IFT-UAM/CSIC-22-150

NIKHEF 2022-024

ZMP-HH/22-22

D-brane and F-theory Model Building

Fernando Marchesano,1 Bert Schellekens2,3 and Timo Weigand4,5

1Instituto de Física Teórica UAM-CSIC, c/Nicolás Cabrera 13-15, 28049 Madrid,
Spain

2NIKHEF Theory Group, Kruislaan 409, 1098 SJ Amsterdam, The Netherlands

3Instituto de Física Fundamental, CSIC, Serrano 123, Madrid 28006, Spain

4 II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee
149,
22607 Hamburg, Germany

5Zentrum für Mathematische Physik, Universität Hamburg, Bundesstrasse 55,
20146 Hamburg, Germany

###### Abstract

We review recent progress in the construction of four-dimensional vacua of
Type II string theory and F-theory which yield the Standard Model of particle
physics (SM) or extensions thereof. In Type II orientifold compactifications
the SM gauge group and chiral spectrum arise from the open string sector of
the theory, namely from stacks of D-branes. The universal features of the
chiral spectrum between various sets of D-branes allow for a general approach
to build realistic models, which can be implemented in different setups. We
describe the realisation of this strategy in Type II Calabi–Yau orientifold
compactifications and Rational Conformal Field Theories, discussing the
specific model building rules and features of each setting. The same
philosophy can be extended to F-theory constructions. These provide new model
building possibilities, as they combine the localisation properties of
D-branes with exceptional gauge groups and their representations.

This is an invited contribution to the Handbook of Quantum Gravity (Eds. C.
Bambi, L. Modesto, and I. L. Shapiro, Springer 2023).

###### Contents

  1. 1 D-branes and Orientifolds
    1. 1.1 D-branes and Chan-Paton Multiplicities
      1. 1.1.1 Oriented Strings: Groups and Representations
      2. 1.1.2 An Oriented String Model
    2. 1.2 From Oriented String Models to Orientifolds
      1. 1.2.1 The Need for O-planes
      2. 1.2.2 Unoriented Strings: Groups and Representations
  2. 2 D-brane Model Building: Generalities
    1. 2.1 Anomalies, Tadpoles and Axions
      1. 2.1.1 Non-abelian Anomalies
      2. 2.1.2 Mixed and Abelian Anomalies
    2. 2.2 The Simplest Examples
      1. 2.2.1 The $Y$-charge
      2. 2.2.2 The Quark Doublet
      3. 2.2.3 Anti-quarks
      4. 2.2.4 The Lepton Doublet
      5. 2.2.5 The Left-handed Positron
      6. 2.2.6 Weak Interaction Anomalies
      7. 2.2.7 The Four Classes of Models
    3. 2.3 Yukawa Couplings
  3. 3 Type II Orientifolds
    1. 3.1 Type IIA Orientifolds
      1. 3.1.1 Intersecting D6-brane Models
      2. 3.1.2 A Simple Model
      3. 3.1.3 Further EFT Features
    2. 3.2 Type IIB Orientifolds
      1. 3.2.1 D-branes at Singularities
      2. 3.2.2 Intersecting D7-branes
  4. 4 Rational Conformal Field Theories
    1. 4.1 Gepner Models
    2. 4.2 Modular Invariant Partition Functions
    3. 4.3 Fusion Rules and Simple Currents
    4. 4.4 Open String CFT
      1. 4.4.1 Boundary and Crosscap States
      2. 4.4.2 Orientifold Partition Functions
      3. 4.4.3 Channel Transformations
      4. 4.4.4 Integrality Conditions
      5. 4.4.5 Gauge Groups
      6. 4.4.6 Completeness, Integrality and Sewing Constraints
      7. 4.4.7 A Solution for C-diagonal Theories
      8. 4.4.8 Simple Current Results
    5. 4.5 RCFT Model Building
  5. 5 F-theory Model Building
    1. 5.1 From $[p,q]$ 7-branes to Exceptional Gauge Algebras
    2. 5.2 F-theory on Elliptic Four-folds
      1. 5.2.1 Geometric Data: Gauge Group, Matter, and Couplings
      2. 5.2.2 Gauge Backgrounds: Stückelberg Terms, Matter Multiplicities
    3. 5.3 Standard Model Constructions
      1. 5.3.1 Direct Standard Model Constructions
      2. 5.3.2 GUT Constructions: Georgi-Glashow $SU(5)$ GUTs

##  1 D-branes and Orientifolds

One of the aims of model building in string theory is to find string vacua
whose spectrum of massless string excitations in four dimensions resembles as
closely as possible the experimentally established and extremely successful
Standard Model of particle physics (SM). Embedding the SM within string theory
as a consistent theory of quantum gravity is more than merely a proof of
principle; it can be viewed as a first step in a more ambitious programme that
hopes to understand some of the mysteries of particle physics from a string
theoretic perspective.

The SM has a gauge group111Throughout this article, we will not distinguish
between the gauge algebra and gauge group unless stated explicitly. In
particular, we will not discern between $O(N)$ and $SO(N)$ D-brane groups, as
the difference cannot be determined by the perturbative arguments that we use.

$SU(3)\times SU(2)\times U(1)$ (1.1)

with matter in the representation

$3\times\left[({\bf 3},{\bf 2},\tfrac{1}{6})+({\bf\bar{3}},{\bf
1},-\tfrac{2}{3})+({\bf\bar{3}},{\bf 1},\tfrac{1}{3}),+({\bf 1},{\bf
2},-\tfrac{1}{2})+({\bf 1},{\bf 1},1)\right]\,.$ (1.2)

We will refer to these multiplets as $Q,u^{c},d^{c},L$ and $e^{+}$,
respectively. They are left-handed fermions. Right-handed fermions in the same
representations do not exist, and for this reason the spectrum is called
chiral. There may also exist non-chiral representations in nature. The most
prominent candidates are singlets $({\bf 1},{\bf 1},0)$, which could play the
role of right-handed neutrinos. They are in fact highly desirable in neutrino
physics, but their existence is still not established. Such singlets are non-
chiral, because their left- and right-handed components couple in the same way
to the Standard Model gauge group. This means in particular that a mass term
can be written down without breaking the Standard Model gauge group, i.e.
without making use of the Standard Model Higgs mechanism. In general, many
other non-chiral particles may exist. They can be in non-trivial Standard
Model representations, and a Dirac or Majorana mass term for these particles
is allowed by the Standard Model symmetries. We do not have any constraints on
how large such a mass could be. However, a common assumption in nearly all
attempts at string phenomenology is to allow such non-chiral particles to
exist in the massless spectrum, and to implicitly assume that by some
unspecified mechanism they acquire a mass, lower than the string scale but
beyond the reach of current experiments. Hence a main goal of string
phenomenology is to find string solutions that reproduce the spectrum (1.2)
chirally. This is just a first step. If that is not possible, the whole idea
is in serious doubt. A few string vacua have been identified in the literature
where this spectrum is indeed realised exactly, but even then the string
solution may differ from the SM in many more details, like for instance the
strength of the couplings.

The vast majority of the literature is about supersymmetric realisations of
the Standard Model, and this is also what we implicitly assume here, unless
stated otherwise. Non-supersymmetric realisations exist, but in general they
have serious stability issues. However, the main feature we focus on here,
which is the gauge group and the chiral fermion spectrum, is anyway the same.

Since 2011 we have known another particle in the “light” (in Planckian units)
spectrum: the Higgs boson. In supersymmetric theories, the minimal way to
accommodate it is to add two chiral supermultiplets to the massless spectrum,
$H_{u}$ and $H_{d}$. These two form a non-chiral pair, in the representation
$({\bf 1},{\bf 2},\tfrac{1}{2})+({\bf 1},{\bf 2},-\tfrac{1}{2})$, and
therefore they can develop a mass term. Hence they are part of the set of non-
chiral particles which string theory should reproduce.

Historically, the goal of finding the chiral Standard Model spectrum within
string theory was first achieved in the framework of compactifications of the
ten-dimensional $E_{8}\times E_{8}$ heterotic string, a theory of closed
strings. Compactification to four dimensions reduces the gauge group rather
naturally to $E_{6}$ with chiral matter in the 78 dimension representation,
and from there one could follow the well-known GUT path down to the Standard
Model, usually via the intermediate groups $SO(10)$ and $SU(5)$. This approach
was first considered in 1984 [1]. Although the “second string revolution” had
been started in that same year by Green and Schwarz with a paper on open
strings [2], that possibility was ignored for more than a decade. The ten-
dimensional open string gauge group, $SO(32)$, looked far less promising with
regard to the SM, and open strings added an extra complication that most –
with the exception of a few courageous ones – preferred to avoid.

###  1.1 D-branes and Chan-Paton Multiplicities

This all changed with the discovery of D-branes. At the endpoints of open
strings, boundary conditions must be imposed on the two-dimensional world-
sheet fields. It had been known for a long time that two kinds of boundary
conditions were possible: Neumann and Dirichlet boundary conditions. If the
former are imposed, the endpoints of the open string move through space-time
at the speed of light. But if Dirichlet conditions are imposed, the endpoints
of the string have a fixed space-time location. This implies the existence of
a special point in space-time, which breaks translation invariance. While this
option was immediately rejected by most people, it was realised around 1989
that one can impose Dirichlet boundary conditions in some directions of space-
time, and Neumann conditions in others [3]. Then the open string endpoints are
fixed in some directions, but can move freely in others. These endpoints then
sweep out a plane, or a membrane, which was called a Dirichlet brane or
D-brane for short. The existence of such a membrane does indeed break
translation invariance in directions orthogonal to it, but not in directions
parallel to it. It was then understood that our universe could live on top of
such a brane, without any contradictions with translation invariance in our
own space-time dimensions.

This suggests a picture where we are living on a four-dimensional space-time
membrane, embedded in the ten-dimensional space-time of string theory. There
would be six uncompactified directions. But this cannot work, because gravity
still detects all of space-time, and would therefore not exhibit the
$\frac{1}{r^{2}}$ behaviour characteristic of Newtonian gravity. Hence the
extra six dimensions must be compactified, but can remain relatively large as
long as limits from fifth-force experiments are respected.222Alternatively,
space in the extra six dimensions may be warped, rather than flat. All
interactions besides gravity are restricted to the brane, and hence impose no
constraints on the extra dimensions. From here more involved scenarios can be
formulated, because one may consider several higher dimensional branes on top
of the four-dimensional space-time that also wrap cycles of the compact
manifold, without constraining the strength of gravity. These D-branes can
then intersect each other in the extra six dimensions, which is a mechanism to
generate a 4d chiral spectrum [4]. This has led to the name “intersecting
D-brane models” [5, 6, 7]. See for instance [8, 9, 10, 11, 12] for reviews on
the subject, to which we refer for the vast original literature, and figure 1
for a pictorial representation of the idea.

Figure 1: Intersecting D-brane World scenario. Figure taken from [10].

Since the early days of string theory, it has been understood that one could
consistently assign multiplicities to the boundaries of open strings. These
are called Chan-Paton multiplicities [13]. The mode expansion of open strings
always contains a massless vector boson, just as the mode expansion of closed
strings always contains a massless rank-2 tensor field, the graviton. This
massless vector boson behaves like a gauge boson. If there is a Chan-Paton
multiplicity $N$, then there are in fact $N^{2}$ such gauge bosons, and by
inspecting their interactions one can verify that they gauge a group $U(N)$.
There can be many distinct D-branes in a theory, each defining a place for
open strings to end on. For simplicity, one may think of them as D-branes
wrapping different cycles on a compactification manifold. Each such brane $a$
can define a Chan-Paton multiplicity $N_{a}$. One may think of that
multiplicity in terms of $N_{a}$ D-branes stacked on top of each other, and
filling our four-dimensional space-time. These are called space-time filling
branes. In this situation, an observer in this space-time sees a gauge group

$U(N_{a})\times U(N_{b})\times U(N_{c})\times\ldots$ (1.3)

####  1.1.1 Oriented Strings: Groups and Representations

Open strings with both ends on the same brane $a$ give rise to a gauge group
$U(N_{a})$. The matter produced by such open strings includes vector bosons in
the adjoint representation of $U(N_{a})$. This immediately suggests the
possibility of open strings having their endpoints on different branes, say
$a$ and $b$. It is clear that the physical particles produced by such strings
must transform as a fundamental representation of $U(N_{a})$ as well as that
of $U(N_{b})$. This is strongly suggested by the multiplicity $N_{a}N_{b}$ of
these states, and can be verified by working out the scattering amplitudes.
Hence what one obtains from such strings are particles in the bi-fundamental
representation $({\bf N_{a}},{\bf N_{b}})$. The mass and spin of these
particles does not follow from this argument alone; we will return to this
later.

The gauge group $U(N_{a})$ has complex representations. Hence the multiplicity
$N_{a}$ can correspond to the representation ${\bf N_{a}}$ or its complex
conjugate, ${\bf\bar{N}_{a}}$. What determines which one of the two we get?
The open strings we are considering here are actually oriented. This defines a
sense of direction along the open string, or in other words, one can
consistently draw an arrow along it. Hence the two endpoints are distinct. Now
we assign the endpoint with an outgoing arrow to ${\bf N_{a}}$ and the one
with an incoming arrow to ${\bf\bar{N}_{a}}$. This is the origin of particles
in the adjoint representation: Open strings with both ends on the same brane
produce a representation in the tensor product of ${\bf N_{a}}$ with
${\bf\bar{N}_{a}}$.

####  1.1.2 An Oriented String Model

As a warm-up exercise let us construct a simple brane configuration that will
turn out to contain the Standard Model. Consider three stacks of $U(3)$
D-branes. Hence the gauge group is

$U(3)\times U(3)\times U(3)\,.$ (1.4)

Now connect each pair of stacks with an oriented string, such that each stack
contains one start- and one endpoint of the oriented strings, as shown in
figure 2.

Figure 2: Trinification, an example of an oriented D-brane model.

If these oriented strings have exactly three chiral (and hence massless)
modes, the resulting spectrum is

$3\times\left[({\bf 3},{\bf\bar{3}},{\bf 1})+({\bf 1},{\bf
3},{\bf\bar{3}})+({\bf\bar{3}},{\bf 1},{\bf 3})\right]\,.$

This spectrum occurs naturally as a step in one of the symmetry breaking paths
from $E_{6}$ Grand Unification to the Standard Model. There are 27 massless
states per family; 10 of them occur as mutually chiral pairs and there are two
right-handed neutrinos. To arrive at the Standard Model, one has to break the
last two factors to $SU(2)\times U(1)$ in a suitable way. This model has
plenty of phenomenological issues, but there is a bigger problem we will have
to deal with first.

###  1.2 From Oriented String Models to Orientifolds

####  1.2.1 The Need for O-planes

It turns out that in addition to branes another ingredient is always needed,
at least in supersymmetric theories: unoriented strings. These are strings,
open or closed, without a definite orientation. If an oriented open string,
with endpoints a and b, traces out a loop through space-time, the loop can
only be closed by linking the a and b boundaries to themselves. The resulting
string diagram is an annulus. But if a string is unoriented, the endpoints a
and b are indistinguishable, so one can also link b to a when closing the
loop. This results in a Moebius strip. Analogously, for orientable closed
strings the one-loop diagram is a torus, but in the case of unorientable
closed strings there is an additional diagram, the Klein bottle.

Constructions using unoriented strings are called “orientifolds” [14, 15], by
analogy to orbifolds. The idea is that one uses world-sheet parity as an
orbifold symmetry. For instance, the Klein bottle amplitude can be thought of
as a closed string sweeping out a closed loop, inverting its orientation
before closing the loop, see [16] for further details. Unorientable strings
enter the discussion not only as a logical possibility providing additional
model building options, but also because without them it is impossible to
construct supersymmetric string vacua with space-time filling D-branes.

