Consider the following 2D conformal field theories:

  • 26D bosonic string theory compactified on the $\Lambda_{24}$ Leech lattice torus.
  • 10D N=1 Superstring Theory (any of the 5 as they are all dual) compactified on the $E_8$ lattice torus.
  • N=(2,0) superconformal 6D theory compactified on the $F_4$ lattice torus.
  • N=4 Super Yang Mills Theory (for some gauge group SU(n)) on the $G_2$ (hexagonal lattice) torus.
  • N=8 Superconformal Chern-Simons Theory in 3D compactified on a line segment.

i.e. Taking (possibly) the most symmetric field theory in these dimensions and compactifying it on a lattice torus which is the most densest lattice (from sphere packing) which is 2 dimensions less.

They all produce a 2D (conformal?) field theory. The first one gives the Monster Conformal Field Theory. Now, it seems that because of there is so much constraints that these other theories should either produce the same Monster CFT or another CFT related to an interesting group?

What is more, if they all can be compactified to the same (or related) 2D conformal field theories, does this imply dualities between all these theories?

To me it seems there is this hierarchy of theories related to densest sphere packings. Which could be related to a hierarchy of groups with the Monster at the top. That's what it looks like but is there any truth in this?

I believe the first case it's call Monstrous Moonshine.

(BTW. I Think this is really a math question but I would guess it is more relevant to theoretical physics).

Note also that all these theories obey the equation $(D-2)\times N = 8$. If we set for 26D bosonic theory $N=\frac{1}{3}$. (But it doesn't really have any supersymmetry).

  • $\begingroup$ do you have any reason to expect these theories to be the same or related? $\endgroup$ – AccidentalFourierTransform May 15 '18 at 23:16
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    $\begingroup$ Well they all can be compactified down to a CFT in 2D on a dense sphere packing lattice. They all have some kind of maximal (super)symmetry in their respective dimensions. The lattices are all sub-lattices of each other. I would expect they are related somehow. Also it doesn't make physical sense for there to be more than one consistent physical theory. $\endgroup$ – zooby May 15 '18 at 23:18
  • $\begingroup$ Also I know that the second case $E_8$ is already an extension of the Monstrous Moonshine conjecture. $\endgroup$ – zooby May 15 '18 at 23:24
  • $\begingroup$ In your 3rd point, the 6d N=(2,0) are classified by ADE ; on the other hand the bosonic string on the Leech lattice is unique. Isn't this an obstacle to the postulate that the theories should be equivalent after compactification ? $\endgroup$ – Antoine May 16 '18 at 18:37
  • $\begingroup$ That is a good point. However, note that all 5 string theories are dual to each other. Under compactification could all lead to a theory that is self-dual? The other theories require a group such as $SU(n)$ but perhaps one could argue that the most symmetric theory is when $n\rightarrow \infty$. All other theories being subsets of this one. $\endgroup$ – zooby May 17 '18 at 15:38

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