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.
Highlights
Approximation to a point in the String Landscape [51]
2Of course, there are stringy world-sheet conformal field theories and other less geometric physical theories which can give rise to quantum theories in Minkowski spacetime, but here we focus on the geometric regime
A perhaps more direct, physical, related conjecture, which includes the possibility of Type B) and C) manifolds is Conjecture 3 Any absolutely stable geometric compactification of superstring/M theory to Minkowski space is based on a special holonomy Ricci flat metric on a compact manifold
Summary
We conclude this introduction with a simple intuitive sketch, mainly aimed at physicists, for why the Cheeger-Gromoll splitting theorem, which provides our basic understanding. Of the structure of Ricci-flat manifolds, is true. The basic idea is that on a compact Ricci flat manifold any harmonic 1-form is parallel with respect to the Levi-Cevita connection, ∇. This provides a one-to-one correspondence between harmonic one-forms and (Abelian) isometries. Each such one-form corresponds to splitting off precisely one flat direction. Similar calculations can be used to show that if V is a Killing vector field and the Ricci tensor vanishes that V is parallel and the corresponding one-form is harmonic.
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