To understand why, note that D-branes carry charge under the higher-form gauge
potentials in the Ramond-Ramond (RR) sector of the superstring theory. Indeed,
the worldvolume of a D$p$-brane enjoys an electric coupling of the form
$S=\mu_{p}\int_{{\rm D}p{\rm-brane}}C_{p+1}+\ldots$ with $\mu_{p}$ the
D$p$-brane charge. In compactifications of string theory to four dimensions, a
D$p$-brane which extends along the visible 3+1 dimensions must fill a
$(p-3)$-dimensional closed subspace – a so-called $(p-3)$-cycle $\Pi_{(p-3)}$
– on the six-dimensional compactification manifold $X_{6}$. Due to their RR
charge, the D$p$-branes act as a source for the $p$-form gauge potential along
the $9-p$ dimensions normal to the D$p$-brane on the compact internal
manifold. One can in fact characterise the charge under $C_{p+1}$ in terms of
the homology class $[\Pi_{(p-3)}]\in H_{(p-3)}(X_{6},\mathbbm{Z})$. This
source is constrained by Gauss’ law: The net charge on a compact space must
vanish. In homological terms this amounts to requiring that a configuration of
several D-branes satisfies

$\sum_{a}N_{a}[\Pi_{(p-3)}^{a}]=0,$ (1.5)

where we have included the multiplicity of D-branes wrapping each internal
cycle. If a D-brane couples to other RR fields due to non-trivial worldvolume
fluxes or curvature corrections, the induced D-brane charge must be cancelled
as well, a condition that can be formulated in a similar fashion to (1.5).

One can see that (1.5) cannot be satisfied in a supersymmetric D-brane
configuration unless additional objects with opposite charge and tension to
D-branes are introduced. The reason is that two mutually BPS D-branes will add
both their charge (as a sum of homology classes) and their tension (as a sum
of positive numbers). Therefore, the total tension in a supersymmetric D-brane
configuration is always a linear function of the D-brane total charge, and a
vanishing-charge condition like (1.5) cannot be compatible with only positive-
tension objects.

From a 4d viewpoint, the problem can be understood in terms of the
cancellation of the 4d dilaton tadpole. The 4d dilaton has positive couplings
(tension) to all D-branes, and hence the one-point functions with a single
dilaton external lines is a sum of positive terms. Hence the 4d dilaton
tadpole cannot cancel with only D-branes.

This requires the introduction of objects of negative charge and tension, the
so-called orientifold $p$-planes, or O$p$-planes for short. As it turns out,
such O$p$-planes are the fixed-point loci of involutions of the form
$\Omega{\cal R}(-1)^{N}$, where $\Omega$ is the worldsheet parity operator
that reverses the orientation of the string as described above and ${\cal R}$
acts as a geometric involution on the compactification space. Furthermore
$(-1)^{N}$ is an operator that makes $\Omega{\cal R}(-1)^{N}$ square to the
identity, and depends on the specific theory under consideration. In practice
an O$p$-plane is specified by a submanifold or a sum of submanifolds
$\Pi_{(p-3)}^{\rm O}$ of $X_{6}$ fixed by the geometric involution ${\cal R}$.
Because this involution must be a symmetry of the compactification, if a
D-brane internal worldvolume is not invariant under ${\cal R}$ there must be
another D-brane located at $\Pi_{(p-3)}^{a^{\prime}}={\cal R}\Pi_{(p-3)}^{a}$,
in order to identify their worldvolume theories. After O$p$-planes are
introduced, the RR tadpole condition (1.5) is modified to

$\sum_{a}N_{a}\left([\Pi_{(p-3)}^{a}]+[\Pi_{(p-3)}^{a^{\prime}}]\right)=Q_{{\rm
O}p}[\Pi_{(p-3)}^{\rm O}],$ (1.6)

where $Q_{{\rm O}p}=2^{p-4}$ is minus the relative charge of an O$p$-plane and
a D$p$-brane wrapping the same $(p-3)$-cycle. Now there is no obstruction to
building a supersymmetric D-brane configuration. The simplest one is to place
all D-branes wrapping homological cycles to the O-planes:
$[\Pi_{(p-3)}^{a}]=[\Pi_{(p-3)}^{\rm O}]$ and $N_{a}=Q_{{\rm O}p}/2$.

To sum up, combining positive tension objects like D-branes in a
supersymmetric fashion necessarily forces us to introduce negative-tension
objects known as O-planes.333Non-supersymmetric setups have additional
options, since one may introduce anti-branes. These objects appear when one
mods out an oriented configuration by an orientation-reversal symmetry of the
theory. From the Type II perspective they appear as non-dynamical objects of
negative tension, whose microscopic nature can only be unveiled in a non-
perturbative framework like F-theory.

####  1.2.2 Unoriented Strings: Groups and Representations

One direct consequence of the presence of O-planes is that new groups and
representations appear. Intuitively, if a D-brane is invariant under the
orientifold projection, the D-brane group becomes real: instead of $U(N)$ it
becomes either $SO(N)$ or $USp(N)$ (with $N$ even). $U(N)$ groups are also
still possible for some branes if their worldvolume is not fixed under the
orientifold involution. One may think of this as two branes mapped to each
other through the orientifold plane, a brane $a$ and its orientifold image
brane $a^{\prime}$. For a more precise discussion of brane groups in
orientifold models see Section 4.4.5.444In the RCFT literature, it is more
common to refer to the orientifold image brane $a^{\prime}$ as the conjugate
brane $a^{c}$. Both terms can be used interchangeably.

All open string states are bi-fundamentals of one or two brane groups. Hence
in D-brane model building one must realise all Standard Model matter as bi-
fundamental representations. In the unoriented case, open strings can connect
to branes passing through an O-plane, and reversing their orientation. This
implies that a bi-fundamental between two unitary branes can be of the form
$({\bf N},{\bf\bar{N}})$ in addition to $({\bf N},{{\bf N}})$. Here ${\bf N}$
denotes the fundamental (or vector) representation of $U(N),SO(N)$ or
$USp(N)$.

Furthermore, the two endpoints of an open string can be a fundamental
representation on the same brane. This allows the existence of rank-2 tensors.
These tensors can be symmetric and anti-symmetric, and will be denoted ${\bf
A}$ or ${\bf S}$ respectively. On unitary branes these tensors are complex
representations of the brane group: one can have ${\bf A}$, ${\bf\bar{A}}$,
${\bf S}$ and ${\bf\bar{S}}$. Rank two tensors can also occur for real groups,
and one can get adjoint representations of unitary groups if the open string
endpoints are on a brane $a$ and its orientifold image $a^{\prime}$.

Only complex representations give rise to chiral matter. This means that if a
theory contains left-handed fermions in the representation $n({\bf N},{\bf
N})+{\overline{n}}({{\bf\bar{N}}},{{\bf\bar{N}}})+m({\bf
N},{{\bf\bar{N}}})+{\overline{m}}({{\bf\bar{N}}},{\bf N})$, this is equivalent
to $(n-{\overline{n}})({\bf N},{\bf N})+(m-{\overline{m}})({\bf
N},{{\bf\bar{N}}})$, up to non-chiral matter. Rank-2 tensors of real groups
and adjoints of unitary groups are chirally irrelevant, and although they may
appear in the massless spectrum of specific D-brane configurations, these are
states that a priori are not protected against becoming massive via a number
of effects.

Despite this richer structure, it turns out that the 4d chiral spectrum
obtained for D-branes in orientifold compactifications is quite universal. To
describe it, one needs to define a chiral index $I_{ab}$ between two D-branes,
that is a bilinear, anti-symmetric tensor of their D-brane charges (not
counting their multiplicity $N_{a}$). The expression for $I_{ab}$ changes from
one model-building setup to another, but it is always of topological nature.
If the D-brane with charge $\Pi^{b}$ is not invariant under the orientifold
action, there will be an orientifold image with charge $\Pi^{b^{\prime}}$ and
a corresponding index $I_{ab^{\prime}}$. Finally, one can also extend this
definition to include an index $I_{aO}$ between a D-brane and the O-plane
content of the compactification. Once that this index has been defined, the
chiral spectrum arising from the open string sector of the compactification
reads as in Table 1.

Non-Abelian gauge group | $\prod_{a}SU(N_{a})$
---|---
Massless $U(1)$s |  $\sum_{a}c_{a}U(1)_{a}$ such that $\sum_{a}c_{a}([\Pi^{a}]-[\Pi^{a^{\prime}}])=0$
Chiral fermions | $\sum_{a<b}\,I_{ab}({\bf N}_{a},{\bf\bar{N}}_{b})\,+\,I_{ab^{\prime}}({\bf N}_{a},{\bf N}_{b})$
| $\frac{1}{2}(I_{aa^{\prime}}-I_{aO}){\bf
S}_{a}\,+\,\frac{1}{2}(I_{aa^{\prime}}+I_{aO}){\bf A}_{a}$
Table 1: Chiral spectrum of a D-brane orientifold compactification, in terms
of the chiral index $I_{ab}$. For simplicity we are assuming that there are no
D-branes with $SO(N)$ or $USp(N)$ gauge group. $\Pi^{a}$ represents the charge
of the D-brane. ${\bf S}_{a}$, ${\bf A}_{a}$ stand for the symmetric and
antisymmetric representations of $U(N_{a})$.

The fact that one has such a universal chiral spectrum for all perturbative
orientifold models allows one to devise model-building strategies that are
independent of their specific realisation, as will be discussed in the next
section. Let us however stress that once one leaves the realm of perturbative
constructions, new types of gauge groups, matter representations and,
consequently, model building possibilities arise. The reason is the appearance
of non-perturbative bound states of strings which can have more than two
endpoints, hence realising, for instance, higher tensor representations or
spinor representations. This will be described in detail in the context of
F-theory in Section 5.

##  2 D-brane Model Building: Generalities

As it turns out, the Standard Model can be built very easily and naturally out
of the limited set of bi-fundamentals and rank 2 tensors available already in
perturbative D-brane models. In this section we outline the systematics
underlying the search for Standard Model like vacua in Type II orientifolds.
This approach can then be applied both in geometric Type IIA or Type IIB
orientifolds (see Section 3) and in conformal field theoretic models (Section
4). Model building in the non-perturbative generalisation described by
F-theory, in particular in the context of Grand Unified Theory (GUT) model
building, is the topic of Section 5.

###  2.1 Anomalies, Tadpoles and Axions

In any chiral model there is one important constraint to be taken into
account: chiral anomaly cancellation. Anomalies cancel automatically in string
theory, provided one satisfies all consistency conditions. The most important
of these is in this context the cancellation of all RR-tadpoles. Note that
there may also be NS-NS tadpoles. They automatically cancel in supersymmetric
models that are free from RR-tadpoles. In non-supersymmetric setups
uncancelled NS-NS tadpoles imply instabilities, which is a serious problem,
but not an inconsistency.

The first step towards building the Standard Model consists of assembling a
set of branes whose spectrum of chiral fermions is the same as that of the SM.
Unless one is extremely lucky, this set does not satisfy tadpole cancellation.
In particular the NS-NS 4d dilaton tadpole can be oversaturated or
undersaturated. In the former case the total contribution of all branes plus
the orientifold plane is positive. Then there is nothing one can do about this
anymore. But if the total contribution is negative, one has the option of
adding some additional branes to the configuration. This means that one
chooses to assign a non-zero Chan-Paton multiplicity to some branes that are
not part of the SM configuration. This must be done in such a way that no
chiral particles are added to the spectrum: preferably no massless particles
at all, or at least no chiral particles charged under the SM gauge group,
dubbed chiral exotics. These additional branes are often referred to as a
hidden sector. Such a sector may have several other uses, such as breaking
supersymmetry or providing dark matter.

$r$ | N | ${{\bf\bar{N}}}$ | S | ${\bf\bar{S}}$ | A | ${\bf\bar{A}}$
---|---|---|---|---|---|---
$D(r)$ | $N$ | $N$ | $\frac{N(N+1)}{2}$ | $\frac{N(N+1)}{2}$ | $\frac{N(N-1)}{2}$ | $\frac{N(N-1)}{2}$
$Q(r)$ | 1 | -1 | 2 | -2 | 2 | -2
$I_{2}(r)$ | $1$ | $1$ | $N+2$ | $N+2$ | $N-2$ | $N-2$
$I_{3}(r)$ | 1 | -1 | $N+4$ | $-N-4$ | $N-4$ | $-N+4$
Table 2: Properties of the fundamental (vector) N, symmetric S, and
antisymmetric A representations of $SU(N)$. $D(r)$ is the dimension of the
representation $r$, $Q(r)$ is the $U(1)$ charge of $r$ under the decomposition
$U(N)=SU(N)\times U(1)$ and $I_{2}(r)$, $I_{3}(r)$ are the quadratic and cubic
anomaly coefficients (also known as Dynkin indices), respectively.

####  2.1.1 Non-abelian Anomalies

Since finding hidden sectors can be very laborious, it helps to eliminate some
SM configurations at an early stage. This means first of all that all non-
abelian anomalies must cancel. Since we are building the Standard Model, one
may think that SM anomaly cancellation ensures this, but this is only partly
true. Indeed, if one has realised the SM spectrum, anomaly cancellation
ensures the absence of $SU(3)_{\rm QCD}$ anomalies.

In QFT, non-abelian anomaly cancellation is a condition on representations of
$SU(N)$, $N\geq 3$. In particular, different representations of $SU(N)$
contribute to the $SU(N)$ cubic anomaly as their cubic anomaly coefficient,
see Table 2, and the sum of their contributions must vanish. In QFT this must
be imposed by hand, or else gauge invariance must be dropped. In string theory
all anomaly cancellations follow from some deeper consistency condition, such
as modular invariance for closed strings. In QFT, the limit $N\geq 3$ arises
because the group $SU(2)$ has pseudo-real representations, so that ${\bf 2}$
is equivalent to ${\bf\bar{2}}$. Furthermore, $SU(1)$ is trivial. But in
string theory $U(3)$, $U(2)$ and $U(1)$ brane stacks are all on the same
footing, and there is no reason to expect a lower limit on $N$.

In open strings the anomaly cancellation condition was first derived by
Bianchi and Morales [17]. As expected, they found that non-abelian anomalies
must cancel for $U(N)$ for all $N$, even if $N=2$ or $N=1$. If a candidate SM
configuration contains $U(2)$ or $U(1)$ branes the conditions must be checked,
and if it not satisfied the configuration can never be realised. We refer to
this class of anomalies as “non-abelian” anomalies using QFT terminology,
although $SU(2)$ is anomaly-free in QFT and $U(1)$ is abelian.

Let us see how the cubic non-abelian anomaly looks like in D-brane models.
Using the spectrum of Table 1 and the anomaly coefficients of Table 2 we find

${\cal A}_{SU(N_{a})^{3}}=\sum_{r\,{\rm
irrep}}I_{3,a}(r)=\sum_{b}N_{b}\left(I_{ab}+I_{ab^{\prime}}\right)-4I_{aO},$
(2.1)

which must vanish for any $SU(N),N\geq 3$, present in the model. In general
one can see that the rhs vanishes when RR tadpole conditions are imposed, by
using the appropriate generalisation of (1.6) and bilinearity of the chiral
index. As anticipated, this occurs even for $N_{a}<3$. The reason is that
otherwise some other anomalies would be left uncancelled, namely mixed and
abelian anomalies.

####  2.1.2 Mixed and Abelian Anomalies

Mixed and abelian anomalies are those that include $U(1)$ symmetries. Using
again the content of Tables 1 and 2 they read:

$\displaystyle{\cal A}_{U(1)_{a}-SU(N_{b})^{2}}=\sum_{r\,{\rm
irrep}}Q_{a}(r)I_{2,b}(r)=\delta_{ab}{\cal
A}_{SU(N_{a})^{3}}+N_{a}\left(I_{ab}+I_{ab^{\prime}}\right),$ (2.2)
$\displaystyle{\cal A}_{U(1)_{a}-U(1)_{b}^{2}}=\sum_{r\,{\rm
irrep}}Q_{a}(r)Q_{b}(r)^{2}=\delta_{ab}N_{a}{\cal
A}_{SU(N_{a})^{3}}+N_{a}N_{b}\left(I_{ab}+I_{ab^{\prime}}\right),$ (2.3)

where we assuming that the abelian factors arise from
$U(N_{a})=SU(N_{a})\times U(1)_{a}$, and $r$ now runs over the irreps with
respect to the groups involved in the anomaly. Here, abusing notation, we have
denoted by ${\cal A}_{SU(N_{a})^{3}}$ the rhs of (2.1), which is a well-
defined quantity even for $N_{a}<3$. It turns out that, in both cases, the
term proportional to $I_{ab}+I_{ab^{\prime}}$ is cancelled by a generalised
Green–Schwarz mechanism. However, the first term has to cancel due to the
D-brane configuration. That is why, in practice, one needs to impose the
condition ${\cal A}_{SU(N)^{3}}=0$ even for $N=1,2$.

