv 0\ \text{mod }2$, whereas
for odd $n$, we need $2a^{p}+a^{q}\equiv 2\ \text{mod }4$. Each of these
generate a $\mathbb{Z}_{2}$-subgroup of the putative center (3.8), which is
the correct presentation of $Z\big{(}Sp(n)\big{)}_{\text{electric}}\cong
Z\big{(}Sp(n)\big{)}$. The mismatch from (3.8) is because not all integer
linear combinations of extended weight junctions can be completed into a
physical string junction with an $\mathfrak{sp}_{n}$ weight. For the magnetic
center symmetry,

$\displaystyle\text{Hom}(Z\big{(}Sp(n)\big{)},\mathbb{Z})\cong
Z\big{(}Sp(n)\big{)}=\text{coweights}/\text{coroots}\cong\frac{\\{k{\bf
w}_{n}^{\vee}\\}}{\text{coroots}}\,,$ (3.16)

the unphysical 5-brane coweight junctions come from the half-integer valued
prongs in

$\displaystyle{\bf
w}^{\vee}_{n}=\sum_{i=1}^{n}\tfrac{i}{2}\bm{\alpha}_{i}^{\vee}=\tfrac{1}{2}\sum_{i=1}^{n}\mathbf{a}_{i}-\tfrac{n}{2}\bm{\omega}_{p}^{\text{O7}^{+}}=\bm{\omega}_{q}^{\mathfrak{sp}_{n}}-\bm{\omega}_{q}^{\text{O7}^{+}}\,,$
(3.17)

which can be made physical by adding
$a^{p}\bm{\omega}_{p}^{\mathfrak{sp}_{n}}+a^{q}\bm{\omega}_{q}^{\mathfrak{sp}_{n}}$
with $a^{q}\equiv 1\ \text{mod 2}$.

In summary, for a physical string junction that ends on an $\mathfrak{sp}_{n}$
stack, which can be uniquely decomposed as

$\displaystyle{\bf j}=\sum_{i}a^{i}{\bf
w}_{i}+a^{p}\bm{\omega}_{p}^{\mathfrak{sp}_{n}}+a^{q}\bm{\omega}_{q}^{\mathfrak{sp}_{n}}$
(3.18)

in terms of the extended weights (3.7) and weight junctions (3.13), the
$\mathbb{Z}_{2}$ charge of the corresponding state under the electric center
symmetry is

$\displaystyle\begin{cases}a^{p}\ \text{mod 2}\,,&\text{$n$ even,}\\\
a^{p}+\tfrac{1}{2}a^{q}\ \text{mod 2}\,,&\text{$n$ odd (then $a^{q}$ must be
even).}\end{cases}$ (3.19)

For a physical 5-brane junction, decomposed into extended weights and
coweights (3.14),

$\displaystyle{\bf j}=\sum_{i}a^{i}{\bf
w}^{\vee}_{i}+a^{p}\bm{\omega}_{p}^{\mathfrak{sp}_{n}}+a^{q}\bm{\omega}_{q}^{\mathfrak{sp}_{n}}\,,$
(3.20)

its magnetic $\mathbb{Z}_{2}$-center charge is

$\displaystyle a^{q}\ \text{mod 2}\,.$ (3.21)

This completes the list of local building blocks of simple gauge algebras that
can be combined into a global model describing 8d supergravity. As we will
discuss now, the consistent combination of the individual brane stacks then
determines the global structure of the gauge dynamics in these models.

##  4 8d string vacua and their global structure from junctions

In this section, we combine the above local descriptions of simple gauge
algebras into a compact setting, to classify all 8d $\mathcal{N}=1$ string
vacua using junctions. These vacua fall into three moduli branches, which have
gauge rank $(2,18)$, $(2,10)$, and $(2,2)$, respectively. As shown in [35],
the gauge symmetries of the effective supergravity descriptions can be
classified by the $SL(2,\mathbb{Z})$ monodromies assoiated with each simple
gauge factor, with a few additional consistency conditions. In theories of
rank $(2,18)$, which enjoy a description as F-theory on an elliptically-
fibered K3 surface, these restrictions are met by 24 $[p,q]$-7-branes with
trivial total monodromy [65, 66, 67]. Via the freezing procedure [30, 32, 33],
one can further obtain all rank $(2,10)$ or $(2,2)$ vacua, if one replaces any
rank $(2,18)$ 7-brane configuration with one or two $\mathfrak{so}_{16+2n}$
stacks with the corresponding frozen $\mathfrak{sp}_{n}$ algebra that contains
an O7+ [35]. Based on this, junctions provide a unified description of the
gauge dynamics, in particular, the gauge group topology, for all these vacua.

Before we dive into the details, let us give a schematic description of this
approach. For a given 7-brane configuration (with or without O7+) in a global
model, the set of characters or cocharacters (i.e., a sublattice of the weight
or coweight lattice that is occupied by dynamical states) correspond to
physical (string or 5-brane) junctions that have zero asymptotic
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge. Since the
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge of a prong
ending on a stack is entirely captured by extended weight junctions, which in
turn encodes the center charge of the (co-)characters represented by the
junction, enumerating all linear combinations of extended weights from
different stacks that add up to zero $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)_{\text{asymp}}$ also enumerates the center charge of
all dynamical gauge charges. In particular, computing the center charges of
all string junctions that have no $\mathfrak{u}(1)$-charges determines $Z(G)$,
and those of 5-brane junctions determine $\pi_{1}(G)\cong{\cal Z}$, where
$G=\widetilde{G}/{\cal Z}$ is the physically realized non-Abelian gauge group,
with simply-connected cover $\widetilde{G}$.

For the rank $(2,18)$ branch, the information about $\mathcal{Z}$ has been
shown to be conveniently encoded in so-called fractional null junctions [49],
which are certain fractional multiples of physical loop junctions
$\bm{\ell}^{N}_{(r,s)}$ around all 7-branes (with trivial total monodromy). As
we will see, this correspondence continues in realizations of rank $(2,10)$
with one O7+-plane, and rank $(2,2)$ theories with two O7+’s. While for rank
$(2,10)$, we can crosscheck the results with those obtained from a dual CHL
string description [52, 34], the junction description provides a prediction
for the gauge group topology of rank $(2,2)$ vacua which are inaccessible via
the heterotic/CHL string. To illustrate the procedure we will explicitly work
out an example for each of the three branches of the 8d moduli space.

###  4.1 Gauge group topology from global null junctions

The construction of supersymmetric 8d theories with a dynamical gravity sector
and rank $(2,18-8k)$ gauge sector requires the identification of a set of
$(24-10k)$ $[p,q]$-7-branes and $k$ O7+-planes with vanishing overall
monodromy. These configurations can then be placed on a $\mathbb{P}^{1}$ which
compactifies the underlying type IIB theory from ten to eight
dimensions101010The number of branes in the setup can also be understood as
the demand that their cumulative gravitational backreaction in terms of the
induced deficit angle adds up to $4\pi$ as appropriate for the 2-sphere
$S^{2}\sim\mathbb{P}^{1}$..

Following the conventions laid out in the previous section, we arrange the
7-branes along a horizontal axis in the perpendicular plane (which is now a
compact $\mathbb{P}^{1}$), and enumerate (from the left to right) the ordinary
$[p,q]$-7-branes and the O7+’s separately. Within the vector space of all
possible junctions (which carries a pairing given by simply linearly-extending
the rules (2.16) and (3.4)),

$\displaystyle
J=\left\\{\sum_{i=1}^{24-10k}a^{i}\mathbf{x}_{[p_{i},q_{i}]}+\sum_{j=1}^{k}\left(b^{j}\bm{\omega}_{p}^{\text{O7}^{+}_{j}}+\tilde{b}^{j}\bm{\omega}_{q}^{\text{O7}^{+}_{j}}\right)\
\big{|}\ a^{i},b^{j},\tilde{b}^{j}\in\mathbb{Q}\right\\}\,,$ (4.1)

string junctions giving rise to the electrically charged states must have
integral number of prongs on 7-branes, with “integrality” on the O7+ being
defined as having even number of prongs. Analogously, magnetically charged
states are described by physical 5-branes with integral number of prongs on
all 7-branes, including the O7+’s. However, since the 7-branes move on a
compact $\mathbb{P}^{1}$, physical junctions must have vanishing asymptotic
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge, i.e., have
no open ends. This means that the _physical_ string / 5-brane junction
lattice, corresponding to dynamical electric / magnetic states of the 8d
supergravity theory, is

$\displaystyle\begin{aligned}
J_{\text{phys}}^{\text{el}}=&\left\\{\begin{array}[]{c}\mathbf{j}=\sum_{i=1}^{24-10k}a^{i}\,\mathbf{x}_{[p_{i},q_{i}]}+\sum_{j=1}^{k}\Big{(}2b^{j}\,\bm{\omega}^{\text{O7}^{+}_{j}}_{p}+2\tilde{b}^{j}\,\bm{\omega}_{q}^{\text{O7}^{+}_{j}}\Big{)}\,,\\\
\text{with}\
a^{i},b^{j},\tilde{b}^{j}\in\mathbb{Z}\,,\quad\sum_{i}a^{i}p_{i}+\sum_{j}2b^{j}=0\,,\quad\sum_{i}a^{i}q_{i}+\sum_{j}2\tilde{b}^{j}=0\end{array}\right\\}\,,\\\
J_{\text{phys}}^{\text{mag}}=&\left\\{\begin{array}[]{c}\mathbf{j}=\sum_{i=1}^{24-10k}a^{i}\,\mathbf{x}_{[p_{i},q_{i}]}+\sum_{j=1}^{k}\Big{(}b^{j}\,\bm{\omega}^{\text{O7}^{+}_{j}}_{p}+\tilde{b}^{j}\,\bm{\omega}_{q}^{\text{O7}^{+}_{j}}\Big{)}\,,\\\
\text{with}\
a^{i},b^{j},\tilde{b}^{j}\in\mathbb{Z}\,,\quad\sum_{i}a^{i}p_{i}+\sum_{j}b^{j}=0\,,\quad\sum_{i}a^{i}q_{i}+\sum_{j}\tilde{b}^{j}=0\end{array}\right\\}\,.\end{aligned}$
(4.2)

Obviously, these lattices are of rank $24-10k-2=22-10k$.

