Are fractals related to string theory?

[xeon]

I have just seen some pretty pictures of flame fractals with reference to string theory. I was just wondering if or where they fit in.

thanks

 

[Coin]

Hm, I'm not very well-informed about string theory but I don't know of anything in mainstream string theory for which fractals are relevant.

However, poking around on google, it turns out there is a specialized subject called "fractal strings". As far as I can tell just from reading what Google turns up (I may not be understanding this correctly!), this is about what happens if you study strings which are "made of" fractals.

Usually when people talk about a "fractal", they mean something which is produced by an iterative process and is self-similar. As far as I can tell "fractal strings" are using the older and more restrictive definition of "fractal", that of something (like a manifold) with "fractional dimension"-- or rather, something for which the two different methods, Hausdorff dimension and topological dimension, of measuring the "dimension" of a manifold or figure turn out to give different values. For example you can imagine something which is floating on the outer edge of the Mandelbrot inkblot. Topologically this outer edge is one-dimensional, a line. But because the shape of the edge is so odd and fractal-y, there is a sense (as measured by the "hausdorff dimension") in which something moving on that edge has more than one dimension of freedom of movement.

Fractal strings seem to be about what happens when rather than studying strings which are properly one-dimensional, you study strings which have fractal dimension-- strings whose Hausdorff dimension is less than one but more than zero, fractional. You could construct a string like this for example by taking a normal one-dimensional string and cutting sections out of it, as is done to construct the Cantor set.

However looking around, it sounds like fractal strings are not really used in physics or physics theories. Their main application appears to be that they are useful as mathematical objects in studying number theory. It sounds like an interesting subject though!

...I don't know if this is what you were asking about or not :)

 

[Thomas Larsson]

There are physically interesting fractals embedded in 2D which are related to conformal field theory, which in turn is relevant for string theory. I once observed that fractal dimensions of several interesting objects in 2D, e.g. percolation cluster (D = 91/48), percolation hull (D = 7/4), self-avoiding walks (D = 4/3) and red links (D = 3/4) fit into the magic formula

D = (100 - n^2)/48, n integer.

This follows immediately from Kac' formula for c = 0 with half-integer values, using D = 2 - 2h. I later learned that Hubert Saleur had found the same formula a few months before me (it was really quite obvious at the time), but it nevertheless paid for a four-year postdoc.

There is of course an glaring limitation with this formula: it only works for fractals embedded in 2D (like a string theory worldsheet), but the physically most interesting fractals live inside 3D. This observation got me starting on higher-dimensionalization of the algebraic structures of string theory.