symmetric monoidal (∞,1)-category of spectra
The concept of a special linear group $SL(2,K)$ with $K = \mathbb{O}$ the octonions does not quite make sense, due to the failure of the octonions to be an associative algebra.
Nevertheless, the pattern of the special linear groups of the other three real normed division algebras suggests that for some suitably adjusted concept of “special linear group”, it should make sense to speak of $SL(2,\mathbb{O})$, after all, and that it should essentially be essentially the spin group $Spin(9,1)$:
exceptional spinors and real normed division algebras
Lorentzian spacetime dimension | $\phantom{AA}$spin group | normed division algebra | $\,\,$ brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\phantom{A}$ $\mathbb{R}$ the real numbers | super 1-brane in 3d |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\phantom{A}$ $\mathbb{C}$ the complex numbers | super 2-brane in 4d |
$6 = 5+1$ | $Spin(5,1) \simeq SL(2, \mathbb{H})$ | $\phantom{A}$ $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) {\simeq}$ “SL(2,O)” | $\phantom{A}$ $\mathbb{O}$ the octonions | heterotic/type II string |
One proposal for making sense of $SL(2,\mathbb{O})$ is due to Hitchin 18
Last revised on March 26, 2019 at 01:06:00. See the history of this page for a list of all contributions to it.