Furthermore, by Hanany–Witten transitions, different elements in these
lattices can represent the same physical junction. Equivalently, we can add
arbitrary multiples of so-called _(global) integer null junctions_ ,

$\displaystyle
J^{N,\text{el/mag}}_{\text{int}}=\left\\{\bm{\delta}_{(r,s)}^{N}\in
J^{\text{el/mag}}_{\text{phys}}\ \big{|}\
\bm{\delta}_{(r,s)}^{N}=\bm{\ell}_{(r,s)}\ \text{loops around \emph{all}
7-branes}\right\\}\,,$ (4.3)

where it is understood that, a priori, there are different integral null
junctions for strings and 5-branes. Because of the compactness, such a loop
can be shrunk to a point “on the other side” of the $\mathbb{P}^{1}$ without
crossing any 7-branes, and thus are physically trivial. However, they would
appear as a non-trivial element in $J_{\text{phys}}$ after pulling them
through the 7-branes, which must therefore be modded out before we can
identify the junction lattice with the physical charge lattice. Note that, by
construction, $\bm{\delta}_{(r,s)}^{N}\in J^{N}_{\text{int}}$ has trivial
pairing with any other $\bm{\delta}^{N}\in J^{N}_{\text{int}}$, as well as no
asymptotic $\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charge
(since they encircle a configuration with trivial overall monodromy).
Moreover, $(\bm{\delta}_{(r,s)}^{N},{\bf j})=0$ for all
$\bm{\delta}_{(r,s)}^{N}\in J_{\text{int}}^{N}$ if and only if ${\bf j}$ has
zero asymptotic charge [38]. Hence, $\bm{\delta}_{(r,s)}^{N}\in
J_{\text{int}}^{N}$ has trivial pairing with all physical junctions,
explaining the prefix “null”. As a notational convention, we shall denote any
loop junctions $\bm{\ell}_{(r,s)}$ with no asymptotic charge by
$\bm{\delta}_{(r,s)}$.

This allows us now to identify the (co-)character lattice $\Lambda_{\text{c}}$
($\Lambda_{\text{cc}}$), the lattice of all electrically (magnetically)
charged states present in the supergravity theory, as

$\displaystyle\Lambda_{\text{c}}\cong
J^{\text{el}}_{\text{phys}}/J_{\text{int}}^{N,\text{el}}\,,\quad\Lambda_{\text{cc}}\cong
J^{\text{mag}}_{\text{phys}}/J_{\text{int}}^{N,\text{mag}}\,,$ (4.4)

which are rank $20-8k$ lattices. Since we mod out a sublattice which is null,
the junction pairing on (4.1) induces a non-degenerate pairing on these
lattices, whose signature can be shown to be $(2,18-8k)$. For $[{\bf
j}_{e}]\in\Lambda_{\text{c}}$ and $[{\bf j}_{m}]\in\Lambda_{\text{cc}}$ with
representatives ${\bf j}_{e},{\bf j}_{m}\in J^{\text{el/mag}}_{\text{phys}}$,
integrality of the Dirac pairing requires $({\bf j}_{e},{\bf
j}_{m})\in\mathbb{Z}$. Moreover, the Completeness Hypothesis for quantum
gravity implies that the two lattices are dual to each other,
$\Lambda_{\text{c}}=(\Lambda_{\text{cc}})^{*}$, i.e., for any $[{\bf j}_{e}]$
there is a $[{\bf j}_{m}]$ such that $({\bf j}_{e},{\bf j}_{m})=1$ and vice
versa. This can be explicitly checked, as we will discuss later.

Now suppose that the 7-branes give rise to the full 8d gauge algebra

$\displaystyle\mathfrak{g}=\bigoplus_{\sigma}\mathfrak{g}_{\sigma}\oplus\mathfrak{u}(1)^{\oplus
r_{A}}\,,\quad\text{with}\enspace
r_{A}=20-8k-\sum_{a}\text{rank}(\mathfrak{g}_{a})\,.$ (4.5)

Since for each 7-brane stack with gauge factor $\mathfrak{g}_{\sigma}$, we
have (co-)weight junctions ${\bf w}^{(\vee)}_{\sigma;i_{\sigma}}$,
$i_{\sigma}=1,...,\text{rank}(\mathfrak{g}_{\sigma})$, we can uniquely (up to
global null junctions) decompose111111Note that each stack $\sigma$ can appear
with a monodromy that is conjugated by $g_{\sigma}\in SL(2,\mathbb{Z})$
compared to the “standard frame” (2.13) or (3.6) chosen in the previous
section, so that the extended weights have $(p,q)$-charges
$g_{\sigma}\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$ for
$\bm{\omega}_{p}^{\sigma}$ and $g_{\sigma}\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)$ for $\bm{\omega}_{q}^{\sigma}$, respectively.

$\displaystyle{\bf
j}_{e\,(m)}=\sum_{\sigma}\left(\sum_{i_{\sigma}}a^{i_{\sigma}}_{\sigma}{\bf
w}^{(\vee)}_{\sigma;i_{\sigma}}+a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}b_{s}\mathbf{x}_{s}\in
J_{\text{phys}}^{\text{el (mag)}}\,,$ (4.6)

where $s$ labels the remaining 7-branes (including potential O7+’s) that are
not part of the non-Abelian stacks, on which the prongs must be integral,
$b_{s}\in\mathbb{Z}$ (or satisfy the corresponding integrality condition on
O7+’s). Since gauge charges under $\mathfrak{g}_{\sigma}$ are carried by the
(co-)weights ${\bf w}^{(\vee)}_{\sigma;i_{\sigma}}$, the states with _only_
Abelian charges live in the subspace orthogonal to the (co-)weights,

$\displaystyle J^{\text{el/mag}}_{A}:=\left\\{P_{A}({\bf j}_{e(m)})\ |\ {\bf
j}_{e(m)}\in J_{\text{phys}}^{\text{el
(mag)}}\right\\}_{\text{phys}}\equiv\left\\{\sum_{\sigma}\left(a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}b_{s}\mathbf{x}_{s}\right\\}\cap
J_{\text{phys}}^{\text{el/mag}}\,,$ (4.7)

where $P_{A}$ is the projection onto the orthogonal complement of the non-
Abelian (co-)weights. In particular, since global null junctions have zero
pairing with all physical junctions, we have
$J_{\text{int}}^{N,\text{el/mag}}\subset J^{\text{el/mag}}_{A}$.

Because the overall $\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$
charge of any physical junction must be zero, only specific linear
combinations of extended weights, and therefore, only (co-)weights of
$\mathfrak{g}_{\sigma}$ with specific center charges, can be completed into a
physical junction in (4.6) with the available singlet branes. If the resulting
string junctions give rise to representations that are all invariant under a
subgroup ${\cal Z}$ of the center, then the gauge group has some non-trivial
global structure.

More precisely, the most general global gauge group structure is

$\displaystyle
G=\frac{[\prod_{\sigma}\widetilde{G}_{\sigma}/\mathcal{Z}]\times
U(1)^{r_{A}}}{\mathcal{Z}^{\prime}}\,,$ (4.8)

where $\widetilde{G}_{\sigma}$ is the simply-connected realization of the
gauge algebra $\mathfrak{g}_{\sigma}$. The finite group $\mathcal{Z}$ embeds
into the overall center $\prod_{\sigma}Z(\widetilde{G}_{\sigma})$ of the non-
Abelian factors with trivial map to the Abelian groups, whereas
$\mathcal{Z}^{\prime}$ does have a non-trivial map into the Abelian sector. We
will now explain how to extract these discrete groups from junctions.

We first focus on the factor $\mathcal{Z}$, which demands that electric states
can appear only in certain irreducible representations under
$\prod_{\sigma}\widetilde{G}_{\sigma}$ that are invariant under ${\cal Z}$.
Equivalently, this can be understood as the existence of magnetic states
$[{\bf j}^{\text{nA}}_{m}]\in\Lambda_{\text{cc}}$ charged only under the non-
Abelian gauge factors, which via the Dirac pairing condition $({\bf
j}_{e},{\bf j}^{\text{nA}}_{m})\in\mathbb{Z}$ enforces the absence of electric
states that are not invariant under ${\cal
Z}\subset\prod_{\sigma}Z(\widetilde{G}_{\sigma})$. Decomposing such a
junction,

$\displaystyle{\bf
j}^{\text{nA}}_{m}=\sum_{\sigma}\left(\sum_{i_{\sigma}}a^{i_{\sigma}}_{\sigma}{\bf
w}^{\vee}_{\sigma;i_{\sigma}}+a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}b_{s}\mathbf{x}_{s}\in
J_{\text{phys}}^{\text{mag}}\,,$ (4.9)

the assumption that this junction is only charged under the non-Abelian
factors implies that the Abelian part,

$\displaystyle\sum_{\sigma}\left(a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}b_{s}\mathbf{x}_{s}\in
J_{A}^{\text{mag}}\otimes\mathbb{Q}\,,$ (4.10)

has also zero pairing with every junction in $J_{A}^{\text{mag}}$. However,
this is only possible if it is proportional to a linear combination of global
integer null junction with rational coefficients, i.e.,

$\displaystyle P_{A}({\bf
j}_{m}^{\text{nA}})=\sum_{\sigma}\left(a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}b_{s}\mathbf{x}_{s}=b_{m}\bm{\delta}_{(r_{m},s_{m})}^{N}\,,\quad\bm{\delta}_{(r_{m},s_{m})}^{N}\in
J_{\text{int}}^{N,\text{mag}}\,,\ b_{m}\in\mathbb{Q}\,.$ (4.11)

Considering such decompositions for all 5-brane junctions ${\bf
j}_{m}^{\text{nA}}\in J_{\text{phys}}^{\text{mag}}$ with no Abelian charge,
one obtains the lattice

$\displaystyle
J^{N,\text{mag}}_{\text{frac}}=\left\\{b_{m}\bm{\delta}_{(r_{m},s_{m})}^{N}\
\bigg{|}\ {\bf
j}_{m}^{\text{nA}}=\sum_{\sigma}\sum_{i_{\sigma}}a^{i_{\sigma}}_{\sigma}{\bf
w}^{\vee}_{\sigma;i_{\sigma}}+b_{m}\bm{\delta}_{(r_{m},s_{m})}^{N}\in
J_{\text{phys}}^{\text{mag}}\right\\}\supset J^{N,\text{mag}}_{\text{int}}\,,$
(4.12)

of what is called (global) fractional null junctions [48, 49]. Let us further
denote the smallest positive integer $n_{m}$ such that
$n_{m}b_{m}\bm{\delta}_{(r_{m},s_{m})}^{N}\in J_{\text{int}}^{N,\text{mag}}$.
Since the prongs on $\mathbf{X}_{s}$ are already integral, due to the
physicality of ${\bf j}_{m}^{\text{nA}}$, $n_{m}$ is also the smallest
positive integer such that
$n_{m}(a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma})\in
J^{\sigma}_{\text{phys, 5-branes}}$ is physical on every non-Abelian stack
$\sigma$. At the same time, according to the discussions around (2.44) and
(3.21), the coefficients $(a^{p}_{\sigma},a^{q}_{\sigma})$ specify an element

$\displaystyle
z_{m}=(z_{\sigma})\in\prod_{\sigma}\frac{J^{\sigma}_{\text{ext}}}{J_{\text{phys}}^{\sigma}\cap
J^{\sigma}_{\text{ext}}}\cong\prod_{\sigma}Z(\widetilde{G}_{\sigma})\,.$
(4.13)

Since the set of all such $z_{m}$ generate the discrete factor ${\cal Z}$ in
(4.8), we find