The generalised Green–Schwarz mechanism works by means of a mixing term of the
longitudinal component of an abelian vector boson with an axion. If such term
is present, the vector boson acquires a mass by absorbing the axion using the
Stückelberg mechanism. This eliminates all vector bosons from the spectrum
that couple to anomalous $U(1)$s, but the mechanism may also affect non-
anomalous $U(1)$s. The result is that only the combinations of $U(1)$s that
appear in Table 1 remain massless, and this may be bad or good: one the one
hand it may make the Standard Model $Y$-boson – and hence the photon –
massive, but on the other hand it may eliminate undesirable non-anomalous
$U(1)$s that occur in most models. Most frequently this occurs with vector
bosons that involve $B-L$, a non-anomalous symmetry in the Standard Model when
three right-handed neutrinos are added.

It is customary to distinguish local and global models. A local model has a
Standard Model brane configuration and no non-abelian anomalies. Constructing
it only requires knowledge of the chiral indices of the participating branes.
If all tadpoles are cancelled and the photon remains massless one speaks of a
global model. Note that checking the latter feature cannot be done in a local
model, as it requires knowledge of the full set of available axions, which
cannot be derived from the brane configuration alone.

###  2.2 The Simplest Examples

In this section we will derive the simplest possible realisations of the
Standard Model spectrum in terms of bi-fundamentals and rank-2 tensors. These
are not explicit realisations, although we will indicate if such realisations
are known. The steps towards an explicit realisation are:

  1. 1.

Find a brane configuration that contains the Standard Model.

  2. 2.

Check absence of cubic $U(N)$ anomalies, including $N=2$ and $N=1$.

  3. 3.

Find a realisation of this configuration in terms of actual branes.

  4. 4.

Check absence of a mass for the $Y$-boson.

  5. 5.

Check tadpole cancellation, or cancel tadpoles by means of a hidden sector.

Here we will limit ourselves to discuss steps 1 and 2. The broadest
exploration of explicit realisations was done in the context of RCFT Gepner
models in [18], where all five steps were considered. Models were collected if
they passed step 4. Here we aim for maximal simplicity: the minimal number of
branes, exact family repetition (so that we can focus on a single family), the
minimal gauge group, and no superfluous non-chiral pairs. Since these guiding
principles are debatable, we will make concessions if necessary.

Perhaps one’s first thought about realising the Standard Model with branes
would be to take a $U(3)$ for QCD, a $U(2)$ or $USp(2)$ brane for the weak
interactions and a $U(1)$ brane for the $Y$-charge.555Note that $USp(2)$ has
the same Lie algebra as $SU(2)$. The same is true for $O(3)$, but open strings
would give matter in the 3-dimensional vector representation of $O(3)$, which
does not occur in the Standard Model. However, associating the $Y$ charge with
a separate $U(1)$ brane cannot work, because the quark-doublet $({\bf 3},{\bf
2},\frac{1}{6})$ would then be a tri-fundamental, which do not occur in open
string models.

####  2.2.1 The $Y$-charge

Hence the $Y$-charge must involve the $U(1)$ factor of $U(3)$, the $U(1)$
factor of $U(2)$ (unless $USp(2)$ is used), and perhaps one or more additional
branes. We label the branes as $a$ (for the QCD brane) $b$ (for the weak
brane), and $c$, $d$, $\dots$ for any additional $U(1)$ branes. The $Y$-charge
generator must then take the form

$Y=(x-\tfrac{1}{3})Q_{a}+(x-\tfrac{1}{2})Q_{b}+\gamma Q_{c}+\delta
Q_{d}+\ldots\,.$ (2.4)

We follow here the conventions of [18], and assume that all branes are unitary
and $Q_{i}$ is the $U(1)$ generator of brane $i$. We normalise these
generators so that a vector representation has charge $+1$. If the coefficient
of any $Q_{i}$ vanishes, one can replace the corresponding brane by a real one
with a symplectic or orthogonal group.

In the following we will examine all possibilities for assigning quarks and
anti-quarks, using the minimal number of branes, and for a single family.

####  2.2.2 The Quark Doublet

The coefficients of $Q_{a}$ and $Q_{b}$ ensure that the quark doublet gets the
correct $Y$-charge, assuming that the bi-fundamental we use for the quark
doublet is $({\bf N},{\bf\bar{N}},0,\ldots)$. Alternatively one could use
$({\bf N},{\bf N},0,\ldots)$. This is just a convention, but it is useful to
choose all multiplets as if we are in an orientable string theory, i.e. a
fundamental representation on one end and an anti-fundamental on the other,
until we do not have a choice anymore. At that point we know that we have
reached a non-orientable configuration. If $x=\tfrac{1}{2}$ then $Q_{b}$ does
not participate in $Y$, and we may use $USp(2)$ instead of $U(2)$ for brane
$b$. This choice makes the configuration non-orientable. Note that in
orientable configurations the coefficients of $Q_{i}$ are not fully determined
by the SM spectrum: if we shift all coefficients by the same amount $\Delta$,
then $\Delta$ cancels out between the two open string ends.

####  2.2.3 Anti-quarks

Next we can try to assign the anti-quarks $u^{c}$ and $d^{c}$. The
representation ${\bf\bar{3}}$ of the antiquark can be obtained in two ways: as
a rank-2 anti-symmetric tensor, from an open string with both ends on the
$U(3)$ brane, or as a ${\bf\bar{N}}$ endpoint of an open string. In the latter
case the other endpoint of that string has to end on another brane. One then
has the four options summarised in Table 3.

Class | $u^{c}$ | $d^{c}$ | $x$ | $\gamma$ | $\delta$ | L | $e^{+}$
---|---|---|---|---|---|---|---
$(i)$ | $({\bf\bar{N}},0,{\bf N})$ | $({\bf\bar{N}},0,{\bf\bar{N}})$ | $\tfrac{1}{2}$ | $-\tfrac{1}{2}$ | – |  $(0,{\bf N},{\bf N})$ or $(0,{\bf\bar{N}},{\bf N})$ | $(0,0,{\bf\bar{S}})$
$(ii)$ | $({\bf\bar{N}},0,{\bf N},0)$ | $({\bf\bar{N}},0,0,{\bf N})$ | $x$ | $x$ | $x-1$ |  $(0,{\bf N},{\bf\bar{N}},0)$ or $(0,{\bf\bar{N}},0,{\bf N})$ | $(0,0,{\bf N},{\bf\bar{N}})$
$(iii)$ | $({\bf A},0,0)$ | $({\bf\bar{N}},0,{\bf N})$ | 0 | 0 | – | $(0,{\bf N},{\bf N})$ | $(0,\bar{\bf A},0)$
$(iv)$ | $({\bf\bar{N}},0,{\bf N})$ | $({\bf A},0,0)$ | $\tfrac{1}{2}$ | $-\tfrac{1}{2}$ | – | $(0,{\bf N},{\bf N})$ | $(0,0,\bar{\bf S})$
Table 3: The four basic quark/anti-quark configurations for one family.
Columns 2-5 define the class. The last two columns specify the possible
completions to a full family, if no further branes are added, and for generic
$x$ in case $(iv)$. Figure 3: The four basic classes from Table 3. Dots
indicate a coupling to the orientifold image brane. $L$ and $L^{\prime}$ are
two possible assignments of the lepton doublet. In case $(i)$ a weak $USp(2)$
group is assumed.

At this point all configurations, except $(ii)$, are non-orientable. In the
quark sector, all models in the literature necessarily have one of these for
structures for a single family. With three families one has the option to make
different choices per family, as long as the coefficients of $Q_{i}$ match.

####  2.2.4 The Lepton Doublet

Now we can try to assign the lepton doublet $({\bf 1},{\bf 2},-\tfrac{1}{2})$.
If we do not add more branes, there is a unique choice in all cases except
$(ii)$: a bi-fundamental $(0,{\bf N},{\bf N})$ between branes $b$ and $c$. In
case $(ii)$ there are two options: Either $(0,{\bf N},{\bf\bar{N}},0)$ or
$(0,{\bf\bar{N}},0,{\bf N})$. In both cases the configuration remains
orientable.

####  2.2.5 The Left-handed Positron

Finally we try to assign the charged lepton $({\bf 1},{\bf 1},1)$. Being an
$SU(2)$ singlet, it must either come from a string that does not end on brane
$b$, or an anti-symmetric tensor on brane $b$. In cases $(i)$ and $(iv)$ this
fixes it uniquely to $(0,0,{\bf\bar{S}})$. In case $(iii)$ the only option is
$(0,{\bf\bar{A}},0)$. In case $(ii)$ a natural choice is $(0,0,{\bf
N},{\bf\bar{N}})$. One could also use rank-2 tensors, but only for special
values of $x$.

####  2.2.6 Weak Interaction Anomalies

The SM contains a quark doublet $({\bf 3},{\bf 2},\frac{1}{6})$ and a lepton
doublet $({\bf 1},{\bf 2},-\tfrac{1}{2})$ per family. The gauge group $SU(2)$
is anomaly free in quantum field theory. If it is realised as $USp(2)$ in
string theory, this is also true. But if it is realised as $U(2)$ in string
theory, there are anomalies not seen in field theory. We have to decide if the
representation denoted “${\bf 2}$” is actual ${\bf 2}$ or ${\bf\bar{2}}$. If
we do that in a single family it is immediately clear that there is no way to
cancel the weak anomalies from the quark and the lepton doublets. There are
four ways out of this:

  1. 1.

If $x=\tfrac{1}{2}$ one may use $USp(2)$ instead of $U(2)$. Then there are no
weak brane anomalies.

  2. 2.

If $x=\tfrac{1}{2}$ one may also use $({\bf N},{\bf N})$ as a quark doublet.
Then one may drop exact family repetition, to write the three doublet as
$2({\bf N},{\bf\bar{N}})+({\bf N},{\bf N})$.

  3. 3.

One may add a rank-2 anti-symmetric anti-tensor. This contributes $+2$ to the
weak anomaly, cancelling the contribution $-3$ of the quark doublet, and $+1$
of the lepton doublet. This adds a singlet chiral particle with charge $1-2x$
to the spectrum. For the most common values of $x$ this is a left-handed
positron or a neutrino.

  4. 4.

If there are two ways of realising a lepton doublet with opposite weak
anomalies, one may add a non-chiral pair. For example suppose $({\bf N},X)$
and $({\bf\bar{N}},Y)$ are both lepton doublets, where $X$ and $Y$ are some
combination of brane representations. Now one may use a combination
$2\times({\bf N},X)+({\bf N},{\bar{Y}})$ to get three times the contribution
of a single lepton doublet, but a net number of only one chiral lepton
doublet. This adds a non-chiral pair to the spectrum with the features of a
Higgsino pair $H_{u}+H_{d}$.

####  2.2.7 The Four Classes of Models

##### Class (i).

This class can be made anomaly free on the weak brane by means of mechanisms
1, 2 or 3. The $c$ brane anomaly cancellation is a bit awkward. The anomaly of
the model as shown in the table would be -3. To cancel it we may add an anti-
symmetric tensor ${\bf A}$ on brane $c$. This has anomaly -3 and ground state
dimension 0, and hence there are no massless states at all in this sector.
Although this may seem a bit weird, explicit realisations of such $U(1)$
branes were found in [18] in fully tadpole-free models.

But there is a more appealing way. We can add a fourth brane with the same
contribution to $Y$ as brane $c$, and realise the left-handed positron as
$(0,0,{\bf\bar{N}},{\bf\bar{N}})$. Then we may add a string $(0,0,{\bf
N},{\bf\bar{N}})$, which has vanishing $Y$ charge, and connect the lepton
doublet to brane $d$ instead of brane $c$. Now all anomalies on branes $c$ and
$d$ cancel. Furthermore, if we use mechanism 1 or 2 on the weak brane we get a
model entirely built out of bi-fundamentals. If we take the weak group as
$USp(2)$ and assume mechanism 1, the full chiral spectrum is:

Figure 4: The “Madrid” configuration, with a weak interaction group $USp(2)$.

$\displaystyle 3$ $\displaystyle\times$ $\displaystyle({\bf N},{\bf N},\
0,0)\quad\,Q$ $\displaystyle 3$ $\displaystyle\times$
$\displaystyle({\bf\bar{N}},0,{\bf N},0)\quad\ u^{c}$ $\displaystyle 3$
$\displaystyle\times$ $\displaystyle({\bf\bar{N}},0,{\bf\bar{N}},0)\quad\
d^{c}$ $\displaystyle 3$ $\displaystyle\times$ $\displaystyle(0,{\bf N},0,{\bf
N})\quad\ L$ $\displaystyle 3$ $\displaystyle\times$
$\displaystyle(0,0,{\bf\bar{N}},{\bf\bar{N}})\quad\ e^{+}$ $\displaystyle 3$
$\displaystyle\times$ $\displaystyle(0,0,{\bf N},{\bf\bar{N}})\quad\ n$

This spectrum is shown in Figure 4. It is the most-studied class of brane
models, first explored in detail in [19]. Numerous examples and variations
have been found in subsequent papers.

A noteworthy feature is the presence of a baryon and a lepton brane: All
quarks attach to brane $a$ and all leptons to brane $d$. In fact, branes $a$
and $d$ have the same intersections with branes $b$ and $c$. One may combine
branes $a$ and $d$ into a $U(4)$ stack, and extend the $U(1)$ group on brane
$2$ to $U(2)$ or $USp(2)$ to get a left-right symmetric model. Combining all
this one obtains a $SU(4)\times SU(2)_{L}\times SU(2)_{R}$ Pati–Salam model,
see [20, 21] and Section 3 for specific realisations of this idea.

The unitary factors of branes $a$ and $d$ are anomalous and therefore broken
by axion mixing. They remain as global baryon number (B) and lepton number (L)
symmetries. The linear combination $B-L$ is anomaly free, and the
corresponding gauge boson may or may not become massive. This can only be
decided by examining the full global model, not just the local configuration.
In [18] global examples were found where the $B-L$ photon acquires a mass.
Furthermore, examples were found with tadpole cancellation without any
additional branes, both with a massless and a massive $B-L$ photon.

Bi-fundamentals between branes $b$ and $c$ are not used to build the fermion
spectrum. But these states have precisely the right quantum numbers to be
identified as supersymmetric Higgs multiplets $H_{d}$ and $H_{u}$. They have
zero lepton number, so that potentially dangerous perturbative couplings such
as $LH_{u}$ are automatically forbidden.

##### Class (ii).

This class is characterised by having separate branes for the $u^{c}$ and
$d^{c}$ endpoints. Models of this kind were first explored in [22, 23].
However, weak anomaly cancellation was not considered in these papers. For
arbitrary values of $x$, only mechanism 4 is available. Using that mechanism
one arrives at a fully orientable brane configuration:

$\displaystyle 3$ $\displaystyle\times$ $\displaystyle({\bf
N},{\bf\bar{N}},0,0)\quad Q$ $\displaystyle 3$ $\displaystyle\times$
$\displaystyle({\bf\bar{N}},0,{\bf N},0)\quad d^{c}$ $\displaystyle 3$
$\displaystyle\times$ $\displaystyle({\bf\bar{N}},0,0,{\bf N})\quad u^{v}$
$\displaystyle 6$ $\displaystyle\times$ $\displaystyle(0,{\bf
N},{\bf\bar{N}},0)\quad L$ $\displaystyle 3$ $\displaystyle\times$
$\displaystyle(0,{\bf N},0,{\bf\bar{N}})\quad L^{c}$ $\displaystyle 3$
$\displaystyle\times$ $\displaystyle(0,0,{\bf N},{\bf\bar{N}})\quad e^{+}$

There is a superfluous non-chiral pair $L+L^{c}$ per family, which has the
quantum numbers of a Higgs pair, but is not distinguished from the lepton
doublet by its quantum numbers. There is no left-handed anti-neutrino.

The trinification model shown in fig. 2 reduces to this model (plus additional
non-chiral states) if one splits the second $U(3)$ to $U(2)\times U(1)$ and
the third to $U(1)^{3}$.

