profile which close to the brane at $z=0$ can be approximated as

$\displaystyle\tau(z)=\tau_{0}+\frac{1}{2\pi i}{\rm
log}\left(\frac{z}{z_{0}}\right)\,.$ (5.2)

Due to the logarithmic branch cut, $\tau(z)$ undergoes a shift $\tau\to\tau+1$
as one encircles the D7-brane at $z=0$. This multi-valuedness is consistent
because it is accompanied by a an $SL(2,\mathbb{Z})$ duality transformation on
all the Type IIB fields, which merely accounts for a change of duality frame.
The $SL(2,\mathbb{Z})$ monodromy induced in this way by a single $[p,q]$
7-brane can be represented by the action of a monodromy matrix of the form

$\displaystyle M_{[p,q]}=\left(\begin{matrix}1-pq&p^{2}\\\
-q^{2}&1+pq\end{matrix}\right)\,.$ (5.3)

Two $[p,q]$ 7-branes are called mutually non-local if their monodromy matrices
cannot be brought into the same form by an $SL(2,\mathbb{Z})$ transformation.

Not all types of mutually non-local $[p,q]$ 7-branes can be consistently
placed on top of each other to form a supersymmetric brane stack without
leading to a drastic degeneration of the theory such as an effective
decompactification. Let us first consider 7-branes in flat space
$\mathbb{R}^{1,9}$. In order to describe the allowed configurations of
coincident $[p,q]$ 7-branes in 10 dimensions, it suffices to consider three
types of mutually non-local $[p,q]$ 7-branes. A possible such generating
system consists of the following three brane types:

$\displaystyle A:[p,q]=[1,0]\,,\qquad B:[p,q]=[3,-1]\,,\qquad
C:[p,q]=[1,-1]\,.$ (5.4)

In the chosen duality frame, an $A$-type brane corresponds to a perturbative
D7-brane. The O7-plane from Type IIB orientifolds is realised as a $BC$-brane
system. Consistently the monodromy matrix $M_{BC}=M_{B}M_{C}$ acts as
multiplication with $(-1)$ on a $(1,0)$ string, which is interpreted as
worldsheet parity. In this way one recovers the perturbative gauge groups
$SU(N)$ (omitting the diagonal $U(1)$ factor) by a configuration of $N$
$A$-type branes, while $SO(N)$ groups correspond to a system of $N$ $A$-type
branes on top of a $BC$ brane system, i.e. on an O7-plane.777For $N<4$, non-
perturbative effects lead to a dynamical separation of the branes. The most
important observation, however, is that in addition to these perturbative
gauge groups, also the exceptional gauge groups $E_{N}$ for $N=6,7,8$ can be
obtained in flat space from brane configurations of the form $A^{N-1}BCC$.

The emergence of exceptional gauge algebras is rooted in the possible
$[p,q]$-strings stretched between the mutually non-local branes in the
configuration. These include, in the present setup, bound states of
$[p,q]$-strings with three endpoints - so-called multi-pronged strings - which
end on three mutually non-local branes, rather than just two branes of the
same type as in perturbative setups. A careful analysis of the possible multi-
pronged strings [110] for the configuration $A^{N-1}BCC$ indeed identifies all
the roots of the exceptional series $E_{N}$, $N=6,7,8$.

The easiest way to classify the allowed configurations of coincident 7-branes
is by passing to the geometrised description of F-theory as pioneered in [103,
111, 112]. The behaviour of the axio-dilaton $\tau$ under an
$SL(2,\mathbb{Z})$ transformation, (5.1), is reminiscent of the transformation
of the modular parameter of an elliptic curve under its modular group. This
motivates interpreting $\tau$ as the complex structure parameter of an
elliptic curve $\mathbb{E}_{\tau}$, which varies holomorphically in the
directions normal to the 7-brane according to (5.2). This structure defines an
elliptic, or more generally a genus-one [113]888The difference is that an
elliptic fibration necessarily has a section. In the sequel we will, for
simplicity, use the term elliptic fibration for both constructions., fibration
over the directions normal to the branes.

The simplest manifestation of this idea is to consider compactifications of
F-theory to eight dimensions with general $[p,q]$ 7-branes filling the
uncompactified dimensions $\mathbb{R}^{1,7}$. Due to the backreation of the
7-branes, the two compact normal directions must be curved to form a
projective sphere $\mathbb{P}^{1}$. The variation of the axio-dilaton over
$\mathbb{P}^{1}$ can be identified with the variation of the complex structure
parameter of an elliptic curve fibered over $\mathbb{P}^{1}$. Supersymmetry
requires that the elliptic fibration defined in this fashion is a Calabi–Yau
manifold, and in this case an elliptically fibered K3 surface. The physical
compactification space from ten to eight dimensions, $\mathbb{P}^{1}$,
represents the base of this elliptic fibration. Note that this physical
compactification space is not Calabi–Yau, but rather carries positive
curvature as a result of the backreaction. The $[p,q]$ 7-branes sit at special
points on the base $\mathbb{P}^{1}$ over which the complex structure $\tau$ of
the elliptic fiber $\mathbb{E}_{\tau}$ degenerates to reflect the singular
behaviour of the axio-dilaton profile (5.2) for $z\to 0$. The monodromies
around the location of a $[p,q]$ 7-brane have a direct geometric
interpretation as monodromies that transform a local basis of one-cycles on
the elliptic fiber into one another as one transports it around the 7-brane
location.

The possible types of monodromies that can consistently occur in this manner
in an effectively eight-dimensional compactification have been classified
geometrically by Kodaira and Néron [114, 115, 116] and are in one-to-one
correspondence with A-D-E type singularities of the elliptic fiber. By
interpreting a monodromy as a product of monodromy matrices of different
$[p,q]$ 7-branes one can translate this geometric classification into a
classification of allowed stacks of coincident 7-branes on $\mathbb{P}^{1}$.

