## How do we solve M-theory?

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 - consider this picture of a "hyperboloid" surface. 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). Here 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.
 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?
 A sphere evolving in time is the boundary of anti de Sitter space.

 Tags m-theory, string theory