Minkowski$_4$ $\times$ $S^2$ solutions of IIB supergravity

TL;DR

This work provides a comprehensive classification of $\mathcal{N}=2$ Minkowski$_4$ solutions of type IIB supergravity with an $SU(2)_R$ symmetry realized by an $S^2$-foliation, recasting the supersymmetry conditions into PDEs across several branches. The analysis yields three main families: Mink$_6$-type from D5 sources, Mink$_5$-type from D7-D5-NS5 intersections, and a novel Mink$_4$ conformal Calabi–Yau sector, plus generalized systems related by dualities. It further delivers explicit examples, including a compact conformal Calabi–Yau solution, infinite AdS$_6$ sectors via cylindrical Laplace equations, and AdS$_5$ structures that reduce to orbifolds of $AdS_5\times S^5$, along with a linear-dilaton background connected to Little String Theory. The results illuminate how brane intersections and fluxes shape Minkowski and AdS vacua via a unifying pure-spinor / PDE framework, with implications for holography and flux compactifications.

Abstract

We classify $\mathcal N = 2$ Minkowski$_4$ solutions of IIB supergravity with an $SU(2)_R$ symmetry geometrically realized by an $S^2$-foliation in the remaining six dimensions. For the various cases of the classification, we reduce the supersymmetric system of equations to PDEs. These cases often accommodate systems of intersecting branes and half-maximally supersymmetric AdS$_{5,6,7}$ solutions when they exist. As an example, we analyze the AdS$_6$ case in more detail, reducing the supersymmetry equations to a single cylindrical Laplace equation. We also recover an already known linear dilaton background dual to the $(1,1)$ Little String Theory (LST) living on NS5-branes, and we find a new Minkowski$_5$ linear dilaton solution from brane intersections. Finally, we also discuss some simple Minkowski$_4$ solutions based on compact conformal Calabi-Yau manifolds.