$\displaystyle{\cal
Z}\cong\frac{J^{N,\text{mag}}_{\text{frac}}}{J^{N,\text{mag}}_{\text{int}}}\,.$
(4.14)

The main advantage of this formula is that we can conveniently compute
$J_{\text{frac}}^{N,\text{mag}}$ from pulling the two generators
$\bm{\delta}^{N}_{a}$ of $J_{\text{int}}^{N,\text{mag}}$ across all 7-brane
stacks, which yields

$\displaystyle\bm{\delta}^{N}_{a}=\sum_{\sigma}\big{(}c_{a;\sigma}^{p}\bm{\omega}_{p}^{\sigma}+c_{a;\sigma}^{q}\bm{\omega}_{q}^{\sigma})+\sum_{s}c_{a;s}\mathbf{x}_{s}\in
J_{\text{phys}}^{\text{mag}}\,.$ (4.15)

Then $J_{\text{frac}}^{N,\text{mag}}$ is generated by $\mathbb{Q}$-linear
combinations,

$\displaystyle\lambda_{1}\bm{\delta}^{N}_{1}+\lambda_{2}\bm{\delta}^{N}_{2}=\sum_{\sigma}\left(\big{(}\lambda_{1}c_{1;\sigma}^{p}+\lambda_{2}c_{2;\sigma}^{p}\big{)}\bm{\omega}_{p}^{\sigma}+\big{(}\lambda_{1}c_{1;\sigma}^{q}+\lambda_{2}c_{2;\sigma}^{q}\big{)}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}\big{(}\lambda_{1}c_{1;s}+\lambda_{2}c_{2;s}\big{)}\mathbf{x}_{s}\,,$
(4.16)

such that
$\big{(}\lambda_{1}c_{1;\sigma}^{p}+\lambda_{2}c_{2;\sigma}^{p}\big{)}$,
$\big{(}\lambda_{1}c_{1;\sigma}^{q}+\lambda_{2}c_{2;\sigma}^{q}\big{)}$ are
integer, and $\big{(}\lambda_{1}c_{1;s}+\lambda_{2}c_{2;s}\big{)}$ satisfies
the physicality condition on $\mathbf{X}_{s}$. As advertised, this procedure
applies indiscriminately to configurations with or without O7+-planes, as long
as the integrality conditions on O7+’s and $\mathfrak{sp}_{n}$ stacks follow
the prescription in Section 3.

The second discrete factor ${\cal
Z}^{\prime}\subset\prod_{\sigma}Z(\widetilde{G}_{\sigma})\times U(1)^{r_{A}}$
in (4.8) correlates the representations under the non-Abelian factors
$\prod_{\sigma}\widetilde{G}_{\sigma}$ of electric states to their
$\mathfrak{u}(1)$ charges, such that their transformation under
$\prod_{\sigma}Z(\widetilde{G}_{\sigma})$ is compensated by a ${\cal
Z}^{\prime}$ subgroup in $U(1)^{r_{A}}$. Analogously as above, this subgroup
can be viewed as being enforced by the presence of magnetic states, now with
non-trivial $U(1)$-charges, and hence have a junction representation $[{\bf
j}_{m}]\in\Lambda_{\text{cc}}$ with

$\displaystyle J_{\text{phys}}^{\text{mag}}\ni{\bf
j}_{m}=\sum_{\sigma}\left(\sum_{i_{\sigma}}a^{i_{\sigma}}_{\sigma}{\bf
w}^{\vee}_{\sigma;i_{\sigma}}+a^{p}_{\sigma}\bm{\omega}_{p}^{\sigma}+a^{q}_{\sigma}\bm{\omega}_{q}^{\sigma}\right)+\sum_{s}b_{s}\mathbf{x}_{s}=\sum_{\sigma}\sum_{i_{\sigma}}a^{i_{\sigma}}_{\sigma}{\bf
w}^{\vee}_{\sigma;i_{\sigma}}+{\bf j}_{A,m}\,,$ (4.17)

where ${\bf j}_{A,m}=P_{A}({\bf j}_{m})\in J_{A}\otimes\mathbb{Q}$ is no
longer a null junction. Nevertheless, as the mismatch of ${\bf j}_{A,m}$ from
being a physical junction is still determined by the coefficients
$(a_{\sigma}^{p},a_{\sigma}^{q})$, we have

$\displaystyle\mathcal{Z}^{\prime}=\frac{P_{A}\left(J_{\text{phys}}^{\text{mag}}\right)}{J_{A}^{\text{mag}}}\,.$
(4.18)

Intuitively, this measures the “fractionality” of the
$\mathfrak{u}(1)$-charges of all magnetic objects (living in
$J_{\text{phys}}^{\text{mag}}$) with respect to the charges of those that are
uncharged under any non-Abelian symmetry (and hence live in
$J_{A}^{\text{mag}})$. Note that, since
$P_{A}(J_{\text{int}}^{N})=J_{\text{int}}^{N}$, the null junctions do not
affect these quotients.

###  4.2 Duality to heterotic and CHL descriptions

Theories of rank $(2,18)$ and $(2,10)$ have a dual construction in terms of
the heterotic and CHL string, respectively, which provides a crosscheck for
the junction description. In both cases, electrically charged states are
identified as elements from a momentum lattice for perturbative string
excitations. For the heterotic string the momentum lattice is the so-called
Narain lattice [68]:

$\displaystyle\Lambda_{\text{Narain}}=(-\text{E}_{8})\oplus(-\text{E}_{8})\oplus
U\oplus U\,,$ (4.19)

with $(-\text{E}_{8})$ the negative of the E8 root lattice and $U$ the
hyperpolic lattice defined by the bi-linear form

$\displaystyle U:\quad\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\,.$ (4.20)

The CHL string is determined by the Mikhailov lattice [69]

$\displaystyle\Lambda_{\text{Mikhailov}}=(-\text{D}_{8})\oplus U\oplus
U=(-\text{E}_{8})\oplus U\oplus U(2)\,,$ (4.21)

with the negative Spin$(16)$ root lattice $(-\text{D}_{8})$. Here, $U(x)$
denotes a lattice of rank two with bi-linear form (4.20) multiplied by $x$.

Following the notation of [13], we will denote them collectively as
$\Lambda_{S}$. Then, if the duality holds, we expect
$\Lambda_{S}=\Lambda_{\text{c}}$. Equivalently, points in the dual momentum
lattice121212Note that $\Lambda_{\text{Narain}}^{*}=\Lambda_{\text{Narain}}$
is self-dual while
$\Lambda_{\text{Mikhailov}}^{*}\neq\Lambda_{\text{Mikhailov}}$ is not.
correspond to physical magnetically charged states and therefore must be
associated to 5-brane junctions (modulo null junctions) in
$\Lambda_{\text{cc}}$. The non-Abelian gauge factors are then specified by an
embedding of the corresponding (negative) root lattice into $\Lambda_{S}$; the
coroot lattice then naturally embeds in the dual lattice $\Lambda^{*}_{S}$. It
is worth noting that the computation of the gauge group topologies from this
data [29, 52, 13] is in a sense complimentary to the junctions approach
outlined above. While in both scenarios, the setting is fully characterized by
the non-Abelian gauge algebras (by specifying either the 7-brane stacks or the
embedding of the (co-)root lattices), the gauge group is concisely encoded in
the projection of the full physical lattice onto the Abelian junctions (see
(4.14) and (4.18)), the methods in [29, 52, 13] extract the gauge group from
the projection onto the (co-)root lattice.

To corroborate the equivalence of the two approaches, we describe in the
following the precise identification of the momentum lattices with string
junctions on 7-brane configurations with zero or one O7+-plane.

#### Narain lattice from junctions

To construct the Narain lattice (4.19) from junctions, it is easiest to find a
7-brane configuration in which the two $(-\text{E}_{8})$ factors are manifest
via the root junctions on two $\mathfrak{e}_{8}$ 7-brane stacks, and make use
of the fact that the lattice structure does not change as we move 7-branes.
Each of these $\mathfrak{e}_{8}$ stacks contains ten 7-branes, leaving a
remaining four branes to specify the compact type IIB background. A convenient
configuration of this sort has been presented in Section 7 of [70], and takes
the form

$\displaystyle\mathbf{A}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}\mathbf{C})\mathbf{X}_{[3,1]}\mathbf{A}^{\prime}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}\mathbf{C})^{\prime}\mathbf{X}^{\prime}_{[3,1]}\,,$
(4.22)

up to possible $SL(2,\mathbb{Z})$ conjugations. In fact, the above
configuration has two identical parts, consisting of
$\mathbf{A}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}\mathbf{C})\mathbf{X}_{[3,1]}$,
each having a trivial SL$(2,\mathbb{Z})$ monodromy.131313Note however that the
Spin cover of the monodromy is non-trivial and given by
$(-1)^{F}\in\text{Mp}(2,\mathbb{Z})$, see e.g. [71, 72, 73]. In addition, by
pushing the twelve branes onto a single stack, one enhances the symmetry to
the so-called double loop algebra $\hat{\mathfrak{e}}_{9}$, whose significance
we will explain further in Section 5. In an F-theory description, one may
interpret this configuration as a stable degeneration limit of the
elliptically-fibered K3 into two $dP_{9}$ surfaces.

Note that for $\mathfrak{e}_{8}$, whose extended weights (2.34) are physical,
roots and weights junctions agree, so the span of all physical string prongs
on each $\mathfrak{e}_{8}$ stack (with decomposition as in (2.35)) contains
two copies of the $(-\text{E}_{8})$-lattice. Next we need to find the factor
$U\oplus U$ in the orthogonal complement of the $\mathfrak{e}_{8}$ root
lattices. A convenient set of generators for these two hyperbolic lattices can
be expressed as141414This is the same result as in [70] (see their Figure 8).

$\displaystyle\begin{split}U:&\quad\big{(}\bm{\delta}_{(1,0)}\,,\bm{\delta}_{(1,0)}+\mathbf{x}_{[3,1]}-\mathbf{x}^{\prime}_{[3,1]}\big{)}\,,\\\
U:&\quad\big{(}\bm{\delta}_{(3,1)}\,,\bm{\delta}_{(3,1)}-\mathbf{a}+\mathbf{a}^{\prime}\big{)}\,.\end{split}$
(4.23)

Here $\bm{\delta}_{(r,s)}=\bm{\ell}_{(r,s)}$ denotes a $(r,s)$-string loop
around one
$\mathbf{A}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}\mathbf{C})\mathbf{X}_{[3,1]}$
configuration, which has no asymptotic charge since this stack has no overall
monodromy. Its orthogonality to the $\mathfrak{e}_{8}$ root lattice is evident
from the fact that this junction has no prongs on any of the two stacks. Note
that, as is evident from Figure 6, such a loop automatically encircles the
other
$\mathbf{A}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}\mathbf{C})\mathbf{X}_{[3,1]}$
configuration. Using Hanany–Witten transition one can rewrite them in terms of
integer strings ending on the brane constituents in the interior (say, on the
left in Figure 6), e.g.,

$\displaystyle\begin{split}\bm{\delta}_{(1,0)}&=3\bm{\omega}^{\mathfrak{e}_{8}}_{p}+\bm{\omega}^{\mathfrak{e}_{8}}_{q}-\mathbf{x}_{[3,1]}\,,\\\
\bm{\delta}_{(3,1)}&=-\mathbf{a}+\bm{\omega}_{p}^{\mathfrak{e}_{8}}\,,\end{split}$
(4.24)

or equivalently for the primed stack.

