# Bimonster and Heterotic String Theory

A. The Bimonster and the Complex Lorentzian Leech Lattice involves a construction that extends Y555 from 16 to 26, and relates to Incidence(P^2/F3) among other things. (13 projective points + 13 projective lines), Complex Lorentzian Leech Lattice as Hyperbolic cell 2 + 24 dimensions, etc.

B. How does this relate to the 26 dimensions of Bosonic String Theory (Where 26 dimensions of the Lorentzian Leech Lattice (IIsub(25,1)) which is a torus in 24 dimensions orbifolded over a 2-point reflection group....and collapses from 26 to 10 dimensions using E8 X E8, one version of heterotic string theory...

The confusion lies in that this (B.) is 10 + 16 = 26 (or 26 dimensions collapsed down to 10), and the Bimonster situation (A.) is 16 nodes of the Dynkin diagram Y555 (or M666) of M X M wreath built up (with some difficulties) to 26 nodes of Inc(P2/F3) if I understand it right.

1. Is this 16 homomorphic (or related) to E8 X E8 in string theory?
2. Is the 2-point reflection group in bosonic string theory the same as the hyperbolic cell discussed in the Bimonster construction? Z2 in E8 X E8 is apparently the identification of the common boundaries of E8 X E8... I am fascinated by both A and B and would like more clarification of their relationships.

Thanks

PGH

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I can't help but add a comment - while the question is probably meant to be read by experts in string theory, it never hurts to formulate it so that a layman (i.e. a mathematician from other field) can at least have a clue about what's going on... – Michal Kotowski Jul 31 '11 at 20:18
@MichalKotowski: I very much doubt that the question is written with string theorists in mind. I, for one, find it hard to read. Judging by this and other questions by the OP, he does not believe in making questions accessible to non-experts. – José Figueroa-O'Farrill Jul 31 '11 at 21:52
My first comment would be: woooot? – Matthias Ludewig Nov 25 '11 at 21:52
@Paul: I would like to add my vote with @Michal's. Please formulate questions so that semi-experts can read them. For example, I am very interested in both representation theory and mathematical physics, and one of the ways I like to learn about related areas is by reading questions on MO. Your question is pretty much useless to me. Conversely, it's clear from your question that R. Borcherds is one expert who likely can help, and I see he has answered below. But perhaps others would also be able to contribute if you provided context, background, definitions, etc. – Theo Johnson-Freyd Nov 26 '11 at 5:02
Yes, I am trying to get my ducks in a row here and supply all the correct definitions, framework, and context. @Percy Paul, please send me your email in your next response, I wasn't able to find a way to contact you. Thanks so much. Paul. – Paul Hjelmstad Nov 26 '11 at 19:53

The connection between the bimonster and moonshine and the 26 dimensions of string theory is still mysterious (at least to me), though there are several intriguing hints that there is something going on.

Some papers discussing this are as follows:

The paper by Miyamoto "21 involutions acting on the Moonshine module" J . Algebra 175 (1995), no. 3, 941–965. gives a relation between the 26 involutions generating the bimonster and the natural module of the monster. His construction involves the 26 dimensional even unimodular Lorentzian lattice that also appears in string theory and moonshine.

Basak "The complex Lorentzian Leech lattice and the Bimonster" J. Algebra 309 (2007), no. 1, 32–56 gives a complex reflection group generated by 26 complex reflections similar to the bimonster, except that the generators have order 3 rather than 2.

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Thanks. I've read the two Basak papers on the Bimonster, I will download the Miyamoto paper, which I may have read but I don't recall it. Note: I am not a string theorist nor an expert on this subject either. I can try to make my posts more accessible, but part of that is due to the limitations of my own understanding on the subject matter. PGH – Paul Hjelmstad Jul 31 '11 at 23:27
@Paul: In my humble opinion, your posts would improve a lot by just a couple of simple changes: defining the notation, using LaTeX and spending some time on the motivation/contextualisation. – José Figueroa-O'Farrill Aug 1 '11 at 14:18

The bimonster acts as the automorphism group of a particular bosonic closed string theory and D-brane states in this theory that preserve a chiral subalgebra transform in representations of the bimonster. See http://arxiv.org/pdf/hep-th/0202074 for more details. This does not involve $Y555$ or the heterotic string, but might be useful in thinking about such connections.

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I've read the papers Dr. Borcherds cited. The Miyamoto paper is interesting in that by removing 4 points and 1 line from the 13 points and lines of the Bimonster (Inc(P^2/F3) you obtain a representation of the Monster, in the 21 involutions of the Affine Plane (3), which has 9 points and 12 lines. I don't understand all of it yet, but apparently these 4 points are on the part of the Bimonster that is identified by the Z2 portion of M X M wr 2 (The wreath product of M with itself). The Monster acts on the LL, which is E8 three times Thanks for the citation - I will try to get to it today. pgh. – Paul Hjelmstad Aug 19 '11 at 17:21
The Monstrous Branes paper is very good also. However, I am stuck on something, I cannot find any definition of the S-modular transform. A question for Dr. Harvey (coauthor of the paper) or whomever has the answer. I need to also go over quite a bit of string theory and brane theory but answering this question would be very helpful. I imagine it relates to S-duality, but I am not sure pgh. PS The short paper "An Elementary Approach to the Monster" by Christopher Simons, and "26 Implies the Bimonster' By J. H. Conway proved to be helpful introductory papers for tackling these longer ones. – Paul Hjelmstad Sep 3 '11 at 23:50