Abstract
We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes, in the framework of β-supergravity. We first provide expressions for the fermionic supersymmetry variations. Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry conditions are then reformulated in terms of pure spinors, similarly to standard supergravity vacua admitting an SU(3)×SU(3) structure in Generalized Complex Geometry. The standard d–H∧ acting on the pure spinors is traded for a generalized Dirac operator $$ \mathcal{D} $$ , depending here on the non-geometric fluxes. Rewriting it with an exponential of the bivector β leads us to discuss the geometrical characterisation of the vacua in terms of a β-twist, in analogy to the standard twist by the b-field. Thanks to $$ \mathcal{D} $$ , we also propose a general expression for the superpotential to be obtained from standard supergravities or β-supergravity, and verify its agreement with formulas of the literature. We finally comment on the Ramond-Ramond sector, and discuss a possible relation to intermediate or dynamical SU(2) structure solutions.
Highlights
(2.28) and (2.29) reduce toeφd ́ ∇q aιa T eφΦ1 ̄ ιae2AΦ1 ́ eφd ́ ∇q aιaT ́ eφ Re Φ2 eAd 3∇q a ̄ ιaeA Re Φ2 Im Φ2
Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry conditions are reformulated in terms of pure spinors, to standard supergravity vacua admitting an SU(3)ˆSU(3) structure in Generalized Complex Geometry
The standard dH^ acting on the pure spinors is traded for a generalized Dirac operator D, depending here on the nongeometric fluxes
Summary
We explained in the Introduction that β-supergravity is, in its current state, only formulated in its NSNS (bosonic) sector. Where notations are defined in the Introduction and appendix A, and ǫ1,2 are the SUSY fermionic parameters, while the upper, lower, sign refers respectively to the number 1, 2 These conventions match those of [25], except for thethere denoted 1, 2 here, and the use here of flat indices. We consider the fermionic SUSY variations of β-supergravity to be given analogously by (2.5), where we replace the vielbein e by e, the fermions ψM1,2, ρ1,2 by ψM1,2, ρ1,2, and use the following derivatives determined in [16]. From (2.5) and (2.6), with aligned vielbeins, we deduce the NSNS contribution to the fermionic SUSY variations of both a type IIA and IIB β-supergravity: it is given by (1.2), that we repeat here for convenience ηCF. ́ ΓABAφ ̄ ΓAηAB βBC BC φ ́ T B ǫ1,2
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