# What is this correspondence between composition algebras over R and superstring theories?

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In the page for superstring theory, Wikipedia states:

Another approach to the number of superstring theories refers to the mathematical structure called composition algebra. In the findings of abstract algebra there are just seven composition algebras over the field of real numbers. In 1990 physicists R. Foot and G.C. Joshi in Australia stated that "the seven classical superstring theories are in one-to-one correspondence to the seven composition algebras".

The paper being cited does not explain this quote in the abstract and is otherwise inaccessible for me.

My understanding is as follows. The seven composition algebras over R are R, C, H, O, split-C, split-H, split-O. The five consistent superstring theories are Type I, Type IIA, Type IIB, SO(32) heterotic, E8×E8 heterotic. The citation implies that there are at least two more.

What are the other two superstring theories, and what is this correspondence?

This post imported from StackExchange MathOverflow at 2023-05-26 20:29 (UTC), posted by SE-user L. E.
What do you mean when you say that the paper is inaccessible? It seems to be freely accessible (at least, I can access it from home) at Foot and Joshi - Nonstandard signature of spacetime, superstrings, and the split composition algebras, which is linked on the Wikipedia page.

This post imported from StackExchange MathOverflow at 2023-05-26 20:29 (UTC), posted by SE-user LSpice
I don't understand what a string theory is, or what it means for a composition algebra to correspond to one, but the paper's justification for the claim seems to be the isomorphisms $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\newcommand\wt{\widetilde}\newcommand\mb{\mathbb}\SL(2, \mb R) \cong \wt\SO(2, 1)$, $\SL(2, \mb C) \cong \wt\SO(3, 1)$, $\SL(2, \mb H) \cong \wt\SO(5, 1)$, $\SL(2, \mb O) \cong \wt\SO(9, 1)$, $\SL(2, \mb C(-1)) \cong \wt\SO(2, 2)$, $\SL(2, \mb H(-1)) \cong \wt\SO(3, 3)$, and $\SL(2, \mb O(-1)) \cong \wt\SO(5, 5)$. I have not checked these isomorphisms.

This post imported from StackExchange MathOverflow at 2023-05-26 20:29 (UTC), posted by SE-user LSpice
I don't understand superstring theory but from what I read here I get the impression that the paper takes the 7 composition algebras and construct over each of them a string theory "classically" rather that they correspond to the "classical" string theories Type I, Type IIA, Type IIB, SO(32) heterotic, E8×E8 heterotic plus some two others although that would be cooler I guess maybe someone can clarify.

This post imported from StackExchange MathOverflow at 2023-05-26 20:29 (UTC), posted by SE-user Dabed
@Emily, I wonder whether it's really appropriate to post SciHub pointers here. It's a bit naughty.

This post imported from StackExchange MathOverflow at 2023-05-26 20:29 (UTC), posted by SE-user Dave Benson
@DaveBenson, according to a moderator, the appropriate action in such a case is to flag.

This post imported from StackExchange MathOverflow at 2023-05-26 20:29 (UTC), posted by SE-user LSpice