M-theory

M-theory (sometimes also called U-theory) is a theory of physics, which is believed to incorporate and generalize the various superstring theories. Edward Witten proposed the existence of this physical model at a conference at USC in 1995, explaining a number of previously observed dualities and sparked a flurry of new research in string theory.

M-theory is not complete. It can be applied in many situations (usually by exploiting string theoretic dualities), but the full theory is not yet known. (The theory of electromagnetism was also in such a state in the mid-19th century. There were separate theories for electricity and magnetism, and although they were known to be related, it was not clear until Maxwell published his equations just what the relationship was.) Witten has suggested that a general formulation will probably require the development of new mathematical language.

Background

It was shown in the early 1990s that the various superstring theories were related by dualities, which allowed physicists to relate the description of an object in one string theory to the description of a different object in another theory. These relationships imply that each of the string theories is a different aspect of a single underlying theory, which has been named "M-theory".

The Theory

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) string theory, and the heterotic $E_{8}xE_{8}$ 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 heterotic $E_{8}xE_{8}$ 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 $E_{8}xE_{8}$ 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 $E_{8}xE_{8}$ 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.

Naming conventions, or What does M stand for?

There are two issues to be dealt with here:

When Witten named M-theory, he didn't specify what the "M" stood for, presumably because he didn't feel he had the right to name a theory which he hadn't been able to fully describe. According to Witten himself, "'M' stands for 'magic,' 'mystery' or 'membrane,' depending on your taste." Cynics have noted that the M might be an upside down "W", standing for Witten. Others have suggested that for now, the "M" in M-theory should stand for Missing.
A more serious problem: The name M-theory is slightly ambiguous. It can be used to refer to both the particular eleven-dimensional theory which Witten first proposed, or it can be used to refer to a kind of Uber theory which looks in various limits like the various string theories. Ashoke Sen has suggested that more general theory could go by the name U-theory, which might stand for Ur, or Uber, or Ultimate, or Underlying, or perhaps Unified. (It might also stand for U-duality, which is both a reference to Sen's own work and a kind of particle physics pun.)

M-theory in the following descriptions refers to the more general theory, and will be specified when used in its more limited sense.

M-theory in various backgrounds

Although no complete description of M-theory (in the more general sense) exists, it can be formulated in certain limits.

Membranes

A (mem)brane is a multidimensional object usually called p-brane referring to its spatial dimensionality p (for example, a string is a 1-brane and a flat surface is a 2-brane). There are different forms of branes: p-brane, D-brane, and black brane. P-branes are membrane-like structures of one to eleven dimensions that arise in equations of M-theory. These branes are said to float in an eleven-dimensional space and contain universes, including our own.

Matrix Theory

Matrix theory is a promising formulation of M-theory.

The Anti-deSitter/Conformal Field Theory Correspondence

See AdS/CFT correspondence.

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