Matrix Theory, U-Duality and Toroidal Compactifications of M-Theory

Abstract

Using U-duality, the properties of the matrix theories corresponding to the compactification of M-theory on $T^d$ are investigated. The couplings of the $d+1$ dimensional effective Super-Yang-Mills theory to all the M-theory moduli is deduced and the spectrum of BPS branes in the SYM gives the corresponding spectrum of the matrix theory.Known results are recovered for $d\le 5$ and predictions for $d>5$ are proposed. For $d>3$, the spectrum includes $d-4$ branes arising from YM instantons, and U-duality interchanges momentum modes with brane wrapping modes.For $d=6$, there is a generalised $þ$-angle which couples to instantonic 3-branes and which combines with the SYM coupling constant to take values in $SL(2,\mathbb{R})/U(1)$, acted on by an $SL(2,\mathbb{Z})$ subgroup of the U-duality group $E_6(\mathbb{Z})$. For $d=4,7,8$, there is an $SL(d+1)$ symmetry, suggesting that the matrix theory could be a scale-invariant $d+2$ dimensional theory on $T^{d+1}\times \mathbb{R}$ in these cases, as is already known to be the case for $d=4$; evidence is found suggesting this happens for $d=8$ but not $d=7$.

28 Pages, Phyzzx Macro. Minor corrections