Explicit realisations of this model have been found [18], but no global
realisations with tadpole cancellation. The models presented in [23] are not
explicit realisations, but hypothetical brane configurations with $x=0$ and
$x=1$. Once $x$ is fixed to these values, it is possible to get left-handed
positrons as anti-symmetric tensors. Note that in classes $(i)$, $(iii)$ and
$(iv)$ the value of $x$ is either 0 or $\tfrac{1}{2}$.

##### Class (iii).

This class can be viewed as a brane realisation of an $SU(5)$ GUT model, with
an extra $U(1)$. One takes an anti-symmetric tensor of $U(5)$ plus an anti-
vector of $U(5)$, giving rise to the familiar ${\bf 10}+{\bf\bar{5}}$. Now one
may split the 5-stack into a 3-stack and a 2-stack. Physically, this may be
realised by separating the two stacks by a certain amount in the compactified
dimensions, or by using entirely different stacks with the same intersections.
To construct a $U(5)$ GUT we need a vector of $U(5)$, which is an open string
with one endpoint on the $U(5)$ stack and its other endpoint on another brane,
labelled $c$. This brane does not contribute the $Y$-charge and hence the
simplest choice is an $O(1)$ brane. The explicit model in the table uses
mechanism 3 to cancel the weak anomaly. This anomaly cancellation is inherited
directly from the $SU(5)$ anomaly cancellation.

This class of models was first studied in [24], but without considering full
tadpole cancellation. The latter problem was addressed in [25], but only
examples with chiral exotics (15 of SU(5)) were found in this paper. The exact
model in the table, with a brane group $U(3)\times U(2)\times O(1)$ has been
found frequently in the search of [18] , and there are even examples with full
tadpole cancellation without any additional branes.

We anticipate that $SU(5)$ GUTs based on D-branes suffer from the absence of a
top quark Yukawa coupling at the perturbative level. This problem is overcome
in non-perturbative realisations of such models as described in Section 5.

##### Class (iv).

This class can be made anomaly-free on the weak brane by using mechanism 3. In
this case the anti-symmetric tensor is a left-handed anti-neutrino. The $c$
brane anomaly cancels because the symmetric anti-tensor contributes -5, and
the strings producing $d^{c}$ and $L$ have a contribution $3+2=5$.

This class contains flipped $SU(5)$ models. In flipped $SU(5)$ $d^{c}$ (as
opposed to $u^{c}$) is realised using an anti-symmetric tensor. Flipped
$SU(5)$ requires an additional $U(1)$, and that $U(1)$ is realised here as a
linear combination of the $U(1)$ from brane $c$ and the phase factor $U(1)$ of
$U(5)$. Orientifold models of this kind were first studied in [26], but also
in the flipped $SU(5)$ case the first examples found had chiral exotics.
Explicit examples of the spectrum shown in the table have been found in the
search of [18]. There are even examples of full tadpole cancellation without a
hidden sector.

###  2.3 Yukawa Couplings

The quarks and leptons discussed above must all acquire a mass from a three-
point coupling with a Higgs boson. The latter should be present in the light
spectrum. It must be a weak doublet, hence a bi-fundamental open string with
one end on the weak brane. Let us call the brane with the other endpoint the
Higgs brane.

In perturbative open string theories, three point couplings are represented by
a disk diagram with three external lines. For these couplings to exist, the
three fields must be bi-fundamentals between branes $(a,b)$, $(b,c)$ and
$(c,a)$. If these branes are complex, the brane charges of each brane must
cancel, in other words there must be an equal number of endpoints ${\bf N}$
and ${\bf\bar{N}}$ on each brane. These charges, the phase factors of the
brane group $U(N)$, are conserved in string perturbation theory, even though
the corresponding $U(1)$s may acquire a mass via the Stückelberg mechanism.
The latter fact means that there are non-perturbative effects, generated by
instantons, that break those symmetries, but these symmetries remain as global
symmetries in perturbation theory, analogously to the baryon number in the
Standard Model.

Figure 5: Three-point open string coupling diagram.

Given this rule for three-point couplings, we can now compute the required
Higgs representation for quarks by tensoring the quark doublet with each quark
singlet, $u$ or $d$. In classes $(iii)$ and $(iv)$ we see immediately that
respectively the up and down Yukawa couplings do not exist perturbatively,
because the quark singlets are realised as anti-symmetric tensors, so we must
rely on three vectors of $U(3)$ coupling to a singlet. This is fine in
$SU(3)$, but not possible in $U(3)$, as first pointed out in [24]. One may try
to generate the missing Yukawa couplings non-perturbatively using instantons
[27, 28, 29, 30, 31], or turn to F-theory models where they arise more
naturally, see Section 5.3.2.

In classes $(i)$ and $(ii)$ a perturbative Yukawa coupling is possible, if one
assigns the Higgs to one of the multiplets denoted $L$ and $L^{\prime}$ in
Figure 3. Then there are a few more issues to worry about: weak brane and
Higgs brane anomaly cancellation, and differences in weak $U(2)$ chirality for
different families. In class $(i)$ this is most easily dealt with by choosing
the Madrid configuration (see Figure 4) with $USp(2)$ as the weak brane group,
and choosing brane $c$ as the Higgs brane. Now one can choose $(0,{\bf N},{\bf
N},0)+(0,{\bf N},{\bf\bar{N}},0)$ as the Higgs system. In class $(ii)$ the
fully orientable configuration discussed above already comes with a Higgs pair
per family, but one of the Higgses has the same quantum numbers as the lepton
doublet, which allows for undesirable couplings.

The discussion of lepton Yukawas goes along the same lines for the latter two
models. Indeed, in the models of Figure 4 quarks and leptons play a symmetric
role. This is not true in the fully orientable model, which lacks a right-
handed neutrino. However, it is not likely that neutrino masses are generated
by the Standard Model Higgs mechanism. It is usually assumed that there is a
Majorana mass component with a different, and necessarily non-perturbative,
origin, in order to understand the smallness of neutrino masses using the see-
saw mechanism [27, 28]. This, as well as many other aspects of Yukawa
couplings in orientifold models, is beyond the scope of this chapter.

##  3 Type II Orientifolds

In this section we describe D-brane model building in specific Type II
compactifications. Our framework will be Calabi–Yau (CY) three-fold
orientifold compactifications,666Our discussion also applies to
compactification backgrounds beyond Calabi–Yau metrics, like six-dimensional
manifolds with $SU(3)$ or $SU(3)\times SU(3)$ structure, that feature a non-
trivial warp factor and internal fluxes. We will however restrict ourselves to
the CY case for simplicity.  at large volume and weak string coupling. This
regime is where most of the model building ideas have been developed in string
theory, because D-branes can be essentially treated as sub-manifolds in a
compactification manifold $X_{6}$, and as a result most of the quantities that
specify the resulting 4d EFT have a simple topological or geometric
realisation. A large fraction of the intuition developed in this setup also
applies to small-volume and strong-coupling compactifications, which will be
dealt with in sections 4 and 5, respectively, and where also new model
building features will arise.

###  3.1 Type IIA Orientifolds

Let us consider type IIA string theory on a background of the form
$X_{4}\times X_{6}$, where $X_{6}$ is a compact Calabi–Yau three-fold $X_{6}$,
with Kähler two-form $J$ and holomorphic three-form $\Omega_{3}$. To this
background we apply an orientifold quotient generated by
$\Omega(-1)^{F_{L}}{\cal R}$, where $\Omega$ is the worldsheet parity reversal
operator, ${F_{L}}$ is the space-time fermion number for the left-movers and
${\cal R}$ an anti-holomorphic involution of $X_{6}$ acting as ${\cal R}J=-J$,
${\cal R}\Omega_{3}=-\overline{\Omega}_{3}$, respectively. The presence of
$(-1)^{F_{L}}$ is important for the orientifold action to square to the
identity. Performing the quotient has two main effects:

  * -

It reduces the supersymmetry on the gravity sector from 4d ${\cal N}=2$ to
${\cal N}=1$.

  * -

It introduces a set of O-planes at the fixed loci of ${\cal R}$ which, as
explained in Section 1, are necessary ingredients for D-brane model building.

Since ${\cal R}$ is an anti-holomorphic involution, its fixed loci are given
by a set of special Lagrangian three-cycles of $X_{6}$ that we collectively
denote as $\Pi_{\rm O6}$, times the four non-compact dimensions $X_{4}$. Thus,
we have a set of O6-planes on $X_{4}\times\Pi_{\rm O6}$. In order to implement
the strategy of Section 2, to this background we add a set of space-time
filling D-branes that give rise to the SM spectrum plus a hidden sector, and
such that the RR tadpole cancellation conditions (1.6) with $p=6$ are
satisfied. The most natural option to build a vacuum is to consider D-branes
that preserve the same supersymmetry as the bulk, because these are BPS
objects that minimise their tension with respect to their RR charge, and then
the cancellation of RR tadpoles implies that the bulk equations of motion are
satisfied. If we focus on single D-branes, there are two types of objects that
satisfy this condition. The first are D6-branes wrapped on a special
Lagrangian three-cycle $\Pi_{3}$, satisfying [32]

${\cal F}+iJ=0,\qquad\text{and}\qquad{\rm Im}\,\Omega_{3}=0,$ (3.1)

where

${\cal F}=B+\frac{\ell_{s}^{2}}{2\pi}F$ (3.2)

is the gauge-invariant D-brane worldvolume field strength, with
$\ell_{s}=2\pi\sqrt{\alpha^{\prime}}$ the string length. In all these
expressions, bulk $p$-forms like $B$, $J$ and $\Omega_{3}$ are implicitly
pulled-back to the D-brane worldvolume, in this case to the three-cycle
$\Pi_{3}$. That is, the D6-brane wraps a special Lagrangian three-cycle with
the same calibration phase as $\Pi_{\rm O6}$, along which it hosts a flat
gauge bundle. These two conditions are referred to as F-flatness and
D-flatness conditions in the literature, because if they are not met there
will be an uncancelled D-term or F-term in the 4d EFT gauge sector,
respectively. The second kind of object are D8-branes wrapped on coisotropic
five-cycles $\Xi$ of $X_{6}$ [33], characterised by the BPS conditions

$\left({\cal F}+iJ\right)^{2}=0,\qquad\text{and}\qquad{\rm
Im}\,\Omega_{3}\wedge{\cal F}=0,$ (3.3)

which can again be interpreted as F-flatness and D-flatness conditions. While
MSSM-like models have been built with coisotropic D8-branes [34] in the
context of general CY geometries it is technically difficult to describe these
objects, and so in the type IIA setting most of the activity has focused on
building models based on D6-branes [8]. In the following we describe the main
features of such models.

####  3.1.1 Intersecting D6-brane Models

Type IIA orientifold models were one of the last frameworks to be explored in
the D-brane model building literature, but they quickly gained a central place
in our current description of this topic. The reason is that the formula for
the chiral index $I_{ab}$ that appears in Table 1 is particularly simple,
which gives us a lot of intuition about the chiral spectrum of these models.
In particular, given $N_{a}$ D6-branes wrapping a three-cycle
$\Pi_{3}^{a}\subset X_{6}$ and a second stack of $N_{b}$ D6-branes on
$\Pi_{3}^{b}$, their chiral index reads

$I_{ab}=[\Pi_{3}^{a}]\cdot[\Pi_{3}^{b}],$ (3.4)

that is, the signed intersection number of the two three-cycles. More
precisely, at each transverse intersection one finds a left-handed 4d ${\cal
N}=1$ chiral multiplet in a bifundamental representation, which is either
$(N_{a},\bar{N}_{b})$ or $(\bar{N}_{a},N_{b})$ depending on the sign of the
intersection, and (3.4) computes the net chiral spectrum in this sector. As
anticipated, this is a topological invariant that only depends on the homology
class of each three-cycle, or in other words of the RR charges of the branes.
Finally, due to the orientifold symmetry, for each D6-brane stack wrapping
$\Pi_{3}^{a}$ there is a similar number of D6-branes wrapping the orientifold
image

$\Pi_{3}^{a^{\prime}}={\cal R}\Pi_{3}^{a}.$ (3.5)

Including these three-cycles and their intersections and identifying them
properly under the orientifold action one arrives at the spectrum of Table 1.
In particular, at transverse intersections between $\Pi_{3}^{a}$ and
$\Pi_{3}^{a^{\prime}}$ that are not on top of $\Pi_{\rm O6}$ we get adjoint
${\bf Adj}={\bf S}+{\bf A}$ representations of $U(N_{a})$, while for those on
top of $\Pi_{\rm O6}$ one either gets a symmetric or an anti-symmetric
representation.

In this setup we can also specify those stacks of $N$ D6-branes that realise
either an $SO(N)$ or $USp(N)$ gauge group. As mentioned in Section 1, these
are D-brane sectors that are left invariant under the orientifold action. More
precisely they are realised by D6-branes on 3-cycles satisfying the property
$\Pi_{3}^{a}=\Pi_{3}^{a^{\prime}}$. There are essentially two kinds of such
three-cycles: those that are left invariant point-wise and those that are only
invariant as a set. Typically, the first kind gives rise to the real gauge
group $SO(N)$ and the second one, which requires an even number of D6-branes,
to $USp(N)$ [35].

The last piece of data needed to realise the content of Table 1 are those
$U(1)$ factors that remain massless after the generalised Green–Schwarz
mechanism and in particular all the $B\wedge F$ couplings have been taken into
account. In the absence of an orientifold projection, these are the
combinations $\sum_{a}c_{a}U(1)_{a}$ with $c_{a}\in\mathbb{Z}$ and such that
the homology class $\sum_{a}c_{a}N_{a}[\Pi_{3}^{a}]$ is trivial in
$H_{3}(X_{3},\mathbb{Z})$. In the presence of O6-planes some of the RR fields
mediating the Green–Schwarz mechanism are projected out, and only the weaker
condition

$\sum_{a}c_{a}N_{a}\left([\Pi_{3}^{a}]-[\Pi_{3}^{a^{\prime}}]\right)=0$ (3.6)

needs to be imposed [36]. Those combinations that do not satisfy (3.6) acquire
a mass via a Stückelberg mechanism, but they remain as perturbative global
symmetries that are only broken by non-perturbative effects. As discussed in
Section 2.3, they constrain the magnitude of those couplings that are not
invariant under them, like certain Yukawa couplings, that can only be
generated non-perturbatively [27, 28, 29, 30, 31]. Finally, if there are
combinations of the form (3.6) with g.c.d$\\{c_{a}\\}=1$ which are $2k$
multiples of some non-trivial element of $H_{3}(X_{3},\mathbb{Z})$, the
corresponding massive $U(1)$ contains a $\mathbb{Z}_{k}$ subgroup that is an
exact gauge symmetry. This will prevent the appearance of certain couplings
even at the non-perturbative level [37].

With these ingredients one may already start discussing explicit examples of
intersecting D6-brane models that realise the model building philosophy of
Section 2. In general, the topological data that one needs are the lattice
$H_{3}(X_{6},\mathbbm{Z})$, the action of the involution ${\cal R}$ on it, the
class $[\Pi_{\rm O6}]$ as well as the intersection product (3.4). Further data
that are important for D6-brane model building are those classes $[\Pi]\in
H_{3}(X_{6},\mathbbm{Z})$ with ${\rm Im}\int_{\Pi}\Omega_{3}=0$ that contain
special Lagrangian representatives. Determining them is the hardest part of
the problem, and hence a large fraction of type IIA orientifold model building
is performed in simple geometries like toroidal orbifolds.

####  3.1.2 A Simple Model

Let us illustrate the general strategy of Section 2 in a simple MSSM-like
model. We focus on building a Class $(i)$ model in the classification of
Section 2.2, following [38, 39]. The first step is to specify the four sets of
D6-branes that host the MSSM-like spectrum, which we do as in Table 4. We
consider two stacks of D6-branes ($b$ and $c$) invariant under the orientifold
projection and such that the gauge group for each of them is $USp(2)\simeq
SU(2)$. For the remaining two stacks ($a$ and $d$) we choose $\Pi_{3}^{a}$ and
$\Pi_{3}^{d}$ to lie in the same homology class, up to a torsion element in
$H_{3}(X_{6},\mathbb{Z})$.

