Exceptional Calabi--Yau spaces: the geometry of $\mathcal{N}=2$ backgrounds with flux

Understand

In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory.

• We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ generalised geometry.
• Such "exceptional Calabi--Yau" geometries are determined by two generalised objects that parametrise hyper- and vector-multiplet degrees of freedom and generalise conventional complex, symplectic and hyper-Kahler geometries.
• The integrability conditions for both hyper- and vector-multiplet structures are given by the vanishing of moment maps for the "generalised diffeomorphism group" of diffeomorphisms combined with gauge transformations.