Keeping in mind that already string theory itself and in fact already quantum field theory itself have only partially been formulated in a precise way, the conjecture is motivated from the fact that with the available knowledge of these subjects – particularly from duality in string theory – one can see indications that there is a kind of commuting diagram of the form
in some sense. The unknown top left corner here has optimistically been given a name, and that is “M-theory”. But even the rough global structure of the top left corner has remained elusive.
The available evidence that there is something of interest consists of various facets of the bottom left and the top right entry of the above diagram, that seem to have a common origin in the top left corner.
Notably, from the black brane-solution structure in 11-dimensional supergravity and from the brane scan one finds that it contains a 2-brane, called the M2-brane, and to the extent that one has this under control one can show that under “double dimensional reduction” this becomes the string. However, it is clear that this cannot quite give a definition of the top left corner by perturbation theory as the superstring sigma-model does for the bottom left corner, because by the very nature of the conjecture, the top left corner is supposed to be given by a non-perturbative strong-coupling limit of the bottom left corner.
One aspect of this is that type IIA string theory with a condensate of D0-branes behaves like a 10-dimensional theory that develops a further circular dimension of radius scaling with the density of D0-branes. (Banks-Fischler-Shenker-Susskind 97, Polchinski 99). See also (FSS 13, section 4.2).
Another hint comes from the fact that the U-duality-structure of supergravity theories forms a clear pattern in those dimensions where one understands it well, giving rise to a description of higher dimensional supergravity theories by exceptional generalized geometry. Now, this pattern, as a mathematical pattern, can be continued to the case that would correspond to the top left corner above, by passing to exceptional generalized geometry over hyperbolic Kac-Moody Lie algebras such as first E10 and then, ultimately E11. The references there show that these are huge algebraic structures inside which people incrementally find all kinds of relations that are naturally identified with various aspects of M-theory. This leads to the conjecture that M-theory somehow is in some way. But it all remains rather mysterious at the moment.
The compactification of M-theory on a torus yields type II string theory – directly type IIA, and then type IIB after T-dualizing. It turns out that the axio-dilaton of the resulting type II-B string theory is equivalently the complex structure-modulus of this elliptic fibration by the compactification torus. This gives a description of non-perturbative aspects of type II which has come to be known as F-theory (see e.g. Johnson 97).
In slightly more detail, write, topologically, for the compactification torus of M-theory, where contracting the first -factor means passing to type IIA. To obtain type IIB in noncompact 10 dimensions from M-theory, also the second is to be compactified (since T-duality sends the radius of to the inverse radius of ). Therefore type IIB sugra in is obtained from 11d sugra compactified on the torus . More generally, this torus may be taken to be an elliptic curve and this may vary over the 9d base space as an elliptic fibration.
Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity vacuum is the input data that F-theory is concerned with.
A schematic depiction of this story is the following:
|M-theory in||F-theory in|
|KK-reduction along elliptic fibration||axio-dilaton is modulus of elliptic fibration|
|IIA string theory in||T-duality||IIB string theory in|
In the simple case where the elliptic fiber is indeed just , the imaginary part of its complex modulus is
First, the KK-reduction of M-theory on yields a type IIA string coupling
Then the T-duality operation along divides this by :
The original insight that gave rise to the conjecture is due to
19:33: “Ten years ago we had the embarrassment that there were five consistent string theories plus a close cousin, which was 11-dimensional supergravity.” (19:40): “I promise you that by the end of the talk we have just one big theory.”
The term “M-theory” occurs somewhere around
For more original references see also at M2-brane.
An early popular account for a general audience is
More technical surveys include
Sophie de Buyl, Kac-Moody Algebras in M-theory, PhD thesis (pdf)
Paul Cook, Connections between Kac-Moody algebras and M-theory PhD thesis (arXiv:0711.3498)
See also the references at exceptional generalized geometry.
In fact, much of the broad structure of M-theory and its relation to the various string theory limits can be seen from the classification of exceptional super L-∞ algebras (such as the supergravity Lie 3-algebra and the supergravity Lie 6-algebra), as discussed in
By passing to automorphism algebras, this reproduces the polyvector extensions of the super Poincaré Lie algebra, which enter the traditional discussion of M-theory, such as the M-theory super Lie algebra (which arises as the symmetries of the M5-brane ∞-Wess-Zumino-Witten theory).