Type II embeddings for $d=6$ Einstein-Maxwell gauged supergravity

TL;DR

This work develops a G-structure and bi-spinor framework to embed $d=6$ Einstein-Maxwell (gauged) supergravity, along with its ungauged and tensor/vector subsectors, into type II string theory. By translating six-dimensional supersymmetry into geometric conditions on a four-dimensional internal manifold and analyzing both cases where the gauge field is and is not part of the internal metric, the authors classify embedding manifolds and uplift formulae, identifying a Toda-like class that governs the full gauged theory and a universal uplift for the ungauged limit. They demonstrate the existence of bounded embeddings within these classes and provide explicit realizations in IIA/IIB (including CY$_2$ and F-theory-type geometries), thereby illustrating concrete pathways to consistent truncations of string theory to minimal 6d supergravities with matter. The results offer a structured program for constructing lower-dimensional gauged supergravities from higher-dimensional string theories and point to future directions such as exploring other gaugings, relaxing the equal-norm spinor assumption, and extending to $d=5$ or $d=4$ theories with minimal supersymmetry. The practical impact lies in enabling controlled string uplifts of phenomenologically interesting 6d theories with vector and tensor multiplets, with potential applications to holography and compactification model-building.

Abstract

Bi-spinor and G-structure methods are used to classify the possible consistent truncations of type II supergravity to $d=6$ Einstein-Maxwell (gauged) supergravity, and its consistent sub-sectors. In the absence of R-symmetry gauging and a tensor multiplet we establish that every supersymmetric Mink$_6$ solution defines an embedding of the $d=6$ theory. Adding a tensor multiplet places restrictions on these embeddings, but embeddings still exist. In the presence of R-symmetry gauging the internal spaces of the embeddings are neither related to Mink$_6$ or AdS$_6$. Under the assumption that the internal space contains a single U(1) isometry housing the $d=6$ gauge field we classify the possible embedding manifolds. We find two classes of embedding for the entire theory, one of which is governed by a Toda-like equation and contains at least one bounded embedding. In the absence of a tensor multiple the classes of embeddings become more permissive, though the PDEs governing them become more complicated in general.