# Is M-theory just a M-yth?

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I would like to know what are the scientific counterarguments to a statement of this type:

"M-theory is a chimera, a non-existent holy grail more wished for than actually proven. It's conjecture is a PR stunt to cover up the fact that there is no unique string theory."

I.e. what are the tangible arguments for the equivalence of all string-theory types under one mother theory and for the mere existence of such a theory if we are not to assume that string theory in fact describes one unified theory of everything. The question also aims to clearly discern what is the difference between the mathematical and physical "theory of everything" content in the various statements in the discussion.

The points I feel should be adressed are:

1. There are various dualities between string theory types. But how does the mere fact of a formal mathematical duality prove that the theories are physically identical? There is a duality between the electric and magnetic intensity in Maxwell vacuum equations; would that mean that the electrical and magnetic field are physically identical? What if a theory is self-dual, does that mean that a sector of the theory is redundant? (As it turns out, in electricity/magnetism the answer is "yes" but I do not see how this necessitates from the discrete duality.)
2. A number of the dualities such as T-duality require a compactified version of a theory to equate between theories. But what about the non-compactified versions, are they also recoverable from the mother theory? Many approaches state that the non-compactified versions are recovered from the compactified ones by taking the typical length $L$ of the compactification and sending $L \to \infty$. But the topological inequivalence is clear. Amongst other things it is clear that for the non-compact theory, the $1/L$ in the dual theory should be substituted not by a "limit infinity" but by some kind of bare infinity (isolated, non-limit, true... whatever is to your terminological liking).
3. More specifically, the D0 branes in IIA and their excitations are interpreted as Kaluza-Klein towers from a higher dimension and the "strong coupling limit of IIA" is basically the definition of M-Theory (as far as I have read). But, apart from blind belief, what compels us to think it is necessary that IIA is a compactification of another theory?
4. Rather obviously, if a strong field theory looks as a compactified theory, the corresponding weak-field effective theory is probably going to look as one too. Hence, a starting point would be to find the corresponding theory which reduces to the weak field limit of IIA under compactification. Such a theory exists, it is 11D supergravity. But a weak-field limit has a myriad of "strong completions" so finding this "weak anticompactification" does not prove anything. Furthermore, the "going around two corners" approach (i.e. assuming weak-fieldization and compactification commutes) raises doubts, especially in the light of the fact that quantization and stability seem to need some very exotic handling in a 11D membrane theory. Is it possible that the weak-field limit of the conjectured M-theory is not 11D supergravity?

LOL PR stunt.

Where did you read this critic?

@conformal_gk The quote is a paraphrase of a few private conversations. I do not have a sharp opinion on the topic but I have admit that after all the hype around string theory and the social pressure this put on the string theorists to present the final "theory of everything", it is natural to ask whether the conjecture forces our expectations on the theory rather than letting the theory speak for itself.

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1)" But how does the mere fact of a formal mathematical duality prove that the theories are physically identical?" In the formulation of this question there are many strange redundancies. A theory is a mathematical structure. Two theories are the same if there exists a natural identification between the corresponding mathematical structures. A given theory is supposed to describe a given physics. By definition two identical theories describe the same  physics and so are "physically identical". In some sense any mathematical result is formal so "formal mathematical duality" should be simply "mathematical duality" or even just "duality" because, as I recalled it, a theory is a mathematical structure and so a duality between theories is obviously "mathematical".

About the Maxwell equations. The electromagnetic duality does not identify the electric and magnetic fields but says that which one is called "electric" or which one is called "magnetic" is arbitrary. Two physicists using both the Maxwell equations but with an opposite convention about what is electric and what is magnetic describe the same physics. The fact that a theory is self-dual does not mean that a sector is redundant. It just means that some labels attached to the theory are in some sense arbitrary. If you think to a theory as a mathematical structure, a self-duality of the theory is simply a non-trivial automorphism. We are not quotienting by this transformation.

2)I am not sure to understand this question. Yes, some dualities only make sense after some compactifications. If it is possible to decompactify both sides of the duality at the same time then we expect the duality to still make sense for the uncompactified theories (the objection about the "topological inequivalence" is in general resolved by some degree of locality of the theories). But in general the point of these dualities is that it is not possible to decompactify both sides at the same time, it is why there are non-trivial, and in this case the question does not make sense.

3)To define M-theory as the "strong coupling limit of IIA string theory" is doing nothing if we have nothing non-trivial to say about this limit. The non-trivial statement is that this theory is 11 dimensional and has 11 dimensional supergravity as its low-energy limit. The question is essentially where does this claim come from.  The study of D0 branes indeed plays an important role here. The point is that type IIA string theory contains an bound state of $n$ $D0$-branes for every integer $n$, each one of mass proportional to $|n|/g$ where $g$ is the IIA coupling constant. In the limit $g \rightarrow 0$, this infinite number of states become massless. Do you know a mechanism to produce an infinite number of massless states ? It is not known how to do such thing in a local quantum field theory or in the known string theories. The only known mechanism to do that is a Kaluza-Klein compactification in the decompactification limit.

In fact there are other independent arguments in favour of this 11 dimensional picture. For example the fact that the low energy type IIA supergravity is the dimensional reduction of the 11 dimensional supergravity and that in this identification the value of the dilaton is directly related to the radius of the 11 dimensional circle. Furthermore the IIA/M relation is only one small piece of the full web of dualities describing  the strong coupling dynamics of the various string theories. For each of them one can give non-trivial arguments and what is remarquable is the consistency of the global picture.

