If I have understood the basics of heterotic strings correctly, one requires a critical radius of compactified dimension to reconcile the apparent difference in spacetime dimension of left and right movers,

$$ R=\sqrt{\alpha} $$

Why would the radius of the compactified 16 dimensional torus be related to the string tension/length etc in such a way? They seem completely unrelated to me.


The 16 toroidal dimensions have a stringy radius because it has to be self-dual under T-duality, $$ R \to \frac{\alpha'}{R}$$ This is needed for these 16 dimensions to be purely left-moving. A fast semi-heuristic way to see it is that the left-moving dimensions obey $$\partial X = 0, \quad \partial_\sigma X = \partial_\tau X$$ If you integrate the latter form of the equation over $\sigma$, you will get $$ \alpha' p = w$$ The 16-momentum is equal to the 16-winding for all the states, in string units. But the momentum and winding belong to lattices that are dual ("inversely proportional") to one another because the momentum spacing goes like $1/R$ while the winding is a multiple of $2\pi R$. So the allowed radius has to be $\sqrt{\alpha'}$ in certain conventions where the factors of $2,\pi$ are properly accounted for.


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