UUITP–56/21

F-theory on 6D Symmetric Toroidal Orbifolds

Finn Bjarne Kohl${}^{\,\text{a}}$, Magdalena Larfors${}^{\,\text{b}}$ and
Paul-Konstantin Oehlmann${}^{\,\text{b}}$

${}^{\text{a}}$ Institute for Theoretical Physics and Mathematical Institute,
Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, 48149
Münster, Germany
${}^{\text{b}}$ Department of Physics and Astronomy, Uppsala University,
Lägerhyddsv. 1, SE-751 20 Uppsala, Sweden

In this work we study F-theory on symmetric toroidal orbifolds that exhibit
roto-translations, which are point group rotations accompanied by fractional
lattice shifts. These geometries admit a rich class of effects, such as
twisted affine folded fibers, multiple fibers, and up to three distinct torus-
fibrations that yield different M/F-theory lifts. We discuss the six-
dimensional physics of the F-theory lifts, which generically host
superconformal subsectors and a IIB axio-dilaton fixed to strong coupling. In
addition we find that these theories exhibit a rich set of $p=0,1,2$ discrete
$p$-form gauge symmetries. We discuss six-dimensional gauge and supergravity
anomalies and match the rank and tensor branch dimension to the Hodge numbers
that were computed using heterotic world sheet techniques.

###### Contents

  1. 1 Introduction
  2. 2 Orbifolds and F-theory
    1. 2.1 Toroidal orbifolds
    2. 2.2 F-theory and SCFTs
    3. 2.3 F-theory on orbifolds
  3. 3 F-theory on symmetric toroidal orbifolds
    1. 3.1 Summary of results
    2. 3.2 F-theory on ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ orbifolds
  4. 4 Summary and conclusion
  5. A F-theory on ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifolds

##  1 Introduction

F-theory is a versatile framework that allows us to construct a large number
of consistent string compactifications in various dimensions. Its main benefit
is the possibility to go beyond perturbative IIB string constructions by
geometrizing the type IIB axio-dilaton field into the complex structure
modulus of an auxiliary torus Vafa_1996_1 ; Morrison:1996pp ; Morrison:1996na
. This modulus, and thus the IIB string coupling, may then vary over the
physical compactification space $B$. This provides a geometric lift of the
compact space to a torus-fibration $X$, which is fixed to be Calabi-Yau in
order to preserve minimal supersymmetry (SUSY). The F-theory torus on the
other hand, becomes fully physical in M-theory on $X$, related to F-theory via
a circle reduction.

This F-theory construction has lead to a vast progress in the construction of
supergravities, superconformal field theories (SCFTs) and little string
theories (LSTs) in various dimensions. In particular, F-theory has be used to
classify SCFTs and LSTs in six dimensions Heckman_2015 ; Bhardwaj_2016  (also
see references in Heckman_2019 ). Moreover, its flexibility has also lead to a
systematic construction of 6D supergravity theories and the explorations of
their symmetries, as described in Kumar:2009ac ; Kumar:2009us ; Park:2011ji ;
Morrison:2012np ; Kumar:2010ru  and references therein.

On the other hand, a particular simple set of compact geometries are those of
toroidal orbifolds. Orbifolds are essentially flat spaces, with curvature
concentrated at certain singular points. These points can often be resolved,
and when this is possible, this connects the orbifold geometries to smooth
Calabi-Yau (CY) spaces. In reverse, orbifolds can often be thought of as
special singular limits of smooth Calabi-Yaus in their Kähler moduli space.
However, as there exist only a couple of hundred 6D orbifolds, in contrast to
the billions of CY threefolds, this interpretation is only true for a very
selected set of CY geometries.

Due to their simplicity, toroidal orbifolds are a classic setting for string
compactifications. As the flatness of the bulk of these space allows an exact
heterotic world-sheet conformal field theory (CFT) description, they have been
used extensively for heterotic string compactifications, starting from the
first examples DIXON1985678 ; DIXON1986285  and moving to large model building
explorations Ibanez:1986tp ; Blaszczyk:2009in ; Lebedev:2006kn . Toroidal
orbifolds have also been considered, but to a lesser extent, as backgrounds
for F-theory. Orbifolds of type
$(\mathbb{T}^{2}\times\mathbb{T}^{2})/{\mathbb{Z}}_{n}$ for $n=2$ provided the
initial example used by Sen Sen:1996vd  to relate 8D F-theory with
perturbative IIB $O7^{-}$ orientifold compactifications, and this
constructions was generalized to non-perturbative IIB vacua by Dasgupta and
Mukhi (for $n=3,4,6$) Dasgupta_1996 . More recently, F-theory on orbifolds
have played a prominent role e.g. in the construction of 4D $\mathcal{N}=3$
SCFTs in non-compact fourfolds Garcia-Etxebarria:2015wns ; Garcia-
Etxebarria:2016erx .

Compactifying F-theory on six dimensional toroidal orbifolds gives rise to 6D
supergravity theories coupled to superconformal sectors Morrison:1996na ;
Gopakumar1996  which have recently been studied in the context of holography
in hayashi2019scfts . The latter reference considered toroidal orbifolds of
type
$(\mathbb{T}^{2}\times\mathbb{T}^{2}\times\mathbb{T}^{2})/{\mathbb{Z}}_{n}\times{\mathbb{Z}}_{m}$,
identifying one of the three tori as the F-theory torus. A key feature of
F-theory compactifications is that they lead to 6D non-cricital strings
obtained from D3 branes that wrap curves inside of the base $B$. For those
strings a holographic IIB dual picture was proposed, that probe the
$Ads_{3}\times\mathbb{S}^{3}\times B$ background Haghighat:2015ega ;
Couzens:2017way . Orbifold bases of the type
$\mathbb{T}^{4}/({\mathbb{Z}}_{n}\times{\mathbb{Z}}_{m})$ are then again
attractive and simple choices, as they are flat apart from the (codimension
two) orbifold fixed points that contribute 6D tensionless strings. The
specific constructions considered in hayashi2019scfts  always lead to
superconformal sectors that originate from the intersection of two D7 brane
stacks, so called superconformal matter Heckman_2015 .

However, the studies just mentioned did not explore all possibilities.
F-theory can be used to construct many more SCFT configurations and there are
many more toroidal orbifolds than the examples just mentioned. In fact, 6D
toroidal orbifolds have been fully classified only relatively recently
Fischer_2013  leading to roughly 520 inequivalent geometries. This
classification includes cases that significantly generalize those cases
considered in hayashi2019scfts , such as about 360 orbifolds with non-Abelian
point group, and multiple Abelian orbifolds that admit roto-translations, i.e.
fractional lattice shifts accompanied with point group rotations. Moreover,
most toroidal orbifolds admit more than one torus-fiber structure and thus
allow multiple F-theory lifts that share a common M-theory dual.

The goal of this work is to initiate a systematic study of F-theory on
toroidal orbifolds and determine the effective supergravity theories resulting
from the compactification. Equivalently, this yields a classification of
$B={\mathbb{T}}^{2}\times{\mathbb{T}}^{2}/\Gamma$ bases, relevant for their
potential holographic study. In order to do so, we follow the general logic
put forward in hayashi2019scfts . In this work, we consider geometries that go
beyond those considered in hayashi2019scfts  in that they can admit a single
${\mathbb{Z}}_{n}$ factor and/or roto-translational elements. Already with
this limited generalization, we find a number of interesting geometric
features that we explore in the F-theory setting, which translate into new
ingredients in the 6D effective theory.

Toroidal orbifolds generically admit a large set of additional discrete
symmetries, due to their singular nature, which can lead to a wide range of
discrete R- and flavor symmetries in heterotic compactifications Nilles:2012cy
.111For a recent account for modular symmetries in orbifolds see Baur:2019kwi
. As those discrete symmetries are inherited from the geometry, it is natural
to expect enhanced discrete symmetries in F-theory physics as well. Indeed, we
observe various discrete zero-, one- and two-form gauge symmetries associated
to the orbifold orders. Related to that are isolated SCFT sectors in the base
that are not of superconformal matter type. Moreover, we find ADE
singularities that are folded by additional freely acting point group elements
and yield twisted affine variants of ADE fibers Braun:2014oya ;
Anderson:2019kmx . Similarly, such twisted elements can lead to so called
multiple fibers Gross97 ; Bhardwaj_2016 ; Anderson:2018heq ; Anderson:2019kmx
.

The relation to the heterotic string and its world-sheet CFT methods plays a
crucial role in this work. In particular we use it, and its implementation
into the C++ Orbifolder Nilles_2012 , to compute stringy Hodge numbers. Those
are required to determine the number of tensor and vector multiplets in the 6D
supergravity theories, and to ensure anomaly cancellation.

This paper is structured as follows: In Section 2 we give a basic introduction
to F-theory from an orbifold perspective as well as a short review of orbifold
geometries and their classification. In Section 3 we present our main results,
summarize the F-theory features that arise from the orbifold geometry, and
discuss the orbifold geometries ${\mathbb{Z}}_{3}$,
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$ and
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-4$ and their physics in some detail.
Some more examples, on ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifolds, are
discussed in Appendix A. In Section 4 we conclude and highlight future
directions and open problems.

##  2 Orbifolds and F-theory

###  2.1 Toroidal orbifolds

Toroidal orbifolds play a distinctive role in string compactifications. Just
like the tori they descend from, the bulk of these spaces is flat, and their
curvature is concentrated at the fixed points of the orbifold action. This
fact provides a lot of control to the study of e.g. the heterotic string
theory on such spaces, as the world-sheet CFT is free and can be solved
exactly. Orbifolds are not isolated geometries, but can be connected, via
resolution Lust:2006zh  of the orbifold singularities, to the set of smooth
Calabi-Yau geometries222Such a resolution, however, might not always be
possible.. Consequently, orbifolds can be viewed as particular singular,
finite-volume, limits of smooth Calabi-Yau threefolds333Oftentimes, cycles in
the geometry can be shrunk further, resulting in Landau-Ginzburg, or hybrid
phases of the geometry..

In this section, we review the basic construction of toroidal orbifolds and
how they are classified. A more detailed discussion of this matter can be
found in Fischer_2013 . In the ensuing sections, we discuss F-theory and how
the orbifold geometry determines the gauge groups, field content and SCFT
sectors of F-theory compactifications.

####  2.1.1 Orbifold geometry

Toroidal orbifolds can be described as the quotient of a torus
${\mathbb{T}}^{2n}$ with some group $G$, called the orbifolding group, or,
equivalently, as the quotient of ${\mathbb{R}}^{2n}\simeq{\mathbb{C}}^{n}$
with the space group $S$

$\displaystyle{\mathbb{O}}_{2n}={\mathbb{T}}^{2n}/G={\mathbb{R}}^{2n}/S={\mathbb{C}}^{n}/S\,.$
(2.1)

The space group is a discrete subgroup of the Poincaré group of
${\mathbb{R}}^{2n}$. A generic element in $S$ is of the form
$g=(\vartheta,\lambda)$ and acts on $q\in{\mathbb{R}}^{2n}$ as

$\displaystyle q\mapsto\vartheta q+\lambda\,,$ (2.2)

and is composed as
$(\vartheta,\lambda)\circ(\omega,\tau)=(\vartheta\omega,\vartheta\tau+\lambda)$.

The subgroup of $S$ that consists of pure translations is the lattice
$\Lambda$ associated to the ${2n}$-torus
${\mathbb{T}}^{2n}={\mathbb{R}}^{2n}/\Lambda$. $\Lambda$ is generated by basis
vectors $e_{\alpha}\in{\mathbb{Z}}^{2n}$, $\alpha=1,...,{2n}$. Such bases are
not unique, which leads to equivalence classes of lattices. To be precise, two
lattices that are invariant under the same finite unimodular group, are said
to be Bravais equivalent. Lattices can be classified in terms of their
symmetry groups, and are, despite ambiguities explained in Fischer_2013 ,
therefore often referred to in terms of Lie algebra root-lattices.

The elements $\vartheta$ of a space group element are discrete rotations and
form the so-called point group $P$. In case $P$ is Abelian an element
$\vartheta$ is characterized by the integral vector $v\in{\mathbb{Z}}^{n}$
which determines the action of the rotation on the $n$ complex planes of
${\mathbb{C}}^{n}$ according to

$\displaystyle(\vartheta)_{ij}=\exp(2\pi\imath\frac{v_{i}}{r})\;\delta_{ij}\,,$
(2.3)

where $r$ denotes the order of the point group. As described below in Section
2.1.2 in order to preserve supersymmetry in string compactifications, the
compact space is required to be Calabi-Yau. As a consequence, the point group
needs to be a discrete subgroup of the $SU(n)$ holonomy group. This criterion
translates to $\sum_{i}v_{i}/r=0\mod{\mathbb{Z}}$.

Note that although the space group is a symmetry of the lattice, it may still
admit shifts by fractional lattice units, yielding so called roto-
translations. In absence of elements with such fractional lattice shifts, the
space group is given by the semi-direct product $S=P\ltimes\Lambda$. In the
generic case the space group is instead described by

$\displaystyle S=\langle\\{G,\Lambda\\}\rangle\,,$ (2.4)

where we recall that the orbifold group $G$ is defined in (2.1). Concluding,
by slight abuse of notation, an element $g=(\vartheta,\lambda)$ in an Abelian
space group will in the following be denoted by the $(n+2n)$-component vector

$\displaystyle\vartheta:\;\;((v_{i}/r)_{i=1,\dots,n}\,|\,(\lambda_{\alpha})_{\alpha=1,\dots,2n})\,.$
(2.5)

As indicated above, the action of the space group on ${\mathbb{R}}^{2n}$ is in
general not free and admits fixed points/lines

$\displaystyle q_{f}=g(q_{f})\,,$ (2.6)

as illustrated for a $\mathbb{T}^{2}/{\mathbb{Z}}_{3}$ fixed point
$q_{f}=2/3\,e_{1}+1/3\,e_{2}$ in Figure 1. Since this equation might have
several solutions in the fundamental domain of the lattice for a choice of a
discrete rotation, one point group element can introduce several fixed points.
Note, that only fixed points in the fundamental domain of the orbifold (cf.
Figure 1) are inequivalent, as we now explain. Using (2.6), one can show, that
each fixed point is in one to one relation with an element $g_{f}$ of the
orbifolding group. However, those fixed points $q_{f}$ and their corresponding
group elements $g_{f}$ might not be independent on the full orbifold, as there
might be additional group elements $h$ upon which fixed points are identified
as

$\displaystyle h(q_{f})=q^{\prime}_{f}\qquad\leftrightarrow\qquad
g_{f}^{\prime}=h\circ g_{f}\circ h^{-1}\,.$ (2.7)

The order $k$ of the fixed point is defined by the order of the associated
point group element, such that $g^{k}=1$. Furthermore, the action induces a
holonomy $\Gamma\subset SU(3)$ at this point, which indicates the presence of
curvature. This explains the already mentioned fact that the curvature of an
orbifold is localized at the singular fixed points. Such orbifold
singularities often admit a crepant resolution to a non-singular Calabi-Yau
manifold, where the singular point is replaced by one, or a chain of,
exceptional divisors (see e.g. Blaszczyk:2011hs ; Lust:2006zh ). This resolved
singular structure will be of crucial importance in F-theory compactifications
on orbifolds, and will therefore be illustrated with two examples in
Subsections 2.1 and 2.1. Beforehand, in the next section, we review the
classification of six dimensional symmetric toroidal orbifolds by Fischer and
collaborators Fischer_2013

Figure 1: Depiction of the action of a pure rotation generated by $\vartheta$
by $120\text{\,}\mathrm{\SIUnitSymbolDegree}$ of a an orbifold fixed point
(pink) in a torus ${\mathbb{T}}^{2}={\mathbb{R}}^{2}/\Lambda_{SU(3)}$. After a
lattice shift by $e_{1}$ the point is transported back onto itself. Note, the
light gray region is the fundamental domain of the lattice, opposed to the
fundamental domain of the orbifold in darker gray, which is smaller due to the
additional identification by the point group elements.

