This question is taken from this MathsOverflow question by "arivero" with it's answer given by "Dimension10 (page creator)".


It is well known the existence of a T-Duality between the two Heterotic String Theories, Type HO String Theory and Type HE String Theory. Beyond the trivial point that both groups have the same dimension (496, which actually is a prerequisite), is there some other mathematical relation between them?

I am thinking in other $ \mathrm{Spin}(N) $ groups whose dimension is a perfect number and that happen to be related to products of manifolds. $ SO(4) $ with $ SU(2) \times SU(2) $, and -I am told- $ SO(8) $ with some variant of $ (S^7 \times S^7) \times G_2 $. It should be nice if all of these were justified by a common construction, but I am happy just with an answer to the $ SO(32) $ case.



Answer 1Edit

The key here are the weight lattices bosonic representations $\Gamma$ of these gauge groups.

As I understand it, the weight lattice of $ E(8) $ is $ \Gamma^8 $, whereas the weight lattice of $ \frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}_2} $ ^ is $ \Gamma^{16} $. The first fact means that the weight lattice of $ E(8)\times E(8) $ is $ \Gamma^{8}\oplus\Gamma^8 $,

Now, an identity, that $ \Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}=\Gamma^{16}\oplus\Gamma^{1,1} $ , which actually allows this T-Duality. Now, this means that it is *this very identity* which allows the identity mentioned in the original post.

So, the answer to your question is "Yes", there is a group-theoretical fact, and that is that $ \Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16}\oplus\Gamma^{1,1} $.

Dimension10 $ \left(\right. $ Talk $ \cdot $ Edits $ \left. \right) $ 16:07, November 7, 2013 (UTC)