SU(3) x SU(3) compactification and mirror duals of magnetic fluxes

TL;DR

This work extends the framework of $SU(3)$-structure compactifications to generalized $SU(3)\times SU(3)$ structures in type II theories, formulated via Hitchin's generalized geometry on the $E=F\oplus F^*$ bundle. It shows how to reorganize ten-dimensional fields into $\mathcal{N}=2$-type multiplets and derives Killing prepotentials that remain manifestly mirror-symmetric under exchange of the two pure spinors $\Phi^+$ and $\Phi^-$. The authors demonstrate that the mirror of Calabi–Yau compactifications with magnetic NS flux is captured within this generalized framework and that non-geometric fluxes (Q, R) are generically required to realize the full charge matrix $\mathcal{Q}$, yielding consistent low-energy theories even when the background is not geometrical. They also connect these results to generalized twisted-torus constructions and to non-geometric superpotentials studied in STW, clarifying how geometric and non-geometric fluxes populate the low-energy spectrum and their impact on moduli stabilization. Overall, the paper provides a comprehensive, mirror-symmetric description of NS/RR fluxes in $SU(3)\times SU(3)$ compactifications and clarifies how non-geometric backgrounds fit into a supersymmetric, geometrically structured formalism with concrete implications for 4D effective theories.

Abstract

This paper analyses type II string theories in backgrounds which admit an SU(3) x SU(3) structure. Such backgrounds are designed to linearly realize eight out of the original 32 supercharges and as a consequence the low-energy effective action can be written in terms of couplings which are closely related to the couplings of four-dimensional N=2 theories. This generalizes the previously studied case of SU(3) backgrounds in that the left- and right-moving sector each have a different globally defined spinor. Given a truncation to a finite number of modes, these backgrounds lead to a conventional four-dimensional low-energy effective theory. The results are manifestly mirror symmetric and give terms corresponding to the mirror dual couplings of Calabi-Yau compactifications with magnetic fluxes. It is argued, however, that generically such backgrounds are non-geometric and hence the supergravity analysis is not strictly valid. Remarkably, the naive generalization of the geometrical expressions nonetheless appears to give the correct low-energy effective theory.