# Why is there a deep mysterious relation between string theory and number theory, elliptic curves, $E_8$ and the Monster group?

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?

• 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. Feb 7 '11 at 14:48
• 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 7 '11 at 15:10
• 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? Feb 8 '11 at 6:37
• 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 - mathoverflow.net/questions/24604/… ) Feb 8 '11 at 7:54
• 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 8 '11 at 14:20

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.

• 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. Aug 9 '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 . Aug 9 '13 at 10:25
• 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. Aug 22 '13 at 22:04
• @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. Aug 23 '13 at 2:15
• @RonMaimon: I know, but I think that is clear (that $E(8)$ is not $E(6)\times SU(3)$. Aug 23 '13 at 4:03