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Supersymmetry with non-geometric fluxes, or a $β$-twist in Generalized Geometry and Dirac operator

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)xSU(3) structure in Generalized Complex Geometry. The standard d-H acting on the pure spinors is traded for a generalized Dirac operator 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 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.