Type II Strings and Generalized Calabi-Yau Manifolds

TL;DR

This paper reformulates supersymmetry conditions for type II string theories on six-manifolds using pure spinors, revealing a mirror symmetry with torsion and linking fluxes to generalized Calabi-Yau structures.

Contribution

It expresses supersymmetry conditions as differential equations on pure spinors, demonstrating mirror symmetry with torsion and connecting fluxes to generalized Calabi-Yau manifolds.

Findings

Supersymmetry conditions are symmetric under exchange of pure spinors.

RR fluxes modify only one of the two key equations.

Type IIB manifolds are always complex, IIA are twisted symplectic.

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.