The classification starts from the notion of a Weierstrass model for the
elliptic fibration: An elliptic curve $\mathbb{E}_{\tau}$ with modular
parameter $\tau$ can be represented as the hypersurface

$\displaystyle P_{W}:=y^{2}-(x^{3}+fxz^{4}+gz^{6})=0$ (5.5)

in the weighted projected space $\mathbb{P}_{2,3,1}$ with homogenous
coordinates $[x:y:z]$. Here $f$ and $g$ are complex parameters which determine
the complex structure or modular parameter $\tau$ as

$\displaystyle
j(\tau)=\frac{4(24f)^{3}}{\Delta}\,,\qquad\Delta=4f^{3}+27g^{2}\,,$ (5.6)

where $j(\tau)$ is the $SL(2,\mathbb{Z})$ invariant Jacobi $j$-function. When
the discriminant $\Delta$ vanishes, the elliptic curve $\mathbb{E}_{\tau}$
degenerates. This description is promoted to a Weierstrass model for the
elliptically fibered K3 surface by allowing $f$ and $g$ – and hence $\tau$ –
to vary suitably over the base $\mathbb{P}^{1}$.999Since $\mathbb{P}^{1}$ is
compact, $f$ and $g$ cannot be globally defined functions, but must rather
represent sections of a certain line bundle, whose degree is fixed uniquely by
the requirement that the elliptic fibration be Calabi–Yau. In fact, $f$ and
$g$ must be sections of $\bar{K}_{\mathbb{P}1}^{4}$ and
$\bar{K}_{\mathbb{P}1}^{6}$, respectively, where $\bar{K}_{\mathbb{P}1}={\cal
O}_{\mathbb{P}^{1}}(2)$ is the anti-canonical bundle on $\mathbb{P}^{1}$. The
possible types of singularities in the elliptic fiber are classified as in
Table 7.

type | ${\rm ord}(f)$ | ${\rm ord}(g)$ | ${\rm ord}(\Delta)$ | sing. | $G$
---|---|---|---|---|---
$I_{0}$ | $\geq 0$ | $\geq 0$ | $0$ | $-$ | $-$
$I_{1}$ | $0$ | $0$ | $1$ | $-$ | $-$
$II$ | $\geq 1$ | $1$ | $2$ | $-$ | $-$
$III$ | $1$ | $\geq 2$ | $3$ | $A_{1}$ | ${SU}(2)$
$IV$ | $\geq 2$ | $2$ | $4$ | $A_{2}$ | ${SU}(3)$
$I_{m}$ | $0$ | $0$ | $m$ | $A_{m-1}$ | ${SU}(m)$
$I_{m-6}^{\ast}$ | $2$ | $3$ | $m$ | $D_{m-2}$ | ${SO}(2m-4)$
$IV^{\ast}$ | $\geq 3$ | $4$ | $8$ | $E_{6}$ | ${E}_{6}$
$III^{\ast}$ | $3$ | $\geq 5$ | $9$ | $E_{7}$ | ${E}_{7}$
$II^{\ast}$ | $\geq 4$ | $5$ | $10$ | $E_{8}$ | ${E}_{8}$
non-min | $\geq 4$ | $\geq 6$ | $\geq 12$ |  |
Table 7: Kodaira table for singular fibers of an elliptic K3 [114, 115, 116].
The non-minimal enhancements in the last row can lie at finite or infinite
distance [117]; in the latter case, one can associate a loop algebra to the
singularity.

###  5.2 F-theory on Elliptic Four-folds

This geometric classification can be extended to higher-dimensional elliptic
fibrations. Relevant for compactifications of F-theory to four dimensions are
elliptic Calabi–Yau fourfolds $Y_{4}$. These automatically preserve 4d ${\cal
N}=1$ supersymmetry at the geometric level. From the Type IIB perspective, the
base $B_{3}$ of the elliptic fibration represents the physical
compactification space, while the additional two directions along the torus
fiber keep track of the type and location of the 7-branes. In this sense,
$B_{3}$ takes the role of the compactification space $X_{6}$ of Section 3.2,
including the effect of the orientifold action.101010The subscript in $B_{3}$
refers to its complex dimension. As was the case for the base of an elliptic
K3 surface, $B_{3}$ has positive curvature due to the 7-brane backreaction.

Such four-dimensional F-theory compactifications depend on two types of data -
the geometry of the elliptic fibration and additional gauge backgrounds.

####  5.2.1 Geometric Data: Gauge Group, Matter, and Couplings

The nature of F-theory as a theory of intersecting 7-branes is reflected in
the singularity structure of the elliptic fiber of $Y_{4}$ along strata of
codimension one, two and three on the base $B_{3}$:

  1. 1.

Non-abelian gauge algebras from codimension-one singularities: The
discriminant locus $\Delta$ on $B_{3}$ is defined as the vanishing locus of
the combination $4f^{3}+27g^{2}$ of the sections $f$ and $g$ that enter the
definition of the Weierstrass model. It represents a holomorphic four-cycle on
$B_{3}$ which is identified with the cycle wrapped by the 7-branes. In
general, the 4-cycle class of $\Delta$ can be decomposed as

$\displaystyle[\Delta]=\sum_{i}a_{i}[\Sigma_{i}]+[\Delta^{\prime}]\,,$ (5.7)

where the holomorphic four-cycle $\Sigma_{i}$ carries a stack of 7-branes
associated with a vanishing order $a_{i}$ in Table 7, while
$[\Delta^{\prime}]$ is the remaining piece of the discriminant with vanishing
order $1$. The non-abelian gauge algebra $G_{i}$ associated with $\Sigma_{i}$
can be read off from Table 7, where in addition one has to take into account
monodromies [118, 119, 120] along $\Sigma_{i}$ which can lead to smaller gauge
algebras than on K3, including all non-simply laced ones. As in the Type IIB
orientifold context, the inverse gauge coupling squared associated with
$G_{i}$ is set by the volume of the wrapped cycle $\Sigma_{i}$.

  2. 2.

Localised matter from codimension-two singularities: Over special curves
$C_{\bf R}$, extra charged massless matter multiplets reside which transform
in some representation ${\bf R}$ of the gauge algebra. These curves $C_{\bf
R}$ correspond to the intersection loci of the 7-branes, including possible
self-intersections. The charged massless matter fields arise from open
$(p,q)$-strings stretched between the intersecting 7-branes. One can
systematically identify the matter curves by searching for enhancements in the
vanishing orders of the Weierstrass model data, which signal an enhancement in
the singularity structure of the elliptic fiber [121, 120].

  3. 3.

