Hidden Symmetries, Central Charges and All That

TL;DR

The article surveys hidden $E_{n(n)}$ symmetries emerging in toroidal reductions of $D=11$ supergravity and their relation to central charges and BPS states, arguing that full M-theory likely requires an extended geometric framework with additional coordinates beyond eleven. It analyzes how central charges decompose into Kaluza-Klein and winding sectors and why toroidal truncations are incomplete without nonperturbative BPS states, motivating the conjectured U-duality as a symmetry of the full theory. It reviews reformulations of $D=11$ supergravity that organize fields into $E_{7(7)}$ or $E_{8(8)}$ covariant structures via generalized vielbeins, while acknowledging that full invariance is obstructed by KK gauge fields, suggesting an underlying exceptional geometry. Finally, it proposes BPS-extended supergravities that couple BPS towers (KKA/KKB) and potentially realize a higher-dimensional description, illustrated by a $D=9$, $N=2$ example with two mass scales and a membrane-based mass formula, highlighting a path toward unifying M-theory, IIA, and IIB within a 12-dimensional-like framework.

Abstract

In this review we discuss hidden symmetries of toroidal compactifications of eleven-dimensional supergravity. We recall alternative versions of this theory which exhibit traces of the hidden symmetries when still retaining the massive Kaluza-Klein states. We reconsider them in the broader perspective of M-theory which incorporates a more extended variety of BPS states. We also argue for a new geometry that may underly these theories. All our arguments point towards an extension of the number of space-time coordinates beyond eleven.