Supersymmetry, Ricci Flat Manifolds and the String Landscape

Understand

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.