Type IIB flux vacua from G-theory II

We find analytic solutions of type IIB supergravity on geometries that locally take the form $\text{Mink}\times M_4\times \mathbb{C}$ with $M_4$ a generalised complex manifold. The solutions involve the metric, the dilaton, NSNS and RR flux potentials (oriented along the $M_4$) parametrised by functions varying only over $\mathbb{C}$. Under this assumption, the supersymmetry equations are solved using the formalism of pure spinors in terms of a finite number of holomorphic functions. Alternatively, the solutions can be viewed as vacua of maximally supersymmetric supergravity in six dimensions with a set of scalar fields varying holomorphically over $\mathbb{C}$. For a class of solutions characterised by up to five holomorphic functions, we outline how the local solutions can be completed to four-dimensional flux vacua of type IIB theory. A detailed study of this global completion for solutions with two holomorphic functions has been carried out in the companion paper [1]. The fluxes of the global solutions are, as in F-theory, entirely codified in the geometry of an auxiliary $K3$ fibration over $\mathbb{CP}^1$. The results provide a geometric construction of fluxes in F-theory.