Non-geometric flux vacua, S-duality and algebraic geometry

TL;DR

The work develops a systematic, algebraic‑geometry–driven framework to construct and solve duality‑invariant (T and S) flux vacua in Type IIB on T^6/(Z_2×Z_2) with O3/O7, by treating the non‑geometric Q flux as the gauge‑subalgebra and the P flux as its second‑cohomology deformation. By embedding Q and P into pairs of six‑dimensional Lie algebras and applying integrability and cohomology conditions, the authors reduce the problem to solving polynomial systems whose solutions correspond to admissible flux configurations and moduli stabilization. They demonstrate, via explicit isotropic examples, that both AdS_4 and supersymmetric Minkowski vacua exist, with some vacua requiring nonzero flux‑induced C_8 tadpoles and others having vanishing tadpoles, thereby mapping flux choices to localized sources. The approach leverages algebraic geometry tools (eg, prime decomposition of ideals) to classify branches of solutions and root alignments of flux‑induced polynomials, providing a constructive route to Minkowski vacua and illuminating the role of S‑duality in shaping non‑geometric flux backgrounds. Overall, the paper offers a concrete, computationally tractable path to stabilizing moduli in duality‑invariant flux vacua and clarifies the interplay between geometry, algebra, and tadpoles in non‑geometric string compactifications.

Abstract

The four dimensional gauged supergravities descending from non-geometric string compactifications involve a wide class of flux objects which are needed to make the theory invariant under duality transformations at the effective level. Additionally, complex algebraic conditions involving these fluxes arise from Bianchi identities and tadpole cancellations in the effective theory. In this work we study a simple T and S-duality invariant gauged supergravity, that of a type IIB string compactified on a $T^6/(Z_2 x Z_2)$ orientifold with O3/O7-planes. We build upon the results of recent works and develop a systematic method for solving all the flux constraints based on the algebra structure underlying the fluxes. Starting with the T-duality invariant supergravity, we find that the fluxes needed to restore S-duality can be simply implemented as linear deformations of the gauge subalgebra by an element of its second cohomology class. Algebraic geometry techniques are extensively used to solve these constraints and supersymmetric vacua, centering our attention on Minkowski solutions, become systematically computable and are also provided to clarify the methods.