Systematics of perturbatively flat flux vacua

TL;DR

This paper systematically analyzes perturbatively flat flux vacua in mirror Calabi-Yau threefolds, classifying models by topology, computing invariants, and exploring non-perturbative effects, revealing new protected vacua and duality-based classes.

Contribution

It provides a detailed classification of PFFV based on topology, introduces the concept of exponentially flat vacua, and constructs new classes using S-duality.

Findings

K3-fibered models have more PFFV than Swiss-cheese models at fixed D3 charge.

Non-perturbative effects significantly reduce the number of trustworthy vacua.

Some PFFV are protected by symmetries, forming exponentially flat flux vacua.

Abstract

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.