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# Why26Whydoesbosonicstringtheoryrequire26spacetimedimensions?

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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 with 10 dimensions, M-theory with 11, 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 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 S^1 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) to this M, sheet); we identify the maps which differs by parametrization (that is how Virasoro comes into game and hence relation with Leonid's question).

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 length(image(S^1)))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...

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# W h y 2 6 ?

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 with 10 dimensions, M-theory with 11, 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 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 S^1 (called WS - world sheet) to this M, we identify the maps which differs by parametrization (that is how Virasoro comes into game and hence relation with Leonid's question).

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 length(image(S^1))). 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...