Figure 6: String junction lattice for rank $(2,18)$ theories.

This accounts for a rank 20 sublattice $(-\text{E}_{8})^{\oplus 2}\oplus
U^{\oplus 2}$ of the full string junction lattice
$J_{\text{phys}}^{\text{el}}$. The remaining two directions are spanned by the
global null junctions $J_{\text{int}}^{N}$, for which the canonical basis is

$\displaystyle\begin{split}&\bm{\delta}_{(1,0)}^{N}=-(3\bm{\omega}_{p}^{\mathfrak{e}_{8}}+\bm{\omega}_{q}^{\mathfrak{e}_{8}})+\mathbf{x}_{[3,1]}-(3\bm{\omega}_{p}^{\mathfrak{e}^{\prime}_{8}}+\bm{\omega}_{q}^{\mathfrak{e}^{\prime}_{8}})+\mathbf{x}_{[3,1]}^{\prime}\,,\\\
&\bm{\delta}_{(0,1)}^{N}=-\mathbf{a}+10\bm{\omega}_{p}^{\mathfrak{e}_{8}}+3\bm{\omega}_{q}^{\mathfrak{e}_{8}}-3\mathbf{x}_{[3,1]}-\mathbf{a}^{\prime}+10\bm{\omega}_{p}^{\mathfrak{e}_{8}^{\prime}}+3\bm{\omega}_{q}^{\mathfrak{e}_{8}^{\prime}}-3\mathbf{x}_{[3,1]}^{\prime}\,.\end{split}$
(4.25)

Since these have trivial pairing with all physical junctions, the lattice
pairing of the above generators (including the $\mathfrak{e}_{8}$ roots and
those of the $U$-lattices) desecend, without modification, to the quotient

$\displaystyle\Lambda_{\text{c}}=\frac{J_{\text{phys}}^{\text{el}}}{J_{\text{int}}^{N,\text{el}}}=\frac{(-\text{E}_{8})^{\oplus
2}\oplus U^{\oplus 2}\oplus
J_{\text{int}}^{N,\text{el}}}{J_{\text{int}}^{N,\text{el}}}\cong\Lambda_{\text{Narain}}\,.$
(4.26)

Since there are no O7+-planes, the 5-brane junction lattices are the same as
their stringy counterparts, so we immediately find

$\displaystyle\Lambda_{\text{cc}}=J_{\text{phys}}^{\text{mag}}/J_{\text{int}}^{N,\text{mag}}\cong
J_{\text{phys}}^{\text{el}}/J_{\text{int}}^{N,\text{el}}\cong\Lambda_{\text{Narain}}=\Lambda_{\text{Narain}}^{*}\,.$
(4.27)

#### Mikhailov lattice from junctions

The Mikhailov lattice describing the momentum lattice for the 8d CHL string is
obtained as follows. We keep one of the $\hat{\mathfrak{e}}_{9}$
configurations unchanged, which still leads to an $(-\text{E}_{8})$ factor in
the string junction lattice. On the other side, we remove a $\mathbf{C}$ brane
from the $\mathfrak{e}_{8}$ stack, but add to it the singlet
$\mathbf{A}$-brane, which leads to an $\mathfrak{so}_{16}$ brane stack, which
we then “freeze” into an O7+-plane:

$\displaystyle\mathbf{A}{\bf
E}_{8}\mathbf{X}_{[3,1]}=\mathbf{A}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}\mathbf{C})\mathbf{X}_{[3,1]}\rightarrow(\mathbf{A}^{8}\mathbf{B}\mathbf{C})\mathbf{C}\mathbf{X}_{[3,1]}\rightarrow{\bf
O7}^{+}\mathbf{C}\mathbf{X}_{[3,1]}\,.$ (4.28)

The resulting complete 7-brane configuration,

$\displaystyle{\bf
O7}^{+}\mathbf{C}\mathbf{X}_{[3,1]}\mathbf{A}^{\prime}(\mathbf{A}^{7}\mathbf{B}\mathbf{C}^{2})\mathbf{X}^{\prime}_{[3,1]}\,,$
(4.29)

and is depicted in Figure 7.

Figure 7: String junction lattice for rank $(2,10)$ theories.

The total rank of the junction lattice is now 14.

Inside the string junction lattice $J_{\text{phys}}^{\text{el}}$ of this
configuration, we now need to identify $U\oplus U(2)$ orthogonal to the
$\mathfrak{e}_{8}$ roots. Given the similarities to the rank $(2,18)$
configuration, a natural choice for the generators would be a variation of
(4.23). While the first set exists also for the frozen configuration, the
second $U$-factor has a generator with a single $\mathbf{a}$-prong, which
would become part of the O7+, and not be physical. Therefore, the set of
generators orthogonal to the E8 root lattce are

$\displaystyle\begin{split}U:&\quad\big{(}\bm{\delta}_{(1,0)}\,,\bm{\delta}_{(1,0)}+\mathbf{x}_{[3,1]}-\mathbf{x}_{[3,1]}^{\prime}\big{)}\,,\\\
U(2):&\quad\big{(}\bm{\delta}_{(3,1)}\,,2\bm{\delta}_{(3,1)}+2\bm{\omega}^{\text{O7}^{+}}_{p}-2\mathbf{a}^{\prime}\big{)}\,,\end{split}$
(4.30)

where the second generator of $U(2)$ is primitive because of the evenness
condition for strings ending on O7+. These have an equivalent representation
with

$\displaystyle\begin{split}\bm{\delta}_{(1,0)}&=2\bm{\omega}_{p}^{\text{O7}^{+}}+\mathbf{c}-\mathbf{x}_{[3,1]}\,,\\\
\bm{\delta}_{(3,1)}&=-2\bm{\omega}^{\text{O7}^{+}}_{p}-2\bm{\omega}^{\text{O7}^{+}}_{q}+2\mathbf{c}\,.\end{split}$
(4.31)

After quotienting out the global null junctions
$J_{\text{phys}}^{N,\text{el}}$, with generators

$\displaystyle\begin{split}&\bm{\delta}_{(1,0)}^{N}=-2\bm{\omega}_{p}^{\text{O7}^{+}}-\mathbf{c}+\mathbf{x}_{[3,1]}-(3\bm{\omega}_{p}^{\mathfrak{e}^{\prime}_{8}}+\bm{\omega}_{q}^{\mathfrak{e}^{\prime}_{8}})+\mathbf{x}_{[3,1]}^{\prime}\,,\\\
&\bm{\delta}_{(0,1)}^{N}=4\bm{\omega}_{p}^{\text{O7}^{+}}-2\bm{\omega}_{q}^{\text{O7}^{+}}+5\mathbf{c}-3\mathbf{x}_{[3,1]}-\mathbf{a}^{\prime}+10\bm{\omega}_{p}^{\mathfrak{e}_{8}^{\prime}}+3\bm{\omega}_{q}^{\mathfrak{e}_{8}^{\prime}}-3\mathbf{x}_{[3,1]}^{\prime}\,,\end{split}$
(4.32)

one finds

$\displaystyle\Lambda_{\text{c}}\cong(-\text{E}_{8})\oplus U\oplus
U(2)=\Lambda_{\text{Mikhailov}}\,.$ (4.33)

In this case it is interesting to also analyze the 5-brane junctions that
correspond to the dual lattice. Here, it is important to remember that
5-branes have different physicality conditions, which allow for an arbitrary
integer number of them to end on the O7+-plane. This does not affect the
$\mathfrak{e}_{8}$ root junctions, and the $U$-factor in (4.30), but does
imply that $\bm{\delta}_{(3,1)}$ in (4.31) is no longer a primitive 5-brane
junction. Instead, it is a multiple of

$\displaystyle\tfrac{1}{2}\bm{\delta}_{(3,1)}=\bm{\delta}_{(\nicefrac{{3}}{{2}},\nicefrac{{1}}{{2}})}=-\bm{\omega}^{\text{O7}^{+}}_{p}-\bm{\omega}^{\text{O7}^{+}}_{q}+\mathbf{c}\,.$
(4.34)

This implies that the there is an overlattice of $U(2)$ in (4.30) inside the
physical 5-brane junction lattice, given by

$\displaystyle
U(\tfrac{1}{2}):\quad\big{(}\tfrac{1}{2}\bm{\delta}_{(3,1)}\,,\bm{\delta}_{(3,1)}+\bm{\omega}^{\text{O7}^{+}}_{p}-\mathbf{a}^{\prime}\big{)}\,.$
(4.35)

Note that the null junction lattice spanned by (4.32) remain primitive as a
sublattice of $J_{\text{phys}}^{\text{mag}}$. Therefore one has for the
magnetic 5-brane junction lattice

$\displaystyle\Lambda_{cc}=(-\text{E}_{8})\oplus U\oplus
U(\tfrac{1}{2})=\Lambda_{\text{Mikhailov}}^{*}\,,$ (4.36)

which precisely coincides with the dual of the Mikhailov lattice (4.21).

###  4.3 A rank $(2,2)$ momentum lattice

8d rank $(2,2)$ string vacua have no known constructions as $T^{2}$\- or
$S^{1}$-reductions of the heterotic or CHL string. However, using the
junctions, we propose an analogue of a momentum lattice description, which can
be applied, in particular, to determine the gauge group topologies of these
theories. To this end, we start with a 7-brane configuration with two O7+’s,
that we obtain from further freezing an $\mathfrak{so}_{16}$ stack on the
primed side of (4.22). The overall brane configuration is then given by

$\displaystyle{\bf O7}^{+}\mathbf{C}\mathbf{X}_{[3,1]}{{\bf
O7}^{+}}^{\prime}\mathbf{C}^{\prime}\mathbf{X}^{\prime}_{[3,1]}\,,$ (4.37)

and is depicted in Figure 8.

Figure 8: String junction lattice for rank $(2,2)$ theories.

