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Why is there a deep mysterious relation between string theory and number theory (Langlands program), elliptic curves, modular functions, the exceptional group $E_8$, and the Monster group as in Monstrous Moonshine?

Surely it's not just a coincidence in the Platonic world of mathematics.

Granted this may not be fully answerable given the current state of knowledge, but are there any hints/plausibility arguments that might illuminate the connections?

twistor59
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    Is this a real question? More importantly, is it a real physics question? Admittedly I don't know the full details of these fields, but this really sounds like it's getting into philosophical territory. – David Z Feb 07 '11 at 14:07
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    At least this question is a bit childish. If anybody had an answer to this, he would publish it with a lot of "celebrations", and we all would know "why", in principle at least. – Georg Feb 07 '11 at 14:48
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    @David you could say that about much of string theory. Of course, it would be helpful if @Arduous could list some references or some more context for what these "deep relations" are. –  Feb 07 '11 at 14:51
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    There are lots of interesting and appropriate questions involving these topics but this broad "why" question is not going to get any kind of reasonable answer. I'd suggest you reword the question to make it a more specific question about some aspect of these relations that you are interested in. – pho Feb 07 '11 at 15:10
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    I actually voted this question thumbs-up. It's a good question and I would like to know the most accurate answer, too. Clearly, the rough sketch of the answer is that string theory just knows about all important and exceptional structures in mathematics. But why does it know them? What is the logic that dictates that "other solutions" of a theory whose main physical goal is "only" to unify the interactions including gravity with quantum mechanics produces all other maths, including maths we used to think was totally abstract? Why did you close this very good question? – Luboš Motl Feb 08 '11 at 06:37
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    I agree with Luboš, the question should remain open. "Arduous" could also try asking at Math Overflow. (P.S. some of the specific connections listed come from the "modular invariance" of string theory, the need for one-loop amplitudes to be invariant under "large" reparametrizations of the world-sheet. This means that modular forms and their properties are relevant - thus Langlands - and also establishes a link to lattices - http://mathoverflow.net/questions/24604/why-are-modular-forms-interesting/24642#24642 ) – Mitchell Porter Feb 08 '11 at 07:54
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    I still think a more specific question would be better, but I can see that there might be some interesting and useful answers so I've voted to reopen. – pho Feb 08 '11 at 14:20
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    Related question on mathoverflow: http://mathoverflow.net/q/58990/13917 – Qmechanic Jul 06 '13 at 17:53
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    This may be inspired by something high-level - good for it. But as worded it's a terrible question. What connection is this? All the question talks about is it being "mysterious," which doesn't bode well for having an answer. Besides, there are many connections, and many more people looking for more connections, between these fields of math. Moreover, the largest sporadic group has enough structure to be "connected" to anything you want. –  Jul 06 '13 at 23:21
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    @ChrisWhite are you really seing this questions should not be reopend because of language/wording nitpicking? It is "mysterious" here does not mean something like magic since the OP knew what he was taling about, it rather means interesting, fascinating, or something along these lines. I still agree with the experts and people knowledgeable in these topics (they are the ones who can judge it correctly!), that the question is good and should get reopend. In such cases, people should more listen to what the experts say and we definitively need such higher level questions ... that are open. – Dilaton Jul 07 '13 at 08:48
  • @Qmechanic could this be migrated to MathOverflow alternatively if reopen here does not succeed (but is should!)? I agree with people in the comments above that it could obtain very in]teresting answers, in particular since Urs Schreiber for example seems to be some kind of back, I have seen him answering questions. – Dilaton Jul 07 '13 at 08:50
  • @Qmechanic could you not help to reopen this question since so many people who are knowledgeable about these topics say that it is a good one in the comments above? – Dilaton Jul 07 '13 at 09:23
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    I remember the days when the eightfold way http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29 was mysterious. please open. – anna v Jul 07 '13 at 11:04
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    @annav if you could vote to reopen too, we would need just one more to have 5 ... :-) – Dilaton Jul 07 '13 at 11:11
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    @dilaton I did vote and it turned from 2 to 3 immediatly, before my comment. I think maybe dimension10 did not do it successfully – anna v Jul 07 '13 at 11:13
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    @annav ah so there is still 2 votes we need :-/. Dimension10 can unfortunately not yet vote to reopen, since he has not yet 3000 rep. I look forward to the time he can join in reopening and closing too, since we definitely need much more people with a good physics knowledge and good judgement (this includes knowing when to better skip if not being knowledgeable enough about the topic too) and who are willing to listen to the comments of experts. – Dilaton Jul 07 '13 at 11:22
  • Vote to remain closed. This question gives no indication that the OP has put any effort into answering his/her own question. (Also, please do not migrate to MathOverflow. This is an open question, and open questions are specifically off-topic there.) – user1504 Jul 07 '13 at 11:57
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    @user1504 I got the impression that at MathOverflow research-level theoretical physics question are recepted better and treated kinder than here, over there (an it certainly would have been allowed to obtain the nice and interesting (even if they are not comprehensive partial) answers it deserves. The effort of the OP (who is gone long time ago!) is irrelevant, if many people (Lubos Motl among them!) deem the question as a good one and are interested in seing answers, and should therefore not be an obstruction for reopen. I disagree with your and Chris's dismissive opinion in this case. – Dilaton Jul 07 '13 at 12:20
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    @user1504 what is the problem with getting an answer by somebody who knows the subject? As I said in my comment when first the su(3) symmetries appeared in the organization of resonances a similar wonder could have been expressed. I trust that Lubos knows a lot more than I and if he finds the question relevant and interesting I would like an answer from him or somebody on that level, so that I can learn something new – anna v Jul 07 '13 at 12:21
  • @annav: I want to see more good research-level questions; two sentence questions asking for an essay from an expert don't count as good in my book. It's telling, I think, that pho chose not to write a response; I doubt there's anyone who knows the subject better. – user1504 Jul 08 '13 at 04:31
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    @user1504 pho and anybody else had not much time to write an answer, as the question was closed only 4 hours after it had been asked. So you can not claim that anybody has actively choosen not to answer the question. Also pho later reconsidered his decision to close and voted to reopen as he said in his last comment. And in my opinion the length of the question has nothing to do with the legitimacy of the question, if it asks about an interesting issue. In addition, this is a conceptual question not homework, so the show your work criterium does not apply. – Dilaton Jul 08 '13 at 07:05
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    user1504 heck I am wrong, the time window anybody could answer was not even 2 hours, so your point about @pho actively choosing not to answer is even complete nonsense... – Dilaton Jul 08 '13 at 08:57
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    Uh... A large number of comments were deleted? The one in whcih Dilaton changed 4 hours to 2 hours, etc. And same with my comment that it should be reopened, +1, etc. – Abhimanyu Pallavi Sudhir Jul 27 '13 at 09:21
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    Urs Schreiber answered this at http://www.physicsoverflow.org/5851 – Arnold Neumaier Jul 12 '15 at 13:23