D6-brane content |  | $3\Pi_{3}^{a}$ | $\Pi_{3}^{b}$ | $\Pi_{3}^{c}$ | $\Pi_{d}$
---|---|---|---|---|---
Gauge group |  | $SU(3)\times U(1)_{a}$ | $USp(2)$ | $USp(2)$ | $U(1)_{d}$
Table 4: Left-right model of intersecting D6-branes. Here
$[\Pi_{3}^{a}]-[\Pi_{3}^{d}]\in{\rm Tor}H_{3}(X_{6},\mathbb{Z})$.

This implies that they will have the same intersection number with any other
three-cycle, and that $U(1)_{B-L}=\frac{1}{3}U(1)_{a}-U(1)_{d}$ will remain
massless. With these choices we only need to specify four intersection
numbers, $I_{ab}=-I_{ac}=3$ and $I_{aa^{\prime}}=I_{aO}=0$, in order to
reproduce the model of Figure 4, or more precisely a left-right extension of
the SM gauge group. If in addition $\Pi_{3}^{a}=\Pi_{3}^{d}$, one arrives at a
Pati–Salam model, whose spectrum is specified in the upper part of Table 6.
This case is particularly simple to realise in a concrete CY geometry because
three-cycles hosting $USp$ gauge groups automatically satisfy the
supersymmetry condition (3.1), and so it only remains to verify that the
cycles $\Pi_{3}^{a}=\Pi_{3}^{d}$ also preserve supersymmetry at some point in
the CY moduli space.

To proceed with the construction of the model one must specify the CY
geometry. As noted, the simplest choices correspond to toroidal orbifolds and,
in this case, to the orbifold $X_{6}=(T^{2}\times T^{2}\times
T^{2})/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ with cohomology
$(h^{1,1},h^{2,1})=(51,3)$, whose O6-plane quotient and D6-brane model
building rules were worked out in [40, 41]. The simplicity of this geometry
stems from the fact that the twisted sector only contains collapsed two-
cycles, and so $H_{3}(X_{6},\mathbbm{Z})$ and its intersection product is
essentially that of $T^{2}\times T^{2}\times T^{2}$. To specify a three-cycle
class one must provide a one-cycle class on each $T^{2}$ factor, that is
specify the following wrapping numbers

$[\Pi_{3}^{a}]=(n_{a}^{1},m_{a}^{1})\times(n_{a}^{2},m_{a}^{2})\times(n_{a}^{3},m_{a}^{3}),\qquad
n_{a}^{i},m_{a}^{i}\in\mathbbm{Z},$ (3.7)

and then the intersection number between two three-cycles is given by

$I_{ab}=[\Pi_{3}^{a}]\cdot[\Pi_{3}^{b}]\,=\,(n_{a}^{1}m_{b}^{1}-m_{a}^{1}n_{b}^{1})\times(n_{a}^{2}m_{b}^{2}-m_{a}^{2}n_{b}^{2})\times(n_{a}^{3}m_{b}^{3}-m_{a}^{3}n_{b}^{3}).$
(3.8)

In a toroidal orientifold geometry such three-cycles must be accompanied by
their images under the orientifold group. In the case at hand the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold group the wrapping numbers (3.7)
are mapped to themselves, and the smallest or fractional D-brane objects
correspond to two copies of (3.7) in certain locations. The orientifold image
is dictated by the anti-holomorphic involution ${\cal R}:z^{i}\to\bar{z}^{i}$,
where $z^{i}$ is the complex coordinate on the $i^{\rm th}$ $T^{2}$. That this
involution is a symmetry of the internal metric restricts the moduli space of
complex structures, with one possibility being that each $T^{2}$ is
rectangular. In that case, which we will assume in the following, the
orientifold image of (3.7) is

$[\Pi_{3}^{a^{\prime}}]=(n_{a}^{1},-m_{a}^{1})\times(n_{a}^{2},-m_{a}^{2})\times(n_{a}^{3},-m_{a}^{3}),$
(3.9)

and the O6-plane class is given by

$\displaystyle[\Pi_{\rm O6}]=$ $\displaystyle
4\Big{[}(1,0)\times(1,0)\times(1,0)+(1,0)\times(0,1)\times(0,-1)$ (3.10)
$\displaystyle\quad+(0,1)\times(1,0)\times(0,-1)+(0,1)\times(0,-1)\times(1,0)\Big{]},$
(3.11)

in fractional three-cycle units. A stack of D6-branes with the wrapping
numbers of one of the components of (3.11) hosts a $USp$ gauge group. Because
such wrapping numbers are invariant under the full orientifold group, one only
needs two D6-branes in the covering space to host a $USp(2)$ gauge group.

With these ingredients one may already build an explicit model, by providing
the wrapping number content of Table 5.

$N_{\alpha}$ |  | $(n_{\alpha}^{1},m_{\alpha}^{1})$ | $(n_{\alpha}^{2},m_{\alpha}^{2})$ | $(n_{\alpha}^{3},m_{\alpha}^{3})$
---|---|---|---|---
$N_{a}=3+1$ |  | $(1,0)$ | $(3,1)$ | $(3,-1)$
$N_{b}=1$ |  | $(0,1)$ | $(1,0)$ | $(0,-1)$
$N_{c}=1$ |  | $(0,1)$ | $(0,-1)$ | $(1,0)$
$N_{h_{1}}=1$ |  | $(-2,1)$ | $(-3,1)$ | $(-4,1)$
$N_{h_{2}}=1$ |  | $(-2,1)$ | $(-4,1)$ | $(-3,1)$
$N_{f}=20$ |  | $(1,0)$ | $(1,0)$ | $(1,0)$
Table 5: Explicit $(T^{2}\times T^{2}\times
T^{2})/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orientifold model realising the
MSSM-like model of Table 4. D6-brane multiplicities are given in fractional
three-cycle units.

Notice that the upper part of the table realises the intersection numbers
needed for the Pati–Salam spectrum, and one finds in addition a minimal Higgs
sector with a non-vanishing $\mu$-term, see Table 6. The lower part of table 5
is the one needed to satisfy the RR tadpole cancellation conditions (1.6). In
principle this sector introduces an additional gauge group
$U(1)^{\prime}\times USp(40)$, with
$U(1)^{\prime}=\frac{1}{3}U(1)_{a}+2[U(1)_{h_{1}}-U(1)_{h_{2}}]$, that we
would like to treat as a hidden sector of the theory. However, the $U(1)$
branes $h_{1}$ and $h_{2}$ intersect with those of the Pati–Salam sector,
generating chiral exotics that naively prevent us from doing so, see [38, 39]
for the resulting spectrum.

Nevertheless, one of the main advantages of the model building strategy
outlined in Section 2 is that this kind of chiral exotics is easily avoidable.
Indeed, because the particle content of the Pati–Salam sector cancels all
cubic $SU(N)$ anomalies by itself (including those for $N=2$ and $N=1$, which
in this particular case are trivial) the chiral exotics that arise from
additional D-brane sectors appear as vector-like from the viewpoint of the
visible sector gauge group. In practice, this means that there is some
direction in moduli space that allows us to get rid of such chiral exotics, as
it is the case in this example. More precisely, by moving in the complex
structure moduli space of the the first $T^{2}$ one can induce a tachyon in
the bifundamental of $U(1)_{h_{1}}\times U(1)_{h_{2}}$ via a D-term potential,
see below. Condensation of this tachyon corresponds to the D-brane
recombination process $h_{1}+h_{2}^{\prime}\to h$, which does not affect the
low energy gauge group, but greatly simplifies the chiral spectrum.

Sector | Matter | $SU(4)\times SU(2)\times SU(2)\times U(1)^{\prime}\times USp(40)$
---|---|---
(ab) | $F_{L}$ | 3(4,2,1)1/3
(ac) | $F_{R}$ | 3(4,1,2)-1/3
(bc) | $H$ | (1,2,2)0
(bh) |  | 2(1,2,1)2
(ch) |  | 2(1,1,2)-2
Table 6: Spectrum of the model of Table 5 after the D6-brane recombination
$h_{1}+h_{2}^{\prime}\to h$, with the subindex indicating the charge under
$U(1)^{\prime}=\frac{1}{3}U(1)_{a}+2U(1)_{h}$. There is no chiral matter under
the gauge group $USp(40)$, which is a hidden sector.

The final result is given in Table 6, where one can see that the lower part of
the spectrum is not chiral under the Pati–Salam gauge group. Given the
simplicity of this construction this is quite an encouraging result, which one
may hope to improve by exploring more general setups. Indeed, applying the
same approach to more involved toroidal orbifolds, general Calabi–Yau
manifolds and RCFT models yields even more realistic models.

####  3.1.3 Further EFT Features

Besides a reasonable chiral spectrum, a realistic model must display a set of
couplings and a non-chiral light spectrum that are compatible with the MSSM or
extensions thereof. In the sequel we outline the computation of these
additional features in the context of intersecting D6-brane models.

The simplest quantity to consider is the gauge kinetic function associated
with each stack of D6-brane. It reads

$f_{a}^{\rm D6}=\frac{1}{2\pi\ell_{s}^{3}}\int_{\Pi_{3}^{a}}e^{-\phi}{\rm
Re}\,\Omega_{3}+iC_{3},$ (3.12)

from where one can compute the gauge couplings of the non-abelian and abelian
gauge factors. In the latter case there will generically be a kinetic mixing
with bulk $U(1)$ gauge symmetries, if present [42].

One may also consider the presence of light or massless non-chiral particles
in the D-brane sector, such as D-brane moduli that appear as ${\cal N}=1$
multiplets in the adjoint. A stack of $N$ D6-branes wrapping a special
Lagrangian three-cycle $\Pi_{3}$ has $b_{1}(\Pi)$ deformations of its
embedding that preserve the special Lagrangian condition, as it follows from
McLean’s theorem. These are complexified by the same amount of Wilson lines,
giving rise to $b_{1}(\Pi)$ ${\cal N}=1$ chiral multiplets in the adjoint
representation of $SU(N)$. Aiming to build models without adjoints leads us to
either consider three-cycles with $b_{1}(\Pi)=0$ or D6-branes with
deformations that are fixed by an F-term potential. There are two sources for
the latter. The first source are elements of $H_{1}(\Pi,\mathbb{Z})$ dual to
two-cycles of $\Pi_{3}$ that are non-trivial in $H_{2}(X_{6},\mathbb{Z})$
[43]. The second one is the superpotential generated by disc worldsheet
instantons ending on one-cycles of $\Pi_{3}$ [44]. While this second source in
principle affects all adjoint fields, it is generically expected to give them
exponentially suppressed masses in the large volume regime in which we are
working. In general, D-brane adjoints redefine the 4d holomorphic variables
that arise from bulk moduli upon dimensional reduction. In this case D6-brane
moduli redefine the CY complex structure moduli see e.g. [45, 46, 47], and
their mass spectrum should be treated in the broader framework of moduli
stabilisation. Such complex structure moduli enter the D-flatness condition
for D6-branes, and can induce Fayet-Iliopoulos terms which break supersymmetry
and may trigger D6-brane recombination [48].

Besides adjoint masses, worldsheet instantons with the topology of a disc are
a source for mass terms for vector-like pairs that arise from the transverse
intersections of a pair of D6-branes, as illustrated in Figure 6, as well as
for Yukawa couplings [7].

a)  b)

Figure 6: Worldsheet instantons as generators of a) masses for vector-like
pairs and b) Yukawa couplings. Figures taken from [10].

These two quantities have a direct interpretation in terms of Kontsevich’s
homological mirror symmetry conjecture, and their computation is a rich
mathematical subject of research. While they are difficult to compute in
general, in simple examples like toroidal orbifold models one can perform the
computation quite explicitly, showing for instance that the Pati–Salam model
described above leads to Yukawa couplings of rank one [21].

What worldsheet instantons cannot generate are couplings that are forbidden by
D-brane $U(1)$ symmetries that become massive due to a Stückelberg mechanism.
In that case, couplings should be generated by D-brane instantons, which in
this case are D2-brane instantons wrapping special Lagrangian three-cycles of
$X_{6}$ [31]. In typical models such couplings include right-handed neutrino
masses and Yukawas forbidden by global $U(1)$ symmetries, as described in
Section 2. Moreover, if any of these couplings is forbidden by the discrete
gauge symmetries that are remnants of the massive $U(1)$s they will not be
generated even at the non-perturbative level. The model-building challenge
then resides in using this structure to forbid unwanted couplings and obtain
those that are necessary phenomenologically, with the appropriate magnitude
given by their suppression factors.

###  3.2 Type IIB Orientifolds

We now turn to Type IIB string theory on the 10d background $X_{4}\times
X_{6}$, where again $X_{6}$ is taken to be a compact Calabi–Yau three-fold
$X_{6}$, with Kähler two-form $J$ and holomorphic three-form $\Omega_{3}$.
There are two different kinds of orientifold projections which are compatible
with a large compactification volume:

$\displaystyle{\rm O3/O7\,projection}:$
$\displaystyle\quad\Omega(-1)^{F_{L}}{\cal R}\quad\text{such that}\quad{\cal
R}J=J,\quad{\cal R}\Omega_{3}=-\Omega_{3},$ $\displaystyle{\rm
O5/O9\,projection}:$ $\displaystyle\quad\Omega{\cal R}\qquad\qquad\text{such
that}\quad{\cal R}J=J,\quad{\cal R}\Omega_{3}=\Omega_{3},$

where ${\cal R}$ is now a holomorphic involution of $X_{6}$. As in the Type
IIA case, this projection reduces the bulk supersymmetry from 4d ${\cal N}=2$
to ${\cal N}=1$ and introduces a series of O-planes at the fixed loci of
${\cal R}$. The difference is that these fixed loci are even-dimensional
submanifolds of $X_{6}$. In the first projection they are given by points or
holomorphic four-cycles, leading to O3 and/or O7-planes, while the second
projection leaves invariant curves or the whole of $X_{6}$. These holomorphic
involutions are much better understood than their anti-holomorphic
counterparts, which has resulted in the construction of models in geometries
beyond toroidal orbifolds. The same remark applies to the space-time D-branes
that host gauge interactions in these models, which now consist of D3, D5, D7
and D9-branes wrapping internal even-dimensional cycles $\Pi_{p-3}$ of
$X_{6}$. The BPS conditions for a single D-brane of this kind read

$\displaystyle\Pi_{p-3}\,\text{holomorphic},\quad{\cal F}^{(2,0)}=0,$
$\displaystyle\text{F-flatness},$ (3.13) $\displaystyle{\rm
Im}\,e^{-i\theta}e^{{\cal F}+iJ}{\sqrt{\hat{A}_{\Pi_{p-3}}}}=0,$
$\displaystyle\text{D-flatness},$ (3.14)

where $\theta=0$ for the O3/O7-projection and $\theta=\pi/2$ in the O5/O9
projection, and $\hat{A}$ is the A-roof genus of the tangent bundle of
$\Pi_{p-3}$, encoding part of the D-brane curvature couplings. As a non-
trivial worldvolume field-strength is allowed, these objects must typically be
understood as D3/D5/D7/D9 bound states [49] or, from a more mathematical
perspective, as coherent sheaves [50, 51]. These abelian BPS conditions can be
generalised to non-abelian D-brane configurations that are allowed for stacks
of several D-branes, featuring a non-abelian field strength ${\cal F}$ and/or
a non-abelian D-brane worldvolume embedding.

Non-abelian D-brane configurations have mostly played a role in models with
D9-branes, due to their analogy with heterotic compactifications. Due to this
similarity, they will not be discussed here, nor will be models with O5/O9
projection. Instead, we will focus on models with O3/O7 projection, which
display a set of features that are very representative of D-brane model
building:

  * -

The SM gauge group or its extension can be localised in a patch of the compact
manifold $X_{6}$. This simplifies its construction, which can be done in two
steps: i) first building a model in a local patch and ii) embedding it into a
compact manifold, see Figure 7. This procedure is known as the bottom-up
approach to model building [52], and it allows one to distinguish between
those physical features that are only sensitive to local data and those that
depend on global data of the construction.

  * -

Due to this localisation, and the fact that 4d gauge couplings are controlled
by the internal volume of D-branes, these models naturally realise the idea of
gauge coupling unification. They in addition permit to decouple the strength
of gauge and gravitational interactions, even to the extent to implement the
large extra dimension scenario [53, 54].