The string junction lattice in this case has rank $6$, whose non-null
directions are isomorphic to

$\displaystyle U\oplus U(2)\,.$ (4.38)

The explicit generators in terms of physical string junctions are given by

$\displaystyle\begin{split}U:&\quad\big{(}\bm{\delta}_{(1,0)}\,,\bm{\delta}_{(1,0)}+\mathbf{x}_{[3,1]}-\mathbf{x}_{[3,1]}^{\prime}\big{)}\,,\\\
U(2):&\quad\big{(}\bm{\delta}_{(3,1)}\,,2\bm{\delta}_{(3,1)}+2\bm{\omega}^{\text{O7}^{+}}_{p}-2\bm{\omega}^{{\text{O7}^{+}}^{\prime}}_{p}\big{)}\,,\end{split}$
(4.39)

where the loop junctions satisfy the same relation as in (4.31). The string
null junctions $J_{\text{phys}}^{N,\text{el}}$ have generators

$\displaystyle\begin{split}&\bm{\delta}_{(1,0)}^{N}=-2\bm{\omega}_{p}^{\text{O7}^{+}}-\mathbf{c}+\mathbf{x}_{[3,1]}-2\bm{\omega}_{p}^{{\text{O7}^{+}}^{\prime}}-\mathbf{c}^{\prime}+\mathbf{x}_{[3,1]}^{\prime}\,,\\\
&\bm{\delta}_{(0,1)}^{N}=4\bm{\omega}_{p}^{\text{O7}^{+}}-2\bm{\omega}_{q}^{\text{O7}^{+}}+5\mathbf{c}-3\mathbf{x}_{[3,1]}+4\bm{\omega}_{p}^{{\text{O7}^{+}}^{\prime}}-2\bm{\omega}_{q}^{{\text{O7}^{+}}^{\prime}}+5\mathbf{c}^{\prime}-3\mathbf{x}_{[3,1]}^{\prime}\,.\end{split}$
(4.40)

The full physical string junction lattice is therefore
$J_{\text{phys}}^{\text{el}}=U\oplus U(2)\oplus
J_{\text{phys}}^{N,\text{el}}$.

As for the rank $(10,2)$ case above, we find that for 5-branes the $U(2)$
turns into a $U(\tfrac{1}{2})$, i.e., $J_{\text{phys}}^{\text{mag}}=U\oplus
U(\tfrac{1}{2})\oplus J_{\text{phys}}^{N,\text{mag}}$. A novel modification
that will affect the gauge group computation is that also the null junctions
are refined. Namely, the generators of $J_{\text{phys}}^{N,\text{mag}}$ are

$\displaystyle\begin{split}\tfrac{1}{2}(\bm{\delta}^{N}_{(1,0)}+\bm{\delta}^{N}_{(0,1)})=\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}})}&=\bm{\omega}_{p}^{\text{O7}^{+}}-\bm{\omega}_{q}^{\text{O7}^{+}}+2\mathbf{c}-\mathbf{x}_{[3,1]}+\bm{\omega}_{p}^{{\text{O7}^{+}}^{\prime}}-\bm{\omega}_{q}^{{\text{O7}^{+}}^{\prime}}+2\mathbf{c}^{\prime}-\mathbf{x}_{[3,1]}^{\prime}\,,\\\
\tfrac{1}{2}(\bm{\delta}^{N}_{(1,0)}-\bm{\delta}^{N}_{(0,1)})=\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},-\nicefrac{{1}}{{2}})}&=-3\bm{\omega}_{p}^{\text{O7}^{+}}+\bm{\omega}_{q}^{\text{O7}^{+}}-3\mathbf{c}-2\mathbf{x}_{[3,1]}-3\bm{\omega}_{p}^{{\text{O7}^{+}}^{\prime}}+\bm{\omega}_{q}^{{\text{O7}^{+}}^{\prime}}-3\mathbf{c}^{\prime}+2\mathbf{x}_{[3,1]}^{\prime}\,.\end{split}$
(4.41)

In summary, after modding out the null junctions, we have

$\displaystyle\Lambda_{\text{c}}=U\oplus
U(2)\,,\quad\Lambda_{\text{cc}}=U\oplus
U(\tfrac{1}{2})=\Lambda_{\text{c}}^{*}\,.$ (4.42)

###  4.4 Examples

Using the techniques outlined in Section 4.1, we can determine the brane
configurations and the resulting gauge group topologies for all 8d ${\cal
N}=1$ string vacua. This is done explicitly for all maximally-enhanced cases
on each branch of the moduli space, as summarized in Appendix C. In the
following we demonstrate the general procedure in specific examples. For
convenience, we focus 8d theories that were discussed in [52, 34] from the
perspective of the heterotic or CHL momentum lattice. The generalization to
rank $(2,2)$ theories is, to our knowledge, the first time in the literature
the global gauge group topology has been computed for these string vacua.

####  4.4.1 A rank $(2,18)$ example

The non-Abelian gauge algebra of the model is given by

$\displaystyle\mathfrak{g}=\mathfrak{so}_{20}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{2}\,,$
(4.43)

which can be generated by the following brane configuration:

$\displaystyle(\mathbf{A}^{10}\mathbf{B}\mathbf{C})\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{4}\mathbf{X}_{[2,5]}^{2}\mathbf{C}^{2}\,.$
(4.44)

Note that for a consistent overall monodromy, the $\mathfrak{su}$ algebras are
not associated to a stack of $\mathbf{A}$-branes, but rather in some
$SL(2,\mathbb{Z})$ conjugated frame. Accordingly, the associated extended
weight junctions summarized in (2.34) need to be conjugated, and are given by

$\displaystyle\begin{split}\mathfrak{so}_{20}\quad(\mathbf{A}^{10}\mathbf{B}\mathbf{C}):&\quad\bm{\omega}_{p}=\tfrac{1}{2}(\mathbf{b}+\mathbf{c})\,,\quad\bm{\omega}_{q}=\tfrac{1}{2}\sum_{i=1}^{10}\mathbf{a}_{i}-3\mathbf{b}-2\mathbf{c}\,,\\\
\mathfrak{su}_{4}\quad(\mathbf{N}^{4}):&\quad\bm{\omega}_{(0,1)}=\tfrac{1}{4}\sum_{i=1}^{4}\mathbf{n}_{i}\,,\\\
\mathfrak{su}_{4}\quad(\mathbf{X}_{[1,3]}^{4}):&\quad\bm{\omega}_{(1,3)}=\tfrac{1}{4}\sum_{i=1}^{4}\mathbf{x}_{[1,3],i}\,,\\\
\mathfrak{su}_{2}\quad(\mathbf{X}^{2}_{[2,5]}):&\quad\bm{\omega}_{(2,5)}=\tfrac{1}{2}\sum_{i=1}^{2}\mathbf{x}_{[2,5],i}\,,\\\
\mathfrak{su}_{2}\quad(\mathbf{C}^{2}):&\quad\bm{\omega}_{(1,1)}=\tfrac{1}{2}\sum_{i=1}^{2}\mathbf{c}_{i}\,,\end{split}$
(4.45)

where $\bm{\omega}_{(p,q)}$ is the extended weight of the corresponding
$\mathfrak{su}$-stack with asymptotic $\left(\begin{smallmatrix}p\\\
q\end{smallmatrix}\right)$-charge. In terms of these extended weights the two
linearly independent integer null junctions are given by

$\displaystyle\begin{split}\bm{\delta}^{N}_{(1,0)}&=-2\bm{\omega}_{p}-4\bm{\omega}_{(0,1)}+4\bm{\omega}_{(1,3)}-2\bm{\omega}_{(2,5)}+2\bm{\omega}_{(1,1)}\,,\\\
\bm{\delta}^{N}_{(0,1)}&=6\bm{\omega}_{p}-2\bm{\omega}_{q}+24\bm{\omega}_{(0,1)}-20\bm{\omega}_{(1,3)}+8\bm{\omega}_{(2,5)}-2\bm{\omega}_{(1,1)}\,.\end{split}$
(4.46)

Notice how, in both junctions, the greatest common divisor of the coefficients
is 2. Therefore, the fractional null junctions are generated by

$\displaystyle
J^{N,\text{mag}}_{\text{frac}}=\Big{\\{}\lambda^{N}_{(1,0)}\,\bm{\delta}^{N}_{(1,0)}+\lambda^{N}_{(0,1)}\,\bm{\delta}^{N}_{(0,1)}\
\Big{|}\
\lambda^{N}_{(1,0)},\lambda^{N}_{(0,1)}\in\tfrac{1}{2}\mathbb{Z}\Big{\\}}\,,$
(4.47)

and the global realization of the non-Abelian gauge group is determined by

$\displaystyle\mathcal{Z}=\frac{J^{N}_{\text{frac}}}{J^{N}_{\text{int}}}=\frac{\\{\tfrac{x}{2}\bm{\delta}_{(1,0)}^{N}\
|\
x\in\mathbb{Z}\\}}{(\bm{\delta}_{(1,0)}^{N})}\times\frac{\\{\tfrac{y}{2}\bm{\delta}_{(1,0)}^{N}\
|\
y\in\mathbb{Z}\\}}{(\bm{\delta}_{(0,1)}^{N})}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}\,.$
(4.48)

Moreover, the coefficients in front of the extended weights in
$\tfrac{1}{2}\bm{\delta}^{N}$ determine, according to (2.44), the generators
of ${\cal Z}\subset Z(Spin(20))\times Z(SU(4))^{2}\times Z(SU(2))^{2}$ to be

$\displaystyle\begin{split}&\tfrac{1}{2}\bm{\delta}_{(1,0)}^{N}\simeq(1,0;2;2;1;1)\in(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\,,\\\
&\tfrac{1}{2}\bm{\delta}_{(0,1)}^{N}\simeq(1,1;0;2;0;1)\in(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\,.\end{split}$
(4.49)

Beyond the non-Abelian gauge factors, the theory contains two gravi-photons
generating two $\mathfrak{u}(1)$ gauge factors. These arise from Abelian
junctions (4.7) that are not null junctions, which for the present model can
be easily determined, from (4.45), to be linear combinations of

$\displaystyle\begin{split}&{\bf
u}_{1}=4(\bm{\omega}_{(0,1)}+\bm{\omega}_{(1,3)}-\bm{\omega}_{(2,5)}+\bm{\omega}_{(1,1)})\equiv
4{\bf v}_{1}\,,\\\ &{\bf u}_{2}=4(\bm{\omega}_{(0,1)}-\bm{\omega}_{q})\equiv
4{\bf v}_{2}\,.\end{split}$ (4.50)

${\bf v}_{a}$, while themselves non-physical, can be made physical by adding
coweight junctions, so lie in $P_{A}(J_{\text{phys}}^{\text{mag}})$. At the
same time, there are no “finer” coweights to make fractions of ${\bf v}_{a}$
physical, so ${\bf v}_{a}$ generate $P_{A}(J_{\text{phys}}^{\text{mag}})$.
Then, the Abelian quotient ${\cal Z^{\prime}}$, according to (4.18), is

$\displaystyle\mathcal{Z}^{\prime}=\frac{P_{A}(J_{\text{phys}}^{\text{mag}})}{J_{A}^{\text{mag}}}=\frac{\\{x{\bf
v}_{1}\ |\ x\in\mathbb{Z}\\}}{({\bf u}_{1})}\times\frac{\\{y{\bf v}_{2}\ |\
y\in\mathbb{Z}\\}}{({\bf u}_{2})}=\mathbb{Z}_{4}\times\mathbb{Z}_{4}\,.$
(4.51)

The generators of ${\cal Z}^{\prime}$ are

$\displaystyle\begin{split}&{\bf v}_{1}\simeq(0,0;1;1;1;1\ |\
e^{i\pi/2};1)\in(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times
U(1)_{1}\times U(1)_{2}\,,\\\ &{\bf v}_{2}\simeq(0,1;1;0;0;0\ |\
1;e^{i\pi/2})\in(\mathbb{Z}_{2}\times\mathbb{Z}_{2})\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times
U(1)_{1}\times U(1)_{2}\,,\end{split}$ (4.52)

where we have used a vertical line to separate the finite groups from the
$U(1)$’s, whose trivial element is $1$.

