-to-mass ratio exceeds that of an extremal black hole. In a stronger version of the WGC these charged states are even expected to populate a sublattice of the entire charge lattice \cite{Cheung:2014vva, Heidenreich:2015nta, Heidenreich:2016aqi,Montero:2016tif} or at least an infinite subtower of states \cite{Andriolo:2018lvp}. Alternatively one can interpret this condition as the statement that there should exist a tower of states for which the net force is repulsive \cite{Palti:2017elp}, \cite{Heidenreich:2019zkl, Lee:2018spm}.

\begin{figure}
  \center
  \includegraphics[width=0.55\textwidth]{mulambda.pdf}
  \caption{Infinite distance limits of the F-theory geometry $C_0\to B_3\to B_2$ (for the class where $C_0$ is a rational curve): The quantity $\lambda\to\infty$ parametrises a finite volume infinite distance limit in which the volumes $\mathcal{V}_{C_0}\to 0$ and $\mathcal{V}_{B_2}\to\infty$. The parameter $\mu$ describes an additional homogeneous rescaling $\mathcal{V}_{B_3}\sim\mu^3$. The plane is separated by the codimension one locus $\mu^2\sim\lambda$ at which a unique emergent heterotic obtained by wrapping a D3 brane on $C_0$ becomes light at the same rate as the KK scale. The region where $\mu^2/\lambda\to 0$ is obstructed by $\alpha'$-corrections, whereas the opposite regime $\mu^2/\lambda\to\infty$ is a decompactification limit where $M_{\rm KK}^2/T_{\rm str}\to0$. Note that $\Delta_0$ and $\Delta_1$ dualise to holomorphic and non-holomorphic threshold corrections in the heterotic frame, see Section \ref{sec_SecFHet}.}
  \label{fig:mu_lambda_plane}
\end{figure}



Consistency with the Emergent String Conjecture requires that in any infinite distance limit which does not correspond to an effective decompactification,
the states satisfying the (sub-) lattice WGC must be the excitations of emergent weakly coupled strings.
This was confirmed in non-perturbative compactifications with eight supercharges in 
six dimensions  \cite{Lee:2018urn, Lee:2018spm, Lee:2019apr} and with four supercharges in four dimensions \cite{Lee:2019jan}.
However, in compactifications to four dimensions quantum corrections have to be taken into account which can in principle obstruct classical infinite distance limits. This  has been shown for $N=2$ theories in the vector multiplet \cite{Lee:2019oct} and the hypermultiplet moduli space \cite{Marchesano:2019ifh, Baume:2019sry}. The issue of quantum corrections in four-dimensional setups with $N=1$ supersymmetry has been addressed in \cite{Cicoli:2018tcq,Gonzalo:2018guu,Xu:2020nlh}. 

In particular the results of \cite{Baume:2019sry} show that (non-)perturbative quantum corrections and tensionless strings go hand-in-hand: The quantum corrections precisely obstruct geometric limits in which the tension of the emergent, critical string would naively drop parametrically below the KK-scale. This would lead to a pathological phase of the theory 
 with a genuine critical four-dimensional string which is not expected to exist on general grounds (see (\ref{phases})). 


 In the present work, we discuss the effect of quantum corrections for the swampland conjectures in four-dimensional theories with ${N}=1$ supersymmetry\footnote{Quantum log corrections to swampland conjectures have been studied in~\cite{Blumenhagen:2019vgj,Blumenhagen:2020dea}.}. 
 We will find a conspiracy of quantum effects to obstruct precisely the pathological limits in (\ref{phases}), while both decompactification limits and emergent string limits
 are unobstructed (see Figure \ref{fig:mu_lambda_plane}). As our second main result we will see how the precise charge-to-mass ratio appearing in the WGC is modified at the quantum level as a consequence of gauge threshold corrections (cf. Figure \ref{fig:WGC_relation_corrections}).
 
