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

Question

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?

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user user1796

Comments

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?

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Luboš Motl

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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/…

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Mitchell Porter

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Related question on mathoverflow: mathoverflow.net/q/58990/13917

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Qmechanic

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I remember the days when the eightfold way en.wikipedia.org/wiki/Eightfold_Way_%28physics%29 was mysterious.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user anna v

Answer 1

To start with, the relation of string theory to complex elliptic curves is clear: these are just pointed, genus one closed Riemann surfaces, and hence are certain string worldsheets. The fact that in constructions such as the "refined Witten genus" it is actually arithmetic elliptic curves (not over the complex numbers but, ultimately, over the ring of integers and hence over the rationals and the p-adics, see at fracture square) that play a role is some deep fact that is vaguely reminiscent of p-adic string theory, only that what presently goes under this headline does not fully live up to what is at issue here. (There is a PO question on this point here).

The true answer to this arithmetic geometry-incarnation of stringy pysics must rest in the function field analogy, which rouhgly says something like if you do algebraic number theory in a single variable -- if you study arithmetic curves -- , then this is analogous to studying complex curves, hence string worldsheets.

To put this in perspective: there is an old motivation from the first pages of the string theory textbooks, which says that where point particle mechanics is about the real line (the worldline) so string theory is about the complex plane (the worldsheet) and hence that the passage from point particles to strings is like the step from real analysis to complex analysis.

Somehow the function field analogy says that this seemingly simple-minded statement is indeed true and much deeper than it might maybe seem. In some way stringy phenomena are visible at the very root of mathematics (number theory) because if you "work with a siingle algebraic variable", then that is already analogous to "working with a single complex variable" hence is analogous to studying complex curves, hence string worldsheets.

Even that statement may still seem far-out at this level.But digging deeper it turns out to work out more and more. For instance 90 per cent of number theory is about picking some such arithmetic curve and then "attaching" to it a zeta-function or theta-function or eta-function or L-function. The deep conjectures of number theory all revolve around this (notably the Langlands correspondence). But looking at this from the point of view of the function field analogy, one finds that all this is analogous to the 3dCS/2dWZW correspondence. I have tried to summarize this a bit in this table here: zeta-functions and eta-functions and theta-functions and L-functions -- table .

There'd be more to say, but I am running out of battery. I gave a talk related to this four weeks back at CUNY, here.

Answer 2

Let me first answer the relation between string theory and $E(8)$ (I don't think I can answer the rest.). A common appearance of $E(8)$ in strings theory, is in the gauge group of Type HE string theory, i.e., in $E(8)\otimes E(8)$. Now, this appears in Type HE string theory because of the fact that it is an even, unimodular lattice. But, it is interesting, for another reason; due to the embedding of the Standard Model Subgroup:

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

That's a lot of embeddings, but notice - The first group here, in the Standard Model subgroup, the second, third, fourth, fifth, are GUT subgroups. And $E(8)$ happens to be the "largest" and "most complicated" of the exceptional lie groups. So a TOE better deal with $E(8)$, somewhere!

I don't know about the relation between monstrous moonshine and string theory, but you can refer to Wikipedia.

There is definitely a connection with number theory. And even more: .

$$1+2+3+4=10$$

Not joking! EM is the curvature of the $U(1)$ bundle . Weak is the curvature of the $SU(2)$ bundles. Strong is the curvature of the $SU(3)$ bundle. Gravity is the curvature of spacetime . I.e. 1D manifold, 2D, 3D, 4D $\implies$ 10 D .

Comments

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.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Philip Gibbs

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@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 .

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Dimensio1n0

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@PhilipGibbs: In fact, why do you think Kaluza - Klein theory is 5-dimensionals?

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Dimensio1n0

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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.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Ron Maimon

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@RonMaimon: Thanks, I added that in too.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Dimensio1n0

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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.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Ron Maimon

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@RonMaimon: I know, but I think that is clear (that $E(8)$ is not $E(6)\times SU(3)$.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Dimensio1n0

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@DImension10AbhimanyuPS: With someone as young as you, people will just assume you don't know anything, so just make sure.

This post imported from StackExchange Physics at 2014-03-07 16:32 (UCT), posted by SE-user Ron Maimon