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added brief pointer to the derivation of $SO(32)$ gauge group via tadpole cancellation, and some references on type I phenomenology. Will add these also to string phenomenology and to GUT, as far as relevant there
My apologies Urs, you can delete this post.
To make up for this comment, I require you now to come up with an actual comment or question that has some intellectual substance.
Re your edited #2: No, actually I cannot delete posts on the nForum.
But what’s going on? You must have had some thought or question, no? Let me know what it is, and then I can try to reply.
I was simply surprised at the size of 32 in a superficial way. I regret essentially spamming this post.
You don’t have to regret a sensation of intellectual surprise where it’s due. On the contrary, that’s healthy curiosity. But you have to verbalize it in a way that allows others to understand how to engange in conversation with you.
As with the number of spacetime dimensions in string theory, it may on first sight seem that also the ranks of gauge groups that appear are large compared to realistic values. But if one steps back and realizes that a priori in field theory, both the spacetime dimension and the rank of the gauge group may be any natural number whatsoever, it is actually striking that string theory exhibits theoretical mechanisms which constrain these possibly humongous numbers to values in the right ballpark, just a bit larger than observed.
And it’s not all that much larger if you dig a bit deeper. For instance the most promising GUT-model compatible with experimental data is still what they call $SO(10)$-GUT, which really has gauge group $Spin(10)$. In view of this a theoretical reason that bounds the ultimate ambient gauge group at rank 32 should not appear that fantastic anymore, and in fact there are realistic breaking patterns from $SO(32)$ to $Spin(10)$ to the Pati-Salam group $SU(5)$ and then finally to the standard model gauge group $S(U(2) \times (3)) \simeq \big(SU(3) \times SU(2) \times U(1)\big) / \mathbb{Z}_6$ (arXiv:1708.02078)
This “in the right ballpark but a bit larger than observed”, both for spacetime dimension and rank of gauge groups, is all the more noteworthy as in quantum field theory a priori the observable spacetime dimension and gauge rank may always be smaller than the one that enters the fundamental Lagrangian, due to possible KK-compactification and symmetry breaking.
So a little bit of reflection reveals that both the number of spacetime dimensions as well as the ranks of gauge groups in string phenomenology are actually surprisingly close to what one could hope for, instead of being surprisingly way off.
That first rough fit notwithstanding, one will of course want to see further details on how the reduction takes place. There are many intersting observations on this. Striking is for instance the computer scan of Gepner model orbifold compactifications here which shows that spontaneous breaking to $SU(5)$-GUT symmetry inhabits a special place in the space of all possibilities.
Another curious observation in this direction I recently highlighted on slide 125 here.
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