Reading off the nongeometric scalar potentials via the topological data of the compactifying Calabi-Yau manifolds

Abstract

In the context of studying the 4D-effective potentials of type IIB nongeometric flux compactifications, this article has a twofold goal. First, we present a modular invariant symplectic rearrangement of the tree level nongeometric scalar potential arising from a flux superpotential which includes the S-dual pairs of nongeometric fluxes ($Q$, $P$), the standard NS-NS and RR three-form fluxes (${F}_{3}$, ${H}_{3}$), and the geometric flux ($\ensuremath{\omega}$). This ``symplectic formulation'' is valid for arbitrary numbers of K\"ahler moduli, and the complex structure moduli which are implicitly encoded in a set of symplectic matrices. In the second part, we further explicitly rewrite all the symplectic ingredients in terms of saxionic and axionic components of the complex structure moduli. The same leads to a compact form of the generic scalar potential being explicitly written out in terms of all the real moduli/axions. Moreover, the final form of the scalar potential needs only the knowledge of some topological data (such as Hodge numbers and the triple-intersection numbers) of the compactifying threefolds and their respective mirrors. Finally, we demonstrate how the same is equivalent to say that, for a given concrete example, various pieces of the scalar potential can be directly read off from our generic proposal, without the need of starting from the K\"ahler and superpotentials.