Effective actions and N=1 vacuum conditions from SU(3) x SU(3) compactifications

TL;DR

We analyze Type II compactifications on general $SU(3)\times SU(3)$ structure manifolds with a wide array of fluxes, including nongeometric ones, showing that the 4d effective theory is a gauged $N=2$ supergravity derived from generalized geometry on $T\oplus T^*$. The moduli spaces of even/odd pure spinors carry Hitchin-based special Kähler metrics, and these metrics reproduce the kinetic terms for internal metric and $B$-field fluctuations; the $N=1$ vacua are obtained from both the $N=2$ action and an $N=1$ truncation, with a direct correspondence to the integrated ten-dimensional pure spinor equations. The $N=2$ data reorganize under $N=2\to N=1$ truncations into a holomorphic superpotential and D-terms, and the resulting F- and D-flatness conditions match the integrated pure spinor constraints, including in the presence of electric and magnetic RR NS/geometric/nongeometric fluxes. Throughout, the twisted Hodge star $*_B$ and the generalized differential $\mathcal{D}$ connect 4d and 10d pictures, offering a coherent path to explicit models (e.g., $O6$-plane truncations) and clarifying how nongeometric fluxes participate in moduli stabilization.

Abstract

We consider compactifications of type II string theory on general SU(3) x SU(3) structure backgrounds allowing for a very large set of fluxes, possibly nongeometric ones. We study the effective 4d low energy theory which is a gauged N=2 supergravity, and discuss how its data are obtained from the formalism of the generalized geometry on T+T*. In particular we relate Hitchin's special Kaehler metrics on the spaces of even and odd pure spinors to the metric on the supergravity moduli space of internal metric and B-field fluctuations. We derive the N=1 vacuum conditions from this N=2 effective action, as well as from its N=1 truncation. We prove a direct correspondence between these conditions and an integrated version of the pure spinor equations characterizing the N=1 backgrounds at the ten dimensional level.