The Geometry of Exceptional Super Yang-Mills Theories

Abstract

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 $Ad{S}_4\otimes S^8$, 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 ($Ad{S}_4\otimes S^7$) and M5-brane ($Ad{S}_7\otimes S^4$) 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 $Ad{S}_4\otimes S^{8n}$ and $Ad{S}_7\otimes S^{8n-3}$, with reduced subspaces $Ad{S}_4\otimes S^{8n-1}$ and $Ad{S}_7\otimes S^{8n-4}$, 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 $Ad{S}_4\otimes S^{23}$, hinting at M2 and M21 branes as solutions of bosonic M-theory, and reduction to $Ad{S}_3\otimes S^{23}$ gives support for Witten's monstrous $AdS$/CFT construction.

15 pages, 2 figures; v3: minor changes, addressed comments from editor