Internal circle uplifts, transversality and stratified G-structures

TL;DR

This paper investigates the complex stratified G-structures in M-theory compactifications on eight-manifolds by analyzing their uplift to nine-manifolds, revealing intricate relations between stabilizers, distributions, and geometric structures.

Contribution

It provides a detailed explanation of the stratified G-structures in eight-manifolds through the uplift to nine-manifolds, establishing a formal dictionary between SU(2) and SU(3) structures.

Findings

The distribution on the nine-manifold can be transverse or non-transverse to the pull-back of the tangent bundle.

The stratified G-structure on the eight-manifold is linked to the intersection properties of distributions.

Explicit relations between SU(2) structures on the eight-manifold and SU(3) structures on the nine-manifold are established.

Abstract

We study stratified G-structures in ${\cal N}=2$ compactifications of M-theory on eight-manifolds $M$ using the uplift to the auxiliary nine-manifold ${\hat M}=M\times S^1$. We show that the cosmooth generalized distribution ${\hat {\cal D}}$ on ${\hat M}$ which arises in this formalism may have pointwise transverse or non-transverse intersection with the pull-back of the tangent bundle of $M$, a fact which is responsible for the subtle relation between the spinor stabilizers arising on $M$ and ${\hat M}$ and for the complicated stratified G-structure on $M$ which we uncovered in previous work. We give a direct explanation of the latter in terms of the former and relate explicitly the defining forms of the $\mathrm{SU}(2)$ structure which exists on the generic locus ${\cal U}$ of $M$ to the defining forms of the $\mathrm{SU}(3)$ structure which exists on an open subset ${\hat {\cal U}}$ of ${\hat M}$, thus providing a dictionary between the eight- and nine-dimensional formalisms.