####  2.1.2 Classification of symmetric toroidal orbifolds

Orbifolds have been studied for a long time in the field of string theory, and
several classifications exist Bailin:1999nk ; Donagi:2008xy ; Forste:2006wq ;
Dillies:2006yb . Of particular relevance is the recent, complete
classification of the symmetric toroidal orbifolds (STO)444In this work we
follow the nomenclature of Fischer_2013  that distinguishes symmetric and
asymmetric orbifolds that correspond to geometric and non-geometric heterotic
string backgrounds.  in six dimensions Fischer_2013 , which classifies these
geometries based on their space groups. This allows one to define three types
of equivalence classes among orbifold geometries

$S\subset$ | $SU(3)$ | $SU(2)$ | $1$ | $\sum$
---|---|---|---|---
$\\#{\mathbb{Q}}$ | 17 | 4 | 1 | 22
$\\#{\mathbb{Z}}$ | 60 | 10 | 1 | 71
$\\#\text{aff.}$ | 138 | 23 | 1 | 162
Table 1: Summary of all orbifold geometries for all Abelian ${\mathbb{Q}}$-,
${\mathbb{Z}}$\- and affine classes split up into their respective (discrete)
holonomy subgroups. Reduced holonomy subgroups lead to enhanced SUSY.

  * •

The ${\mathbb{Q}}$-equivalence determines the point group $P$ in the space
group $S$. Two space groups are ${\mathbb{Q}}$-equivalent iff for their point
groups $P_{1}$ and $P_{2}$

$\displaystyle\exists\;V\in\text{GL}({2n},{\mathbb{Q}}):$ $\displaystyle
V^{-1}P_{1}V=P_{2}\,.$ (2.8)

Orbifolds that are representatives of one ${\mathbb{Q}}$-class have the same
holonomy group. However, Bravais-inequivalent lattices are not distinguished
in this classification.

  * •

${\mathbb{Z}}$-equivalence distinguishes lattices $\Lambda$ in $S$. Two space
groups are ${\mathbb{Z}}$-equivalent iff for their point groups $P_{1}$ and
$P_{2}$

$\displaystyle\exists\;U\in\text{GL}({2n},{\mathbb{Z}}):$ $\displaystyle
U^{-1}P_{1}U=P_{2}\,.$ (2.9)

The ${\mathbb{Z}}$-classification provides a refinement of the
${\mathbb{Q}}$-classification, i.e. each ${\mathbb{Q}}$-class may contain
several ${\mathbb{Z}}$-classes.

  * •

Affine-equivalence distinguishes different orbifolding groups in the $S$, in
terms of different fractional lattice shifts, so called roto-translations. Two
space groups $S_{1}$ and $S_{2}$ belong to the same affine class iff

$\displaystyle\exists\;(f:\;{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n}):$
$\displaystyle f^{-1}S_{1}f=S_{2}\,,$ (2.10)

with $f=(A,t)$ consisting of translations $t$ and linear mappings $A$.
This equivalence is of special importance in this work as the effect of
different non-trivial affine classes on the physical theory is investigated.

In Ref. Fischer_2013 , these equivalence relations are used to classify and
count inequivalent orbifolds, grouped in terms of the amount of supersymmetry
preserved in compactification of the heterotic string to four dimensions. We
summarize this classification, for Abelian point groups, in Table 1. For a
more detailed analysis of the classification the reader is referred to the
original work Fischer_2013 .

In string compactifications, the geometry of the internal space determines the
theory in lower dimensions, and so these different equivalence classes
determine different aspects of the physical theory. Whereas the
${\mathbb{Q}}$-class determines the amount of SUSY that is preserved in the
compactified theory, transforming between different representatives of one
affine class corresponds to moving in the moduli space of the respective
compactification.

Note, that in the classification as well as in the present work geometries of
both simply-connected and non-simply connected geometries appear. The
fundamental group of toroidal orbifolds is given by

$\displaystyle\pi_{1}=S/\expectationvalue{F}\,,$ (2.11)

where $\expectationvalue{F}$ denotes the span of non-freely acting space group
elements. In case $\expectationvalue{F}=S$ (which holds for most affine
classes), $\pi_{1}$ is trivial. A non-trivial fundamental group exists, if
there are freely acting space group elements, that are not decomposable into
the set of non-freely acting elements. In the context of the heterotic string
this indicates non-local symmetry breaking. In the present work the orbifold
discussed in Section 3.2 is an example for an orbifold with non-trivial
fundamental group. As we will see later, similarly as discussed in
Anderson:2018heq ; Anderson:2019kmx  such geometries lead to F-theory
description in terms of a covering theory modulo additional superconformal
sectors.

In the following, adopting the nomenclature of the classification, six
dimensional STOs are referred to in terms of these three classes as
${\mathbb{Z}}_{n}\times{\mathbb{Z}}_{m}-X-Y$, where
${\mathbb{Z}}_{n}\times{\mathbb{Z}}_{m}$ specifies the point group, that is
the ${\mathbb{Q}}$-class, and $X$ and $Y$ determine the ${\mathbb{Z}}$\- and
affine class, respectively.

####  2.1.3 Stringy Hodge numbers of orbifolds

A key component in the analysis of F-theory compactifications are the Kähler
and complex structure moduli of the orbifold geometries $X_{\text{orb}}$.
Strictly speaking, these topological invariants are not defined for singular
geometries but only for the smooth Calabi-Yau A short-cut to determining these
numbers, that bypasses the resolution of the geometry, is provided by the free
heterotic CFT in the standard embedding. This refers to a particular solution
of the Bianchi identity and Hermitian Yang-Mills equation of the heterotic
compactification, where the gauge connection is equated with the $SU(3)$ spin
connection. The observable 4D gauge group is then the $SU(3)$ commutant in
$E_{8}\times E_{8}$, that is $E_{6}\times E_{8}$, and we can use this
information to determine the Hodge numbers. Decomposing the broken $E_{8}$
adjoint as

$\displaystyle\mathbf{248}\to(\mathbf{1},\mathbf{8})+(\mathbf{27},\mathbf{3})+(\overline{\mathbf{27}},\overline{\mathbf{3}})+(\mathbf{78},\mathbf{1})\,.$
(2.12)

we observe that $\mathbf{27}$ and $\mathbf{\overline{27}}$ come with a
$\mathbf{3}$ and $\bar{\mathbf{3}}$ representations of the $SU(3)$ tangent
bundle $T_{X}$ of the CY $X$. Hence, one can count the $\mathbf{27}$,
$\mathbf{\overline{27}}$ representations555Recall that the
$\overline{\mathbf{3}}$ is the double antisymmetric of the fundamental of
$SU(3)$. as

$\displaystyle\mathbf{27}:\text{dim}(H^{1}(X,T_{X}))=h^{(1,1)}(X)\,,\quad\overline{\mathbf{27}}:\text{dim}(H^{1}(X,\wedge^{2}T_{X}))=h^{(1,2)}(X)\,,$
(2.13)

which are thus the Hodge number of the threefold. These target space fields
also correspond to the (a,c) and (c,c) chiral primaries of the heterotic
$(2,2)$ world sheet theory. Such operators are preserved in different phases
of the target space geometry, as explicitly given via the GLSM construction
Witten:1993yc .

Hence if we can compute the (anti-)fundamentals of $E_{6}$ of the heterotic
target space theory in the standard embedding, we know the Hodge numbers also
in the smooth phase. Luckily, we have such a method as the heterotic world
sheet theory is a free CFT at the orbifold point and the spectrum can thus be
computed exactly. This has conveniently been implemented in the C++ Orbifolder
Nilles_2012 . This information also allows to deduce the amount of twisted and
untwisted moduli in terms of string states that are, respectively, located at
the fixed points and freely propagating bulk states. These states correspond
to the deformations of the fixed points and the parent tori respectively.

Figure 2: This depicts the location of the fixed points (pink) of the orbifold
$({\mathbb{T}}^{2}\times{\mathbb{T}}^{2})/{\mathbb{Z}}_{3}$. The lattice
$\Lambda=SU(3)\times SU(3)$ is depicted in terms of its generating vectors
(black) and the fundamental domain of the orbifold is marked gray.

For all the aforementioned Abelian orbifolds, this information is summarized
in Appendix C.1 of Fischer_2013 .

We exemplify those considerations with a two- and three-fold example.

#### Geometry of ${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}$:

Orbifolds of type ${\mathbb{T}}^{2n}/{\mathbb{Z}}_{3}$ have been extensively
studied in the string theory literature since DIXON1985678 . As all four
dimensional non-trivial Calabi-Yau orbifolds are K3 surfaces, this example is
a singular limit of an (elliptic) K3 as e.g. considered by Sen in Sen:1996vd
and Dasgupta_1996 .

To be explicit, this geometry has one point group generator of the form

$\displaystyle\begin{array}[]{cccc|ccccc}\vartheta:&&\left(\right.1/3&-1/3&\;0&0&0&0\left.\right)\end{array}\,.$
(2.15)

This acts on a lattice $\Lambda=SU(3)\times SU(3)$ which is the only lattice
invariant under a ${\mathbb{Z}}_{3}$ action. The action of this element
induces three fixed locations in each of the two tori giving rise to in total
nine fixed points. This structure is depicted in Figure 2. The order of these
fixed points is three, requiring two blow-ups at each fixed point. The
resolved geometry is depicted in Figure 3 where the black circle denotes the
base $\mathbb{P}^{1}$, the dark blue circle the generic F-theory fiber and the
light blue circles the blown up singularities. There we find over each of the
three base loci an $E_{6}$ type fiber which required $3\times 6=18$ resolution
divisors. This geometry is in fact an extremal K3 surface, which can be seen
as follows: The Picard number $\rho$ counts the subpart of $h^{(1,1)}(K3)$
that is realized as divisors modulo algebraic equivalence. Extremal K3s are
rigid, i.e. they have no other deformations but the Kähler moduli associated
to divisors. Since $h^{(1,1)}(K3)=20$, it then follows that $\rho=20$. This is
exactly what we have in our case since $\rho=2+18=20$, where the $2$ is the
contribution of fiber and base.

We can double check this observation from the (4,4) heterotic orbifold CFT
perspective as analyzed in Honecker:2006qz . The standard embedding requires
us to identify the $SU(2)$ gauge group inside one $E_{8}$ with the spin
connection. This breaks the heterotic six dimensional target space gauge group
as $E_{8}\rightarrow E_{7}$. Each $\mathbf{56}$ half-hypermultiplet in 6D
represents a deformation in the Picard group of the K3. For this geometry
there are two half-hypers in the bulk, and two $\mathbf{56}$ at each of the
nine fixed points Honecker:2006qz . In total we have the twisted and untwisted
Picard numbers

$\displaystyle(\rho_{\text{untw}},\rho_{\text{tw}})=(2,18)\,,$ (2.16)

which shows that we have correctly identified the singularities. A notable
benefit of extremal elliptic K3s is, that they are classified in shimada2001
which we will also use in order to discuss the F-theory physics in Section
2.3.1.

Figure 3: Depiction of the resolution of the orbifold singularities in
$({\mathbb{T}}^{2}\times{\mathbb{T}}^{2})/{\mathbb{Z}}_{3}$. Light blue
circles depict the ${\mathbb{Z}}_{3}$ resolution divisors. The ambient tori
over fiber and base also become $\mathbb{P}^{1}$’s depicted as blue and black
circles.

#### Geometry of ${\mathbb{T}}^{6}/{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$

Turning to six dimensional STOs, there exists the option to have not only one
point group factor, but two (and accepting non-Abelian orbifolds also three).
As an example, take the orbifold of
${\mathbb{T}}^{6}={\mathbb{R}}^{6}/\Lambda_{SU(3)^{3}}$ modulo two
${\mathbb{Z}}_{3}$ quotients. The generators $\vartheta$ and $\omega$ of the
two quotients can be chosen to act according to

$\displaystyle\begin{array}[]{ccccc|ccccccc}\vartheta:&&\left(\right.\;\;0&1/3&-1/3&\;0&0&0&0&0&0\left.\right)&\\\
\omega:&&\left(\right.1/3&0&-1/3&\;0&0&0&0&0&0\left.\right)&.\end{array}$
(2.19)

Figure 4: This depicts the location of the fixed points (pink) of the orbifold
$({\mathbb{T}}^{2}\times{\mathbb{T}}^{2}\times{\mathbb{T}}^{2})/{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$
grouped in sectors according to the group element that fixes them. The lattice
$\Lambda=SU(3)\times SU(3)\times SU(3)$ is depicted in terms of its generating
vectors (black) and the fundamental domain of the orbifold is marked gray.

The generated inequivalent group elements induce fixed points, that are
depicted in Figure 4. The fixed points may be grouped according to the element
that fixes them. There are 27 fixed points associated to the group element
$\vartheta\omega$ that are localized in all three tori (i.e. they are co-
dimension three). In contrast to these, the remaining inequivalent group
elements all have fixed planes, that is its action leaves an entire torus
fixed. The fixed points associated to these group elements are referred to as
co-dimension two. Analyzing these sectors separately, one realizes that each
of them resembles the orbifold
${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}\times{\mathbb{T}}^{2}$, where the first
factor is the four dimensional orbifold discussed in the previous chapter. The
second factor is either the first, second or third torus of
${\mathbb{T}}^{6}={\mathbb{T}}^{2}\times{\mathbb{T}}^{2}\times{\mathbb{T}}^{2}$
for each of the three group elements. Note, that the fixed points in these
three sectors mutually intersect in the co-dimension three fixed points
associated to $\vartheta\omega$ discussed above.

Regarding the Hodge numbers, one finds no contributions to $h^{(2,1)}$.
Furthermore, there are three untwisted states contributing to $h^{(1,1)}$. The
$27$ fixed points in the sector associated to $\vartheta\omega$ of co-
dimension three contribute each one state twisted state to $h^{(1,1)}$. Due to
the states associated to the remaining sectors one finds Fischer_2013

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,78)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(0,0)\,.$
(2.20)

###  2.2 F-theory and SCFTs

In this section we summarize the basic concepts of F-theory, and how this
framework allows to construct 6D superconformal field theories (SCFTs). A more
thorough summary can be found in one of the many reviews in the literature
(see e.g. weigand2018tasi ; Heckman_2019 ). F-theory means that we enhance the
IIB compactification space $B_{n}$ by an auxiliary torus, which we refer to as
the F-theory torus in the following, whose complex structure describes the IIB
axio-dilaton coupling $\tau=C_{0}+ig_{\text{IIB}}$ modulo $SL(2,\mathbb{Z})$,
which is a symmetry of the type IIB theory. This geometrization allows $\tau$
to vary from point to point over the compactification space, as described by a
holomorphic fibration. The total space $X_{n+1}$ hence obtains the structure

$\displaystyle\begin{array}[]{cc}{\mathbb{T}}^{2}\rightarrow&X_{n+1}\\\
&\downarrow\pi\\\ &B_{n}\end{array}\,.$ (2.24)

The power of F-theory thus lies in the possibility tp olet the axio-dilaton
vary, which allows to explore strong coupling regimes.

When compactified on a circle, F-theory is dual to M-theory on $X_{n+1}$,
where the volume of the elliptic fiber becomes physical. In order to preserve
minimal SUSY of the M-theory compactification, $X_{n+1}$ is fixed to be
Calabi-Yau. This perspective gives a dual definition of F-theory, by thinking
of it as M-theory on the CY $(n+1)$-fold upon decompactifying the elliptic
fiber. This process we call the F-theory lift with respect to a specified
torus fiber of a given $(n+1)$-fold. The power of the construction is, that we
are able to keep track and read off the IIB axio-dilaton $\tau$ in a SUSY
preserving way, for cases when it is neither constant nor fixed to small
couplings. This is important when adding D7 branes, that act as defects that
source $SL(2,{\mathbb{Z}})$ valued monodromies for the axio-dilaton $\tau$.
While perturbative D7 branes shift $\tau$ by the $T$ generator of
$SL(2,{\mathbb{Z}})$, this generalization also allows for strong coupling
branes that shift $\tau$ by the $S$ generator.

When moving onto the locus of a D7 brane, the axio-dilaton $\tau$ diverges and
in turn the F-theory ${\mathbb{T}}^{2}$ becomes singular. These singularities
admit an ADE classification which, in type IIB language, describes the D7
worldvolume gauge algebra $G$. This connection is beautifully reproduced in
geometry, where the intersection matrix of the resolved fiber is precisely the
(affine) ADE Cartan matrix. In 8D, such singularities are schematically of
form

$\displaystyle
Y_{2}=(\mathbb{T}^{2}\times\mathbb{C})/\Gamma_{n}\,,\qquad\Gamma_{n}\subset
SU(2)\,\quad\text{ with }(\Gamma_{n})^{n}=1\,.$ (2.25)

E.g. for $\Gamma_{3}=(e^{2\pi i/3},e^{-2\pi i/3})$ this yields three order
three fixed points in the ${\mathbb{T}}^{2}$ direction and the origin of the
$\mathbb{C}$ factor. Each are resolved by two $(-2)$ curves in
$\mathbb{T}^{2}$, leading to an affine $E_{6}$ over the origin of
$\mathbb{C}$.

