text:
\section{Introduction and Summary}

Local symmetries are the guiding principle for formulating
field theories as well as the theory of gravity.
While manifest in the classical theory these symmetries can, however,
be broken at the quantum level and lead to inconsistencies by the violation
of essential current conservation laws on the quantum level. Such
inconsistencies manifest themselves already
at one-loop level and are known as anomalies \cite{Anomaly-reviews}.
In particular, four-dimensional quantum field theories can admit
anomalies which signal the breaking of the gauge symmetry transformations
acting on chiral fermions. The cancellation of these anomalies is thus
crucial to determine consistent theories and imposes constraints on the
spectrum and couplings of the theory. Anomalous transformations
of the chiral fermions either cancel among each other or require
the implementation of a generalized Green-Schwarz mechanism \cite{Green:1984sg,Sagnotti:1992qw}.
In the latter case the one-loop anomalies are
canceled by a tree-level diagram involving
a $U(1)$ gauged axion-like scalar.
In this work we study the manifestation of the
anomaly cancellation mechanisms in four-dimensional F-theory
compactifications.

To extract four-dimensional observed physics from string theory one is aiming
to find a compactification scenario that describes a very broad class of
consistent string vacua and naturally incorporates the ingredients of the
Standard Model of particle physics and its extensions. One
promising scheme is to consider F-theory compactifications with space-time
filling seven-branes supporting non-Abelian gauge groups.
F-theory geometrizes the complexified coupling constant
of Type IIB string theory as the complex structure modulus
of an additional two-torus. Much of the in general non-perturbative physics of
seven-branes is encoded
in the singular geometry of a torus-fibered compact manifold. Requiring
$\mathcal{N}=1$ supersymmetry in the four-dimensional effective theory enforces this
space to be an elliptically  fibered Calabi-Yau fourfold $X_4$.
The complex three-dimensional base $B_3$ of $X_4$ is the physical
compactification space of Type IIB string theory.
The singularities of this fourfold along co-dimension one
loci in the base signal the presence of non-Abelian gauge groups, while
additional sections of the elliptic fibration signal extra $U(1)$
gauge group factors \cite{Vafa:1996xn,Morrison:1996pp}.

A non-trivial four-dimensional chiral spectrum is only induced if the
seven-branes also support gauge fluxes on their internal world volume.
The interplay of geometry, fluxes and the low-energy
effective action of the four-dimensional gauge theory is crucial
in the study of anomaly cancellation. However, the fact that F-theory does
not posses a low-energy effective action in twelve dimensions prevents one
from deriving the four-dimensional low-energy effective theory directly.
To nevertheless study
F-theory effective physics one has to use the M-theory dual description in one
dimension lower. In fact, M-theory on the same Calabi-Yau fourfold $X_4$ is
dual to F-theory on $X_4 \times S^1$. The new modulus, the radius of
$S^1$, is in this identification related to
the inverse volume of the torus fiber in M-theory. In this duality,
the familiar F-theory limit of a shrinking volume of the torus-fiber maps to
the decompactification limit of the $S^1$ in which one grows a large extra
dimension and recovers four dimensional F-theory physics. All
F-theory objects can then be followed through this three-dimensional
duality and can be studied directly in the dual M-theory. In this work
we will mainly be interested in questions about the low energy physics
of F-theory, most prominently anomalies, and it will be one of our results to
reformulate and answer these questions in M-theory.

At low energies the theory of interest is an effective
three-dimensional (3D) $\mathcal{N}=2$ gauge theory with a number of chiral multiplets
coupled to supergravity. An important property of the 3D
$\mathcal{N}=2$ gauge theories is that they admit, in contrast to a
$\mathcal{N}=1$ gauge theory in four dimensions (4D), a Coulomb branch where
the non-Abelian gauge group $G$ breaks to its maximal torus,
$G \rightarrow U(1)^{r}$, with massless 3D gauge fields $A^\Lambda$,
$\Lambda=1,\ldots,r=\text{rank}(G)$. In this work we will
mainly analyze the theory on this 3D Coulomb branch and understand corrections
to the effective action both from the F-theory perspective and from the
M-theory perspective.

There is one essential difference between the two
descriptions of the 3D theory, that is crucial for this discussion.
By reducing F-theory from 4D to 3D we can describe the 3D $\mathcal{N}=2$
gauge theory also away from the Coulomb branch. The
Coulomb branch is understood purely field theoretically by giving a vacuum
expectation value (vev) to the scalars $\zeta^\Lambda$ along the Cartan
generators of $G$ in the 3D vector
multiples coming from the direction of the 4D vector along the $S^1$. Fields
become massive due to
this vev and at low energies have to be integrated out quantum
mechanically correcting the effective theory on the Coulomb branch. In
contrast, although M-theory as a fundamental theory should also describe the
full 3D $\mathcal{N}=2$ gauge theory, it is only explicitly known how to
describe the 3D Coulomb branch. The description is given by M-theory on the
smooth $\hat{X}_4$ fourfold with all singularities in $X_4$ resolved inducing
shrinkable rational curves, i.e.~two-spheres $\mathbb{P}^1$, in the geometry.
Geometrically, the resolution process corresponds to giving a vev
in the field theory perspective with $\zeta^\Lambda$ being related to the
volume of these shrinkable $\mathbb{P}^1$'s.
Furthermore, at large volume of $\hat{X}_4$ it
is consistent to consider the long wavelength approximation of
eleven-dimensional supergravity and dimensionally reduce it on $\hat{X}_4$ to 3D.
In this description certain microscopic M-theory
degrees of freedom corresponding to certain wrapped M2-branes are massive.
However, these states have already been integrated out consistently in the
eleven-dimensional supergravity, which
follows from the validity and consistency of the supergravity approximation at
low energies. The corrections to the 3D low-energy theory on the Coulomb
branch are thus visible as \textit{classical} effects in the dimensional
reduction on $\hat{X}_4$. Morally speaking, the information about the massless
microscopic states on $X_4$ has been traded for the new shrinkable $\mathbb{P}^1$'s
on $\hat{X}_4$, that have not been present on $X_4$.
These corrections are the same as those that arise
\textit{quantum mechanically} in F-theory and it will be one key observation
of this paper to map these corrections in the M-/F-theory duality in 3D.

The geometrically massive modes that have to be integrated out on the 3D
Coulomb branch and are relevant for our discussion of anomalies are from an F-
and M-theory point of view
\begin{itemize}
	\item massive 3D W-bosons in F-theory from the breaking
	$G\rightarrow U(1)^r$, or massive M2-branes on shrinking
	$\mathbb{P}^1$'s in $\hat{X}_4\rightarrow X_4$ over co-dimension one in $B_3$
	in M-theory.
    \item charged 3D fermions massive on the 3D Coulomb branch in F-theory, or
    massive M2-branes on $\mathbb{P}^1$'s fibered over co-dimension two in $B_3$,
    i.e.~curves from intersections of seven-branes in F-theory later denoted
    as matter curves, in M-theory.
    \item massive Kaluza-Klein states of 4D charged fermions in the reduction
    of F-theory from 4D to 3D, or M2-branes wrapping a shrinking $\mathbb{P}^1$ over
    a matter curve once and multiply the elliptic fiber of $\hat{X}_4$.
\end{itemize}

The 3D couplings that are corrected by integrating out these massive states
and that will be of most use for our discussion of 4D
anomalies in F-theory are the 3D Chern-Simons (CS) terms for the Abelian
vector fields $A^\Lambda$ on the 3D Coulomb branch,
\begin{equation}
	S^{(3)}_{CS}=-\frac{1}
	{2}\int \Theta_{\Lambda\Gamma}A^\Lambda \wedge F^\Gamma\,.
\end{equation}
Again, the crucial point is that from the F-theory perspective these
CS-terms are purely quantum mechanically and
generated only at one-loop of massive fermions charged under the vectors
$A^\Lambda$. In contrast  on the M-theory
side they are generated classically by $G_4$-flux on $\hat{X}_4$. Thus, the
CS-terms still carry the signature of the states that have been integrated
out in F-theory but are efficiently calculated in M-theory. More precisely
we will use in this work as a tool to derive the chiral
index $\chi(\mathbf{R})$ of 4D charged matter in a representation
$\mathbf{R}$ the observation of \cite{Grimm:2011fx} that
certain $G_4$-fluxes induce particular classical M-theory CS-terms
which are induced in F-theory by one-loop diagrams of the 3D massive
charged fermions from the 4D chiral matter multiplets reduced on $S^1$.
The identification of classical M- and one-loop F-theory CS-terms
$\Theta_{\Lambda\Sigma}^M$ respectively $\Theta_{\Lambda\Sigma}^F$ we will
employ takes the form
\begin{equation}
\frac12\Theta_{\Lambda\Sigma}^M=\frac14\int_{\hat{X}_4}G_4\wedge\omega_\Lambda\wedge\omega_\Sigma
\equiv-\frac12\sum_{\mathbf{R}}\chi(\mathbf{R})\sum_{q\in \mathbf{R}}q_\Lambda
q_\Sigma \ \text{sign}(q\cdot \zeta)=-\Theta^{F}_{\Lambda\Sigma}\,,
\end{equation}
where $\omega_{\Lambda}$, $\omega_{\Sigma}$ denote $(1,1)$-forms on
$\hat{X}_4$ generated by resolving the singularities in $X_4$ and the
sum is taken over all 4D matter representations $\mathbf{R}$ with Dynkin
labels $q$. The sign-function is applied to the scalar product of the
charges and the Coulomb branch parameters,
$q\cdot \zeta=q_\Lambda\, \zeta^\Lambda$.
In general, M-theory $G_4$-flux on the fully resolved Calabi-Yau
$\hat{X}_4$ that is consistent with the F-theory limit is the description
of seven-brane gauge flux in F-theory. Thus, as is known
that gauge fluxes on seven-branes induce chirality of 4D matter in Type
IIB $G_4$-flux induces chirality in F-theory. The relation of such fluxes
to the 4D matter spectrum has been recently studied intensively
\cite{Donagi:2008ca,Beasley:2008dc,Beasley:2008kw,Hayashi:2008ba,Braun:2011zm,Marsano:2011hv,Krause:2011xj,Grimm:2011fx,Krause:2012yh,Kuntzler:2012bu}.

Besides using the CS-terms in 3D as tools for determining 4D chirality we
will discover the intricate relations between different 3D CS-terms that are
implied by 4D anomaly cancellation. Let us summarize our findings where we
distinguish F-theory compactifications without Abelian gauge symmetries and
those with Abelian gauge factors. In the first case it is
required that the non-Abelian gauge anomaly vanishes which puts one constraint
on the spectrum for each gauge group factor $G_{(I)}$ in $G$ whereas in the second
case anomaly cancellation is more involved due to mixed anomalies.
The cancellation of these anomalies may require a Green-Schwarz (GS) mechanism
in F-theory. Then fluxes on space-time filling seven-brane have to both induce
a non-trivial chiral spectrum and corresponding gaugings of the axions for a
working GS-mechanism. Anomaly cancellation is then
tightly linked with tadpole cancellation that imposes constraints on the brane
configuration and allowed fluxes. For setups at weak string coupling this link has
been discussed and reviewed, for example, in \cite{Cvetic:2001nr,MarchesanoBuznego:2003hp,Blumenhagen:2005mu,Blumenhagen:2006ci,Plauschinn:2008yd,Cvetic:2011vz}.

In contrast to the weakly coupled Type IIB theory, where brane fluxes are constrained by tadpole conditions,
for $G_4$ fluxes lifting from M-theory to F-theory it is expected that all anomalies are canceled
and the Green-Schwarz mechanism is implemented automatically. This can be anticipated
since in M-theory no additional consistency conditions on $G_4$, apart from the for
anomaly cancellation irrelevant M2-brane tadpole matching,
have to be imposed. In particular the seven-brane tadpole is
canceled geometrically in F-theory by considering a compact Calabi-Yau
fourfold $X_4$ and 5-brane tadpoles are canceled by considering closed fluxes $dG_4=0$
signalling the absence of a net M5-brane charge. Therefore, by
consistency of the underlying M-theory compactification also the low-energy effective action
should be consistent.

From anomaly cancellation in combination with F-/M-theory duality in 3D we
discover the following constraints for and links to 3D CS-terms:
\begin{itemize}
	\item Without $U(1)$-factors in 4D, the CS-terms for
	the vectors $A^\Lambda$ encoding chiralities have to obey non-trivial
	relations such that the cubic non-Abelian anomalies are canceled.
	\item With $U(1)$-factors in 4D, the CS-terms for the 4D chiral indices
	      have to be related to the CS-terms determining the gaugings of the
	      4D axions so that all mixed anomalies are canceled.
	\item We show the cancellation of 4D mixed Abelian-gravitational anomalies in
	      3D using F-/M-theory duality. We discover that the 4D mixed
	      Abelian-gravitational anomaly is the coefficient $\Theta^{\text{loop}}_{0m}$
	      of the 3D Chern-Simons term $A^0 \wedge F^m$ for the
	      Kaluza-Klein vector $A^0$ and the corresponding 3D $U(1)$ vector field $A^m$,
	\begin{equation}
	\Theta^{\text{loop}}_{0m}=-\frac1{12}\sum_{\underline{q}}
	n(\underline{q})q_m\,,
	 \end{equation}
	 where the sum is taken over all $U(1)$-charges $
	 \underline{q}$ under the 4D $U(1)$-vectors $A^m$ and $n(\underline{q})$ is
	 the number of fermions with a given charge $\underline{q}$.
	 We obtain this result since $\Theta_{0m}$ is one-loop induced in F-theory
	with the full Kaluza-Klein tower of 4D charged fermions in
	the loop. F-/M-theory duality implies that this expression is related to
	the classical flux-integral of $G_4$ in M-theory, which is precisely the
	tree-level contribution to the anomaly from the GS-mechanism,
	\begin{equation}
		\Theta^{\text{loop}}_{0m}\equiv-\frac{1}{2}\Theta_{0m}=\frac{1}{4}
	 K^\alpha\Theta_{\alpha m}\,.
	\end{equation}
	Here $K^\alpha$ is a vector determining a direction in the space of 4D
	axions and $\Theta_{m\alpha}$ is the $G_4$-flux induced gauging of the
	$\alpha$th axion  under $A^m$.
\end{itemize}
In addition, we discuss in general terms the construction of $U(1)$
symmetries in F-theory on fourfolds by considering a non-trivial
Mordell-Weil group of rational sections\footnote{We neglect the torsion
subgroup here for simplicity.} of $\hat{X}_4$. This construction has been
applied successfully on elliptically fibered threefolds in
\cite{Morrison:1996pp,Aspinwall:1998xj,Park:2011ji,Morrison:2012ei} to
study non-simply laced gauge groups and $U(1)$'s as well as Abelian
anomalies in six-dimensional F-theory. The discussion
we provide in this note is a first step to being able to construct also
four-dimensional F-theory compactifications with more than one $U(1)$ and
a matter sector with more general $U(1)$-charges extending the analysis of
\cite{Grimm:2010ez}.

Furthermore, for consistent $G_4$-fluxes in the F-theory limit, we
discover geometric relations that need to be satisfied in any
resolved elliptically fibered fourfold  $\hat{X}_4$,
\begin{eqnarray}  \label{eq:geo_rel}
	\frac{1}{3}\sum_{S_{\mathbf{R}}}\sum_{c\,\subset S_\mathbf{R}}(S_\mathbf{R}\cdot [G_4])(c\cdot D_\Lambda)(c\cdot D_\Sigma)
	(c\cdot D_\Gamma)&=&\frac{1}{2}[G_4]\cdot D_{(\Gamma}\cdot
    \pi_*(D_\Lambda\cdot D_{\Sigma)})\ ,\\
    \frac1{3}\sum_{S_{\mathbf{R}}}\sum_{c\,\subset S_\mathbf{R}}(S_\mathbf{R}\cdot [G_4])(c\cdot D_\Lambda)&=&[G_4]\cdot
    [c_1(B_3)]\cdot D_\Lambda\, .
\end{eqnarray}
Here $G_4$ is the $G_4$-flux in M-theory and $S_{\mathbf{R}}$ is a
four-cycle denoted the matter surfaces obtained by fibering the shrinking
$\mathbb{P}^1$'s in  $\hat{X}_4$ that are the weights of a representation
$\mathbf{R}$ of $G$ over matter curves. The divisors
$D_\Lambda$, $D_\Sigma$ and $D_\Gamma$ are the exceptional divisors
obtained by fibering the shrinkable curves $\mathbb{P}^1$ over
co-dimension one loci in the base $B_3$, i.e.~the seven-brane divisors in
$B_3$. The curves $c$ are any shrinkable holomorphic curve in the fiber
$S_{\mathbf{R}}$. The map $\pi_*$ is induced by the
projection to the base $\pi\,:\,\hat{X}_4\rightarrow B_3$ of the elliptic
fibration of $\hat{X}_4$ and $c_1(B_3)$ denotes its first Chern-class.
These geometric relations are indeed valid for concrete
resolutions as we show for several examples with non-Abelian gauge groups.

We note that the geometric relations \eqref{eq:geo_rel} and the
corresponding statements about anomaly cancellation should even apply in a
broader context than considered in this paper. In particular one expects
that \eqref{eq:geo_rel} is valid for more general $G_4$-flux that
is not necessarily given by a sum of products of $(1,1)$-forms,
i.e.~$G_4$-flux that is not vertical. A phenomenologically interesting
example of these fluxes is the F-theory analog of hypercharge flux $F_Y$
considered in \cite{Marsano:2010sq,Dudas:2010zb,Palti:2012dd}
in the context of anomalies. In order to understand anomaly cancellation
also in these cases, it is crucial to note that the right hand side of
\eqref{eq:geo_rel} is in general only non-trivial for vertical $G_4$-flux
and vanishes otherwise. Constraints on the spectrum then arise from the
vanishing of the left-hand side of this equality. A better understanding
of the global geometry of $\hat X_4$ and the matter surfaces
$S_{\mathbf R}$ might reveal that these constrains are also automatic
fulfilled for concrete resolved Calabi-Yau fourfolds with no further
restriction on the $G_4$-flux. It would be interesting to investigate this
further.

Finally,  we compare our analysis of anomaly cancellation in F-theory to anomaly cancellation
in Type IIB Calabi-Yau orientifold compactifications with O7-planes and intersecting D7-branes
\cite{MarchesanoBuznego:2003hp,Blumenhagen:2005mu,Blumenhagen:2006ci,Plauschinn:2008yd}. We show that in general the Type IIB anomaly cancellation
reduces to the F-theory anomaly cancellation when projecting to the sector of geometrically massless $U(1)$'s
in Type IIB. This demonstrates on the levels of anomalies that the geometrically massless $U(1)$'s correspond directly
to the $U(1)$'s engineered in F-theory, whereas the geometrically massive $U(1)$'s in Type IIB are captured by the
residual discriminant of the elliptic fibration in F-theory, cf.~\cite{Grimm:2011tb} for a more general discussion
of geometrically massive $U(1)$'s.

The paper is organized as follows. In section \ref{sec:anomlies+cancellation} we recall some basic facts about anomaly cancellation in
4D and introduce the Green-Schwarz mechanism. In order to determine the Green-Schwarz counter terms in
F-theory we discuss the geometric structure of resolved Calabi-Yau fourfolds used in the
duality of M-theory to F-theory in section \ref{sec:AnomaliesInFGUTs}. It will be crucial to allow for
$U(1)$ gauge group factors and their geometric F-theory manifestation in our analysis.
In section \ref{sec:ChiralityInF} we turn to the analysis of one-loop Chern-Simons terms in the
3D effective theory. We recall how they encode the 4D chiral index for the matter spectrum
and show that they can also capture 4D mixed Abelian/gravitational anomalies.
Anomaly cancellation in F-theory is discussed in section \ref{sec:anomalycancellationinF} were we also derive the
geometric conditions \eqref{eq:geo_rel}. We contrast the F-theory analysis with the description
of anomaly cancellation in weakly coupled Type IIB setups with D7-branes and O7-planes
in section \ref{sec:orientifold_anomalies}. Explicit examples of resolved Calabi-Yau fourfolds are introduced
in section \ref{sec:examples}. Our work has two appendices supplementing additional information
about 4D anomalies and $SU(5)$ representations.

\section{Four-Dimensional Anomaly Cancellation}
\label{sec:anomlies+cancellation}

In this section we present the basic techniques for characterizing
anomalies of gauge symmetries and their cancellation via the
generalized Green-Schwarz mechanism in a four dimensions. More details
can be found in appendix \ref{app:ReviewOfAnomalies} and the reviews \cite{Anomaly-reviews}.

We will consider a 4D $\mathcal{N}=1$ supersymmetric effective theory
with a vector multiplet transforming in the adjoint of a general
non-Abelian gauge group,
\begin{equation} \label{eq:Gsplit}
	G=G_{(1)}\times\cdots \times G_{(n_G)}\times U(1)_1\times\cdots\times U(1)_{n_{U(1)}}\,.
\end{equation}
Here the group factors $G_{(I)}$ denote $n_G$ arbitrary simple Lie-groups
and we allow for a number $n_{U(1)}$ of $U(1)_m$-factors.
In the following we will use indices
\begin{equation}
   I, J= 1,\ldots, n_G\ , \qquad \quad m,n =1,\ldots, n_{U(1)}\ ,
\end{equation}
to label non-Abelian and Abelian group factors.
We consider matter in chiral multiplets that transform in
representations denoted
\begin{equation}
	\mathbf{R}=(\mathbf{r}^1,\ldots,
	\mathbf{r}^{n_G})_{\underline{q}}\,
\end{equation}
of $G$. Here $\mathbf{r}^I$ denote the representations of the
non-Abelian factors of $G_{(I)}$ and $\underline{q}=(q_1,
\ldots ,q_{n_{U(1)}})$ denotes the corresponding $U(1)$-charges
arranged as a column vector. Furthermore, since we are considering
a theory with $\mathcal{N}=1$ supersymmetry we have one
gravity multiplet containing a single gravitino.

In general an anomaly of a symmetry denotes the effect that a
symmetry of the classical theory is not promoted to a symmetry of
the quantum theory. An anomaly of a local symmetry, i.e.~a gauge
symmetry, spoils the consistency of the quantum theory due to the
quantum mechanical violation of current conservation laws. Thus in a
consistent quantum theory anomalies of gauge symmetries have to be absent.
The breakdown of a gauge symmetry is encoded
in a gauge invariant and regulator independent way in the anomaly
polynomial. In  four dimensions this polynomial is a cubic
polynomial in the gauge field strength of the gauge group $G$
and the 4D Riemann tensor. Thus, it is a formal six-form denoted by
$I_6$. The polynomial $I_6$
is the sum of the anomaly polynomials for all fields contributing
to the anomaly. These are massless Weyl-fermions, gravitinos and
self-dual tensors and their corresponding anomaly polynomials have been
worked out in every space-time dimension in the seminal work of
\cite{AlvarezGaume:1983ig}. The theory is anomaly free if the full anomaly
polynomial vanishes identically, i.e.~if all
coefficients of the various monomials in the field strength and the
Riemann tensor are zero. As we will discuss next, the full anomaly
polynomial consistent of $I_6$ from the quantum anomalies in the matter
sector and another contribution from a tree-level effect, the Green-
Schwarz mechanism.

In four dimensions, the possible anomalies are gauge and mixed anomalies
only, since pure gravitational anomalies are absent
by symmetry. For the fields in the standard $\mathcal{N}=1$ supergravity
theory described above the anomaly polynomial reads
\begin{equation}
\label{eq:I6general}
	I_6=\sum_{\mathbf{R}}n(\mathbf{R})I_{1/2}(\mathbf{R})\,,
\end{equation}
where $I_{1/2}(\mathbf{R})$ denotes the anomaly polynomial
\eqref{I12-4d} of a left-chiral Weyl fermion that
occurs with multiplicity $n(\mathbf{R})$ in the 4D spectrum.

In general, the anomaly polynomial \eqref{eq:I6general} of the matter
sector does not need to vanish identically if it is a sum of factorizable
contributions.
In this case, the residual anomalies can be canceled by a certain
tree-level mechanism known as the Green-Schwarz mechanism
\cite{Green:1984sg}. We consider the higher derivative effective
action, referred to as the Green-Schwarz counter terms,
\begin{equation} \label{eq:GSterm}
     S^{(4)}_{\rm GS} = -\frac{1}{8}\int \frac{2}{\lambda_I}
     b_I^\alpha \rho_{\alpha}  \
     \text{tr}_{\textbf{f}}\,( F^I\wedge F^I)+2 b_{mn}^{\alpha}
     \rho_{\alpha}\   F^m\wedge F^n  -  \frac{1}{2}
     a^\alpha \rho_\alpha\ \text{tr}( R\wedge R )\ .
\end{equation}
Here $F^I$ and $F^m$ denote the field strengths in the adjoint of
$G_{(I)}$ respectively of the $m$-th $U(1)$ and $\text{tr}_{\mathbf{f}}$
denote the traces in the corresponding fundamental representations. Furthermore, we have used
$\lambda_I=2c_{G_{(I)}}/V(\mathbf{adj})=\frac{2}{\langle \alpha_0,\alpha_0\rangle}$ with $c_{G_{(I)}}$ the
dual Coxeter number of $G_{(I)}$ and $V(\mathbf{adj})$ defined in
\eqref{trF^3}, whereas $\langle \alpha_0,\alpha_0\rangle$ denotes the length squared of the root of maximal length $\alpha_0$.
The real scalar fields $\rho_\alpha$ are axions that are gauged by the
Abelian vectors $A^m$ of $G$ as
\begin{equation} \label{eq:rhoGauging}
   \mathcal{D} \rho_\alpha = d \rho_\alpha +
   \Theta_{\alpha m}\, A^m\,.
\end{equation}
The combinations of axions $\rho_\alpha$ in the various terms in
\eqref{eq:GSterm} are determined by parameters $b^\alpha_I$,
$b^\alpha_{mn}$ and $a^\alpha$, which can be determined
by the underlying microscopic theory together with the matrix
$\Theta_{\alpha m}$ in \eqref{eq:rhoGauging}. We will
explicitly determine these parameters for F-theory compactifications
in section \ref{sec:GStermsInF}.
The gauging \eqref{eq:rhoGauging} induces an anomalous variation
of the action \eqref{eq:GSterm} that lifts to a contribution to the
anomaly polynomial. In factorizable situations this can cancel a
non-vanishing $I_6$ in \eqref{eq:I6general} from the matter sector of the
theory.

Before we continue by writing down the anomaly conditions we pause for a
brief discussion of additional terms in the effective action that can have
an anomalous variation. As discussed in \cite{Anastasopoulos:2006cz} there
can be generalized Chern-Simons terms in four dimensions of the form
$E_{mnk}A^m\wedge A^n\wedge F^k$ for the $U(1)$'s respectively non-Abelian
terms $E_{mI}A^m\wedge \omega_3^I$ with $\omega_3^I$ denoting the
Chern-Simons form of $\text{tr}(F^I\wedge F^I)$. As discussed also in
\cite{Anastasopoulos:2006cz} these terms depend on the chosen
regularization scheme and it is possible to work in a scheme where all
these terms are absent. Equivalently these terms parametrize the
ambiguities, namely the exact forms, in the descend equations
that is used to relate the unambiguous, i.e.~scheme-independent anomaly
polynomial $I_6$, with the anomalous and scheme dependent counter terms in
the quantum effective action. Thus the generalized
Chern-Simons terms are irrelevant for anomaly cancellation and will be
neglected in the rest of our discussion.