Yukawa couplings from codimension-three singularities: At the intersection of
two or more matter curves, Yukawa interactions between the matter multiplets
are localised [122, 123, 124, 125, 126]. The strength of the couplings depends
on the overlap of the wavefunctions of the participating massless modes. The
physics rationale behind these couplings is completely analogous to the Type
II orientifold setting (see Section 2.3). Geometrically, to each such Yukawa
point one can associate a higher singularity type in the fiber, and the
resulting Yukawa couplings follow from the group theoretically allowed triple
couplings within the associated higher symmetry group. Note that the higher
singularity types characterising the singularity enhancements over curves and
points do not correspond to gauge algebras in the four-dimensional effective
action.

The discussion so far has focused on the non-abelian part of the gauge algebra
and its charged matter and their Yukawa type couplings. This data can, to a
certain extent, be described already locally by analysing the gauge theory
along a stack of 7-branes together with additional matter on curves where
other branes intersect the brane stack. The local approach amounts to zooming
into the neighbourhood of one of the components $\Sigma_{i}$ of the
discriminant. This is analogous to the local approach described in the
perturbative context in Section 3.2. The benefits of such a local approach in
F-theory have been advocated in particular in and following [122, 123, 124,
126].

Understanding non-Cartan abelian gauge symmetries, on the other hand, requires
going beyond a local analysis of a given brane stack. As in the Type II
orientifolds reviewed in Section 3, non-Cartan abelian gauge symmetries depend
on global, rather than local, data: The diagonal abelian gauge symmetries
associated with the $U(N)$ gauge groups can acquire a Stückelberg mass, and
only certain linear combinations of $U(1)$s from different brane stacks remain
as massless $U(1)$s. Similarly, the existence of a massless non-Cartan $U(1)$
depends on global geometric properties of the elliptic fibration in F-theory.

The underlying construction builds on the definition of F-theory via duality
with eleven-dimensional M-theory. Compactification of M-theory on the same
elliptic fourfold $Y_{4}$ gives a theory in three dimensions, which is
identified with the compactification of F-theory on $Y_{4}$ times an
additional circle $S^{1}_{\rm M}$ (see e.g. the reviews [104, 108] for details
and references). In particular, this approach admits a detailed derivation of
the F-theory effective action, as explained in [127]. To treat the M-theory
compactification on $Y_{4}$ in supergravity, one must resolve the
singularities in the elliptic fiber of $Y_{4}$ over $\Delta$. Let us denote
the four-fold after the resolution as $\hat{Y}_{4}$. At the abelian level, the
gauge potentials in M-theory arise from the expansion of the M-theory three-
form in terms of harmonic 2-forms of $\hat{Y}_{4}$,

$\displaystyle
C_{3}=\sum_{\alpha=1}^{h^{1,1}(\hat{Y}_{4})}A_{\alpha}\wedge\omega_{\alpha}+\ldots$
(5.8)

for $\omega_{\alpha}$ a basis of $H^{1,1}(\hat{Y}_{4})$. Not all of these
three-dimensional gauge potentials correspond to gauge fields in the four-
dimensional F-theory. It turns out that only two types of 2-forms give rise to
a gauge field in three dimensions which is associated with a 7-brane gauge
field in F-theory:

First, the resolution process from $Y_{4}$ to $\hat{Y}_{4}$ induces a set of
exceptional divisors $E_{\alpha_{i}}$ on $\hat{Y}_{4}$ which are fibered by
rational curves over the four-cycles $\Sigma_{i}$. Here
$\alpha_{i}=1,\ldots,{\rm rk}(G_{i})$ runs over the generators of the Cartan
subalgebra of the Lie algebra $G_{i}$. The associated gauge potentials are the
abelian Cartan gauge fields associated with the non-abelian gauge sector
[128].

Non-Cartan U(1) gauge potentials on the other hand require a different source
of harmonic 2-forms: These are provided by so-called rational sections of
$\hat{Y}_{4}$ [112, 129, 128]. A rational section embeds the base $B_{3}$ into
$\hat{Y}_{4}$ as a divisor, and the different ways of doing so gives
additional independent dual harmonic 2-form classes. The set of rational
sections $S_{a}$ forms a finitely generated abelian group, the Mordell–Weil
group ${\rm MW}(\hat{Y}_{4})$, whose free part is in one-to-one correspondence
with non-Cartan $U(1)_{a}$ gauge fields in F-theory. By taking a certain
linear combination of the two-forms associated with the extra section, the
zero-section and other 2-form classes pulled back from $B_{3}$ one obtains an
element $\sigma(S_{a})\in H^{1,1}(\hat{Y}_{4})$ (called the image of the
Shioda map) with the property that

$C_{3}=\sum_{a=1}^{{\rm rk}({\rm
MW}(\hat{Y}_{4}))}A_{a}\wedge\sigma(S_{a})+\ldots$ (5.9)

yields the non-Cartan gauge potentials $A_{a}$ [130, 131, 132]. More details
and a guide to the vast literature on this topic are provided in the reviews
[108, 109].

A subtle point concerns that fate of the diagonal $U(1)$ gauge groups which
play such an important role for model building in Type II orientifolds as
detailed in Sections 2 and 3. For instance, in Type IIB orientifolds, the
diagonal $U(1)$ gauge field in the $U(N)$ group can acquire a Stückelberg mass
even before the effect of gauge fluxes is taken into account. We will refer to
such $U(1)$s as geometrically massive. Depending on the geometric details, the
$U(1)$ gauge group is broken to a $\mathbb{Z}_{k}$ gauge group [37]. If $k\geq
2$, this manifests itself in a corresponding selection rule on the allowed
couplings, while for $k=1$ no such selection rules survive and the
geometrically massive $U(1)$ cannot be detected at the level of the effective
action. In F-theory, a geometrically massive $U(1)$ is realised directly in
terms of its remnant discrete $\mathbb{Z}_{k}$ subgroup. Such discrete
symmetries are associated with multi-sections which occur on genus-one
fibrations not possessing a zero-section [113]. On the other hand, if a
geometrically massive $U(1)$ does not leave behind a $\mathbb{Z}_{k}$ symmetry
for $k\geq 2$, it only manifests itself at worst in certain
$\mathbb{Q}$-factorial terminal singularities over curves in the base [133,
134].