This geometric correspondence guarantees the very same gauge algebra to appear
in M-theory, when compactified on the same (singular) geometry, as it is dual
to F-theory reduced on a circle. In M-theory the torus-fiber and its
resolution is physical and the volume of the resolution curves are given by
Kähler moduli. Expanding the M-theory 3-form along those $(1,1)$-forms
associated to Kähler moduli, contributes vector multiplets that yield the
$U(1)^{\text{rk}(G)}$ Cartan algebra. At the singular point this Cartan
subalgebra is enhanced to the full non-Abelian group, that originates from M2
branes that wrap the collapsing fibral divisors.

####  2.2.1 6D anomaly cancellation

Any consistent quantum theory must be free of gauge and gravity anomalies,
which can be achieved via the GSSW mechanism Green:1984bx ; Sagnotti:1992qw .
In the following we are mostly interested in 6D $\mathcal{N}=(1,0)$
supergravity theories, where the anomaly cancellation conditions are
particularly tight. In the following, we briefly review those constraints, and
refer to taylor2011tasi  or Kumar:2010ru ; Johnson_2016  for more details. We
will then discuss how these conditions are encoded in the geometry of the
compactification.

Any supergravity theory must be free of the irreducible gravitational anomaly,
which translates to the condition

$\displaystyle H-V+29T=273\,,$ (2.26)

where $H,V$ and $T$ are the numbers of 6D massless hyper, vector and tensor
multiplets, respectively. Then there are the reducible gauge and gravitational
anomaly cancellation conditions

$\displaystyle a\cdot a$ $\displaystyle=9-T$ (2.27) $\displaystyle a\cdot b$
$\displaystyle=-\frac{\lambda}{6}\left(A_{adj}-\sum_{\mathbf{R}}n[\mathbf{R}]A_{\mathbf{R}}\right)$
(2.28) $\displaystyle b\cdot b$
$\displaystyle=-\frac{\lambda^{2}}{3}\left(C_{adj}-\sum_{\mathbf{R}}n[\mathbf{R}]C_{\mathbf{R}}\right)$
(2.29) $\displaystyle 0$
$\displaystyle=B_{adj}-\sum_{\mathbf{R}}n[\mathbf{R}]B_{\mathbf{R}}\;,$ (2.30)

where the right hand side are the anomalies, which should be cancelled by a GS
contribution on the left.666Additional sets of conditions arise when $U(1)$
gauge factors are present. We leave these out for brevity, and refer the
interested reader to e.g. Appendix A of Klevers:2014bqa . On the right hand
side, $n[\mathbf{R}]$ is the number of hypermultiplets in a representation
$\mathbf{R}$ in the theory, and $A_{\mathbf{R}},B_{\mathbf{R}},C_{\mathbf{R}}$
are Casimir coefficients of representations $\mathbf{R}$ of gauge algebras777
In this work we do not distinguish gauge groups and algebras. If those
structures do no coincide, we remark so explicitly.  $G$. On the left hand
side, the coefficients $a$ and $b$ are $SO(1,T)$ vectors, and $\cdot$ denotes
the scalar product with the $SO(1,T)$ invariant pairing $\Omega_{ij}$. This
tensor determines the GS axion couplings as well as the coupling of BPS
strings to the tensor multiplets in the theory with the condition to be uni-
modular Seiberg:2011dr , that is $\text{Det}(\Omega)=-1$.

All field theory quantities above can be matched to the geometric quantities
of F-theory on $X_{3}$. In particular the pairing $\Omega$ can be expressed as
the intersection pairing on the base $B_{2}$

$\displaystyle\Omega_{ij}=\omega_{i}\cdot\omega_{j}$ (2.31)

with $\omega\in H^{2}(B_{2},\mathbb{Z})$ being an integral basis of 2-cycles.
Further, $b$ and $a$ correspond to divisors $\mathcal{Z}$ of the codimension
one singularities and the anticanonical class $c_{1}(B_{2})$ of the base
respectively. The $h^{(1,1)}(B_{2})=1+T$ dimensional base cohomology lifts to
the vertical divisor part of the the threefold $X_{3}$. The Shioda-Tate-Wazir
formula tells us then, that all other Kähler moduli of the threefold $X_{3}$
are of fibral type or sections. Thus, they count ADE resolution divisors and
generators of a free Mordell-Weil (MW) group, respectively. It is by now well
established that the former class lifts to ADE gauge algebras and the second
to $U(1)$ factors F-theory. Summarizing we can write the Kähler moduli as

$\displaystyle h^{(1,1)}(X)=2+\text{rk}(G)+T\,,$ (2.32)

which is required in order to match with the gauge algebra in M-theory on
$X_{3}$.

From the M-theory dual, we can also deduce that the Cartan generators of the
ADE singularities get enhanced to the full adjoint valued gauge algebra that
support $V=\text{dim}(ADE)$ vectors in 6D by M2 branes that wrap the
respective collapsed curve. Similarly, additional hypermultiplets in the
adjoint representation appear, if the D7 brane wraps a curve with genus $g>0$
Witten:1996qb . Finally, we also need to know the number $h^{(2,1)}(X_{3})$ of
complex structure moduli. These contribute uncharged neutral hypermultiplets

$\displaystyle H_{\mathbf{1}}=1+h^{(2,1)}(X_{3})\,,$ (2.33)

that are important to cancel the gravitational anomaly (2.26). In addition
there are massless charged hypermultiplets $H_{\mathbf{R}}$ that carry
representations $\mathbf{R}$ under the gauge algebra $G$. From the
F-theory/IIB perspective these originate from intersections of D7 brane
stacks, i.e. codimension two loci, where bifundamental matter resides. However
in many cases such a perturbative matter interpretation is not applicable.
This is the case, whenever the brane collisions leads to non-minimal
singularity in the F-theory torus, which can only be avoided upon one or
multiple blow-up of the respective collision points in the base. Such
configurations are called superconformal matter DelZotto:2014hpa  and their
occurrence highlights the presence of tensionless strings in the theory.

####  2.2.2 SCFTs as orbifolds

The F-theory construction gives a convenient way to geometrically engineer 6D
supergravity, LSTs Ganor:1996mu  and SCFTs Heckman_2015 ; Heckman_2019 . The
latter ones are characterized via their base in terms of two-dimensional
orbifolds of the type

$\displaystyle B_{2,\text{SCFT}}=\mathbb{C}^{2}/\Gamma_{B}\,,\qquad\text{ with
}\quad\Gamma_{B}\subset U(2)\,.$ (2.34)

This base is not required to be Calabi-Yau itself but only the elliptic total
space. When adding the elliptic fiber, one can uplift the quotient into a
discrete subgroup of $SU(3)$ e.g as

$\displaystyle
X_{3}=(\mathbb{C}^{2}\times\mathbb{T}^{2})/\Gamma\,,\quad\text{with}\quad\Gamma=(\Gamma_{B},-\det(\Gamma_{B}))\subset
SU(3)\,,$ (2.35)

such that $X$ is an genus-one fibered threefold. A simple example is that of
$\Gamma_{B}=(e^{2\pi i/3},e^{2\pi i/3})$. Such a quotient admits a single
minimal resolution in the base performed via a $(-3)$ curve at the origin
10.1007/BFb0097582 ; Mumford:1961vq ; Morrison:2012np . Putting a Weierstrass
model over the geometry, one finds a minimal singularity of type $IV$, that is
an $SU(3)$ in the F-theory context. This can also be seen, from the quotient
perspective, by enhancing $\Gamma_{B}$ to $\Gamma_{3}=(e^{2\pi i/3},e^{2\pi
i/3},e^{2\pi i/3})$ which yields the singular local threefold

$\displaystyle X_{3}=(\mathbb{C}^{2}\times\mathbb{T}^{2})/\Gamma_{3}\,,$
(2.36)

which admits three fixed codimension three fixed points in the
$\mathbb{T}^{2}$ direction. Each fixed point can be resolved via a single
resolution divisor, which yields the desired $SU(3)$ fiber over the $(-3)$
curve, which we depict as

$\displaystyle\overset{SU(3)}{3}\,.$ (2.37)

Several other configurations exist, with non-trivial actions on the non-
compact $\mathbb{C}$ directions, in particular when
$\Gamma=\mathbb{Z}_{n}\times\mathbb{Z}_{m}$ that have been explored e.g. in
DelZotto:2015rca . E.g. for $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ with the two
generators written as

$\displaystyle\begin{array}[]{ccccc|ccccccc}\vartheta:&&\left(\right.\;\;0&1/3&-1/3&\;0&0&0&0&0&0\left.\right)&\\\
\omega:&&\left(\right.1/3&0&-1/3&\;0&0&0&0&0&0\left.\right)&.\end{array}$
(2.40)

can be thought of two local $\mathbb{Z}_{3}$, quotients, that lead to two
$E_{6}$ resolutions, as described in the section before, that intersect at the
origin of the F-theory base. This configuration is that of $(E_{6},E_{6})$
conformal matter, with tensor branch

$\displaystyle[E_{6}]\;1\overset{SU(3)}{3}1\;[E_{6}]\,.$ (2.41)

This configuration can be anticipated by noting that there are three more
twisted sectors given by the combinations $\vartheta\omega$ and
$\vartheta^{2}\omega$.888 $\vartheta\omega^{2}$ is simply the inverse of the
second sector and admits no new independent fixed points. The former one
yields the $(-3)$ sector with the $SU(3)$ fiber over it while the other is
empty and just yields the two $(-1)$ curves with no fiber enhancement. We will
encounter such configurations in various combinations throughout the examples
in the next sections.

###  2.3 F-theory on orbifolds

The discussion in the previous section now allows us to analyse F-theory
compactifications on the geometries introduced in Section 2.1.1. We start with
the geometry ${\mathbb{T}}^{6}$ and decompose it into three two-tori. If the
orbifold action respects this decomposition there are a priori three different
torus-fibration structures. From the M-theory perspective, each torus allows a
lift to F-theory. Note however, that not all of those lifts might be
inequivalent. This happens, if there is a permutation symmetry of the orbifold
torus actions. Examples are given by six dimensional STOs with trivial affine
class and orbifold generators of the same order. Turning to STOs that exhibit
two generators of different order and/or non-trivial affine class destroys
this symmetry and lead to inequivalent F-theories.

In more in general however not all STOs in the classification of Fischer_2013
can serve as an F-theory background due to the lack of at least a single
torus-fibration structure. The authors of Ref. Fischer_2013  dubbed this
property factorizability. Factorizability can in general be obstructed by non-
diagonal action of both the lattice as well as the point group. As in this
work Abelian point groups are considered, only a lattice action which mixes
the three complexified (torus-)coordinates can prevent F-theory lifts.

####  2.3.1 F-theory on ${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}$

As our first example we consider F-theory on
${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}$. The geometry and in particular its fixed
points structure has been discussed in Section 2.1.3. Compactifying F-theory
on this four dimensional geometry yields an 8-dimensional supergravity. In
fact this theory admits two equivalent F-theory lifts when taking either of
the two ambient tori as the F-theory fiber. In that picture, the F-theory base
is ${\mathbb{T}}^{2}/{\mathbb{Z}}_{3}$, which is topologically a
$\mathbb{P}^{1}$ with three marked points, where the F-theory torus is
singular. From the resolution Figure 3 we already know that there is an
(affine) $E_{6}$ fiber sitting at each of the three base loci, leading to a
$E_{6}^{3}$ gauge algebra. This structure in fact might be viewed as a torus
compactification of IIB strings with a ${\mathbb{Z}}_{3}$ orientifold
Dasgupta_1996 . An important observation made there is that the IIB axio-
dilaton is fixed to a large value, since the F-theory torus lattice is of
$SU(3)$ algebra type which fixes $\tau=e^{2\pi i/3}$.

As noted before, this geometry is an extremal K3 which in fact is also unique.
Thus, there is only one way to construct this geometry, which we do via the
following Weierstrass model. In terms of the base $\mathbb{P}^{1}$ coordinates
$u_{0},u_{1}$ this model is given by the Weierstrass equation

$\displaystyle Y_{2}:\qquad
y^{2}=x^{3}+yz^{3}\left(u_{0}u_{1}(u_{0}+u_{1})\right)^{2}\,,$ (2.42)

with Weierstrass functions

$\displaystyle(f,g,\Delta)=(0,\frac{u_{0}^{4}u_{1}^{4}(u_{0}+u_{1})^{4}}{4},\frac{27u_{0}^{8}u_{1}^{8}(u_{0}+u_{1})^{8}}{16})\,.$
(2.43)

The Kodaira classification tells us, that there are three type $IV^{*}$
singularities at $u_{0}=0,u_{1}=0$ and $u_{0}=-u_{1}$, which each support an
$E_{6}$ singularity. Remarkably, the above geometry admits an enhanced
Mordell-Weil group of order three. This group is generated by the zero-section
at $s_{0}:(x,y,z)=(1,1,0)$ and the additional section at
$s_{1}:(x,y,z)=(0,0,1)$, which is a point of order three Aspinwall:1998xj . As
this model is unique, there is no other way of setting up an elliptic K3 with
three $E_{6}$ fibers. This MW factor leads to a gauged center
${\mathbb{Z}}_{3}$ 1-form symmetryAspinwall:1998xj ; Mayrhofer:2014opa ;
Gaiotto:2014kfa , which gives a $E_{6}^{3}/{\mathbb{Z}}_{3}$ in the 8D
supergravity theory.999Those gaugings can in general lead to 8D anomalies,
which are absent in cases they are realized by MW torsion of the elliptic K3
Cvetic:2020kuw  as similarly seen in the dual heterotic Narrain
compactification Font:2021uyw ; Fraiman:2021soq .

####  2.3.2 F-theory on
${\mathbb{T}}^{6}/{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$

We move on to exemplify F-theory on a six dimensional toroidal orbifold,
reviewing the results of hayashi2019scfts . We do so on a
${\mathbb{T}}^{6}/{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ geometry which we
discussed in detail in Section 2.1. As prepared around equation (2.19), this
geometry has three fixed point sectors of co-dimension two, that are of type
${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}\times{\mathbb{T}}^{2}$, as well as one
sector that exhibits co-dimension three fixed points.

We choose the third torus101010Note that the geometry admits a permutation
symmetry among all three tori, thus leading to equivalent F-theory lifts. as
F-theory fibre for concreteness, and torus one and two as the base part. Then
the fixed points associated to the constructing elements $\vartheta$ and
$\omega$ are co-dimension one singularities of the base which are the loci of
D7 branes in the IIB picture. The discussion here is analogous to the
${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}$ orbifold, where the gauge group factors
are $E_{6}$ for each base fixed point. Furthermore, one observes that these
singularities intersect mutually in nine points. These points are the fixed
points in the sectors associated to the $\vartheta\omega$ and
$\vartheta\omega^{2}$ element. As discussed in Section 2.2, these yields SCFT
sectors namely the $(E_{6},E_{6})$ conformal matter whose tensor branch we
recall as

$\displaystyle[E_{6}]\;1\overset{SU(3)}{3}1\;[E_{6}]\,.$ (2.44)

The above superconformal matter is exactly what we expect for each of the 9
$(E_{6},E_{6})$ collisions, that we summarized in in Figure 5.

Figure 5: Depiction of the intersection structure of the six $E_{6}$ gauge
algebra factors in the
${\mathbb{T}}^{6}/{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$ orbifold. At the
nine intersection points, there is $(E_{6},E_{6})$ conformal matter,
illustrated as a circle.

We summarize the non-Abelian gauge algebra and the full massless spectrum of
this compactification (on the tensor branch) as

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&E_{6}^{6}\times SU(3)^{9}\\\
\hline\cr H&1\\\ \hline\cr V&6\times\mathbf{78}+9\times\mathbf{8}\\\ \hline\cr
T&27+1\\\ \hline\cr\end{array}\,.$ (2.49)

Note that the spectrum matches the stringy Kähler moduli, computed via the
heterotic theory: The three untwisted Kähler moduli translate into volume
moduli for the fiber and base plus one additional tensor. The 81 twisted
Kähler moduli are distributed as the $36$ $E_{6}^{6}$ Cartan generators, 18
from those of $SU(3)^{9}$ and finally the $27$ tensors of the conformal matter
points.