Adding up the two contributions to the total anomaly polynomial from the
matter sector, $I_6$ in \eqref{eq:I6general}, and from the Green-Schwarz
mechanism, $I_6^{\text GS}$ in \eqref{I6-GS}, and requiring
the sum to vanish, the conditions for cancellation read, as
reviewed in more detail in appendix
\ref{app:ReviewOfAnomalies},
\begin{eqnarray}
	\text{tr}_\mathbf{f} (F^I)^3&:\qquad
        & \sum_\mathbf{\mathbf{r}^I} n(\mathbf{r}^I)V(\mathbf{r}^I)=0\,, \label{eq:Anomalies:purenonAb}\\
	F^m F^n F^k&:\qquad
        & \frac{1}{6}\sum_{\underline{q}}n(\underline{q})q_{(m} q_n q_{k)}=\frac1{4}b_{(mn}^\alpha\, \Theta^{\phantom{\mathcal{A}}}_{k) \alpha} \label{eq:Anomalies:pureAb}  \\
        F^m\text{tr}_{\mathbf{f}}(F^I)^2&:\qquad
        & \frac12\sum_{r^I}\sum_{\underline{q}} n(\mathbf{r}^I_{\underline{q}})U(\mathbf{r}^I)q_m
=\frac{1}{4\lambda_I} b_I^\alpha\, \Theta_{\alpha m} \label{eq:Anomalies:AbnonAb} \\
        F^m\text{tr}R^2&:\qquad
       &\frac{1}{48}\sum_{\underline{q}}n(\underline{q})q_m=-\frac1{16}a^\alpha\Theta_{m \alpha}\,, \label{eq:Anomalies:Abgravitational}
\end{eqnarray}
where in the second line we have symmetrized in the indices $m,n,k$.
Here, we have to sum over each representation $\mathbf{r}^I$ of
the simple group $G_{(I)}$ and over all possible $U(1)$-charges
$\underline{q}$ with which the representation $\mathbf{r}^I$
occurs in the 4D spectrum. In addition, we introduced
$n(\mathbf{r}^I_{\underline{q}})$ denoting the number of chiral
multiplets in the representation $\mathbf{r}^I_{\underline{q}}$
and the number of chiral multiplets $n(\underline{q})$ with
charges $\underline{q}$. The latter can be written as
\begin{equation}
	n(\underline{q})=\sum_{\mathbf{r}_{\underline{q}'}}n(\mathbf{r}^I_{\underline{q}'})
	\text{dim}(\mathbf{r}^I)\delta_{\underline{q}\,\underline{q}'}\,,
\end{equation}
where $\delta_{\underline{q}\,\underline{q}'}=1$ if $\underline{q}=\underline{q}'$ and
zero otherwise.
Furthermore, we made use of the group theory relations
\begin{equation} \label{eq:traceRelations}
     \text{tr}_{\mathbf{r}^I} F^3 = V(\mathbf{r}^I)  \text{tr}_{\mathbf{f}}
     F^3\,,\qquad
	\text{tr}_{\mathbf{r}^I}
	F^2=U(\mathbf{r}^I)\text{tr}_{\mathbf{f}}F^2\,.
\end{equation}
The conditions in the order in which they appear in
\eqref{eq:Anomalies:purenonAb}--\eqref{eq:Anomalies:Abgravitational}
are the purely non-Abelian anomaly, that has to cancel by itself,
the purely Abelian anomaly and the two mixed anomalies, the
Abelian-non-Abelian and Abelian-gravitational anomalies.

\section{Fluxes and Green-Schwarz-Terms in F-Theory}
\label{sec:AnomaliesInFGUTs}

In this section we prepare the ground for our analysis of anomalies in
four-dimensional F-theory compactifications. We start in section
\ref{sec:F+MresolvedFF} with a brief review of the geometry of smooth
fourfolds $\hat{X}_4$, obtained by resolving the singularities
of the elliptic fibration of $X_4$, and describe the construction
of $G_4$-flux on $\hat{X}_4$. Recalling the duality of
3D M-theory compactifications on $\hat X_4$ with F-theory on $X_4\times S^1$
will be crucial. This 3D perspective on 4D F-theory physics is the backbone of most
considerations in this paper. In section \ref{sec:flux_review} we discuss
the conditions on $G_4$-fluxes in M-theory to be viable fluxes in F-theory.
Finally, we show that
4D Green-Schwarz terms in F-theory are determined by the M-theory
compactification geometry $\hat{X}_4$ in section \ref{sec:GStermsInF}.

\subsection{F-theory as M-theory \& the geometry of resolved fourfolds}
\label{sec:F+MresolvedFF}

An F-theory compactification to four dimensions is specified by
an, in general singular, elliptically fibered Calabi-Yau fourfold
$\pi:\,X_4\rightarrow B_3$ over a K\"ahler threefold base $B_3$.
We will consider cases in which the fibration has at least one
section $\sigma$ and a number of rational sections $\hat{\sigma}_m$.

Since there exists no twelve-dimensional
low-energy effective action of F-theory the 4D physics of F-theory
compactifications has to be extracted via its M-theory
dual. More precisely, one considers the 4D theory on
an additional $S^1$ and pushes the resulting 3D theory onto
its Coulomb branch. In the dual picture this 3D
theory is described by compactifying M-theory on the smooth
Calabi-Yau fourfold $\hat{X}_4$, that is obtained by resolving all
singularities in $X_4$. Since we will make extensive use of these
two dual perspectives, we summarize them
schematically as
\begin{equation} \label{eq:3dDuality}
	\text{F-theory on }X_4 \quad \rightarrow \quad \begin{array}{c}
	                                                \text{F-theory on }X_4\times S^1\\
                                                        \text{in 3D Coulomb branch}
	                                               \end{array} \ \equiv\ \text{M-theory on }  \hat{X}_4\,.
\end{equation}

In general the identification of F-theory on $X_4 \times S^1$ with
M-theory on $X_4$ will also hold at the origin of the Coulomb branch
where the non-Abelian gauge group is restored.
This provides a microscopic definition of F-theory in the UV by M-theory.
However, due to our poor understanding of the microscopics of M-theory
the equality in \eqref{eq:3dDuality} is most explicitly evaluated on the Coulomb
branch. On the M-theory side this corresponds to
using the resolved $\hat{X}_4$ in the compactification of
11D supergravity to three dimensions.
The resulting low energy effective theory is valid below the Kaluza-Klein scale and the
energy scales defined by wrapped M2-branes on (shrinkable)
cycles in $\hat{X}_4$. Using 11D supergravity on
$\hat X_4$ the M2-branes and Kaluza-Klein modes have effectively been integrated out.
On the F-theory side, W-bosons and matter fields in 4D F-theory precisely arise from such
M2-brane states. For the matching \eqref{eq:3dDuality} yet to work, we thus have to go to
the 3D Coulomb branch where these states become massive and have been integrated
out in the IR. In addition we have to choose the circle radius $S^1$ to be
in a regime so that Kaluza-Klein modes and winding modes of F-theory are above the cut-off scale.
Only then the light degrees of freedom in the 3D effective theories of
F-theory on $X_4\times S^1$ and M-theory on $\hat{X}_4$ in \eqref{eq:3dDuality} match
in the IR and the identification of the effective actions can be performed.

There are further motivations to consider compactifications on the
resolved fourfold~$\hat X_4$ that should be stressed here. Firstly,
from a mathematical point of view
the singularities of the fourfold $X_4$ can be
classified on the smooth $\hat{X}_4$ by analyzing the local
resolution geometry. Resolution of
the singularities leads to new exceptional divisors $E_i$ in $\hat{X}_4$, whose
intersections specify the type of the original singularities.
Secondly, the resolved fourfold allows to include $G_4$-fluxes, i.e.~topologically
non-trivial background values of the M-theory
three-form field strength. Such fluxes are elements
of the cohomology $H^4(\hat{X}_4,\mathbb{Z}/2)$, where half-integrality
can be consistent with the quantization condition discussed in \eqref{G4half} below.
This cohomology group also contains new classes due to the resolution of $\hat X_4$.
Precisely the fluxes arising in the expansion with respect to these forms correspond to seven-brane
gauge fluxes and are the key to understand chiral anomalies and their cancellation
as we will discuss in more detail in the following.

Having a closer look at the geometry of a resolved fourfold $\hat X_4$ one encounters four
different types of divisors. We denote a basis of divisors and their Poincar\'e
dual two-forms by $D_A$ and $\omega_A$ with $A=1,\ldots,h^{1,1}(\hat{X}_4)$.
In the following we discuss the intersection numbers of these divisors and two-forms in
detail. They are denoted by
\begin{equation} \label{eq:fourfoldintersect}
   \mathcal{K}_{ABCD} = D_A\cdot D_B \cdot D_C \cdot D_D = \int_{\hat X_4} \omega_A \wedge \omega_B \wedge \omega_C \wedge \omega_D\ .
\end{equation}
The four types of divisors and their Poincar\'e dual two-forms in $\hat X_4$ are
\begin{equation} \label{eq:basisH^11}
	D_A=(B,D_\alpha,D_i,D_m)\,,\qquad \omega_A=( \omega_{\hat 0},\omega_\alpha,\omega_i,\omega_m)\,,
\end{equation}
that we characterize as follows:
\begin{itemize}
\item The zero section $B$: The single divisor $B$ is the zero section of the elliptic fibration of $\hat{X}_4$. It is the class of the
base $B_3$ with its Poincar\'e dual $\omega_{\hat 0}$\footnote{The hat on the index $0$ is introduced since, as we will discuss in detail later,
it is more natural for the  description of the 3D effective action to redefine $B$ and $\omega_{\hat{0}}$.}.
The section obeys the intersection property
\begin{equation} \label{eq:B^2}
	B\cdot B=-c_1(B_3)\cdot B\,,
\end{equation}
as can be seen from application of the adjunction formula.

\item The vertical divisors $D_\alpha$: There are $h^{(1,1)}(B_3)$ divisors $D_\alpha=\pi^{-1}(D_\alpha^{\rm b})$, with
dual two-forms $\omega_\alpha$, that are lifted from divisors $D_\alpha^{\rm b}$ of the base $B_3$ to the fourfold
$\hat{X}_4$ and are thus inherited from the singular fourfold $X_4$.

\item The Cartan divisors $D_i$: There are
$\text{rank}(G)-n_{U(1)}$ divisors $D_i$ with their dual two-forms $\omega_i$ that are related to the exceptional divisors $E_i$
resolving the singularities in the elliptic fibration of $X_4$. The $D_i$ are denoted as Cartan divisors and their intersections encode the
types of singularities  in $X_4$ that correspond to the non-Abelian
gauge symmetry in F-theory.

\item Rational sections $\hat{\sigma}_m$ and $U(1)$ Cartan divisors $D_m$: In
general there are $n_{U(1)}$ extra
divisor classes $D_m$, denoted the Cartan divisors of the $m$th $U(1)$
symmetry in F-theory, with Poincar\'e duals $\omega_m$. Geometrically
these are related, as discussed below, to a non-trivial Mordell-Weil group of
rational sections $\hat{\sigma}_m$ of the elliptic fibration of $\hat{X}_4$
\cite{Morrison:1996pp,Aspinwall:1998xj}. For our purposes we are mostly interested in the intersection
properties of these sections. Most notably, the $\hat{\sigma}_m$ obey a
relation like \eqref{eq:B^2},
\begin{equation} \label{eq:defPropSections}
	\hat{\sigma}_m^2\cdot D_\alpha\cdot D_\beta=-[c_1(B_3)]\cdot\hat{\sigma}_m\cdot D_\alpha\cdot D_\beta
\end{equation}
for all $m$ and all vertical divisors $D_\alpha$, $D_\beta$,
where $[\cdot]$ indicates Poincar\'e duality.
\end{itemize}

Let us next explain the intersection properties of these four different
types of divisors. As we will see in this discussion, these intersections
reflect on the one hand the geometry of the elliptic fibration
of $\hat{X}_4$ and on the other hand  the physical structure of the 3D
effective theory. These intersection properties can be checked
by performing an explicit resolution, employing compact toric methods
in
\cite{Candelas:1996su,Candelas:1997eh,Blumenhagen:2009yv,Grimm:2009yu,Cvetic:2010rq,Chen:2010ts,Knapp:2011wk,Braun:2011ux}
and the local methods and their extensions in
\cite{Esole:2011sm,Marsano:2011hv}.

We begin with the divisor $B$. Instead of stating its intersections we first
perform a basis change.
It is was noted in \cite{Grimm:2011sk,Park:2011ji,Bonetti:2011mw} that it is
necessary in the reduction of M-theory on $\hat{X}_4$ to
shift the 3D M-theory fields for the correct identification by the duality
\eqref{eq:3dDuality} with 4D F-theory fields obtained by the
reduction on a circle to 3D. Furthermore, we will find in this work that
this redefinition is the key to discover a simple
interpretation of the mixed Abelian-gravitational anomalies in the three-
dimensional effective theory.
The coordinate shift of the 3D fields can be translated into a redefinition of the basis \eqref{eq:basisH^11} as
\begin{equation} \label{eq:omega0tilde}
	\tilde{\omega}_0=\omega_{\hat 0}+\frac12 c_1(B_3)\,,\qquad \tilde{B}=B+\frac12 [c_1(B_3)]\,,
\end{equation}
with $\omega_\alpha,\omega_\Lambda$ respectively $D_\alpha,D_\Lambda$
unchanged. The brackets $[\cdot]$
indicate the Poincar\'e dual cycle of the cohomology class $c_1(B_3)$.
The interesting intersection properties of this new basis are
\begin{equation} \label{eq:newIntsTildeB}
	\tilde{B}^2=\frac14 [c_1(B_3)]^2\,,\qquad  \mathcal{K}_{00 \alpha \beta}=0\,,
\end{equation}
where we stress that the intersection numbers with indices $0$ without a hat
are invoking $\tilde \omega_0$.
The first equation follows from \eqref{eq:B^2} for the section $B$,
and the second equation is a consequence of the first and
\eqref{eq:tripleintersect}.

Next we turn to the intersection numbers of the vertical divisors $D_\alpha$. By the fibration structure of $\hat{X}_4$ the intersections of
three and four $D_\alpha$ are given by
\begin{equation} \label{eq:tripleintersect}
   \mathcal{K}_{\hat{0} \alpha \beta \gamma} = \mathcal{K}_{\alpha \beta \gamma} = D^{\rm b}_\alpha \cdot D^{\rm b}_\beta \cdot D^{\rm b}_\gamma\ , \qquad \qquad \mathcal{K}_{\alpha \beta \gamma \delta} = 0\ ,
\end{equation}
where we introduced the triple intersections $\mathcal{K}_{\alpha \beta \gamma}$ of the divisors $D^{\rm b}_\alpha$  in the base $B_3$.

Let us next turn to the intersection numbers involving the Cartan divisors $D_i$.
We consider stacks of non-Abelian seven-branes wrapped on divisors $S^{\rm b}_{(I)}$ in the base $B_3$.
Each such divisor class can be expanded in the basis $D_\alpha^{\rm b}$ as
\begin{equation} \label{eq:def-Calpha}
   S_{(I)}^{\rm b} = \delta^\alpha_{(I)} D_\alpha^{\rm b} \ ,
\end{equation}
where $\delta^\alpha_{(I)}$ are constant coefficients.
In order to incorporate the split \eqref{eq:Gsplit} of $G$ into simple Lie-groups
we divide the $D_i$ as
\begin{equation}
   D_i = ( D_{i_I})\ , \qquad \quad i_I = 1,\ldots \text{rank}(G_{(I)})\ ,
\end{equation}
where the index $i_I$ labels the Cartan divisors for the group factor $G_{(I)}$.
We say that a singularity in $X_4$ at codimension one in the base $B_3$ is of type $G_{(I)}$, if the subset of
divisors $D_{i_I}$ resolving that particular
singularity intersect on $\hat{X}_4$ as
\begin{equation}
\label{eq:Cartan}
	 D_{i_I}\cdot D_{j_J}\cdot D_\alpha\cdot D_\beta=-\delta_{IJ}\mathcal{C}^{(I)}_{i_Ij_I} S_{(I)}\cdot B\cdot D_\alpha\cdot D_\beta
              \ ,  \qquad \quad  D_{i_I}\cdot D_\alpha\cdot D_\beta\cdot D_{\gamma}=0
        \,,
\end{equation}
where no sum is taken over $I,J$.
Here we introduced, starting with the divisors $S_{(I)}^{\rm b}$ in
\eqref{eq:def-Calpha}, vertical
divisors $S_{(I)} = \pi^{-1}(S_{(I)}^{\rm b})$ defined on the fourfold
$\hat X_4$. Thus, they are related to the $S_{(I)}^{\rm b}$ in the base
by $S_{(I)}\cdot B=S_{(I)}^{\rm b}$. The matrices $\mathcal{C}^{(I)}_{i_Ij_I}$
characterize the type of the singularity over $S_{(I)}^{\rm b}$ and can agree
with the inner product of coroots of simple Lie-groups including the ones of
ADE type. For simply laced Lie-groups, i.e.~precisely for ADE groups,
the coroot inner product agrees with the Cartan matrix
$\mathcal{C}^{(I)}_{i_Ij_I}=C^{(I)}_{i_Ij_I}$ since all roots have length
$2$ and $\lambda_I$. In general we have the relation
\begin{equation}
	\mathcal{C}^{(I)}_{i_Ij_I}
	=\frac{2}{\lambda_{I}\langle \alpha_J,\alpha_J\rangle}C_{i_Ij_I}^{(I)}\,,
\end{equation}
where $\alpha_{i_I}$ are the simple roots of the Lie-algebra with inner
product $\langle\cdot,\cdot\rangle$.
This motivates the name Cartan divisors of $G_{(I)}$ for the
divisors $D_{i_I}$ since these divisors can be identified with the
negative of the simple roots of an ADE gauge group\footnote{For a non-simply laced Lie-
group the $D_{i_I}$ are to be identified with the negative of the simple
coroots $\alpha^{\vee}_{i_I}=\frac{2}{\langle\alpha_{i_I},
\alpha_{i_I}\rangle}\alpha_{i_I}$,
i.e.~the simple roots of the coroot lattice.},
$-\alpha_i^{(I)}$. In other words, the $D_{i_I}$ span the negative of the
root lattice of $G_{(I)}$ that consequently is embedded
into the K\"ahler cone of $\hat{X}_4$, more precisely the complement of
the K\"ahler cone of the singular fourfold $X_4$ in $\hat{X}_4$, that is
denoted the relative K\"ahler cone. Note that given the $D_{i_I}$ one can
reverse the logic and use their intersections to unambiguously define the
$S_{(I)}^{\rm b}$ by equation \eqref{eq:Cartan} as discussed in section \ref{sec:GStermsInF}.

The vertical divisors $S_{(I)} $ on the fourfold $\hat X_4$ are related to the inverse images $\hat{S}_{(I)}$
of the $S^{\rm b}_{(I)}$ under the projection to the base on the \textit{singular} fourfold $X_4$
by the shift
\begin{equation}
\label{eq:Shat}
	\hat{S}_{(I)}=S_{(I)}-\sum_{i_I} a_{i_I} D_{i_I}\,,
\end{equation}
where $a_{i_I}$ are the dual Coxeter labels of $G_{(I)}$ and no sum is taken over $I$. We note that due
to this shift the divisors $\hat{S}_{(I)}$ are
not elements in the base $B_3$ in the resolved fourfold
$\hat{X}_4$.  The divisors $D_{i_I}$ project onto the divisors $\pi(\hat{S}_{(I)})$ in $B_3$
in the blow-down map $\hat{X}_4\rightarrow X_4$.

Finally we discuss the intersections of the Cartan divisors $D_m$ of the
$U(1)$-factors. First we introduce analogously to \eqref{eq:def-Calpha}
divisors $S_{(m)}^{\rm b}$ that indicate the location of the seven-branes
supporting the $U(1)$'s in the base $B_3$. We expand
\begin{equation} \label{eq:def-Calpham}
  S_{(m)}^{\rm b} = \delta^\alpha_{(m)} D_\alpha^{\rm b}
\end{equation}
and introduce the corresponding vertical divisors
$S_{(m)}=\pi^{-1}(S^{\rm b}_{(m)})$. Next we construct divisors
$\tilde D_m$ starting from a given basis of rational sections $\hat{\sigma}_m$
by the Shioda map \cite{ShiodaI,ShiodaII}
\begin{equation} \label{eq:ShiodaMap}	\tilde{D}_m = \hat{\sigma}_m-\tilde{B}-(\hat{\sigma}_m\cdot\tilde{B}\cdot\mathcal{C}^\alpha)
	\eta^{-1\beta}_\alpha	D_\beta+
	\sum_{I}(\hat{\sigma}_m\cdot D_{i_I}\cdot \mathcal{C}^\alpha)
	(\delta^\beta_{(I)}\eta_{\beta}^{\,\ \alpha})^{-1}(C_{(I)}^{-1})^{i_Ij_I}D_{j_I}\ ,
\end{equation}
where we denoted the inverse Cartan matrix of $G_{(I)}$
by $C_{(I)}^{-1}$. We are not summing over $\alpha$, but rather fix
one particular value $\alpha$ so that
$\delta^\beta_{(I)}\eta_\beta^{\,\ \alpha}\neq0$ respectively a different
$\alpha$ so that $\delta^\beta_{(n)}\eta_\beta^{\,\ \alpha}\neq0$ to evaluate
\eqref{eq:ShiodaMap}.\footnote{We note that this step intrinsically introduces
the exceptional curves $c$ which are the negative of a simple
root $-\alpha_{i_I}$ of $G_{(I)}$ respectively the Cartans of the $m$th $U(1)$,
\begin{equation}
	 c_{-\alpha_{i_I}}\equiv D_{i_I}\cdot \mathcal{C}^\alpha
	(\delta^\beta_{(I)}\eta_{\beta}^{\,\ \alpha})^{-1}\,,\quad c_{m}\equiv D_{m}\cdot \mathcal{C}^\alpha
	(\delta^\beta_{(m)}\eta_{\beta}^{\,\ \alpha})^{-1}
\end{equation}
This is clear by calculating $c_{-\alpha_{i_I}}\!\!\cdot D_{j_J}$ and
$c_{m}\cdot D_n$ yielding $-\delta_{IJ}C^{(I)}_{ij}$ respectively $-1$ for $m=n$ and an in general model-dependent number for $n\neq m$.}
We also have to introduce a basis of $h^{1,1}(B_3)$
vertical four cycles $\mathcal{C}^\alpha$ in $\hat{X}_4$ as
\begin{equation}
    \mathcal{C}^\alpha = \pi^{-1}(\mathcal{C}^\alpha_{\rm b})\ ,
\end{equation}
with Poincar\'e dual four-forms $\tilde \omega^\alpha$.
These are inherited from curves $\mathcal{C}_{\rm b}^\alpha$ in $B_3$\footnote{\label{footnt1}Technically  these curves are formed by
finding the linearly independent combinations of intersections of two $D^{\rm b}_\alpha$ in the base as $D^{\rm b}_\alpha\cdot
D^{\rm b}_\beta=\tilde{\mathcal{K}}_{\alpha\beta\gamma}\mathcal{C}_{\rm b}^\gamma$, where we introduced the three-point function
${\mathcal{K}}_{\alpha\beta\gamma}=\tilde {\mathcal{K}}_{\alpha\beta\delta}\eta_{\gamma}^{\ \, \delta}\,$
on $B_3$. The metric $\eta$ is defined in \eqref{eq:intmatrix}.}. In general, these curves have a full rank intersection matrix
\begin{equation} \label{eq:intmatrix}
	\eta_{\alpha}^{\ \, \beta} = D^{\rm b}_\alpha\cdot \mathcal{C}_{\rm b}^\beta\ ,
\end{equation}
with the divisors $D^{\rm b}_\alpha$ in $B_3$. Finally, the Cartan divisors $D_m$ are obtained
from $\tilde D_m$ by a simple basis transformation which diagonalizes the intersection
numbers $\tilde D_m \cdot \tilde D_n \cdot D_\alpha \cdot D_\beta$ in the indices $m,n$
as in \eqref{eq:intsD_m}.

The Shioda map \eqref{eq:ShiodaMap} is a map from the Mordell-Weil group to
$H_6(\hat{X}_4)$ and constructed such that the following intersections vanish,
\begin{equation} \label{eq:def_properties_Shioda}
	\tilde{D}_m\cdot D_\alpha \cdot D_\beta\cdot D_\gamma=0\,,\quad
	\tilde{D}_m\cdot \tilde{B} \cdot \mathcal{C}^\alpha=0\,,\quad
	\tilde{D}_m\cdot c=0\,,
\end{equation}
for all exceptional curves $c$ introduced in the resolution of the
singularities of type $G_{(I)}$\footnote{More precisely, these are curves that
are associated to the roots in the root lattice of any $G_{(I)}$.} in
$\hat{X}_4$. The first relation follows since
$\hat{\sigma}_m\cdot D_\alpha \cdot
D_\beta\cdot D_\gamma=\tilde{B}\cdot D_\alpha \cdot D_\beta\cdot D_\gamma$ as
both are sections and the second and third relation are obvious from
\eqref{eq:ShiodaMap} and \eqref{eq:tripleintersect}, \eqref{eq:Cartan}.
The Shioda map has been applied for the
construction of $U(1)$-symmetries in six-dimensional F-theory
compactifications \cite{Morrison:1996pp,Aspinwall:1998xj,Park:2011ji,Morrison:2012ei} on
elliptically fibered
Calabi-Yau threefolds with rational sections. The map
\eqref{eq:ShiodaMap} is the natural extension of the conventional Shioda map
to Calabi-Yau fourfolds. Both $\tilde{D}_m$ and $D_m$ define
$U(1)$-symmetries in F-theory. However, the definition
of $D_m$ ensures in addition that the $D_m$ do not mutually intersect,
whereas the intersections of the $\tilde{D}_m$ can be in general non-diagonal.
This is clear since the $\tilde{D}_m$ are fibrations of the curves $c_m$,
see footnote \ref{footnt1},
over divisors in the base $B_3$.

Then the divisors $D_m$, $S_{(m)}^{\rm b} $ describe four-dimensional $U(1)$
gauge symmetries with the following intersection properties, that are in
complete analogy with
\eqref{eq:Cartan},
\begin{equation} \label{eq:intsD_m}
	D_m\cdot D_n\cdot D_\alpha\cdot D_\beta=-\delta_{mn}S_{(n)}\cdot B\cdot D_\alpha\cdot D_\beta\,,\quad \quad
        D_m\cdot D_{i_I}\cdot D_\alpha\cdot D_\beta=0\,,\quad
        D_m\cdot D_\alpha\cdot D_\beta\cdot D_{\gamma}=0 \,.
\end{equation}
The second and third equalities are a direct consequence of the
defining properties \eqref{eq:def_properties_Shioda} of the Shioda map,
the definition of $D_m$,
and the fact that the $D_i$ are fibrations of shrinking curves $c$ over
divisors in the base $B_3$.
Given a set of $D_m$ the first equation in \eqref{eq:intsD_m} can be
viewed as the defining equation for the $S^{\rm b}_{(m)}$
and the vertical divisors  $S_{(m)}$. We will show
in concrete examples how to construct a basis of $D_m$ and the divisors
$S_{(m)}$ obeying the properties \eqref{eq:intsD_m}. We note that the
association of divisors $S_{(m)}$ to $U(1)$'s in F-theory leads to a new
perspective on the interpretation of $U(1)$'s in F-theory.

That the conditions \eqref{eq:intsD_m} have to hold can also be inferred
physically from the analysis of the 3D gauge kinetic terms
and their F-theory lift by extending the discussion \cite{Grimm:2010ks} to
include $U(1)$-gauge symmetries. In
this context the first relation ensures that the gauge kinetic function of
the $U(1)$'s is diagonal, which is always achievable in field theory, and
the second that the gauge couplings is also diagonal between the $U(1)$'s
and the non-Abelian group $G_{(I)}$.

