kdv said:
Unfortunately, few string theorists visit these pages so it's hard to have good discussion on this topic. The vast majority of the posts are on sociology (the people doing the physics rather than physics itself), not physics and when it's on physics, it's almost exclusively on non-string theory (especially lqg) approaches. Anyone knows of a good forum where string theory is discussed at introductory and intermediate levels?
Although I personally find lqg much more interesting than string theory (that is why I read this forum
) I too am curious whether there exists any forum or discussion group suitable for asking string theory questions in. There is that sci.physics.strings group but it seems to have been dead for a long time, the one time I attempted to post a question there I got no response.
It's a very interesting fact that a quantum anomaly restricts the number of spacetime dimensions and I would myself love to see more discussion on this from knowledgeable people. The only way I know to get this result (for the bosonic string) is buried under so much maths that the physical interpretation is completely obscure.
So, the math here is well beyond me as well. But! In my generally-futile attempts to understand what string theorists are saying, something I've consistently noticed about string theory is that even fiendishly complicated effects in string theory often turn out to be possible to describe in very simple terms when described in terms of some topological property of the background geometry. (Of course, topology is itself so complicated that this may not give much of a better intuition of what is happening!) As far as I can tell the only reason why these simple topological descriptions of string situations tend to not get mentioned more often is because they are generally not the most useful way of describing things if one wants to do calculations. But, if you catch string theorists off guard they will sometimes leak them
.
I bring this up because Urs Shreiber posted an
interesting thing in a recent discussion on Not Even Wrong:
...Bosonic 2d CFTs of central charge 26 correspond to effective target spaces which are 26-dimensional manifolds only in a tiny subset of the space of all such CFTs, namely those that are entirely of the naive sigma-model type with large flat dimensions.
Supersymmetric 2d CFTs of central charge 15 correspond to effective target spaces which are 10-dimensional manifolds only in a tiny subset of the space of all such CFTs, namely those that are entirely of the naive sigma-model type with large flat dimensions...
So perturbative string theory does not predict that spacetime is 10-dimensional. What it does predict (essentially as its fundamental hypothesis!) is that spacetime is the effective target geometry of a 2dSCFT of central charge 15. That’s all.
10-dimensional manifolds appear here only in the most simple minded examples. Claiming that string theory predicts 10-dimensional spacetime is exactly like claiming that general relativity predicts flat empty Minkowski spacetime. No, it does not. This just happens to be the most simple solution that comes to mind...
So, why is this interesting? Well, CFT is Conformal Field Theory, which is a tool important for doing calculations in string theory. I am not clear on its exact use but google turns up lots of statements
like:
the perturbative expansion of string theory happens to be given by 2D CFT
(Incidentally the wikipedia article on
non-critical strings, which you may want to read, kind of makes it sound like when we talk about the "2DCFT" of a string theory, we really just mean the
worldsheet.)
Meanwhile, Wikipedia describes
"Central Charge" like:
In theoretical physics, a central charge is an operator Z that commutes with all the other symmetry operators... In string theory, in the first quantized formalism, these operators also have the interpretation of
winding numbers (
topological quantum numbers) of various strings and branes.
So, it sounds like Dr. Shreiber's hinted here at a way of understanding, without having to understand the whole messy anomaly cancellation thing, exactly what property it is that ultimately makes 10 an okay number of dimensions for a string theory background geometry, but not 4: Specifically it sounds like it is required to be using a 2DCFT with a central charge (winding number?) of 15, and ten dimensions is the easy way to do that.
...So, I don't know if this helps or just makes things more confusing! But if I were deadset on understanding why some numbers of geometry are acceptable for string theory and not others, I would maybe focus on trying to understand on exactly what the central charge of a 2DCFT is and why it makes a difference for that central charge to be 15.
