 equations
\begin{align}
 G_{\mu\nu}&=
 \frac{1}{2}\left(\partial_{\mu}\phi\partial_{\nu}\phi-
 \frac{1}{2}g_{\mu\nu}\left(\nabla\phi\right)^2\right)
 +\frac{e^{-\phi}}{8}\left(2F_{\mu\alpha}F_{\nu}^{~\alpha}
 -\frac{1}{2}g_{\mu\nu}F^2\right)\notag\\
 &+\frac{e^{-2\phi}}{12}\left(3H_{\mu\alpha\beta}H_{\nu}^{~\alpha\beta}
 -\frac{1}{2}g_{\mu\nu}H^2\right),
\end{align}
as well as the equations of motion for the dilaton field
\begin{equation}
 {\rm d}(\star {\rm d}\phi)=-\frac{e^{-\phi}}{4}F\wedge \star F-
 e^{-2\phi}H\wedge \star H,
 \label{phi}
\end{equation}
which is written in this way for later
convenience. The Maxwell equation is
\begin{equation}
 {\rm d}\star (e^{-\phi}F)= \star e^{-2\phi}H\wedge F,
 \label{f}
\end{equation}
while the $H$ field obeys
\begin{equation}
 {\rm d}\star (e^{-2\phi}H)=0.
 \label{h} 
\end{equation}

In the following section, we introduce the concept of 
flux and distinguish it from charge.
We comment on how the fluxes of the Abelian fields 
we investigate are related to circular
bundles and two-bundles. The Maxwell field leads 
to two fluxes electric and magnetic.
The higher gauge field $H$ determines a flux as well,
which by analogy is of magnetic type.
Equation (\ref{h}) shows that $H$ can be effectively 
replaced by a dual scalar field. This field
is the axion. Comparing the axion and dilaton allow for
a fourth flux to be defined which is of electric type.

The backgrounds we consider share the global
structure of the Taub--NUT vacuum solution,
as well as many of its local similarities. 
These dilatonic Taub--NUT backgrounds are 
described in detail in section \ref{nuts}
In section \ref{fluxq} we present the 
flux quantization which arises in these configurations 
due to nontrivial topological structures
which underline them. Finally, in section \ref{disc}
we summarize our results and compare them with previous
research. We also comment on how the solutions we have studied
fit into a larger class of solutions.
Some of our results are particular to the special cases presented
here and some are valid in general.

\section{Charges and Fluxes}

In this section, we introduce the concepts of charge and flux
that apply to differential forms of any degree, 
representing field strengths and currents.
We are interested in describing fluxes which resemble those of
Maxwell theory. Maxwell's equations are statements about
 a differential two-form $F$, which represents the field strength. 
The gauge symmetry of these fields can be encoded in U(1)-bundles,
and the study of their topological invariants has led to
many interesting physical effects. In string theory, similar fields
arise which are also Abelian in nature, i.e.,
\begin{equation}
 C_{n}\rightarrow C_{n}+{\rm d}B_{n-1},
\end{equation}
however, they lead to more complicated bundle structures, $n$-bundles.
In a principal $n$-bundle, the fiber space corresponds to an $n$-group.
Since the U(1) structure can be restated in terms of spacetime 
cohomology,  a similar approach can be performed for field strengths
of higher rank. This is the main idea of our method and for which we
follow the concepts presented in \cite{ao}. Therein,
the distinction is stressed between charges and 
fluxes as they are classified by distinct homological structures of spacetime. 
Charges are produced by currents and fluxes are produced by fields. Of course,
both are related but we will remark their differences in the following analysis. 

This work focuses on a dilatonic Taub--NUT space \cite{jm} that arises
from a truncation of four dimensional low-energy effective heterotic string theory. 
The solution has a scalar dilaton field, 
an electromagnetic field, and a Kalb--Ramond-like field.
Since currents are physically absent we focus only on fluxes.
The equations of motion are more complicated than in the case of
Maxwell's theory, because of string theory intricacies,
but they allow for an appropriate interpretation
for these topological methods to hold.

To distinguish charges from fluxes, let us first consider the action
\begin{equation}
 \int\limits_{M} e^{-\phi}F\wedge \star F+A\wedge \star j,
\end{equation}
defined over a spacetime $M$. This action together with $F={\rm d}A$ yield
\begin{subequations}
\begin{align}
 {\rm d}F&=0,\label{maxwell0}\\ 
 {\rm d}\star(e^{-\phi}F)&=\star j,\label{maxwell}
\end{align} 
\end{subequations}
which are Maxwell's equations in the presence of a dilaton field $\phi$. 
We can immediately conclude that $\star j$ is a conserved current. However,
physical currents are also localized. So we have both
\begin{subequations}
\begin{align}
 {\rm d}\star j&=0,\label{dstarj}\\ 
 \star j|_{\partial M}&=0.\label{starj}
\end{align} 
\end{subequations}
Here, we have in mind a manifold which includes a spacetime and its asymptotic boundary.
In other words, Eq.\ (\ref{starj}) is a boundary condition. This condition may be relaxed
in a compact manifold; however, the analysis of this case is beyond the scope of the present work.
The charge on a (hyper)surface $S\subset M$ is defined by
\begin{equation}
 Q_E(S)=\int\limits_{S}\star j,
\end{equation}
for $\partial S\subset \partial M$. If we now consider $C\subset M$ and
$\beta\subset \partial M$, then conditions (\ref{dstarj}-\ref{starj}) imply
\begin{equation}
 Q_E(S+\partial C+\beta)-Q_E(S)=
 \int\limits_{\partial C}\star j+\int\limits_{\beta}\star j=
 \int\limits_{C}{\rm d}\star j=0,\label{qe}
\end{equation}
where we have used Stokes' theorem.
We conclude that the electric charge is an invariant of the
relative homology of spacetime modulo its boundary.

As is well-known, the nontrivial geometry of spacetime in electrovacuum
can lead to solutions without current, but with a nontrivial
field strength. This is an indication of the existence of a field flux
without charge. Electric flux is a simpler object than electric charge;
it is defined only 
through the field without reference to currents, i.e.,
\begin{equation}
 \Phi_E(\beta)=\int\limits_{\beta}\star F,
\end{equation}
which is is an invariant of the absolute homology of the
spacetime boundary. Likewise, magnetic flux is linked to the
absolute homology of spacetime itself according to the definition
\begin{equation}
 \Phi_M(b)=\int\limits_{b} F,
\end{equation}
where $\partial b=0$. Of course, charges do lead to fluxes as
can be concluded from Eq.\ (\ref{maxwell}), i.e., 
\begin{equation}
 Q_E(S)=\int\limits_{S}\star j=
 \int\limits_{S}{\rm d}\star (e^{-\phi}F)=
 \int\limits_{\partial S}\star F,
 \label{chf}
\end{equation}
whenever $\phi|_{\partial M}=0$. In Eq.\ (\ref{chf}) 
we used Stokes' theorem just as we did in relation (\ref{qe}).
Once more we emphasize that we are interested here
only in electric and magnetic type fluxes. Currents will be absent
and hence charges are null.

Finally, we mention that the equations of motion that
interest us intermingle the fields, but they can be written in a way
closely following Maxwell's Eqs.\ (\ref{maxwell0}-\ref{maxwell}).
The role of the currents are effectively played by the
field content that behaves as in Eqs.\ (\ref{dstarj})--(\ref{starj}).


\section{NUT Spaces from String Theory}
\label{nuts}

The first dilatonic Taub--