I do not think it is possible really believe or experimentally check (now), but all modern physical doctrines suggest that out world is NOT 4-dimensional, but higher. 
The least sophisticated candidate - bosonic string theory says that out world is 26 dimensional (it is not realistic due to presence of tachion, and so there are [super strings](http://en.wikipedia.org/wiki/Super_string) with 10 dimensions, [M-theory](http://en.wikipedia.org/wiki/M-theory) with 11, [F-theory](http://en.wikipedia.org/wiki/F-theory) with 12).


Let us do not care about physical realities and ask: what mathematics stands behind the fact that 26 is the only dimension where bosonic string theory can live ? 
Definitely there is some mathematics e.g. 26 in [that MO question](https://mathoverflow.net/questions/6923/complexes-of-representations-with-complementary-central-charges) is surely related.


Let me recall the bosonic string theory background. 
Our real world is some Riemannian manifold M which is called TS (target space).
We consider the space of all maps from the circle to M, actually we need to consider how
the circle is moving inside M, so we get maps from $S^1\times [0~ T]$ to M ( here $S^1\times [0~ T]$ is called WS - world sheet); 
we identify the maps which differs by parametrization (that is how Virasoro comes into game
and hence relation with [Leonid's question](https://mathoverflow.net/questions/6923/complexes-of-representations-with-complementary-central-charges)). 


That was pretty mathematical, but now ill-defined physics begin - we need integrate
over this infinite-dimensional space of maps/parametrizations with measure
corresponding to exp( i/h volume\_{2d}(image(WS))).
This measure is known NOT to exist mathematically, but somehow this does not stop
physists they do what they call regularization or renormalization or something like
that and 26 appears... 



Why does bosonic string theory require 26 spacetime dimensions?
In addition to Chris Gerig's operator-language approach, let me also show how this magical number appears in the path integral approach.


Let $\Sigma$ be a compact surface (worldsheet) and $M$ a Riemannian manifold (spacetime). The string partition function looks like
$$Z\_{string}=\int\_{g\in Met(\Sigma)}dg\int\_{\sigma\in Map(\Sigma,M)}d\sigma\exp(iS(g,\sigma)).$$
Here $Met(\Sigma)$ is the space of Riemannian metrics on $\Sigma$ and $S(g,\sigma)$ is the standard $\sigma$-model action $S(g,\sigma)=\int\_{\Sigma} dvol\_\Sigma \langle d\sigma,d\sigma\rangle$. In particular, $S$ is quadratic in $\sigma$, so the second integral $Z\_{matter}$ does not pose any difficulty and one can write it in terms of the determinant of the Laplace operator on $\Sigma$. Note that the determinant of the Laplace operator is a section of the determinant line bundle $L\_{det}\rightarrow Met(\Sigma)$. The measure $dg$ is a 'section' of the bundle of top forms $L\_g\rightarrow Met(\Sigma)$. Both line bundles carry natural connections.


However, the space $Met(\Sigma)$ is enormous: for example, it has a free action by the group of rescalings $Weyl(\Sigma)$ ($g\mapsto \phi g$ for $\phi\in Weyl(\Sigma)$ a positive function). It also carries an action of the diffeomorphism group. The quotient $\mathcal{M}$ of $Met(\Sigma)$ by the action of both groups is finite-dimensional, it is the moduli space of conformal (or complex) structures, so you would like to rewrite $Z\_{string}$ as an integral over $\mathcal{M}$.


Everything in sight is diffeomorphism-invariant, so the only question is how does the integrand change under $Weyl(\Sigma)$. To descend the integral from $Met(\Sigma)$ to $Met(\Sigma)/Weyl(\Sigma)$ you need to trivialize the bundle $L\_{det}\otimes L\_g$ along the orbits of $Weyl(\Sigma)$. This is where the critical dimension comes in: the curvature of the natural connection on $L\_{det}\otimes L\_g$ (local anomaly) vanishes precisely when $d=26$. After that one also needs to check that the connection is actually flat along the orbits, so that you can indeed trivialize it.


Two references for this approach are D'Hoker's lectures on string theory in "Quantum Fields and Strings" and Freed's "Determinants, Torsion, and Strings".