This theory satisfies all 6D supergravity and gauge anomaly cancellation
conditions of Section 2.2.1 which we discuss next (more details can be found
in hayashi2019scfts  and thesis ). Here we only state the results: first, the
irreducible gravitational anomaly cancels provided the data above. Second, the
reducible gravitational anomaly as well as mixed anomalies can be cancelled
for which we first give the reduced intersection matrix

$\displaystyle\Omega=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\oplus\left[\left(\begin{array}[]{ccc}-1&1&0\\\ 1&-3&1\\\
0&1&-1\end{array}\right)\right]^{\oplus 9}\,,$ (2.55)

where the curves supporting the $E_{6}$ gauge group factors are in the classes
of the first two components of $\Omega$, e.g.

$\displaystyle b_{(E_{6})_{1}}=(1,0,\left\\{-2,-1,-1\right\\}^{\oplus
3},\\{0\\}^{\oplus(3\times 6)})\,.$ (2.56)

The remaining nine factors correspond to the $(E_{6},E_{6})$ conformal matter
(see equation (2.44)), e.g. the $(-3)$-curve in the first factor

$\displaystyle b_{112}^{(-3)}=(0,0,\\{0,1,0\\},\\{0\\}^{\oplus(3\times 8)})$
(2.57)

Note, that this intersection pairing is unimodular and its signature agrees
with $(+,-)=(1,T)=(1,28)$ as required.

The components of the canonical class can be found by adjunction and requiring
the correct genera of the appearing curves. This results in hayashi2019scfts
in the Ansatz

$\displaystyle
a_{i}=\sum\nolimits_{j}(\Omega^{-1})_{ij}(2g_{j}-2-(C_{j})^{2})\,,$ (2.58)

where $g_{j}$ and $(C_{j})^{2}$ denote the genus and self-intersection,
respectively, of the $j$’th curve. This evaluates here to

$\displaystyle a=(-2,-2,\left\\{2,1,2\right\\}^{\oplus 9})\,.$ (2.59)

Using these representations as well as the intersection pairing (2.55) one can
show the vanishing of the anomalies reviewed in Section 2.2.1 and thereby
verify the consistency of this 6D theory.

Finally we also remark, that the above geometry is as also consistent with a
global $\mathbb{Z}_{3}$ torsion factor that acts diagonally on the various
group factors. This is in fact also compatible with the
$(E_{6},E_{6})/{\mathbb{Z}}_{3}$ non-simply connected conformal matter
Dierigl:2020myk . This suggests that the actual global gauge group should
rather be $(E_{6}^{6}\times SU(3)^{9})/\mathbb{Z}_{3}$.

Figure 6: Depiction of the affine Dynkin diagram of $E_{6}$. The nodes are
associates to the roots $\alpha_{i}$ of $E_{6}$ and their Kac-labels. The node
associtated to $\alpha_{0}$ is the affine node.

Before moving on to the main part we want to comment on the unusual fiber
structure that appears in the ${\mathbb{T}}^{4}/{\mathbb{Z}}_{3}$,
${\mathbb{T}}^{6}/{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ but also all other
orbifold geometries. The main remark is, that the generic torus fiber for all
orbifolds represents inner $\mathbb{P}^{1}$’s in the affine Dynkin diagram. In
the examples here, the generic torus fiber is the unique root with Kac label
3, see Figure 6. This is at odds with degenerate elliptic fibrations, where
the generic fiber is indeed given by the affine node with Kac label 1,
consistent with the fact that they are intersected with the zero-section. In
this way the presence of a zero-section marks one of these outer nodes as the
affine one. However, in the orbifolds there is a ${\mathbb{Z}}_{3}$
permutation symmetry among the three fibers and hence none of them can be
distinguished from the others.

Therefore, we have only two options: First we might have no section, in which
case we expect a genus-one fibration with three-sections only. The expectation
for F-theory on genus-one fibrations on the other hand is, that they encode a
discrete ${\mathbb{Z}}_{n}$ gauge symmetry essentially given by the order of
the $n$-sections Braun:2014oya ; Morrison:2014era ; Anderson:2014yva ;
Klevers:2014bqa ; Mayrhofer:2014haa ; Mayrhofer:2014laa ; Cvetic:2015moa ;
Cvetic:2018bni  and also the review Cvetic:2018bni .

In the case at hand, we would expect a ${\mathbb{Z}}_{3}$ gauge symmetry. This
explanation however, is at odds with the fact that there is no massless matter
which could be charged under the discrete group. Moreover, there are no
complex structure moduli that would allow us to un-Higgs the
${\mathbb{Z}}_{3}$ to a $U(1)$ symmetry which is in conflict with common
quantum gravity lore. The second option is, that we must have three sections
that are permuted by the ${\mathbb{Z}}_{3}$ symmetry. Indeed, we argued
already that this model should have a ${\mathbb{Z}}_{3}$ MW torsion, which
modifies the global gauge group which therefore seems to be the correct
interpretation. In various other examples, that we are going to discuss in the
next sections, there will be further geometric features that rather point the
absence of a section and hence some ${\mathbb{Z}}_{n}$ discrete gauge symmetry
in the F-theory supergravity theory.

##  3 F-theory on symmetric toroidal orbifolds

Having established the geometries of symmetric toroidal orbifolds (STOs) and
the framework of F-theory in the last section, we want to present the main
results of our analysis. We start by giving an overview over the STOs, and
their F-theory lifts, that we have considered. Many of those geometries lead
to new features that have not been present in orbifold compactifications of
F-theory in the literature Morrison:1996na ; Dasgupta_1996 ; Gopakumar1996 ;
hayashi2019scfts . Finally, this section serves to give a detailed discussion
of three example geometries that highlight those features.

###  3.1 Summary of results

####  3.1.1 Orbifold statistics

The classification of STOs Fischer_2013  comprises orbifolds in 60 different
${\mathbb{Q}}$–classes, that have a holonomy group which is contained in
$SU(3)$. Of these, 22 are Abelian and 38 non-Abelian. As already reviewed in
Section 2.1.2, these ${\mathbb{Q}}$-classes can be subdivided in
${\mathbb{Z}}$\- and affine classes, which are summarized in Table 1 for the
Abelian subset. In total there are 162 affine classes of orbifolds with
Abelian point group in the mentioned ${\mathbb{Q}}$-classes, whereof 138 have
a holonomy group which is contained in $SU(3)$ but not in $SU(2)$.

As explained in Section 2.3 not all of these geometries give sensible F-theory
vacua, as the fibration structure of F-theory compactifications requires the
factorizability of “the F-theory torus”. Taking into account the requirement
that the group action111111As noted in Section 2.3, the action of Abelian
point groups always factorizes all three tori, as this defines them. In this
case only a non-factorizable lattice action can obstruct F-theory lifts. needs
to factorize, of the naively $3\times 138=414$ models only $125$, that is
$30.2\text{\,}\mathrm{\char 37\relax}$, yield sensible F-theory
compactifications. Note, that not all of these F-theory models are actually
different. As encountered in Section 2.3, different choices of the F-theory
torus, that is different lifts, might give the same model, if the orbifold is
symmetric under exchange of these tori.

In this paper, we have considered 16 different STO geometries in various
${\mathbb{Q}}$-classes, but trivial ${\mathbb{Z}}$-class and their $26$
inequivalent F-theory lifts as summarized in Table 2. In there, we make use of
the nomenclature and specification of the STO geometries and computation of
stringy Hodge numbers as given in Fischer_2013 . While this is a small
fraction of the 125 orbifold geometries that may be used in F-theory
compactifications, it allows us to initiate a study of interesting geometric
features. In particular, we explore the effect of non-trivial affine classes
on the compactification of F-theory. This uncovers new features, described in
more detail in the next sections, that hints at a rich structure of F-theory
models beyond what is already established in the literature. Let us emphasise
again that we only take the first step in this exploration: to determine the
consistent F-theory model for the full set of STOs, requires a more systematic
approach than we undertake here.

$\\!{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ |  $\\!\\!h^{(1,1)},h^{(2,1)}$ |  lift |  $G$ | $T$ |  $T_{(2,0)}$ |  $H_{c}$
---|---|---|---|---|---|---
$-1-1$ |  $51,3$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{8}SO(8)_{i}$ | $17$ |  - |  -
$-1-2$ |  $19,19$ |  ${\mathbb{T}}^{2}_{1}$ |  $\prod_{i=1}^{8}SO(5)_{i}$ | $1$ |  $\\!8\times\mathcal{A}_{1}$ |  $\oplus_{i=1}^{8}\mathbf{10}_{i}$
|  |  ${\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{2}SO(8)_{i}$ | $9$ |  - |  $\oplus_{i=1}^{2}\mathbf{28}_{i}$
$-1-3$ |  $11,11$ |  ${\mathbb{T}}^{2}_{1}$ |  $\prod_{i=1}^{4}SO(5)_{i}$ | $1$ |  $\\!8\times\mathcal{A}_{1}$ |  $\oplus_{i=1}^{4}\mathbf{10}_{i}$
|  |  ${\mathbb{T}}^{2}_{2}$ |  Ø | $9$ |  - |  -
|  |  ${\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{2}SO(8)_{i}$ | $1$ |  $\\!8\times\mathcal{A}_{1}$ |  $\oplus_{i=1}^{2}\mathbf{28}_{i}$
$-1-4$ |  $3,3$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  Ø | $1$ |  $\\!8\times\mathcal{A}_{1}$ |  -
$\\!{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{4}$ |  |  |  |  |  |
$-1-1$ |  $61,1$ |  ${\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{2}(E_{7})_{i}\times SO(8)$
$\times\prod_{j=1}^{4}[(SO(7))\times Sp(1)_{a}\times Sp(1)_{b}]_{j}$ | $21$ |  - |  $\oplus_{j=1}^{4}[1/2(\mathbf{8},\mathbf{2}_{a})\oplus 1/2(\mathbf{8},\mathbf{2}_{b})]_{j}$
|  |  ${\mathbb{T}}^{2}_{2}$ |  $\prod_{i=1}^{10}SO(8)_{i}$ | 19 |  - |  -
$\\!{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{6}$ |  |  |  |  |  |
$-1-1$ |  $51,3$ |  ${\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{3}$ |  $E_{8}\times F_{4}\times SO(8)$
$\times\prod_{i=1}^{4}[G_{2}\times Sp(1)]_{i}$ | 21 |  - |  $\oplus_{i=1}^{4}[1/2(\mathbf{7},\mathbf{2})\oplus 1/2(\mathbf{1},\mathbf{2})]_{i}$
|  |  ${\mathbb{T}}^{2}_{2}$ |  $\prod_{i=1}^{8}SO(8)_{i}$ | $17$ |  - |  -
$-1-2$ |  $31,7$ |  ${\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{2}(F_{4})_{i}\times\prod_{j=1}^{2}(G_{2})_{j}$
$\times\prod_{k=1}^{2}SU(3)_{k}$ | $13$ |  - |  $\mathbf{26}_{1}\oplus_{j=1}^{2}\mathbf{7}_{j}$
$\\!{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ |  |  |  |  |  |
$-1-1$ |  $84,0$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{6}(E_{6})_{i}\times\prod_{j=1}^{6}SU(3)_{j}$ | $28$ |  - |  -
$-1-2$ |  $24,12$ |  ${\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2}$ |  $E_{6}\times\prod_{i=1}^{3}SU(3)_{i}$ | $10$ |  - |  $\mathbf{78}$
|  |  ${\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{6}(G_{2})_{i}$ | $10$ |  - |  $\oplus_{i=1}^{6}\mathbf{7}\oplus 9\times 2\times\mathbf{1}$
$-1-3$ |  $18,6$ |  ${\mathbb{T}}^{2}_{1}$ |  $E_{6}\times\prod_{i=1}^{3}SU(3)_{i}$ | $4$ |  $\\!3\times\mathcal{A}_{2}$ |  $\mathbf{78}$
|  |  ${\mathbb{T}}^{2}_{2}$ |  $\prod_{i=1}^{3}SU(3)_{i}$ | $10$ |  - |  -
|  |  ${\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{3}(G_{2})_{i}$ | $10$ |  - |  $\oplus_{i=1}^{3}\mathbf{7}_{i}$
$-1-4$ |  $12,0$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{3}SU(3)_{i}$ | $4$ |  $\\!3\times\mathcal{A}_{2}$ |  -
$\\!{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{6}$ |  |  |  |  |  |
$-1-1$ |  $73,1$ |  ${\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{3}$ |  $E_{8}\times E_{6}\times G_{2}$
$\times\prod_{i=1}^{3}(F_{4})_{i}$
$\times\prod_{j=1}^{3}SU(3)_{j}$
$\times\prod_{k=1}^{3}[G_{2}\times Sp(1)]_{k}$ | $28$ |  - |  $\oplus_{k=1}^{3}[1/2(\mathbf{7},\mathbf{1})\oplus 1/2(\mathbf{1},\mathbf{2})]_{k}$

$\\!{\mathbb{Z}}_{6}\times{\mathbb{Z}}_{6}$ |  $\\!\\!h^{(1,1)},h^{(2,1)}$ |  lift |  $G$ | $T$ |  $T_{(2,0)}$ |  $H_{c}$
---|---|---|---|---|---|---
$-1-1$ |  $84,0$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{4}(E_{8})_{i}\times\prod_{j=1}^{3}(F_{4})_{j}$
$\times\prod_{k=1}^{6}(G_{2}\times Sp(1))_{k}$
$\times SU(3)$ | $34$ |  - |  $\oplus_{k=1}^{6}[1/2(\mathbf{7},\mathbf{2})\oplus 1/2(\mathbf{1},\mathbf{2})]_{k}$
$\\!{\mathbb{Z}}_{4}\times{\mathbb{Z}}_{4}$ |  |  |  |  |  |
$-1-1$ |  $90,0$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{4}(E_{7})_{i}\times\prod_{j=1}^{2}[SO(7)\times Sp(1)]_{j}$
$\times\prod_{k=1}^{4}[Sp(1)_{a}\times SO(7)\times Sp(1)_{b}]_{k}$ | $30$ |  - |  $\oplus_{j=1}^{2}[1/2(\mathbf{8},\mathbf{2})]_{j}$
$\oplus_{k=1}^{4}\left[1/2(\mathbf{2}_{a},\mathbf{8},\mathbf{1})\right.$
$\left.\oplus 1/2(\mathbf{1},\mathbf{8},\mathbf{2}_{b})\right]_{k}$
$\\!{\mathbb{Z}}_{6}\times{\mathbb{Z}}_{6}$ |  |  |  |  |  |
$-1-1$ |  $84,0$ |  $\\!\\!{\mathbb{T}}^{2}_{1},\,{\mathbb{T}}^{2}_{2},\,{\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{4}(E_{8})_{i}\times\prod_{j=1}^{3}(F_{4})_{j}$
$\times\prod_{k=1}^{6}[G_{2}\times Sp(1)]_{k}$
$\times SU(3)$ | $34$ |  - |  $\oplus_{k=1}^{6}[1/2(\mathbf{8},\mathbf{2})]_{j}$
$\\!{\mathbb{Z}}_{6}-II$ |  |  |  |  |  |
$-1-1$ |  $35,11$ |  ${\mathbb{T}}^{2}_{1}$ |  $\prod_{i=1}^{3}(F_{4})_{i}\times\prod_{j=1}^{4}(G_{2})_{j}$ | $13$ |  - |  $\oplus_{i=1}^{3}\mathbf{26}_{i}\oplus_{j=1}^{4}\mathbf{7}_{j}$
|  |  ${\mathbb{T}}^{2}_{2}$ |  $\prod_{i=1}^{2}(E_{6})_{i}\times\prod_{j=1}^{4}SU(3)_{j}$ | $13$ |  - |  $\mathbf{78}_{1}$
|  |  ${\mathbb{T}}^{2}_{3}$ |  $\prod_{i=1}^{5}SO(8)_{i}$ | $13$ |  - |  $\mathbf{28}_{1}$
Table 2: This table summarizes the massless matter content of F-theory
compactifications of the various STOs. The first two columns show the STO
class and their Hodge-numbers. The four columns to the right provide for the
specified choices of F-theory fibre, the gauge groups of the theory $G$ (Ø
denotes no gauge group), the number of tensor multiplets $T$ as well as the
representations of the charged hyper multiplets at the tensor branch. When
multiple fibers are present we highlight the $(2,0)$ tensors in the sixth
column. The sub index of the representation specifies the factor of $G$ under
which the multiplets are charged, i.e. if $G=G_{1}\times G_{2}$, then
$\mathbf{R}_{1}$ means $(\mathbf{R},\mathbf{1})$. We also do not depict
possible discrete gauge factors.

####  3.1.2 Summary of new F-theory features

The STOs that we are considering throughout admit a couple of interesting
geometric features which translates also into their F-theory physics. In the
following we briefly summarize those which allowed us to compile Table 2.