Finally, the global structure of the 7-brane gauge group, as opposed to the
gauge algebra, is encoded in the torsional part of the Mordell–Weil group
[129, 135] in F-theory, with additional subtleties appearing in presence of
abelian gauge algebra factors [136, 109].111111As throughout this article, we
will mostly not distinguish between the gauge group and the algebra in the
sequel.

Apart from the gauge group supported on 7-branes, four-dimensional F-theory
compactifications exhibit a gauge sector on spacetime-filling D3-branes as
well as an abelian gauge sector in the Type IIB Ramond-Ramond sector [127,
137]. These extra gauge sectors cannot support chiral charged matter.

####  5.2.2 Gauge Backgrounds: Stückelberg Terms, Matter Multiplicities

In addition to this purely geometric structure, the gauge background affects
both the gauge algebra and the matter spectrum. This is in complete analogy to
the effect of gauge background in Type IIB orientifolds, see Section 3.2. For
simplicity we will only consider abelian gauge backgrounds. Part of the
information is encoded in the background value of the gauge field strengths
along the compactified dimensions. In the language of M-theory, this
corresponds to a background for the M-theory field strength $G_{4}=dC_{3}$,

$\displaystyle G_{4}=\sum_{\alpha_{i}=1}^{{\rm
rk}(G_{i})}F_{\alpha_{i}}\wedge[E_{\alpha_{i}}]+\sum_{a=1}^{{\rm rk}({\rm
MW}(\hat{Y}_{4}))}F_{a}\wedge[\sigma(S_{a})]\,.$ (5.10)

Here $F_{\alpha_{i}}$ and $F_{a}$ take values in $H^{1,1}(B_{3})$ and
parametrise the internal part of the Cartan and non-Cartan $U(1)$ field
strengths, respectively. The flux must satisfy the following constraints:

  * -

The flux background is subject to a D3-brane tadpole equation, which can be
elegantly written as [138]

$\displaystyle n_{\rm D3}+\frac{1}{2}\int_{\hat{Y}_{4}}G_{4}\wedge
G_{4}=\chi(\hat{Y}_{4})\,.$ (5.11)

The Euler characteristic of $\hat{Y}_{4}$, $\chi(\hat{Y}_{4})$, accounts for
curvature contributions to the D3-brane tadpole on the 7-branes, and $n_{\rm
D3}$ is the number of spacetime-filling D3-branes.

  * -

The flux must obey the Freed-Witten quantisation condition
$G_{4}+\frac{1}{2}c_{2}(\hat{Y}_{4})\in H^{2}(\hat{Y}_{4},\mathbb{Z})$, where
$c_{2}$ represents the second Chern class [139, 140, 141]. Whenever
$\frac{1}{2}c_{2}(\hat{Y}_{4})$ is not by itself an integer class, this
condition implies that the gauge background must be non-vanishing in a
consistent compactification.

  * -

The flux induces a D-term supersymmetry condition involving the Kähler moduli,
which is satisfied if

$\int_{B_{3}}J\wedge
F_{A}\wedge\pi_{\ast}(\omega_{A}\wedge\omega_{B})=0\,,\qquad\quad\omega_{A}\in\\{[E_{\alpha_{i}}],[\sigma(S_{a})]\\}\,.$
(5.12)

This is essentially the condition (3.14). The F-term condition (3.13)
translates into the condition $G_{4}\in H^{2,2}_{\rm vert}$, the primary
vertical subspace [142], which is automatically fulfilled for the choice
(5.10).

The gauge background must also be specified at the level of the underlying
three-form potential in M-theory rather than merely the field strength. The
gauge background is fully specified by an element of the Déligne cohomology
group [143], which can partly be parametrised by the Chow group of
$\hat{Y}_{4}$ [144, 145].

The gauge background plays at least three different roles in the model
building context in F-theory:

  * -

A Cartan gauge background breaks the non-abelian gauge algebra to a subgroup
involving abelian gauge factors. This effect will be discussed in more detail
in the next section.

  * -

Cartan and non-Cartan gauge fluxes both in general induce Stückelberg mass
terms for the abelian gauge symmetries (along with a D-term potential for the
Kähler moduli). If one collectively denotes by $\omega_{A}$ the two-forms
$[E_{\alpha_{i}}]$ and $\sigma(S_{a})$, then the Stückelberg mass matrix
$M_{AB}$ for the $U(1)_{A}$ gauge potentials is proportional to

$\displaystyle
M_{AB}\propto\int_{\hat{Y}_{4}}G_{4}\wedge\omega_{A}\wedge\omega_{B}\,.$
(5.13)

For Standard Model constructions, it must therefore be checked whether the
hypercharge $U(1)_{Y}$ remains massless in presence of gauge backgrounds.

  * -

The gauge background determines the multiplicities of the massless chiral
superfields charged under the gauge algebra. In particular, the chiral index
can be expressed as an integral of the form [126, 146, 147, 148, 149]

$\displaystyle\chi({\mathbf{R}})=n_{\bf
R}-n_{\bf\bar{R}}=\int_{S_{\mathbf{R}}}G_{4}\,,$ (5.14)

where $S_{\mathbf{R}}$ is a complex surface on $\hat{Y}_{4}$ which can be
attributed to every representation of the gauge group as detailed in [108] and
references therein. This is the F-theory version of the expression (3.19). The
amount of vector-like matter, i.e. the multiplicities $n_{\bf R}$ and
$n_{\bf\bar{R}}$ rather than merely their difference, is likewise encoded in
the gauge background, but is sensitive to the finer information contained in
the Déligne cohomology group [144, 145].

###  5.3 Standard Model Constructions

There are two different classes of Standard Model realisations in F-theory:

####  5.3.1 Direct Standard Model Constructions

The first approach is to directly engineer, in the geometry of the elliptic
four-fold, Standard Model quivers of a form similar to the ones reviewed in
Section 2. In such constructions, the gauge background controls the
multiplicities of the massless charged matter fields and provides a mass term
for additional abelian gauge fields other than hypercharge, if present.
According to our discussion of the previous section, a geometric realisation
of a gauge algebra of the form

$\displaystyle G=SU(3)\times SU(2)\times\prod_{a}U(1)_{a}$ (5.15)

requires

  * •

a fibral singularity of either Kodaira Type $I_{3}$ or Type IV without
monodromy along a divisor $\Sigma_{SU(3)}$,

  * •

a fibral singularity of Kodaira Type $I_{2}$ or Type III, or of Type IV with
monodromy, along a divisor $\Sigma_{SU(2)}$,

  * •

additional $U(1)_{a}$ gauge group factors as a consequence of extra rational
sections of the fibration.