Consequently, the global form of the gauge group is given by

$\frac{\left[\left(Spin(20)\times SU(2)^{2}\times
SU(4)^{2}\right)/\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\right)\right]\times
U(1)^{2}}{\mathbb{Z}_{4}\times\mathbb{Z}_{4}}\,,$ (4.53)

in perfect agreement with the heterotic analysis in [34].

####  4.4.2 A rank $(2,10)$ example

Since the brane configuration above already containes an $\mathfrak{so}_{16}$
stack we can simply reinterpret this as an O7+-plane, leading to the brane
configuration

$\displaystyle(\mathbf{A}^{2}{\bf
O7}^{+})\mathbf{N}^{4}\mathbf{X}_{[1,3]}^{4}\mathbf{X}_{[2,5]}^{2}\mathbf{C}^{2}\,.$
(4.54)

This setup has a non-Abelian gauge algebra given by

$\displaystyle\mathfrak{g}=\mathfrak{sp}_{2}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{4}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{2}\,,$
(4.55)

leading to a model with rank $(2,10)$ dual to a specific CHL background.
Except for the $\mathfrak{sp}$ factor, the extended weights are the same as in
(4.45). For the $\mathfrak{sp}$ algebra one has

$\displaystyle\mathfrak{sp}_{2}\quad(\mathbf{A}^{2}{\bf
O7}^{+}):\quad\bm{\omega}_{p}=\bm{\omega}_{p}^{\text{O7}^{+}}\,,\quad\bm{\omega}_{q}=\tfrac{1}{2}(\mathbf{a}_{1}+\mathbf{a}_{2})-\bm{\omega}_{p}^{\text{O7}^{+}}+\bm{\omega}_{q}^{\text{O7}^{+}}\,.$
(4.56)

Formally, the global null junctions $\bm{\delta}^{N}_{(p,q)}$ are the same as
in (4.46), except that the $\bm{\omega}_{p,q}$ appearing there are now the
extended weight junctions of $\mathfrak{sp}$. Again, we can divide both by 2,
obtaining the fractional 5-brane junctions

$\displaystyle
J^{N,\text{mag}}_{\text{frac}}=\Big{\\{}\lambda^{N}_{(1,0)}\,\bm{\delta}^{N}_{(1,0)}+\lambda^{N}_{(0,1)}\,\bm{\delta}^{N}_{(0,1)}\
\Big{|}\
\lambda^{N}_{(1,0)},\lambda^{N}_{(0,1)}\in\tfrac{1}{2}\mathbb{Z}\Big{\\}}\,,$
(4.57)

implying ${\cal Z}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, with generators

$\displaystyle\begin{split}&\tfrac{1}{2}\bm{\delta}_{(1,0)}^{N}\simeq(0;2;2;1;1)\in\mathbb{Z}_{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}=Z(Sp(2)\times
SU(4)^{2}\times SU(2)^{2})\,,\\\
&\tfrac{1}{2}\bm{\delta}_{(0,1)}^{N}\simeq(1;0;2;0;1)\in\mathbb{Z}_{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}=Z(Sp(2)\times
SU(4)^{2}\times SU(2)^{2})\,,\end{split}$ (4.58)

where the entry for $\mathbb{Z}_{2}=Z(Sp(2))$ is determined only by the
coefficient in front of $\bm{\omega}_{q}$, see (3.21).

In a similar way, there are two $\mathfrak{u}(1)$ generators that formally are
the same as in (4.50). Since $({\bf u}_{1},{\bf v}_{1})$ have no prongs on the
$\mathfrak{sp}$ stack, we get a $\mathbb{Z}_{4}$ factor in ${\cal
Z}^{\prime}$, with generator

$\displaystyle{\bf v}_{1}\simeq(0;1;1;1;1\ |\
e^{i\pi/2};1)\in\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}\times\mathbb{Z}_{2}^{2}\times
U(1)_{1}\times U(1)_{2}\,.$ (4.59)

Moreover, since ${\bf v}_{2}$ has an order-4 prong that is not on the O7+,
physicality conditions do not change the fact that we obtain another
$\mathbb{Z}_{4}$ factor in ${\cal Z}^{\prime}$, now with generator

$\displaystyle{\bf v}_{2}\simeq(1;1;0;0;0\ |\
1;e^{i\pi/2})\in\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}\times\mathbb{Z}_{2}^{2}\times
U(1)_{1}\times U(1)_{2}\,.$ (4.60)

To summarize, the global gauge group of this rank $(2,10)$ model is

$\frac{\left[\left(Sp(2)\times SU(2)^{2}\times
SU(4)^{2}\right)/\left(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\right)\right]\times
U(1)^{2}}{\mathbb{Z}_{4}\times\mathbb{Z}_{4}}\,,$ (4.61)

agreeing with the CHL result computed in [34].

####  4.4.3 A rank $(2,2)$ example

The rank $(2,2)$ moduli branch has six special points with non-Abelian
symmetry enhancements [35]. We have enumerated the 7-brane configurations for
all of these, as well as the resulting gauge group topologies in Appendix C.4.

It turns out that there is only one whose non-Abelian gauge group is non-
simply-connected. For illustration, we consider this example in more detail.
The brane configuration is

${\bf O7}^{+}{{\bf O7}^{+}}^{\prime}\mathbf{B}^{2}\mathbf{C}^{2}\,,$ (4.62)

where the monodromy of the ${{\bf O7}^{+}}^{\prime}$ is $SL(2,\mathbb{Z})$
conjugated to be $\left(\begin{smallmatrix}7&16\\\
-4&-9\end{smallmatrix}\right)$.151515This can be viewed as freezing both
$\mathfrak{so}_{16}$ algebras of the rank $(2,18)$ configuration
$(\mathbf{A}^{8}\mathbf{B}\mathbf{C})(\mathbf{X}_{[2,-1]}^{8}\mathbf{B}\mathbf{X}_{[3,-1]})\mathbf{B}^{2}\mathbf{C}^{2}$.
Notice that, since the extended weights of O7+ generate all
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges, whose
parity is invariant under $SL(2,\mathbb{Z})$ conjugation, we use the canonical
basis $\bm{\omega}_{p,q}^{{\text{O7}^{+}}^{\prime}}$ for the ${{\bf
O7}^{+}}^{\prime}$, which have $\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)$ prongs, respectively.

The non-Abelian gauge algebra of (4.62) is
$\mathfrak{su}_{2}\oplus\mathfrak{su}_{2}$, with extended weights

$\displaystyle\begin{split}\mathfrak{su}_{2}\quad(\mathbf{B}^{2}):&\quad\bm{\omega}_{(1,-1)}\equiv\bm{\omega}_{b}=\tfrac{1}{2}(\mathbf{b}_{1}+\mathbf{b}_{2})\,,\\\
\mathfrak{su}_{2}\quad(\mathbf{C}^{2}):&\quad\bm{\omega}_{(1,1)}\equiv\bm{\omega}_{c}=\tfrac{1}{2}(\mathbf{c}_{1}+\mathbf{c}_{2})\,.\end{split}$
(4.63)

The physical 5-brane null junctions $J_{\text{phys}}^{N,\text{mag}}$ (see
(4.41)) are then

$\displaystyle\begin{split}\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}})}&=\bm{\omega}_{p}^{\text{O7}^{+}}-\bm{\omega}_{q}^{\text{O7}^{+}}+\bm{\omega}_{p}^{{\text{O7}^{+}}^{\prime}}-\bm{\omega}_{q}^{{\text{O7}^{+}}^{\prime}}-2\bm{\omega}_{b}\,,\\\
\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},-\nicefrac{{1}}{{2}})}&=-3\bm{\omega}_{p}^{\text{O7}^{+}}+\bm{\omega}_{q}^{\text{O7}^{+}}-7\bm{\omega}_{p}^{{\text{O7}^{+}}^{\prime}}+5\bm{\omega}_{q}^{{\text{O7}^{+}}^{\prime}}+8\bm{\omega}_{b}+2\bm{\omega}_{c}\,,\end{split}$
(4.64)

from which we find that the fractional null junctions
$J_{\text{frac}}^{N,\text{mag}}$ is generated by

$\displaystyle\tfrac{1}{2}\big{(}\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}})}+\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},-\nicefrac{{1}}{{2}})}\big{)}=\bm{\delta}^{N}_{(\nicefrac{{1}}{{2}},0)}=-\bm{\omega}^{\text{O7}^{+}}_{p}-3\bm{\omega}^{{\text{O7}^{+}}^{\prime}}_{p}+2\bm{\omega}^{{\text{O7}^{+}}^{\prime}}_{q}+3\bm{\omega}_{b}+\bm{\omega}_{c}\,.$
(4.65)

It corresponds to the generator

$\displaystyle(3\ \text{mod 2},1\ \text{mod
2})=(1,1)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}=Z(SU(2)\times SU(2))$ (4.66)

of ${\cal
Z}=J_{\text{frac}}^{N,\text{mag}}/J_{\text{phys}}^{N,\text{mag}}=\mathbb{Z}_{2}$.

Together with the null junctions, the Abelian junction lattice
$J_{A}^{\text{mag}}$ is generated by

$\displaystyle{\bf u}_{1}=2{\bf
v}_{1}=2\big{(}-\bm{\omega}_{p}^{{\text{O7}^{+}}}+\bm{\omega}_{q}^{{\text{O7}^{+}}}+\bm{\omega}_{b}\big{)}\,,\quad{\bf
u}_{2}=2{\bf
v}_{2}=2\big{(}-\bm{\omega}_{p}^{{\text{O7}^{+}}}-\bm{\omega}_{q}^{{\text{O7}^{+}}}+\bm{\omega}_{c}\big{)}\,.$
(4.67)

Clearly, this leads to ${\cal Z}^{\prime}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$,
with generators

$\displaystyle\left.\begin{array}[]{c}{\bf v}_{1}\simeq(1;0\ |\ -1;1)\\\ {\bf
v}_{2}\simeq(0;1\ |\
1;-1)\end{array}\right\\}\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times
U(1)_{1}\times U(1)_{2}=Z\big{(}SU(2)_{\mathbf{B}}\times
SU(2)_{\mathbf{C}}\times U(1)_{1}\times U(1)_{2}\big{)}\,.$ (4.70)

The full gauge group is thus

$\displaystyle\frac{[SU(2)\times SU(2)]/\mathbb{Z}_{2}\times
U(1)^{2}}{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}\,.$ (4.71)

##  5 9d vacua via affine 7-brane stacks

Recently, it was argued that one can recover any 9d ${\cal N}=1$ string vacuum
with gauge rank $(1,17)$ from F-theory on a suitably degenerated K3 geometry
that lies at infinite distance in the complex structure moduli space [27, 28].
As shown in these works, such decompactification limits have a particularly
convenient description in terms of $[p,q]$-7-branes and junctions that realize
affine algebras. In the following, we demonstrate how the methods from the
previous sections naturally apply also to these limiting configurations, and
compute the 9d gauge group topologies for rank $(1,17)$ vacua.

