Demystifier said:
Come on, is it possible that no string theorist here is able to answer these elementary questions?
Surely any string physicst can demystify this for Demystifier ;-)
The way you discussed dualities is correct, namely as different descriptions or parametrizations (in terms of different degrees of freedom) of one and the same theory. There is no reason for an underlying more general therory; we just talk about _one single_ theory.
In string theory, the word duality is sometimes used in a confusing way. For example, it is said that 10d type IIA strings would by dual to M-theory (defined here as the membrane theory whose low-energy limit gives 11d supergravity). But the more precise statement is that type II A strings arise from a compactification of M-theory on a circle in the limit where this circle shrinks to zero size (which kills the "eleventh" dimension). So the two theories, 10d strings and 11d M-theory, are not dual to one and the same theory, rather one theory or the other theory arises when we adjust a parameter (the radius R of the circle) appropriately. In other words, they arise as different singular limits of a continuous family of theories, parametrized by a parameter R.
The story is most interesting "in between" these limits R=0 (type IIA str) and R=infinity (M-th). Let's consider a fixed value, say R=1. Then what theory do we have here? Obviously just one single one, but it has two different dual interpretations. In the M-theory language, one has for example Kaluza-Klein states which are the quantized momentum modes of the circle. In type IIA language, these very same states have the interpretation as non-pertubative D0-branes.
So this is exactly a duality as you describe it, namely between different descriptions or parametrizations of one and the same theory. The unification aspect lies in that both type IIA strings and M-theory arise as different limits (R=0 or infinity) of one continuous interpolating family of theories. So this family as a whole may be viewed as unifying theory, being neither just type II strings nor M-theory, but including also all theories "in between". So, how to call such a unifying theory? Traditionally the whole family is often also called M-theory, and this is a main source of confusion; namely one could have called it type IIA string with equal rights.
And actually all other string theories are connected in analogous dual ways with each other; there are continuous interpolating parameter families connecting them (typically geometrical parameters that give the size of a compactification manifold, or dually, quantum coupling constants). So all string theories can be viewed as different singular limits of one big deformation family of … something. And how to call this something? Well you guessed it -- in lack for a better name it is often just called M-theory as well. But as I said, this is somewhat misleading as M-theory should, strictly speaking, used for the limiting theory in 11d only.
Now the biq question is, what is the nature of the one big deformation family whose various weak-couling limits give the various strings in 10d or M-theory in 11d? Is there one "underlying" mysterious M-theory with unknown fundamental degrees of freedom, that we didn't discover yet? There are many different opinions on that.
My personal take is that there are no more fundamental degrees of freedom, and the big fat blob of continuous deformation families is all there is. I like the following analogy with a well-known mathematical picture: imagine an abstract topologically non-trivial manifold M (this represents the big blob of theories). Physical degrees of freedom correspond to choosing a local coordinate patch anywhere on M, and expanding quantities like the metric locally in these patches corresponds to choosing physical degrees of freedom to write down a lagrangian etc. Depending on where we are on M, particular classes of coordinates are more suitable than others (corresponding to choosing suitable local, weakly coupled physical degrees of freedom; these can be strings, membranes, particles…).
But as is well-known, as M is topologically non-trival, there don't exist choices of coordinates that would be globally valid, ie, anywhere on M. Rather, all what we can do is to cover M by local coordinate patches or charts, and make sure that all those patches overlap in a globally consistent manner. For phyiscs this would translate into the claim that there exists no globally valid description of the whole blob in terms of local, weakly coupled degrees of freedom: an underlying universal local physical theory does not exist in the same spirit in which global coordinates on M do not exist. Whenever you sit down and write a lagragian for some choice of physical degrees of freedom, you already made a choice for a particular coordinate patch; outside a certain domain of convergence, your choice of fields will become ill-suited (non-local, strongly coupled) and your formalism will break down. Then it may be a good idea to switch to a new set of "coordinates" that are adapted to the neighboring coordinate patch that you entered.
Dualities in the strict sense (where one keeps all parameters fixed), would correspond to differend choices of coordinates (or fields) at the given point on M. This is particular interesting for regions that lie in the common intersection of coordinate patches, namely then duality transforations can map quite "different" physical degrees of freedom into each other (eg, as mentioned above, type IIA D0 branes into Kaluza-Klein Modes of M-theory).
So, if we can't find any globally valid coordinate system on M (and thus no set of physical degrees of freedom for writing a lagrangian everywhere on the blob), what can we then do with all of that? Well, the point is that the manifold M does exist and has many properties that are coordinate independent, for example its topology. In fact when studying manifolds, mathematician rarely write down coordinates for a given manifold, rather they formulate their computations as much as possible in a coordinate- (and thus patch-) independent way; differential forms are an easy example for that.
For physics, the analogy is obvious but highly speculative. A coordinate free description of the big blob would correspond to a fully background independent formulation of string theory. It would avoid using any local degrees of freedom, as these are tied to choices of coordinate patches, which are choices of backgrounds. So it would be a topologial theory with few or even no physical degrees of freedom, but capture, somehow, the big blob, or space of theories, as a whole. Perhaps it would have some critical points on M and "condense" on those; or perhaps no points on M would be preferred. The idea would be that then, when expanded around a given but non-trivial choice of vaccuum, physical degrees of freedom are generated "out of nothing", eg, by spontaneous symmetry breaking.
Again: there latter remarks are highly speculative and others may have different opinions.
