How do we solve M-theory?

Schreiberdk

Hi there PF.

I am a bit curious of how we are going to solve M-theory. Do we just need to find a Lagrangian, describing all of the dynamics of the theory? or is more needed? And how do one find such a Lagrangian? From what principles do we derive it?

I am no expert on string theory, I have just been reading a lot about it and i was curious of this problem :)

Schreiber

mitchell porter

To discover the fundamental structure of the theory, I think the single most promising path to take is to explore the "M/ABJM duality", which is a version of the "AdS/CFT correspondence". The ABJM model is a 2+1-dimensional field theory which is believed to be exactly, "holographically" equivalent to M theory on a particular 11-dimensional background (AdS4 x S7). Solve that model and you have solved a sector of M theory.

rogerl

M-theory can't be solved because we didn't include some essential feature. In the 1950's for example. It was taboo to talk about branes or higher dimensional world and stuff. You would be labelled as a crackpot if you mentioned those to physicists, now all physicists talk about it. We have a very similar situation now. There is another field that is currently label as crackpottry but will be part of physics in maybe 50 to 100 years from now. It is then that we can solve for M-theory or at least its equivalent if nature doesn't choose strings as an option.

Kevin_Axion

This is probably the least relevant answer you can have to the question asked.

mitchell porter's explanation and idea looks like your answer.

Schreiberdk

I have recently been looking at the Bagger-Lambert-Gustavsson Action, which has been proposed as a candidate Lagrangian describing M-theory. How does that fit in with M/ABJM duality? Or is it describing something completely different?

MTd2

2+1 dimensions can have an arbitrary statistics because it is not constrained by lorentz invariance. Is that attempted at ABJM?

mitchell porter

"ABJM with SU(2) X SU(2) = BLG with A4 3-product." (I'm just quoting that from a talk.) ABJM has led to more results, but BLG's use of 3-algebras might be a clue to M-theory's deep structure.

By the way, this thesis is an excellent discussion of BLG. Also look out for the discussion of "M-theory objects" on pages 9 and 19, where the M-wave, M-KK-monopole, and M9-brane are listed alongside the more familiar M2-brane and M5-brane.
I thought it was because you have the braid group, not just the permutation group, describing particle exchange. Anyway, there are people constructing anyons in ABJM (paper, talk).

MTd2

I am talking about anyons because it seems a more general construction than just supersymmetry. I mean, maybe we won't see supersymmetry in any scale, because even if superstrings is a correct theory, in principle, they are just limits of a more general construction.

Schreiberdk

So if I have gotten this right, you need to generalize the ABJM model to AdS4 x S7 spacetime, so that we got a model describing multi M2-brane dynamics.

And the approach of the ABJM model has some connection to Chern-Simons matter theory?

Does this mean, that a final field theory, describing M-theory, could be on the form of a Chern-Simons field theory in 11 dimensions?

mitchell porter

You don't generalize it or modify it. The ABJM model (with parameter k=1) is already completely equivalent to M-theory on AdS4 x S7. It is supposed to encode the entire dynamics of M theory on that 11-dimensional background, even though it is a 3-dimensional field theory. This is what holography means in string theory.

The better known example is the equivalence between d=4 N=4 super-Yang-Mills theory and Type IIB string theory on AdS5 x S5. The four-dimensional field theory completely contains the ten-dimensional string theory. You get the fifth, AdS dimension by continuing fields from the boundary - the boundary of AdS space is a flat space one dimension lower; N=4 SYM lives on that flat space, and it provides boundary conditions for what happens in the AdS interior. Where the other five, compact dimensions come from is a lot more abstract. Apparently N=4 SYM has a six-continuous-parameter degenerate ground state (a moduli space), you can define bosonic excitations in this space of ground states, and these spontaneously self-assemble themselves into the S5 geometry (and a string or brane extended in this S5 space must similarly be made of these moduli). There's probably also another path to the extra 5 dimensions in terms of the worldvolume theories of D3-branes in the AdS space.

edit: N=4 SYM has six scalar fields and these moduli are probably just their VEVs.

d=3 ABJM should give rise to d=11 M-theory in an entirely analogous fashion: the 3-dimensional ABJM space-time is the boundary of an AdS4, and the other seven dimensions, and structures in them, should emerge from the ABJM fields.
It is a Chern-Simons theory, a particular superconformal Chern-Simons theory - or rather, a two-parameter class of them, where the parameters N and k in the field theory describe brane content and topology in M-theory. (The seven-dimensional space is the quotient S7/Z_k - so when k=1 it's just S7 - and N is the number of M2-branes.)

Schreiberdk

As I mentioned above, I am no expert in string theory or M-theory. As a matter of fact I am a high school student, who is going to university after the summer:) I do though get quite a lot of sense out of your answers Mitchell :D thank you for your clarifications.

Though i would like to ask, what happened to the BFSS and IKKT models? Are they of no significance now, since the line of research is now focusing on AdS/CFT duality? Or am I completly wrong ? :)

mitchell porter

AdS spaces are dominating theoretical research because that duality is so fruitful. The matrix models, which describe flat space, are still studied, but they just haven't yielded as many insights. And by the way, what we need for the real world is string theory on de Sitter space, and there is no agreement at all on what the fundamental approach there should look like.

Schreiberdk

Is there some way, that one can find a duality between AdS and dS spaces? In analogy to the way that different string and SUGRA theories relate to each other through S, T and U-dualities?

MathematicalPhysicist

People don't even know what is M-theory, so solving something which isn't defined seems to be like pushing it.

Schreiberdk

In solving I dont mean like solving an equation, but rather finding a Lagrangian describing the full dynamics of the theory :)

MathematicalPhysicist

First, you should define what is M-theory, besides what I am hearing that all five superstring theories and supergravity combine might have some common ground, I haven't heard of any such commong ground explicitly defined.

Schreiberdk

But can one make a spacetime duality connecting the dS and AdS spaces? or is that just bollocks?

I also know that there is a similar approach to AdS/CFT called dS/CFT, but does dS/CFT make any of the predictions that AdS/CFT does or is telling something completely different?