We can summarize equations \eqref{eq:Cartan} and \eqref{eq:intsD_m} in a more compact way in terms of the
quartic intersections \eqref{eq:fourfoldintersect} as
\begin{equation} \label{eq:vanishingInts_0}
		\mathcal{K}_{i_I\, j_J\, \alpha \beta} =- \delta_{IJ}\, \mathcal{C}^{(I)}_{ij} \delta^\gamma_{(I)}   \mathcal{K}_{\gamma \alpha \beta} \,,\qquad\mathcal{K}_{mn\alpha \beta} = -\delta_{mn} \delta^\gamma_{(m)} \mathcal{K}_{\gamma \alpha \beta}\ , \qquad
        \mathcal{K}_{\Lambda \alpha \beta \gamma} = \mathcal{K}_{i_I m \alpha \beta}= 0 \ .
\end{equation}
Here $\mathcal{K}_{\alpha \beta \gamma}$ are the triple intersections \eqref{eq:tripleintersect} in the base $B_3$ and $\delta^\gamma_{(I)}$ respectively
$\delta^\gamma_{(m)}$ restrict to the $I$'s seven-brane stack defined in \eqref{eq:def-Calpha} respectively the $m$'s $U(1)$ seven-brane
defined in \eqref{eq:def-Calpham}.
We also introduced for later convenience the notation
\begin{equation}
   D_\Lambda=(D_i,D_m)\ ,\qquad \quad  \omega_\Lambda=(\omega_i,\omega_m)\ , \qquad \Lambda \in\{i,m\}
\end{equation}
unifying all divisors associated to gauge symmetries in M- and F-theory.
To complete the discussion of intersection relations
let us also note that on $\hat{X}_4$ one has
\begin{equation} \label{eq:vanishingInts_1}
 \mathcal{K}_{\hat 0 \Lambda A B}=  B \cdot D_{\Lambda} \cdot D_A \cdot D_B  = 0\,.
\end{equation}

\subsection{Four-form fluxes in M-theory to F-theory duality} \label{sec:flux_review}

Next we construct the four-form flux $G_4$ on
$\hat{X}_4$. The flux $G_4$ is an element in the fourth cohomology
group $H^{4}(\hat{X}_4,\mathbb{Z}/2)$ due to the quantization
condition \cite{Witten:1996md}
\begin{equation} \label{G4half}
	G_4+\frac12 c_2(\hat{X}_4)\,\,\in\,\, H^{4}(\hat{X}_4,\mathbb{Z})\,.
\end{equation}
These conditions have been investigated in the F-theory context in
\cite{Collinucci:2010gz,Collinucci:2012as}.
Splitting into Hodge types, there are two different types of fluxes due to the even complex
dimension of $\hat{X}_4$ \cite{Greene:1993vm,Mayr:1996sh,Klemm:1996ts}. The first type are
$G_4$-fluxes in the vertical cohomology group
$H_V^{(2,2)}(\hat{X}_4,\mathbb{Z}/2)$ that is generated by the product of two forms in
$H^{2}(\hat{X}_4,\mathbb{Z})$. Thus fluxes in $H_V^{(2,2)}(\hat{X}_4,\mathbb{Z}/2)$ can be specified as
\begin{equation} \label{eq:verticalFluxGeneral}
	G_4=m^{AB}\omega_A\wedge \omega_B\,,
\end{equation}
for appropriate constant coefficients $m^{AB}$. Note that it is
crucial here to know the
cohomology $H^{2}(\hat{X}_4,\mathbb{Z})$ explicitly which is not
straightforward for the singular
geometry $X_4$. These vertical fluxes are crucial in generating
chirality in F-theory as discussed in section \ref{sec:ChiralityInF}.

The second type are fluxes $G_4$ in the normal space to the
vertical cohomology. They lie in
the horizontal cohomology $H^{4}_H(\hat{X}_4,\mathbb{Z}/2)$ that is
obtained from complex structure
variations of the $(4,0)$-form on $\hat{X}_4$. Physically they give
rise to a non-trivial classical flux superpotential \cite{Gukov:1999ya}, that corresponds in weak coupling
to D7-brane and Type IIB flux superpotentials. See
\cite{Lust:2005bd,Grimm:2009ef,Grimm:2009sy,Jockers:2009ti,Alim:2010za,Klevers:2011xs,Braun:2011zm,Intriligator:2012ue}
for a list of some works on
the physical interpretations of the flux superpotential in F-theory and studies of properties
of horizontal $G_4$-fluxes.

It can be shown that the vertical fluxes
\eqref{eq:verticalFluxGeneral} induce Chern-Simons terms for
vectors $A^A$ in the 3D effective action that are obtained by reducing the M-theory three-form $C_3$. For more details on the complete reduction of M-theory as well as on the lift back to 4D F-theory see \cite{Grimm:2010ks}.
Reducing $C_3$ in the M-theory compactification on $\hat{X}_4$ with
respect to the forms $\omega_A$ introduced in \eqref{eq:basisH^11}
we expand into 3D vectors $A^A$ as
\begin{equation}
\label{eq:C3expansion}
	C_3=A^A\wedge \omega_A=A^0\wedge \tilde \omega_0+ A^\alpha \wedge \omega_\alpha +A^\Lambda\wedge \omega_\Lambda\, ,
\end{equation}
with $\tilde \omega_0$ defined in \eqref{eq:omega0tilde}.
From counting indices we thus obtain
$1+h^{1,1}(B_3)+\text{rank}(G)
=h^{1,1}(\hat{X}_4)$ Abelian $U(1)$ vector fields. The vectors $A^\Lambda$ are the
remaining massless vectors on the 3D Coulomb branch, that are,
from the F-theory perspective in \eqref{eq:3dDuality},
the gauge fields in the maximal torus of the non-Abelian gauge group $G$. The additional vectors
$A^\alpha$ are the 3D dual to the imaginary part of the K\"ahler moduli of $B_3$.
It will be important for us that $A^0$ is identified with the
Kaluza-Klein vector from reducing the 4D metric of the F-theory effective
action on $S^1$, also denoted as the 3D graviphoton. It arises as the
component $A^0_\mu \propto g_{\mu 3}$, where the index $3$ indicates the
$S^1$-direction.

Performing the dimensional reduction of the eleven-dimensional action
in a background with the $G_4$-flux \eqref{eq:verticalFluxGeneral} one
obtains a 3D Chern-Simons action for the vectors $A^A$ of the form \cite{Haack:2001jz}
\begin{equation} \label{eq:theta_AB}
	S^{(3)}_{CS}=-\int \frac{1}
	{2}\Theta_{AB}A^A \wedge F^B\,,\qquad
	\Theta_{AB}=\frac12\int_{\hat{X}_4}G_4\wedge \omega_A\wedge
	\omega_B\,,
\end{equation}
where we used the conventions of \cite{Grimm:2010ks}.
Here we employ in the definition of the Chern-Simons levels $\Theta_{AB}$ the shifted basis \eqref{eq:omega0tilde}.

In addition, these flux integrals $\Theta_{AB}$ in general induce in the
3D M-theory effective action St\"uckelberg gaugings of the complexified
K\"ahler moduli $T_A$ associated to the divisors $D_A$ as
\begin{equation} \label{eq:gaugingT_A}
	DT_A=dT_A+i\Theta_{AB}A^B\,.
\end{equation}
In this context, the couplings $\Theta_{AB}$ play the role of the
``embedding tensors''\footnote{We have chosen the conventions of
\cite{Grimm:2011sk} for the normalization of the $T_A$ so that no numerical factors of 2 appear in \eqref{eq:gaugingT_A}.
Then $T_\alpha$
agrees with 4D Type IIB K\"ahler moduli and is related to 3D K\"ahler moduli $t_\alpha$ as $t_\alpha=2 T_\alpha$.}.
We note that for the purpose of anomaly cancellation these gaugings are
essential since the imaginary part of some $T_\alpha$ will play
the role of an axion with an anomalous gauge transformation under
the $U(1)$ gauge symmetries in the theory. This
will lead, as we will discuss in section \ref{sec:GStermsInF}, to a 4D
generalized Green-Schwarz mechanism.

The Chern-Simons levels \eqref{eq:theta_AB} are key objects to study the physics of the vertical fluxes
\eqref{eq:verticalFluxGeneral} in F-theory.
In order to have a clear 4D F-theory interpretation we have to impose additional conditions
on the $G_4$-flux in M-theory
that we summarize in terms of the flux integrals $\Theta_{AB}$ in
\eqref{eq:theta_AB}. We require the following integrals to vanish
\cite{Marsano:2011hv,Grimm:2011sk},
\begin{eqnarray} \label{eq:vanishingThetas}
	\Theta_{0\alpha}=\Theta_{\alpha\beta}=\Theta_{i\alpha}=0\,,\\
	\Theta_{00}=\Theta_{0i}=0\,.\nonumber
\end{eqnarray}
We emphasize that these conditions have to be evaluated in the
basis \eqref{eq:omega0tilde} that relates the fields of the M-theory
reduction in this basis correctly to the circle-reduced 4D fields.
The conditions \eqref{eq:vanishingThetas} on the $G_4$-flux are imposed in
addition to the conventional M-theory conditions on allowed $G_4$-flux. We
will show in section \ref{sec:strucOfFlux+anomalies}
by evaluating the $\Theta_{AB}$ for a general $G_4$-flux of the form
\eqref{eq:verticalFluxGeneral}
that these can be always satisfied by restricting the flux numbers
$m^{AB}$. We will exemplify this
even further for concrete examples in section \ref{sec:examples}.

The requirement $\Theta_{i\alpha}=0$ can be understood readily in the
effective field theory.
By imposing these conditions the gaugings of the $T_\alpha$
by the gauge fields $A^i$ are absent according to \eqref{eq:gaugingT_A}.
These gaugings would break the
non-Abelian part in the gauge group $G$ in the corresponding
F-theory compactification,
that we want to retain e.g.~as a GUT group
for phenomenological applications.
As we will discuss below in section \ref{sec:ChiralityInF}, the
non-vanishing $\Theta_{\Lambda\Sigma}$ encode the chirality
of charged matter in 4D.
It is important to stress that we do not require a vanishing of the Chern-Simons levels
\begin{equation}
  \Theta_{0m} = \frac12\int_{\hat X_4} \tilde \omega_0 \wedge \omega_m \wedge G_4\ ,
\end{equation}
where, as in \eqref{eq:vanishingThetas}, one uses the redefined
$\tilde \omega_0$ given in \eqref{eq:omega0tilde}.
As we will discuss in more detail in section \ref{sec:Theta0m},
the non-vanishing components $\Theta_{0m}$
are crucial in the study of mixed Abelian-gravitational anomalies. Clearly, due to
the fact that $B \cdot D_m = 0$, as stated already in \eqref{eq:vanishingInts_1}, the integral
$\int_{\hat{X}_4}\omega_{\hat 0} \wedge \omega_m \wedge G_4$ vanishes trivially
for all fluxes $G_4$. However, this is not generically true for
$\Theta_{0m}$ as discussed below.

Let us note that one might think that the $\Theta_{0m}$ are on an equal footing with the $\Theta_{0i}$ on
the 3D Coulomb branch of the circle reduced theory. However, as we will show in section \ref{sec:Theta0m}
a non-vanishing $\Theta_{0m}$ is generated in the IR from integrating out
Kaluza-Klein states of 4D charged fermions while the $\Theta_{0i}$ are also zero at one loop.
We will deduce that $\Theta_{0m}$ is
precisely given by the 4D mixed Abelian-gravitational anomaly.

\subsection{Green-Schwarz terms in F-theory}
\label{sec:GStermsInF}

In the following we will determine all Green-Schwarz counter terms
in \eqref{eq:GSterm}. More precisely, we will outline the steps
to derive the coefficients $b^{\alpha}_I$, $b_{mn}^\alpha$
and $a^\alpha$ in 4D F-theory compactifications. We
demonstrate that these coefficients are completely determined in terms
of the intersections on the resolved fourfold $\hat{X}_4$ as well as
by the canonical divisor, or equivalently the first Chern class, of the base $B_3$. A similar analysis for 4D compactifications without
$U(1)$ factors can be found in \cite{Grimm:2012yq}.

We start our discussion by noting that the prefactors of the first two
terms in \eqref{eq:GSterm} are the imaginary parts of the 4D
gauge coupling functions for the gauge fields $F^I$ of each non-Abelian
factor $G_{(I)}$ respectively for the $F^m$ of each $U(1)_m$. These gauge
couplings are given by the volume of the divisors $S_{(I)}^{\rm b}$ and
$S_{(m)}^{\rm b}$ in the base $B_3$
introduced after \eqref{eq:def-Calpha} and \eqref{eq:def-Calpham}. This
can be argued by performing the M-theory reduction on $\hat{X}_4$ in the
non-Abelian case or by using intuition from weak coupling results in Type
IIB, where the relevant divisors are those wrapped by D7-branes. In all
these cases the divisors are determined uniquely on $\hat{X}_4$ from their
defining intersection properties
\eqref{eq:Cartan} and \eqref{eq:intsD_m}. The association
of divisors wrapped by seven-branes to $U(1)$'s in F-theory might be
unexpected and new since $U(1)$'s, as discussed in section
\ref{sec:F+MresolvedFF} above, are not simply related to singularities in
the elliptic fibration of $X_4$, but to rational sections of the elliptic
fibration, i.e.~a non-trivial Mordell-Weil group. Mathematically, the
relation of these rational sections to divisors
in $B_3$ is more subtle but straightforwardly formulated in terms of the
intersections \eqref{eq:intsD_m}
of the $D_m$ as elucidated further in the following.

We start with the determination of the coefficients $b^\alpha_I$. First we
note that the axions $\rho_\alpha$ in \eqref{eq:GSterm} are in F-theory
the imaginary parts of the K\"ahler moduli $T_\alpha$ associated to the
divisors $D_\alpha$ in \eqref{eq:basisH^11}.
Then, the coefficients $b^\alpha_I$ are just given as the coefficients
in the expansion \eqref{eq:def-Calpha}, and one finds
\begin{equation} \label{eq:matchC-b}
     b_I^\alpha = \delta^\alpha_{(I)}\ .
\end{equation}
Next we determine by the same logic the coefficients $b_{mn}^\alpha$ in \eqref{eq:GSterm}.
As mentioned before this physically amounts to define seven-branes supporting
$U(1)$-symmetries in F-theory.
As in \eqref{eq:def-Calpham} we denote by $S^{\rm b}_{(m)}$ the divisors that support
$U(1)$-gauge factors.
Thus, we see from the first equation
in \eqref{eq:intsD_m} that the coefficients $b_{mn}^\alpha$ must be diagonalizable,
and we identify
\begin{equation} \label{eq:match_b_mn}
   b_{mn}^\alpha = \delta_{mn} \delta_{(m)}^\alpha \ ,
\end{equation}
where the coefficients $\delta_{(m)}^\alpha$ were introduced in
\eqref{eq:def-Calpham}. Finally, we determine the coefficients $a^\alpha$.
We just state the final result here and refer to
\cite{Sadov:1996zm,Bonetti:2011mw,Grimm:2012yq} for the derivation in six-
and four-dimensional F-theory. The coefficients read
\begin{equation}  \label{eq:defa^alpha}
  a^\alpha = K^\alpha\ , \qquad  \quad c_1(B_3) = - K^\alpha \omega_\alpha\ ,
\end{equation}
where we expanded the first Chern class into a basis restricted to $B_3$.

To end this section let us introduce an alternative
way to present the coefficients $b^\alpha_I, b^\alpha_{mn},a^\alpha$ in terms of intersection numbers of
geometrical objects. In order to do so we have to use the basis of $h^{1,1}(B_3)$ curves $ \mathcal{C}^\alpha_{\rm b}$
in $B_3$ and their intersection matrix $\eta_{\alpha}^{\ \, \beta}$ introduced in \eqref{eq:intmatrix} as well as
the induced four-cycles $ \mathcal{C}^\alpha$ in $\hat X_4$
with Poincar\'e dual four-forms $\tilde \omega^\alpha$.
Furthermore, we introduce the  push-forward $\pi_*$ from $H_{4}(\hat{X}_4)$ to $H_{2}(B_3)$ in homology
induced by the projection $\pi$. Its action on surfaces $S$, and by Poincar\'e duality
also on four-forms $\tilde{\omega}$ in $H^{4}(\hat{X}_4)$, is defined as
\begin{equation}	\label{eq:push-forward}
\pi_*(S)=D^{\rm b}_\beta\eta^{-1\beta}_\alpha\ (\mathcal{C}^\alpha\cdot S)_{\hat X_4}\,,\qquad \pi_*(\tilde{\omega})=\omega^{\rm b}_{\beta}\eta^{-1\beta}_\alpha\int_{\hat{X}_4}\tilde{\omega}\wedge \tilde{\omega}^\alpha \,,
\end{equation}
where $\omega^{\rm b}_{\alpha}$ is a two-form in $H^{(1,1)}(B_3)$ dual to $D^{\rm b}_{\alpha}$ and where we introduced the
inverse $\eta^{-1}$ of the intersection matrix \eqref{eq:intmatrix}.
Using these definitions and the equations of section \ref{sec:F+MresolvedFF} it is straightforward
to infer
\begin{eqnarray}\label{eq:determingb_A}
	\delta^\alpha_I C^{(I)}_{i_Ij_I}  &=&- \eta^{-1\alpha}_\gamma \ D_{i_I}\cdot D_{j_I}\cdot \mathcal{C}^\gamma  =-  \eta^{-1\alpha}_\gamma\ \pi_*(D_{i_I}\cdot D_{j_I})\cdot \mathcal{C}^\gamma_{\rm b}  \,,\\
     \label{eq:determingb_m}
	\delta^\alpha_m&=&- \eta^{-1\alpha}_\beta \ D_m\cdot D_m\cdot \mathcal{C}^\beta=- \eta^{-1\alpha}_\beta\ \pi_*( D_m\cdot D_m)\cdot \mathcal{C}_{\rm b}^\beta\ ,\\
	\label{eq:determininga}
        a^\alpha & = & \eta^{-1\alpha}_\beta\ B \cdot B \cdot \mathcal{C}^\beta =  \eta^{-1\alpha}_\beta\ \pi_*(B \cdot B) \cdot \mathcal{C}_{\rm b}^\beta\ ,
\end{eqnarray}
where no sum is performed over $I$ and we have used that $B$ satisfies $B\cdot B= -[c_1(B_3)]\cdot B$. This way of presenting the Green-Schwarz coefficients will be
particularly useful when translating the anomaly conditions into purely geometric conditions involving
$G_4$ in section \ref{sec:anomalycancellationinF}. It also facilitates the determination of the
coefficients $b^\alpha_I, b^\alpha_{mn},a^\alpha$ if only the Cartan divisors $D_i$ and the sections $D_m$ are known.

\section{One-loop Chern-Simons Terms and Their F-Theory Interpretation}
\label{sec:ChiralityInF}

In this section we study the 3D Chern-Simons terms in the duality \eqref{eq:3dDuality}
between F-theory on $X_4\times S^1$ and M-theory on the smooth fourfold
$\hat{X}_4$. We describe the matching of the one-loop Chern-Simons term in the circle
compactification of F-theory with the classical flux-induced Chern-Simons term in M-theory.
In section~\ref{sec:ChiralityFormula3D} we concentrate on the Chern-Simons terms
for the 3D gauge fields inherited from 4D gauge fields. We recall that
in the circle reduction such terms are generated at one-loop after
integrating out charged matter that acquired a mass on the 3D Coulomb
branch. A matching with the flux-induced Chern-Simons term in M-theory
allows to infer the 4D chiral index counting the net number of chiral
fermions. In section~\ref{sec:Theta0m} we focus on a new
one-loop Chern-Simons term that has not been considered so far in three
dimensions. It involves the Kaluza-Klein vector $A^0$, and we will show
explicitly that it is induced at one loop by integrating out massive
Kaluza-Klein modes of the charged fermions. It will be linked with the 4D
gauge-gravitational anomalies in section~\ref{sec:anomalycancellationinF}.

\subsection{4D Chirality formulas from the 3D Coulomb branch}
\label{sec:ChiralityFormula3D}

We begin our discussion by first stating the expected form of the 4D
chiral index formula for charged chiral matter in a representation
$\mathbf{R}$. In F-theory on a singular elliptic Calabi-Yau fourfold $X_4$
charged chiral matter is induced by seven-brane flux which maps to
vertical $G_4$-flux. The chiral index of charged matter in a
representation $\mathbf{R}$ of the gauge group $G$ is given by the flux
integral
\cite{Donagi:2008ca,Hayashi:2008ba,Braun:2011zm,Marsano:2011hv,Krause:2011xj,Grimm:2011fx,Krause:2012yh,Kuntzler:2012bu}
\begin{equation} \label{eq:chi(R)}
	\chi(\mathbf{R})= \int_{S_{\mathbf{R}}}G_4\,.
\end{equation}
In this general form, without specifying a construction of the fluxes and
the so-called matter surfaces $S_{\mathbf{R}}$, the expression \eqref{eq:chi(R)} is
not surprising. If seven-brane fluxes have an $G_4$-flux image in M-theory then only
a linear expression that vanishes for $G_4=0$ is conceivable.
However, it is important to stress that for our construction
both $G_4$ and $S_{\mathbf{R}}$ are both naturally defined on the smooth
M-theory fourfold $\hat{X}_4$, not on the singular fourfold $X_4$.
This might be counter-intuitive, since in an M-theory compactification
on a smooth space no massless charged matter appears in the effective theory.
Nevertheless, following the strategy of \cite{Grimm:2011fx}, we argue next that
the formula \eqref{eq:chi(R)} can be derived by consideration of the three-dimensional $\mathcal{N}=2$
effective gauge theory and its dual formulation in terms of F-theory on $S^1$ on the one hand and M-theory on
$\hat{X}_4$ on the other hand.

We start on the F-theory side with a 4D $\mathcal{N}=1$ supergravity theory with
gauge group $G$ and chiral matter in a representation
$\mathbf{R}$. Then we compactify this theory on $S^1$ and move onto the Coulomb
branch of the resulting 3D gauge theory. This breaks the non-Abelian 4D gauge
symmetry to its maximal torus, $G\rightarrow U(1)^{\text{rank}(G)}$, with Abelian
gauge fields $A^{\Lambda}$. The Coulomb branch parameters are given by the VEVs
of scalars $\zeta^\Lambda$ in the 3D $\mathcal{N}=2$ vector multiplets that are the
components $A^\Lambda_3$ of the 4D vectors
along the $S^1$, i.e.~the holonomies
\begin{equation}
    \zeta^\Lambda=\int_{S^1}A^\Lambda\,.
\end{equation}
Simultaneously the chiral matter receives mass-terms that are
proportional to the Coulomb branch parameters $\zeta$. In the 3D effective action in
the IR at energy scales below $\zeta^\Lambda$ these massive fields have to be integrated
out and generate Chern-Simons terms for the Abelian vectors at one loop,
\begin{equation} \label{eq:CSterms}
	S^{(3)}_{CS}=\int \Theta^{\text{loop}}_{\Lambda\Gamma}A^\Lambda\wedge F^\Gamma\, ,
\end{equation}
where we use the conventions of \cite{Kao:1995gf}.
We note that these terms are classically absent, i.e.~not generated in the compactification
from 4D to 3D. In order to prepare for section \ref{sec:Theta0m} it is
important to mention that in a Kaluza-Klein theory also excited modes
of the 4D matter fields along $S^1$ are
charged under $A^\Lambda$ and can, in principle, induce a one-loop
contribution in \eqref{eq:CSterms}. In the following we will first discuss
the one-loop term without these excited Kaluza-Klein modes and then
comment on their inclusion.

Let us consider the one-loop Chern-Simons term induced by Kaluza-Klein
zero modes of the charged matter fields that became massive in the Coulomb
branch. We denote the real mass of the matter fermion $f$ by $m_f$. The
Coulomb branch masses $m_f$ are given by
\begin{equation} \label{eq:qdotxi}
	m_f = q_f \cdot
	\zeta \equiv \sum_{\Gamma=1}^{\text{rank}(G)}(q_f)_\Gamma
	\zeta^\Gamma\,,
\end{equation}
where $(q_f)_\Lambda$ denotes the charge of the fermion $f$ under the $U(1)$ vector field
$A^\Lambda$.
The loop integral expression for the Chern-Simons level is \cite{Niemi:1983rq,Redlich:1983dv,Aharony:1997bx}
\begin{equation} \label{eq:theta_LambdaSigma}
	\Theta^{\text{loop}}_{\Lambda\Sigma}=\frac{1}
	{2}\sum_f(q_f)_\Lambda(q_f)_\Sigma \ \text{sign}(m_f)=\frac12\sum_{\mathbf{R}}n(\mathbf{R})\sum_{q\in \mathbf{R}}q_\Lambda q_\Sigma \ \text{sign}(q\cdot
	\zeta)\,,
\end{equation}
with the sum taken over all fermions $f$ charged under $A^\Sigma$. In the
second equality we split this sum
over fermions into a sum over representations $\mathbf{R}$ and then over
charges $q$ in that representation, where $n_\mathbf{R}$ denotes the
multiplicity of $\mathbf{R}$. Thus, we see that the Chern-Simons terms are
only proportional to the 4D chiralities
$\chi(\mathbf{R})=n(\mathbf{R})-n(\mathbf{R}^*)$ since the weights of the
complex conjugate representations $\mathbf{R}^*$ are
$q(\mathbf{R}^*)=-q(\mathbf{R})$ and since
\eqref{eq:theta_LambdaSigma} is an odd function in the charges $q$.

In a next step we can consider the contributions of excited Kaluza-Klein
modes. These can be equally charged under $A^\Lambda$ and hence contribute
to the Chern-Simons term.The mass of the $n$th excited fermionic mode $f$
is given by
\begin{equation}
   m_n^f = q_f \cdot \zeta + \frac{n}{r}\ .
\end{equation}
This expression has to be used in the one-loop Chern-Simons level
\eqref{eq:theta_LambdaSigma}.
In the following we will consider the mass hierarchy
\begin{equation} \label{mass_hierarchy}
  m_0^f = q_f \cdot \zeta \ <\ \frac{1}{r} = m_{\rm KK}\ ,
\end{equation}
such that the Coulomb branch mass scale is below the Kaluza-Klein scale.
In this case one can drop the contribution $q_f \cdot \zeta$ for all
excited modes $n\neq0$, since the sign $\text{sign}(m^f_n)$ is
determined by the contribution $n/r$ alone. For each Kaluza-Klein level
$n>0$ one finds that there is a positive term that is canceled by a term
arising from the level $-n$. This pairwise cancellation can be inferred
physically from the fact that a Chern-Simons term arises from parity
violation and the excited modes do not violate parity. Therefore,
we conclude that only the zero modes contribute non-trivially to this
coupling. As we will see in section \ref{sec:Theta0m} the situation
changes once we consider Chern-Simons terms involving the Kaluza-Klein
vector under which the excited modes are charged.

The above argument shows that a certain linear combination
of the CS-couplings in \eqref{eq:CSterms} yields the chiral index
$\chi(\mathbf{R})$ of matter in the representation $\mathbf{R}$,
\begin{equation} \label{eq:chi(R)3D}
\chi(\mathbf{R})=t_{\mathbf{R}}^{\Lambda\Sigma}\Theta_{\Lambda\Sigma}\,,
\end{equation}
where the matrix $t_{\mathbf{R}}^{\Lambda\Sigma}$ roughly first projects
the sum in \eqref{eq:theta_LambdaSigma} to a particular representation
$\mathbf{R}$ and then cancels the sum over all charges $q$ in this
representation $\mathbf{R}$.

