Geometry of exceptional super Yang-Mills theories

Some time ago, Bars found $D=11+3$ supersymmetry and Sezgin proposed super Yang-Mills theory (SYM) in $D=11+3$. Using the “magic star” projection of $e_{8(-24)}$, we show that the geometric structure of SYM’s in $12+4$ and $11+3$ space-time dimensions descends to the affine symmetry of the space ${\mathrm{AdS}}_4\otimes S^8$. By reducing to transverse transformations along maximal embeddings, the near-horizon geometries of the M2 brane (${\mathrm{AdS}}_4\otimes S^7$) and M5 brane (${\mathrm{AdS}}_7\otimes S^4$) of M-theory are recovered. Utilizing the recently introduced “exceptional periodicity” (EP) and exploiting the embedding of semisimple rank-3 Jordan algebras into rank-3 T-algebras of special type yields the spaces ${\mathrm{AdS}}_4\otimes S^{8n}$ and ${\mathrm{AdS}}_{8n-1}\otimes S^5$ with reduced subspaces ${\mathrm{AdS}}_4\otimes S^{8n-1}$ and ${\mathrm{AdS}}_{8n-1}\otimes S^4$, respectively. As such, EP describes the near-horizon geometries of an infinite class of novel exceptional SYM’s in $(8n+3)+3$ dimensions that generalize M-theory for $n=1$. Remarkably, the $n=3$ level hints at M2 and M21 branes as solutions of bosonic M-theory and gives support for Witten’s monstrous AdS/CFT construction.