  * -

One can easily combine these models with additional ingredients that implement
mechanisms for moduli stabilisation at large volume, like background fluxes
[55], in order to build more sophisticated models, see e.g. [56].

In the following we will discuss two classes of models that illustrate how
these features are realised: D-branes at singularities and intersecting
D7-branes. The latter can be thought of as a particular case of the F-theory
constructions of Section 5, which also incorporate these attractive features.

Figure 7: Two-step procedure for building $4d$ models based on D3-branes at
singularities. Figure taken from [52].

####  3.2.1 D-branes at Singularities

Strictly speaking, D-branes at singularities do not correspond to large volume
models that can be treated in the 10d supergravity regime. They are engineered
in neighbourhoods of the compact manifold $X_{6}$ that display a singular
geometry, obtained for instance from collapsing some of its cycles.
Nevertheless, one can analyse this system by directly quantising open strings
in a such a singular geometry, following the techniques initiated in [57, 58],
and then embed the resulting gauge sector into a fully-fledged
compactification, along the lines of the bottom-up philosophy described above.

The simplest example of this class of models is given by D3-branes at orbifold
singularities. A stack of $N$ D3-branes in this flat space yields a 4d ${\cal
N}=4$ $U(N)$ gauge theory and therefore a non-chiral gauge sector. The non-
chiral nature of the gauge sector persists if the D3-brane is placed at any
smooth point of a compactification manifold $X_{6}$, since the effect of
curvature and background fluxes can at best lead to a massive deformation of
4d ${\cal N}=4$ SYM. The only way to obtain a chiral spectrum is to place the
D3-brane on top of a singular geometry, like the one obtained by an orbifold
quotient of the form $\mathbbm{C}^{3}/\Gamma$ with fixed points. Let us for
simplicity consider the cyclic orbifold group $\Gamma=\mathbbm{Z}_{k}$
generated by an action on $\mathbbm{C}^{3}$ of the form

$(z^{1},z^{2},z^{3})\mapsto\left(e^{2\pi i\frac{a_{1}}{k}}z^{1},e^{2\pi
i\frac{a_{2}}{k}}z^{2},e^{2\pi i\frac{a_{3}}{k}}z^{3}\right),\qquad
a_{i}\in\mathbb{Z}.$ (3.15)

In order for spinors to be well-defined in this background one needs to
require that $\sum_{i}a_{i}\in 2\mathbb{Z}$. Then one can embed $\Gamma$ into
$SU(4)$ as ${\rm diag}(e^{2\pi i\frac{b_{0}}{k}},e^{2\pi
i\frac{b_{1}}{k}},e^{2\pi i\frac{b_{2}}{k}},e^{2\pi i\frac{b_{3}}{k}})$ with
$b_{i}\in\mathbb{Z}$ and $\sum_{i}b_{i}=0\mod k$, such that
$a_{1}=b_{2}+b_{3}$, $a_{2}=b_{3}+b_{1}$ and $a_{3}=b_{1}+b_{2}$, and quantise
closed strings in this background [59]. To embed this singularity into a CY
geometry one must impose local $SU(3)$ holonomy, which amounts to
$a_{1}+a_{2}+a_{3}=0\mod k$. Then one can assume $b_{0}=0$ and $b_{i}=-a_{i}$.

Placing a stack of $N$ D3-branes at the fixed point of (3.15) yields at gauge
sector that is an orbifold projection of the initial 4d ${\cal N}=4$ $U(N)$
gauge theory. The result depends on how the orbifold generator acts on the
D3-brane Chan-Paton degrees of freedom, which is specified by an element of
$U(N)$ of the form

$\gamma={\rm
diag}\,\left(\mathbbm{1}_{N_{0}},\omega\mathbbm{1}_{N_{1}},\dots,\omega^{k-1}\mathbbm{1}_{N_{k-1}}\right),\qquad\omega=e^{\frac{2\pi
i}{k}},$ (3.16)

with $\sum_{a=0}^{k-1}N_{a}=N$. The 4d ${\cal N}=4$ vector multiplet gets
projected out to those Chan-Paton degrees of freedom $\lambda$ invariant under
the adjoint action $\lambda\mapsto\gamma\lambda\gamma^{-1}$, while for the
three adjoint chiral multiplets $\Phi^{i}$ only the modes invariant under
$\lambda\mapsto e^{-2\pi i\frac{a_{i}}{k}}\gamma\lambda\gamma^{-1}$ survive.
This results in the following spectrum:

$\displaystyle\text{Vector multiplet}:$
$\displaystyle\quad\prod_{a=0}^{k-1}U(N_{a}),$ (3.17)
$\displaystyle\text{Chiral multiplets}:$
$\displaystyle\quad\sum_{a=0}^{k-1}\left[({\bf
N}_{a},{\bf\bar{N}}_{a+a_{1}})+({\bf N}_{a},{\bf\bar{N}}_{a+a_{2}})+({\bf
N}_{a},{\bf\bar{N}}_{a+a_{3}})\right],$

with a set of Yukawa couplings that arise from truncation of the parent ${\cal
N}=4$ superpotential $W=\operatorname{Tr}(\Phi^{1}[\Phi^{2},\Phi^{3}])$.
Notice that this is a particular case of the general spectrum of Table 1, with
the chiral index $I_{ab}$ determined by the orbifold twists $a_{i}$, and
without the presence of those representations that arise due to the
orientifold projection. Indeed, while orientifold planes are a necessary
ingredient of the global construction, a singularity can be located at a point
$p\in X_{6}$ away from any O-plane. Then the orientifold projection simply
requires that there is an identical singularity with similar D-brane content
located at ${\cal R}p$.

This chiral spectrum has a limited capacity of family replication, which
occurs when two or more orbifold twists $a_{i}$ are equal mod $k$. This sets
an upper bound of three families and makes the orbifold $\mathbbm{Z}_{3}$ with
twist $\\{a_{i}\\}=(1,1,-2)$ particularly attractive [52, 60]. One may take
$N_{0}=3$, $N_{1}=2$ and $N_{2}=1$, yielding a SM gauge group with hypercharge
$U(1)_{Y}=U(1)_{0}/3+U(1)_{1}/2+U(1)_{2}$ and a partial SM chiral spectrum.
The spectrum can then be completed by considering a stack of D7-branes going
through the orbifold singularity and with a non-trivial action of the orbifold
group on their Chan-Paton degrees of freedom [52].

Besides the particularities of each model, there is a series of general
features common to all of them that are worth mentioning:

  * -

These models directly realise gauge coupling unification at the
compactification scale because the coupling of all the gauge groups in (3.17)
is given by the 4d dilaton. When introducing D7-branes these may carry their
own gauge group, but their gauge couplings are suppressed with respect to the
local ones by the volume of the four-cycle wrapped by the D7-brane, and so
from the viewpoint of the local model they are treated as flavour branes. In
general, the role of gauge groups coming from D7-branes can only be determined
upon the global completion of the local model.

  * -

There is always an anomaly-free massless $U(1)$ combination given by

$U(1)_{\rm diag}=\sum_{a=0}^{k-1}\frac{U(1)_{a}}{N_{a}},$ (3.18)

which plays the role of hypercharge in the $\mathbbm{C}^{3}/\mathbbm{Z}_{3}$
model discussed above. Other $U(1)$ symmetries are typically anomalous and
acquire a mass of the order of the string scale via a Green–Schwarz mechanism
[61].

  * -

The gauge and chiral content of these local models can be encoded in a quiver
diagram, in which each node represents a gauge group and a set of arrows
connecting them the bifundamental representations. A D3-brane on a given node
is dubbed fractional D3-brane, and the set of D3-branes that add up to the
regular representation in (3.16) is identified as a bulk D3-brane that can be
separated from the singularity and probe the bulk of $X_{6}$. One can also
extend the diagram to incorporate flavour D7-branes and their associated
chiral content, see [62] for a short review.

  * -

When choosing (3.16), or more generally the D-branes in the quiver, one has to
observe the local RR tadpole conditions. These form a subset of the whole set
of tadpole conditions of the compactification which is only sensitive to
fractional D3-brane and D7-brane charges. As usual satisfying these conditions
implies that the non-abelian and mixed anomalies cancel [52].

Additionally, this simple orbifold setup can be generalised in a number of
ways, which give rise to more and more sophisticated models, and which we
briefly summarise in the following:

  * -

One may study more general orbifold groups $\Gamma$ such as
$\mathbb{Z}_{k}\times\mathbb{Z}_{m}$, or non-abelian subgroups of $SU(3)$ like
for instance $\Gamma=\Delta_{27}$, which also features family triplication
[52].

  * -

One may consider non-orbifold toric singularities such as conifold or del
Pezzo singularities. In this case the gauge theory data are more efficiently
encoded in a so-called dimer diagram [63], which is a tiling of $T^{2}$. While
these models provide more flexibility, certain features like the upper bound
on three families remain generic [64].

  * -

One may explore orientifolded singularities. In particular certain
orientifolded del Pezzo singularities give rise to realistic SM spectra
without the need of D7-branes [65], using the full spectrum of Table 1. Note
that this result is in agreement with the general model building philosophy of
Section 2, in the sense that one can build a realistic spectrum that avoids
the necessity of additional D-brane sectors by cancelling all anomalies
locally.

Finally, it is worth mentioning the efforts to perform systematic embeddings
of local models of D-branes at singularities into global compactifications,
mostly using toric geometry techniques, see e.g. [66, 67, 68].

####  3.2.2 Intersecting D7-branes

Models of D-branes at singularities are a particular class of a broader set,
which in the generic regime without collapsed cycles correspond to models of
intersecting D7-branes. In this case the basic object is a D7-brane wrapping a
holomorphic four-cycle $\Pi_{4}\subset X_{4}$, threaded by a non-trivial
worldvolume field strength ${\cal F}$ defined as in (3.2). The full spectrum
between two D7-branes wrapping 4-cycles $\Pi_{4}^{a}$ and $\Pi_{4}^{b}$ must
be computed in terms of Ext groups [69], but in order to determine the chiral
index of Table 1 one may use the Riemann-Roch-Hirzebruch theorem to arrive at
the expression [70]:

$I_{ab}=\int_{X_{6}}[\Pi_{4}^{a}]\wedge[\Pi_{4}^{b}]\wedge\left(c_{1}(F_{a})-c_{1}(F_{b})\right)\,.$
(3.19)

Here $[\Pi_{4}^{a}]$ is the two-form Poincaré dual to the divisor class of
$\Pi_{4}^{a}$, and $c_{1}(F_{a})$ is the first Chern class of the quantised
piece of the worldvolume flux ${\cal F}$ threading $\Pi_{4}^{a}$, viewed as an
element of $H^{2}(X_{6},\mathbbm{Z})$. This formula extends to orientifold
images by using that $(\Pi_{4}^{a^{\prime}},F_{a^{\prime}})=({\cal
R}\Pi_{4}^{a},-{\cal R}F_{a})$, and to the orientifold as
$I_{aO}=2\int_{X_{6}}[\Pi_{4}^{a}]\wedge[\Pi_{\rm O7}]\wedge c_{1}(F_{a})$.
With these expressions at hand, the strategy to build models works quite
similarly to the Type IIA case, and for instance one may reproduce the
Pati–Salam sector of Table 6 by using three stacks of intersecting branes [38,
39].

A natural arena for D7-brane model building is in the context of local models,
following the general philosophy outlined above. The gauge group is now
localised on a (fluxed) non-trivial four-cycle $\Pi_{4}$ which can be
collapsed by moving in moduli space, and that may host chiral matter either
via self-intersection or via the intersection with flavour D7-branes.
Contracting such a four-cycle leads to a singularity of the type described
above with the wrapped D7-branes becoming fractional D3-branes [71]. While
this shows that the two classes of Type IIB models under discussion are
secretly similar to each other, in practice the model building possibilities
are quite different. The reason is that the spectrum of BPS D-branes at the
singular point and at volumes large compared to the string scale are quite
different. In this sense both classes of models should be considered
separately.

Still, they have a number of similarities. If the volume of the contractible
four-cycle $\Pi_{4}$ is large enough, one may achieve an approximate gauge
coupling unification. This is because the gauge coupling constant associated
to a D7-brane gauge group is set by its volume and its worldvolume flux, and
in the regime of dilute flux densities the former is the dominant
contribution. This suggests GUT model building as an obvious model building
option, and in particular $SU(5)$ GUTs, with a 10 representation coming from
intersection with an O7-plane [70]. These models suffer from problems similar
to the ones mentioned in Section 2.3, in the sense that the massive $U(1)$
symmetry within $U(5)$ forbids the generation of the top Yukawa coupling ${\bf
10}\,{\bf 10}\,{\bf 5}_{H}$ at the perturbative level. This motivates applying
the same model building philosophy but in the more general context of F-theory
GUTs, to be discussed in Section 5. Given the similarities between F-theory
and D7-brane GUTs, we will refrain from further discussing the latter, and
refer the reader to [70, 60] for details.

##  4 Rational Conformal Field Theories

A conformal field theory (CFT) is a field theory with invariance under the
conformal group. Here we are only interested in two-dimensional CFT’s living
on the worldsheet of strings. The properties (such as spectra and correlation
functions) of such a theory can be described in terms of the algebra of
conformal field theory, the Virasoro algebra. This algebra is characterised by
a number $c$ called the central charge. For $c<1$ the representation theory of
the Virasoro algebra is well-known [72]. It has a discrete series of unitary
representations for certain rational values of $c$, accumulating at $c=1$. For
each $c$ in the series there is a finite number of representations,
characterised by a conformal weight denoted as $h$. Conformal field theories
with a finite number of representations are called rational conformal field
theories, or RCFT for short.

Conformal field theory encompasses all world-sheet descriptions of
perturbative string theory, including geometric orbifolds and orientifolds,
like the ones discussed in the previous section. But in those cases the
geometric language provides the more powerful description. Using RCFTs will
allow us to go into uncharted territory not easily accessible geometrically.

In addition to the Virasoro algebra, generated by currents of spin 2, other
algebras may be present. They may be generated by currents of spin 1 (affine
Lie algebras, often called Kac-Moody algebras), higher spin (called $W$
algebras), spin-$\tfrac{1}{2}$ (free fermions) and spin-$\tfrac{3}{2}$
(superconformal algebras). As a general rule, the extra symmetry makes
representations larger, and reduces their number. In particular, the number of
representations may become finite, producing an RCFT. The set of generators,
acting on either the right- or left-moving modes of the theory, is called the
chiral algebra.

###  4.1 Gepner Models

Compactified closed superstring theories are often described geometrically, as
strings propagating in a space with $D$ flat dimensions, and $10-D$ dimensions
rolled up on a torus, an orbifold or a Calabi–Yau manifold. But there is an
alternative description in terms of a suitable CFT on the worldsheet.

To characterise the compactified sector of a superstring with four
uncompactified dimensions we need superconformal field theories with a total
central charge of 9, and two supersymmetries. This is because the
uncompactified theory is defined by world-sheet fields $X^{I}$ and $\Psi^{I}$,
$I=1,\ldots,6$. The free world-sheet theory has ${\cal N}=2$ supersymmetry,
and its conformal field theory has a total central charge of $6+3=9$ (each
word-sheet boson contributes 1, each fermion $\tfrac{1}{2}$). The compactified
theory has to mimic these properties to preserve the consistency of the
theory. In short, we need a $c=9$, ${\cal N}=2$ superconformal field theory.

The ${\cal N}=2$ superconformal field theories also have a discrete series,
this time accumulating at $c=3$. The values of $c$ are

$c=\frac{3k}{k+2}\,.$ (4.1)

These ${\cal N}=2$ minimal models are superconformal RCFTs, but they do not
have the required central charge of 9. This can be solved by “tensoring"
several copies of them, in such a way that the sum of their central charges is
9. There are 168 solutions $(k_{1},\ldots,k_{n})$ to the equations

$\sum_{i}\frac{3k_{i}}{k_{i}+2}=9\,.$ (4.2)

One needs 4 to 9 copies, for example $(3,8,8,8)$ or $(1,1,1,1,1,1,1,1,1)$.
This method was first used by D. Gepner [73] to construct compactified
heterotic strings, and hence these tensor products are often referred to as
Gepner models. Here we are using them to describe the compactified sector of a
Type II superstring.

