I think string theory is doing what it can to model our universe. But the math becomes so extremely complicated and esoteric that people like me get a headache after just reading a couple of sentences. And I'm talking about the math now.

" In the late 70s and very early 80s bosonic string theory had been constructed, initially with the hope it might explain the interaction between quarks. This was because the force between two quarks seemed to act like a string or tube between them, if you pulled them further apart the force attracting them increased, as if connected by a bit of elastic string.

Eventually QCD won out, using the notion of colour confinement and gluons to construct 'flux tubes'. String theory then entered a bit of a lull because bosonic string theory had problems with tachyons, as well as being totally unphysical because it lacked fermions. Someone then realised that if you put in fermions, a number of amazing things happened. The tachyons disappeared and the theory was supersymmetric. Previously, supersymmetry (which had been developed for usual QFT around the same time as bosonic string theory) had to be tagged on by hand, you could have unequal numbers of fermions and bosons. In string theory, if you had both kinds of particles, you had to have equal numbers!

However, there are varying degrees of supersymmetry. N=0 means 'no supersymmetry'. N=1 means 'a little bit' (in very non-technical terms) and it goes all the way up to N=8, "that's ****loads of supersymmetry!". The bigger the value of N, the more extra particles there were predicted, as if the particles we can currently see are a smaller and smaller tip of an iceberg. N=0, we can see all the iceberg, N=1 we see about half of it, N=8 we only see the very edge of it. The value of N also alters the particle content. N>1 is excluded by experiments because it doesn't account for the left/right handed nature of the Weak force (known as 'chirality'). Usual constructions of superstring theory had N=2. A bit of a problem.

Another problem was that superstring theory had lots of dimensions, 10 of them (9 space, 1 time). But we only see 4 (3+1) so the only way that makes sense is if 6 of them are too small for us to see with experiments. In QFT (and relativity in general), energy and length are inversely proportional (if you set c=G=1, your units are such that units of mass = 1/(units of length)), so that in order to access smaller and smaller length interactions, you need more and more energy. This makes sense if you think about how we need bigger and bigger accelerators to probe smaller and smaller objects. So if these dimensions hadn't appeared yet, they must be very small. Experiments say smaller than 10^(-15) metres, theory says as small as 10^(-35) metres! We'd need an accelerator larger than the universe to get that kind of energy!

So how to get past these two problems? Well one problem fixes the other. We have to make 6 of the dimensions small, 'compactified'. But how we do that affects the symmetry of our theory and so alters the amount of supersymmetry. If you compactify your 6 extra dimensions in the simplest way, onto 6 circles (ie a 6d torus, T^6), then you quadruple the amount of supersymmetry your theory has. So a 10d string theory with N=2 goes to a 4+6d string theory with N=8. Bugger, that didn't help.

Well perhaps our ideas about compactification need to be refined a little? Perhaps we need to make the small dimensions a little more complicated. This is where holonomy comes in. We want a theory with N=1 supersymmetry. This is akin to saying that we want a unique spinor in our theory under parallel transport around a closed loop in our space-time. In n dimensions, parallel transport around a closed loop acts like an operator on the spinor, specifically like a membler of a subgroup, G, of SO(n). In flat space-time G=SO(n). However, in complicated spaces you get G < SO(n). It turns out that in order to get a working theory, you need an even number of dimensions in your compact space, so n=2m, and to get the right kind of supersymmetry you need an holonomy group G = SU(m).

Through some rather unpleasant differential geometry (see Joyce), you can prove that a manifold with SU(m) holonomy is Kahler and Ricci flat and has vanishing 1st Chern Class and that's known as a Calabi-Yau space (since Calabi predicted and Yau proved those properties are equivalent). Thus 3 dimensional (3 complex dimensions) Calabi Yau spaces form the core of string theory interest in compact dimensions. Orbifolded and orientifolded tori are a kind of step in the right direction. A general Calabi Yau doesn't have anywhere near the nice description that tori do and you'd have to work through a huge number of cases. At present the orbi/orientifold constructions allow us to see the kind of problems which will arise in full Calabi Yau models and learn how to solve some of them in a nicer environment.

The difference between an orbifold and an orientifold is that an orientifold has had an extra quotienting process applied to it. Rather than just an orbifold group G, an orientifold also have the group (-1)^(F_L)Os quotiented out. (-1)^(F_L) is the 'left fermionic oscillation counter'. It's +1 if there's even numbers of fermionic oscillators going 'to the left' and -1 if odd. O is the worldsheet parity operator and s is an involution (ie a Z_2 map). Sometimes orbifolding isn't enough or you want to have additional structure."

On a 'a quick history of compactification' by AlphaNumeric, who actually works in this field.

And just reading this makes me realize how easily one can lose oneself in this type of advanced mathematics. But as I said, I have great hope and faith in mathematics and I think they are on to something. Another thing I wonder on is how they see it, as a linear field of study, or as a non-linear.

Simply expressed, how do we expect non-linearity to come out of linearity?