Considering the same terms in the M-theory reduction, we have noted in section \ref{sec:F+MresolvedFF}
that classical Chern-Simons terms for the vectors $A^A$ are given by the
flux-integrals \eqref{eq:theta_AB}. By M-/F-theory duality \eqref{eq:3dDuality} these have to be
identified precisely with the loop-generated CS-terms
\eqref{eq:theta_LambdaSigma} as \cite{Grimm:2011fx}
\begin{equation}	 \label{eq:matching1loop+classical}
	-\Theta^{\text{loop}}_{\Lambda\Sigma}\equiv\frac12\Theta_{\Lambda\Sigma}
	=\frac14\int_{\hat{X}_4}G_4\wedge\omega_\Lambda\wedge\omega_\Sigma\,.
\end{equation}
It is important to note that also the sign-function in
\eqref{eq:theta_LambdaSigma} is then determined by the
geometry by a simple rule. A charge vector $\underline{q}$ is positive, i.e.~$\text{sign}(q\cdot \zeta)=1$, if the corresponding shrinkable curve
in M-theory lies in the Mori cone of $\hat{X}_4$. Accordingly, we call
it negative, $\text{sign}(q\cdot \zeta)=-1$, if it does not lie in the
Mori cone. Therefore, in combination with \eqref{eq:chi(R)3D} we obtain
the F-theory chiral index \eqref{eq:chi(R)} from matching 3D CS-terms.
Furthermore, the matter surfaces $S_{\mathbf{R}}$
are identified as the Poincar\'e duals of
\begin{equation}
\big[S_{\mathbf{R}}\big]=t_{\mathbf{R}}^{\Lambda\Sigma}\omega_\Lambda\wedge\omega_\Sigma\,,
\end{equation}
where the brackets $[\cdot]$ indicate the action of Poincar\'e duality on $\hat{X}_4$..

\subsection{Graviphoton Chern-Simons terms with Kaluza-Klein modes}
\label{sec:Theta0m}

In this section we show that a mixed Chern-Simons term for the 3D graviphoton $A^0$ and the 4D $U(1)$-vectors
$A^m$ is generated at one-loop when integrating out all Kaluza-Klein states of 4D charged fermionic matter.
The fact that Chern-Simons terms involving the Kaluza-Klein vector are induced
by one-loop diagrams with excited Kaluza-Klein modes running in the loop was
first noted in a five-dimensional context  \cite{Bonetti:2011mw,Bonetti:2012fn}. It was
shown in these works that the five-dimensional Chern-Simons terms capture the
information about six-dimensional gravitational anomalies.

In the following we present the one loop computation in the
three-dimensional setting. We find that the only non-vanishing
Chern-Simons term involving the 3D graviphoton $A^0$ is
\begin{equation} \label{eq:graviphotonCS}
	S^{(3)}_{CS}=\int \Theta^{\text{loop}}_{0m} A^0\wedge F^m\,,
	\qquad\Theta^{\text{loop}}_{0m}=-\frac1{12}\sum_fq_m^f=-\frac1{12}\sum_{\underline{q}} n(\underline{q})q_m\,.
\end{equation}
The coefficient $\Theta^{\text{loop}}_{0m}$ is precisely the mixed
Abelian-gravitational anomaly encountered in
\eqref{eq:Anomalies:Abgravitational}. In fact, we argue that
the relevant 3D loop diagrams generating $\Theta^{\text{loop}}_{0m}$ can
be understood as a dimensional reduction of the 4D anomalous triangle
diagram for the mixed Abelian-gravitational anomalies. In addition, by the
relation \eqref{eq:matching1loop+classical} between the Chern-Simons
levels also to classical flux integrals \eqref{eq:theta_AB}, we can
actually deduce in 3D the 4D cancellation condition
\eqref{eq:Anomalies:Abgravitational} of mixed Abelian-gravitational
anomalies. Indeed, anticipating the property
$\Theta_{0m}=-\frac12 K^\alpha\Theta_{\alpha m}$ of
the classical intersections on $\hat{X}_4$ that we will derive in
\eqref{eq:ThetaABcalculated}, \eqref{eq:ThetaAB_abel_2} in section
\ref{sec:strucOfFlux+anomalies}, we obtain the anomaly condition
by the M-/F-theory duality \eqref{eq:3dDuality} as
\begin{equation}	 \label{eq:3DDerivationMixedAnomaly}
	\frac1{3}\sum_{\underline{q}} n(\underline{q})q_m=-4\Theta^{\text{loop}}_{0m}\equiv2\Theta_{0m}=
	 -K^\alpha\Theta_{\alpha m}\,.
\end{equation}
This agrees precisely with \eqref{eq:Anomalies:Abgravitational} using the identification $a^\alpha=K^\alpha$ made in
\eqref{eq:defa^alpha}. In contrast to the non-Abelian anomalies, this is a direct derivation of 4D mixed anomaly cancellation
from 3D Chern-Simons terms.

We begin by understanding diagrammatically the connection of 4D anomalies to the Chern-Simons-levels
$\Theta_{0m}$ and the relevance of KK-states running in the 3D loops.
The 4D mixed Abelian-gravitational anomaly is encoded in the one-loop triangle diagram in figure
\ref{fig:triangleDiagram}, that leads to a violation of the corresponding current conservation law at the
quantum level.
\begin{figure}[ht!]
	\begin{center}
	\includegraphics[width=0.35\textwidth]{MixedAnomalyCut}
	\end{center}
	\caption{One-loop triangle diagram responsible for the Abelian-gravitational anomaly. }
	\label{fig:triangleDiagram}
\end{figure}
Here the external lines are given by two gravitons $g_{\mu\nu}$ and one
$U(1)$-vector $A_\mu^m$. Massless charged Weyl fermions carrying
$U(1)$-charges under the $A_\mu^m$ run in the loop.
The vertex rules are attached to each vertex, where $k^\rho$ denotes the
4-momentum of the fermions in the loop and we have used
$\sigma^\mu=(-\mathbf{1},\sigma^i)$ with $\sigma^i$ denoting the
Pauli-matrices.

For the dimensional reduction of the four-dimensional theory to three
dimensions, one direction of 4D Minkowski space, say the third direction,
is replaced by a circle of radius $r$. Thus one metric
component reads $g_{33}=r^2$, that we interpret as a constant background
field in 3D. From the isometries of $S^1$ we get in addition a new 3D
vector $A_\mu^0\sim g_{\mu 3}$, the graviphoton. The heuristic dimensional
reduction of the diagram \ref{fig:triangleDiagram} follows analogous to
the situation in \cite{Bonetti:2012fn}. We replace in the diagram
\ref{fig:triangleDiagram} one external graviton by the background field
$r^2$ and the second graviton by the graviphoton as in figure
\ref{fig:loopForTheta0m}.
\begin{figure}[ht!]
	\begin{center}
	\includegraphics[width=0.45\textwidth]{CSTermA0AmCut}
	\end{center}
	\caption{One-loop diagram with KK-states in the loop generating the 3D Chern-Simons level $\Theta_{0m}$. }
	\label{fig:loopForTheta0m}
\end{figure}
We also introduced the momentum $k^3$ along the circle that is quantized as $k^3=\frac{n}{r}$. This gives a qualitative
idea of the 3D vertex rules, however, we refer to the remainder of this section for a more thorough derivation.

The background field $r$ acts like a mass term and together with the fact
that only KK-states are charged under the graviphoton, this enforces that all KK-states of the 4D
massless charged fermions run in the loop. Thus, effectively we obtain infinitely many one loop diagrams,
one for each KK-state, with only two external lines, namely the Abelian gauge field $A^m_\mu$ and the
graviphoton $A^0_\mu$. Each single of these infinitely many loop-diagrams is of the form of the
loop-diagrams considered in \cite{Niemi:1983rq,Redlich:1983dv,Aharony:1997bx}. Thus, it is clear that a
3D Chern-Simons term of the form \eqref{eq:graviphotonCS} is induced from the 4D mixed anomaly. The main
subtlety that remains is to evaluate the infinite sum over KK-states appropriately. This sum has to be
regularized by zeta function regularization and will precisely account for the prefactor in \eqref{eq:graviphotonCS}.

We conclude this diagrammatic discussion by excluding further 3D
Chern-Simons terms by a similar logic.
As can be seen from the comparison to 4D anomaly graphs, neither the
Chern-Simons level $\Theta_{00}$,
nor the levels $\Theta_{0i}$ can be induced at one loop. This is clear
since the former would arise from the
dimensional reduction of the gravitational anomaly, that is identically
zero in 4D, and the latter from the
mixed non-Abelian-gravitational anomaly, i.e.~either
$\text{tr}R^2\text{tr}F$ or $\text{tr}R\text{tr}F^2$.
However, these are identically zero by $\text{tr}F=0$ due to  the
traceless condition of the generators of the
non-Abelian gauge group in $G$ respectively the reality of the
representations of the 4D Lorentz group
implying $\text{tr}R=0$. The vanishing of $\Theta_{00}$ and $\Theta_{0i}$ can also be inferred directly in
the 3D effective theory. For example, for $\Theta_{00}$ one can invoke a
similar argument to the one of section \ref{sec:ChiralityFormula3D},
namely that all Kaluza-Klein modes pairwise cancel because of the sign-
function in the loop-correction \eqref{eq:theta_LambdaSigma}.

Let us now come to the quantitative discussion and the derivation of the Chern-Simons
levels \eqref{eq:graviphotonCS}.
The Kaluza-Klein ansatz for the 4D metric on $\mathbb{R}^{(1,2)}\times S_r^1$ takes the form
\begin{equation} \label{eq:KKansatzg}
	g^{(4)}_{MN}(x^\mu,y)=\begin{pmatrix} g_{\mu\nu}+r^2A^0_\mu A^0_\nu & -r^2A^0_\mu\\
						-r^2A^0_\nu &  r^2
	\end{pmatrix}
\end{equation}
where $y$ denotes an angular coordinate of period $2\pi$, that is related to the coordinate on
the circle of radius $r$  by $ry$ and $M,N$ denote 4D Minkowski indices for the following discussion
to avoid confusion. We denote the 3D Minkowski coordinates by $x^\mu$ and introduced the 3D metric
$g_{\mu\nu}$ and the graviphoton $A_\mu^0$. For simplicity we assume no dynamics of the radial mode
$r$, i.e.~$r$ is a constant. We note that the reduction ansatz
\eqref{eq:KKansatzg} implies the vielbein
\begin{equation}
	e^a=\tilde{e}^a_\mu dx^\mu\,,\quad e^3=r(dy-A^0_\mu dx^\mu)\,,
\end{equation}
where we indicated a split $e_M^A=(e_M^a,e_M^3)$ of the 4D vierbein
$e_M^A$, $A=0,\ldots, 3$ into the 3D dreibein
$e_\mu^a$, $a=0,1,2$, and a 3D one-form $e^3$. For completeness we
introduce the inverse metric $g_{(4)}^{MN}$
and the inverse vierbein $e_A^M=e_{A N}g_{(4)}^{NM}$ reading
\begin{equation} \label{eq:KKansatzg-1}
	g_{(4)}^{MN}=\begin{pmatrix} g^{\mu\nu} & (A^0)^{\mu}\\
						(A^0)^{\nu} &  \frac{1}{r^2}+A^0_\kappa (A^0)^{\kappa}
			     \end{pmatrix}\,,\qquad e_a=\tilde{e}^\nu_a\frac{\partial}{\partial x^\nu}
                             +\tilde{e}_a^\nu A^0_\nu \frac{\partial}{\partial y}\,,\quad e_3=\frac1r\frac{\partial}{\partial y}
\end{equation}
where we raise and lower the indices $A$ and $a$ by the flat metric.

Next we specify the reduction ansatz for the 4D gauge theory. We start
with vectors $A_\mu$ with KK-ansatz
\begin{equation} \label{eq:KKansatzA}
	A(x^\mu,y)=A_\mu dx^\mu+ \zeta r(dy- A^0_\mu dx^\mu )=A_\mu dx^\mu+ \zeta e^3\,,
\end{equation}
where $\zeta$ denotes the adjoint valued scalar in 3D.  Note that
\eqref{eq:KKansatzg} and \eqref{eq:KKansatzA} imply
that both $\zeta$ and $A^0$ have mass-dimension one by noting that $dy$
and the 3D metric have mass-dimension zero.
For 4D fermions $\Psi$ charged under the 4D $U(1)$-gauge symmetries, we specify their KK-ansatz as
\begin{equation} \label{eq:KKansatzPsi}
	\Psi(x,y)=\sum_n \Psi_n(x)e^{iy n}\,.
\end{equation}
The KK-tower of these states will be running in the loop and generate the
Chern-Simons term $\Theta_{0m}$ as we will show next. It is interesting to
note that while $T_\alpha$ transforms under the $U(1)$ as
dictated by the gauge covariant derivative \eqref{eq:gaugingT_A}
the fermionic partners will have an ordinary derivative
\cite{Wess:1992cp}, $D_M=\partial_M$.

Considering massless 4D fermions the only terms in the 4D effective theory for the fermions $\Psi$
that are relevant for our discussion are the kinetic term. In the UV and considering a single fermion for
simplicity this is the standard kinetic term
for a Weyl fermion reading
\begin{equation} \label{eq:kineticWeyl}
	\mathcal{L}_{\text{Weyl}}^{(4)}=-\frac{i}{\kappa_4^2}\bar{\Psi}\sigma^A e_A^M D_M \Psi\,,
\end{equation}
where $D_M=\partial_M+iA_M$ is the 4D covariant derivative, $\sigma^A=(-\mathbf{1},\sigma^i)$  and $\kappa_4^2$ the 4D Planck mass. We use \eqref{eq:KKansatzg-1}, \eqref{eq:KKansatzA} and \eqref{eq:KKansatzPsi} to reduce
this to three dimensions. After integration over $y$  we obtain
\begin{equation} \label{eq:3DKKTheory}
	\mathcal{L}_{\text{KK}}^{(3)}=
	 \sum_{n=-\infty}^\infty \left[-i\bar{\Psi}_n\sigma^a \tilde{e}^\mu_a \mathcal{D}_\mu\Psi_n +\bar{\Psi}_n\sigma^3
	 (\frac{n}{r}+q\cdot\zeta)\Psi_n\right]\,,
\end{equation}
where we used the shorthand notation \eqref{eq:qdotxi}.
Furthermore, we have assumed in addition that we are readily on the 3D Coulomb branch by switching on VEVs for
the scalars $\zeta$ along the Cartan directions of $G$ breaking it into its maximal torus $U(1)^{\text{rank}(G)}$.
The $U(1)$-charges under the remaining massless gauge fields on the Coulomb branch $A^\Lambda$ are
denoted by  $q_\Lambda$. The covariant derivative for the KK-fermions is given by
\begin{equation}
	  \mathcal{D}_\mu\Psi_n=(\partial_\mu+iq_\Lambda A^\Lambda_\mu +inA_\mu^0)\Psi_n\,.
\end{equation}
From this we read off the 3D mass and the 3D charges\footnote{The 3D theory \eqref{eq:3DKKTheory}
without a dynamical dilaton in \eqref{eq:KKansatzA} is automatically in the 3D Einstein frame since the factor $r$ is
absorbed into the definition of the 3D Newtons constant multiplying the entire 3D Lagrangian. We have set it to one
for convenience.}
of the KK-fermions that are discretely
labeled by the integral momentum number $n$ along the $S^1$ as
\begin{equation} \label{eq:KKcharge+mass}
	q_0=n\,,\qquad m_n^f=\frac{n}{r}+q^f\cdot \zeta\,,
\end{equation}
where we label by $f$ all the $U(1)$-charges in the case that we have more than one fermion.
We see that the masses of the KK-states are offset from zero by
the mass $m_0^f=\zeta\cdot q^f$ of the zero mode $\Psi_0$ on the Coulomb
branch, see \eqref{eq:qdotxi}.
The expressions \eqref{eq:KKcharge+mass} straightforwardly generalize for
a more complicated
constant background metric in the kinetic term \eqref{eq:kineticWeyl} for
several fermions. However, the diagram we want to compute has two
vertices and two fermion propagators and the metric and normalizations
drop out.

To determine the loop-induced Chern-Simons level $\Theta_{0m}$, we have to
calculate the loop-diagram depicted in figure \ref{fig:loopForTheta0m}. As
we have argued above heuristically, we have to integrate out all massive
fermions coupling to $A^0$ and $A^m$. Indeed, we see from
\eqref{eq:3DKKTheory} that the whole KK-tower of fermions with charge
and mass \eqref{eq:KKcharge+mass} couples and runs in the loop. The vertex
rules are as anticipated in figure \ref{fig:loopForTheta0m}. The idea is
now to apply at each KK-mass-level $m^f_n$ the general formula
\eqref{eq:theta_LambdaSigma} with the replacement of
$\text{sign}(\zeta\cdot q^f)$ by $ \text{sign}(m^f_n)$.
Then we use $q_0=n$ to obtain the one-loop correction to the Chern-Simons
level as
\begin{equation} \label{eq:Theta0m}
	\Theta^{\text{loop}}_{0m}=\frac12\sum_f\sum_{n=-\infty}^\infty  n q^f_m\, \text{sign}\Big(\zeta\cdot q^f+\frac{n}{r}\Big)\,.
\end{equation}

This sum is in general divergent and can be regulated using zeta function regularization.
We recall the definition of the Riemann zeta function by the Dirichlet series
\begin{equation} \label{eq:Riemannzeta}
	\zeta(s)=\sum_{k=0}^\infty k^{-s}\,,\qquad \text{for }\ \ \text{Re}(s)> 1\,,
\end{equation}
In order to evaluate the infinite sum \eqref{eq:Theta0m} we need the following identity that holds for $\vert x\vert<1$,
\begin{equation} \label{eq:basicIdentity}
	\sum_n n \,\text{sign}\Big(x+n\Big)=
	\sum_n n \,\text{sign}(n)
	=2\sum_{n=1}^\infty n=2\zeta(-1)=-\frac{1}{6}\,.
\end{equation}
We note that the first equality holds since $x$ lies in the interval
$[0,1)$ and this allows us to replace $\text{sign}(n+x)=\text{sign}(n)$.
Then in the second equality we split the sum into positive and negative
$n$, that yields due to the sign$(n)$ the same sum. Finally we used the
well-known result $\zeta(-1)=-\frac{1}{12}$ from analytic
continuation of the Riemann zeta function \eqref{eq:Riemannzeta} for
$\text{Re}(s)<1$.

Now we are prepared to perform the sum over $n$ in \eqref{eq:Theta0m}. We
focus on each summand in the sum over fermions $f$ independently.
Furthermore, we identify $x\equiv r m_0^f=r \zeta\cdot q^f$
in \eqref{eq:basicIdentity}.  We consider the effective theory in the IR
at an energy scale, that is sufficiently smaller than
the mass-scale $m_{\text{KK}}$ of KK-states and the mass scale $m_0^f$ on
the Coulomb branch. However, in order to
apply the field theory analysis of
\cite{Niemi:1983rq,Redlich:1983dv,Aharony:1997bx} and the result
\eqref{eq:theta_LambdaSigma} the masses $m^f_0$  of the massive fermions
have to be smaller than the KK mass scale $m_{\text{KK}}$
as in \eqref{mass_hierarchy}.
Since the mass scale $m_0^f$ depends on the position $\zeta^\Lambda$ on
the 3D Coulomb branch, it can be made parametrically small. In this case
we can trust the loop result \eqref{eq:theta_LambdaSigma}.\footnote{The
effective field theory description used in \cite{Aharony:1997bx}
breaks down for $x\geq 1$ since interactions with KK-states become as
relevant as interactions with the $\Psi_0$.
To find a relation similar to \eqref{eq:basicIdentity} when $x>1$
one uses the floor function $\lfloor x \rfloor$ that rounds down a given
real number $x$. Then one obtains  $\sum_n n \,\text{sign}\Big(x+n\Big)=
\zeta(-1,-\lfloor x\rfloor)+\zeta(-1,\lfloor x\rfloor)
-\lfloor x\rfloor=-\frac16-\lfloor x\rfloor(\lfloor x\rfloor+1)$,
where the Hurwitz zeta function $\zeta(s,q)$ has been evaluated. One sees
that the Chern-Simons terms incrementally jump by $2$ for each fermion
with mass $m_0^f$ crossing the threshold $m_{\text{KK}}$.}
We then have $x\equiv r m_0^f<1$ and can apply \eqref{eq:basicIdentity} to
readily obtain the Chern-Simons levels as
\begin{eqnarray} \label{eq:Theta0mfinal}
	\Theta^{\text{loop}}_{0m}=-\frac1{12}\sum_fq_m^f=-\frac1{12}\sum_{\underline{q}} n(\underline{q})q_m\,.
\end{eqnarray}
Here we replaced the sum over individual fermions $f$ by a sum over
charges $q$ where $n(\underline{q})$ denotes their multiplicities. We note that \eqref{eq:Theta0mfinal} is an odd function in the charges and
as a result only sensitive to the chiral index 4D $\chi(\underline{q})=n(\underline{q})-n(-\underline{q})$
since 4D vector-like pairs of fermions cancel out.

We see that \eqref{eq:Theta0mfinal} is precisely the mixed Abelian-gravitational anomaly on the left hand side
in \eqref{eq:Anomalies:Abgravitational}. This is what one expects since, by reversing the logic
from the beginning of this section, in the limit $\zeta\rightarrow 0$ we have $x=0$ and sending $r\rightarrow\infty$
all KK-states become massless and we have to recover the 4D anomaly result.

\section{Anomaly Cancellation In F-Theory} \label{sec:anomalycancellationinF}

In this section we will discuss anomaly cancellation in F-theory. This
involves relating, via the general formulas
\eqref{eq:Anomalies:purenonAb}--\eqref{eq:Anomalies:Abgravitational} for
anomaly cancellation in section \ref{sec:anomlies+cancellation}, the
Green-Schwarz counter terms discussed in section \ref{sec:GStermsInF} to
the 4D chiralities obtained from 3D Chern-Simons terms following section
\ref{sec:ChiralityInF}. This will connect seemingly physically different
object, the geometric intersections on $\hat{X}_4$ and flux integrals,
with each other. The main challenge of this section will be to understand
these relations directly from analyzing the geometrical structure of the
resolution fourfold $\hat{X}_4$ on the one hand, and 3D Chern-Simons terms
on the other hand.
Anomaly cancellation in 6D F-theory compactifications has been under
intense investigation as reviewed in \cite{Taylor:2011wt}.

\subsection{The geometric structure of F-theory anomaly cancellation}

We begin with an outline of the general geometric relations
imposed on $\hat{X}_4$ by anomaly cancellation in F-theory. As we
will argue in this section these relate intersections
of resolution divisors and holomorphic curves over the matter curves
$\Sigma_{\mathbf{R}}$ in the base $B_3$ on the one side to certain flux
integrals on $\hat{X}_4$ on the other side. These geometric relations that
we will discover by imposing 4D anomaly cancellation will be very similar
to those found in 6D F-theory on Calabi-Yau threefolds \cite{Park:2011ji}.
The crucial point in the analysis in 4D will
be the necessity of the inclusion of $G_4$-flux on both sides of the
discovered relations.

The 4D anomaly cancellation conditions
\eqref{eq:Anomalies:purenonAb}--\eqref{eq:Anomalies:Abgravitational} can
be directly translated into the geometry of the resolved fourfold
$\hat{X}_4$. We denote the holomorphic curves in $\hat{X}_4$, that resolve
singular elliptic fibers over codimension two\footnote{These
curves are characterized by the fact that they are isolated over
the codimension two curves $\Sigma_{\mathbf{R}}$ in $B_3$ with
moduli space given by  $\Sigma_{\mathbf{R}}$. Curves
corresponding to Yukawa couplings are isolated at points, i.e.~do
exhibit the moduli space of a point.} in the base $B_3$
and that thus lie in the weight lattice of $G$, by $c$. Then,
the 4D charge of a matter particle obtained from a wrapped
M2-brane on $c$ under $A^\Lambda$ is given by
\begin{equation} \label{eq:M2charge}
	q_\Lambda=\int_{c}\omega_\Lambda\,.
\end{equation}
This can be seen from reducing the electric coupling of the
M2-brane to $C_3$ along $c$. Then the anomaly constraints relate
sums over these holomorphic curves in the weight lattice of $G$ to
certain flux integrals of $G_4$. It is important to emphasize that the
following geometric relations only hold if the $G_4$-flux obeys all the
conditions \eqref{eq:vanishingThetas} and not for generic $G_4$-flux in
M-theory. The anomaly cancellation conditions
\eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:AbnonAb}, using
\eqref{eq:M2charge} to express the charges, then translate into
\begin{eqnarray} \label{eq:anomalyCondCohomology_1}
    \frac13\sum_{S_{\mathbf{R}}}\sum_{c\,\subset S_\mathbf{R}}\int_{S_\mathbf{R}}G_4\int_{c}\omega_{(\Lambda}\int_{c}\omega_\Sigma
    \int_{c}\omega_{\Gamma)}&=&\frac{1}{2}\int_{\hat{X}_4}G_4\wedge
    \omega_\alpha\wedge\omega_{(\Gamma} \eta^{-1\,\alpha}_\beta\int_{S^\beta}
    \omega_\Lambda\wedge\omega_{\Sigma)}       \nonumber \\
    &=&\frac{1}{2}\int_{\hat{X}_4}G_4\wedge
    \pi_*(\omega_{(\Lambda}\wedge\omega_\Sigma)\wedge\omega_{\Gamma)}
    \,,
\end{eqnarray}
where we indicate a symmetrization of indices $\Lambda$, $\Sigma$ and
$\Gamma$ by $(\cdot)$. The cancellation condition
\eqref{eq:Anomalies:Abgravitational} of the Abelian-gravitational anomaly
yields the geometric relation
\begin{eqnarray} \label{eq:anomalyCondCohomology_2}
    \frac1{3}\sum_{S_{\mathbf{R}}}\sum_{c\,\subset S_\mathbf{R}}\int_{S_\mathbf{R}}G_4\int_{c}\omega_\Lambda&=&\int_{\hat{X}_4}G_4\wedge
    \omega_\alpha\wedge\omega_\Lambda \eta^{-1\,\alpha}_\beta\int_{S_{\rm b}^\beta}
    c_1(B_3)\nonumber\\
    &=&\int_{\hat{X}_4}G_4\wedge
    c_1(B_3)\wedge\omega_\Lambda\,.
\end{eqnarray}
In both equations we split the sum over curves $c$ into a sum over matter
surfaces $S_\mathbf{R}$ and a sum over curves $c$ lying inside a
particular $S_{\mathbf{R}}$. In addition we used the topological metric
$\eta_{\alpha}^{\ \beta}$ respectively its inverse
$\eta_{\alpha}^{-1\,\beta}$ on $B_3$ as introduced in \eqref{eq:intmatrix}
and from the first to the second line in each equation the push-forward
$\pi_*$ to $H^{(1,1)}(B_3)$ defined in \eqref{eq:push-forward}.

