Classification of Modular Symmetries in Type IIB Flux Landscape

TL;DR

The work develops a systematic framework to classify modular symmetries in the Type IIB flux landscape by constructing period vectors on toroidal orbifolds and enforcing consistency with duality symmetries. A central innovation is the Scaling duality, a generalized S-transformation, which, together with duality constraints from the mass spectrum, fixes the otherwise ambiguous third-cohomology basis and its associated period vector. The authors show that the period vector transformations reduce to subgroups like $\bar{\Gamma}_0(2)$ (and its semidirect extension with a $\mathbb{Z}_2$) when paired with a Scaling duality, providing a concrete mechanism to classify modular symmetries of the untwisted complex-structure modulus $U$. Applying this to the $T^6/\mathbb{Z}_4$ orientifold, they derive explicit flux expansions, superpotential coefficients, and F-term equations, demonstrating how modular transformations act on the flux data to preserve the effective theory structure. The results offer a principled path to map the vacuum structure of the type IIB flux landscape across various toroidal orbifolds and guide future analyses of tadpole constraints and Dirac quantization.

Abstract

In this work, we study modular symmetries in type IIB flux landscape by investigating symplectic basis transformations of period vectors on toroidal orbifolds. To fix explicit cycles of a third-cohomology basis regarding the untwisted complex structure modulus, which is necessary to construct the period vectors, we find that the following two symmetries are required for the period vectors: (i) ``Scaling duality '' which is a generalized $S$-transformation of $PSL(2, \mathbb{Z})$ and (ii) the modular symmetries to be consistent with symmetries derived from mass spectra of the closed string in type IIB string theory. Furthermore, by considering flux quanta on the cycles, we explore type IIB flux vacua on toroidal orientifolds and flux transformations under the modular symmetries of the period vectors.