 
 
\begin{figure}\center
  \includegraphics[width=\textwidth]{WGC.pdf}
  \caption{Left: Neglecting corrections away from the infinite distance point the (sub-)lattice WGC is fulfilled by a (super-)extremal tower of heterotic string excitations. Center: The gauge coupling is corrected away from the infinite distance point such that the former WGC states do not marginally lie above the classical bound (here, they lie below, as in the example of Section \ref{sec_WGC1loop}.). Right: Taking into account corrections to the relation $M_n^2\sim n T_{\rm str}$ the tower of formerly extremal states satisfies the repulsive force condition $|F_{\rm Coulomb}|\geq|F_{\rm Grav}|+|F_{\rm Yuk}|$ although it is still sub-extremal with respect to the classical extremality bound. We expect corrections to black hole solutions to be such that (super-)extremality of the string states is restored (red line).}
  \label{fig:WGC_relation_corrections}
\end{figure}

 
 
 
 To obtain these results, we reconsider the setup of \cite{Lee:2019jan} and investigate the role of perturbative $\alpha'$-corrections in infinite distance limits in the K\"ahler moduli space of F-theory compactifications on Calabi--Yau fourfolds.\footnote{For a discussion of infinite distance limits in the complex structure moduli space of Calabi--Yau fourfolds in the context of the swampland conjectures, see \cite{Grimm:2019ixq}.} Our starting point for this investigation are the leading perturbative  corrections  at order $(\alpha')^2$ in the form computed in \cite{Grimm:2013gma, Grimm:2013bha,Grimm:2017pid,Weissenbacher:2019mef, Weissenbacher:2020cyf}.
 The first part of this paper is thus dedicated to investigating whether such quantum corrections obstruct certain classical infinite distance limits similar to what happens in $N=2$ setups. 

Our main interest is in infinite distance limits that do not correspond to decompactification limits and thus reduce to a weakly-coupled string theory as displayed in Figure~\ref{fig:fibration}. To make contact with the WGC we further need a gauge theory to become weakly-coupled at the infinite distance point. As analysed in \cite{Lee:2019jan} such limits can only be attained if the base of the elliptically-fibered Calabi-Yau fourfold  admits a rational fibration. The emergent string then corresponds to a D3-brane wrapped on the rational fiber and is dual to a critical heterotic string. In the case of a $U(1)$ gauge theory, a sublattice of the string excitations captured by the elliptic genus classically satisfy the WGC \cite{Lee:2019jan}. Since also the asymptotic geometry of the moduli space receives perturbative corrections a particularly interesting question is whether this classical statement remains true even after the inclusion of quantum effects. 

Quantum corrections should also be a necessary ingredient when analysing the swampland conjectures upon  moving slightly away from the infinite distance points towards the interior of the moduli space. The general expectation is that swampland conditions such as the WGC should be still valid in the interior of the moduli space. However, quantities such as the charge-to-mass ratio of extremal black holes  receive corrections encoded in the corrections to the F-theory moduli space geometry slightly away from the strict emergent string limit. Analysing the structure of the F-theory corrections then allows us to also make a statement about the WGC slightly inside the quantum moduli space. 



\begin{figure}
\center
  \begin{tikzpicture}
      \node at (0,0) {\includegraphics[width=0.25\textwidth]{Fibration2.pdf}};
      \node at (-1,-1.4) {$\mathbb{R}^{1,3}$};
      \node at (-1.5,0.9) {$B_2$};
      \node at (0.8,2.85) {$C_0$};
    \end{tikzpicture}
  \caption{We consider F-theory compactifications on $\mathcal{E}_\tau\to Y_4\to B_3$. As in \cite{Lee:2019jan}, infinite distance limits with emergent strings feature a fibration structure $C_0\to B_3\to B_2$, such that the fiber $C_0$ shrinks relative to $B_2$. The role of the emergent string is played by a D3 brane wrapped on $C_0$. The curve $C_0$ can either be a torus leading to a Type II string, or a rational curve leading to a heterotic string.}
  \label{fig:fibration}
\end{