Is superstrings on the $E_8$ torus dual to bosonic string theory on the Leech lattice torus?

Question

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.

Answer 1

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).