Note that in direct Standard Model constructions which cannot be unhiggsed to
a theory with a GUT group, $\Sigma_{SU(3)}$ and $\Sigma_{SU(2)}$ lie in
different $B_{3}$ homology classes.

One linear combination of the abelian group factors must correspond to
hypercharge $U(1)_{Y}$, while the orthogonal linear combinations must acquire
a Stückelberg mass by a suitable choice of gauge background. This approach to
Standard Model building is comparable in spirit to the constructions in
perturbative Type II orientifolds even though the F-theory framework is
considerably more general and includes brane configurations which are not
realisable in perturbative constructions. In particular, the engineering of
the $SU(3)$ and $SU(2)$ factors via Kodaira fibers of Type IV or Type III has
no perturbative analogue as it involves mutually non-local 7-brane stacks.

The first realisation of this approach in [150, 151] specialises a fibration
with Mordell–Weil group of rank two constructed in [152, 153] to accommodate
$SU(3)\times SU(2)\times U(1)_{a}\times U(1)_{b}$ (with the non-abelian part
realised via $I_{3}$ and $I_{2}$ fibers), where in realistic constructions one
linear combination of the abelian factors must be massive through the gauge
background. Instead of an additional massive $U(1)$, dimension four proton
decay is prohibited in the construction of [154] by a discrete symmetry group
$\mathbb{Z}_{2}$ (matter parity). In [155], one of the toric elliptic fibers
dubbed $F_{11}$ in [156] automatically encodes $(SU(3)\times SU(2)\times
U(1))/\mathbb{Z}_{6}$ without additional symmetries to protect proton decay;
this construction admits a plethora of chiral three-general models [157, 158].
A systematic analysis of $(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_{6}$
models (containing these as special subcases) has been undertaken in [159,
160, 161]. Non-perturbative Standard Model with Kodaira fibers of Type IV and
III are investigated in [162]; in such scenarios, for instance the $SU(3)$
group can be non-higgsable [163, 164], while the incorporation of the abelian
factors is to date less well understood.

####  5.3.2 GUT Constructions: Georgi-Glashow $SU(5)$ GUTs

The second approach is via Grand Unified Theories (GUTs) and was initiated in
[122, 123, 124, 126]. This approach makes full use of the non-perturbative
nature of the F-theory construction because unlike perturbative Type II
orientifolds [165, 70], F-theory models admit a natural engineering also of
the top quark Yukawa couplings with order one coefficients [122, 123, 124,
125, 126]. The general idea is to geometrically engineer a GUT group $G$
containing $SU(3)\times SU(2)\times U(1)_{Y}$ and to break $G$ to the latter
by a suitable choice of gauge background.

In the context of F-theory, GUT groups $G=SU(5)$ (beginning with [122, 123,
124, 126]), $SO(10)$ [166] and $E_{6}$ [167, 168, 169] have been studied in
detail (see also the reviews [105, 106] for further references). In the sequel
we will illustrate the key ideas in the context of Georgi-Glashow $SU(5)$ GUT
theories.

In Georgi-Glashow $SU(5)$ GUTs, the Standard Model gauge algebra is embedded
into $SU(5)$,

$\displaystyle SU(3)\times SU(2)\times U(1)_{Y}\subset SU(5)\,.$ (5.16)

In an ${\cal N}=1$ supersymmetric framework, the charged matter content of the
MSSM organises into three generations of chiral multiplets transforming as the
${\bf\bar{5}_{m}}$ and ${\bf 10}$ representations of $SU(5)$:

$\displaystyle{\bf\bar{5}_{m}}$ $\displaystyle\rightarrow$
$\displaystyle({d^{c}},L)$ (5.17) $\displaystyle{\bf 10}$
$\displaystyle\rightarrow$ $\displaystyle({Q},{u^{c}},{e^{+}})\,.$ (5.18)

The MSSM Higgs doublet is part of an additional vector-like pair of chiral
multiplets, $H_{u}$ and $H_{d}$, which arise from the decomposition

$\displaystyle{\bf
5}_{H}\rightarrow(T_{u},H_{u})\,,\qquad{\bf\bar{5}}_{H}\rightarrow(T_{d},H_{d})\,.$
(5.19)

The triplets $T_{u}$ and $T_{d}$ do not exist in the MSSM and must be
sufficiently heavy by the process of doublet-triplet splitting such that they
are not only unobservable at the massless level but also do not induce
dangerous dimension-five proton decay operators in the low-energy effective
theory. Finally, extensions of the MSSM by right-handed neutrinos $\nu^{c}$
contain extra $SU(5)$ singlets.

The MSSM Yukawa couplings are inherited from the two possible Yukawa couplings
of the $SU(5)$ theory:

$\displaystyle{\bf 10}\,{\bf 10}\,{\bf 5}_{H}$ $\displaystyle\rightarrow$
$\displaystyle{Q}\,{u^{c}}\,H_{u}$ (5.20) $\displaystyle{\bf
10}\,{\bf\bar{5}_{m}}\,{\bf\bar{5}}_{H}$ $\displaystyle\rightarrow$
$\displaystyle{Q}\,{d^{c}}\,H_{d}+{e^{+}}\,L\,H_{d}\,.$ (5.21)

By contrast, Yukawa couplings of the form ${\bf
10}\,{\bf\bar{5}_{m}}\,{\bf\bar{5}_{m}}$ or ${\bf
10}\,{\bf\bar{5}}_{H}\,{\bf\bar{5}}_{H}$ would lead to phenomenologically
unacceptable dimension-four proton decay operators and must hence be
suppressed by additional selection rules which distinguish between the
${\bf\bar{5}_{m}}$ and the ${\bf\bar{5}}_{H}$ representation.

This general framework can be embedded into F-theory as follows:

GUT group, matter, Yukawas

The GUT group $SU(5)$ can be realised on a stack of 7-branes encoded in a
Kodaira Type $I_{5}$ singularity in the elliptic fiber over a divisor
$\Sigma_{SU(5)}$ on the base $B_{3}$. Alternatively, the $SU(5)$ GUT group can
also by itself be embedded into a higher gauge group which is broken
accordingly by a gauge background, as studied systematically in [170, 171].

