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

TL;DR

This paper introduces the concept of exceptional Calabi--Yau geometries as a generalization of traditional Calabi--Yau spaces for flux backgrounds in supergravity, linking solutions to integrable structures in exceptional generalised geometry.

Contribution

It defines exceptional Calabi--Yau geometries for flux backgrounds, connecting Killing spinor solutions with integrable structures in $E_{7(7)}\times\mathbb{R}^+$ generalised geometry, and explores their properties and examples.

Findings

Solutions correspond to integrable structures in generalised geometry.

Exceptional Calabi--Yau geometries unify complex, symplectic, and hyper-Kahler geometries.

Explicit examples and moduli space structures are provided.

Abstract

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. 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 $O(d,d)\times\mathbb{R}^+$ generalised geometry.