Systematics of perturbatively flat flux vacua

In this article, we present a systematic analysis of the so-called perturbatively flat flux vacua (PFFV) for the mirror Calabi-Yau (CY) $3$-folds ($\tilde{X}_3$) with $h^{1,1}(\tilde{X}_3) =2$ arising from the Kreuzer-Skarke database of the four-dimensional reflexive polytopes. We consider the divisor topologies of the CY $3$-folds for classifying the subsequent models into three categories; (i) models with the so-called Swiss-cheese structure, (ii) models with the $K3$-fibered structure, and (iii) the remaining ones which we call as models of "Hybrid type". In our detailed analysis of PFFV we find that for a given fixed value of the D$3$ tadpole charge $N_{\text{flux}}$, the $K3$-fibered mirror CY $3$-folds have significantly larger number of such PFFV as compared to those which have Swiss-cheese structure, while the Hybrid type models have a mixed behavior. We also compute the Gopakumar-Vafa invariants necessary for fixing the flat valley in the weak string-coupling and large complex-structure regime by using the non-perturbative effects, which subsequently reduce the number of physically trustworthy vacua quite significantly. Moreover, we find that there are some examples in which the PFFV are protected even at the leading orders of the non-perturbative effects due to the some underlying symmetry in the CY geometry, which we call as "Exponentially flat flux vacua". We also present a new class of PFFV using the $\mathcal{S}$-duality arguments.