The concept of a special linear group SL(2,K)SL(2,K) with K=𝕆K = \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,𝕆)SL(2,\mathbb{O}), after all, and that it should essentially be essentially the spin group Spin(9,1)Spin(9,1):

exceptional spinors and real normed division algebras

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string

One proposal for making sense of SL(2,𝕆)SL(2,\mathbb{O}) is due to Hitchin 18


  • Nigel Hitchin, SL(2)SL(2) over the octonions, Mathematical Proceedings of the Royal Irish Academy. Vol. 118. No. 1. Royal Irish Academy, 2018 (arXiv:1805.02224)

Last revised on March 26, 2019 at 01:06:00. See the history of this page for a list of all contributions to it.