Frozen

TL;DR

This work analyzes the duality between frozen M-theory ALE singularities and F-theory 7-branes on a circle, showing that a frozen M-theory singularity corresponds to a circle-compactified F-theory setup with a transverse rotation and monodromy described by $r = n/d$ (mod 1) and $g^d$. It connects the discrete flux to the commutant gauge algebra $\mathfrak{h}_r$ and to outer automorphisms of the elliptic fibration, arguing that Kodaira's classification encodes the resulting structure. A central result is that the only completely frozen F-theory singularity is the O7+ plane, and any (partially) frozen 7-brane must reduce to an O7+ with an integral number of D7-branes; no frozen half-D7 on O7+ is allowed. This work narrows the landscape of frozen singularities in F-theory, providing a framework for further study of genus-one fibrations without sections and their physical implications.

Abstract

We revisit the duality between ALE singularities in M-theory and 7-branes on a circle in F-theory. We see that a frozen M-theory singularity maps to a circle compactification involving a rotation of the plane transverse to the 7-brane, showing an interesting correspondence between commuting triples in simply-laced groups and Kodaira's classification of singular elliptic fibrations. Our analysis strongly suggests that the O7+ plane is the only completely frozen F-theory singularity.