Why string dualities imply unifying M-theory?

[Demystifier]

For a harmonic oscillator, there is a simple duality that exchanges position and momentum. However, it does not imply that the harmonic oscillator and the dual harmonic oscillator are just special cases of some more general unifying theory.

The electro-magnetic duality relates electric and magnetic monopoles. However, it does not imply that electric and magnetic monopoles must be only special cases of some more general unifying theory.

In 2 dimensions, there is a remarkable duality between the sine-Gordon and the Thirring model. However, it does not imply that these two models are just special cases of some more general unifying theory.

Given these well understood examples, then why, for the god sake, the string dualities do imply (or at least suggest) that all these string theories should be just special cases of a unifying but mysterious M (or F) theory? Is it just because there are many, and not only one, duality? Or just because otherwise string theory would not be cool enough? Or is there a more rational argument? What is wrong if we just say: OK, there are 5 string theories mutually related by dualities to each other, at low energies there is also duality to a yet another theory - 11 dimensional supergravity, and that's it?

 

[javierR]

A duality in the sense of electric-magnetic and generalizations of it to non-abelian gauge theories *is* saying there is one theory, and that one model is convenient for the description in one phase of this theory, while a dual model is convenient in another phase (they are not simply mappings to each other). In particular, in these examples, a strongly-coupled field theory of "electric" states can be equivalent to a weakly-coupled theory of "magnetic" states (can you say which version should be considered fundamental?). The string dualities also relate the various models (heterotic, type I and II) as different "phases" of a 10-dimensional theory. For example, the strongly coupled type IIB string is conjectured to be equivalent to the weakly coupled IIA string...neither one is more fundamental than the other. But what this 10D theory could be was a puzzle. But in addition, it was found that the strongly coupled IIA string looks like 11-dimensional supergravity, with 2- and 5-branes, at "long wavelengths" (similarly for the heterotic E8xE8 string). 11D supergravity is a unique theory with a single supermultiplet (and happens to be the highest dimension for a maximum-spin-2 field theory to be supersymmetric). So it's natural to say that the fundamental theory is this higher dimensional one, though it's shorter-wavelength nature is unknown...we can obtain from it the IIA and E8xE8 string theories, which are related to the other string theories via various dualities.

 

[Demystifier]

A duality in the sense of electric-magnetic and generalizations of it to non-abelian gauge theories *is* saying there is one theory, and that one model is convenient for the description in one phase of this theory, while a dual model is convenient in another phase (they are not simply mappings to each other).

And what that ONE theory would be? One would expect that in such a simple system the explicit analog of M-theory should be known explicitly. Is it?

If we are not able to find the explicit unifying theory for such a simple case, then how can we even hope to do that for the much much more complicated case of strings?

 

[Demystifier]

Come on, is it possible that no string theorist here is able to answer these elementary questions?

 

[Demystifier]

I am still waiting for an answer.

 

[atyy]

Wouldn't that be Maxwell's equations?

 

[MTd2]

You could ask that to LM. He is very pissed off that people are not looking at this kind of deep questions.

 

[MTd2]

Demystifier, there are simple cases indeed solved:

http://en.wikipedia.org/wiki/Matrix_string_theory

http://arxiv.org/abs/hep-th/9612115

Check its citations too:

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+EPRINT+HEP-TH/9612115

Specially these recent ones:

http://arxiv.org/abs/0902.4102
http://arxiv.org/abs/0902.2417
http://arxiv.org/abs/0902.1333

 

[suprised]

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.

 

[Naty1]

Brian Greene in FABRIC OF THE COSMOS discusses conceptual ideas involving M theory and its variants...and hints, I think, at an answer...

(I suspect a real answer lies in the inability to compute precise answers in any string theory, the lack of a unifying concept against which to formulate ideas such as an equivalence principle or an uncertainty principle...and perhaps in the subtely of the mathematics...)

Seems like it was Ed Witten's paper that discovered the M theory and the follow on work that was initially important was a Witten and Petr Horava collabaration...must be more advanced work by now...that was 1995...

According to Greene, this paper IS very convincing..anybody know which one it is?

I recall reading that during the 1970's and 1980's string theorists missed the tenth space dimension...Witten apparently "found" it along with M theory...that very small tenth dimension was previously hidden from theorists approximate solutions

Brian Greene notes that the five different versions sometimes permit translating (reformulating) an impossibly difficult question to comparatively simple one.

Also Witten's work led to the realization that there were ingredients beyond strings...branes had been overlooked along with the additional dimension...

Then I think people discovered sticky branes with strings attached and with stringy characteristics...(unlike closed loop gravitons)...

so the physical meaning of the mathematics was unclear for some time and likely portions still are unclear...seems like the perturbation theory solutions truned out to be a bit vague...

Polchinski showed how mathematical methods developed to study one dimensional branes (strings) could be used to study higher dimensional objects, p-branes...

And along comes Lisa Randall with her large dimensional theories...

xxxxxxxxxxxxxxxxxxxxxxxxx

So it's not like there have been a fixed group of Einstein field equations awaiting interpretation...which arguably took close to 50 years to fully understand

 

[atyy]

For a harmonic oscillator, there is a simple duality that exchanges position and momentum. However, it does not imply that the harmonic oscillator and the dual harmonic oscillator are just special cases of some more general unifying theory.

Mark Van Raamsdonk has a fascinating speculative essay in which he mentions a dual for the harmonic oscillator in section 5.1 (http://arxiv.org/abs/0907.2939).

 

[Fra]

I'm not sure if I got Demystifiers real point but as far as I see this, there is a rational reason for expecting a more unifying theory that has little to do wether you agree with it or not.

Suppose

(A) one takes the common realist position that the laws of physics are observer independent - ie the laws of physics are unique are must look the same to all observers, and each observer should infere the same laws from his experiments.

(B) Add to that the string theory conjecture. ie. that you are convinced that the string thinking or "string inference" (it's a term I made up, since I think it's speaking) is necessarily correct and does correspond to reality.

Then, given that this "string inference" actually gives rise to not a unique theory, but to several theories, that moreover are parameterized by a landscape of parameters. Then (A) implies that if (B) is right, then there must exist a yet unknown formulation of (B) which removes the redundance and ambigoussness of the prediction of the theory, so that the physics becomes unique.

I think there is a rationale there. However to me (A) is not fundamental, it's more emergent since the set of observers which we rightfully expect agreement of laws, are not fixed. Clearly the set of observers must have evolved along with the universe. And I sure don't find (B) near plausible either.

But I think that the hope of string theory is with this yet unknown master theory.

The question is more wether this is at all the right direction to walk. Since I reject (B) and has an emergent view of (A) this "string problem" don't exist for me, but I still see the logic in it if you accept their premises.

/Fredrik

 

[Demystifier]

You could ask that to LM. He is very pissed off that people are not looking at this kind of deep questions.

You mean Lubos Motl? I don't like his style of arguing, to put it mildly.

 

[Demystifier]

Thank you all for your comments!
Especially to suprised whose comments I find very illuminating and demystifying.

 

[Demystifier]

Mark Van Raamsdonk has a fascinating speculative essay in which he mentions a dual for the harmonic oscillator in section 5.1 (http://arxiv.org/abs/0907.2939).

Seems interesting, I'll read it.