I know Type IIA/B, Type I, HO, & HE are related through the T and S Dualities. However, how does SUGRA factor in here? What exactly is 11D SUGRA’s significance in M-Theory?
Some seem to suggest it may be the lower energy bound, or maybe even a background field on which the 5 above string theories relate. I have no idea though.
This below linked question is similar to, but not exactly, what I was looking for: Is M-Theory and 11D Supergravity the same thing?
Taken out of the BBS-book and ignoring all the technical details..
S-duality explains how three of the five original superstring theories behave at strong coupling. The question that remains is the following
What happens to the other two superstring theories; namely type-IIA and E$_8 \times$ E$_8$ the coupling is large?
$\underline{Answer}$: They grow an $11^{th}$ dimension of size g$_s$l$_s$. This new dimension is a circle in the type IIA case and a line interval in the heterotic case.
When the eleventh dimension is large, one is outside the regime of perturbative string theory, and new techniques are required. Most importantly, a new type of quantum theory in 11 dimensions, called M-theory, emerges.
At low energies, it is approximated by a classical field theory called 11-dimensional supergravity, but M-theory is much more than that. The relation between M-theory and the two superstring theories previously mentioned, together with the T and S dualities, imply that the five superstring theories are connected by a web of dualities. A schematic figure is shown below
There are techniques for identifying large classes of superstring and M- theory vacua and describing them exactly, but there is not yet a succinct and compelling formulation of the underlying theory that gives rise to these vacua. Such a formulation should be completely unique, with no adjustable dimensionless parameters or other arbitrariness. Many things that we usually take for granted, such as the existence of a space-time manifold, are likely to be understood as emergent properties of specific vacua rather than identifiable features of the underlying theory. If this is correct, then the missing formulation of the theory must be quite unlike any previous theory. Usual approaches based on quantum fields depend on the existence of an ambient space-time manifold. It is not clear what the basic degrees of freedom should be in a theory that does not assume a space-time manifold at the outset.
There is an interesting proposal for an exact quantum mechanical description of M-theory, applicable to certain space-time backgrounds, that goes by the name of Matrix theory. Matrix theory gives a dual description of M- theory in flat 11-dimensional space-time in terms of the quantum mechanics of N × N matrices in the large N limit. When n of the spatial dimensions are compactified on a torus, the dual Matrix theory becomes a quantum field theory in n spatial dimensions (plus time). There is evidence that this conjecture is correct when n is not too large. However, it is unclear how to generalize it to other compactification geometries, so Matrix theory provides only pieces of a more complete description of M-theory.