Additional non-Cartan $U(1)$ or discrete gauge symmetries may be realised via
the geometric mechanisms described in the previous section. At the geometric
level, such extra symmetries require a further tuning of the Weierstrass
model. The extra $U(1)$ gauge symmetries, if present, would have to be massive
by a flux induced Stückelberg mechanism.

For definiteness, we will focus in the sequel on constructions with a
geometrically tuned $SU(5)$ symmetry associated with a Kodaira Type $I_{5}$
fiber. Such a singularity can be conveniently engineered by applying Tate’s
algorithm [119, 172], as reviewed in the $SU(5)$ GUT context in [106]. The
$SU(5)$ charged matter multiplets are localised on curves on $\Sigma_{SU(5)}$
where the GUT brane stack intersects other 7-branes. The two relevant
representations, ${\bf R}={\bf 10}$ and ${\bf R}={\bf 5}$, arise on curves
$C_{\bf R}$ where the Kodaira type of the fibers enhance as follows:

$\displaystyle I_{5}\,(A_{4})$ $\displaystyle\rightarrow$ $\displaystyle
I_{6}\,(A_{5}):\qquad C_{\bf 5}$ (5.22) $\displaystyle I_{5}\,(A_{4})$
$\displaystyle\rightarrow$ $\displaystyle I_{1}^{\ast}\,(D_{5}):\qquad C_{\bf
10}\,.$ (5.23)

In the parantheses we display the the symmetry groups associated with these
enhanced Kodaira singularities (see Table 7). In a Weierstrass model with an
$I_{5}$ singularity over a divisor, both types of higher enhancements occur
generically without further tuning. From a Type IIB orientifold point of view,
$C_{\bf 5}$ is the intersection of the GUT brane stack with another 7-brane
away from an orientifold plane, while $C_{\bf 10}$ represents the intersection
of the GUT brane stack with its orientifold image on top of an O7-plane. In
absence of extra massive $U(1)$ or discrete gauge groups differentiating
between ${\bf 5_{m}}$ and ${\bf 5}_{H}$, both representations generically
reside on the same matter curve, while extra such symmetries lead to a
corresponding splitting of the matter curves. See also the discussion of Class
$(iii)$ models in Section 2.2.7.

At the intersection points of the matter curves, the overlap of the matter
wavefunctions gives rise to the Yukawa couplings allowed by gauge symmetry.
The intersection points are characterised by further singularity enhancements:

$\displaystyle(I_{6},I_{1}^{\ast})$ $\displaystyle\rightarrow$
$\displaystyle{\rm IV}\,(E_{6}):\qquad{\bf 10}\,{\bf 10}\,{\bf 5}$ (5.24)
$\displaystyle(I_{6},I_{1}^{\ast})$ $\displaystyle\rightarrow$ $\displaystyle
I_{2}^{\ast}\,(D_{6}):\qquad\,\,{\bf 10}\,{\bf\bar{5}}\,{\bf\bar{5}}$ (5.25)

The second type of enhancement corresponds to a symmetry group $SO(12)$ and
the associated couplings are present also in perturbative Type II
orientifolds; the enhancement to a symmetry group $E_{6}$, on the other hand,
cannot be achieved perturbatively. The existence of such couplings is a
trademark of mutually non-local $[p,q]$ 7-branes in F-theory.

Note that without extra $U(1)$ or discrete gauge symmetries, the
${\bf\bar{5}_{m}}$ and the ${\bf 5}_{H}$ representations are also
indistinguishable at the level of the Yukawa couplings. In particular, this
means that the phenomenologically unacceptable coupling ${\bf
10}\,{\bf\bar{5}_{m}}\,{\bf\bar{5}_{m}}$ or ${\bf
10}\,{\bf\bar{5}}_{H}\,{\bf\bar{5}}_{H}$ cannot be avoided without such
selection rules [173].

GUT breaking

There are two different ways to break the $SU(5)$ GUT group to the Standard
Model gauge group: Either via a dynamically generated vacuum expectation value
for a GUT Higgs or by a topological gauge background. The first mechanism
faces the challenge of explaining the origin of the Higgs potential required
for the GUT breaking. The smallest representation in which the GUT Higgs field
can occur is the ${\bf 24}$ representation, and there are two candidates for
such a GUT Higgs: If the divisor $\Sigma_{\rm SU(5)}$ is non-rigid inside
$B_{3}$, there arise $h^{0}(\Sigma_{\rm SU(5)},K_{\Sigma_{\rm SU(5)}})$
massless chiral multiplets in the ${\bf 24}$, whose bosonic components
represent geometric deformation moduli of the brane stack. The Higgsing of
$SU(5)\to SU(3)\times SU(2)\times U(1)_{Y}$ amounts to a geometric deformation
of the brane stack into an intersecting brane model. The end result can
equivalently be interpreted as a direct Standard Model construction in which
the $SU(3)$ and $SU(2)$ brane stacks lie in the same homology class.
Alternatively, if $h^{1}(\Sigma_{\rm SU(5)})>0$ there exists a corresponding
number of continuous Wilson line moduli, whose VEV can likewise break the
$SU(5)$ gauge group.

In view of the difficulty of accounting for the symmetry breaking potential
dynamically, the second, topological mechanism of GUT breaking is particularly
attractive. The $SU(5)$ group can be broken to the Standard Model gauge group
by an internal gauge background for the hypercharge $U(1)_{Y}$, the abelian
subgroup of $SU(5)$ associated with a generator $T_{Y}={\rm
diag}(\frac{1}{3},\frac{1}{3},\frac{1}{3},-\frac{1}{2},-\frac{1}{2})$. This
results in the following breaking pattern:

$\displaystyle SU(5)$ $\displaystyle\rightarrow SU(3)\times SU(2)\times
U(1)_{Y}$ (5.26) $\displaystyle{\bf 24}$ $\displaystyle\rightarrow({\bf
8},{\bf 1},0)\oplus({\bf 1},{\bf 3},0)\oplus({\bf 1},{\bf 1},0)\oplus({\bf
3},{\bf 2},\tfrac{5}{6})\oplus({\bf\bar{3}},{\bf 2},-\tfrac{5}{6})$
$\displaystyle{\bf 10}$ $\displaystyle\rightarrow({\bf 3},{\bf
2},\tfrac{1}{6})\oplus({{\bf\bar{3}}},{\bf 1},-\tfrac{2}{3})+({\bf 1},{\bf
1},1)$ $\displaystyle{\bf 5}$ $\displaystyle\rightarrow({\bf 3},{\bf
1},-\tfrac{1}{3})\oplus({\bf 1},{\bf 2},\tfrac{1}{2})\,.$

The ${\bf 24}$ here refers to matter fields propagating on the 7-brane stack
along $\Sigma_{SU(5)}$, so-called bulk matter. If $\pi_{1}(\Sigma_{SU(5)})$ is
non-trivial, the $U(1)_{Y}$ gauge background can be taken to be flat,
corresponding to a discrete Wilson line. This discrete version of the
continuous GUT breaking via Wilson line moduli is studied in [174].

