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We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes,in the framework of $\beta$-supergravity. We first provide expressions for thefermionic supersymmetry variations. Specifying a compactification ansatz tofour dimensions, we deduce internal Killing spinor equations. Thesesupersymmetry conditions are then reformulated in terms of pure spinors,similarly to standard supergravity vacua admitting an SU(3)xSU(3) structure inGeneralized Complex Geometry. The standard d-H acting on the pure spinors istraded for a generalized Dirac operator D, depending here on the non-geometricfluxes. Rewriting it with an exponential of the bivector $\beta$ leads us todiscuss the geometrical characterisation of the vacua in terms of a$\beta$-twist, in analogy to the standard twist by the b-field. Thanks to D, wealso propose a general expression for the superpotential to be obtained fromstandard supergravities or $\beta$-supergravity, and verify its agreement withformulas of the literature. We finally comment on the Ramond-Ramond sector, anddiscuss a possible relation to intermediate or dynamical SU(2) structuresolutions.

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