##### Multiple F-theory lifts and 5D dualities

Most of the STOs have multiple factorizable torus structures. In fact, since
the construction descends from
$\mathbb{T}^{2}\times\mathbb{T}^{2}\times\mathbb{T}^{2}$, we have three
F-theory lifts at our disposal whenever the orbifold action is factorizable.
STOs $X$ with multiple fibration structures are interesting as they yield the
same 5D theory upon circle compactification, where they are dual to M-theory
on $X$. In such theories, the distribution into 6D Vector and tensors is
reshuffled and related theories satisfy

$\displaystyle\text{rank}(G)+T=h^{(1,1)}(X)-2.$ (3.1)

Such multiple fibration structures are a very frequent feature of e.g. smooth
CY geometries. In fact, the three fibration structures we observe, are a
comparably low number, when compared e.g. with classifications in smooth CICY
Anderson:2017aux  or toric hypersurface Huang:2018esr  threefolds. This is due
to the fact that the orbifold describes a CY at a particularly singular point
in its Kähler moduli space. Indeed, we find that generically all but the three
torus Kähler moduli are taken to zero. In the smooth geometry however, one
expects to find many more linear combinations of those divisor classes that
can give a torus fiber, or in more technical terms a Kollar divisor.

##### Identifications in fibre and base singularities

An important new class of geometries are those, that admit additional shifts
accompanied with some point group action, also known as roto-translations. The
first case of such generators are those, where the shift appears in the same
torus as the orbifold. An order three example is given by

$\displaystyle\begin{array}[]{c}\vartheta:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.0&1/3&-1/3&\;0&0&0&0&2/3&1/3\left.\right)\end{array}\,,$
(3.4)

that acts on the local geometry

$\displaystyle
X_{3}=(\mathbb{C}\times\mathbb{T}_{2}^{2}\times\mathbb{T}^{2}_{3})/\vartheta=\mathbb{C}\times(\mathbb{T}^{2}_{2}\times\mathbb{T}^{2}_{3})/\vartheta\,.$
(3.5)

The above geometry admits nine fixed points in the
$\mathbb{T}_{2}^{2}\times\mathbb{T}_{3}^{2}$ directions. Focusing on
$\mathbb{T}^{2}_{3}$, the three fixed points are given as

$\displaystyle z_{f_{1}}=1/3\,e_{5},\quad z_{f_{2}}=1/3\,e_{6},\quad
z_{f_{3}}=2/3\,(e_{5}+e_{6}).$ (3.6)

Note that the three fixed points are at different positions of the torus, due
to the roto-translational element. The relevant insight is when considering
another non-trivial orbifolding element given as

$\displaystyle\begin{array}[]{c}\omega:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.1/3&\;0&-1/3&\;0&0&0&0&1/3&\,\,2/3\left.\right)\end{array}\,.$
(3.9)

This element leads to its own set of fixed point, but, more importantly, it
also conjugates the fixed points of the the $\vartheta$ sector. In particular
it acts as a cyclic translation of fixed points in the $\mathbb{T}^{2}_{3}$
direction as

$\displaystyle\omega(z_{f_{i}})=z_{f_{i+1}}\,.$ (3.10)

Without $\omega$, the resolution of those fixed points in the
${\mathbb{T}}^{2}_{3}$ torus leads to an $E_{6}^{(1)}$ type fiber just as in
the K3 example. The $\omega$ elements however, mods out the ${\mathbb{Z}}_{3}$
cyclic permutation symmetry of the three fixed points, which extends to a
quotient by the outer automorphism of affine $E_{6}^{(1)}$ fiber. The folded
geometry results in a $D_{4}^{(3)}$ twisted affine fiber as depicted in Figure
7.

Identifications of fixed points also appear e.g. in $\mathbb{Z}_{n}$ orbifolds
when $n$ is non-prime and leads to fibers with non-simply laced groups.
However, the free action we have encountered above always acts on (any choice
of) the affine node as well and thus leads to a twisted affine type of fiber.
Such actions though can not respect a zero-section and thus are associated to
genus-one fibrations Braun:2014oya .

$z_{f_{1}}$$z_{f_{2}}$$z_{f_{3}}$$\mathbf{\uparrow}$$\mathbf{\omega}$$\mathbf{\omega}$
Figure 7: A depiction of the three ${\mathbb{Z}}_{3}$ fixed points of
$\vartheta$ and their identification by the element $\omega$. In the left we
show the reduction of the fundamental domain and its action on the singular
$\mathbb{P}^{1}$. On the right we depict the action on the resolved fibers
twisting the $E_{6}^{(1)}$ affine fiber to $D_{4}^{(3)}$.

Following the prescription in Braun:2014oya , such fibers are associated to a
twisted version of $G_{2}$ in the F-theory physics. In total such fixed points
come in three copies due to the fixed points in the ${\mathbb{T}}_{2}^{2}$
direction. Hence in total we identify this as a $G_{2}^{3}$ configuration.
Note that this is locally, like the extremal $K3$ considered in Section 2.1.3
where the $E_{6}^{3}$ is reduced to $G_{2}^{3}$ by the quotient.

This geometry also admits a second fibration in the $\mathbb{T}^{2}_{2}$
direction. Here we find regular affine $E_{6}^{(1)}$ fiber over each of the
fixed points in the $\mathbb{T}^{2}_{3}$ direction, which without the $\omega$
element also leads to a $E_{6}^{3}$ theory. The quotient action reduces this
multiplicity however by three reducing the lift to a single $E_{6}$ theory.
Note that we have considered only a subsector of the theory, which in the full
compactification is given by the ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$
orbifold, discussed in detail in Section 3.2.1.

##### Freely acting generators and $(2,0)$ tensors

The final class of roto-translations that is of relevance for us appears, when
point group rotations and shifts appear in different planes. An example is
given by the following order three space group element

$\displaystyle\begin{array}[]{c}\tilde{\vartheta}:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.0&1/3&-1/3&\;2/3&1/3&0&0&0&0\left.\right)\end{array}\,,$
(3.13)

which acts on the local geometry as

$\displaystyle
X_{3}=(\mathbb{T}^{2}\times\mathbb{C}^{2})/\tilde{\vartheta}\,.$ (3.14)

The important observation to make is, that $\tilde{\vartheta}$ acts freely on
$\mathbb{T}^{2}\times\mathbb{C}^{2}$ and thus the total space $X_{3}$ is
smooth. The F-theory interpretation however is more interesting as the
physical base is of the local form $B_{2}=\mathbb{C}^{2}/\mathbb{Z}_{3}$ and
hence admits a codimension two crepant orbifold singularity at the origin, see
Figure 8. This singularity however is avoided in the total space by the
accompanied torus shift of $\tilde{\vartheta}$.

Fibers of this type have been discussed in the little string theory context
Bhardwaj_2016  and in compact cases in Anderson:2018heq ; Anderson:2019kmx ;
Oehlmann:2019ohh 121212See Gross97  where multiple fibers appear in the
finiteness proof of elliptic threefolds.  and are called multiple fibers as
the torus fiber becomes a multi-copy over the origin of the base. As the
action of $\tilde{\vartheta}$ is free on the fiber, it can not respect the
zero-section. As a consequence, the fibration is no longer elliptic (it does
not have a section), and is only of genus-one type as discussed before. Hence
again, we are led to the conclusion that we should expect a discrete gauge
symmetry of the order of the quotient in F-theory Braun:2014oya . However, in
addition to this gauge group structure there is the important contribution of
the $\mathbb{C}^{2}/{\mathbb{Z}}_{n}$ singularity to the spectrum. Such
isolated singularities locally lead to $\mathcal{A}_{n-1}$ $(2,0)$
superconformal theories DelZotto:2014fia . The non-trivial fiber action gauges
the theory under the ${\mathbb{Z}}_{n}$ gauge symmetry. From the perspective
of anomalies, each such sector contributes like

$\displaystyle\mathcal{A}_{n-1}\quad\sim\quad(1H+1T)\times(n-1)\,,$ (3.15)

that is a neutral hyper and a tensor multiplet. It is important to remark
though, that the actual tensor branch of those theories differs from a simple
$A_{n}$ type theory by the appearance of additional discrete charged singlets
over the tensors Anderson:2018heq ; Anderson:2019kmx .

Figure 8: Illustration of a multiple fiber cause by $\tilde{\vartheta}$. While
$\tilde{\vartheta}$ leads to an $\mathbb{C}^{2}/{\mathbb{Z}}_{n}$ orbifold
base, (pink), the simultaneous translation in $\mathbb{T}^{2}_{F}$ renders the
total space smooth.

For computational purposes however, those sectors contribute exactly the
degrees of freedom of a regular $(2,0)$ theory as discussed above, which is
sufficient to cancel all supergravity anomalies.

##### Discrete higher form symmetries

We close this section by noting that the above geometries generically admit a
rich structure of discrete symmetries as it is typical e.g. in heterotic
string compactifications on such spaces. In particular, as discussed above,
many orbifold geometries are genus-one fibrations, which hints at the
possibility for non-trivial ${\mathbb{Z}}_{n}$ discrete gauged symmetries.
Such symmetries include discrete 0-form symmetries but there are generically
higher form discrete gauge symmetries present as well. In fact, a gauged
center 1-form symmetry generated will typically be generated by a finite MW
group, such as in the examples discussed in Section 3.2 and Appendix A. In the
${\mathbb{T}}^{6}/{\mathbb{Z}}_{3}$ case we even find a free MW group that
leads to (non-Higgsable) $U(1)$ gauge factors. Moreover, we also expect
discrete 2-form gauged symmetries $G$ present. Such a discrete symmetry $G$
appears Braun:2021sex  whenever the sublattice of collapsed divisors
$\Lambda_{S}$ related to codimension two orbifold fix points in the base $B$
is non-primitively embedded into that of the full base $\Lambda_{B}$. As shown
in Braun:2021sex  this can be expressed as the following torsion group, given
as

$\displaystyle G=\text{Tor}\left(\Lambda_{B}/\Lambda_{S}\right)\,.$ (3.16)

Generically, such a torsion element $\Gamma_{B}$ is present, whenever the
singular base can be written as $B_{\text{sing}}=B/\Gamma_{B}$. Physically,
such 2-form symmetries couple to the light self-dual strings that are
supported from D3 branes that wrap the collapsing divisors that span
$\Lambda_{S}$. These light BPS strings exhibit a global 2-form symmetry
$G_{S}=\text{ord}(\Gamma_{B}\cdot k)$ which is generically broken. The
presence of the above torsional $\Gamma_{B}$ element however shows that a
remnant that acts diagonal on the SCFT centers is still present as a gauge
symmetry.

Since the bases we are considering here, can generically be represented as

$\displaystyle B=\mathbb{T}^{4}/\Gamma_{B}\,,$ (3.17)

it follows, that we generically have a non-trivial discrete 2-form symmetry of
order ${\mathbb{Z}}_{\text{ord}(\Gamma_{B})}$, that acts on the respective
light strings. We remark, that such actions are trivial, if the respective
SCFT sectors admit only trivial 2-form symmetries, as in the case of the
superconformal matter, or E-string theories. This is due to the fact, that the
determinant of the intersection pairing of such configurations is equal to
one. For configurations such as the $(E_{6},E_{6})$ conformal matter, as
discussed in Section 2.2.2 this can be readily satisfied. On the other hand,
SCFTs that collapse to orbifold singularities do admit a 2-form symmetry, such
as ${\mathbb{Z}}_{n}$ for non-Higgsable clusters on $(-n)$ curves.

###  3.2 F-theory on ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ orbifolds

In this subsection compactifications of F-theory on several orbifolds are
discussed illustrating the effects listed above. These examples generalize the
ones given in Section 2.3 in different ways. First the different lifts that
can be performed in the orbifold ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$
are analyzed. As this is the first non-trivial affine class of
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ orbifolds on the ordinary $SU(3)^{3}$
lattice, it is the natural way to generalize the
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$ example of Section 2.3 and of
hayashi2019scfts . One can observe some of the effects exclusive to non-
trivial affine classes discussed in the previous subsection here, such as
twisted affine fibres.

As a second example we discuss the maximal (fourth) affine class of
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1$ orbifolds. This orbifold can
equivalently be seen as a freely acting quotient of the single quotient
orbifold ${\mathbb{Z}}_{3}-1-1$. This relation is elaborated on.

####  3.2.1 F-theory on ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$

We discuss now the first minimal generalization of the
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ orbifold by a tranlational lattice
shift into the two orbifold elements. The two generators are now given as
Fischer_2013

$\displaystyle\begin{array}[]{c}\vartheta:\\\
\omega:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.\;\;0&1/3&-1/3&\;0&0&0&0&2/3&\,\,1/3\left.\right)\\\
\left(\right.1/3&0&-1/3&\;0&0&0&0&1/3&\,\,2/3\left.\right)\end{array}\,.$
(3.22)

The induced fixed points are depicted in Figure 9. These yield the following
stringy Hodge numbers which we can compute via the heterotic string

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,21)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(0,12)\,.$
(3.23)

As the action of the orbifolding group on the three tori is not symmetric in
the three tori anymore compared to the trivial affine class, there are two
inequivalent F-theory lifts. These are discussed in the following.

Figure 9: Illustration of the fixed point structure of the orbifold class
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$. The fixed points are divided
into three groups, according to their constructing element ($\vartheta$,
$\omega$,$\vartheta\omega$), and represented by their location in the
respective torus (pink, red, and purple). Arrows between fixed point locations
indicate identification.

##### F-theory lift along ${\mathbb{T}}^{2}_{3}$

Choosing this lift, the fractional lattice shift acts only in the F-theory
torus. Thus the structure of the physical base qualitatively remains the same
as in ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$. There are three fixed tori
in the sector associated to $\vartheta$ mutually intersecting three fixed tori
in the sector associated to $\omega$. The intersections are located at the
fixed points associated to $\vartheta\omega$, which are of co-dimension three
in the orbifold. Examining the fibre over the fixed tori, one realizes, that
the shift identifies the three fixed points, which are of order three. This is
due to the fact that the fundamental domain of the geometry is reduced by a
three-volution. The F-theoretic interpretation of this is a folding of the
affine Dynkin diagram of $E_{6}$ to $G_{2}$ by modding its outer automorphism.

Let us remark that the folding through which these $G_{2}$ arise, is different
than that in Ref. hayashi2019scfts . In their work, the $G_{2}$ factors arise
through an ordinary folding of an affine $D_{4}^{(1)}$ Dynkin diagram, where
three nodes get identified while the affine node stays intact. Here, they
emerge through the affine Dynkin diagram of $E_{6}$ in a folding including the
affine node. Note that, as discussed in Section 3.1.2, such a folding can only
be realized, when roto-translations are present. Hence twisted affine foldings
are an exclusive feature of orbifolds of non-trivial affine classe.

The associated base curves, furthermore, have compared to the orbifold with
trivial affine class, discussed in Section 2.3 reduced self-intersection
$(-3)$. The singularity at intersections of these curves is non-minimal and is
resolved by the blow-up of one $(-1)$-curve in the base, yielding
$(G_{2},G_{2})$ superconformal matter which is just an E-string131313One might
wonder, whether the freely acting element $\vartheta\omega^{2}$ could lead to
contributions of multiple fibers as explained in Section 3.1.2). This is not
the case, as the base fixed points are the same as in the $\vartheta\omega$
sector, which are already accounted for.. To be explicit about anomaly
cancellation, which is reviewed in Section 2.2.1, the irreducible
gravitational anomaly is cancelled

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&(G_{2})^{6}\\\ \hline\cr
H&6\times\mathbf{7}+18\times\mathbf{1}+12+1\\\ \hline\cr
V&6\times\mathbf{14}\\\ \hline\cr T&9+1\\\ \hline\cr\end{array}\,.$ (3.28)

Note that the above identification of gauge group and tensors matches the
$2\cdot 6+10+2=24$ Kähler parameters in equation (3.23). The origin of the
different multiplets is the following. The vectors obviously are given by the
adjoints of the $G_{2}$ gauge algebra. The tensors are geometrically given by
the nine blow-up modes at the intersections $(G_{2},G_{2})$ as well as the
untwisted one from the ${\mathbb{T}}^{2}\times{\mathbb{T}}^{2}$. Furthermore,
there are different contributions to the hyper multiplets. There are twelve
neutral hypers due to the complex structure moduli, the universal neutral
hyper, as well as 54 charged hyper multiplets. $6\times 6=36$ of those are
$\mathbf{7}$-plets of the $G_{2}$ gauge groups having charge dimension six and
cancelling additionally the gauge anomaly on the $(-3)$-curves. The remaining
$18$ hypers are discrete singlets located at the codimension two loci in the
base. Note that we were required to add those additional discrete charged
singlets in order cancel the gravitational anomaly. This is also motivated by
the fact, that we expect a ${\mathbb{Z}}_{3}$ discrete gauge that we expect to
be accompanied by the twisted affine $G_{2}$ fibers as those are only possible
in genus-one fibrations Braun:2014oya .