We can use Poincar\'e duality to
rewrite the relation \eqref{eq:anomalyCondCohomology_1} and
\eqref{eq:anomalyCondCohomology_2} more compactly in homology
as intersections of divisors and holomorphic curves $c$ in the weight
lattice of $G$
\begin{eqnarray} \label{eq:anomalyCondHomology}
	\frac{1}{3}\sum_{S_{\mathbf{R}}}\sum_{c\,\subset S_\mathbf{R}}(S_\mathbf{R}\cdot [G_4])(c\cdot D_{(\Lambda})(c\cdot D_\Sigma)
	(c\cdot D_{\Gamma)})&=&\frac{1}{2}[G_4]\cdot D_{(\Gamma}\cdot
    \pi_*(D_\Lambda\cdot D_{\Sigma)})\ ,\\
    \frac1{3}\sum_{S_{\mathbf{R}}}\sum_{c\,\subset S_\mathbf{R}}(S_\mathbf{R}\cdot [G_4])(c\cdot D_\Lambda)&=&[G_4]\cdot
    [c_1(B_3)]\cdot D_\Lambda\, , \nonumber
\end{eqnarray}
where we again symmetrized the indices $\Lambda$, $\Sigma$ and $\Gamma$.

Both \eqref{eq:anomalyCondCohomology_1} and the first equation contains of
\eqref{eq:anomalyCondHomology} capture all three different
types of gauge anomalies, i.e.~the purely Abelian as well as non-Abelian
anomalies and the mixed Abelian-non-Abelian anomalies. This can be
seen by choosing $\Gamma, \Sigma, \Lambda$ as the indices of the
Cartan divisors $D_{i_I}$ of $G_{(I)}$ and noting that the right
hand side vanishes because of $\Theta_{i\alpha}=0$ in
\eqref{eq:vanishingThetas} yielding the purely non-Abelian anomaly
condition \eqref{eq:Anomalies:purenonAb}. The mixed
Abelian-non-Abelian anomaly \eqref{eq:Anomalies:AbnonAb} is obtained
by choosing  $\Gamma, \Sigma, \Lambda=i,j,m$. In order to see a matching
of the Lie algebra structure we expand following \cite{Park:2011ji} the
Cartan generators in the GS-terms \eqref{eq:GSterm} involving
$\text{tr}(F^I\wedge F^I)$ in a coroot basis $\mathcal{T}_{i_I}$.
Since the anomalies have to hold for all Cartan generators and exploiting
$\frac{1}{\lambda_I}\text{tr}(\mathcal{T}_{i_I}\mathcal{T}_{j_I})
=\mathcal{C}^{(I)}_{i_Ij_I}$ in the GS-terms we obtain a match with the
right hand side of \eqref{eq:anomalyCondCohomology_1},
\eqref{eq:anomalyCondHomology}, where we employ \eqref{eq:Cartan} .
The purely Abelian anomaly \eqref{eq:Anomalies:pureAb}
is manifest for the choice  $\Gamma, \Sigma, \Lambda=m,n,k$. The second
equation summarizes all mixed Abelian-gravitational anomalies
\eqref{eq:Anomalies:Abgravitational} as is evident by choosing
$\Lambda=m$.

As we have noted already in the introduction the
conditions \eqref{eq:anomalyCondHomology} might generally hold also for
$G_4$-flux that are not wedges of two two-forms as in
\eqref{eq:verticalFluxGeneral}. These more general $G_4$-flux can
correspond to seven-brane fluxes trivial in the ambient space. For these
fluxes the right-hand side of  \eqref{eq:anomalyCondHomology} induced by
the gaugings vanishes and the left side might pose a non-trivial
constraint on the spectrum.

\subsection{The structure of F-theory fluxes and Green-Schwarz terms}
\label{sec:strucOfFlux+anomalies}

Next we discuss the general structure of the  $G_4$-flux
obeying the F-theory constraints \eqref{eq:vanishingThetas}. A very
general, presumably the most general, vertical $G_4$-flux on $\hat{X}_4$
that can meet all conditions listed in section \ref{sec:flux_review} takes
the form
\begin{equation} \label{eq:generalG4}
	G_4= \widehat G_4+ F^{(m)}\wedge \omega_m\,.
\end{equation}
Here the first term $\widehat G_4$ denotes a non-Abelian flux
related to the non-Abelian gauge groups $G_{(I)}$. It only involves the
two-forms $\omega_i, \omega_\alpha, \omega_0$ introduced in
\eqref{eq:basisH^11}.
The second part are Abelian fluxes that are of the form of wedge products
of a two-form $F^{(m)}$ in $B_3$ with each $\omega_m$ dual
to the $U(1)$-Cartan divisors $D_m$ in \eqref{eq:basisH^11}. These fluxes
can be thought of as the lift of internal world-volume fluxes $F^{(m)}$ on
each $U(1)$ seven-brane to $G_4$-flux.
The fluxes $F^{(m)}$ being $(1,1)$-forms on
$B_3$ can be expanded, after introducing in general arbitrary flux numbers
$f^{(m)\alpha}$, as
\begin{equation}
	F^{(m)}=f^{(m)\alpha}\, \omega_\alpha\,.
\end{equation}

The validity of the generic splitting \eqref{eq:generalG4} of $G_4$ into
Abelian and non-Abelian fluxes can be motivated by moving in the
complex structure moduli space of the Calabi-Yau fourfold. For this
argument let us begin with a fourfold $\hat {Y}_4$ yielding a purely
non-Abelian gauge group $G_{(I)}$ without any $U(1)$-factors. Then only
the first flux in \eqref{eq:generalG4} is present, of which we assume that
it obeys all consistency conditions \eqref{eq:vanishingThetas} on
$\hat {Y}_4$. Upon specializing the complex structure
of $\hat {Y}_4$ we can unhiggs a number of $U(1)$ symmetries by forming
and resolving a singularities in $\hat {Y}_4$. This leads to a new
fourfold $\hat{X}_4$ with extra divisors $D_m$. Then we can just pull-back
the non-Abelian flux $\widehat G_4$ from $\hat {Y}_4$ to $\hat{X}_4$ and
obtain, by virtue of the second and third equation in \eqref{eq:intsD_m},
a valid flux that we again denoted by $\widehat G_4$. This pull-back flux
is still the most general non-Abelian flux $\widehat G_4$ we can construct
on $\hat{X}_4$, since $\widehat G_4$ by definition involves only the
$\omega_i$, $\omega_\alpha$ and $\omega_0$ and since in the transition
from $\hat{Y}_4 $ to $\hat{X}_4$ no any new Cartan divisors $D_i$ nor new
base divisors $D_\alpha$ were induced.
Thus, new fluxes on $\hat{X}_4$ that were not available  on $\hat{Y}_4$
are of the form $F^{(m)}\wedge \omega_m$\footnote{Although physically more
elusive and not present in examples considered below, fluxes of the form
$\omega_i\wedge\omega_m$ and $\omega_m\wedge\omega_n$ for more than one
$U(1)$ are mathematically not excluded in general.}, where $F^{(m)}$ is
a general two-form on $B_3$ as we will see below.

\subsubsection{Non-Abelian fluxes only}

We start by working out the conditions imposed by \eqref{eq:vanishingThetas} on the non-Abelian flux
$\widehat G_4$ in \eqref{eq:generalG4} as well as the gaugings induced by it. We expand $\widehat G_4$ that in a basis of
cohomology of $H^{4}_V(\hat{X}_4)$ as
\begin{equation} \label{eq:G4expansion}
	\widehat G_4 =N^{ij}\, \omega_i\wedge\omega_j+f^{i\alpha}\, \omega_i\wedge\omega_\alpha
	+\tilde{N}_{\alpha}\, \tilde{\omega}^\alpha+N^{\alpha}\, \omega_0\wedge\omega_\alpha\,,
\end{equation}
where the $\tilde{\omega}^\alpha$ as before denote pull-backs from
$H^{4}(B_3)$.
The coefficients $f^{i\alpha}$ have the interpretation
of a seven-brane two-form flux $F_2^i$ in the direction of the i-th Cartan
generator of $G$. The fluxes $N^{ij}$ are non-Abelian in nature as they
are associated to products of two Cartan divisors.

With the expansion \eqref{eq:G4expansion} of $\widehat{G}_4$ we readily
evaluate the constraints \eqref{eq:vanishingThetas}.
Employing the conditions \eqref{eq:vanishingInts_0},
\eqref{eq:vanishingInts_1}, \eqref{eq:Cartan},
\eqref{eq:intsD_m}  and \eqref{eq:newIntsTildeB} we obtain after some
algebra the following,
\begin{eqnarray} \label{eq:ThetaABcalculated}
	\Theta_{00}&=&K^\alpha K^\beta\Theta_{\alpha\beta}\,,\qquad\qquad\qquad\qquad\quad\ \ \,\Theta_{0\alpha}
	=\frac12(\tilde{N}_\alpha+N^\gamma K^{\beta}\mathcal{K}_{\alpha\beta\gamma}- K^\beta
	\Theta_{\alpha\beta})\,,\nonumber\\
	\Theta_{0\Lambda}&=&-\frac12 K^\alpha \Theta_{\alpha \Lambda}\,,\qquad\qquad\qquad\qquad\quad\ \
	\Theta_{\alpha\beta}=\frac12(N^\gamma-N^{ij}C_{ij}^\gamma)\mathcal{K}_{\alpha\beta\gamma} 	\,,\nonumber\\
	\Theta_{\alpha i}&=&\frac12(N^{j_1 j_2}\mathcal{K}_{ij_1j_2\alpha}-
	f^{j\beta}C_{ij}^\gamma \mathcal{K}_{\alpha\beta\gamma})\,,\qquad \quad
	\Theta_{\alpha m}=\frac12 N^{ij}\mathcal{K}_{ijm\alpha}\,,\nonumber\\
	\Theta_{ij}&=&\frac12(N^{i_1i_2}\mathcal{K}_{i_1i_2ij}+f^{i_1\alpha}\mathcal{K}_{i_1ij\alpha}
	-\tilde{N}_\alpha C_{ij}^\alpha)\,,\,\,\,
	\Theta_{im}=\frac12(N^{i_1i_2}\mathcal{K}_{i_1i_2im}+f^{j\alpha}\mathcal{K}_{ijm\alpha})\,,\nonumber\\
	\Theta_{mn}&=&\frac12(N^{ij}\mathcal{K}_{ijmn}+f^{i\alpha}\mathcal{K}_{imn\alpha}-\tilde{N}_\alpha b_{mn}^\alpha) \,.
\end{eqnarray}
Here we used the definitions \eqref{eq:fourfoldintersect},
\eqref{eq:matchC-b}, \eqref{eq:match_b_mn} and
$\mathcal{K}_{\alpha\beta\gamma}$ denotes the triple intersections on
$B_3$ introduced in \eqref{eq:tripleintersect} and the $K^\alpha$
were defined by the expansion $c_1(B_3)=-K^\alpha\omega_\alpha$. We also
introduced the shorthand notation $C^\gamma_{ij}=C_{ij}^{(I)}b^\gamma_I$.

Indeed, we can solve the constraints \eqref{eq:vanishingThetas} explicitly
using the expressions \eqref{eq:ThetaABcalculated}.
We obtain three sets of conditions from
$\Theta_{\alpha\beta}=\Theta_{0\alpha}=\Theta_{\alpha i}=0$
that we solve to determine the flux numbers $N^\alpha$, $\tilde{N}_\alpha$
and $f^{i \alpha}$ in terms of the $N^{ij}$,
\begin{eqnarray} \label{eq:vanishingThetasExplicit}
	&N^{\alpha}=N^{ij}C_{ij}^\alpha\,,\qquad\tilde{N}_{\gamma}
	= -\frac12 (N^\gamma+N^{ij}C_{ij}^\gamma)K^\beta\mathcal{K}_{\alpha\beta\gamma}
	=-N^{ij}C_{ij}^\gamma K^\beta\mathcal{K}_{\alpha\beta\gamma}\,,&\nonumber\\
	&f^{i\alpha}=N^{i_1i_2}\mathcal{K}_{i_1i_2j\alpha}(C^{-1}_{(I)})^{j i}\mathcal{K}_{S_{(I)}^{\rm b}}^{\beta\alpha}\,.&
\end{eqnarray}
Here we have introduced the inverse $\mathcal{K}_{S_{(I)}^{\rm b}}^{\beta\alpha}$ of the intersection form on the
seven-brane divisor $S^{A}_{\rm b}$ as well as the inverse of the
corresponding Cartan matrix $C^{-1}_{(I)}$.

It is satisfying to see that $\Theta_{0m}=-\frac12 K^\alpha \Theta_{\alpha m}$ is implied by \eqref{eq:ThetaABcalculated}. Thus, $\Theta_{0m}$ is
precisely the coefficient of the
Green-Schwarz counter term relevant for the cancellation of mixed Abelian-
gravitational anomalies in \eqref{eq:Anomalies:Abgravitational}
respectively \eqref{eq:anomalyCondCohomology_2}.
This is precisely what we expect from the discussion of section
\ref{sec:Theta0m} where we related
the Chern-Simons level $\Theta_{0m}^{\text{loop}}$ induced by loops of KK-
fermions to precisely the mixed anomalies.
As we argued there further, the matching of the 3D Chern-Simons levels
$\Theta_{0m}\equiv \Theta_{0m}^{\text{loop}}$
by M-/F-theory duality \eqref{eq:3dDuality} is a prove of the 4D anomaly
cancellation condition.

Before switching to the discussion of Abelian fluxes let us conclude by
mentioning that by virtue of section \ref{sec:ChiralityInF} the flux
integrals $\Theta_{\Sigma\Lambda}$ in the next to last and last line in
\eqref{eq:ThetaABcalculated} encode the 4D chiralities. By means of
\eqref{eq:vanishingThetasExplicit} the chiralities only depend on the
flux numbers $N^{ij}$. It would be interesting to explicitly express
these in terms of the $N^{ij}$.

\subsubsection{Inclusion of Abelian fluxes}

We begin by checking that Abelian fluxes of the form alluded to in \eqref{eq:generalG4} can be added
to a given non-Abelian flux $\widehat G_4$. For this we check that fluxes of the form
$F^{(m)} \wedge \omega_m $ with $F^{(m)}$ obtained from elements in $H^{(1,1)}(B_3)$
obey all constraints in \eqref{eq:vanishingThetas}.

As before we work out explicitly the flux integrals $\Theta_{AB}$.
Using again the intersections \eqref{eq:intsD_m} and
\eqref{eq:vanishingInts_0}, \eqref{eq:vanishingInts_1} as well as the
shifted form $\tilde{\omega}_0$ defined in \eqref{eq:omega0tilde} we
obtain \begin{equation} \label{eq:ThetaAB_abel_1}
	\Theta_{00}=\Theta_{0\alpha}=\Theta_{0i}=\Theta_{\alpha\beta}=\Theta_{\alpha i}=0\,,\quad
\end{equation}
and
\begin{eqnarray} \label{eq:ThetaAB_abel_2}
        &\Theta_{0 m}=-\frac12 K^\alpha\Theta_{\alpha m}\ ,\qquad \quad \Theta_{\alpha m}=-\frac12 f^{(n)\beta} b_{nm}^\gamma\mathcal{K}_{\alpha\beta\gamma}\ ,&\nonumber\\
	& \Theta_{ij}=\frac12 f^{(m)\alpha}\, \mathcal{K}_{\alpha mij}\ ,\quad \Theta_{im}=\frac12 f^{(n) \alpha} \mathcal{K}_{\alpha mnj}\ ,\quad \Theta_{mn}=\frac12 f^{(p) \alpha} \mathcal{K}_{\alpha p mn}\ .&
\end{eqnarray}
We see that all conditions \eqref{eq:vanishingThetas} are automatically obeyed. We note that we again have
$\Theta_{0 m}=-\frac12 K^\alpha\Theta_{\alpha m}$ in agreement with the discussion in section \ref{sec:Theta0m},
as in the non-Abelian case discussed above. As expected the new fluxes contribute to the $\Theta_{\Sigma\Lambda}$
and thus will generate additional 4D chiralities.

\section{Anomalies and GS-terms in Type IIB Orientifolds} \label{sec:orientifold_anomalies}

In this section we
analyze the connection of F-theory anomaly cancellation to
anomaly cancellation in Type IIB orientifold setups at weak string coupling.
We first recall anomaly cancellation for compactifications with
D7-branes and O7-planes in subsection \ref{sec:anom_review} by
largely following \cite{Plauschinn:2008yd}. The discussion of anomalies
and their cancellation
in Type II orientifold setups has been studied significantly in
the past, as can be found in the reviews \cite{Cvetic:2001nr,MarchesanoBuznego:2003hp,Blumenhagen:2005mu,Blumenhagen:2006ci}.
This comparison to F-theory is performed in subsection \ref{sec:F-match},
by relating the gaugings and Green-Schwarz counter terms of both setups.
We discuss the complications arising due to additional `geometrically
massive' $U(1)$ gauge
symmetries that are present in orientifold setups but integrated out in
F-theory. A detailed comparison $G_4$-flux in M-theory dual to F-theory
compactifications with $SU(N)$ and $SU(N)\times U(1)$ and D7-brane fluxes
in the Type IIB orientifold limit has been addressed in
\cite{Krause:2012yh}.

\subsection{Anomaly cancellation in Type IIB orientifolds} \label{sec:anom_review}

We begin with a review of anomaly cancellation in
Type IIB orientifolds with intersecting D7-branes.
We consider Abelian or non-Abelian D7-branes on divisors $S^{\rm IIB}_{(\mathcal{A})}$
in the orientifold covering space $Y_3$ and denote the orientifold images of
these D7-brane divisors by $S^{\rm IIB}_{(\mathcal{A}')}$. The orientifold planes are
located at the fix-point set of the orientifold involution denoted by
$S^{\rm O7}$. The number of D7-branes on a stack are denoted by $N_{\mathcal{A}}$.
This includes cases where $N_\mathcal{A} =1$ yielding a D7-brane with a $U(1)$ gauge
group. In the following we will only focus on $U(N_{\mathcal{A}})$
groups for simplicity, such that the 4D gauge group is given by
\begin{equation} \label{G_ori}
  G_{\rm ori} = U(N_1) \times\ldots  \times  U(N_{{\widehat n_G}}) \ , \qquad
  {\mathcal{A}} =1,\dots, \widehat n_G\ .
\end{equation}
The cases $SO(N_{I})$ and $Sp(N_{I})$ should work out similarly and it would
be interesting to perform such an analysis using F-theory.

The Type IIB orientifold matter spectrum arises from strings stretching
between D7-branes. Four representations are present:
(1) the bi-fundamental representation
$\mathbf{F}_{\bar \mathcal{A} \mathcal{B}} = (\bar N_{\mathcal{A}},N_{\mathcal{B}})$ corresponding to strings
stretching from a D7-brane on $S_{(\mathcal{A})}^{\rm D7}$ to a D7-brane on
$S_{(\mathcal{B})}^{\rm D7}$ with $\mathcal{A} \neq \mathcal{B}$;
(2) the bi-fundamental representation $\mathbf{F}_{\mathcal{A} \mathcal{B} } =
(N_{\mathcal{A}},N_{\mathcal{B}})$ for strings stretching between $S_{(\mathcal{A}')}^{\rm D7}$ and
$S_{(\mathcal{B})}^{\rm D7}$ with $\mathcal{A} \neq \mathcal{B}$;
(3) the symmetric representation $\mathbf{S}_\mathcal{A}$, (4) the anti-symmetric
representation $\mathbf{A}_\mathcal{A}$.
The latter two arise from strings stretching between the brane and its
orientifold image. For evaluating the anomaly cancellation conditions it will
be crucial to give the $U(1)$-charges $q_{\mathcal{A}}(\mathbf{R})$
under the $\mathcal{A}$th $U(1)$ in \eqref{G_ori} of the various representations.
Together with the dimension of the representations they are summarized in
Table \ref{table:orientifold_sprectrum}.

The chiral indices in the orientifold setup are denoted by $\chi(\mathbf{R})$.
For the intersecting D7-brane setup one has the relations
\begin{equation} \label{eq:index_relations}
 \chi(\mathbf{F}_{\mathcal{A} \mathcal{B}}) = \chi(\mathbf{F}_{\mathcal{B} \mathcal{A}}) \ ,     \quad
 \chi(\mathbf{F}_{\bar \mathcal{A} \mathcal{B}})= - \chi(\mathbf{F}_{\bar \mathcal{B} \mathcal{A}})\ , \quad
  \chi(\mathbf{F}_{\bar \mathcal{A} \mathcal{A}}) =0\,,\quad \chi(\mathbf{F}_{\mathcal{A}\cA})=\chi(\mathbf{S}_\mathcal{A})+\chi(\mathbf{A}_\mathcal{A})\, .
\end{equation}
The first condition is trivially true since exchanging brane and image, which
is a symmetry of the theory, exchanges the fundamental representations.
The second condition uses the
fact that the bi-fundamental $(\bar{N}_{\mathcal{A}},N_{\mathcal{B}})$ is the
complex conjugate of $(N_{\mathcal{A}},\bar{N}_{\mathcal{B}})$ and the chiral index changes
sign under the exchange of a representation and its complex conjugate. The
third equality is true since since the representation
$\mathbf{F}_{\bar{\mathcal{A}}\mathcal{A}}$ is real and the last equality makes use of the
group theoretical decomposition $N_\mathcal{A}\otimes N_\mathcal{A}=\mathbf{S}_\mathcal{A}\oplus \mathbf{A}_\mathcal{A}$. To simplify our notation we will introduce the
abbreviations
\begin{equation}
  \chi_{\mathcal{A} \mathcal{B}} = \chi(\mathbf{F}_{\mathcal{A} \mathcal{B}})\ ,\qquad
  \chi_{\bar \mathcal{A} \mathcal{B}} = \chi(\mathbf{F}_{\bar \mathcal{A} \mathcal{B}})\ ,\qquad
  \chi_{(\mathcal{A})} = \chi(\mathbf{S}_\mathcal{A})\ ,\qquad
  \chi_{[ \mathcal{A}]} = \chi(\mathbf{A}_\mathcal{A})\ .
\end{equation}
The chiralities  in Type IIB are calculated explicitly from
a simple index depending on the D7-brane flux.
In the following we will focus on D7-branes with $U(1)$-fluxes $\mathcal{F}^{\mathcal{A}}$
only, which are a non-trivial background on $S^{\rm IIB}_\mathcal{A}$ in the $U(1)$-directions
of \eqref{G_ori}. The indices then read
\begin{equation}  \label{eq:I_abTypeIIB}
	I_{\mathcal{A}\mathcal{B}}=\int_{S^{\rm IIB}_{(\mathcal{A})}\cdot S^{\rm IIB}_{(\mathcal{B})}} (\mathcal{F}^{\mathcal{A}}-\mathcal{F}^{\mathcal{B}})\,
\end{equation}
for two D7-brane divisors $S^{\rm IIB}_{(\mathcal{A})},$ $S^{\rm IIB}_{(\mathcal{B})}$ with
their intersection denoted by $S^{\rm IIB}_{(\mathcal{A})}\cdot S^{\rm IIB}_{(\mathcal{B})}$.
For convenience we have set the string length $\ell_s=1$.
In terms of these
indices the chiralities are given by
\begin{eqnarray} \label{eq:chiTypeIIB}
	\chi_{\bar\mathcal{A} \mathcal{B}} & =& I_{\mathcal{A}\mathcal{B}}\,,\qquad \chi_{\mathcal{A}\mathcal{B}}=I_{\mathcal{A}'\mathcal{B}}\,,\\
	\chi_{(\mathcal{A})} &=&\tfrac{1}{2}(I_{\mathcal{A}'\mathcal{A}}-2 I_{O7 \mathcal{A}})\,,\quad \chi_{[\mathcal{A}]}=\tfrac{1}{2}(I_{\mathcal{A}'\mathcal{A}}+2 I_{O7 \mathcal{A}})\, . \nonumber
\end{eqnarray}
Here $\mathcal{A}'$ denotes the orientifold image of a divisor $S^{\rm IIB}_{(\mathcal{A})}$
and $I_{O7 \mathcal{A}}$ is obtained from \eqref{eq:I_abTypeIIB}  by
using the orientifold divisor $S_{\rm O7}$ and noting that the are
no fluxes on an O7-plane.
We note that from \eqref{eq:chiTypeIIB} and the symmetry properties of \eqref{eq:I_abTypeIIB}
we immediately obtain the relations \eqref{eq:index_relations}, where we here and
in the following have to make use of
\begin{equation} \label{eq:orientifoldAction}
	\mathcal{F}^{\mathcal{A}'} \wedge [S^{\rm IIB}_{\mathcal{A}'}]=- \sigma^* (\mathcal{F}^\mathcal{A}\wedge [S^{\rm IIB}_{\mathcal{A}}])\,, \qquad
	A^\mathcal{A}_{U(1)} = - \sigma^* A^{\mathcal{A}'}_{U(1)}\,.
\end{equation}
The map $\sigma$ denotes the orientifold involution and $A^{\mathcal{A}}_{U(1)}$
are the $U(1)$-gauge fields in the setup.
\begin{table}[h!]
\centering
 \begin{tabular}{|c |c |c |c|c|c|c|}
 \hline
 \rule[-.2cm]{0cm}{0.7cm} $\mathbf{R}$ & rep.  & $U(1)_\mathcal{C}$-charge  &   $ \text{dim}(\mathbf{R})$ &   $\quad V(\mathbf{R}) \quad $  &   $U(\mathbf{R})  $  &   $ \chi(\mathbf{R})$    \\ \hline \hline
 \rule[-.2cm]{0cm}{0.7cm}   $\mathbf{F}_{\bar \mathcal{A} \mathcal{B}} $  &  $(\bar N_\mathcal{A}, N_\mathcal{B})_{\underline{q}}$ & $(\underline{q})_\mathcal{C}=-\delta_{\mathcal{C} \mathcal{A}}  + \delta_{\mathcal{C} \mathcal{B}}$  & $N_\mathcal{A} N_\mathcal{B}$  & $(-1,1)$ & $(1,1)$ &  $\chi_{\bar \mathcal{A} \mathcal{B}}$  \\
 \rule[-.2cm]{0cm}{0.7cm}   $\mathbf{F}_{\mathcal{A} \mathcal{B} }$ &  $(N_\mathcal{A}, N_\mathcal{B})_{\underline{q}}$ &  $(\underline{q})_\mathcal{C}=\delta_{\mathcal{C} \mathcal{A}}  + \delta_{\mathcal{C} \mathcal{B}}$  & $N_\mathcal{A} N_\mathcal{B}$ & $(1,1)$ & $(1,1)$ &  $\chi_{\mathcal{A} \mathcal{B}}$  \\
 \rule[-.2cm]{0cm}{0.7cm}   $\mathbf{S}_\mathcal{A}$ & $\text{Sym}(N_\mathcal{A})_{\underline{q}}$ & $(\underline{q})_\mathcal{C}=2 \delta_{\mathcal{C} \mathcal{A}}$ & $\tfrac12 N_\mathcal{A} (N_\mathcal{A} + 1)$ & $N_A+4$ & $N_A+2$ &  $\chi_{( \mathcal{A})}$ \\
 \rule[-.2cm]{0cm}{0.7cm}   $\mathbf{A}_\mathcal{A}$ & $\Lambda^2(N_\mathcal{A})_{\underline{q}}$ & $(\underline{q})_\mathcal{C}=2 \delta_{\mathcal{C} \mathcal{A}}$ &  $\tfrac12 N_\mathcal{A} (N_\mathcal{A} - 1)$& $N_A-4$ & $N_A-2$ &  $\chi_{[ \mathcal{A}]}$ \\
 \hline
\end{tabular}
\caption{A list of the matter representations arising from intersecting D7-
branes. The second column denotes the representations of $SU(N_\mathcal{A})\times
SU(N_\mathcal{B})\times U(1)^{\hat{n}_G}$ respectively of $SU(N_\mathcal{A})\times
U(1)^{\hat{n}_G}$ and the third column denotes the charges under the $\mathcal{C}$th
$U(1)$. Finally we summarize the dimension of $\mathbf{R}$ in the fourth, the
values of the cubic Casimir and the index in \eqref{eq:traceRelations} in the fifth respectively
sixth columns and the chiral indices in the last.}
\label{table:orientifold_sprectrum}
\end{table}

