mathematics - Number of dimensions in string theory and possible link with number theory

This question has led me to ask somewhat a more specific question. I have read somewhere about a coincidence. Numbers of the form $8k + 2$ appears to be relevant for string theory. For k = 0 one gets 2 dimensional string world sheet, For k = 1 one gets 10 spacetime dimensions and for k = 3 one gets 26 dimensions of bosonic string theory. For k = 2 we get 18. I don't know whether it has any relevance or not in ST. Also the number 24, which can be thought of as number of dimensions perpendicular to 2 dimensional string world sheet in bosonic ST, is the largest number for which the sum of squares up to 24 is itself a square. $(1^2 + 2^2 + ..+24^2 = 70^2)$

My question is, is it a mere coincidence or something deeper than that?

Answer

There is definitely something deep going on, but there is not yet a deep understanding of what it is. In math the topology of the orthogonal group has a mod 8 periodicity called Bott periodicity. I think this is related to the dimensions in which one can have Majorana-Weyl spinors with Lorentzian signature which is indeed $8k+2$. So this is part of the connection and allows both the world-sheet and the spacetime for $d=2,10$ to have M-W spinors. The $26$ you get for $k=3$ doesn't have any obvious connection with spinors and supersymmetry, but there are some indirect connections related to the construction of a Vertex Operator Algebra with the Monster as its symmetry group. This involves a $Z_2$ orbifold of the bosonic string on the torus $R^{24}/\Lambda$ where $\Lambda$ is the Leech lattice. A $Z_2$ orbifold of this theory involves a twist field of dimension $24/16=3/2$ which is the dimension needed for a superconformal generator. So the fact that there are $24$ transverse dimensions does get related to world-sheet superconformal invariance. Finally, the fact you mentioned involving the sum of squares up to $24^2$ has been exploited in the math literature to give a very elegant construction of the Leech lattice starting from the Lorentzian lattice $\Pi^{25,1}$ by projecting along a null vector $(1,2, \cdots 24;70)$ which is null by the identity you quoted. I can't think of anything off the top of my head related to $k=2$ in string theory, but I'm sure there must be something.