M-theory: Encyclopedia II - M-theory - Basics

M-theory - Basics

It was believed before 1995 that there were exactly five consistent superstring theories, which are called, respectively, the Type I string theory, the Type IIA string theory, the Type IIB string theory, the heterotic SO(32) (the HO string) theory, and the heterotic E₈×E₈ (the HE string) theory. As the names suggest, some of these string theories are related to each other. In the early 1990s, string theorists discovered that these relations were so strong that they could be thought of as an identification. The Type IIA string theory and the Type IIB string theory are connected by T-duality; this means, essentially, that the IIA string theory description of a circle of radius R is exactly the same as the IIB description of a circle of radius 1/R.

This is a profound result. First, it is an intrinsically quantum mechanical result; the identification is not true classically. Second, because we can build up any space by gluing circles together in various ways, it would seem that any space described by the IIA string theory can also be seen as a different space described by the IIB theory. This means that we can actually identify the IIA string theory with the IIB string theory; any object which can be described with the IIA theory has an equivalent although seemingly different description in terms of the IIB theory. This means that the IIA theory and the IIB theory are really aspects of the same underlying theory. It might be said at this point that we have reduced our count of fundamental string theories by one.

There are other dualities between the other string theories. The heterotic SO(32) and the heterotic E₈×E₈ theories are also related by T-duality; the heterotic SO(32) description of a circle of radius R is exactly the same as the heterotic E₈×E₈ description of a circle of radius 1/R. There are then really only three superstring theories, which might be called (for discussion) the Type I theory, the Type II theory, and the heterotic theory.

There are still more dualities, however. The Type I string theory is related to the heterotic SO(32) theory by S-duality; this means that the Type I description of weakly interacting particles can also be seen as the heterotic SO(32) description of very strongly interacting particles. This identification is somewhat more subtle, in that it identifies only extreme limits of the respective theories. String theorists have found strong evidence that the two theories are really the same, even away from the extremely strong and extremely weak limits, but they do not yet have a proof strong enough to satisfy mathematicians. However, it has become clear that the two theories are related in some fashion; they appear as different limits of a single underlying theory.

At this point, there are only two string theories: The heterotic string theory (which is also the type I string theory) and the Type II theory. There are relations between these two theories as well, and these relations are in fact strong enough to allow them to be identified.

This last step, however, is the most difficult and most mysterious. It is best explained first in a certain limit. In order to describe our world, strings must be extremely tiny objects. So when one studies string theory at low energies, it becomes difficult to see that strings are extended objects—they become effectively zero-dimensional (pointlike). Consequently, the quantum theory describing the low energy limit is a theory which describes the dynamics of particles moving in spacetime, rather than strings. Such theories are called quantum field theories. However, since string theory also describes gravitational interactions, one expects the low-energy theory to describe particles moving in gravitational backgrounds. Finally, since superstring string theories are supersymmetric, one expects to see supersymmetry appearing in the low-energy approximation. These three facts imply that the low-energy approximation to a superstring theory is a supergravity theory.

The possible supergravity theories were classified by W. Nahm in the 1970s. In 10 dimensions, there are only two supergravity theories, which are denoted Type IIA and Type IIB. This is not a coincidence. The Type IIA string theory has the Type IIA supergravity theory as its low-energy limit. Likewise, the Type IIB string theory gives rise to Type IIB supergravity. More interestingly, however, the heterotic SO(32) and heterotic E₈×E₈ string theories also reduce to Type IIA and Type IIB supergravity in the low-energy limit. This suggests that there may indeed be a relation between the heterotic/Type I theories and the Type II theories.

In 1995, Edward Witten outlined the following relationship: The Type IIA supergravity (corresponding to the heterotic SO(32) and Type IIA string theories) can be obtained by dimensional reduction from the single unique eleven-dimensional supergravity theory. This means that if one studied supergravity on an eleven-dimensional spacetime that looks like the product of a ten-dimensional spacetime with another very small one-dimensional manifold, one gets the Type IIA supergravity theory. (And the Type IIB supergravity theory can be obtained by using T-duality.) However, eleven-dimensional supergravity is not consistent on its own. It does not make sense at extremely high energy, and likely requires some form of completion. It seems plausible then, that there is some quantum theory—which Witten dubbed M-theory—in eleven-dimensions which gives rise at low energies to eleven-dimensional supergravity, and is related to ten-dimensional string theory by dimensional reduction. Dimensional reduction to a circle yields the Type IIA string theory, and dimensional reduction to a line segment yields the heterotic SO(32) string theory.

Taking seriously the notion that all of the different string theories should be different limits and/or different presentations of the same underlying theory, then the concept of string theory must be expanded. But little is known about this underlying theory. The bonus is that all of the different string theories may now be thought of as different limits of a single underlying theory.

Adapted from the Wikipedia article "Basics", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki