Hassaan Saleem

This is a post about compactifications in superstring theory and M theory. This post is related to my second Ph.D. project. A little familiarity with the Ricci tensor and the word “gauge group” is assumed but I tried to make it self-contained.

So let’s start with superstring theory including type II theories and heterotic string theories. These theories live in 10 spacetime dimensions and if we want to have four large spacetime dimensions, then we need to compactify the rest of 6 dimensions (i.e. find a suitable 6-dimensional manifold).

So, how to find a suitable 6-dimensional manifold? A reasonable condition to impose is to have N=1 supersymmetry in the large four dimensions (called 4D N=1 SUSY). Why N=1 supersymmetry and not more? Well, if we have more than N=1 supersymmetry, then we won’t be able to get the standard model because we won’t have chiral spinors.

So, if we impose 4D N=1 SUSY, then it turns out that the small 6D manifold should have a spinor that is “covariantly constant”. It means that the covariant derivative of this spinor should be zero. To proceed, we need to introduce a term named “holonomy”.

If we have a manifold then we can consider a vector (named V) at a point and parallel transport this vector to bring it back to the starting point after transporting it on a loop (see the picture below). It will come back as a vector (named P(V)). Now, for a certain kind of manifold called “Riemannian manifolds with Levi Civita connection” the length of this vector wouldn’t change but P(V) will be rotated with respect to V.

Now, since the length of the vector is unchanged, the transformation that takes V to P(V) should be an SO(N) transformation where N is the dimension of the manifold. However, the set of transformations that take V to P(V) for different loops forms what we call the holonomy group. For generic manifolds (like spheres), the holonomy group is SO(N) but for special manifolds, the holonomy group may be a subgroup of SO(N).

Now, let’s come back to the compactifications. We need a 6D manifold that has a covariantly constant spinor. Now, if we turn off some fields in string theory called the flux fields, then the string equations of motion imply that the Ricci tensor of the 6D manifold should be zero (we say that such a manifold is Ricci flat). It turns out that this implies that the holonomy group of this 6D manifold should be SU(3). Such manifolds are called Calabi Yau 3 folds (denoted as CY3).

Ok so now we have done the heavy lifting. Let’s come to M theory now. M theory lives in 11 spacetime dimensions. If we compactify M theory to have 4 large spacetime dimensions, then we need a small 7-dimensional manifold. Again, if we require 4D N=1 SUSY, then we again get the condition that the 7D manifold is Ricci flat. It turns out that such a 7D manifold has a very special holonomy group called the G2 group (it is one of the groups called the exceptional groups).

Now, it turns out that working with G2 manifolds is much harder than working with Calabi Yau manifolds. One of the reasons is that it is not guaranteed that such manifolds exist. This is not the case with Calabi Yau manifolds because a theorem named Yau’s theorem guarantees their existence.

So we have to construct G2 manifolds explicitly. Once such construction was done by D.D.Joyce. He started with a 7-dimensional torus and then he did a procedure called “orbifolding”. This procedure creates singularities. Joyce repaired these singularities using techniques from algebraic geometry. So, these Joyce manifolds are smooth (i.e. without singularities).

It turns out that if we do an M theory compactification on a small and smooth G2 manifold, then the physics that comes out of it has a particular feature i.e. it gives us Abelian gauge groups (unlike the standard model). To get nonabelian gauge groups (like the standard model has) we need to have small G2 holonomy manifolds that have different kinds of singularities.

The problem is that such manifolds have not yet been found. There is a certain kind of manifold called G2 cones which have a certain kind of singularities called the conical singularities but it is not small (the technical term is that it is not compact i.e. noncompact).