1 Answers1

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I'll answer the relation between string theory and $E(8)$ -- a common appearance of $E(8)$ in string theory is in the gauge group of Type HE string theory $E(8)\times E(8)$ (see here for an explanation why). But it's interesting physically because it embeds the standard model subgroup.

$$SU(3)\times SU(2)\times U(1)\subset SU(5)\subset SO(10)\subset E(6)\subset E(7)\subset E(8)$$

Indeed, the ones in between are GUT subgroups, and $E(8)$ happens to be the "largest" of the exceptional lie groups.

Wikipedia has some things to say about the connections to monstrous moonshine, I'm not familiar with it. See [1], [2] re: the connections to number theory. Another example is how "1+2+3+4=10" demonstrates a 10-dimensional theory's ability to explain the four fundamental fources -- EM is the curvature of the $U(1)$ bundle, the weak force is the curvature of the $SU(2)$ bundle, the strong is the curvature of the $SU(3)$ bundle and gravity is the curvature of spacetime.

[Archiving Ron Maimon's comment here in case it gets deleted --]

There is another point, that E(8) is has embedded E6xSU(3), and on a Calabi Yau, the SU(3) is the holonomy, so you can easily and naturally break the E8 to E6. This idea appears in Candelas Horowitz Strominger Witten in 1985, right after Heterotic strings and it is still the easiest way to get the MSSM. The biggest obstacle is to get rid of the MS part--- you need a SUSY breaking at high energy that won't wreck the CC or produce a runaway Higgs mass, since it seems right now there is no low-energy SUSY.

Abhimanyu Pallavi Sudhir
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    SO(10) is not a subgroup of U(5). Why would a TOE need E(8) just because it is the largest exceptional group? The 1,2,3,4 numerology is rather weak since you are just looking at groups with these numbers in them that appear in very different ways. – Philip Gibbs - inactive Aug 09 '13 at 10:19
  • @PhilipGibbs: Fixed the SO(10) U(5) probem . The $E(8)$ logic was supposed to be intuitive . The 1,2,3,4 thing isn't numerology, it isn't so different, by the way . – Abhimanyu Pallavi Sudhir Aug 09 '13 at 10:25
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    There is another point, that E(8) is E6xSU(3), and on a Calabi Yau, the SU(3) is the holonomy, so you can easily and naturally break the E8 to E6. This idea appears in Candelas Horowitz Strominger Witten in 1985, right after Heterotic strings and it is still the easiest way to get the MSSM. The biggest obstacle is to get rid of the MS part--- you need a SUSY breaking at high energy that won't wreck the CC or produce a runaway Higgs mass, since it seems right now there is no low-energy SUSY. – Ron Maimon Aug 22 '13 at 22:04
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    @DImension10AbhimanyuPS: ok, but you shouldn't write what I said, which is technically wrong--- E8 is not E6xSU(3), it's a simple group, but it has an embedded E6xSU(3) and fills in the off-diagonal parts with extra crud that's broken when you have SU(3) gauge fluxes which follow the holonomy of the manifold. The precise decomposition is described in detail in Green Schwarz and Witten, which has a nice description of E8. – Ron Maimon Aug 23 '13 at 02:15
  • @RonMaimon: I know, but I think that is clear (that $E(8)$ is not $E(6)\times SU(3)$. – Abhimanyu Pallavi Sudhir Aug 23 '13 at 04:03
  • @DImension10AbhimanyuPS: With someone as young as you, people will just assume you don't know anything, so just make sure. – Ron Maimon Aug 23 '13 at 06:04