Exceptional Calabi-Yau spaces: the geometry of N=2 backgrounds with flux

© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim In this paper we define the analogue of Calabi–Yau geometry for generic D=4, 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) ×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-Kähler 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. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to D=5 and D=6 flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in O(d, d) ×R + generalised geometry.