Does $SO(32) \sim_T E_8 \times E_8$ relate to some group theoretical fact?

It is well known the existence of a T duality between the two heterotic string theories, $SO(32) \sim_T E_8 \times E_8$. 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 SO(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.

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Given how rare perfect numbers are, and how comparatively common T-duality is, are you sure that whatever pattern you seek is only supported at the perfect numbers? – Theo Johnson-Freyd Mar 6 '11 at 3:47
Small correction: it's actually not $SO(32)$ but a different $\mathbb{Z}/2\mathbb{Z}$ quotient of $Spin(32)$, and the relation goes via the weight lattices of the corresponding groups. Each lattice gives rise to a 16-dimensional torus and Milnor observed that these tori are isospectral (a postdiction of T-duality?). – José Figueroa-O'Farrill Mar 6 '11 at 13:51
@Theo I could expect a more general pattern for any SO(4n), but these "square constructions" seem to be more specific of perfect numbers, and even for the "SO(8)" case it is not as pretty as in string theory. – arivero Mar 6 '11 at 23:47
From the link between perfect numbers and Mersenne primes, Phys.Rev.D60:087901,1999 (hep-th/9904212v1) looks for cancelation of polygonal anomaly. Also, some hints are Weinberg $SO(2^13)$ in dim 26 and Bern-Dunbar SO(4) in four dimensions. But these results are all based on string theory technicalities, not pure group theory nor other branch of pure math. – arivero 0 secs ago – arivero Mar 7 '11 at 9:59

The answer to this question can be found in Lubos Motl's answer to this question of mine on Physics.SE.

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 $\Gamma^{8}\oplus\Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16}\oplus\Gamma^{1,1}$.

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These rank-16 lattices also famously have the same theta function (and even the same Siegel thetas for ranks 2 and 3). No idea if that has any significance for this question in string theory, though... – Noam D. Elkies Sep 16 '15 at 2:28
@NoamD.Elkies I guess it does have significance: theta functions do turn up in string theory in surprising places. Urs Schreiber is one person I know (personally) who might know. – David Roberts Sep 16 '15 at 6:05

It's worth noting this fact. Suppose $L$, $L'$ are even unimodular Euclidean lattices, like $\mathrm{E}_8= \Gamma^8$ or $\Gamma^{16}$ or any direct sum of these, but also the Leech lattice and 23 others in 24 dimensions, and at least 80,000,000,000,000,000 others in 32 dimensions, etc. Let $\Gamma^{1,1}$ be the even unimodular Lorentzian lattice in 2 dimensions. Then

$$L \oplus \Gamma^{1,1} \cong L' \oplus \Gamma^{1,1}$$

The reason is that there's only one even unimodular Lorentzian lattice in dimensions $8n + 2$, up to isomorphism (and none in other dimensions).

So, the fact that $\mathrm{E}^8 \oplus \mathrm{E}^8$ and $\Gamma^{16}$ become isomorphic when you take their direct sum with $\Gamma^{1,1}$ is an instance of this general pattern.

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