nLab ninebrane group

Contents

Contents

Idea

The low degrees in the Whitehead tower of the orthogonal group are closely linked to the physics of branes appearing in string theory/M-theory. Accordingly the low stages are called

\cdots \to fivebrane group \to string group \to spin group \to special orthogonal group \to orthogonal group

To which extent and in which sense this matching with brane physics continues is not entirely clear. Hisham Sati gives arguments that co-killing the next integer homotopy group π 11(O)\pi_{11}(O) corresponds to the M9-brane. Therefore one might call the corresponding stage in the Whitehead tower the ninebrane group. This suggestion appeared in (Sati-Schreiber-Stasheff 08)

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Whitehead tower of orthogonal grouporientationspin groupstring groupfivebrane groupninebrane group
higher versionsspecial orthogonal groupspin groupstring 2-groupfivebrane 6-groupninebrane 10-group
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

References

Last revised on May 26, 2022 at 22:52:10. See the history of this page for a list of all contributions to it.