Next we evaluate the anomalies on the left-hand side of
\eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:Abgravitational}
for the 4D orientifold theory.
One evaluates for the weak coupling representations employing
the last relation in \eqref{eq:index_relations}
and Table \ref{table:orientifold_sprectrum} that
\begin{eqnarray} \label{eq:AnomaliesOrie}
   \text{tr}_\mathbf{f} (F^\mathcal{A})^3:&
        & A^{F^3}_\mathcal{A} = \sum_{\mathcal{C} \neq \mathcal{A}} N_\mathcal{C} \big(\chi_{\bar \mathcal{C} \mathcal{A}}
        +  \chi_{ \mathcal{C} \mathcal{A}}  \big)  + (N_\mathcal{A} + 4) \chi_{(\mathcal{A})} + (N_\mathcal{A}-4)
        \chi_{[ \mathcal{A}]} \,,   \nonumber \\[0cm]
   F^\mathcal{A}_{U(1)} (F^\mathcal{B}_{U(1)})^2 :&
        &\tfrac12 N_\mathcal{A} \big[ N_\mathcal{B}   \big(\chi_{\bar \mathcal{B} \mathcal{A}} +
        \chi_{ \mathcal{B} \mathcal{A}}\big) (1-\delta_{\mathcal{A}\mathcal{B}})\\&&
         +  \tfrac{1}{3}\delta_{\mathcal{A}\mathcal{B}} \big(\sum_{\mathcal{C} \neq \mathcal{A}} N_\mathcal{C} \big(\chi_{\bar \mathcal{C} \mathcal{A}}
        +  \chi_{ \mathcal{C} \mathcal{A}}  \big)  + 4(N_\mathcal{A} + 1) \chi_{(\mathcal{A})} +4 (N_\mathcal{A}-1)
        \chi_{[ \mathcal{A}]}    \big) \big]\nonumber\\
          &&=\tfrac12 N_\mathcal{A} \big[ N_\mathcal{B}   \big(\chi_{\bar \mathcal{B} \mathcal{A}} +
        \chi_{ \mathcal{B} \mathcal{A}}\big)    +  \tfrac13\delta_{\mathcal{A}\mathcal{B}} A^{F^3}_\mathcal{A} \big]\,,\nonumber \\[.2cm]
  F^\mathcal{A}_{U(1)}\text{tr}_\mathbf{f} (F^\mathcal{B})^2: &
        & \tfrac12\big[  N_\mathcal{A} \big(\chi_{\bar \mathcal{B} \mathcal{A}}
         +  \chi_{ \mathcal{B} \mathcal{A}} \big) (1-\delta_{\mathcal{A}\mathcal{B}})
         +  \delta_{\mathcal{A} \mathcal{B}} \big( \sum_{\mathcal{C} \neq \mathcal{A}} N_\mathcal{C} \big(\chi_{\bar \mathcal{C} \mathcal{A}}
        +  \chi_{ \mathcal{C} \mathcal{A}}  \big)  \\
        &&+ 2(N_\mathcal{A} + 2) \chi_{(\mathcal{A})} + 2(N_\mathcal{A}-2)
        \chi_{[ \mathcal{A}]} \big)\big]=\tfrac12 \big[N_\mathcal{A} \big(\chi_{\bar \mathcal{B} \mathcal{A}}
         +  \chi_{ \mathcal{B} \mathcal{A}} \big)+\delta_{\mathcal{A}\mathcal{B}} A^{F^3}_\mathcal{A}  \big]\,, \nonumber   \\[.2cm]
     F^\mathcal{A}_{U(1)} \text{tr}R^2:&
        &  \tfrac{1}{48} N_\mathcal{A} \Big[ A^{F^3}_\mathcal{A} - 3 \chi_{(\mathcal{A})} + 3 \chi_{[\mathcal{A}]}\Big]=\tfrac{1}{48} N_\mathcal{A} \Big[ A^{F^3}_\mathcal{A} +6I_{O7\mathcal{A}}\Big]\ .
\end{eqnarray}
Here $F^\mathcal{A}$ denotes the field strength of $SU(N_{\mathcal{A}})$
and $F^\mathcal{A}_{U(1)}$ are the $U(1)$ field strengths of the
Abelian factors appearing in \eqref{G_ori}.
Note that the contribution $\delta_{\mathcal{A} \mathcal{B}}$ in the second and third condition
arises from bi-fundamental matter from all intersections with D7-branes with the
$\mathcal{A}$-stack together with the symmetric and anti-symmetric representations
arising from the $\mathcal{A}$-stack itself. We have conveniently written the
result using $A^{F^3}_\mathcal{A}$.

These anomalies are canceled by a GS-mechanism in 4D when imposing
D7-brane and D5-brane tadpole cancellation. For general Type
IIB Calabi-Yau orientifold setups with D7-branes this has been shown
in \cite{Plauschinn:2008yd}. Since $ \text{tr}_\mathbf{f} (F^\mathcal{A})^3$
is non-factorizable one has
\begin{equation}
  \text{D7-/D5-tadpole cancellation}\qquad \Rightarrow\qquad   A^{F^3}_\mathcal{A}=0\ .
\end{equation}
The remaining conditions
then read
\begin{eqnarray} \label{eq:AnomaliesOrie_2}
   F^\mathcal{A}_{U(1)} (F^\mathcal{B}_{U(1)})^2&:\quad
        & \tfrac{1}{2} N_\mathcal{A}  N_\mathcal{B} \, \big[   I_{\mathcal{B} \mathcal{A}} +
        I_{ \mathcal{B}' \mathcal{A}} \big]\,, \\[.2cm]
  F^\mathcal{A}_{U(1)} \text{tr}_\mathbf{f} (F^\mathcal{B})^2&:\quad
        & \tfrac12 N_\mathcal{A} \, \big[I_{\mathcal{B} \mathcal{A}}
         +  I_{ \mathcal{B}' \mathcal{A}} \big]\,, \nonumber   \\[.2cm]
       F^\mathcal{A}_{U(1)} \text{tr}R^2&:\quad
        &  \tfrac{1}{8}   N_\mathcal{A}\, I_{O7 \mathcal{A}} \ ,
      \nonumber
\end{eqnarray}
where we have used the relations \eqref{eq:chiTypeIIB} and
\eqref{eq:orientifoldAction}. This concludes the evaluation of the
one-loop anomaly.

Next we will recall the derivation of the gaugings and
Green-Schwarz counter terms. There are two different types of axions
relevant in the Green-Schwarz mechanism
in Type IIB orientifolds. Firstly, there are orientifold-even axions
$\rho_\alpha$ arising
from the expansion of the R--R-form $C_4$ into a basis of $H^4_+(Y_3)$.
Secondly, there are orientifold-odd axions in the expansion of the
R--R-form $C_2$ into a basis of $H^2_-(Y_3)$. They appear in the
Green-Schwarz-terms of Type IIB as
\begin{equation} \label{eq:GStermIIB}
	S_{\rm GS}^{(4)}=-\frac{1}{4}\int (b^\alpha_\mathcal{A}\rho_\alpha + b_{a\,\mathcal{A}}c^a)\text{tr}(F^\mathcal{A}\wedge F^\mathcal{A})
	                       + (b^{\alpha}_{\mathcal{A}\mathcal{B}}\rho_\alpha
	                       + b_{a\,\mathcal{A}\mathcal{B}}c^a)F^\mathcal{A}_{ U(1)}\wedge F^\mathcal{B}_{U(1)}-\frac{1}{4}a^\alpha \rho_\alpha \text{tr}(R\wedge R)\,,
\end{equation}
where we identified
$I \cong \mathcal{A}$, $m,n \cong \mathcal{A},\mathcal{B}$ and used $\lambda_\mathcal{A}=1$ for
$SU(N_\mathcal{A})$ in \eqref{eq:GSterm}.
The coupling coefficients $b_{\mathcal{A}}$, $b_{\mathcal{A}\mathcal{B}}$ and $a$ determined in
the following.

In order to obtain the gaugings and GS-terms in weakly coupled Type IIB
setups one has to expand the D7-brane and O7-plane
world-volume actions.
The resulting gaugings of these axions can be extracted from St\"uckelberg couplings arising
from  $C_4$ and $C_6$ in the D7-brane actions and are given by \cite{Jockers:2004yj}
\begin{equation} \label{eq:TypeIIBgaugings}
 \nabla \rho_\alpha = d\rho_\alpha + \Theta_{\alpha \mathcal{A}} \, A^{\mathcal{A}}_{U(1)}\ , \qquad
	\nabla c^a  =  dc^a + \Theta^{a}_{\ \mathcal{A}}\, A^{\mathcal{A}}_{U(1)}\ , \end{equation}
with constant gaugings specified by
\begin{equation} \label{eq:ThetacAalphaTypeIIb}
  \Theta_{\alpha \mathcal{A}}=2N_{\mathcal{A}}
	\int_{S_{(\mathcal{A})}^{\rm IIB}}\mathcal{F}^{\mathcal{A}}\wedge
	\omega_\alpha
	\ ,
	\qquad
   \Theta^{a}_{\ \mathcal{A}} = - 2N_{\mathcal{A}} \delta_{(\mathcal{A})}^{a}\, .
\end{equation}
Here the factor of $2$ in the first equation appears since the same
coupling is generated
twice, by the D7-brane on $S_{(\mathcal{A})}$ and by its image brane on
$S_{(\mathcal{A}')}$. The $A^\mathcal{A}_{U(1)}$ are the $U(1)$'s in \eqref{G_ori}
that comprise combinations of the brane and image brane $U(1)'s$.
The D7-brane field strength
$F^{(\mathcal{A})}_{\rm D7}$ has a flux
part $\mathcal{F}^{(\mathcal{A})}$ on $S^{\rm IIB}_{(\mathcal{A})}$, where we focus
on fluxes in the $U(1)_{\mathcal{A}}$-direction only. We stress that the gaugings
of $c^a$ are purely geometrical and independent of any brane flux. They
contain the coefficients $ \delta_{(A)}^{a}$
arising in the expansion of the D7-brane locus $S_{(\mathcal{A})}^{\rm IIB}$ as
\begin{equation} \label{eq:Sexp_ori}
	\tfrac12(S_{(\mathcal{A})}^{\rm IIB} +S_{(\mathcal{A}')}^{\rm IIB} )  = \delta_{(\mathcal{A})}^{\alpha}D_\alpha^{\rm IIB}\, ,\qquad  \tfrac12(S_{(\mathcal{A})}^{\rm IIB} - S_{(\mathcal{A}')}^{\rm IIB})  =  \delta_{(\mathcal{A})}^{a}D_a^{\rm IIB}\,,\qquad
	S_{\rm O7}=\delta_{\rm O7}^{\alpha}D_\alpha^{\rm IIB}\, ,
\end{equation}
where $D_\alpha^{\rm IIB},\ D_a^{\rm IIB}$ is a basis of $H_4^+(Y_3)$,
$H_4^-(Y_3)$, respectively. The factor of $2$ in the second equation
in \eqref{eq:ThetacAalphaTypeIIb} is due to the factor $\frac12$ in
\eqref{eq:Sexp_ori}.
For completeness we have displayed here the expansion of the O7-plane
locus as well.

The GS-counter terms can be extracted
from the Chern-Simons world-volume actions of D7-branes and O7-planes.
One obtains the GS-terms \eqref{eq:GSterm} for both the $C_4$ axions
$\rho_\alpha$ with the coefficients $b^\alpha_{\mathcal{A}}$, $b_{\mathcal{A}\mathcal{B}}^\alpha$
and $a^\alpha$ and for the $C_2$ axions $c^a$ with  coefficients
$b_{a\,\mathcal{A}}$, $b_{a\,\mathcal{A}\mathcal{B}}$ and $a^\alpha$ as
\begin{eqnarray} \label{eq:bs_asTypeIIB}
	   b^\alpha_{\mathcal{A}} \!\!\!&=&\!\!\! 2\delta^\alpha_{(\mathcal{A})}\,, \qquad b_{a\, \mathcal{A}} =	2\int_{S_{(\mathcal{A})}^{\rm IIB}}\mathcal{F}^{\mathcal{A}}\wedge
	\omega_a\, , \qquad b_{\mathcal{A}\mathcal{B}}^\alpha =  N_\mathcal{A}\delta_{\mathcal{A}\mathcal{B}}\, b^\alpha_{ \mathcal{A}}\,, \qquad
         b_{a\, \mathcal{A} \mathcal{B}}  = N_\mathcal{A} \delta_{\mathcal{A} \mathcal{B}}\, b_{a\, \mathcal{A}} \,\nonumber \\
          a^\alpha\!\! \! &=&\! \! \!
	-\tfrac{1}{6}\big(2\delta_{\rm O7}^\alpha+\sum_\mathcal{A}
	N_\mathcal{A}\delta_{(\mathcal{A})}^\alpha\big)=-\delta_{O7}^\alpha\, .
\end{eqnarray}
As before the factor of $2$ in $b_{a\,\mathcal{A}}$ is due to the two identical
contributions from $S_{(\mathcal{A})}$ and the
image cycle $S_{(\mathcal{A}')}$.
Here the first equality for $a^\alpha$ can be inferred by the reduction of
the D7-brane world-volume theory on $S_{(\mathcal{A})}^{\rm IIB}$  and the image
cycle $S_{(\mathcal{A}')}^{\rm IIB}$ as well as the O7-plane action on
$S_{\rm O7}$ using \eqref{eq:Sexp_ori} and
$\mu_{\rm O7}=-8\mu_{\rm D7}$ on the orientifold double cover
\cite{Grimm:2012yq}.
For the second equality we use the Type IIB  D7-brane tadpole, formulated
again on the double cover, reading
\begin{equation}
	8S^{O7}=\sum_\mathcal{A} N_{\mathcal{A}}(S^{\rm IIB}_{(\mathcal{A})}+S^{\rm IIB}_{(\mathcal{A}')})\,.
\end{equation}

Finally we can formulate the anomaly cancellation conditions of a Type IIB
orientifold. This means that the anomalies \eqref{eq:AnomaliesOrie} have
to equal the GS-terms as required in
\eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:Abgravitational}.
With \eqref{eq:bs_asTypeIIB},  \eqref{eq:ThetacAalphaTypeIIb} and \eqref{eq:I_abTypeIIB}  these take
the form
\begin{align}
	& F^\mathcal{A}_{U(1)}\, (F^\mathcal{B}_{U(1)})^2: &\qquad  &
         \tfrac{1}{4}b^\alpha_{\mathcal{B}\cB} \Theta^{\phantom{\alpha}}_{\mathcal{A}\alpha} + \tfrac14 b^{\phantom{a}}_{a\, \mathcal{B} \mathcal{B}} \Theta^a_{\ \mathcal{A}} =  \tfrac12 N_\mathcal{A} N_\mathcal{B} \, \big[I_{\mathcal{B} \mathcal{A}}
         +  I_{ \mathcal{B}' \mathcal{A}} \big] \\[.2cm]
  &  F^\mathcal{A}_{U(1)}\text{tr}_\mathbf{f} (F^\mathcal{B})^2:&\qquad &
        \tfrac{1}{4}b^\alpha_{\mathcal{B}} \Theta^{\phantom{\alpha}}_{\mathcal{A} \alpha} + \tfrac14 b^{\phantom{a}}_{a\,  \mathcal{B}} \Theta^a_{\ \mathcal{A}}
        = \tfrac12 N_\mathcal{A} \, \big[I_{\mathcal{B} \mathcal{A}}
         +  I_{ \mathcal{B}' \mathcal{A}} \big] \,, \nonumber   \\[.2cm]
&   F^\mathcal{A}_{U(1)} \text{tr}R^2 :&\qquad  &
         -\tfrac{1}{16}a^\alpha\Theta_{\alpha\mathcal{A}}=
        \tfrac{1}{8}   N_\mathcal{A}\, I_{O7 \mathcal{A}}\, .\nonumber
\end{align}
We note that there is no symmetrization over the indices $\mathcal{A}$, $\mathcal{B}$ in
the first equation since the
Green-Schwarz terms for the $U(1)$-field strengths
$F_{ U(1)}^\mathcal{A}$ in \eqref{eq:GStermIIB} are diagonal in $\mathcal{A}$, $\mathcal{B}$
due to \eqref{eq:bs_asTypeIIB}.
This shows agreement of the Green-Schwarz terms with the anomalies
\eqref{eq:AnomaliesOrie_2}. Thus, anomaly cancellation in Type IIB
with D7-branes and O7-planes with canceled tadpoles is guaranteed.

\subsection{Comparison with F-theory anomaly cancellation} \label{sec:F-match}

The complication in matching the F-theory setup with the corresponding weak
coupling compactification arises from
the fact that the $U(1)$'s in each of the $U(N_\mathcal{A})=U(1)_\mathcal{A}\times
SU(N_\mathcal{A})$ can be massive due to a geometric St\"uckelberg mechanism
\cite{Jockers:2004yj} as investigated recently in \cite{Grimm:2011tb}.
More precisely, the geometric gaugings \eqref{eq:TypeIIBgaugings},
\eqref{eq:ThetacAalphaTypeIIb} of the additional R--R two-form axions
$c^a$ arising from cycles negative under the
orientifold projection can render the combinations
$\Theta^a_{\ \mathcal{A}}A_{U(1)}^\mathcal{A}$ of $U(1)$'s massive even in the absence
of fluxes. This implies that in general
\begin{equation} \label{G_ori_reduction}
   G \subset G_{\rm ori}\ ,\qquad n_{ U(1)} \leq \widehat n_{G}\ ,
\end{equation}
where $G$ is the F-theory gauge group
$G= SU(N_1) \times \ldots  \times SU(N_{n_G}) \times U(1)_1 \times \ldots
U(1)_{n_{U(1)}}$, and $n_G$ is the number of $U(N_\mathcal{A})$ factors in
\eqref{G_ori} with $N_\mathcal{A}>1$.

We denote the geometrically massive and massless $U(1)$'s by
\begin{eqnarray}
   \text{geom. massive:} \quad &A^M,& \quad M = 1,\ldots, \text{rank}(\Theta^a_{\ \ \mathcal{A}})\, ,\\
   \text{geom. massless:}\quad &A^m,& \quad m = 1, \ldots,  n_{U(1)} \ , \nonumber
\end{eqnarray}
where $n_{U(1)} \equiv \widehat n_G -  \text{rank} (\Theta^a_{\ \ \mathcal{A}})$.
To perform the comparison with F-theory we thus
have to perform a basis transformation of the
$U(1)$-gauge fields $A^\mathcal{A}_{U(1)}$ to separate $A^M$ and $A^m$
in the Lagrangian.
Thus, we use the ansatz
\begin{equation} \label{eq:projections}
   A^{\mathcal{A}}_{U(1)} = \pi_m^{\mathcal{A}} A^m + \pi_M^\mathcal{A} A^M \, , \qquad \quad
   \pi^{\mathcal{A}}_m \Theta^{a}_{\ \mathcal{A}} = 0 \, ,
\end{equation}
where $\text{rank}(\pi_m^{\mathcal{A}})=n_{U(1)}$ and $\text{rank}(\pi_m^{\mathcal{A}}) +
\text{rank}(\pi_M^{\mathcal{A}}) = \widehat n_{G}$.
The vanishing condition tells us that the geometrically massless $U(1)$'s
are in the kernel of the map $\Theta^{a}_{\ \mathcal{A}}$.

The map to F-theory can now be formulated using the matrices $\pi_m^{\mathcal{A}} ,\pi_M^\mathcal{A}$.
We identify the $A^m$ with the $U(1)$-vectors in
\eqref{eq:C3expansion} in F-theory.
The F-theory gaugings and Green-Schwarz terms are captured by
\begin{equation}
   \Theta_{m \alpha} = \pi^\mathcal{A}_m \Theta_{\mathcal{A} \alpha}\ , \qquad  b^\alpha_m = \pi^\mathcal{A}_m b^\alpha_\mathcal{A}\ .
\end{equation}
Using these identifies we can project the orientifold Green-Schwarz terms to the
geometrically massless modes as
\begin{align} \label{proj_GSterms}
	& F^m_{U(1)}\, (F^n_{U(1)})^2: &\qquad  &
         \pi^{\mathcal{A}}_m   \pi^{\mathcal{B}}_n \big(    b^\alpha_{(\mathcal{A}\mathcal{B}} \Theta^{\phantom{\alpha}}_{\mathcal{B})\alpha} +  b^{\phantom{a}}_{a\, (\mathcal{A} \mathcal{B}} \Theta^a_{\ \mathcal{B})} \big)
          =  b^\alpha_{(m n} \Theta^{\phantom{\alpha}}_{n)\alpha}   \\[.2cm]
  &  F^m_{U(1)}\text{tr}_\mathbf{f} (F^I)^2:&\qquad &
      \pi^{\mathcal{A}}_m   \big( b^\alpha_{I} \Theta^{\phantom{\alpha}}_{\mathcal{A} \alpha} +  b^{\phantom{a}}_{a\,  I} \Theta^a_{\ \mathcal{A}} \big)
        =  b^\alpha_{I} \Theta^{\phantom{\alpha}}_{m \alpha} \,, \nonumber   \\[.2cm]
&   F^m_{U(1)} \text{tr}R^2 :&\qquad  &
         \pi^{\mathcal{A}}_m a^\alpha\Theta_{\alpha\mathcal{A}}= a^\alpha\Theta_{\alpha m }   \, ,\nonumber
\end{align}
where we have repeatedly used the condition $\pi^{\mathcal{A}}_m \Theta^{a}_{\ \mathcal{A}} = 0$ for the geometrically
massless $U(1)$'s.
In order to complete the match we note that $a^\alpha$ agrees for the
orientifold and F-theory setup.
The Kodaira constraint, ensuring that $c_{1}(\hat X_4)=0$, that takes at weak coupling the form
\begin{equation}
	\label{eq:Kodaira}
	12 c_1(B_3)=[\Delta]=2[S^{\text{O7}}]+\sum_{I}
	N_{\text{D7}_{I }}[S^{\text{IIB}}_{(I)}]+\sum_{m}
	[S^{\text{IIB}}_{(m)}] +  [\Delta'] \,.
\end{equation}
Here we split the discriminant $[\Delta]$ into a sum over D7-branes and
O7-planes. It is important to note that the sum over D7-branes includes
stacks with $N_{\text{D7}_I}$ D7-branes and single D7-branes corresponding
to massless $U(1)$-symmetries. We claim that there is a residual
discriminant $\Delta'$ corresponding to D7-branes with no associated
geometrically massless $U(1)$'s. Thus, both geometrically massless and
massive $U(1)$'s are captured by $\Delta$ in F-theory.

To conclude this section, we stress that the F-theory effective action
obtained from a Calabi-Yau reduction of M-theory only includes the subset
$A^m$ of $U(1)$'s that are geometrically massless but can become massive due to fluxes.
Following the arguments of section \ref{sec:anomalycancellationinF}, together with
the explicit checks for a number of examples in section~\ref{sec:examples},
we claim that this F-theory effective action is free of anomalies.
The F-theory setup comprises a consistent reduction of the orientifold setup
for scales where the geometrically massive $U(1)$'s are
already integrated out. We have recalled that anomalies are canceled
in orientifold compactifications with both types of $U(1)$'s due to
tadpole cancellation. While we have not explicitly integrated out
the geometrically massive $U(1)$'s one expects an anomaly free
theory after this process. Indeed, the resulting theory should match
the F-theory effective action determined via M-theory.
We have then shown that the Green-Schwarz terms of the
projected orientifold setup \eqref{proj_GSterms} indeed match the Green-Schwarz terms
in F-theory determined in section \ref{sec:AnomaliesInFGUTs}.
To extend also the F-theory setup it would be very interesting to include
the geometrically massive $U(1)$'s as suggested in \cite{Grimm:2011tb}.
In this work it was argued that the geometrically massive $U(1)$'s can be included in
the M-theory to F-theory limit when extending the M-theory reduction
to include certain non-closed forms.

\section{4D Anomaly-Free F-Theory Compactifications}
\label{sec:examples}

In this section we study anomaly cancellation in concrete
examples. We consider compact elliptically fibered Calabi-Yau fourfolds
realized as toric hypersurfaces. We begin by showing anomaly cancellation
for a fourfold with base $B_3=\mathbb{P}^2\times \mathbb{P}^1$ and giving
rise to an $SU(5)$ gauge symmetry in section \ref{sec:X4withSU5}.
Then we include $U(1)$ symmetries by the method of the $U(1)$-restricted
Tate model in section \ref{sec:X4withSU5xU1}. We consider two fourfolds,
one with base $B_3=\mathbb{P}^2\times \mathbb{P}^1$ and another one with
$B_3=\text{Bl}_x(\mathbb{P}^3)$. In both cases we show that all anomalies
are cancelled by an implementation of the Green-Schwarz mechanism.

\subsection{F-theory with $SU(5)$ gauge symmetry}
\label{sec:X4withSU5}

Let us consider a concrete F-theory compactification to 4D with an $SU(5)$
gauge symmetry. The corresponding elliptically fibered Calabi-Yau fourfold
$X_4$ has to be constructed so that an $SU(5)$ singularity in the elliptic
fibration is formed over a divisor $S$ in the base $B_3$. In this simple
case and assuming enhancement over codimension two in $B_3$ to $SU(6)$ and
$SO(10)$ matter arises in the representations $\mathbf{5}$, $\mathbf{10}$
and its conjugate representations $\bar{\mathbf{5}}$,
$\overline{\mathbf{10}}$. Upon switching on suitable chirality inducing
$G_4$-flux a chiral 4D matter spectrum is induced.

For this theory we evaluate the anomaly condition
\eqref{eq:anomalyCondHomology}. Since the other anomalies are trivial,
only the anomaly cancellation condition for the
$SU(5)$ gauge anomaly poses a non-trivial constraint on the spectrum.
As discussed above this anomaly is non-factorizable and
can thus not be canceled by a GS-mechanism but has to cancel by
itself. The anomaly condition then reads
\begin{equation} \label{F^3-anomaly-SU5}
	\chi(\mathbf{5})+\chi(\mathbf{10})=0\,,
\end{equation}
which is the well-known fact that in any consistent
$SU(5)$ GUT theory one must have an equal number of
$\mathbf{5}$'s and $\overline{\mathbf{10}}$'s. In evaluating
\eqref{F^3-anomaly-SU5} we employed the general trace relations
\eqref{eq:traceRelations} that we evaluated explicitly for $SU(5)$ using
\eqref{F^3-trace-relations}.