In order to show the cancellation of the reducible gravitational anomaly it is
convenient to find the intersection matrix of base divisors. A priori it is
given by

$\displaystyle\tilde{\Omega}=(-3)^{\oplus 6}\oplus(-1)^{\oplus 9}\,,$ (3.29)

with appropriate off-diagonal entries denoting intersecting curves. However,
due to the correspondence of the intersection matrix with the charge lattice,
one requires that the signature of $\tilde{\Omega}$ is $(+,-)=(1,T)=(1,10)$,
and hence that its rank must be eleven. Similar as explained in
hayashi2019scfts , one in fact finds zero directions in $\tilde{\Omega}$ and
one can find a intersection pairing with correct signature by the
identification of the three fixed tori in one sector into one class as

$\displaystyle C$ $\displaystyle=C_{i}+\sum\nolimits_{j}C_{ij}\,,$ (3.30)
$\displaystyle C^{\prime}$
$\displaystyle=C^{\prime}_{j}+\sum\nolimits_{i}C_{ij}\,,$ (3.31)

where $C^{(\prime)}_{i}$, $i=1,2,3$, denote the curves corresponding to the
fixed tori and $C_{ij}$, the resolution divisors. In the new basis the
intersection pairing is block-diagonal as

$\displaystyle\Omega=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\oplus(-1)^{\oplus
9}\,,$ (3.32)

and all zero-eigenvalue directions are absent. This intersection matrix
fulfills all requirements such as the correct signature, unimodularity and
also the gauge anomalies as well as the reducible gravitational anomaly can be
verified. Therefore, one needs to find the representation of the canonical
class in terms of the basis. As explained in Section 2.3, this can be found
using hayashi2019scfts

$\displaystyle
a_{i}=\sum\nolimits_{j}(\Omega^{-1})_{ij}(2g_{j}-2-(C_{j})^{2})\,,$ (3.33)

to be

$\displaystyle a=(-2,-2,(-1)^{\oplus 9})\,.$ (3.34)

The agreement with these various constraints shows that the theory, in
particular its massless spectrum, which is deduced above, actually constitutes
a consistent supergravity theory in 6 dimensions.

##### F-theory lift along ${\mathbb{T}}^{2}_{1}$

In order to analyze the other inequivalent lift, one chooses either the first
or the second torus to be the F-theory fibre (in the following the first is
chosen). As a result, the shift acts in one of the two base tori. This has in
fact immense consequences on the intersection structure of the base. First of
all, there is only one fixed torus left associated to $\omega$. Recall, in the
orbifold ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$ there were three tori
fixed under $\omega$. These get identified here by the action of the
additional shift. Furthermore, there were three more fixed tori associated to
$\vartheta\omega^{2}$ that intersected the former ones. This orbifolding group
element acts, however, freely here.

Beyond that there is the sector associated to $\vartheta\omega$, where three
inequivalent fixed points of co-dimension two are found in the base. In the
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-1$ model, these where located at the
intersection points of co-dimension one loci. Due to the modified action of
the generators, the co-dimension two and co-dimension loci are separated. This
results in the fact, that the co-dimension one locus, which supports an
$E_{6}$ singularity, is actually a fixed torus. As this is a curve of genus-
one we thus expect the contribution of an additional adjoint charged
hypermultiplet of $E_{6}$. Then one notices, that in the $\vartheta$ and
$\omega$ sector, we have fixed points purely located in the base that yield
superconformal matter sectors. These sectors can be assembled to three
$(-1)(-3)(-1)$ clusters, that host an $SU(3)$ over the middle ones.

In the following it will be verified that the above analysis in fact describes
a consistent supergravity theory. From the considerations above, the massless
spectrum is given as

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&E_{6}\times SU(3)^{3}\,,\\\
\hline\cr H&\mathbf{78}+12+1\\\ \hline\cr V&6\times\mathbf{14}+\mathbf{78}\\\
\hline\cr T&9+1\\\ \hline\cr\end{array}\,.$

Let us explain the spectrum in some more detail. As the gauge content of this
theory is described by

$\displaystyle G=E_{6}\times SU(3)^{3}\,,$ (3.35)

there are $V=78+3\times 8=102$ vector multiplets. The number of tensor
multiplets is geometrically counted by the volume moduli of the nine
resolution divisors located at the three co-dimension two fixed points in the
base as well as the universal tensor. Hence,

$\displaystyle T=9+1=10\,.$ (3.36)

Moreover, there are $H=72+12+1=85$ hyper multiplets in this theory. $12+1$ of
those are neutral and are due to the complex structure moduli and the
universal hyper. The other $72$ are in the adjoint representation of $E_{6}$
(which has dimension $78$ and charge dimension $72$) and are required from
gauge anomaly cancellation of a $E_{6}$ on a genus-one curve.

Again as above, in order to quantitatively check the cancellation of the
reducible gravitational anomaly and the gauge anomalies, the intersection
pairing of base divisors is studied. Here it is given by a block-diagonal
combination of the transcendental factor

$\displaystyle\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\,,$ (3.37)

as well as three factors of the $(-1)(-3)(-1)$ chain which coincides with
$(E_{6},E_{6})$ superconformal matter

$\displaystyle\begin{pmatrix}-1&1&0\\\ 1&-3&1\\\ 0&1&-1\end{pmatrix}\,.$
(3.38)

This intersection pairing, again, satisfies all conditions such as
unimodularity and the correct signature. For completeness, the representation
of the canonical class in terms of this basis is given by

$\displaystyle a=(-2,-2,\\{2,1,2\\}^{\oplus 3})\,.$ (3.39)

One might wonder how one recovers the curve that supports the $E_{6}$ gauge
group. In this basis it is in fact given by

$\displaystyle b_{E_{6}}=(3,3,(-3,-2,-3)^{\oplus 3})\,.$ (3.40)

This fulfills all the properties, that one requires. Cancellation of the gauge
anomaly requires that a curve that supports an $E_{6}$ singularity has genus-
one and vanishing self intersection. By adjunction one finds that also its
intersection with the canonical class must vanish. Furthermore, it cannot
intersect any curve of self-intersection lower that $(-2)$, as this would
yield additional superconfomal matter, which is not found here. All these
properties can be verified using the intersection pairing described above.
Note also, that the $E_{6}$ in this new basis in fact does intersect all the
$(-1)$ curves of the superconformal matter points.

####  3.2.2 F-theory on ${\mathbb{Z}}_{3}-1-1$ and
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-4$

In this subsection, we investigate F-theory on the ${\mathbb{Z}}_{3}$ orbifold
together with ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-4$ as the later one
can be expressed simply as a freely acting quotient of the former one. In the
F-theory context, such quotients have been discussed e.g. in Anderson:2018heq
; Anderson:2019kmx  and together with their effects on the F-theory physics.

#### F-theory on the ${\mathbb{Z}}_{3}$ orbifold

We start by analyzing F-theory on the one-quotient ${\mathbb{Z}}_{3}$
orbifold. For historical reasons, this geometry is special as it was the first
that has been discussed in the heterotic orbifold literature DIXON1985678 ;
DIXON1986285  where the massless spectrum has been computed using the CFT
methods. This geometry does in fact admit a couple of special features, which
we will also encounter in the F-theory physics. The space group generator is
given as

$\displaystyle\begin{array}[]{c}\vartheta:\end{array}\;\;\begin{array}[]{ccc|cccccc}\left(\right.1/3&1/3&1/3&\;0&0&0&0&0&0\left.\right)\end{array}\,.$
(3.43)

First we note, that the underlying
$\mathbb{T}^{2}\times\mathbb{T}^{2}\times\mathbb{T}^{2}$ basis vectors $e_{i}$
are fixed to an $SU(3)^{3}$ root lattice form to admit the ${\mathbb{Z}}_{3}$
quotient symmetry, i.e. the complex structure of all ambient tori is fixed to
$\tau=e^{2\pi i/3}$. When choosing each of the three, as the F-theory torus we
find that the theory is stuck at strong coupling. In fact, this theory is
fully rigid, in that it does not admit any complex structure moduli. Such
theories are interesting in that they do not allow to further symmetry
enhancement via unhiggsings or E-string transitions. In this sense, such rigid
theories yield endpoint theories that are non-enhance-able.

The quotient action yields a single twisted sector with $3\times 3\times 3=27$
fixed points of co-dimension three in the total space, see Figure 10. When
computing the stringy Hodge numbers via the (2,2) heterotic string CFT, we
note that something special happens for this case, as opposed to other
orbifolds: The standard embedding on the heterotic side forces the $SU(3)$
gauge bundle to be identified with the CY holonomy, which in our case is
exactly the discrete ${\mathbb{Z}}_{3}$ center. Hence, for this case, the
heterotic gauge group is $E_{6}\times SU(3)$. This is significant, as the
untwisted Kähler moduli come in copies of $(\mathbf{27,3})$ bifundamentals for
each ambient torus. We thus end up with nine untwisted Kähler modul, six more
than one might expect. In addition there is one $(\mathbf{27,1})$ twisted
sector field and hence Kähler modulus for each of the 27 fixed points. As
noted before, there are no $\overline{\mathbf{27}}$, that we associated with
complex structure moduli. We summarize the stringy Hodge numbers as

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(9,27)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(0,0)\,.$
(3.44)

Choosing any of the three equivalent F-theory we end up with a base of the
form

$\displaystyle
B=(\mathbb{T}^{2}\times\mathbb{T}^{2})/\theta_{B}\,,\quad\theta=(e^{i2\pi
i/3},e^{i2\pi i/3})\,.$ (3.45)

The non-crepant singularities have been analyzed already in Section 2.2 and
they are simply isolated (-3) NHCs, shrunk to the singular point. This is also
consistent with the three twisted Kähler moduli we find for each base
singularity, that yield the tensor and $SU(3)$ resolution divisors.

Figure 10: An illustration of the fixed points in the ${\mathbb{Z}}_{3}-1-1$
orbifold. Their location in the three tori is denoted pink.

We hence expect in total nine $SU(3)$ NHCs in the theory. However, now we also
need to take the untwisted Kähler moduli into account. As noted before, these
originate from the orbifold invariant Kähler-forms $J\sim J_{ab}\;du^{a}\wedge
d\bar{u}^{\bar{b}}$, with $a,b=1,2,3$ and $u$ being the complexified tori
coordinates. The diagonal terms, with $a=b$ are the actual torus moduli. The
other ones are relative rotations of these tori that appear as off-diagonal
deformations in the metric.

In terms of the vectors $e_{i}$ of the tori Casas:1991ac  these deformations
can be written as the angles

$\displaystyle\alpha_{i,j}=\frac{e_{i}\cdot e_{j}}{|e_{i}||e_{j}|}\,,$ (3.46)

with $|e_{i}|=R_{i}$. The orbifold fixes nine of those angles and only six are
free parameters. A choice for a parametrization is given as
$\alpha_{1,3},\alpha_{1,4},\alpha_{1,5},\alpha_{1,6},\alpha_{3,6},\alpha_{3,6}$.
When writing the deformations of the metric in terms of the moduli above, we
get

$\displaystyle e_{i}\cdot
e_{j}=\left(\begin{array}[]{cccccc}R_{1}^{2}&-\frac{1}{2}R_{1}^{2}&R_{1}R_{3}\alpha_{1,3}&R_{1}R_{3}\alpha_{1,4}&R_{1}R_{5}\alpha_{1,5}&R_{1}R_{5}\alpha_{1,6}\\\
-\frac{1}{2}R_{1}^{2}&R_{1}^{2}&R_{1}R_{3}\alpha_{2,3}&R_{1}R_{3}\alpha_{1,3}&R_{1}R_{5}\alpha_{2,5}&R_{1}R_{5}\alpha_{1,5}\\\
\cline{3-6}\cr
R_{1}R_{3}\alpha_{1,3}&R_{1}R_{3}\alpha_{2,3}&\vrule\lx@intercol\hfil
R_{3}^{2}\hfil\lx@intercol&-\frac{1}{2}R_{3}^{2}&R_{3}R_{5}\alpha_{3,5}&R_{3}R_{5}\alpha_{3,6}\\\
R_{1}R_{3}\alpha_{1,4}&R_{1}R_{3}\alpha_{1,3}&\vrule\lx@intercol\hfil-\frac{1}{2}R_{3}^{2}\hfil\lx@intercol&R_{3}^{2}&R_{3}R_{5}\alpha_{4,5}&R_{3}R_{5}\alpha_{3,5}\\\
R_{1}R_{5}\alpha_{1,5}&R_{1}R_{5}\alpha_{2,5}&\vrule\lx@intercol\hfil
R_{3}R_{5}\alpha_{3,5}\hfil\lx@intercol&R_{3}R_{5}\alpha_{4,5}&R_{5}^{2}&-\frac{1}{2}R_{5}^{2}\\\
R_{1}R_{5}\alpha_{1,6}&R_{1}R_{5}\alpha_{1,6}&\vrule\lx@intercol\hfil
R_{3}R_{5}\alpha_{3,6}\hfil\lx@intercol&R_{3}R_{5}\alpha_{3,5}&-\frac{1}{2}R_{5}^{2}&R_{5}^{2}\end{array}\right)\,.$
(3.53)

In the depiction above, we have singled out the metric deformations along the
first torus direction, as this is going to be our F-theory torus
choice141414The $\alpha_{2,3},\alpha_{2,4},\alpha_{2,5}$ can be rewritten in
terms of other basis angles using
$\alpha_{i,j}=-(\alpha_{i+1,j}+\alpha_{i,j+1})$. . Here we find, when taking
the F-theory lift, that is $R_{1}\rightarrow\infty$, we decouple the four off-
diagonal moduli $\alpha_{1,3},\alpha_{1,4},\alpha_{1,5},\alpha_{1,6}$. Hence
these moduli must be associated to horizontal divisors, and as they do not
correspond to codimension one (singular) fibers, they must correspond to the
Kähler classes that generate a rank four free Mordell-Weil group and thus a
$U(1)^{4}$ gauge group in F-theory.

On the other hand, there are still the four relative moduli
$\alpha_{3,5},\alpha_{3,6}$ of the two base
$(\mathbb{T}^{2}\times{\mathbb{T}}^{2})/\theta_{B}$ that are dual to other
base divisors and hence yield two additional tensor multiplets. When putting
everything together, the F-theory gauge group is given as

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&SU(3)^{9}\times U(1)^{4}\\\
\hline\cr H&1\\\ \hline\cr V&9\times\mathbf{8}+4\times\mathbf{1}\\\ \hline\cr
T&9+3\\\ \hline\cr\end{array}\,,$

consistent with the gravitational anomaly.

However, it remains an open task to find the complete consistent intersection
matrix of the resolved base. In this structure, there will be nine curves of
self intersection $(-3)$, that do not intersect each other, as well as the
transcendental factor. The remaining two directions associated to the two off-
diagonal Kähler-moduli of the fibre have negative signature (as the signature
of the entire intersection matrix is supposed to be $(+,-)=(1,T)=(1,12)$) and
are supposed to intersect the other divisors in a manner, such that the
resulting intersection matrix is unimodular151515A resolution of this orbifold
in terms of a GLSM construction can be found in Blaszczyk:2011hs , which
however is limited to the diagonal Kähler parameters only. In that resolution,
the fiber is described as a generic cubic, that is a genus-one fibration with
three-sections only.. While we were not able to show this unimodularity, we
are still confident that the analysis of the massless spectrum of F-theory on
this orbifold is valid due to the reasons mentioned above.

We close by commenting on the various additional symmetries that are present.
First we have noted already that the symmetry of the fixed points suggests a
gauged 1-form symmetry, as in the K3 example, which implies a global
$SU(3)^{9}/{\mathbb{Z}}_{3}$ global gauge group. Moreover, following the
general arguments of Section 3.1.2, the F-theory lift admits a
${\mathbb{Z}}_{3}$ discrete 2-form gauge symmetry. As observed the nine
collapsed fixed points of the
${\mathbb{T}}^{2}\times{\mathbb{T}}^{2}/{\mathbb{Z}}_{3}$ base all admit self-
intersection $-3$ and have hence a ${\mathbb{Z}}_{3}$ 2-form global symmetry.
The diagonal ${\mathbb{Z}}_{3}$ action of the global ${\mathbb{Z}}_{3}^{9}$
global symmetry of the SCFT sectors.