Gepner models are by no means the only way to construct superconformal RCFTs
with $c=9$, but they are the only ones that are both non-trivial and
mathematically under control, so that relevant quantities are computable.
Therefore they present an excellent theoretical laboratory for investigating
both closed and open string models.

###  4.2 Modular Invariant Partition Functions

After choosing an RCFT, a second important choice to be made is that of the
modular invariant partition function (MIPF), which enters the discussion when
the world-sheet diagram is a torus. In closed string theories this diagram
encodes the closed string spectrum, in terms of a partition function

$\begin{split}\int{\cal D}Xe^{-S_{E}(X)}&={\rm Tr}\ e^{2\pi
i\tau(L_{0}-1)}e^{-2\pi i\bar{\tau}(\bar{L}_{0}-1)}\,,\end{split}$ (4.3)

where the left-hand side denotes – symbolically – the path integral. The
complex parameter $\tau$ denotes the modular parameter of the torus, which
describes its shape. By expanding in powers of

$q=e^{2\pi i\tau}\ \ \hbox{and}\ \ \bar{q}=e^{-2\pi i\bar{\tau}}$ (4.4)

one can read off the multiplicities of the closed string spectrum. Only states
with the same power of $q$ and $\bar{q}$ are physical.

The right-hand side of (4.3) can be expanded in terms of characters of the
(extended) conformal field theory,

$\chi_{i}={\rm Tr}\mid_{i}\ e^{2\pi i\tau(L_{0}-1)}\,,$ (4.5)

where the trace is over all states in the representation built on ground state
$i$ by the action of the Virasoro generators $L_{n}$ and the generators of
some extension of the Virasoro algebra. The characters have the following
expansion

$\chi_{i}(\tau)=q^{h_{i}-c/24}\sum_{n}d_{n}q^{n}\,,$ (4.6)

where $d_{n}$ are non-negative integers.

The number of such representations will in general be infinite, but if the
extension of the symmetry algebra is large enough we have an RCFT, and then
the number is finite. The character label 0 corresponds to the vacuum
representation. It contains the vacuum and all states obtained from it by the
action of the Virasoro generators and all generators of the extended algebra
(if any), modulo states of zero norm (null states).

The torus partition function has the following character expansion:

${\rm Tr}\ e^{2\pi i\tau(L_{0}-1)}e^{-2\pi
i\bar{\tau}(\bar{L}_{0}-1)}=\sum_{ij}Z_{ij}\chi_{i}(\tau)\bar{\chi}_{j}(\bar{\tau})\,.$
(4.7)

Here the coefficients $Z_{ij}$ must be non-negative integers, subject to the
constraint of modular invariance. This constraint is a consequence of the fact
that there are infinitely many parametrisations of the torus that must all
give the same result. These reparametrisations are generated by discrete
transformations of the parameter $\tau$:

$\displaystyle T:$ $\displaystyle\tau\rightarrow\tau+1\,,$ (4.8)
$\displaystyle S:$ $\displaystyle\tau\rightarrow-\frac{1}{\tau}\,.$ (4.9)

These transformations are represented as matrices $S$ and $T$ on the set of
characters $\chi_{i}$, and the conditions for modular invariance are therefore

$\left[S,Z\right]=\left[T,Z\right]=0\,.$ (4.10)

Furthermore $Z_{00}$, the multiplicity of the vacuum state, must be equal to
1. In closed string theory this has the consequence that there is precisely
one graviton in the spectrum.

There has been a lot of work on finding solutions to the conditions for
$Z_{ij}$, but this has been completed only for a few extended algebras.
Notable examples are the unextended Virasoro algebra with $c<1$ as discussed
above, and the $SU(2)$ Kac-Moody algebra [74]. Furthermore there are known
classes of general solutions that are valid for any RCFT: the charge
conjugation invariant $Z_{ij}=C_{ij}$, the diagonal invariant
$Z_{ij}=\delta_{ij}$, conformal embeddings [75, 76] and simple current
invariants [77]. Here $C_{ij}$ is a bijection that corresponds to charge
conjugation in the world-sheet theory, acting on the ground states. It is
known that the diagonal invariant is always a solution to the conditions of
modular invariance at one loop, but does not always define a consistent CFT
[78]. On the other hand, the charge conjugation invariant always defines a
consistent CFT, and should be viewed as the canonical definition of the
theory. We refer to this case as “C-diagonal".

###  4.3 Fusion Rules and Simple Currents

The fusion rules of an RCFT indicate how many couplings exist when two
representations $[i]$ and $[j]$ are combined. It can be formally written as

$[i]\times[j]=\sum_{k}N_{ij}^{~{}~{}k}[k]\ ,$ (4.11)

where $[i]$ denotes an (extended) CFT representation. A simple current [77,
79] is a special representation $J$ with the particular feature that just one
term (labelled $Ji$) exists on the right-hand side:

$[J]\times[i]=[Ji]\,.$ (4.12)

The set of representations generated by the action of $J$ on $[i]$ is called
the orbit of $[i]$.

The reason simple currents enter the story is that they allow us to construct
a large number of modular invariant partition functions. This works roughly as
follows. The action of the simple currents on themselves defines an abelian
discrete group. Take any subgroup of that discrete group. Now take a set of
generators of that subgroup. On that basis of generators one defines a matrix
$X$. This is a matrix of rational numbers [80] computed from a simple
equation, which occasionally has no solution, but generally has a number of
solutions that grows exponentially with the size of the basis. We will omit
the details here, but the main point is that for any subgroup of the simple
current group one has many modular invariants.

The simple current group of ${\cal N}=2$ minimal model is fairly large. For
$k$ odd it is $\mathbb{Z}_{4k}$ and for $k$ even it is
$\mathbb{Z}_{2}\times\mathbb{Z}_{2k}$. But much more importantly, in a tensor
product one gets a product of all these groups. For example, the combination
$(3,8,8,8)$ yields
$\mathbb{Z}_{12}\times(\mathbb{Z}_{2})^{4}\times(\mathbb{Z}_{16})^{4}$. These
discrete groups have a huge number of subgroups, and hence for every Gepner
model we get a huge number of MIPFs. This boosts the number of available
Gepner models from 168 (for just the C-diagonal MIPF) to 5403.

###  4.4 Open String CFT

To use RCFTs for open string model building one needs a description of ends of
open strings. In a worldsheet description they sweep out worldsheets with
boundaries. The most general worldsheet for oriented closed strings is a
Riemann surface of arbitrary genus $g$, which is a torus with $g$ handles. One
can attach tubes to act as external closed strings. To get all open string
diagrams, one can make holes in those surfaces with the topology of a circle.
Finally, one can add strips to the edges of the holes to act as external open
strings.

In section (1.2.1) we have seen that O-planes are needed in order to build
consistent open string models (at least in the supersymmetric case). Just as
D-branes are described by means of boundaries of the surface, O-planes are
described by means of another topological feature, a crosscap. A crosscap is
added to a surface by making a hole, as above, but identifying the opposite
points of the boundary circle to each other in an orientation reversing way.
Hence an ant crawling on one side of the circle finds itself on the other side
after crossing the crosscap. One can add more than one crosscap, but not all
resulting surfaces are topologically distinct. A sphere with two holes is
topologically an annulus; a sphere with one hole and a crosscap is equivalent
to a Moebius strip, and a sphere with two crosscaps is a Klein bottle.

Now we have to determine the behaviour of the CFT near the edges of the
surface or in the prasence of a crosscap

####  4.4.1 Boundary and Crosscap States

Any surface with boundaries or crosscaps has a double cover which only has
handles, and on which one defines a closed, oriented conformal field theory.
This CFT is the starting point for constructions of open (and unoriented)
strings, which were referred to as “open descendants” of the closed string
theories in [81]. The presence of boundaries and crosscaps is described by
boundary and crosscap “state”, which are not really states themselves, but in
fact non-normalisable linear combinations of states in the closed string
Hilbert space.

Here we will assume that the entire chiral algebra remains unbroken at the
boundary or by a crosscap, ignoring the interesting possibility of breaking
part of the closed string symmetries. The condition that a symmetry is not
broken by a boundary or a crosscap is

$(W_{n}+(-1)^{h_{W}}\tilde{W}_{n})|B\rangle=0\ ,\ \ \ \ \ \ \
(W_{n}+(-1)^{h_{W}+n}\tilde{W}_{n})|C\rangle=0\,,$ (4.13)

where $W_{n}$ is a mode of a chiral current, $\tilde{W}_{n}$ a mode of an
anti-chiral current and $h_{W}$ its conformal weight; $|B\rangle$ a boundary
state and $|C\rangle$ a crosscap state. A basis for the solutions to these
conditions is formed by the Ishibashi states [82]

$|B_{i}\rangle=\sum_{I}|I\rangle_{i}\otimes U_{B}|I\rangle_{i^{c}}\ ,\ \ \ \ \
\ \ |C_{i}\rangle=\sum_{I}|I\rangle_{i}\otimes U_{C}|I\rangle_{i^{c}}\ .$
(4.14)

Here the $i$ labels a representation of the chiral algebra and $i^{c}$ its
charge conjugate. The sum is over all states in the representation, and
$U_{B}$ and $U_{C}$ are operators satisfying

$\tilde{W}_{n}U_{B}=(-1)^{h_{W}}U_{B}\tilde{W}_{n}\ ,\ \ \ \ \ \ \
\tilde{W}_{n}U_{C}=(-1)^{h_{W}+n}U_{C}\tilde{W}_{n}\,.$ (4.15)

Any boundary state must be a linear combination of these Ishibashi states,
i.e.

$|B_{a}\rangle=\sum_{i}B_{ia}|B_{i}\rangle\ ,\ \ \ \ \ \ \
|C\rangle=\sum_{i}\Gamma_{i}|C_{i}\rangle\,.$ (4.16)

It turns out that in general one can allow for several boundary states,
labelled by a boundary label $a$, but for only one crosscap state for a given
theory. For a given MIPF $Z_{ij}$ more than one crosscap state may exist, but
one cannot mix them.

A choice of a set of boundary labels $a$, and a set of coefficients $B_{ai}$
and $\Gamma_{i}$ form part of the data that define an open string CFT.
Although more is required to specify all correlation functions on arbitrary
surfaces, this information is sufficient to compute the one-loop diagrams
without external lines that contribute to the open and closed string partition
functions. The relevant string diagrams are those with vanishing Euler number.
From these diagrams we can compute the spectrum of the theory.

####  4.4.2 Orientifold Partition Functions

In the presence of boundaries and crosscaps there are four topologically
distinct surfaces with vanishing Euler number: the torus, the Klein bottle,
the annulus and the Möbius strip. These contributions can be expanded in
(bi)linears of characters:

$\displaystyle{\cal T}=\sum_{ij}Z_{ij}\chi_{i}(\tau)\chi_{j}^{*}(\tau)\,,$
$\displaystyle\;\;\;\;{\cal K}=\sum_{i}K^{i}\chi_{i}(2it)\,,$ (4.17)
$\displaystyle{\cal A}=\sum_{ab}N_{a}N_{b}A^{i}_{ab}\chi_{i}(it/2)\,,$
$\displaystyle\;\;\;\;{\cal
M}=\sum_{a}N_{a}M^{i}_{a}e^{-i\pi(h_{i}-c/24)}{\chi}_{i}\left(\tfrac{1}{2}(1+it)\right)\,.\;\;\;$
(4.18)

Here $\tau$ is the modular parameter of the torus, as before, and $t={\rm
Im}\tau$. As discussed in section 4.2, the torus defines the oriented closed
string partition function. Likewise, the sum of the torus and the Klein bottle
defines the unoriented closed string partition function. The annulus is an
open string loop, and defines. together with the Möbius strip, the open string
partition function. The Klein bottle amplitude and the Möbius strip act as an
orientifold projection on the closed and open string spectrum respectively.

####  4.4.3 Channel Transformations

The diagrams that describe the spectrum are computed in the transverse
channel, in which closed strings propagate between two boundaries, a boundary
and a crosscap, or two crosscaps. In Figure 8 this is illustrated for the
simplest case: a diagram of closed strings propagation from boundary $a$ to
boundary $b$ is transformed to a diagram where open strings with endpoints $a$
and $b$ propagate in a closed loop. This is called the direct channel. By the
rules of string perturbation theory this is the same diagram, but with two
different parametrisations. To transform from one parametrisation to another
one has to interchange the world-sheet space and time directions. This can be
done by means of the transformation $\tau\rightarrow-\frac{1}{\tau}$, which
acts on the characters as a matrix $S$.

Figure 8: Channel transformation.

The analogous figure for non-orientable surfaces is harder to draw, but the
direct channel for closed strings propagating between a boundary and a
crosscap is a Moebius strip, and the direct channel for propagation between
two crosscaps is a Klein bottle. The required transformations are also
different. Here we will just give the result. The details and many references
may be found in [16].

The three basic transverse channel amplitudes are obtained by sandwiching the
closed string propagators between boundary and crosscap states:

Transverse Annulus: $\displaystyle N_{a}N_{b}\langle B^{c}_{a}|e^{i\tau
H}|B_{b}\rangle\,,$ (4.19) Transverse Moebius strip: $\displaystyle
N_{a}\left[\langle B^{c}_{a}|e^{i\tau H}|C\rangle+\langle C^{c}|e^{i\tau
H}|B_{a}\rangle\right]\,,$ (4.20) Transverse Klein bottle: $\displaystyle
N_{a}\langle C^{c}|e^{i\tau H}|C\rangle\,.$ (4.21)

Here $H$ is the closed string Hamiltonian: $H=2\pi(L_{0}+\tilde{L}_{0}-c/12)$,
and $\tau$ is a real number representing the length of the cylinder. The
subscript “$c$” indicates that a CPT conjugate state is to be used. The
integers $N_{a}$ are the Chan-Paton multiplicities. One can express these
amplitudes in terms of characters of the representation $i$. By means of a
transformation of the parameter $\tau$ one can then compute the corresponding
amplitudes in the direct channel (the open and closed string loop channels).
In the case of the Klein bottle and the annulus this transformation acts on
the characters as the modular transformation matrix $S$, whereas in the case
of the Moebius strip one uses the matrix $P=\sqrt{T}ST^{2}S\sqrt{T}$, with
$\sqrt{T}$ defined as $\exp{i\pi(L_{0}-c/24})$. Then one arrives at the
following expressions:

Direct Annulus:
$\displaystyle\tfrac{1}{2}N_{a}N_{b}A^{i}_{~{}ab}\chi_{i}(\tfrac{1}{2}\tau)\,,$
(4.22) Direct Moebius strip:
$\displaystyle\tfrac{1}{2}N_{a}M^{i}_{a}\hat{\chi}_{i}(\tfrac{1}{2}+\tfrac{1}{2}\tau)\,,$
(4.23) Direct Klein bottle: $\displaystyle\tfrac{1}{2}K^{i}\chi_{i}(2\tau)\,.$
(4.24)

Here $\hat{\chi}_{i}\equiv T^{-\tfrac{1}{2}}\chi_{i}$, and the parameter
$\tau$ is purely imaginary. The coefficients are

$\displaystyle A^{i}_{~{}ab}$
$\displaystyle=\sum_{m}S^{i}_{~{}m}B_{ma}B_{mb}\,,$ (4.25) $\displaystyle
M^{i}_{~{}a}$ $\displaystyle=\sum_{m}P^{i}_{~{}m}B_{ma}\Gamma_{m}\,,$ (4.26)
$\displaystyle K^{i}$ $\displaystyle=\sum_{m}S^{i}_{~{}m}\Gamma_{m}\Gamma_{m}\
\,.$ (4.27)

####  4.4.4 Integrality Conditions

To interpret these expressions in terms of state counting it is clearly
important that all the relevant coefficients be non-negative integers. Indeed,
in all cases discussed here $Z_{ij}$ and $A^{i}_{ab}$ are explicitly non-
negative integers, and $K^{i}$ and $M^{i}_{a}$ are integers. But that is not
sufficient, because the actual state multiplicities are sums and differences
of these numbers. Note first of all that the argument of the Klein bottle
term, $2it$, coincides precisely with the terms in the expansion of
$\chi_{i}(\tau)\chi_{j}^{*}(\tau)$ for the physical states: those with the
same powers of $q$ and $\bar{q}$. Hence this term may alter the multiplicity
of the physical states. It is the total multiplicity that must be a non-
negative integer. The vacuum representation, $i=0$, has $Z_{00}=1$ and
$K_{0}=1$. As stated before, this representation produces the graviton, which
should have multiplicity 1. Hence we find that we must divide the entire
partition function by 2 to get the right multiplicity. To get correct
multiplicities for all the other closed string states, we need

$K^{i}=Z_{ii}\ {\rm mod}\ 2\ \hbox{ and}\ |K_{i}|\leq Z_{ii}\ ,$

which is indeed satisfied in all known case. Hence the multiplicity of the
closed string states is given by

$\frac{1}{2}(Z_{ii}+K_{i})\,.$

Note that it is possible for closed string states to be “projected out”, i.e.
removed from the spectrum, if $K_{i}=-Z_{ii}$.

