Aspects of U-duality in matrix theory Original Research Article

We explore various aspects of implementing the full M-theory U-duality group Ed+1, and thus Lorentz invariance, in the finite N matrix theory (DLCQ of M-theory) describing toroidal IIA-compactifications on d-tori: (1) We generalize the analysis of Elitzur et al. (hep-th/9707217) from Ed to Ed+1 and identify the highest weight states unifying the momentum and flux Ed-multiplets into one Ed+1-orbit. (2) We identify the new symmetries, in particular the Weyl group symmetry associated to the (d + 1)th node of the Ed+1 Dynkin diagram, with Nahm-duality-like symmetries (N-duality) exchanging the rank N of the matrix theory gauge group with other (electric, magnetic, …) quantum numbers. (3) We describe the action of N-duality on BPS bound states, thus making testable predictions for the Lorentz invariance of matrix theory. (4) We discuss the problems that arise in the matrix theory limit for BPS states with no top-dimensional branes, i.e. configurations with N = 0. (5) We show that N-duality maps the matrix theory SYM picture to the matrix string picture and argue that, for d even, the latter should be thought of as an M-theory membrane description (which appears to be well defined even for d > 5). (6) We find a compact and unified expression for a U-duality invariant of Ed+1 for all d and show that in d = 5, 6 it reduces to the black hole entropy cubic E6- and quartic E7-invariants respectively. (7) Finally, we describe some of the solitonic states in d = 6, 7 and give an example (a ‘rolled-up’ Taub-NUT 6-brane) of a configuration exhibiting the unusual 1gs3 behaviour.