Another possibility which is available more generally is to consider a
hypercharge gauge background [175, 126] characterised by a non-trivial line
bundle $L_{Y}$ on $\Sigma_{SU(5)}$ with $c_{1}(L_{Y})=\frac{1}{2\pi}F_{Y}\in
H^{2}(\Sigma_{SU(5)})$. The hypercharge flux is subject to a number of
conditions in order for the gauge breaking mechanism to meet some basic
phenomenological criteria:

  * -

A hypercharge gauge background induces a Stückelberg mass for the hypercharge
gauge boson unless $F_{Y}$ is cohomologically trivial on $B_{3}$ [176]. This
condition can be written, at the level of cohomology, as

$\displaystyle\iota_{!}c_{1}(L_{Y})=0\,,$ (5.27)

where $\iota:\Sigma_{\rm SU(5)}\to B_{3}$ denotes the embedding of the divisor
$\Sigma_{\rm SU(5)}$ into the base and the Gysin map
$\iota_{!}:H^{2}(\Sigma_{\rm SU(5)})\to H^{4}(\Sigma_{\rm SU(5)})$ is defined
by first taking the Poincaré dual, then pushing forward at the level of
homology and finally taking the Poincaré dual again. As follows from (5.13),
the 4-form flux $G^{Y}_{4}$ associated with such a line bundle must take value
in the so-called remainder piece $H^{2,2}(\hat{Y}_{4})_{\rm rem}$ [177] in the
orthogonal decomposition

$\displaystyle H^{2,2}(\hat{Y}_{4})=H^{2,2}(\hat{Y}_{4})_{\rm hor}\oplus
H^{2,2}(\hat{Y}_{4})_{\rm vert}\oplus H^{2,2}(\hat{Y}_{4})_{\rm rem}\,,$
(5.28)

where the first two summands refer to the primary horizontal and primary
vertical subspaces of $H^{2,2}(\hat{Y}_{4})$, respectively.

  * -

The second condition on the gauge background arises by demanding that there
occur no exotic massless states in the representation $({\bf 3},{\bf
2},\frac{5}{6})+c.c$ from the decomposition (5.26) of the ${\bf 24}$
representation. These so-called lepto-quarks are absent only if all cohomology
groups of $L_{Y}^{\pm 5/6}$ on $\Sigma_{SU(5)}$ vanish,

$\displaystyle H^{i}(L^{\pm 5/6}_{Y})=(0,0,0)\,,$ (5.29)

which is a strong constraint. The power of $\tfrac{5}{6}$ reflects the
$U(1)_{Y}$ charge of the lepto-quarks. It turns out [123, 126] that rather
than an integral line bundle $L_{Y}$, one can consider a fractionally
quantised line bundle ${\cal L}_{Y}$ with the property that the lepto-quarks
are counted by the cohomology group $H^{i}({\cal L}^{\pm 1}_{Y})$, $i=0,1,2$.
The vanishing of these cohomology groups can be achieved for instance by
taking ${\cal L}_{Y}=E_{i}-E_{j}$ on a del Pezzo surface dPn with exceptional
curve classes $E_{i}$, $i=1,\ldots n$.

  * -

The Standard Model matter should appear in complete GUT multiplets. This means
that

$\displaystyle\int_{C_{\bf 10}}F_{Y}=0=\int_{C_{\bf\bar{5}_{m}}}F_{Y}\,.$
(5.30)

The requirement (5.30) is only a necessary condition which guarantees that the
chiral index of the MSSM representations do not differ within a GUT multiplet.
This must in fact be ensured also at the vector-like level.

Apart from being topological, one of the benefits of hypercharge GUT breaking
is that the doublet-triplet splitting problem can be solved by a suitable
restriction of the hypercharge gauge background to the Higgs curve. This will
be discussed momentarily.

On the other hand, the hypercharge flux breaking affects precision gauge
coupling unification via flux dependent subleading corrections to the gauge
kinetic function, as pointed out first in [126, 178]. This effect may be
counter-balanced by the appearance vector-like exotics for instance at
intermediate scales [179].

Proton decay and selection rules

A closer analysis of this and related effects and more generally of the
detailed realisation of the matter spectrum requires specifying the additional
selection rules invoked to prevent the phenomenologically excluded proton
decay operators by which GUT models are typically inflicted. As a minimal
requirement, the extra symmetry must distinguish the ${\bf 5_{m}}$ and ${\bf
5}_{H}$ representations and forbid the dangerous dimension four operators
${\bf 10}\,{\bf\bar{5}_{m}}\,{\bf\bar{5}_{m}}$ and ${\bf
10}\,{\bf\bar{5}}_{H}\,{\bf\bar{5}}_{H}$ while allowing for the Yukawa
couplings ${\bf 10}\,{\bf\bar{5}_{m}}\,{\bf\bar{5}}_{H}$ and ${\bf 10}\,{\bf
10}\,{\bf 5}_{H}$. Up to an overall normalisation, and assuming for now that
the charge assignments do not distinguish between the three matter families,
these conditions are met by an extra (massive) $U(1)$ gauge symmetry with
charge assignments

$\displaystyle{\bf 10}_{1}\,,\qquad({\bf\bar{5}_{m}})_{-q}\,,\qquad({\bf
5}_{H})_{-2}\,,\qquad({\bf\bar{5}}_{H})_{q-1}\,,\qquad q\neq\frac{1}{2}\,,$
(5.31)

or a suitable discrete subgroup thereof.

