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  IX. M-Theory

Maps between string theories

We saw in a previous section that certain string theories that looked superficially different, turned out to be merely different descriptions of the same physics. We will be more precise about these statements in the present section. There are in fact five seemingly different string theories. They go by the name of Type I, Type IIA and Type IIB and Heterotic E8 and Heterotic SO32. The first has a reasonable name, the second and third are two sides of the same coin, and the fourth and the fifth are mixtures of the Type II and the bosonic string theory we discussed up until now. The bosonic string theory is not in this list because it does not incorporate supersymmetry. All five string theories that we listed are supersymmetric. The two heterotic string theories and the two type II theories are linked by T-duality, in a way very similar to the one described in the section on duality. Type I and heterotic SO32 are linked by an S-duality, and type I is an unoriented version of type IIB. Thus, all string theories are linked -- the question of which string theory is realized in nature and the embarrassment of choice is partly lifted, since all five string theories turn out to be linked by various dualities. But, as usual, new theoretical insight raises new theoretical questions.

Is there an S-duality for the Type IIB string ? Yes, and the dual string theory turns out to be type IIB again. The IIB superstring is self-dual. What about the S-dual for type IIA ? Actually, we don't know the dual of type IIA string theory, but we do know many if its properties. It has to be a theory that lives in eleven dimensions, which partly explains why we don't know it; none of the string theories we encountered thus far lives in eleven dimensions. We only know a partial description of an eleven dimensional supersymmetric theory that includes gravity, called eleven dimensional supergravity. That is part of the formulation of the S-dual of type IIA string theory, but we don't know much more than that.

A general description

Thus, all five string theories are linked by chains of dualities. They seem to be part of a bigger whole, that should also include the S-dual of type IIA string theory, and perhaps a lot more. The theory that would describe this big whole, we call M-theory, although we do not have a good description of it yet. We can sum up many of its properties (like the fact that they have to contain all string theories and the equivalence relations between them), but the formulation of the theory is difficult. M-theory remains a challenge for us, and we may have to pass down the problem for the next generation to solve it.