U-duality and M-theory

This work is intended as a pedagogical introduction to M-theory and to its maximally supersymmetric toroidal compactifications, in the frameworks of 1 1D supergravity, type II string theory and M(atrix) theory. U-duality is us ed as the main tool and guideline in uncovering the spectrum of BPS states. We review the 11D supergravity algebra and elementary 1/2-BPS solutions, d iscuss T-duality in the perturbative and non-perturbative sectors from an a lgebraic point of view, and apply the same tools to the analysis of U-duali ty at the level of the effective action and of the BPS spectrum, with a par ticular emphasis on Weyl and Borel generators. We derive the U-duality mult iplets of BPS particles and strings, U-duality invariant mass formulae for 1/2- and 1/4-BPS states for general toroidal compactifications on skew tori with gauge backgrounds, and U-duality multiplets of constraints for states to preserve a given fraction of supersymmetry. A number of mysterious stat es are encountered in D less than or equal to 3, whose existence is implied by T-duality and 11D Lorentz invariance. We then move to the M(atrix) theo ry point of view, give an introduction to Discrete Light-Cone Quantization (DLCQ) in general and DLCQ of M-theory in particular. We discuss the realiz ation of U-duality as electric-magnetic dualities of the Matrix gauge theor y, display the Matrix gauge theory BPS spectrum in detail, and discuss the conjectured extended U-duality group in this scheme. (C) 1999 Elsevier Scie nce B.V. All rights reserved.