Since the affinization is characterized entirely by the $SL(2,\mathbb{Z})$
monodromy, a natural proposition is that these configurations also describe 9d
uplifts when we include O7+-planes. Indeed, (after resolving an ambiguity by
string dualities) this straightforwardly reproduces the landscape of 9d rank
$(1,9)$ vacua [52], including their global gauge group structures. Moreover,
applying the same reasoning to configurations with two O7+’s, we consistently
find two branches of 9d rank $(1,1)$ vacua [54], which are only connected
through circle-reductions to 8d.

The key ingredient that enters the description for all ranks are 7-brane
stacks realizing an _affine_ Lie algebra $\widehat{\mathfrak{e}}_{k}$, which
we will now briefly recall.

As found in [74, 75, 70], the $\mathfrak{e}_{n}$ and
$\tilde{\mathfrak{e}}_{n}$ algebras can be enhanced to their affine versions,
by including a specific 7-brane on top:

$\displaystyle\begin{split}\widehat{{\bf E}}_{n\geq 1}=&\underbrace{{\bf
A}^{n-1}{\bf B}{\bf C}^{2}}_{{\bf E}_{n}}{\bf X}_{[3,1]}={\bf A}^{n-1}{\bf
B}{\bf C}{\bf B}{\bf C}\,,\\\ \widehat{\tilde{{\bf E}}}_{n\geq
0}=&\underbrace{{\bf A}^{n}{\bf X}_{[2,-1]}{\bf C}}_{\tilde{\bf E}_{n}}{\bf
X}_{[4,1]}\,.\end{split}$ (5.1)

Note that for $n\geq 2$, these are equivalent up to 7-brane moves and
$SL(2,\mathbb{Z})$ conjugations [75]. It is straightforward to check that, in
this $SL(2,\mathbb{Z})$-frame, they have monodromy

$\displaystyle M(\widehat{\bf E}_{n})=M(\widehat{\tilde{\bf
E}}_{n})=\begin{pmatrix}1&9-n\\\ 0&1\end{pmatrix}\,.$ (5.2)

The hallmark of these stacks is the existence of a special loop junction
$\bm{\delta}_{(1,0)}\equiv\bm{\delta}$ around them (with no asymptotic
charge), satisfying
$(\bm{\delta},\bm{\delta})=(\bm{\delta},\bm{\alpha}_{i})=0$, with
$\bm{\alpha}_{i}$ the root junctions of ${\bf E}_{n}$ or $\tilde{\bf E}_{n}$.
By a Hanany–Witten transition, one finds the equivalent presentation [70]

$\displaystyle\bm{\delta}_{\widehat{\text{E}}}={\bf x}_{[3,1]}-{\bf b}-{\bf
c}_{1}-{\bf c}_{2}={\bf b}_{2}+{\bf c}_{2}-{\bf b}_{1}-{\bf
c}_{1}\,,\quad\bm{\delta}_{\widehat{\tilde{\text{E}}}}={\bf x}_{[4,1]}-2{\bf
c}-{\bf x}_{[2,-1]}\,.$ (5.3)

Representation theoretically, $\bm{\delta}$ plays the role of the imaginary
root required for the affinization of $\mathfrak{e}_{n}$ or
$\tilde{\mathfrak{e}}_{n}$. They generate an infinite dimensional Kac–Moody
algebra with roots $\\{\bm{\alpha}+k\bm{\delta}\,|\,k\in\mathbb{Z}\\}$, where
$\bm{\alpha}$ is any root of $\mathfrak{e}_{n}$ or $\tilde{\mathfrak{e}}_{n}$.
When we seperate the affinizing $\mathbf{X}$-branes in (5.1) from the
$\mathfrak{e}_{n}$ or $\tilde{\mathfrak{e}}_{n}$ stacks, these junctions, as
strings, give rise to BPS states with masses proportional to $k$. In the
affine limit, we thus obtain an infinite tower of massless BPS states.
Physically, string junctions of these type give rise to an infinite tower of
massless BPS states.

A special extension exists for $n=8$. Here, by adding an ${\bf A}$-brane from
the left to the $\widehat{\bf E}_{8}$ or $\widehat{\tilde{\bf E}}_{8}$, the
monodromy becomes $\mathbb{1}\equiv M(\widehat{\bf
E}_{9})=M(\widehat{\tilde{\bf E}}_{9})$. This would give rise to two
independent towers of massless BPS states from loops of
$\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}0\\\ 1\end{smallmatrix}\right)$ string junctions,
which lead to the double loop enhancement of $\mathfrak{e}_{8}$. These special
enhancements reflect a decompactification to 10d [27, 28]. For discussions of
9d vacua, we will not consider such double loop brane-stacks, but the
necessary constituent branes form one half of the rank $(2,18)$ configuration
(4.22) that correspond geometrically to the singular fibers of a $dP_{9}$
surface.

Among the various types of infinite distance limits of F-theory compactified
on K3 surfaces, those describing decompactification from 8d to 9d are captured
by so-called Kulikov models of type III.a [27, 28]. In these geometries, the
complex structure moduli have been tuned such that the K3 degenerates into a
collection of intersecting elliptic and/or rational surfaces. While we refer
to those references for details, the relevant fact about these deformations is
that they correspond to brane motions which generate one or two 7-brane stacks
carrying an $\widehat{\mathfrak{e}}_{n}$ or
$\widehat{\tilde{\mathfrak{e}}}_{n}$ algebra (with $n\leq 8$). The tower of
massless states from the imaginary root may then be identified with the
momentum states of a Kaluza–Klein (KK) tower on a circle whose size becomes
infinite at the infinite distance limit. In the case with two affine stacks,
the individual imaginary root junctions turn out to be identical in the global
setting, consistent with having just one KK-tower [28].

###  5.1 Global structure of 9d vacua of rank 17

As for the classification of 8d vacua, one can also categorize all brane
configurations with such affine stacks. Then, if the non-Abelian brane stacks
correspond to the algebra $\mathfrak{h}\oplus\widehat{{\cal E}}_{n}$ or
$\mathfrak{h}\oplus\widehat{{\cal E}}_{n}\oplus\widehat{{\cal E}}_{m}$ (where
${\cal E}=\mathfrak{e}$ or $\tilde{\mathfrak{e}}$) for some finite semi-
simple, simply-laced algebra $\mathfrak{h}$, the associated non-Abelian gauge
algebra in 9d is $\mathfrak{h}\oplus{\cal E}_{n}$ or $\mathfrak{h}\oplus{\cal
E}_{n}\oplus{\cal E}_{m}$, respectively [27, 28]. This reproduces, e.g., all
the maximally enhanced non-Abelian algebras (i.e., with rank 17) determined in
the dual heterotic frame [29].

To also analyze the gauge group topologies in this description, we need to
examine the full junction lattice, including the branes away from the affine
stack. An important detail here is that the overall gauge rank reduces by 2 as
we decompactify from 8d to 9d, corresponding to the re-interpretation of the
KK-states (which become massless) and the decoupling of the winding states
(which become infinitely heavy) as we increase the size of the
compactification circle. In the momentum lattice description of the 8d and 9d
theories of rank $(2,18)$ and $(1,17)$, respective, we have

$\displaystyle\Lambda^{\text{het}}_{\text{8d}}=\Lambda^{\text{het}}_{\text{9d}}\oplus
U\quad\Rightarrow\quad\Lambda^{\text{het}}_{\text{9d}}\cong\Lambda^{\text{het}}_{\text{8d}}/U\,,$
(5.4)

with $U$ the rank 2 hyperbolic lattice that is spanned by the KK and winding
states. Since the momentum lattice is equivalently described by junctions
$J_{\text{phys}}^{\text{el}}=J_{\text{phys}}^{\text{mag}}\cong\Lambda_{\text{8d}}^{\text{het}}\oplus
J_{\text{phys}}^{N}$, with the KK-tower being generated by the junction
$\bm{\delta}$, there must exist another non-null junction $\bm{\epsilon}$ that
generates this $U$ factor with $\bm{\delta}$, i.e., satisfying

$\displaystyle(\bm{\delta},\bm{\epsilon})=1\,,\quad(\bm{\delta},\bm{\delta})=(\bm{\epsilon},\bm{\epsilon})=(\bm{\delta},{\bf
j})=(\bm{\epsilon},{\bf j})=0\,,$ (5.5)

for ${\bf j}$ any (co-)weight or (co-)root junction, or a non-null generator
of the Abelian junctions $J_{A}$. Such a $\bm{\epsilon}$-junction always
exists, but the details depend on the specific configuration.

Since the junction lattice reproduces the 9d momentum lattice, it must also
encode the global structure of the gauge group. In particular, it allows us to
use the intuition in terms of fractional null junctions to re-derive the
results of [29]. Let us demonstrate this for 9d models with maximally enhanced
non-Abelian symmetries, for which there are two classes of 8d brane
configurations [28].