#### F-theory on the ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-4$ orbifold

Having established the F-theory physics of the ${\mathbb{Z}}_{3}$ orbifold
before we are adding another ${\mathbb{Z}}_{3}$ quotient factor that acts
freely on the geometry and coincides with the
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-4$ geometry. In order to see this,
the latter geometry is discussed briefly. This STO is in the same
${\mathbb{Q}}$\- and ${\mathbb{Z}}$-class as the ordinary
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ orbifold considered for example in
hayashi2019scfts  and the ${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$ in
Section 3. This geometry however admits maximal affine-class, that is exhibits
the maximal amount of fractional lattice shifts. The two generators can be
chosen to be161616This reproduces the group generators in the form as used in
Fischer_2013 , as $\vartheta=\tilde{\vartheta}\tilde{\omega}$ and
$\omega=\tilde{\vartheta}$, with
$\displaystyle\begin{array}[]{c}\,\tilde{\vartheta}:\\\
\tilde{\omega}:\end{array}\;\;\begin{array}[]{ccc|cccccc}\left(\right.\;\,0&1/3&-1/3&\;2/3&1/3&2/3&1/3&1/3&2/3\left.\right)\\\
\left(\right.1/3&0&-1/3&\;2/3&1/3&2/3&1/3&1/3&2/3\left.\right)\end{array}\,.$
(3.58)

$\displaystyle\begin{array}[]{c}\vartheta:\\\
\omega:\end{array}\;\;\begin{array}[]{ccc|cccccc}\left(\right.1/3&1/3&1/3&\;0&0&0&0&0&\;\,0\;\,\left.\right)\\\
\left(\right.\;\,0&1/3&-1/3&\;2/3&1/3&2/3&1/3&1/3&2/3\left.\right)\end{array}\,.$
(3.63)

The first factor is as in the ${\mathbb{Z}}_{3}$ orbifold and the second
$\omega$ realizes the free action. Using the free action, we can relate
indices in the geometry: If the action is free, indices like Euler numbers get
simply divided by the order of the quotient. Hence in our case we have

$\displaystyle\chi_{{\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}}=\frac{\chi_{{\mathbb{Z}}_{3}}}{3}\,.$
(3.64)

and therefore relate also the stringy Hodge numbers to

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,9)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(0,0)\,.$
(3.65)

The above Hodge numbers also agree with the spectrum in the heterotic (2,2)
CFT.

Figure 11: An illustration of the fixed points in the
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-4$ orbifold. Their location in the
three tori is denoted pink and identification is indicated by arrows.

The free generator $\omega$ relates all of the 27 fixed points and groups them
into nine inequivalent ones as depicted in Figure 11 and also removes six of
the off-diagonal untwisted Kähler moduli leaving only the three ambient tori
radii. Lets now discuss the F-theory physics of the resulting geometry. First,
there are only three $(-3)$ $SU(3)$ NHCs cluster in the base left. With
respect to the untwisted moduli, we find that the additional quotient $\omega$
fixes all residual off-diagonal Kähler modul $\alpha_{i,j}$ in (3.53). Thus
the quotient removes all four $U(1)$ factors and two tensors of the base.
Before we can summarize the spectrum we note, that the
${\mathbb{Z}}_{3}\times\mathbb{Z}_{3}-1-1-4$ is non-simply connected as a
manifold171717This property is also preserved in the resolution phase.. Hence
from the considerations of Section 3.1.2 and Anderson:2018heq ;
Anderson:2019kmx  we expect $\omega$ to lead to multiple fibers. Indeed, when
viewed from an F-theory perspective, the $\omega$ element acts as an
additional $\mathbb{Z}_{3}$ orbifold in the (singular) base, and while it
identifies a couple of fixed points in the $\vartheta$ sector it also
introduces three new ${\mathbb{Z}}_{3}$ fixed points in the base. Those fixed
points count as three $\mathcal{A}_{2}$ theories with local
$\mathcal{N}=(2,0)$ supersymmetry and thus, like $3\times 2\cdot(H+T)$-plets.
Note again, that those fixed points appear only in the base geometry but not
in the total space and hence admit no associated $h^{(1,1)}$ contributions.

We are now in the position to summarize the massless F-theory spectrum to

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&SU(3)^{3}\\\ \hline\cr H&1\\\
\hline\cr V&3\times\mathbf{8}\\\ \hline\cr T&3+1\\\ \hline\cr
T_{2,0}&3\times\mathcal{A}_{2}\\\ \hline\cr\end{array}\,.$

The above spectrum spectrum is consistent with the gravitational anomaly,
which notably requires the additional fixed point contribution in the base.
Similar, as in the covering geometry, we expect the various curves to be glued
together to lead to an uni-modular intersection pairing.

Now we can also discuss the various discrete symmetries in the theory. As
discussed before, the existence of a multiple fiber is a smoking gun for a
genus-one fibration. Indeed, compatibility requires to have (at least) a
three-section and thus it follows that there is a discrete ${\mathbb{Z}}_{3}$
0-form gauge symmetry. Again, there are no massless charged hypermulitplets in
the theory, but the $(2,0)$ SCFT sector as explicitly discussed in
Anderson:2018heq . In addition we find, that the base is a quotient of the
form

$\displaystyle
B_{2}=(\mathbb{T}^{2}\times\mathbb{T}^{2})/(\vartheta_{B}\times\omega_{B})\,.$
(3.66)

Indeed, this implies a $\mathbb{Z}_{3}\times{\mathbb{Z}}_{3}$ discrete 2-form
gauge symmetry that acts diagonally on the ${\mathbb{Z}}_{3}^{6}$ global
2-form symmetries of the $(-3)^{3}\times\mathcal{A}_{2}^{3}$ SCFT sectors.

##  4 Summary and conclusion

Symmetric toroidal orbifold geometries are distinguished loci in the landscape
of Calabi-Yau geometries, in that they are essentially flat, apart from the
orbifold fixed points. This simplicity makes the geometries also very
attractive for F-theory compactifications. Moreover, all six-dimensional
symmetric toroidal orbifolds have been classified rather recently Fischer_2013
and therefore allows us to initiate a systematic study of F-theory on these
geometries.

Our analysis highlights a couple of significant features: First STOs admit up
to three inequivalent torus fibrations. This allows to obtain multiple
different F-theory lifts from the same geometry that yield the same M-theory
dual. Second, in case with non-trivial affine classes the orbifold rotation is
accompanied by fractional lattice shifts, so called roto-translations. Those
shifts identify fixed points, which lead to twisted affine fibers in the
F-theory torus and/or reduced amounts of SCFTs when they act in the base.
Moreover, they can lead to exotic SCFT sectors supported by multiple-fibers.
In particular we find that the bases, that generically are of
$\mathbb{T}^{4}/\Gamma$ type, can host isolated superconformal theories at the
fixed points. These superconformal points include in particular non-Higgsable
clusters at the origin of their tensor branch. These cluster admit, by
definition, no flavor symmetry. Our study enlarges the class of configurations
that have been explored in hayashi2019scfts , where the SCFT sectors were
always of superconformal matter type, that exhibit non-trivial (gauged) flavor
symmetry.

Third, we find that the IIB axio-dilaton is constant and fixed to strong
coupling. In fact, many of the geometries are completely rigid, as they do not
admit any complex structure deformations. The resulting supergravity can
therefore be viewed as an endpoint theory that cannot admit further gauge or
SCFT enhancements. Fourth, orbifolds admit a vast amount of discrete
symmetries that impact the underlying effective string compactification, such
as enhanced flavor or R-symmetries in the heterotic case. Correspondingly we
also find enhanced discrete gauge symmetries in F-theory on the same spaces.
These in particular include 0-, 1-, and 2-form discrete gauge symmetries.

The heterotic theories were very useful in various other ways for this works
as well. The $(2,2)$ worldsheet CFT in Fischer_2013  helped to obtain the
stringy Hodge numbers of the orbifold geometries. We required those in order
to show 6D supergravity anomaly cancellation as well as to cross-check the
gauge group rank and number of tensor multiplets.

While this work initiates a complete classification of F-theory on symmetric
toroidal orbifolds, various questions and directions for future work remain.
This includes in particular a better understanding of theories with twisted
affine fibers and a complete derivation of the anomaly lattices for some of
the geometries considered here. The next goal is then the completion of the
study of F-theory on Abelian orbifolds for all affine-classes.

To complete our study we would then turn to the new class of non-Abelian
orbifolds and their F-theory lifts which we expect to lead to various other
potentially more exotic theories. All stringy Hodge numbers for all 331 non-
Abelian orbifolds have been computed Fischer:2013qza  as well, such that we
can simply run our program there too. Moreover we have left an analysis of the
little string degenerations, that are readily obtained from decompactifying
one of the base tori, for future work. This direction is particularly
appealing, as the various torus-fibration structures which relate T-dual
little string theories, are directly build into the STOs.

This program on the other hand, can further be generalized to toroidal four-
or even five-folds which, however, are not classified so far. F-theory on
those spaces lead to four and two dimensional theories. Those geometries admit
a couple of additional interesting features, such as terminal singularities
that do not exhibit resolutions Arras:2016evy  or deformations. The heterotic
world sheet CFT however is still well defined, such that we can compute the
corresponding massless states at the fixed points, that correspond to their
deformation spaces in the standard embedding Font:2004et . In 4D we would then
expect to find supergravity theories coupled to the (local) $\mathcal{N}=3$
theories constructed in Garcia-Etxebarria:2015wns ; Garcia-Etxebarria:2016erx
.

## Acknowledgements

We thank Fabian Ruehle and Damian Kaloni Mayorga Pena for discussions and
collaboration on related work. We further thank Michele del Zotto and Patrick
Vaudrevange for comments and valuable discussions. F.B.K wants to thank Raimar
Wulkenhaar for supporting this project. The work of F.B.K. is financed by the
Research Training Group 2149 funded by the Deutsche Forschungsgemeinschaft as
well as supported by the Cluster of Excellence Mathematics Münster181818Funded
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Project-ID 427320536 – SFB 1442.. The research of M.L. is financed by
Vetenskapsradet under grant number 2020-03230. The work of P.K.O. is supported
by a grant of the Carl Trygger Foundation for Scientific Research and the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 851931).

##  Appendix A F-theory on ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifolds

In order to illustrate our analysis as well as some issues that still remain,
we provide the analysis of ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifolds
on a lattice $\Lambda=SU(2)^{6}$. New features that are described in the main
section are also found here as well as challenges for future studies.

###  ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-1$ orbifold

In order to be complete and for later reference, F-theory on the standard
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifold, that is the trivial affine
class is very briefly reviewed here. The complete analysis can be found in
hayashi2019scfts .
The action of the two generators of the space group

$\displaystyle\begin{array}[]{c}\vartheta:\\\
\omega:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.\;\;0&1/2&-1/2&\;0&0&0&0&0&0\left.\right)\\\
\left(\right.1/2&0&-1/2&\;0&0&0&0&0&0\left.\right)\end{array}\,$ (A.5)

gives rise to three fixed point sectors, each containing $16$ fixed tori (see
Figure 12). The stringy Hodge numbers are given as

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,48)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(3,0)\,.$
(A.6)

As this structure is symmetric in the three tori, there is just one
inequivalent lift. In this lift one finds $4+4$ fixed tori in the base
distributed among two sectors mutually intersecting in the the fixed points of
co-dimension two in the third sector. The F-theoretic interpretation of
degenerated fibre over the fixed tori yields $SO(8)$ gauge group factors. The
intersection of these base curves hence gives rise to $(SO(8),SO(8))$
superconformal matter. These singular loci of co-dimension one in the base are
resolved by the blow-up of one $(-1)$-curve in the base.

Figure 12: An illustration of the fixed points (pink) in the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-1$ orbifold.

The massless matter spectrum of theory is summarized as follows

$\displaystyle\begin{array}[]{|c|c|}\hline\cr
G&(SO(8))^{4}\times(SO(8))^{4}\\\ \hline\cr H&3+1\\\ \hline\cr
V&(4+4)\times\mathbf{28}\\\ \hline\cr T&17\\\ \hline\cr\end{array}\,.$ (A.11)

As there are no charged hyper multiplets, the total number of hyper multiplets
is given by the $H=h^{(2,1)}+1=3+1=4$. The $17$ tensors can be explained by
the volume moduli of the exceptional divisors at the $(SO(8),SO(8))$
superconformal matter loci and the universal mode.

The resulting intersection pairing, described in detail inhayashi2019scfts ,
yields cancellation of all anomalies and hence a consistent theory. Before
moving on to the next theory, we also want to remark here, that we can view
the geometry as the collision of two K3s that come from the Sen configuration
of ${\mathbb{T}}^{2}\times{\mathbb{T}}^{2}/{\mathbb{Z}}_{2}$ that yield an
$SO(8)^{4}$ configuration each. These K3s are the unique configurations 2147
in 2005math……5140S  that admit a ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ MW
group. Hence we expect that our configuration admits a
$SO(8)^{8}/{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ form. This is again
compatible with the supreconformal matter configurations found as well
Dierigl:2020myk . Those considerations are again consistent with the presence
of the four order two points that sit at at the precise locations, where the
orbifold fixed points are.

###  ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-2$ orbifold

Turning to the fist non-trivial affine class, a shift is introduced into the
generator $\vartheta$ resulting in

$\displaystyle\begin{array}[]{c}\vartheta:\\\
\omega:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.\;\;0&1/2&-1/2&\;0&1/2&0&0&0&0\left.\right)\\\
\left(\right.1/2&0&-1/2&\;0&0&0&0&0&0\left.\right)\end{array}\,.$ (A.16)

The fixed point structure is depicted in Figure 14.

Figure 13: Illustration of the Schoen manifold as the fibre product of two del
Pezzo surfaces $dP_{9}$.

As already indicated by the Hodge numbers

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,16)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(3,16)\,,$
(A.17)

this orbifold is in fact a singular limit of the Schoen manifold Schoen1988 .
This geometry has been analysed in the heterotic orbifold context in
nibbelink2013schoen  as well as in the F-theory context in Gopakumar1996 .

The Schoen manifold is a fibre product of two rational elliptic surfaces
$S_{1}$ and $S_{2}$. The $S_{i}$ are described by a torus fibration
$\pi_{i}:\;S_{i}\to{\mathbb{P}}^{1}$ such that

$\displaystyle X_{Schoen}=\\{(x_{1},x_{2})\in S_{1}\times
S_{2}:\;\pi_{1}(x_{1})=\pi_{2}(x_{2})\\}\,.$ (A.18)

This structure is illustrated in Figure 13. As a result of this construction,
one finds the cohomology group
$H^{2}(X_{Schoen},{\mathbb{Z}})\simeq\text{Pic}(X_{Schoen})$ to be

$\displaystyle
H^{2}(X_{Schoen},{\mathbb{Z}})=\frac{H^{2}(S^{1},{\mathbb{Z}})\oplus
H^{2}(S_{2},{\mathbb{Z}})}{H^{2}({\mathbb{P}}^{1},{\mathbb{Z}})}\,.$ (A.19)

In terms of the resolved orbifold in the present convention, the base of fibre
product is torus one. Along this direction the two factors of the product are
glued together such that

$\displaystyle x_{1}\in S_{1}:\;x_{1}=(z_{1},z_{3})\,,$ $\displaystyle
x_{2}\in S_{2}:\;x_{2}=(z_{1},z_{2})\,,$ (A.20)

and $\pi_{i}(x_{i})=z_{1}$, where $z_{i}$, $i=1,2,3$, are the complexified
coordinates of the three tori, which are subject to the orbifolding group
identifications.

In the following we discuss the different F-theory lifts of this geometry. As
it is symmetric in torus ${\mathbb{T}}^{2}_{2}$ and ${\mathbb{T}}^{2}_{3}$ or
equivalently $S_{1}$ and $S_{2}$, there are two inequivalent lifts.

##### F-theory lift along ${\mathbb{T}}^{2}_{1}$

In this lift, the shift only acts in the F-theory fibre, that is the common
${\mathbb{P}}^{1}$ is lifted.

As the base of the F-theory fibration is not effected by the shift, it remains
almost the same as in the previous example
${\mathbb{Z}}_{1}\times{\mathbb{Z}}_{1}-1-1$ without shift. In both fixed
point sectors ($\omega$ and $\vartheta\omega$) there are four fixed tori, that
mutually intersect.

