Type II Strings and Generalized Calabi-Yau Manifolds

TL;DR

The authors address how to extend mirror symmetry to Type II flux compactifications on six-manifolds with SU(3) structure by formulating the supersymmetry conditions as differential constraints on a pair of pure spinors $\varphi_1 = e^{iJ}$ and $\varphi_2 = \Omega$. They show that a twisted generalized Calabi–Yau structure emerges, with one pure spinor being twisted-closed under $d_H = d + H\bullet$ (parity matching the RR flux), while RR fluxes back-react only on the other equation, yielding a framework in which IIA manifolds are twisted symplectic and IIB manifolds are complex. Mirror symmetry is realized as the exchange of the two pure spinors, consistent with flux parity, and the results unify geometric and flux data via generalized complex geometry. The work opens questions on tadpoles, deformations of twisted operators, and moduli spaces, but suggests that Hitchin’s generalized Calabi–Yau concepts provide a natural language for flux compactifications and their spectra.

Abstract

This is a short version of hep-th/0406137. 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 e^{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: e^{iJ} is closed under the action of the twisted exterior derivative in IIA theory, and similarly Omega is closed in IIB. This means that supersymmetric SU(3)-structure manifolds are always complex in IIB while they are twisted symplectic in IIA. Modulo a different action of the B-field, these are all generalized Calabi-Yau manifolds, as defined by Hitchin.