Generalized structures of N=1 vacua

TL;DR

This paper characterizes N=1 vacua in type II string theories using generalized complex geometry, revealing conditions that describe the internal manifold as a twisted generalized Calabi-Yau with implications for supersymmetry preservation.

Contribution

It introduces a unified framework using pure spinors to describe N=1 vacua, generalizing previous equations from N=2 and topological string theories.

Findings

Internal space is a twisted generalized Calabi-Yau manifold.

Supersymmetry conditions are expressed as simple differential equations.

RR-fields act as integrability defects for the pure spinor structure.

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.