Figure 14: An illustration of the fixed points (pink) in the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-2$ orbifold.

The shift in the fibre identifies fixed points and thus resolution divisors,
compared to ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-1$. Former $D_{4}$-type
fibres are folded into $B_{2}^{(2)}\simeq C_{2}^{(2)}$-type. This folding is a
twisted affine folding, as explained in Section 3.1.2, and yields the gauge
group supported by the fixed point divisors are $SO(5)\simeq Sp(2)$. Note that
the singularity at the intersection of these divisors does not exceed the
minimality bound and yields therefore no conformal matter.

Beyond this, there are fixed points, when $\vartheta$ is restricted to the
base tori. As explained above in Section 3.1.2, this yields SCFT sectors in
the theory with ${\mathcal{N}}=(2,0)$ SUSY, which also contribute to the
massless spectrum. In our analysis of the geometry the number of these
sectors, however, does not match with what anomaly cancellation requires. The
restricted element $\vartheta|_{\text{base}}$ acts as

$\displaystyle\vartheta|_{\text{base}}:\;\;\;\left(\begin{array}[]{cc|cccc}1/2&-1/2&\;0&0&0&0\end{array}\right)\,,$
(A.22)

which has $4\times 4=16$ fixed points. Requiring the cancellation of the
irreducible gravitational anomaly with the data provided below, restricts the
number of ${\mathcal{A}}_{1}$ factors to $8$. This mismatch is likely due to
an additional geometric effect that reduces the number of
${\mathcal{N}}=(2,0)$ SCFT sectors that are found from the fixed points of
$\vartheta|_{\text{base}}$ either by an identification of divisors in the base
or an effect similar to that encountered in the analysis of the lift of torus
${\mathbb{T}}^{2}_{3}$ of the orbifold
${\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}-1-2$. This issue can be analyzed in
detail in the resolved phase of the geometry, which goes beyond the scope of
this work.

Concluding, the massless sector of the theory is summarized as

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&SO(5)^{4}\times SO(5)^{4}\\\
\hline\cr H&(4+4)\times\mathbf{10}+19+1\\\ \hline\cr
V&(4+4)\times\mathbf{10}\\\ \hline\cr T&1\\\ \hline\cr
T_{2,0}&8\times{\mathcal{A}}_{1}\\\ \hline\cr\end{array}\,.$ (A.28)

Note, there is charged matter, that contributes to the number of hypers, which
is in the adjoint representation of $SO(5)$. The appearance of these
multiplets is consistent as the base divisors associated to the fixed points
of $\omega$ and $\vartheta\omega$ are genus-one curves. Cancellation of the
gauge anomaly of $SO(5)$ on such curves requires exactly one adjoint
multiplet. Finally we also note, that the identification of the rank $8\times
2$ gauge group and its single tensor is consistent with the $19$ Kähler
parameters. Note again that the $\mathcal{A}_{1}$ theories do not contribute
to those, but are again a hallmark for a ${\mathbb{Z}}_{2}$ discrete 0-form
gauge symmetry.

##### F-theory lift along ${\mathbb{T}}^{2}_{2}$

Lifting either torus two or three yields an F-theory fibration, where the
shift acts in one of the two base tori. In terms of the fibre product
(equation (A.18)), the fibre of one of the two factors chosen to be F-theory
fibre. Note, that this lift has already been discussed in Gopakumar1996 . It
is nevertheless briefly discussed for the sake of completeness.

The fixed point structure yields two genus-one curves in the base over which
the fibre is of $D_{4}$ type. This exhausts the amount of co-dimension one
singularities in the base. Comparing to the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ orbifold with trivial affine class,
one realizes that the amount of fixed tori in the $\omega$-sector is reduced
by a pairwise identification of fixed points in the base. Furthermore, the
fixed tori due to the action of $\vartheta$, that mutually intersected the
former, do not appear here, as $\vartheta$ acts freely on the orbifold as well
as on the base.

Note, that this analysis is consistent with the fact, that the resolved
orbifold is a Schoen manifold. There we have two $dP_{9}$ surfaces with
homology lattice $\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\oplus E_{8}$. These
are glued along the common ${\mathbb{P}}^{1}$ whose homology class is encoded
in the transcendental $U$ lattice part, as described in the beginning of this
section. This suggests, that in the F-theory compactification that follows the
fibration structure of the fibre product, one finds a gauge group which is
either $E_{8}$ or a rank preserving subgroup thereof. Consistently, the
(adjoint) higgsing of $E_{8}$ to $SO(8)\times SO(8)$ yields two $\mathbf{28}$
adjoint multiplets matching with gauge anomaly cancellation of $SO(8)$ on a
genus-one curve. To summarize we have the massless spectrum

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&SO(8)^{2}\\\ \hline\cr
H&2\times\mathbf{28}+19+1\\\ \hline\cr V&2\times\mathbf{28}\\\ \hline\cr
T&9\\\ \hline\cr\end{array}\,.$ (A.33)

Note that the nine tensor multiplets are geometrically realized by the
universal mode as well as the volumes of the resolution divisors of the
singularities associated to the eight fixed points of the action of $\omega$.
Again, the rank $2\times 4$ gauge group, including the $9$ tensors is
consistent with the 19 Kähler parameters found in the geometry.

###  ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-3$ orbifold

Turning to the second non-trivial affine class, two shifts are introduced into
the generator $\vartheta$ yielding the three inequivalent orbifolding group
elements

$\displaystyle\begin{array}[]{c}\vartheta:\\\ \omega:\\\
\vartheta\omega:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.\;\;0&1/2&-1/2&\;0&1/2&\;0&0&0&1/2\left.\right)\\\
\left(\right.1/2&0&-1/2&\;0&0&\;0&0&0&\;\;0\;\left.\right)\\\
\left(\right.1/2&1/2&\;0&\;0&1/2&\;0&0&0&1/2\left.\right)\end{array}\,.$
(A.40)

The resulting geometry has Hodge numbers

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,8)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(3,8)\,.$
(A.41)

Note that the action of the orbifolding group is symmetric in torus one and
three. This suggests that there are only two inequivalent lifts,
${\mathbb{T}}^{2}_{1}$ (or ${\mathbb{T}}^{2}_{3}$) and ${\mathbb{T}}^{2}_{2}$.
This is, naively, also reflected in the fixed point structure (see Figure 15).
However, in the identification of fixed points due to the shifts there appears
to be a choice. The only remaining fixed points are due to the action of
$\omega$, as the action of the other orbifolding group elements on the
orbifold is smooth. The action naively has $4\times 4=16$ fixed points.
However, conjugation yields that there are in fact only $8$.191919This agrees
with $h^{(1,1)}$ obtained Fischer_2013  or through data of the orbifolder
Nilles_2012 . This identification by the action of the shift inevitably causes
a asymmetry between torus one and three, as the identification only acts in
one of these tori. As shown below, depending on the choice of identification
and lift, we find theories consistent with the cancellation of the irreducible
gravitational anomaly with either two $SO(8)$ or four $SO(5)=Sp(2)$.

In the following all possibilities of choice of F-theory fibre are discussed,
supposing the fixed point structure illustrated in Figure 15.

Figure 15: An illustration of the fixed points (pink) in the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-3$ orbifold.

##### F-theory lift along ${\mathbb{T}}^{2}_{2}$

In this lift, the shift acts in both base tori and there is no ambiguity.
Descending from the ${\mathbb{T}}^{2}_{2}$ lift of the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-2$ orbifold, this model is also
discussed in Gopakumar1996 .

As both $\vartheta$ and $\vartheta\omega$ act freely on the orbifold, there
are no fixed points of co-dimension one in the base. In fact, $G=\emptyset$
and, furthermore,

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&\emptyset\\\ \hline\cr
H&11+1\\\ \hline\cr V&-\\\ \hline\cr T&9\\\ \hline\cr\end{array}\,.$ (A.46)

As in the lift of ${\mathbb{T}}^{2}_{2}$ of the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-2$ orbifold, the tensors are due to
the eight fixed points of the action of $\omega$ and the universal mode. Also
the 11 Kähler moduli are identified with the nine tensors and generic fiber
and base class.

##### F-theory lift along ${\mathbb{T}}^{2}_{3}$

Lifting torus three to be F-theory fibre and identifying fixed points in the
base yields two fixed tori. The degeneration of the fibre suggests, that one
finds $SO(8)$ gauge group factors here. These are actually genus-one curves,
as one observes that there are no fixed points of co-dimension two in the
base, that intersect the fixed tori. To anticipate the analysis of the gauge
anomaly on these curves, one finds two adjoint charged hyper multiplets of the
$SO(8)$ in the spectrum.

Although this exhausts the fixed points of the total space, we again encounter
some fixed point elements coming from freely acting elements of the space
group $\vartheta\omega$, which, as mentioned above, in fact do not intersect
the fixed tori. As described in Section 3.1.2, over these locations in the
base, one observes a multiple fibre giving rise to ${\mathcal{N}}=(2,0)$ SCFT
sectors of the theory. As there are $4\times 4/2$ such loci due to
identification, these ${\mathcal{A}}_{1}$ sectors yield further multiplets,
equivalent to $8\times 30=240$ hypers.

The remaining orbifolding group element acts freely on the orbifold as well as
the base.

Counting multiplets, one finds cancellation of the irreducible gravitational
anomaly as well as gauge anomalies, as

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&SO(8)^{2}\\\ \hline\cr
H&2\times\mathbf{28}+11+1\\\ \hline\cr V&2\times\mathbf{28}\\\ \hline\cr
T&1\\\ \hline\cr T_{2,0}&8\times{\mathcal{A}}_{1}\\\ \hline\cr\end{array}\,.$
(A.52)

In order to show cancellation of the reducible gravitational anomaly the
entire intersection pairing is required. This, however, still remains an open
task due issues connected to the unimodularity of the embedding of
${\mathcal{A}}_{1}$ into the theory.

##### F-theory lift along ${\mathbb{T}}^{2}_{2}$

In this case, the identification of fixed points takes place in the F-theory
fibre, which gives rise to twisted affine foldings of Dynkin diagrams in the
F-theory interpretation of the geometry.

Investigating the F-theory base, one finds four inequivalent fixed tori
associated to the orbifolding group element $\omega$. These are, further, not
intersected by any singular structure in the geometry, yielding a genus-one
curve in the base. The fibre here is a folding of the affine Dynkin diagram
$D_{4}$ to a twisted affine Dynkin diagram $B_{2}^{(2)}\simeq C_{2}^{(2)}$,
yielding an $SO(5)\simeq Sp(2)$ gauge group contribution. Another example of a
twisted affine folding induced by a roto-translation. Beyond that analogously
to the lift discussed above, eight ${\mathcal{N}}=(2,0)$ SCFT sectors of type
${\mathcal{A}}_{1}$ are found.

To conclude, the massless sprectrum of this theory given by

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&SO(5)^{4}\\\ \hline\cr
H&4\times\mathbf{10}+11+1\\\ \hline\cr V&4\times\mathbf{10}\\\ \hline\cr
T&1\\\ \hline\cr T_{2,0}&8\times{\mathcal{A}}_{1}\\\ \hline\cr\end{array}\,.$
(A.58)

This spectrum cancels the irreducible gravitational anomaly as well as gauge
anomalies. The verification of the cancellation of the reducible gravitational
anomaly again is left for future studies.

###  ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}-1-4$ orbifold

Analyzing the maximal affine class of ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
orbifolds on a lattice $\Lambda=SU(2)^{6}$, one introduces a third shift by
setting

$\displaystyle\begin{array}[]{c}\vartheta:\\\ \omega:\\\
\vartheta\omega:\end{array}\;\;\;\begin{array}[]{ccc|ccccccc}\left(\right.\;\;0&1/2&-1/2&\;0&1/2&\;0&0&0&1/2\left.\right)\\\
\left(\right.1/2&0&-1/2&\;0&0&\;0&1/2&0&\;\;0\;\left.\right)\\\
\left(\right.1/2&1/2&\;0&\;0&1/2&\;0&1/2&0&1/2\left.\right)\end{array}\,,$
(A.65)

yielding a geometry with Hodge numbers

$\displaystyle(h^{(1,1)}_{\text{untw.}},h^{(1,1)}_{\text{tw.}})=(3,0)\,,$
$\displaystyle(h^{(2,1)}_{\text{untw.}},h^{(2,1)}_{\text{tw.}})=(3,0)\,.$
(A.66)

The action of all those generators on the total space orbifold is smooth.
Irrespective of the chosen lift the only structure remaining here are eight
fixed points of the restriction of one of the orbifolding group elements to
the base. As in the above examples, this yields $8\times{\mathcal{A}}_{1}$
$(2,0)$ subsector, consistent with the cancellation of the irreducible
gravitational anomaly via the following massless spectrum

$\displaystyle\begin{array}[]{|c|c|}\hline\cr G&\emptyset\\\ \hline\cr
H&3+1\\\ \hline\cr V&-\\\ \hline\cr T&1\\\ \hline\cr
T_{2,0}&8\times{\mathcal{A}}_{1}\\\ \hline\cr\end{array}\,.$ (A.72)

## References

  * (1) C. Vafa, _Evidence for F-theory_ , _Nuclear Physics B_ 469 (1996) 403–415.
  * (2) D. R. Morrison and C. Vafa, _Compactifications of F theory on Calabi-Yau threefolds. 2._ , _Nucl. Phys. B_ 476 (1996) 437 [hep-th/9603161].
  * (3) D. R. Morrison and C. Vafa, _Compactifications of F theory on Calabi-Yau threefolds. 1_ , _Nucl. Phys. B_ 473 (1996) 74 [hep-th/9602114].
  * (4) J. J. Heckman, D. R. Morrison, T. Rudelius and C. Vafa, _Atomic classification of 6d scfts_ , _Fortschritte der Physik_ 63 (2015) 468–530.
  * (5) L. Bhardwaj, M. Del Zotto, J. J. Heckman, D. R. Morrison, T. Rudelius and C. Vafa, _F-theory and the classification of little strings_ , _Physical Review D_ 93 (2016) .
  * (6) J. J. Heckman and T. Rudelius, _Top down approach to 6d scfts_ , _Journal of Physics A: Mathematical and Theoretical_ 52 (2019) 093001.
  * (7) V. Kumar, D. R. Morrison and W. Taylor, _Mapping 6D N = 1 supergravities to F-theory_ , _JHEP_ 02 (2010) 099 [0911.3393].
  * (8) V. Kumar and W. Taylor, _String Universality in Six Dimensions_ , _Adv. Theor. Math. Phys._ 15 (2011) 325 [0906.0987].
  * (9) D. S. Park, _Anomaly Equations and Intersection Theory_ , _JHEP_ 01 (2012) 093 [1111.2351].
  * (10) D. R. Morrison and W. Taylor, _Classifying bases for 6D F-theory models_ , _Central Eur. J. Phys._ 10 (2012) 1072 [1201.1943].
  * (11) V. Kumar, D. R. Morrison and W. Taylor, _Global aspects of the space of 6D N = 1 supergravities_ , _JHEP_ 11 (2010) 118 [1008.1062].
  * (12) L. Dixon, J. Harvey, C. Vafa and E. Witten, _Strings on orbifolds_ , _Nuclear Physics B_ 261 (1985) 678.
  * (13) L. Dixon, J. Harvey, C. Vafa and E. Witten, _Strings on orbifolds (ii)_ , _Nuclear Physics B_ 274 (1986) 285.
  * (14) L. E. Ibanez, H. P. Nilles and F. Quevedo, _Orbifolds and Wilson Lines_ , _Phys. Lett. B_ 187 (1987) 25.
  * (15) M. Blaszczyk, S. Groot Nibbelink, M. Ratz, F. Ruehle, M. Trapletti and P. K. S. Vaudrevange, _A Z2xZ2 standard model_ , _Phys. Lett. B_ 683 (2010) 340 [0911.4905].
  * (16) O. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P. K. S. Vaudrevange et al., _A Mini-landscape of exact MSSM spectra in heterotic orbifolds_ , _Phys. Lett. B_ 645 (2007) 88 [hep-th/0611095].
  * (17) A. Sen, _F theory and orientifolds_ , _Nucl. Phys. B_ 475 (1996) 562 [hep-th/9605150].
  * (18) K. Dasgupta and S. Mukhi, _F-theory at constant coupling_ , _Physics Letters B_ 385 (1996) 125–131.
  * (19) I. n. García-Etxebarria and D. Regalado, _$\mathcal{N}=3$ four dimensional field theories_, _JHEP_ 03 (2016) 083 [1512.06434].