There are then two general possibilities: The assignment $q=3$ corresponds to
the so-called $U(1)_{X}$ symmetry, under which the charges of $H_{u}$ and
$H_{d}$ merely differ by a sign. In a geometric realisation of such an
additional symmetry (and no additional symmetries on top), $H_{u}$ and $H_{d}$
form a vector-like pair localised on a single curve $C_{{\bf 5}_{H}}$, and
altogether such models have three different types of localised matter curves,
$C_{{\bf 10}_{1}}$, $C_{({\bf 5_{m}})_{-3}}$ and $C_{({\bf 5}_{H})_{-2}}$
[173]. The first globally consistent realisations of this model with three
chiral generations of MSSM matter have been constructed in [148].

The $U(1)_{X}$ charge assignment has the disadvantage that it does not forbid
dimension-five proton decay operators. To prevent these from being generated,
the additional $U(1)$ (or, for that matter, discrete $\mathbb{Z}_{k}$)
selection rule must distinguish also between $H_{u}$ and $H_{d}$ other than
just by an overall sign of the charge, which fixes $q\neq 3$. A $U(1)$ gauge
symmetry with this property is said to be of Peccei-Quinn (PQ) type [123,
126]. The different charge assignments have been studied intensively in the
F-theory literature [180, 181, 179, 182, 183], to which we refer for details
and further references.

Note that the above discussion assumes that the extra selection rule does not
distinguish between the three different families of matter within the same
MSSM representation. Without this requirement a plethora of new symmetry
patterns opens up. At a phenomenological level, distinguishing between
families can explain the hierarchical structure of Yukawa couplings via the
Froggatt–Nielsen mechanism [184, 185].

Gauge background

The hypercharge flux must be complemented by additional gauge background which
controls the multiplicities of charged matter, in particular the chiral index
(5.14). At the level of 4-form flux, the flux background takes the form

$\displaystyle G_{4}=G_{4}^{\rm c}+G_{4}^{Y}\,,$ (5.32)

where $G_{4}^{Y}\in H^{2,2}_{\rm rem}(\hat{Y}_{4})$ denotes the hypercharge
flux background and $G_{4}^{\rm c}$ is the part of the flux background which
is blind to the GUT group breaking. To obtain the correct chiral index of MSSM
matter, the flux must satisfy the following conditions:

$\displaystyle\int_{S_{{\bf
R}_{\vec{q}}}}G_{4}^{Y}=0\,,\qquad\sum_{\vec{q}}\int_{S_{{\bf
R}_{\vec{q}}}}G_{4}^{\rm c}=3\,,\qquad{\bf R}={\bf 10},\,{\bf\bar{5}_{m}}\,.$
(5.33)

Here ${S_{{\bf R}_{\vec{q}}}}$ is the matter surface associated with
representation ${\bf R}_{\vec{q}}$ of $U(1)$ charge vector
$\vec{q}=(q_{1},\ldots,q_{n})$, where we are allowing for $n$ additional
massive abelian gauge groups. The first condition is essentially (5.30) and
ensures that all representations appear in complete GUT multiplets.

As for the Higgs sector, let us specialise for definiteness to a single extra
massive $U(1)$ gauge group with charges (5.31). If $q=3$, one requires that at
the chiral level

$\displaystyle\int_{S_{{\bf 5}_{H}}}G_{4}^{Y}=0\,,\qquad\int_{S_{{\bf
5}_{H}}}G_{4}^{\rm c}=0\,,$ (5.34)

but the gauge background must give rise to one vector-like pair of massless
$H_{u}$ and $H_{d}$ fields while both triplets $T_{u}$ and $T_{d}$ must be
absent at the massless level. This means that the restriction of $G_{4}^{\rm
c}$ and $G_{4}^{Y}$ must describe a line bundle of trivial curvature whose
cohomology groups are compatible with this vector-like spectrum. If, on the
other hand, the abelian symmetry is of PQ type ($q\neq 3$), then doublet
triplet splitting can already be imposed at the chiral level:

$\displaystyle\int_{{S_{{T_{u}}}}}G_{4}^{Y}+G_{4}^{\rm
c}=0=\int_{{S_{{T_{d}}}}}G_{4}^{Y}+G_{4}^{\rm c}\,,$ (5.35)
$\displaystyle\int_{{S_{{H_{u}}}}}G_{4}^{Y}+G_{4}^{\rm
c}=1=\int_{{S_{{H_{d}}}}}G_{4}^{Y}+G_{4}^{\rm c}\,.$

Here we have split the matter surface into a surface associated with the
triplets and the doublets in the decomposition (5.19). Both these surfaces
share the same base curve, but differ in the fiber. On top of these
conditions, no vector-like pairs of states must be generated by the flux
background to generate the exact massless MSSM matter content.

The second condition of (5.33) controlling the chiral matter content was for
the first time realised in globally consistent SU(5) GUTs in [148]. The
conditions (5.34) have been exemplified in simple toy models in the literature
[186, 187]. Their counterpart in PQ-type models, (LABEL:HiggsU1PQ-splitting),
has not yet been realised, as of this writing, in globally consistent F-theory
models where the hypercharge flux satisfies in addition the important
constraint (5.27). At the same time, considerations in the weak coupling limit
suggest that this should be achievable in principle [186].

Yukawa couplings and Flavour hierarchies

The Yukawa couplings can in principle be computed locally by evaluating the
overlap of the matter wavefunctions at the intersection points of the matter
curves. The local structure of the Yukawa couplings favours a mass hierarchy
among the different families whose wavefunctions overlap at the same point.
For details and a guide to the literature, we refer to [188, 189, 190, 191,
192, 193]. A remaining challenge for the future, however, is to connect such
local computations to the data of the globally defined F-theory model. An
alternative approach to explaining flavour hierarchies is via the
Froggatt–Nielsen mechanism [184, 185].

Acknowledgements

We thank I. García-Etxebarria, F. Quevedo, A. Sagnotti and A. M. Uranga for
discussions. FM is supported through the grants CEX2020-001007-S and
PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF A
way of making Europe. TW is supported in part by Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 Quantum
Universe 390833306 and by Deutsche Forschungsgemeinschaft through a German-
Israeli Project Cooperation (DIP) grant “Holography and the Swampland”.

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