In the first class, the non-Abelian algebra (with the place holder ${\cal
E}=\mathfrak{e}$ or $\tilde{\mathfrak{e}}$) is

$\mathfrak{g}_{\text{8d},\infty}=\mathfrak{su}_{18-m-n}\oplus\widehat{{\cal
E}}_{m}\oplus\widehat{{\cal
E}}_{n}\quad\Rightarrow\quad\mathfrak{g}_{\text{9d}}=\mathfrak{su}_{18-m-n}\oplus\mathcal{E}_{m}\oplus\mathcal{E}_{n},\quad
m,n\in\\{0,1,3,\dots,8\\}\,,$ (5.6)

whose brane configurations (together with the $U$-lattice generators) are

$\displaystyle\begin{split}m\geq n\geq
1:&\quad\mathbf{A}^{18-m-n}(\overbrace{\mathbf{A}^{m-1}\mathbf{X}_{[n-10,1]}\mathbf{X}_{[n-8,1]}^{2}\mathbf{X}_{[n-6,1]}}^{\widehat{{\bf
E}}_{m}})(\overbrace{\mathbf{A}^{n-1}\mathbf{B}\mathbf{C}^{2}\mathbf{X}_{[3,1]}}^{\widehat{{\bf
E}}_{n}})\,,\\\ &\quad(\bm{\delta},\bm{\epsilon})=(\bm{\delta}^{R}_{(1,0)},\ \
(n-5)\bm{\delta}^{R}_{(1,0)}+{\bm{\ell}}^{R}_{(0,1)}+\mathbf{x}_{[n-6,1]}-\mathbf{x}_{[3,1]})\,,\end{split}$
(5.7a)
$\displaystyle\begin{split}m=1,n=0:&\quad\mathbf{A}^{17}(\overbrace{\mathbf{X}_{[10,-1]}\mathbf{X}_{[8,-1]}^{2}\mathbf{X}_{[6,-1]}}^{\widehat{{\bf
E}}_{1}})(\overbrace{\mathbf{X}_{[2,-1]}\mathbf{C}\mathbf{X}_{[4,1]}}^{\widehat{\tilde{{\bf
E}}}_{0}})\,,\\\ &\quad(\bm{\delta},\bm{\epsilon})=(\bm{\delta}^{R}_{(1,0)},\
\
-17\bm{\omega}_{A}+\mathbf{x}_{[10,-1]}+2\mathbf{x}_{[4,1]}-\mathbf{c})\,,\end{split}$
(5.7b) $\displaystyle\begin{split}m>n=0;\ m\neq
1:&\quad\mathbf{A}^{18-m}(\overbrace{\mathbf{A}^{m}\mathbf{X}_{[11,-1]}\mathbf{X}_{[8,-1]}\mathbf{X}_{[5,-1]}}^{\widehat{\tilde{{\bf
E}}}_{m\neq
1}})(\overbrace{\mathbf{X}_{[2,-1]}\mathbf{C}\mathbf{X}_{[4,1]}}^{\widehat{\tilde{\bf
E}}_{0}})\,,\\\ &\quad(\bm{\delta},\bm{\epsilon})=(\bm{\delta}^{R}_{(1,0)},\ \
-\bm{\delta}^{R}_{(1,0)}+\mathbf{x}_{[5,-1]}-2\mathbf{x}_{[2,-1]}-\mathbf{c})\,.\end{split}$
(5.7c)

where $\bm{\delta}^{R}$ and $\bm{\ell}^{R}$ are loop junction that encircle
counterclockwise around the second affine stack. Note that, because the
$\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$ loop is mutually-
local with respect to the ${\bf A}$-branes, it is evident that, by pulling
$\bm{\delta}_{(1,0)}$ across these, one obtains a loop junction around the
other affine stack, showing explicitly that imaginary roots of each affine
stack are identical.

The second class has non-Abelian gauge algebras

$\mathfrak{g}_{\text{8d},\infty}=\mathfrak{so}_{34-2k}\oplus\widehat{{\cal
E}}_{k}\quad\Rightarrow\quad\mathfrak{g}_{\text{9d}}=\mathfrak{so}_{34-2k}\oplus{\cal
E}_{k}\,,\quad 0\leq k\leq 8,k\neq 2\,,$ (5.8)

whose brane configurations (and $U$-lattice generators) are

$\displaystyle\begin{split}k=1,3,\dots,8:&\quad(\overbrace{\mathbf{A}^{17-k}\mathbf{X}_{[k-10,1]}\mathbf{X}_{[k-8,1]}}^{\textbf{D}_{17-k}})\mathbf{X}_{[k-8,1]}(\overbrace{\mathbf{A}^{k-1}\mathbf{B}\mathbf{C}^{2}\mathbf{X}^{(1)}_{[3,1]}}^{\widehat{\textbf{E}}_{k}})\mathbf{X}^{(2)}_{[3,1]}\,,\\\
&\quad(\bm{\delta},\bm{\epsilon})=(\bm{\delta}^{R}_{(1,0)},\ \
\bm{\delta}^{R}_{(1,0)}+\mathbf{x}^{(1)}_{[3,1]}-\mathbf{x}^{(2)}_{[3,1]})\end{split}$
(5.9a)
$\displaystyle\begin{split}k=0:&\quad(\overbrace{\mathbf{A}^{17}\mathbf{X}_{[10,-1]}\mathbf{X}^{(1)}_{[8,-1]}}^{\textbf{D}_{17}})\mathbf{X}^{(2)}_{[8,-1]}(\overbrace{\mathbf{X}_{[2,-1]}\mathbf{C}\mathbf{X}_{[4,1]}}^{\widehat{\tilde{\textbf{E}}}_{0}})\mathbf{X}_{[3,1]}\,,\\\
&\quad(\bm{\delta},\bm{\epsilon})=(\bm{\delta}^{R}_{(1,0)},\ \
-2\bm{\delta}^{R}_{(1,0)}+\mathbf{x}^{(2)}_{[8,-1]}-3\mathbf{x}_{[2,-1]}-2\mathbf{x}_{[1,1]})\,,\end{split}$
(5.9b)

again, with the imaginary root junction $\bm{\delta}^{R}_{(1,0)}$ being the
$\left(\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right)$-loop around the
affine stack to the right.

By separating the ${\bf X}$-brane responsible for the affinization from each
affine stack, we obtain a genuine 8d configuration. For these configurations,
we can apply the same procedure as in the previous section, and construct the
global fractional null junctions that encode to the cocharacters of the 8d
gauge symmetry. Since the affinization (5.4) mods out by physical junctions
that are orthogonal to the root lattices of the 9d gauge factors, it does not
affect the coefficients of global null junctions in front of the extended
weights. Therefore, the fractional null junctions are the same for the 8d
configuration as for its affinized version.

As a concrete example, consider (5.9a) with $k=7$, which in 9d gives rise to
$\mathfrak{g}_{9d}=\mathfrak{so}_{20}\oplus\mathfrak{e}_{7}$. Separating the
singlet brane responsible for the affinization,

$\displaystyle(\underbrace{\mathbf{A}^{10}\mathbf{X}_{[-3,1]}\mathbf{X}_{[-1,1]}}_{\textbf{D}_{10}^{\prime}})\,\mathbf{X}_{[-1,1]}\,(\underbrace{\mathbf{A}^{6}\mathbf{B}\mathbf{C}^{2}}_{\textbf{E}_{7}})\,\mathbf{X}^{(1)}_{[3,1]}\,\mathbf{X}^{(2)}_{[3,1]}\
,$ (5.10)

we find the $\mathfrak{so}_{20}$-stack to have monodromy
$\left(\begin{smallmatrix}-1&6\\\ 0&-1\end{smallmatrix}\right)$ in this
$SL(2,\mathbb{Z})$ frame, with extended weight junctions
$\bm{\omega}^{\prime}_{p,q}$ carrying asymptotic
$\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$-charges as follows:

$\displaystyle\bm{\omega}_{p}^{\prime}:\,\,\left(\begin{smallmatrix}1\\\
0\end{smallmatrix}\right)\,,\qquad\bm{\omega}_{q}^{\prime}:\,\,\left(\begin{smallmatrix}-2\\\
1\end{smallmatrix}\right)\,.$ (5.11)

The two global null junctions $\bm{\delta}^{N}_{(1,0)}$ and
$\bm{\delta}^{N}_{(0,1)}$ can be then expressed in terms of the extended
weights $\bm{\omega}^{\prime}_{p,q}$ of $\mathfrak{so}_{20}$ and
$\bm{\omega}_{p,q}$ of $\mathfrak{e}_{7}$ as

$\displaystyle\begin{split}&\bm{\delta}^{N}_{(1,0)}=-2\bm{\omega}_{p}^{\prime}-\mathbf{x}_{[-1,1]}-5\bm{\omega}_{p}-\bm{\omega}_{q}+\mathbf{x}^{(1)}_{[3,1]}+\mathbf{x}^{(2)}_{[3,1]}\,,\\\
&\bm{\delta}^{N}_{(0,1)}=2\bm{\omega}_{p}^{\prime}-2\bm{\omega}_{q}^{\prime}+5\mathbf{x}_{[-1,1]}+17\bm{\omega}_{p}+3\bm{\omega}_{q}+\mathbf{x}^{(1)}_{[3,1]}+\mathbf{x}^{(2)}_{[3,1]}\,.\end{split}$
(5.12)

It is easy to see that the fractional null junctions are then multiples of

$\displaystyle\tfrac{1}{2}(\bm{\delta}^{N}_{(1,0)}+\bm{\delta}^{N}_{(0,1)})=-\bm{\omega}_{q}^{\prime}+2\mathbf{x}_{[-1,1]}+6\bm{\omega}_{p}+\bm{\omega}_{q}+\mathbf{x}_{[3,1]}^{(1)}+\mathbf{x}_{[3,1]}^{(2)}\,.$
(5.13)

By (2.44), this corresponds to the central element

$\displaystyle(0,1;1)\in\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{2}=Z(Spin(20)\times
E_{7}),$ (5.14)

which leads to the 9d non-Abelian gauge group $[Spin(20)\times
E_{7}]/\mathbb{Z}_{2}$.

To determine the full gauge group, including the gravi-photon $U(1)$, we must
first find the non-null generator of the Abelian junction lattice that is
orthogonal to the $U$-lattice spanned by $(\bm{\delta},\bm{\epsilon})$. In
this example, it can be easily determined (by avoiding prongs on the
$\mathfrak{e}_{7}$ stack or the $\mathbf{X}_{[3,1]}$ branes),

$\displaystyle{\bf u}=2{\bf
v}=2(\bm{\omega}_{p}^{\prime}+\bm{\omega}_{q}^{\prime}-\mathbf{x}_{[-1,1]})\,,$
(5.15)

which immediately gives ${\cal Z}^{\prime}=\mathbb{Z}_{2}$, with generator

$\displaystyle{\bf v}\simeq(1,1;0\ |\ e^{i\pi})\in Z(Spin(20)\times
E_{7}\times U(1))\,.$ (5.16)

Therefore, the 9d gauge group is

$\displaystyle\frac{[Spin(20)\times E_{7}]/\mathbb{Z}_{2}\times
U(1)}{\mathbb{Z}_{2}}\,.$ (5.17)

By analogous computations, we compute the non-Abelian gauge groups of all
models with maximally enhanced non-Abelian symmetry (summarized in Table 1),
which agree with results from the heterotic picture [29].

$(\mathfrak{g}_{\text{9d}},\pi_{1}(G_{\text{9d}}))$ | FNJ | $(\mathfrak{g}_{\text{9d}},\pi_{1}(G_{\text{9d}}))$ | FNJ
---|---|---|---
$\mathfrak{g}_{\text{8d},\infty}=\mathfrak{su}_{18-m-n}\oplus\widehat{\mathfrak{e}}_{m}\oplus\widehat{\mathfrak{e}}_{n}$
$(\mathfrak{e}_{8}\oplus\mathfrak{e}_{8}\oplus\mathfrak{su}_{2},-)$ | - | $(\mathfrak{e}_{6}\oplus\mathfrak{su}_{3}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{9},\mathbb{Z}_{3})$ | $\bm{\delta}^{N}_{(0,1)}/3$
$(\mathfrak{e}_{8}\oplus\mathfrak{e}_{7}\oplus\mathfrak{su}_{3},-)$ | - | $(\mathfrak{e}_{6}\oplus\mathfrak{su}_{2}\oplus\mathfrak{su}_{11},-)$ | -