Generalized structures of N=1 vacua

TL;DR

This work rewrites the problem of ${\cal N}=1$ vacua in type II string theory with fluxes as a problem in generalized complex geometry. By employing an ${\cal SU}(3)\times{\cal SU}(3)$ structure on the generalized tangent bundle $T\oplus T^*$, the authors derive two pure spinor equations, $(d+H\wedge)\Phi_1=0$ and $(d+H\wedge)\Phi_2=F_{RR}$, whose interpretation unifies twisted generalized Calabi–Yau geometry with RR-induced integrability defects. The analysis shows that, in general, the internal manifold is a complex–symplectic hybrid (IIA/IIB asymmetries appear in the type of generic structures), and that the RR flux acts as a defect to the integrability of the second pure spinor. In addition, the results connect these geometric conditions to topological models, discuss mirror-symmetry implications, and outline practical consequences for moduli counting and the search for compact AdS/Minkowski vacua, subject to Bianchi identities and flux equations of motion.

Abstract

We characterize N=1 vacua of type II theories in terms of generalized complex structure on the internal manifold M. The structure group of T(M) + T*(M) being SU(3) x SU(3) implies the existence of two pure spinors Phi_1 and Phi_2. The conditions for preserving N=1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N=2 and topological strings. They are (d + H wedge) Phi_1=0 and (d + H wedge) Phi_2 = F_RR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second.