The Geometry of Exceptional Super Yang-Mills Theories

Some time ago, Sezgin, Bars and Nishino have proposed super Yang-Mills theories (SYM's) in $D=11+3$ and beyond. Using the "Magic Star" projection of $\mathfrak{e}_{8(-24)}$, we show that the geometric structure of SYM's in $11+3$ and $12+4$ space-time dimensions is recovered from the affine symmetry of the space $AdS_{4}\otimes S^{8}$, with the $8$-sphere being a line in the Cayley plane. By reducing to transverse transformations, along maximal embeddings, the near horizon geometries of the M2-brane ($AdS_{4}\otimes S^{7}$) and M5-brane ($AdS_{7}\otimes S^{4}$) are recovered. Generalizing the construction to higher, generic levels of the recently introduced "Exceptional Periodicity" (EP) and exploiting the embedding of semi-simple rank-3 Jordan algebras into rank-3 T-algebras of special type, yields the spaces $AdS_{4}\otimes S^{8n}$ and $AdS_{7}\otimes S^{8n-3}$, with reduced subspaces $AdS_{4}\otimes S^{8n-1}$ and $AdS_{7}\otimes S^{8n-4}$, respectively. Within EP, this suggests generalizations of the near horizon geometry of the M2-brane and its Hodge (magnetic) duals, related to $(1,0)$ SYM's in $(8n+3)+3$ dimensions, constituting a particular class of novel higher-dimensional SYM's, which we name exceptional SYM's. Remarkably, the $n=3$ level gives $AdS_{4}\otimes S^{23}$, hinting at M2 and M21 branes as solutions of bosonic M-theory, and reduction to $AdS_{3}\otimes S^{23}$ gives support for Witten's monstrous $AdS$/CFT construction.