4)The fact that "a weak-field theory limit has a myriad of completions" is just not correct in general. Even in standard quantum field theory, to find a UV completion of an IR effective field theory is already a non-trivial problem which seems to have no solution in Lagrangian terms in spacetime dimension $d>4$ and requires non-abelian gauge fields in $d=4$. But it is true that once these restrictions are understood there can be many possible choices. But it is certainly not what we know for theories containing gravity. To find a UV completion of an IR field theory containing gravity is just difficult and before the proposal of the existence of M-theory the only known examples where string theories. For example the only known quantum theory whose low energy limit is type IIA supergravity theory is type IIA string theory and the fact that it is possible to UV complete type IIA supergravity is a kind of miracle from the field theory point of view: it seems that one needs string theory to do it. It is why many string theorists thought in the 80s that 11 dimensional supergravity was just an ordinary non-renormalizable field theory without UV completion. It is the arguments given in 3) which suggests the existence of a UV completion of 11 dimensional supergravity and we call this theory M-theory, which does not seem to be a local quantum field theory or to have a perturbative string theory description. If there were many known quantum gravity theories in 11 dimensions, we could ask which one is the strong coupling limit of type IIA string theory and so should be called M-theory. But it would be an embarassement of riches and we are not in this situation. Rather the fact that we have arguments showing that the strong coupling limit of type IIA theory is 11 dimensional implies the non-trivial fact that there exists a 11 dimensional quantum gravity.

There is no expected problem of commutativity in the double limit weak energy/compactification simply because the Kaluza-Klein modes are massive (because we are compactifying on a circle of given radius. Things are different for more general compactifications on possibly singular spaces) and so do not affect the low energy dynamics. I don't know what is the needed "very exotic handling in a 11D membrane theory". As the membranes are massive, they do not enter in the low energy discussion anyway.

The fact that at low energies M-theory reduces to 11 dimensional supergravity is imposed by supersymmetry. 11 dimensional supergravity is the only field theory with at most two derivatives in the Lagrangian with $N=1$ $11d$ supersymmetry, as the type IIA supergravity is the only field theory with at most two derivatives in the Lagrangian with $N=(1,1)$ $10d$ supersymmetry.

answered May 30, 2015 by (5,000 points)
edited May 30, 2015 by 40227

Thanks, this is great.
1) I just realized the el-mag example is self-explanatory: the non-vacuum equations are actually not self-dual, they correspond to the interchange of magnetic monopoles and electric charges. I.e. the duality says that the theory of electromagnetism with charges is physically equivalent to the theory of electromagnetism with monopoles (which is true).

2) Do you perhaps have a nice reference on examples how the topological inequivalence is handled? As I naively imagine it, one side of the duality using T-duality may be decompactified while the other is ultracompactified, i.e. the extra dimension is completely eliminated. I do not understand how two theories with different dimensionality can be equivalent directly (i.e. not through the limiting process).

3) The question is: do we need a mechanism to generate an infinite number of massless states? They are, after all, generated by the IIA theory itself. Is there an inconsistency in the strong-coupling limit of IIA which needs to be resolved? Or is it just "elegance" or "convenience" which leads us to the conjecture of M-theory?

4) I do not have a reference but from various conversations I have obtained the (maybe mistaken) idea that extended objects of higher dimensions than strings bring with themselves pathologies upon quantization. "Nobody knows how to quantize membranes" (p. 10 here), hence I assumed the correspondence of the M-theory dynamics to IIA (aka "compactification") as well as the correspondence to a "usual" effective low energy quantum field theory might be very non-trivial. Again, my naive picture is that even in the direction M-theory $\to$ IIA there is an additional "effectivization" deforming the alternative quantization process. Not knowing the nature of this "effectivization" and it's limit-commutation is my main concern. But the last paragraph of your post indeed strengthens the case for M-theory.

1)Exactly.

2)In T-duality, a theory 1 on a circle of radius R is equivalent to a theory 2 on a circle of radius 1/R. In that case it is not possible to decompactify both sides at the same time. In some sense the duality should still be true in the limit R goes to zero but it does not make really sense because we do not have a good control of the theory 1 on a extremely small circle. Rather the point of the T-duality is to give a dual description under control of this limit.

3) In some sense this is similar to 2). One could just say that the theory with infinitely many states is just the strong coupling limit of type IIA. But the point is that we would like a "better" description of this limit. There is no real reason why such dual description should exist except probably the general philosophy that when a system becomes strongly coupled, often we have some emergent degrees of freedom which are themself weakly coupled and so that it is better to base the theory on them. It just happens that it seems that in the case of the strong coupling limit of type IIA, the Kaluza-Klein description is a well-known mechanism which seems to give the expected limit. In fact, at the time of the discoveries of the various string dualities, some particular strong coupling limits appeared like nothing known before (some theories of tensionless strings). In such cases one just has to work harder to understand what happens.

4)It is true that there is no known quantum worldsheet description of the membranes and I agree that it is certainly a gap in our understanding of M-theory. But this is irrelevant for the low energy limit question because the membranes do not appear in the discussion anyway. If I want to compute the low energy limit of some compactification of a string theory on a smooth compact space, the fact that I have a good wordsheet description of the string does not help, it just does not enter in the discussion.

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