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Two important unimodular lattices are $E_8$ and the Leech lattice.

  1. One can take 10D superstring theory and compactify it over the $E_8$ torus.

  2. One can also take 26D bosonic string theory and compactify it over the Leech Latice $\Lambda_{24}$.

In both cases one ends up with a 2 dimensional theory.

(Due to the various dualities each of the 10D superstring theories is probably dual to each other when compactified down to 2 dimensions.)

The question is then whether these pair of 2D field theories one ends up with are equivalent in some way. Yes, one started with N=1 supersymmetry and has fermions but in 2D the distinction between bosons and fermions is less important (due to e.g. bozonization). Also with heterotic string theory one can think of it as the left hand modes moving in 26 dimensions anyway.

We know the second one has conncections with the Monster group. So either the first one is equivalent and also had conncections with the Monster group or it would be connected to some other group.

So the question is:

"Is there a duality between a 10D superstring theory on $E_8$ torus with 26D bosonic string theory on the Leech lattice torus".

I think the easiest way to disprove this would be to compare the degrees of freedom of the lowest energy level particles.

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The answer is no. The reason is supersymmetry.

No matter how do you compactify the theory of the bosonic string; tachyons sit ubiquitously in the spectrum.

On the other hand, the five $d=10$ superstring theories are tachyon free; this property is preserved under compactification on a flat torus.

The point is that dualities can't relate quantum, consistent, stable, UV-complete and anomaly free backgrounds (namely, supersymmetric string compactifications) with theories that, indeed, doesn't fully exist as quantum mechanical systems ($d=26$ bosonic string theory).

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    $\begingroup$ Interesting. Here's a crazy thought: In the 1+1 dimensional theory "bozonization" in which fermions and bosons can be exchanged may add an extra symmetry cancelling out the tachyons. Or perhaps the tachyons get restricted to the compacted dimensions. Strange things can happen.... $\endgroup$
    – zooby
    Oct 10, 2020 at 11:40
  • $\begingroup$ I can't see how bosonization can induce new symmetries; I'm not aware of a single example where your speculation works, Do You have an example or an argument?. Also the tachyons cannot be restricted to the internal directions; the noncompact directions are also unstable and there is no way to project out the tachyons. $\endgroup$
    – Ramiro Hum-Sah
    Oct 11, 2020 at 23:46
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    $\begingroup$ No, I don't have any examples. Just a hunch that 2 dimensions strange things can happen, like 2D CFTs. In fact even stranger things might happen if you compactified both theories down to 1 dimension or even 0 dimensions. My hunch is that in 2D the requirements for a consistent "string-like" theory might be so great that there is only one. It's just a hunch though. $\endgroup$
    – zooby
    Oct 12, 2020 at 0:16
  • $\begingroup$ Oh, I understand :) Yeah, in principle is difficult to prove that your idea does cannot take place. Indeed, there is something really interesting about your question. Both lattices, the root lattice of $E_{8}$ and the Leech one, seem to suggest that "useful" 2D CFTs can be constructed with string theories as their starting point. The problem is that, the precise meaning of the intrinsic meaning, or the correct usage of such constructions from a purely stringy theoretical is far from obvious. $\endgroup$
    – Ramiro Hum-Sah
    Oct 12, 2020 at 4:24
  • $\begingroup$ I agree with the statement that something non-trivial and truly exciting is happening between those lattices and string theory. I adventure the following speculation: Maybe an "heterotic"-like version of your setup could work. Construct an heterotic string with left movers (bosonic string ones) compactified on the Leech lattice and the right ones (superstring) compactified on the torus associated to the Leech lattice of $E_{8}$. I cannot see any obvious argument about why that construction shouldn't work. $\endgroup$
    – Ramiro Hum-Sah
    Oct 12, 2020 at 4:34

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