Uncovering the Solution to M-Theory: A Lagrangian Approach?

[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 http://arxiv.org/abs/0909.1580" , 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]

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.

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]

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).

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

 

[mitchell porter]

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?

"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, http://arxiv.org/abs/1012.2707" 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.

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

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 (http://arxiv.org/abs/0910.5536" ).

 

[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]

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.

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.

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

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 completely 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 don't mean like solving an equation, but rather finding a Lagrangian describing the full dynamics of the theory :)

 

[MathematicalPhysicist]

In solving I don't mean like solving an equation, but rather finding a Lagrangian describing the full dynamics of the theory :)

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?

 

[mitchell porter]

The dS/CFT correspondence is very similar, it's basically what you get if you say "let's do AdS/CFT in de Sitter space", but unlike AdS/CFT, you run into problems for which there is no agreement about the solution.

I don't know if this will help you visualize things or just confuse you, but - http://en.wikipedia.org/wiki/File:HyperboloidOfOneSheet.png" . See how it has two sets of lines, horizontal and vertical (the horizontal lines run around the surface and form circles). You can actually get a simple picture of dS space and of AdS space from this image, depending on which set of lines you interpret as "time". If the vertical direction is time, it's de Sitter space; if the horizontal direction is time, it's anti de Sitter space. In each case, the other set of lines is "space".

So for de Sitter space, you start out with a big circle that gets small and then big again, while for anti de Sitter space you have to look sideways at the diagram, and think of a big hyperbola (one of the vertical lines) as space, and then time goes in a circle. This doesn't mean that time in anti de Sitter space is actually cyclic - I'm using this diagram just as a visualization aid. You could think of an anti de Sitter space with infinite time as being wrapped left-to-right around the hyperboloid infinitely many times. (If you have access to a copy of Roger Penrose's Road to Reality, you can see what I'm talking about in figures 28.7 and 28.8.)

Now, the way the holographic principle works in string theory is that fields on the boundary of space-time determine the behavior of strings in the "bulk" or interior of the space-time. So let's see where the boundary is in this picture of dS and AdS. Space in the AdS model is a hyperbola - one of the vertical lines running up and down the hyperboloid surface - and so at each moment, AdS space has a "boundary" corresponding to the ends of the hyperbola - two points at infinity. Then this evolves over time, so the boundary is really two timelike lines at infinity.

But the dS space is a circle. The only boundaries are in the infinite past and the infinite future - at the top and the bottom of the hyperboloid surface. So the boundary for dS space is a circle in the infinite past and another circle in the infinite future.

For this whole discussion, I've been talking about dS and AdS for one space dimension and one time dimension. That can be a little misleading for AdS, because the spatial boundary looks disconnected (two points). http://www.achtphasen.net/media/users/achtphasen/pwsup5_11-03.jpg" is a picture of AdS space for two space dimensions and one time dimension. The two space dimensions really form an infinite "hyperbolic plane", but it has been squeezed into a circle for the picture. The Escher image of bats getting smaller and smaller towards the boundary of the circle is there to remind you of what it's like inside the space.

Here, the boundary of AdS at one moment is a circle, and the space-time boundary of AdS as a whole is a cylinder. I said back in comment #10 that "the boundary of AdS space is a flat space one dimension lower". A cylinder is flat in the "intrinsic" sense of Riemannian geometry (which is the space-time geometry used in physics since relativity). To make a picture of it, we have to place it in 3-dimensional space and curve it, but if you look at distances and angles which are confined to the 2-dimensional boundary of the cylinder, it's just like a plane.

Anyway, the message is that the boundary of an AdS space is a flat space with a time direction, but the boundary of a dS space is two "spheres" that are purely spatial. (A circle, which was the dS boundary we found above, is a "1-sphere" in topological notation, since its boundary is a line which is 1-dimensional. An ordinary sphere is a 2-sphere, because its surface is 2-dimensional, and it would show up as the past or future boundary of a dS space with two space dimensions, and the relationship repeats in higher dimensions.) The fact that you don't have a time dimension in the dS boundary already tells you that the extrapolation of boundary fields into the interior can't work in the same way as it does in AdS/CFT. Lots of people have tried to make it work, with partial success, but there still just isn't a coherent accepted picture of how dS/CFT functions.

 

[Schreiberdk]

Can't one say, that the sphere of the de-Sitter space boundary is evolving in time, and therefore it does contain a time dimension?

 

[mitchell porter]

A sphere evolving in time is the boundary of anti de Sitter space.