$ \mathcal{N} $ = 2 vacua in generalized geometry

Mariana Graña,Francesco Orsi

doi:10.1007/jhep11(2012)052

Abstract

Abstract We find the conditions on compactifications of type IIA to four-dimensional Minkowski space to preserve $ \mathcal{N} $ = 2 supersymmetry in the language of Exceptional Generalized Geometry (EGG) and Generalized Complex Geometry (GCG). In EGG, off-shell $ \mathcal{N} $ =2 supersymmetry requires the existence of a pair of SU(2)R singlet and triplet algebraic structures on the exceptional generalized tangent space that encode all the scalars (NS-NS and R-R) in vector and hypermultiplets respectively. We show that on shell $ \mathcal{N} $ = 2 requires, except for a single component, these structures to be closed under a derivative twisted by the NS-NS and R-R fluxes. We also derive the corresponding GCG-type equations for the two pairs of pure spinors that build up these structures.

Highlights

Supergravity of the form M1,9 = R1,3 × M6 require the internal manifold M6 to be CalabiYau [1]
We find the conditions on compactifications of type IIA to four-dimensional Minkowski space to preserve N = 2 supersymmetry in the language of Exceptional Generalized Geometry (EGG) and Generalized complex geometry In Generalized (Complex) Geometry (GCG)
The paper is organized as follows: in section 2 we introduce the necessary concepts of generalized complex geometry

Summary

Generalized complex geometry

In Generalized (Complex) Geometry [4, 5], one constructs algebraic structures on the generalized tangent bundle T M ⊕ T ∗M These structures appear in compactifications of type II theories as they are constructed from the tensor product of two internal spinors. Where the plus and minus refer to spinor chirality, and φ is the dilaton, which defines the isomorphism between the spinor bundle and the bundle of forms. We finish this section by mentioning that the 6d annihilator space of an O(6, 6) pure spinor can be thought as the holomorphic bundle of a generalized almost complex structure (GACS) J , which is a map from T M ⊕ T ∗M to itself such that it satisfies the hermiticity condition (J tηJ = η) and J 2 = −112. The GACS can be obtained from the pure spinor by [10, 11]7

Exceptional generalized geometry

Comparing to equations coming from Killing spinors

Discussion