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To get the 26 dimensions is sort of intuitive (in a handwavey sort of way). Basically we solve:

$$(D-2)\frac{1}{2}(1+2+3+4+...)=-1$$

Where $1+2+3+..$ times $\frac{1}{2}\hbar$ are the ground energy for the different modes of a string which each have $D-2$ degrees of freedom to vibrate in. This sums to $-\frac{1}{12}$. This all has to sum to $-1$ in order that the graviton is massless. (Although because it all sums to -1 it means the lowest state, the tachyon has mass $-1$).

What is the equivalent equation for superstring theory? I notice if you take all the half integers because it now includes fermions:

$$\frac{1}{2}+1+\frac{3}{2}+2+\frac{5}{2}+...=-\frac{1}{6}$$

Then if you add these to the the previous sum $-\frac{1}{12}-\frac{1}{6}=-\frac{1}{4}$. Which if you put this in you get $D=10$. But of course, this is just a random guess.

Just wondered if there's a more-or-less intuitive explanation which can be explained in a paragraph of less.

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