Supersymmetric backgrounds from generalized Calabi–Yau manifolds

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 Kähler form e iJ and the holomorphic form Ω. 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 Ω 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.

Exceptional Calabi–Yau spaces: the geometry of backgrounds with flux

“…Focussing for the moment on , backgrounds, in the case of NS‐NS flux such a reformulation has already appeared under the guise of generalised complex geometry . One considers a generalised tangent bundle , admitting a natural metric.…”

String Theory Compactifications

“…22 Harmonic forms are those that satisfy A = 0. It is not hard to show that a form is harmonic exactly if it is both closed and coclosed…”

“…In order to do so, we will use a formalism that geometrically covariantizes the symmetries of string theory and that is called generalized complex geometry. Generalized complex geometry was first proposed by Hitchin and his students in [6,7] in order to describe complex and symplectic geometry in a unifying formalism, before being utilized to describe flux compactifications of string theory [22,23]. In this section we introduce the mathematical formalism and discuss its relevance in string theory in the subsequent section.…”

Calculation of X-ray Magnetic Circular Dichroism for Sr-Substituted LaCoO3 by Discrete-Variational Hartree–Fock–Slater Method

“…In addition, one may also consider background fluxes. As shown in [13], in any supersymmetric compactification on a SU(3)-structure manifold one can define an hermitean metric, a normalizable supersymmetry generator η + , its complex conjugate η − , and a couple of differential forms J and Ω such that the relations (2.8) still hold. One can then check that the computations carried out in [8] also apply to this case, and that the supersymmetry equations (2.9) and (2.10) still apply to this more general type IIB backgrounds with fluxes and/or torsion.…”

“…, ζ h 0,2 }. 13 When we introduce the background flux G 3 , the geometrical moduli space of the D7-brane should be reduced to M F (S 4 ). Locally, this reduced moduli space would look like a submanifold M f (S 4 ) defined by imposing the equations (3.13) on the former moduli space.…”