Towards the Non-Perturbative Completion of 4d N=1 Effective Theories of Gravity

We show that four-dimensional $\mathcal N=1$ effective theories of gravity obtained from string compactifications require a non-perturbative completion, as additional light states of non-perturbative origin must be incorporated in the small volume regime to obtain a consistent low-energy description. This completion becomes concrete in subsectors that locally exhibit enhanced supersymmetry, where the enhancement predicts the existence of additional light degrees of freedom absent in the perturbative description. Motivated by analogous setups in six-dimensional $\mathcal N=(1,0)$ theories, we focus on the K\"ahler moduli space of F-theory compactifications on Calabi--Yau fourfolds to four dimensions, where shrinkable curves not intersected by 7-planes realize such supersymmetry-enhanced subsectors. Guided by the enhanced supersymmetry, we use F-theory to identify the degrees of freedom missing in the small volume regime of the perturbative Type IIB description. A consistent embedding of these local subsectors into a four-dimensional $\mathcal N=1$ theory of gravity requires an appropriate inclusion of complex structure moduli and spacetime-filling D3-branes. We also discuss supersymmetry enhancement in the complex structure sector and study how a heterotic dual description gives a unifying picture of the different F-theory sectors with enhanced supersymmetry. Finally, we comment on cases without local supersymmetry enhancement.