We demonstrate that the anomaly condition \eqref{F^3-anomaly-SU5} is
automatically obeyed in a concrete example of an F-theory
compactification with $SU(5)$ gauge symmetry. We consider the F-theory
compactification on a Calabi-Yau fourfold $X_4$ elliptically fibered over
the base $B_3=\mathbb{P}^2\times \mathbb{P}^1$.
We realize it as a toric hypersurface in the five-dimensional toric
ambient space $\mathcal{W}=\mathbb{P}(\mathcal{O}\oplus \mathcal{L}^2\oplus \mathcal{L}^2)$, where the
line bundle $\mathcal{L}$ is given by the anti-canonical bundle
$K^{-1}_{B_3}$ of $B_3$, $\mathcal{L}=K^{-1}_{B_3}$.
By specializing the complex structure of a generic elliptic fibration over $B_3$, we generate
a single $SU(5)$ singularity over a divisor $\hat{S}$ in $B_3$,
that we choose to be the zero-section of the $\mathbb{P}^1$, i.e.~$\hat{S}=\mathbb{P}^2$.
We readily blow-up this $SU(5)$ singularity torically by blowing up the toric ambient space
$\mathcal{W}$. This induces four new, exceptional divisors $D_i$.
The Calabi-Yau hypersurface in this new toric space is the smooth Calabi-Yau fourfold $\hat{X}_4$.
The toric data specifying this smooth fourfold reads
\small
\begin{equation} \label{P1P2}
\begin{array}{|rrrrr|c|}
\hline
\multicolumn{5}{|c|}{\text{Points}}& D_A\\
\hline
\hline
   -1 &  0 & 0  & 0 & 0 &  \phantom{\int^X}\hspace{-1.5Em}\tilde{X}\\
     0 & -1 & 0 & 0 & 0 & Y\\
     3 &  2 &  0 & 0 & 0 & B\\
     3 &  2 & 1 & 0 & 0 &  \hat{S}\\
     3  & 2 & -1 & 0 & 0 & S\\
     3 &  2 & 0 & 1 & 1 & H\\
     3  & 2 & 0 &-1 & 0 & H\\
     3 &  2 & 0 & 0 &-1 & H\\
     2 &  1 & 1 & 0 & 0 & D_1\\
     1 & 1 & 1 & 0 &0 & D_4\\
     1  &  0 & 1 & 0 &0 & D_2\\
     0  & 0 & 1 & 0 & 0 & D_3\\
     \hline
\end{array}
\end{equation}
\normalsize
Each line denotes a point in $\mathbf{Z}^5$
representing the vertices defining the polyhedron of the toric ambient
space $\mathcal{W}$. We introduced the generators of the divisor group,
that are associated to these vertices. By linear equivalences,
only $D_A=(B,S,H,D_1,D_2,D_3,D_4)$ are independent. Both $\tilde{X}$ and
$Y$ as well as $S=\hat{S}+\sum_i D_i$, according to \eqref{eq:Shat} with
$a_i=1$, are linear combinations of these generators. The basis of
divisors on the base $B_3$ is given by $D_\alpha=(S,H)$.
It can easily be checked that the $D_i$ are the Cartan divisors of $SU(5)$
obeying the relation \eqref{eq:Cartan} for the Cartan matrix of $SU(5)$
and with the divisor $S$ as introduced in \eqref{P1P2}.

Omitting the details and referring to \cite{Grimm:2011fx} for the technicalities, the polyhedron
defined by \eqref{P1P2} admits three different Calabi-Yau phases out of 54 different star-triangulations of the toric ambient space $\mathcal{W}$. Picking one particular triangulation,
it can be seen that the relative
Mori cone of $\hat{X}_4$\footnote{We note that we have to form the Mori cone of $\hat{X}_4$. It
is found by intersecting the various Mori cones of the different triangulations of the toric
ambient space that lead to the same Calabi-Yau phase.} of effective curves contains some of the weights of
the $SU(5)$ representations $\mathbf{R}=\mathbf{5},\overline{\mathbf{10}}$ and their complex
conjugates. Thus, we see that indeed 6D matter arises precisely in these representations.
The 4D matter is induced by turning on a suitable $G_4$-flux meeting all the consistency conditions
\eqref{eq:vanishingThetas}. Choosing the triangulation denoted as phase I
in \cite{Grimm:2011fx} we start with the ansatz \eqref{eq:G4expansion}
without Abelian fluxes, since there are no $U(1)$-symmetries by
construction, and satisfy the equations
\eqref{eq:vanishingThetasExplicit}. We obtain the
one-parameter family of allowed flux that takes the form
\begin{equation} \label{eq:GfluxSU5}
	G_4(\lambda)=\lambda(8D_2D_4-4D_2^2-2D_2D_3+3D_3^2+9D_3 H)\,,
\end{equation}
with quantized flux parameter $\lambda$. We have suppressed the use of
Poincar\'e duality in our notation of $G_4(\lambda)$
for the sake of the readability of this expression.

Next we introduce the four multiplicities $n(\mathbf{R})$ of the representations $\mathbf{R}=\mathbf{5},\,\mathbf{10},\,\overline{\mathbf{10}}\,,\overline{\mathbf{5}}$ as free variables and
solves for them by  the matching \eqref{eq:matching1loop+classical} of 3D
Chern-Simons terms. One finds that as expected only the chiralities
$\chi(\bar{\mathbf{5}})=n(\bar{\mathbf{5}})-n(\mathbf{5})$
and $\chi(\mathbf{10})=n(\mathbf{10})-n(\overline{\mathbf{10}})$ can be determined as
\begin{equation}
	\chi(\bar{\mathbf{5}})=\chi(\mathbf{10})=162\lambda\,.
\end{equation}
This immediately confirms that the non-Abelian $SU(5)$ anomaly condition
\eqref{F^3-anomaly-SU5}
is obeyed. Since this is the only possible anomaly the constructed $SU(5)$
GUT is anomaly-free.

\subsection{F-theory with $SU(5)\times U(1)$ gauge symmetry}
\label{sec:X4withSU5xU1}

As a more involved example we consider now an F-theory compactification with
gauge group $SU(5)\times U(1)$. Anomaly cancellation in this theory is
richer due to the possibility of both mixed and purely Abelian anomalies that
might require a non-trivial GS-mechanism as discussed in section \ref{sec:anomlies+cancellation}.

We will construct the singular fourfold leading to $SU(5)\times U(1)$
gauge symmetry by starting with a singular fourfold with only an $SU(5)$
singularity. Then we apply the method of the $U(1)$-restricted Tate model
\cite{Grimm:2010ez} to generate a new section of the elliptic fibration of
the fourfold. This means that the complex structure of the
given fourfold hast to be restricted even further so that not
only an $SU(5)$ is generated, but also the $I_1$ locus self-intersects in
a curve. This additional singularity results in a new section and a
corresponding $U(1)$ gauge symmetry after resolution.
For this gauge group, four-dimensional matter can arise in representations
$\mathbf{5}_q$, $\mathbf{10}_{q'}$ and their complex conjugates. In
general there can be matter in the same $SU(5)$ representation but with
different $U(1)$-charge.

In this section we will consider two concrete fourfolds, that do not
differ by their fiber singularities but the choice of base $B_3$. The
first example will be the fourfold of the last section \ref{sec:X4withSU5}
with base $B_3=\mathbf{P}^2\times \mathbf{P}^1$ on which
we apply the method of the $U(1)$-restricted Tate model. In this case we
will demonstrate the structure of the the $G_4$-flux \eqref{eq:generalG4},
i.e.~the split into a non-Abelian and Abelian flux, explicitly. As
a second example we consider the elliptically fibered fourfold
with $SU(5)\times U(1)$ singularities over the base
$B_3=\text{Bl}_x(\mathbf{P}^3)$, the blow-up of $\mathbf{P}^3$ at a
generic point $x$.

\subsubsection{An $SU(5)\times U(1)$ singularity over $B_3=\mathbf{P}^2\times \mathbf{P}^1$}
\label{sec:SU5U1overP2xP1}

The first concrete example we consider is the elliptically fibered
Calabi-Yau fourfold $X_4$ with base $B_3=\mathbf{P}^2\times \mathbf{P}^1$
considered in the last section \ref{sec:X4withSU5}.

In addition
to the $SU(5)$ singularity we generate by a further specialization of the
complex structure a self-intersection of the $I_1$-locus.
As before we study the geometry $\hat{X}_4$ obtained after the toric
blow-up of all singularities by blowing up the toric ambient space
$\mathcal{W}$. Because of the extra singularity, we have to perform five
blow-ups that yield five new, exceptional divisors, the four
Cartan divisors $D_i$ of the $SU(5)$  and a single new exceptional divisor
$X$ from resolving the curve of self-intersection of the $I_1$ locus.
The Calabi-Yau hypersurface in this new toric space is again a smooth
Calabi-Yau fourfold $\hat{X}_4$. The toric data of the blown-up ambient
space is that of \eqref{P1P2} augmented by a single new point $[-1\ -1\ 0\ 0\ 0\vert X]$, where as before the first column denotes the
vertex and the second the associated divisor $X$. The basis of divisors on $\hat{X}_4$ is that
of the last section with the one new divisor $X$,
\begin{equation} \label{eq:divsOnSU5xU1P2xP1}
	B\,,\quad D_\alpha=(H,S)\,,\quad D_i=(D_1,D_2,D_3,D_4)\,,\quad X\,.
\end{equation}
Here, the divisors are as before the zero section of the elliptic
fibration $B$, the vertical divisors $D_\alpha$ and the four Cartan
divisors $D_i$ of $SU(5)$. The new toric divisor $X$ introduces a new
section of the elliptic fibration of $\hat{X}_4$. We confirm this by
checking in a toric calculation the  necessary condition
\eqref{eq:defPropSections} for a section, namely
\begin{equation}
	X^2\cdot \mathcal{C}^\alpha=-[c_1(B_3)]\cdot X\cdot
	\mathcal{C}^\alpha\,.
\end{equation}
We have evaluated this in a basis of vertical four-cycles
$\mathcal{C}^{\alpha}=(S\cdot H, H^2)$ on $\hat{X}_4$. From the section
$X$ we then construct a divisor of type
$D_m$ by applying the Shioda map \eqref{eq:ShiodaMap} to $X$. This divisor
thus induces a $U(1)$-gauge symmetry in the effective theory by the
reduction of $C_3$ in \eqref{eq:C3expansion} along it. We note that the
Shioda map agrees with the construction of a divisor of type $D_m$ used in
\cite{Grimm:2010ez}.

It is instructive to perform the Shioda map for $X\equiv \hat{\sigma}_m$
in two steps. We first introduce a divisor \cite{Grimm:2010ez}
\begin{equation} \label{eq:defD_5}
	D_5=X-B-c_1(B_3)\,,
\end{equation}
which takes into account only the first three terms in
\eqref{eq:ShiodaMap}.
Here we have used that the intersection $X\cdot B=0$ on $\hat{X}_4$, as
confirmed in an explicit toric calculation, which
immediately implies $X\cdot B\cdot \mathcal{C}^\alpha=0$.
Then, we obtain a divisor of type $D_m$, denoted by
$B_X$ for reasons discussed in the next paragraph, as
\begin{equation} \label{B_X_example2}
	B_X=-2 D_1-4 D_2-6 D_3-3 D_4-5 D_5=-4C^{5i}_{SO(10)} D_i\,.
\end{equation}
This is the fourth term of the Shioda map, where we have changed the
normalization so that $B_X\equiv -5D_m$ in \eqref{eq:ShiodaMap} for later
convenience. Indeed we take again the above basis
of vertical four-cycles $\mathcal{C}^{\alpha}$ and of vertical
divisors $D_\alpha$ of \eqref{eq:divsOnSU5xU1P2xP1} to calculate the
following intersection matrix \eqref{eq:intmatrix} and the intersections
with $X\cdot D_i$,
\begin{equation} \label{eq:intmatrix_example2}
	\eta_\alpha^{\,\ \beta}=\delta_{\alpha}^\beta\,,\qquad X\cdot D_i\cdot \mathcal{C}^{\alpha}=\delta_{3i}\delta^\alpha_2\,.
\end{equation}
Thus since only $X\cdot D_3\cdot H^2=1$ is non-vanishing we choose
$\alpha=2$ when evaluating \eqref{eq:ShiodaMap} and note that since
$S_{(1)}=S$ in \eqref{eq:def-Calpha} we obtain
$(\delta_{(1)}^\beta\eta_{\beta}^{\,\ 2})=\delta_{(1)}^2=1$.
Then we evaluate the inverse $C_{(1)}^{-1}$ of the Cartan matrix $C_{(1)}$
of $SU(5)$ of which we need according to the second equality in
\eqref{eq:intmatrix_example2} the third line $(C_{(1)}^{-1})^{3i}$. We
finally obtain
\begin{equation}
	D_m=D_5+\frac{1}{5}(2D_1+4D_2+6D_3+3D_4)\,,
\end{equation}
which clearly agrees with \eqref{B_X_example2} up to a factor of $-5$.

We note that the coefficients of $B_X$ in \eqref{B_X_example2} also agree
with the entries $(C^{-1}_{SO(10)})^{5i}$ of the inverse of the Cartan matrix
$C_{SO(10)}$ of $SO(10)$. This is pointing to an underlying $SO(10)$
structure hidden in the construction of the divisor $B_X$ via the Shioda map.
This has been noted earlier, but differently motivated and without reference
to the Shioda map, in \cite{Grimm:2010ez} as we outline next.
First, one notes that the divisor $D_5$ defined in \eqref{eq:defD_5} intersects
with the Cartan divisors $D_i$, $i=1,2,3,4$, of the $SU(5)$ singularity
almost like the Cartan divisors of an underlying $SO(10)$
singularity. Indeed, one obtains in a concrete toric calculation the
intersections of the $D_i$ of $SU(5)$ and $D_5$ as
\begin{equation} \label{eq:almostSO10ints}
	D_i\cdot D_j\cdot D_\alpha\cdot D_\beta= -(\tilde{C}_{SO(10)})_{ij}S\cdot B\cdot D_\alpha\cdot D_\beta\,,\quad
	\tilde{C}_{SO(10)}=\begin{pmatrix}
		2 &-1&0&0&0\\
		-1 &2& -1& 0&0\\
		0&-1&2&-1&-1\\
		0&0&-1&2&0\\
		0&0&-1&0&*
	\end{pmatrix}\,.
\end{equation}
Here $i,j=1,\ldots,5$ and $\tilde{C}_{SO(10)}$ agrees with the Cartan matrix
of $SO(10)$ up to the entry $(\tilde{C}_{SO(10)})_{55}$. This entry is the
intersection of $B_5^2$ with the product of two vertical divisors
$D_\alpha\cdot D_\beta$, which is not $-2$, as required for an $SO(10)$, but
actually depends on the choice of $D_\alpha\cdot D_\beta$. Geometrically,
this fact is, however, not surprising since there is only an $SU(5)$
singularity over $S$ and the enhancement to $SO(10)$ happens only over
co-dimension two on the $\mathbf{10}$ matter curve.
We see that the construction of the divisor $B_X$ in \eqref{B_X_example2}
reflects this underlying $SO(10)$ structure up to the mentioned mismatch.
In this interpretation $B_X$ agrees with the $U(1)$-factor $U(1)_X$ in
the group theoretical breaking $SO(10)\rightarrow SU(5)\times U(1)_X$.
This analogy between geometry and group theory has been the motivation for
the notation $B_X$ in the literature \cite{Grimm:2010ez,Krause:2011xj}.

Finally we double-check in an explicit toric calculation that $B_X$ is
indeed of type $D_m$, i.e.~that it obeys the intersection properties
\eqref{eq:intsD_m}. The first condition is easily proven in general by
noting that the construction of $B_X$ agrees precisely with conventional
group theory embedding of $U(1)_X$ into $SO(10)$ as noted before.
Thus, it also follows directly from group theory arguments that
$B_X$ commutes with the Cartan divisors $D_i$, $i=1,\ldots, 4$, of $SU(5)$
which in geometrical terms directly implies the first condition in
\eqref{eq:intsD_m}. In addition, we confirm this expectation
from group theory by an explicit toric computation.
Next we have to find the divisor $S^{(1)}$ of \eqref{eq:intsD_m}, namely
the vertical divisor supporting the seven brane inducing the
$U(1)$-symmetry in F-theory. In the Calabi-Yau phase specified later in
this section we explicitly find the solutions $b^\alpha_{1}$ to
\eqref{eq:determingb_m} that determine $S^{(1)}$ completely to
\begin{equation} \label{D_U(1)expansion}
	S^{(1)}=150 H+70 S\,,
\end{equation}
where we expanded in the basis of divisors \eqref{eq:divsOnSU5xU1P2xP1} on
$B_3$. Thus we have confirmed explicitly that the divisor $B_X$ is indeed
of type $D_m$ fulfilling the intersection properties \eqref{eq:intsD_m}.
In addition we checked that the intersections \eqref{eq:vanishingInts_0}
and \eqref{eq:vanishingInts_1} vanish as expected.

In order to determine the occurring matter representations we have to choose a
Calabi-Yau phase and to analyze its Mori cone.
The blown-up toric ambient has 108 different star-
triangulations that are grouped into six inequivalent Calabi-Yau phases. The
Calabi-Yau phase we consider is directly related to the Calabi-Yau phase
chosen in section \ref{sec:X4withSU5} by the geometric transition induced by
blowing down $X$ and deforming away the $I_1$ self-intersection.
We specify it by its charge vectors, i.e.~the Mori cone, that we construct by
intersecting the Mori cones of the ambient toric variety corresponding to the
same Calabi-Yau phase. It reads
\begin{equation}
\label{eq:MoriConeSU5U1P2xP1}
\begin{array}{|r|rrr|rrrrr|}
\hline
&\multicolumn{8}{c|}{\text{Mori cone generators}}\\
\hline
\hline
	&\phantom{\int^X}\hspace{-1Em}\ell^1 &\ell^2 &\ell^3 &\ell^4 &\ell^5 &\ell^6 &\ell^7 &\ell^8\\ \hline
     \phantom{\int^X}\hspace{-1Em}\tilde{X} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
     Y & 0 & 0 & 0 & 1 &-1 & 1 & 0 & 0\\
     B &-3 & 1 &-2 & 0 & 0 & 0 & 0 & 0\\
\hat{S} & 0 &-2 & 1 & 0 & 0 & 0 & 1 & 0\\
     S & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
     H & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
     H & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
     H & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
     \hline
   D_1 & 0 & 1 & 0 & 0 & 0 & 0 &-2 & 1\\
   D_2 & 0 & 0 & 0 & 0 & 1 &-1 & 1 &-1\\
   D_3 & 0 & 0 & 0 & 0 &-1 & 0 & 0 & 1\\
   D_4 & 0 & 1 & 0 & 0 & 0 & 1 & 0 &-1\\
     X & 0 & 0 & 0 &-1 & 1 & 0 & 0 & 0\\
     \hline
\end{array}
\end{equation}

Here, we noted the corresponding divisor classes in the first row. The three vectors $\ell^1$, $\ell^2$ and $\ell^3$ correspond to the two curves in the
base $\mathbf{P}^2\times \mathbf{P}^1$ and the class of the elliptic fiber. The five vectors
$\ell^4$ to $\ell^8$ encode the rational curves originating from the resolution process. They
shrink to zero in the blow-down $\hat{X}_4\rightarrow X_4$ and constitute the generators
of the relative Mori cone \cite{Intriligator:1997pq,Grimm:2011fx}, the
complement of the Mori cone of $X_4$ in $\hat{X}_4$. Physically we can
wrap M2-branes over these curves that correspond to the W-bosons, i.e.~the
roots, of $SU(5)$. The roots further split into weights of matter
representations corresponding to new $\mathbf{P}^1$'s
isolated over higher co-dimension loci in the base $B_3$. Thus, these
exceptional $\mathbf{P}^1$'s naturally take values in the weight lattice
of $SU(5)$, with which we identify the relative Mori cone
\cite{Grimm:2011fx}.

The matter representations can be read off from the Mori cone generators
\eqref{eq:MoriConeSU5U1P2xP1}. Performing the basis transformation from $X$ to $B_X$
and focusing on the relative Mori cone and the last five entries of the charge vectors
$\ell^{i}$, $i=4,\ldots,8$, we obtain the vectors
\begin{equation} \label{eq:relMCSU5xU1P2xP1}
	\begin{array}{|r|rrrrr|}
\hline
	&\phantom{\int^X}\hspace{-1Em}\ell^4 &\ell^5 &\ell^6 &\ell^7 &\ell^8\\
	\hline
		D_1&-2 & 0 & 0 & 0 & 1\\
		D_2&1 & 0 & 1 &-1 &-1\\
		D_3&0 & 0 &-1 & 0 & 1\\
		D_4&0 & 0 & 0 & 1 &-1\\
		B_X&0 & 5 &-3 & 1 &-1\\
		\hline
	\end{array}
\end{equation}
The rows are now precisely the $U(1)$-charges of an M2-brane wrapping the respective curves
corresponding to $\ell^i$ with respect to the Cartan divisors $D_i$, that are identified with the
negative of the simple roots $-\alpha_i$, and $B_X$. The columns are then
just the Dynkin labels of the $SU(5)$ representation with $U(1)_X$-charge given by the last entry.
We refer to appendix \ref{app:SU5reps} for the Dynkin labels the $\mathbf{5}$ and $\mathbf{10}$
representations.
We immediately identify the Dynkin labels of the following representations with the charge vectors,
\begin{equation} \label{eq:repsSU5xU1P2xP1}
	\sum_i\ell^i\subset \overline{\mathbf{5}}_{2}\,,\quad\ell^5\subset \mathbf{1}_{5}\,,\quad \ell^6\subset \overline{\mathbf{5}}_{-3}\,,\quad \ell^7\subset \mathbf{10}_{1}\quad \ell^8\subset \overline{\mathbf{10}}_{-1}\,.
\end{equation}
The other weights of the respective representations, with some
of them lying in the Mori cone, can be constructed by adding the simple roots $\alpha_i$.

Having determined the spectrum \eqref{eq:repsSU5xU1P2xP1} of
six-dimensional matter we have to construct appropriate $G_4$-flux that
potentially induces 4D chiral matter. We start with a general vertical
flux of the general form \eqref{eq:verticalFluxGeneral} and choose the
flux quanta $m^{AB}$ such that \eqref{eq:vanishingThetas} are obeyed. In
fact, for the Calabi-Yau fourfold at hand, there are only eleven
independent surfaces $S$, i.e.~only eleven independent
combinations $\omega_A\wedge\omega_B$ and as many flux parameters
$m^{AB}$. We choose the basis of surfaces given by
\begin{equation}
	D_3^2,\ D_2\cdot D_3,\ D_2^2,\ D_2\cdot D_4,\ D_4^2,\ D_3\cdot H,\ H^2,\ \hat{S} \cdot H ,\ B\cdot H,
	\ B_X\cdot H,\ B_X\cdot S\,.
\end{equation}
In addition to the nine
independent surfaces on the fourfold of the last section
\ref{sec:X4withSU5} with only an $SU(5)$-singularity
we have found two new surfaces $B_X\cdot H,\, B_X\cdot S$.
The allowed $G_4$-flux then reads
\begin{equation} \label{eq:GfluxSU5xU1P2xP1}
	G_4=G^{\text{nA}}_4(\lambda)+(\alpha H+\beta S)\cdot B_X\,,
\end{equation}
where as before we suppressed the use of Poincar\'e duality. The flux
$G^{\text{nA}}_4(\lambda)$ denotes the one-
parameter non-Abelian flux \eqref{eq:GfluxSU5} already present on the Calabi-Yau fourfold before the formation
of an $U(1)$-singularity and $\alpha$, $\beta$ denote two quantized flux numbers.
We note that \eqref{eq:GfluxSU5xU1P2xP1} is precisely of the form
\eqref{eq:generalG4} with one non-Abelian flux part $G_4(\lambda)$ induced
from the fourfold without the $U(1)$-symmetry and
thus independent of the divisor $B_X$ and a second, Abelian flux of the
form $F\wedge[B_X]$ with $F$ denoting a general two-form flux on the base
$B_3$. Fluxes of this type have been considered previously
in \cite{Braun:2011zm,Marsano:2011hv,Krause:2011xj,Krause:2012yh}.

Next we calculate the 4D chiralities using M-/F-theory duality for the CS-terms
\eqref{eq:matching1loop+classical}. We calculate both the loop-induced CS-terms
$\Theta_{\Lambda\Gamma}^{\text{loop}}$ and the classical
flux integrals $\Theta_{\Lambda\Gamma}$. For the former we only have to determine the sign-function in \eqref{eq:theta_LambdaSigma}, since we have already identified the charges in
\eqref{eq:repsSU5xU1P2xP1}.
As explained in section \ref{sec:ChiralityInF} this is achieved by just testing whether a curve associated
to a given charge $\underline{q}$ lies in the Mori cone of $\hat{X}_4$. In this case we call
$\underline{q}$ positive, otherwise negative. This allows us to readily evaluate the loop-induced
CS-levels, suppressing the superscript for brevity, as
\begin{equation}
\small
\begin{array}{lll}
	\Theta_{11}=
	-2\Theta_{12}=-2\Theta_{23}=-\Theta_{44}=\chi(\bar{\mathbf{5}}_{-3}) +\chi(\bar{\mathbf{5}}_{2})
	+\chi(\overline{\mathbf{10}}_{-1})\,,&&\\[0.3Em] \Theta_{24}= \Theta_{25}=\Theta_{33}=
	\frac{1}{2}\Theta_{45}=\chi(\overline{\mathbf{10}}_{-1})\,,&\hspace{-3.7cm}\Theta_{22}=\chi(\bar{\mathbf{5}}_{-3})
	+ \chi(\bar{\mathbf{5}}_{2}) \,,&\hspace{-1.8cm}\Theta_{35}=3\chi(\bar{\mathbf{5}}_{-3})
	- 2\chi(\bar{\mathbf{5}}_{2})\,,\\[0.3Em]
	\Theta_{34}=\frac12(\chi(\bar{\mathbf{5}}_{-3}) + \chi(\bar{\mathbf{5}}_{2})
	- \chi(\overline{\mathbf{10}}_{-1}))\,,&\hspace{-3.7cm}\Theta_{55}=\frac92\chi(\bar{\mathbf{5}}_{-3}) + 2\chi(\bar{\mathbf{5}}_{2}) + \frac{25}{2}\chi(\mathbf{1}_{5})\, &
\end{array}
\end{equation}
with all other CS-levels vanishing and invoking symmetry of the matrix $\Theta_{\Lambda\Gamma}$. As expected the CS-levels are functions only of the 4D chiralities $\chi(\mathbf{R})$.
Similarly, we evaluate the classical CS-levels \eqref{eq:theta_AB} to obtain
 \begin{equation}
 \small
	\Theta_{\Lambda\Sigma}=\begin{pmatrix}
	 0 &  0 & 0 & 0 & 0\\
     0 &-162\lambda + 3\alpha & 0 & 162\lambda - 3\alpha & 162\lambda - 3\alpha\\
     0 & 0 &162\lambda - 3\alpha & -162\lambda + 3\alpha&-486\lambda - 141\alpha\\
     0 & 162\lambda - 3\alpha & -162\lambda + 3\alpha & 0 & 324\lambda - 6\alpha\\
     0 & 162\lambda - 3\alpha & -486\lambda - 141\alpha & 324\lambda - 6\alpha& -729\lambda + 5751\alpha +6750\beta
	\end{pmatrix}\,
\end{equation}
depending on the three flux quanta. Equating the two expressions according
to \eqref{eq:matching1loop+classical} we solve for the 4D chiralities as
\begin{eqnarray} \label{eq:chiralitiesSU5xU1P2xP1}
	\chi(\bar{\mathbf{5}}_{-3})&=&162\lambda + 27\alpha\,,\quad \chi(\bar{\mathbf{5}}_{2})=-30\alpha\,, \nonumber\\ \chi(\overline{\mathbf{10}}_{-1})&=&-162\lambda + 3\alpha,\, \quad\chi(\mathbf{1}_{5})=-465\alpha - 540\beta
\end{eqnarray}

We are now prepared to determine the anomalies for the spectrum
\eqref{eq:chiralitiesSU5xU1P2xP1}. Evaluating the left hand side of \eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:Abgravitational} we
obtain
\begin{eqnarray} \label{eq:anomaliesSU5xU1P2xP1}
	\text{tr}_\mathbf{f} F^3\!&\!\!:\!\!&\!\chi(\overline{\mathbf{10}}_{-1}) + \chi(\bar{\mathbf{5}}_{2})
	+\chi(\bar{\mathbf{5}}_{-3})=0\,,\\
	F^m F^n F^k\!&\!\!:\!\!&\!\frac{1}
{6}(-10\chi(\overline{\mathbf{10}}_{-1})+40\chi(\bar{\mathbf{5}}_{2})
-135 \chi(\bar{\mathbf{5}}_{-3})+125\chi(\mathbf{1}_5))\nonumber\\&&\nonumber=-3375\lambda-10500\alpha-11250\beta\nonumber\\
F^m\text{tr}_{\mathbf{f}}(F^I)^2\!&\!\!:\!\!&\!
\frac12(-3\chi(\overline{\mathbf{10}}_{-1})+2\chi(\bar{\mathbf{5}}_{2}) -3 \chi(\bar{\mathbf{5}}_{-3}))
=-75\alpha\nonumber\\
F^m\text{tr}R^2\!&\!\!:\!\!&\!\frac{1}{48}(-10\chi(\overline{\mathbf{10}}_{-1})+10\chi(\bar{\mathbf{5}}_{2})-15\chi(\bar{\mathbf{5}}_{-3})+5\chi(\mathbf{1}_{5}) )\nonumber\\&&=-\frac{1}{8}(135\lambda+510\alpha+450\beta)\,,\nonumber
\end{eqnarray}
where we made use of the group theory relations \eqref{F^3-trace-relations} and \eqref{F^2trace-relations} in
appendix \ref{app:ReviewOfAnomalies} for $SU(5)$. We note that since the non-Abelian anomaly is
automatically canceled, we can potentially obtain a non-anomalous theory by a GS-mechanism.

