# Supersymmetry, Ricci Flat Manifolds and the String Landscape

## TL;DR

This paper explores the mathematical and physical conditions under which superstring and M-theory compactifications can preserve supersymmetry, focusing on Ricci flat manifolds and their holonomy properties, and discusses stability issues.

## Contribution

It formulates a necessary mathematical conjecture linking Ricci flat manifolds' holonomy to supersymmetry in string/M-theory compactifications and analyzes stability of non-simply connected cases.

## Key findings

- All stable Ricci flat manifolds likely have special holonomy in dimensions below eleven.
- Many non-simply connected Ricci flat manifolds are unstable due to bubble of nothing instabilities.
- The landscape of Minkowski compactifications is predominantly supersymmetric under certain conditions.

## Abstract

A longstanding question in superstring/$M$ theory is does it predict supersymmetry below the string scale? We formulate and discuss a necessary condition for this to be true; this is the mathematical conjecture that all stable, compact Ricci flat manifolds have special holonomy in dimensions below eleven. Almost equivalent is the proposal that the landscape of all geometric, stable, string/$M$ theory compactifications to Minkowski spacetime (at leading order) are supersymmetric. For simply connected manifolds, we collect together a number of physically relevant mathematical results, emphasising some key outstanding problems and perhaps less well known results. For non-simply connected, non-supersymmetric Ricci flat manifolds we demonstrate that many cases suffer from generalised Witten bubble of nothing instabilities.