U-Duality and M-Theory

U-Duality and M-Theory This work is intended as a pedagogial introdution to M-theory and to its maximally supersymmetric toroidal compactifiations, in the frameworks of 11D supergravity, type II string theory and M(atrix) theory . U-duality is used 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, discuss T-duality in the perturbative and non-perturbative sectors from an algebraic point of view, and apply the same tools to the analysis of U-duality at the level of the effective action and of the BPS spectrum, with a particular emphasis on Weyl and Borel generators. We derive the U-duality multiplets of BPS particles and strings, U-duality in variant mass formulae for 1/2- and 1/4-BPS states for general toroidal compactifiations on skew toric with gauge backgrounds, and U-duality multiplets of constraints for states to preserve a given fraction of supersymmetry. A number of mysterious states are encountered in D ≤ 3 , whose existence is implied b y T-duality and 11D Lorentz invariance. We then move to the M(atrix) theory point of view, give an introduction to Discrete Light-Cone Quantization (DLCQ) in general and DLCQ of M-theory in particular. We discuss the realization of U-duality as electri-magnetic dualities of the Matrix gauge theory , display the Matrix gauge theory BPS spectrum in detail, and discuss the conjectured extended U-duality group in this scheme.