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It is known that in F-theory compactifications on CY 4-folds one can get gauge groups with very large ranks. The largest single factor* gauge group for compact CY 4-folds I found in the literature is the mind-blowing SO(7232) in this paper: http://arxiv.org/abs/hep-th/9706226 . One can imagine that there exist 4D compactifications of F-theory, which don't have Heterotic duals, with gauge groups that are even bigger in rank. Now, my question is whether there exists an upper bound on the ranks of single factors* in product gauge groups in either F-theory compactifications on CY 4-folds or in G2 holonomy compactifications of M-theory? An order of magnitude answer, if known, would be good enough :) .

(*) Thanks to @Luboš Motl for correcting my question!

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Stringpheno, you are slightly confused about - and you underestimate - the gauge group of the model you linked to. Its rank is not 7232/2, as it would be if the gauge group were SO(7232). The full gauge group of the model actually has rank 302,896 - about 100 times higher than what you say - and SO(7232) is just one of the 251 factors of the gauge group.

I don't remember the exact number but I think that this gauge group is still far from being the world record when it comes to the rank of gauge groups in F-theory geometric engineering, the term you may want to search for if you're interested in these fun constructions. I believe that there are 6-dimensional compactifications with 8 supercharges that admit even simple groups (or at least products of a smaller number of very large factors) as gauge groups which are larger than the product above.

There can clearly exist "infinite" gauge groups but if there is a limit for finite ranks is unknown to me. I believe that these very high gauge groups only appear in F-theory, not in M-theory on G2 manifolds, and it's arguably not just due to the limited creativity of the geometric engineers.

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Dear @Luboš Motl ! I probably should have specified that I meant to ask about the largest rank of a single factor in a product gauge group, so I'll edit my question accordingly. Do you have a specific reason to believe why in the G2 case there should be an upper bound? –  stringpheno Feb 14 '11 at 14:19
    
Hi @stringpheno, I didn't claim any qualitative difference between G2 and F-theory vacua. I just wrote that I think that high ranks that are comparable to millions don't appear in G2 vacua, but the upper bound on (finite) ranks may exist in both cases although the bound for the F-theory case may be very high. This is a pure evaluation of the papers I have seen but I am almost sure that the workers in that field would offer you a semi-intuitive reason. –  Luboš Motl Feb 14 '11 at 14:49
    
Dear @Luboš Motl, thank you very much for the input! –  stringpheno Feb 14 '11 at 15:18
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