The second line in (4.17) gives rise to open string states, projected by the
Moebius amplitude. In this case the coefficients satisfy, in all known
theories

$M^{i}_{a}=A^{i}_{aa}\ {\rm mod}\ 2\hbox{ and}\ |M^{i}_{a}|\leq A^{i}_{aa}\,.$

The natural interpretation is that diagonal states (open strings between the
same boundaries) are projected to obtain the following multiplicities

$\frac{1}{2}(N_{a}N_{a}A^{i}_{aa}+N_{a}M^{a}_{i})\,,$

whereas the off-diagonal terms are not affected by the Moebius amplitude. In
the simplest case, $A^{i}_{aa}=1$ and $M^{a}_{i}=\pm 1$, one gets dimensions
of symmetric and anti-symmetric tensors. Further work is needed to demonstrate
that these particles do indeed couple in that manner.

####  4.4.5 Gauge Groups

The gauge bosons come from the identity character. Its massless state is a
space-time vector. In the closed string a left-right combination of two
vectors gives the graviton, the dilaton and the $B_{\mu\nu}$ Kalb-Ramond
field. In the open string sector, the vector comes with multiplicity

$\frac{1}{2}(A^{0}_{aa}N_{a}N_{a}+M^{0}_{a}N_{a})\,.$

It turns out that the matrix $A^{0}_{ab}$ in the space of boundaries is a
bijection: it has the property that for any a there is precisely one label $b$
with $A^{0}_{ab}=1$, and $A^{0}_{ab}=0$ for all other $b$. The label $b$ for
which $A^{0}_{ab}=1$ is called the complex conjugate boundary, and is denoted
$a^{c}$. If $a=a^{c}$ the boundary is called self-conjugate or real. In that
case, $M^{0}_{a}=\pm 1$. The number of gauge bosons is either
$\frac{1}{2}N_{a}(N_{a}+1)$, suggesting that the gauge group is $USp(N_{a})$,
or $\frac{1}{2}N_{a}(N_{a}-1)$, suggesting that it is $SO(N_{a})$. Further
studies of the amplitudes are necessary to verify that this interpretation is
correct, because from these arguments we only obtained multiplicities. Note
that in the symplectic case we must have $N_{a}$ even.

For complex boundaries, those with $a\not=a^{c}$, we get a vector boson
multiplicity
$\frac{1}{2}N_{a}N_{a}\left(A^{0}_{aa^{c}}+A^{0}_{a^{c}a}\right)=(N_{a})^{2}$,
since $M^{0}_{a}=0$. This suggests a group $U(N_{a})$, which is indeed
correct. Hence now we can get, in principle, the same brane configurations as
in the oriented case, but with the additional option of symmetric and anti-
symmetric tensors. If $b$ is not equal to $a$ or $a^{c}$ we get a bi-
fundamental.

####  4.4.6 Completeness, Integrality and Sewing Constraints

Now we have to determine the coefficients $B_{ia}$ and $\Gamma_{i}$. Indeed,
the first task is to determine the set of labels $a$, which determine the set
of boundaries, and hence, in geometric language, the number of D-branes at our
disposal. The answer is provided by a conjecture called the completeness
condition for boundaries, formulated in [83]. It states essentially that
$B_{ia}$ is an invertible matrix, so the number of labels $a$ is equal to the
number of labels $i$. The latter number can be inferred from (4.14). Note that
this definition pairs states $i$ with their charge conjugates $i^{c}$. The
closed string states that propagate in the transverse channel are those with
$Z_{ii^{c}}\not=0$, and they have multiplicity $Z_{ii^{c}}$. Hence the number
of labels $i$, also known as Ishibashi labels, is equal to
$\sum_{i}Z_{ii^{c}}={\rm Tr}ZC$. By the completeness conjecture the number of
labels $a$ is the same. It follows that the boundary coefficients $B_{ia}$
depend on the choice of partition function $Z_{ij}$ of the closed string
theory.

The boundary and crosscap coefficients are subject to the integrality
conditions described above, but more importantly by sewing constraints. These
constraints follow from the requirement that different ways to build a Riemann
surface from three-point functions by “sewing” must yield the same answer. See
[84] for a discussion of these constraints for closed strings, [85] and [86]
for open strings and [87] for unoriented strings.

####  4.4.7 A Solution for C-diagonal Theories

The first solution to these constraints in the oriented case was found by
Cardy [88]. He considered the C-diagonal modular invariant, and gave the
following formula for the boundary coefficients:

$B_{ia}=\frac{S_{ia}}{\sqrt{S_{0i}}}\,,$ (4.28)

where $S$ is the matrix defined in (4.9). Note that this explicitly satisfies
the completeness condition, because $S$ is a square matrix. Substituting this
in (4.25) we get

$A^{i}_{~{}ab}=\sum_{m}\frac{S^{i}_{~{}m}S_{ma}S_{mb}}{S_{m0}}=N^{i}_{~{}ab}\
,$ (4.29)

where in the last step we used the Verlinde formula for fusion multiplicities
[89]. This shows that also the integrality conditions are satisfied, because
fusion coefficients are integer. In fact, the purpose of Cardy’s work was to
understand the Verlinde formula, but while doing so he proposed and initiated
the field of boundary CFT. This was picked up almost immediately [90] by the
Tor Vergata (Rome II) group, who did a lot of pioneering work in this area.

They also considered unoriented surfaces, a subject not treated by Cardy.
After some earlier work, in [91] they proposed the following formula for the
crosscap coefficients for the case $Z=C$:

$\Gamma_{i}=\frac{P_{0i}}{\sqrt{S_{0i}}}\ ,$ (4.30)

which uses the $P$-matrix introduced above.

####  4.4.8 Simple Current Results

Now attempts started to obtain similar results for general simple current
modular invariants. This culminated about a decade later in a completely
general formula [92] for all simple current MIPFs. This paper contains
references to all the earlier partial results obtained by various groups. The
formula for boundary coefficients is

$B_{(i,J)[j,\psi]}=\sqrt{\frac{|{\cal G}|}{|{\cal S}_{i}||{\cal
C}_{j}|}}{\frac{\alpha(J)S^{J}_{ij}}{\sqrt{S_{0i}}}}\psi(J)\ ,$ (4.31)

and the crosscap formula is

$\Gamma_{(i,J)}={\frac{1}{|{\cal G}|}}\sum_{K\in{\cal
G}}\eta(K)P_{Ki}\delta^{J0}\,.$ (4.32)

We only present these results here to give a flavour of what is involved. The
simple currents form a subgroup ${\cal G}$ under fusion. Multiplicities
$Z_{ij}>1$ occur when a field $i$ is a fixed point of the simple current
action: $Ji=i$. The simple currents that fix $i$ for a subgroup of ${\cal G}$
called the stabiliser of $i$ and denoted ${\cal S}_{i}$. Consequently, the
Ishibashi labels $(i,J)$ are equipped with a degeneracy label $J\in{\cal
S}_{i}$, and $i$ is a label such that $Z_{ii^{c}}=1$. The boundary states are
labelled by orbit representatives $i$ (that is, one representative of each
orbit $(i,Ji,J^{2}i,\ldots)$), and their degeneracy is governed by the size of
another discrete group, ${\cal C}_{j}\subset{\cal S}_{j}$. The discrete group
characters $\psi(J)$ are used as degeneracy labels for the boundaries. The
matrices $S^{J}$ are modular transformation matrices of a “fixed point CFT”,
an algebraic structure associated with the fixed points, intuitively
introduced in [93] and more rigorously defined in [94]. For further details,
such as the precise definition of ${\cal C}_{j}$, the phase $\alpha(J)$, and
the signs $\eta(J)$ we refer the reader to [92].

The boundary coefficients are uniquely defined as soon as the torus partition
function $Z_{ij}$ is known, but for the crosscap there is a variety of
possibilities. First of all there is a “Klein bottle current” $K$, and
secondly the signs $\eta(J)$ must satisfy a condition that may have several
solutions. Taking into account these choices for different crosscaps increases
the total number of distinct, unoriented Gepner CFT models to 49304. However
only 33012 of them have non-zero tension. The remaining ones are not usable
for building supersymmetric orientifold models.

###  4.5 RCFT Model Building

We are now ready to apply this machinery to model building. In the situation
of interest, the relevant CFT is the usual world-sheet theory of the
superstring (or non-supersymmetric fermionic string) in four flat dimensions,
combined with a non-trivial superconformal CFT to describe the six
“compactified” dimensions. The characters are products of superstring
characters and internal characters. The former determine the space-time
properties of the string excitations, in particular spin and chirality,
whereas the latter contribute to the counting of states. The gauge
representations of the physical states can be read off from the Chan-Paton
labels of the string state under consideration.

Now we start with a choice of an RCFT (in practice always a Gepner model), a
choice of a MIPF, a choice of a crosscap, and a choice of three or four
boundary labels $(a,b,c)$ and $d$, depending on the configuration we want to
realise, as discussed in Section 2.2. Usually $a$ is taken to be the QCD label
and $b$ is the weak label. We can already make sure that $N_{a}=3$ and
$N_{b}=2$ do not oversaturate the dilaton tadpole. Now we compute the annulus
coefficients $A^{i}_{ab}$, $A^{i}_{ac}$, $A^{i}_{bc}$ etc. for any two chosen
labels and check if the chiral intersections (as defined in Table 1) match the
required spectrum.

In case of success, we now check if the putative $Y$ boson remains massless
after taking into account axion mixing. This requires checking all axions in
the full closed string spectrum. At the same time we may check if any other
$U(1)$ vector bosons in the spectrum acquire a mass. If they do not, this is a
phenomenological issue which we will have to deal with later because it is not
solved at the level of the RCFT.

The next step is to cancel all tadpoles by finding a suitable hidden sector,
as explained in Section 2.1. This can be very time-consuming, because it
requires considering all subsets of all branes not used in the Standard Model
configuration. In practice, this can usually not be done exhaustively.

There is still one more check to be made: the absence of global anomalies, as
discussed in [95].

The first work along these lines appeared in [96]. In this paper six-
dimensional theories were studied, and examples with chiral spectra were
found. In [97], building on [98], the first chiral spectra were found in four
dimensions. Then in [99, 100] a general search was undertaken for the Madrid
configuration shown in Figure 4 and some of its variations.

In [18] a different approach was taken. Rather than searching for specific
pre-selected brane configurations, these authors searched for any combinations
of brane labels that yields the Standard Model, in a rather generous
definition of the latter. This includes enlarged gauge groups, non-chiral
pairs of quarks and leptons as chiral brane matter (i.e. chiral matter that
becomes non-chiral if only the SM group is considered) and gauged flavour
symmetries. The result essentially demonstrated that anything one could
theoretically propose as a Standard Model configuration will likely be
realisable if one has a large enough scope of brane models to start with.

One may think that this scope can be extended significantly by moving to other
RCFTs, but in practice this is not easy. For most RCFTs we simply do not have
all the relevant data available. It is often easy to get the spectrum modulo
integers, but a lot harder to get the exact spectrum, as required. Furthermore
any RCFT building blocks other than ${\cal N}=2$ minimal models (one example
are the Kazama-Suzuki models [101]) have much smaller simple current groups,
and hence far fewer MIPFs. There might exist vast numbers of exceptional (not
simple current related) MIPFs in some cases, but a general formalism to
compute their boundary and crosscap coefficients is not available. Free
fermion orientifolds have been considered, but are a far less fertile area
[102]. So it seems that Gepner models are in a sense the optimal possibility.

##  5 F-theory Model Building

F-theory was introduced in [103] as a non-perturbative formulation of Type IIB
compactifications with 7-branes. Its importance for model building is owed to
its generality, including the fact that it allows for the construction of
gauge sectors which enjoy an embedding into the exceptional Lie group $E_{8}$.
This property makes it a particularly natural framework to study Grand Unified
Theories (GUTs) in string theory and distinguishes it from its perturbative
cousins discussed in the previous sections, which are based on gauge groups
$U(N)$, $SO(N)$ and $USp(N)$. Hence, F-theory combines the attractive features
of model building with D-branes - the localisation of gauge degrees of freedom
on branes, which in principle invite a local approach to model building within
a certain realm of questions - with the appearance of exceptional gauge
symmetry as in heterotic $E_{8}\times E_{8}$ string theory, which bears its
fruit in the context of GUT model building. The goal of this programme is to
solve some of the outstanding model building challenges faced by four-
dimensional SUSY GUTs in the higher-dimensional brane-world framework provided
by F-theory.

In addition, F-theory offers a formulation of D-branes in terms of the
geometry of so-called elliptic (or more generally genus-one) fibrations; many
involved questions of brane dynamics are hence translated into entirely
geometric questions and oftentimes have a clear answer in algebraic or
arithmetic geometry.

General introductions to F-theory are provided for instance in [104, 105, 106,
107, 108, 109], to which we refer for details and the original references. In
the sequel, after briefly presenting some of the technical foundations of
F-theory, we focus on its role for particle physics model building.

###  5.1 From $[p,q]$ 7-branes to Exceptional Gauge Algebras

The starting point for F-theory is Type IIB string theory. Type IIB string
theory contains in its massless spectrum higher-form Ramond-Ramond (RR) gauge
potentials of even degree, $C_{2p}$, with $p=0,\ldots,4$. The 2-form $C_{2}$
couples electrically to a D1-brane, which is a string-like soliton of tension
$T_{\rm D1}=\frac{2\pi}{\ell_{s}^{2}}\frac{1}{g_{s}}$ in perturbative string
theory. The Neveu-Schwarz 2-form potential $B_{2}$, on the other hand, couples
electrically to the fundamental, or F1-, string of tension $T_{\rm
F1}=\frac{2\pi}{\ell_{s}^{2}}$. These two types of strings can form BPS bound
states: A bound state of $p$ F1-strings and $q$ D1-branes is called a $(p,q)$
string, and it exists as a BPS bound state for $p$ and $q$ co-prime integers.
Type IIB string theory enjoys a weak-strong coupling duality, which maps an
F1- or $(1,0)$-string into a D1- or $(0,1)$-string, and vice versa. The theory
is believed to be invariant, at the non-perturbative level, under an
$SL(2,\mathbb{Z})$ duality transformation, which in particular acts on the
axio-dilaton $\tau=C_{0}+\frac{i}{g_{s}}$ and the two-form potentials as

$\displaystyle\tau\to\frac{a\tau+b}{c\tau+d}\,,\qquad\left(\begin{matrix}C_{2}\\\
B_{2}\end{matrix}\right)\to M\left(\begin{matrix}C_{2}\\\
B_{2}\end{matrix}\right)\,,\qquad M=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL(2,\mathbb{Z})\,.$ (5.1)

Of central interest for the formulation of F-theory are the Type IIB 7-branes.
A D7-brane, often called $[1,0]$-brane in the sequel, is, by definition, a
7-brane on which an F1-string can end; $SL(2,\mathbb{Z})$ duality implies that
there must then exist also another type of 7-brane, a so-called $[0,1]$-brane,
on which a D1-string can end. More generally one defines a $[p,q]$ 7-brane as
a 7-brane on which a $(p,q)$-string can end.

The fact that a 7-brane is an object of real codimension two in ten dimensions
implies a rather severe backreaction on the supergravity background due to its
tension and RR charge. If one considers a single D7-brane and introduces the
complex coordinate $z=x^{8}+ix^{9}$ in the complex plane spanned by the two
directions normal to the brane, the axio-dilaton $\tau$ acquires a varying