We will show next that the chirality inducing $G_4$-flux \eqref{eq:GfluxSU5xU1P2xP1} indeed induces the appropriate
gaugings for a working GS-mechanism. We evaluate the $G_4$-flux induced
gaugings of the K\"ahler moduli associated to the divisors
$D_\alpha=(H,S)$ in \eqref{eq:divsOnSU5xU1P2xP1} with respect to the $U(1)$ corresponding to $B_X$ as
\begin{equation}\label{eq:gaugingsSU5xU1P2xP1}
	\Theta_{m\alpha}=(\Theta_{m 1},\Theta_{m2})=( -90\lambda - 140\alpha - 300\beta,-300\alpha)\,.
\end{equation}
Next we note that the canonical bundle of the base $B_3$ is given by
$K_{B_3}=\mathcal{O}(-3H-2S)$.
Recalling that the $SU(5)$-singularity is located over the
divisor $S$ by construction and the $U(1)$-divisor $S^{(1)}$ has been determined in \eqref{D_U(1)expansion},
we obtain the contribution of the GS-terms to the anomalies on the right hand side of
\eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:Abgravitational} as
\begin{eqnarray}
	F^m F^n F^k&:&\frac1{4}(150\,\Theta^{\phantom{\mathcal{A}}}_{m 1}+70\, \Theta_{m2})=-3375\lambda-10500\alpha-11250\beta\\
F^m\text{tr}_{\mathbf{f}}(F^I)^2&:&\frac{1}{4\lambda_I} \Theta_{m2}
=-75\alpha\nonumber\\
F^m\text{tr}R^2&:&-\frac1{16}(-3\Theta_{m 1}-2\Theta_{m 2})=-\frac{1}{8}(135\lambda+510\alpha+450\beta)\,,\nonumber
\end{eqnarray}
where we used $\lambda_I=1$ for $SU(N)$. By comparing this with the anomalies
\eqref{eq:anomaliesSU5xU1P2xP1}
we see that the anomaly cancellation conditions
\eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:Abgravitational} are obeyed
and the theory is anomaly-free.

\subsubsection{An $SU(5)\times U(1)$ singularity over $B_3=\text{Bl}_x(\mathbb{P}^3)$}

In this section we will consider another F-theory compactification on an
elliptically fibered fourfold with an $SU(5)\times U(1)$
gauge symmetry and with base $B_3=\text{Bl}_x(\mathbb{P}^3)$,
i.e.~the blow-up of $\mathbb{P}^3$ at a point $x$. Since this discussion
will be very similar to the one in the last section, section
\ref{sec:SU5U1overP2xP1}, we will be as brief as possible here. We
refer the reader to \cite{Grimm:2011fx} for a more detailed discussion of
the geometry.

We directly start with the smooth fourfold $\hat{X}_4$ that is obtained by resolving all singularities
in the singular fourfold $X_4$ with an $SU(5)$-singularity over a divisor $S$ and a codimension two singularity from the self-intersection of the $I_1$-locus. The fourfold $\hat{X}_4$ is realized as a toric Calabi-Yau
hypersurface in a five-dimensional toric ambient space defined by a polyhedron with the following vertices
\small
\begin{equation} \label{P3BU}
\begin{array}{|rrrrr|c|}
\hline
\multicolumn{5}{|c|}{\text{Points}}& D_A\\
\hline
\hline
   -1 &  0 & 0  & 0 & 0 &  \phantom{\int^X}\hspace{-1.5Em}\tilde{X}\\
     0 & -1 & 0 & 0 & 0 & Y\\
     3 &  2 &  0 & 0 & 0 & B\\
     3 &  2 & 1 & 1 & 1 &  H\\
     3 &  2 & -1 & 0 & 0 & H+S\\
     3 &  2 & 0 & -1 & 0 & H+S\\
     3  & 2 & 0 &0 & -1 & H\\
     3 &  2 & 1 & 1 &0 & \hat{S}\\
     2 &  1 & 1 & 1 & 0 & D_1\\
     1 & 1 & 1 & 1 &0 & D_4\\
     1  &  0 & 1 & 1 &0 & D_2\\
     0  & 0 & 1 & 1 & 0 & D_3\\
     -1 & -1 & 0 & 0 &0 & X\\
     \hline
\end{array}
\end{equation}
\normalsize
Here we present as before the coordinates of the vertices as the
five-dimensional row vectors and the corresponding toric divisors.
The polyhedron admits 348 star-triangulations that correspond to twelve
independent Calabi-Yau phases when restricted to the anti-canonical
divisor in the toric ambient space.

A basis of divisors with all other divisors in \eqref{P3BU} by linear equivalences reads
\begin{equation} \label{eq:divsOnSU5xU1P3BU}
	B\,,\quad D_\alpha=(H,S)\,,\quad D_i=(D_1,D_2,D_3,D_4)\,,\quad X\,
\end{equation}
with $B$ being the base, $D_\alpha$ the vertical divisors, the $D_i$ denoting the Cartan divisors of $SU(5)$
and $X$ the exceptional divisor generated by the blow-up in the construction of the $U(1)$-restricted Tate model. We readily check the necessary condition
\eqref{eq:defPropSections} for $X$ being a section.
The divisor $S$ is related to the divisor $\hat{S}$ as $S=\hat{S}+\sum_i D_i$. A divisor of type $D_m$
is constructed from $X$ as before in section \ref{sec:SU5U1overP2xP1} by applying the Shioda map \eqref{eq:ShiodaMap}. Again, we first construct $D_5=X-B-\left[c_1(B_3)\right]$, that intersects
the Cartan divisors $D_i$ as in \eqref{eq:almostSO10ints} resembling an $SO(10)$ singularity, and then form
$B_X=-2D_1-4D_2-6D_3-3D_4-5B_5$. Here we choose a basis of vertical
four-cycles given by $\mathcal{C}^\alpha=(H\cdot S,S^2)$ and evaluate in the
basis of vertical divisors $D_\alpha$ and Cartan divisors $D_i$ in
\eqref{eq:divsOnSU5xU1P3BU}
\begin{equation}
	\eta_{\alpha}^{\,\ \beta}=\begin{pmatrix}
		0 & 1\\
		1& -2
	\end{pmatrix}\,,\quad X\cdot D_3\cdot \mathcal{C}^1=1\,,\quad X\cdot D_3\cdot \mathcal{C}^2=-2\,,
\end{equation}
with all other intersections $X\cdot D_i\cdot \mathcal{C}^\alpha=0$. Thus, we can choose any $\alpha$ to evaluate \eqref{eq:ShiodaMap} and obtain  $D_m=-\frac{1}{5}B_X$.
It can then be shown as in the last section that $B_X$ obeys the intersections
properties \eqref{eq:intsD_m} and thus generates a $U(1)$-symmetry in the low-energy effective action.

In particular we use \eqref{eq:intsD_m} and \eqref{eq:determingb_m} to determine in a concrete toric calculation in
a Calabi-Yau phase specified later in this section  the
coefficients $b^\alpha_{1}$ in \eqref{eq:determingb_m}. The corresponding divisor  $S^{(1)}$
reads
\begin{equation} \label{D_U(1)expansion_2}
	S^{(1)}=200 H+120 S\,.
 \end{equation}
In addition, we check explicitly that the intersection properties \eqref{eq:intsD_m} are obeyed.

Again we determine the 4D chiral matter by first looking at the Mori cone \eqref{eq:MoriConeSU5U1P3BU}
to determine the 6D matter representations. For the Calabi-Yau phase we consider it reads
\begin{equation}
\label{eq:MoriConeSU5U1P3BU}
\begin{array}{|r|rrr|rrrrr|}
\hline
&\multicolumn{8}{c|}{\text{Mori cone generators}}\\
\hline
\hline
	&\phantom{\int^X}\hspace{-1Em}\ell^1 &\ell^2 &\ell^3 &\ell^4 &\ell^5 &\ell^6 &\ell^7 &\ell^8\\ \hline
     \phantom{\int^X}\hspace{-1Em}\tilde{X} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
     Y & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 1\\
     B &-3 &-1 & 1 & 0 & 0 & 0 &0 & 0\\
     H & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
    H+S& 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
    H+S& 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
     H & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
\hat{S}& 1 &-1 &-2 & 0 & 0 & 1 & 0 & 0\\
     \hline
   D_1 & 0 & 0 & 1 & 0 & 1 &-2 & 0 & 0\\
   D_2 & 0 & 0 & 0 &-1 &-1 & 1 & 1 &0\\
   D_3 & 0 & 0 & 0 & 0 & 1 & 0 &-1 & 0\\
   D_4 & 0 & 0 & 1 & 1 &-1 & 0 & 0 &0\\
     X & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\
     \hline
\end{array}
\end{equation}
As one can see by analyzing this Mori cone as in
\eqref{eq:relMCSU5xU1P2xP1} and \eqref{eq:repsSU5xU1P2xP1}
in the last section, cf.~\cite{Grimm:2011fx} for details, the occurring
matter representations are as before given by
\begin{equation} \label{eq:repsSU5xU1P3BU}
	\overline{\mathbf{10}}_{-1}\,,\quad \overline{\mathbf{5}}_{-3}\,,\quad\overline{\mathbf{5}}_{2}\,,\quad\mathbf{1}_{5}\, .
\end{equation}
Next we construct the chirality inducing vertical $G_4$-flux starting with
the ansatz \eqref{eq:verticalFluxGeneral}. There are 15 independent
four-forms in $H^{(2,2)}_V(\hat{X}_4)$ that are Poincar\'e dual to the
surfaces
\begin{equation}
	H\cdot D_i\,,\ S\cdot D_i\,,\ D_4^2\,,\ D_3\cdot D_4\,,\ B\cdot \hat{S}\,,B^2\,,\ H\cdot S\,,\ B_X\cdot
	H\,,\  B_X\cdot S\,,
\end{equation}
where $i$ runs from $i=1,\ldots, 4$. Accordingly, we have only 15
independent flux numbers $m^{AB}$ that are constrained further
by the conditions \eqref{eq:vanishingThetas}. We obtain the $G_4$-flux
meeting all these constraints as
\begin{eqnarray} \label{eq:GfluxSU5xU1P3BU}
	G_4&=&\lambda[ c_1(B_3)\cdot(D_1+2D_2-2D_3-6D_4)+5 D_4\cdot (D_3+2D_4)]+(\alpha H+\beta S)\cdot B_X\nonumber\\
	&\equiv & G_4^{\text{nA}}(\lambda)+(\alpha H+\beta S)\cdot B_X
\end{eqnarray}
where we implicitly apply Poincar\'e duality. Again we identified
a non-Abelian flux $G_4^{\text{nA}}(\lambda)$ depending on a single flux
parameter $\lambda$ and two Abelian fluxes with flux numbers $\alpha$,
$\beta$ in accordance with the general structure \eqref{eq:generalG4}.

As a next step, we determine the 4D chiralities by the matching
\eqref{eq:matching1loop+classical} of the CS-levels in 3D. We evaluate
first the one-loop CS-terms \eqref{eq:theta_LambdaSigma} yielding
\begin{equation}
\small
\begin{array}{lll}
\Theta_{11}=-2\Theta_{12}=-2\Theta_{23}=-\Theta_{44}=\chi(\bar{\mathbf{5}}_{-3}) + \chi(\bar{\mathbf{5}}_{2})  +\chi(\overline{\mathbf{10}}_{-1})\,,&
\\[0.3Em] \Theta_{24}=\Theta_{25}=\Theta_{33}=\frac12\Theta_{45}=\chi(\overline{\mathbf{10}}_{-1}) \,, & \hspace{-3.7cm}\Theta_{22}=\chi(\bar{\mathbf{5}}_{-3}) + \chi(\bar{\mathbf{5}}_{2})\,,& \hspace{-2cm}\Theta_{35}=3\chi(\bar{\mathbf{5}}_{-3}) - 2\chi(\bar{\mathbf{5}}_{2})\,, \\[0.3Em]
\Theta_{34}=\frac12\chi(\bar{\mathbf{5}}_{-3}) + \frac12\chi(\bar{\mathbf{5}}_{2})  - \frac12\chi(\overline{\mathbf{10}}_{-1}) \,,& \hspace{-3.7cm}\Theta_{55}=\frac{9}{2}\chi(\bar{\mathbf{5}}_{-3}) + 2\chi(\bar{\mathbf{5}}_{2})  + \frac{25}{2}\chi(\mathbf{1}_5)  \,, &
\end{array}
\end{equation}
where we suppressed the superscript for brevity.
All other CS-levels vanish and the remaining ones are determined by symmetry of the matrix
$\Theta^{\text{loop}}_{\Lambda\Gamma}$. As expected the CS-levels are functions only of the 4D chiralities $\chi(\mathbf{R})$.
Then, we evaluate the classical CS-levels \eqref{eq:theta_AB} to obtain
 \begin{equation}
 \small
	\Theta_{\Lambda\Sigma}=\begin{pmatrix}
	 0 &  0& 0 &0 &  0\\
	 0&92\lambda+ 3\alpha - 2\beta &0&-92\lambda - 3\alpha + 2\beta&-92\lambda - 3\alpha + 2\beta\\
    0 &0 &-92\lambda - 3\alpha + 2\beta2 &92\lambda + 3\alpha - 2\beta&256\lambda- 111\alpha + 34\beta\\
    0 &-92\lambda - 3\alpha + 2\beta&92\lambda + 3\alpha - 2\beta &0&-184\lambda - 6\alpha + 4\beta\\
0 & -92\lambda - 3\alpha + 2\beta&256\lambda - 111\alpha + 34\beta&-184\lambda - 6\alpha + 4\beta& 404\lambda + 3891\alpha + 5886\beta
	\end{pmatrix}\,
\end{equation}
that are functions of the three flux quanta. Identifying the two
expressions as in \eqref{eq:matching1loop+classical} we determine the 4D
chiralities as
\begin{eqnarray} \label{eq:chiralitiesSU5xU1P3BU}
	\chi(\bar{\mathbf{5}}_{-3})&=&-88\lambda + 21\alpha - 6\beta\,,\quad \chi(\bar{\mathbf{5}}_{2})=-4\lambda - 24\alpha + 8\beta\,, \nonumber\\ \chi(\overline{\mathbf{10}}_{-1})&=&92\lambda + 3\alpha- 2\beta ,\, \quad\chi(\mathbf{1}_{5})=-315\alpha - 470\beta
\end{eqnarray}

We can now evaluate the anomalies of the F-theory compactification
under consideration. Inserting the chiralities
\eqref{eq:chiralitiesSU5xU1P3BU} into the left hand side of the anomaly
constraints
\eqref{eq:Anomalies:purenonAb}-\eqref{eq:Anomalies:Abgravitational} we
obtain, using again the group theory relation of appendix
\ref{app:ReviewOfAnomalies},
\begin{eqnarray}	\label{eq:anomaliesSU5xU1P3BU}
	\text{tr}_\mathbf{f} F^3\!&\!\!:\!\!&\!\chi(\overline{\mathbf{10}}_{-1}) + \chi(\bar{\mathbf{5}}_{2})
	+ \chi(\bar{\mathbf{5}}_{-3})=0\,,\\
	F^m F^n F^k\!&\!\!:\!\!&\!\frac{1}
{6}(-10\chi(\overline{\mathbf{10}}_{-1})+40\chi(\bar{\mathbf{5}}_{2}) -135 \chi(\bar{\mathbf{5}}_{-3})+125\chi(\mathbf{1}_5))=1800\lambda-7200\alpha-9600\beta\nonumber\\
F^m\text{tr}_{\mathbf{f}}(F^I)^2\!&\!\!:\!\!&\!\frac12(-3\chi(\overline{\mathbf{10}}_{-1})+2\chi(\bar{\mathbf{5}}_{2}) -3 \chi(\bar{\mathbf{5}}_{-3}))
=-10\lambda-60\alpha+20\beta\nonumber\\
F^m\text{tr}R^2\!&\!\!:\!\!&\!\frac{1}
{48}(-10\chi(\overline{\mathbf{10}}_{-1})+10\chi(\bar{\mathbf{5}}_{2})-15\chi(\bar{\mathbf{5}}_{-3})+5\chi(\mathbf{1}_5) )=\frac{1}{2}(15\lambda-90\alpha-90\beta)\,,\nonumber
\end{eqnarray}
The anomaly is obviously completely factorized and thus
of the appropriate form to be canceled by the GS-counter terms
\eqref{eq:GSterm}.

Next we will show that the chirality-inducing $G_4$-flux induces the appropriate gaugings of 4D axion so that
the GS-mechanism works to cancel the anomalies.
We obtain the gaugings induced by the $G_4$-flux \eqref{eq:GfluxSU5xU1P3BU} explicitly as
\begin{equation} \label{eq:gaugingsSU5xU1P3BU}
	\Theta_{m\alpha }=(\Theta_{m1},\Theta_{m2})=(60\lambda - 240\beta,-40\lambda - 240\alpha + 80\beta)\,
\end{equation}
in the basis $D_\alpha=(H,S)$. The determination of the GS-terms on the right hand side of the anomaly condition is then completed
by noting that the $SU(5)$-singularity is supported over $S$, the $U(1)$-divisor has the expansion \eqref{D_U(1)expansion_2}
and the canonical bundle of the base $B_3$  reads $K_{B_3}=\mathcal{O}(-4H-3S)$. Thus, we obtain using $\lambda_A=1$ for $SU(N)$
\begin{eqnarray}
	F^m F^n F^k&:&\frac1{4}(200\,\Theta_{m 1}+120\, \Theta_{m2})\\
	&& =
	1800\lambda-7200\alpha-9600\beta\\
F^m\text{tr}_{\mathbf{f}}(F^I)^2&:&\frac{1}{4\lambda_I} \Theta_{m2}
=-10\lambda - 60\alpha + 20\beta\nonumber\\
F^m\text{tr}R^2&:&-\frac1{16}(-4\Theta_{m 1}-3\Theta_{m 2})=\frac{1}{2}(15\lambda-90\alpha-90\beta)\,.\nonumber
\end{eqnarray}
Comparing this to \eqref{eq:anomaliesSU5xU1P3BU} we immediately see that by the GS-mechanism all
anomalies are canceled and we obtain an anomaly-free effective theory.

\section{Conclusions}

In this work we have addressed anomaly cancellation in four-dimensional
$\mathcal{N}=1$ F-theory compactifications via their three-dimensional dual
$\mathcal{N}=2$ M-theory reductions. The M-theory compactification was performed
on a resolved Calabi-Yau fourfold and included a non-trivial $G_4$-flux
background. To match the M-theory configuration the four-dimensional
F-theory setup was dimensionally reduced on an extra circle. The
three-dimensional effective theories have been compared in the Coulomb
branch and we argued that it is crucial to properly integrate out massive
charged matter and excited Kaluza-Klein modes in the circle
compactification. We focused on globally consistent chiral
compactifications with both non-Abelian and Abelian $U(1)$ gauge
symmetries due to space-filling seven-branes.

The investigation of some key geometric properties of divisors in
resolved Calabi-Yau fourfolds was our first focus. We gave a detailed
account of the intersection structure and the appropriate identifications
to lift the 3D M-theory effective action to a 4D F-theory setup. This
included an extended introduction to the geometry of seven-branes with
$U(1)$ gauge symmetries, for example, introducing the Shioda map.
Also the geometric definition of the $U(1)$ vectors $A^0,A^\alpha$,
corresponding to the Kaluza-Klein vector and the duals of the K\"ahler
moduli axions in the 4D to 3D reduction, turned out to be crucial
in the discussion of 4D anomalies from a 3D point of view.

The intriguing interplay of geometry and couplings in the effective theory
becomes particularly apparent when aiming to demonstrate that
F-theory compactifications automatically cancel all 4D gauge and
mixed gauge-gravitational anomalies. Two ingredients are necessary:
(1) The F-theory compactification has to be defined via a singular
Calabi-Yau fourfold that admits a Calabi-Yau resolution of, at least,
all gauge group and matter singularities. (2) A $G_4$-flux has to be
defined on the resolved Calabi-Yau fourfold in such a way that it can be
lifted to a 4D F-theory compactification, i.e.~it has to obey the
vanishing conditions \eqref{eq:vanishingThetas}.

To explicitly study anomaly cancellation in F-theory we had to
explore the map between M-theory and F-theory and the corresponding
3D and circle compactified 4D effective actions.
We argued that it suffices to concentrate on deriving
the coefficients of the 3D Chern-Simons terms and the coefficients of
the four-dimensional topological terms $F\wedge F$ and $R\wedge R$.
The 3D Chern-Simons terms in the M-theory reduction are induced by
$G_4$-flux. On the circle compactified F-theory side, they admit two
very different origins. They arise either from classical 4D gaugings,
or are induced at one loop by integrating out matter fields
that are massive in the 3D Coulomb branch or that arise as Kaluza-Klein
modes.

We particularly focused on Chern-Simons terms involving the
Kaluza-Klein vector $A^0$  of the 4D metric. While this term has been
previously neglected we showed that it can be generated at one loop by
integrating out Kaluza-Klein modes of 4D chiral matter. If this matter is
charged under an $U(1)$ vector $A^m$ descending from four dimensions a
one-loop Chern-Simons term $\Theta_{0m}\, A^0 \wedge F^m$ is induced with
$\Theta_{0m}$ precisely measuring the 4D mixed Abelian-gravitational
anomaly. On the M-theory side, geometry predicts that this coefficient is
equal to the $U(1)$ gaugings contracted with the coefficient of the
topological term $R\wedge R$ given by the first Chern class of the base
$B_3$. This is precisely the matching enforced by the generalized
Green-Schwarz mechanism. Hence, this term can be either used to justify
anomaly cancellation for an F-theory setup with
a given spectrum or used to infer information about an unknown spectrum
after specifying the $G_4$-flux.

The mixed Abelian-gravitational anomaly allows for the most direct match
of the 3D and 4D data. The study of gauge anomalies is complicated by
the fact that the chiral index, counting the net number of matter
fields in a given representation, has to be extracted from the one-loop
result of the 3D Chern-Simons terms. We demonstrated that this can be done
for a number of explicit examples. Indeed, in these geometrically realized
setups one can check complete anomaly cancellation for each $G_4$-flux
lifting to F-theory. Our examples included gauge groups $SU(5)$ and
$SU(5)\times U(1)$, all relevant for GUT model building. Needless to say
our strategy is not fixed to specific examples and anomaly
cancellation will be guaranteed generally. An inversion of the
one-loop result for the Coulomb branch Chern-Simons couplings in terms of
the chiral indices would shed additional light on the mechanisms ensuring
anomaly-freedom in F-theory.

Turning the story around, one can also use anomaly cancellation to derive
a set of very non-trivial conditions on the geometry of resolved
Calabi-Yau fourfolds. These have been given in
\eqref{eq:anomalyCondHomology} and display conditions on the
intersections of the resolution spheres supporting M2-branes in the
M-theory Coulomb branch. We have shown that these equations are satisfied
for our examples, but a direct general geometrical proof would be
desirable. Further support from a physical point of view for automatic
anomaly cancellation in F-theory can also be given in the weak string
coupling limit. We have recalled that in this Type IIB limit the
cancellation of D7-brane and induced D5-brane tadpoles suffices to
guarantee the absence of 4D anomalies. Absence of anomalies in F-theory
is then simply a consequence of the fact that the construction of the
elliptically fibered Calabi-Yau fourfold ensures the cancellation of
seven-brane tadpoles, while the choice of a harmonic $G_4$ flux guarantees
the absence of 5-brane tadpoles.

In our analysis of the weakly coupled setup we also noted that general
D7-brane stacks can support geometrically massive $U(1)$ gauge symmetries.
In the F-theory effective action derived via M-theory these $U(1)$ gauge
bosons and corresponding St\"uckelberg axions are already integrated
out and do not appear in the low energy spectrum. We have shown that the
projection to the reduced low energy spectrum allows to consistently match
Type IIB anomaly cancellation and F-theory anomaly cancellation.
It would be interesting to include these additional
massive $U(1)$ also into the F-theory setup and extend the analysis
accordingly.

Let us end by some additional observations and remarks.
It is interesting to recall that 4D anomalies are only induced in the
presence of $G_4$-flux and their cancellation employs the Green-Schwarz
mechanism only in the presence of gauged $U(1)$ symmetries. In contrast,
the coefficient $b^\alpha_I, b^\alpha_m$ and
$a^\alpha$ in front of the 4D topological terms $F\wedge F$ and
$R\wedge R$ can be extracted generally for any F-theory geometry without
making reference to the flux $G_4$.
Also anomalies are automatically satisfied for any $G_4$-flux that lifts
to F-theory.  Following \cite{Grimm:2012yq} it is thus natural to
conjecture that the geometric constraints are actually stronger than the
4D low-energy constraints. The $G_4$ fluxes can be used to
probe these geometric constraints and translate them into actual 4D
anomaly cancellation conditions. It would be
interesting to further investigate the theoretical underpinnings of these
observations.

Let us close by emphasizing that in this work only the match of discrete
data of topological couplings in 4D and 3D have been of importance. Indeed
one expects that this information is most uncontrollably extracted for a
dimensional reduction and can be traced through dualities.
For a complete study of the effective action, however, one will also need
to face the computation of less protected $\mathcal{N}=1$ data such as the K\"ahler
potential. While the classical matching can be found in
\cite{Grimm:2010ks} it remains to show how corrections can be reliably
computed in the M-theory to F-theory duality.

\subsubsection*{Acknowledgments}

We gratefully acknowledge discussions with Massimo Bianchi, Federico Bonetti, Yi-Zen Chu, Babak Haghighat, Jim Halverson,
Stefan Hohenegger, Andreas Kapfer, Jan Keitel, Albrecht Klemm, Eran Palti, Daniel Park, Erich Poppitz, Raffaele Savelli,
Sakura Sch\"afer-Nameki, Wati Taylor,
and Timo Weigand. We like to especially thank Eran Palti and Sakura Sch\"afer-Nameki  for
enlightening discussions about anomaly cancellation in F-theory including hypercharge flux.
The authors like to thanks the CERN Theory group and the Simons Center at Stony Brook
for hospitality and support. T.G.~also acknowledges hospitality and support
by the UPenn Theory group at Philadelphia. This work is supported in part
by DOE grant DE-SC0007901, the Fay R. and Eugene L. Langberg Endowed Chair
and the Slovenian Research Agency (ARRS).

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