Supersymmetric Backgrounds from Generalized Calabi-Yau Manifolds

TL;DR

The paper develops a unified description of type II flux compactifications on six-manifolds using a pair of pure spinors, $e^{iJ}$ and $\Omega$, revealing a mirror-symmetric structure that exchanges the two spinors and the RR parity. NS flux twists the geometry into Hitchin’s generalized Calabi–Yau framework, while RR fluxes enter through Clifford-product couplings that preserve the symmetry and lead to a representation-by-representation classification of SUSY vacua. In particular, IIB backgrounds are always complex, whereas IIA backgrounds are twisted-symplectic, and the RR contributions generally affect only one of the two pure-spinor equations. The core results are the explicit pure-spinor differential equations (the p1–p4 system), their interpretation in generalized complex geometry, and the connection to topological strings via a decoupled RR sector and a proposed pure-spinor superfield formalism for a unified superpotential. This framework provides a powerful, symmetric language for classifying and constructing $\mathcal{N}=1$ flux vacua in type II theories with torsion.

Abstract

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form exp(iJ) and the holomorphic form Omega. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: exp(iJ) is closed under the action of the twisted exterior derivative in IIA theory, and similarly Omega is closed in IIB. Modulo a different action of the B-field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.