U-duality and M-Theory

TL;DR

This paper provides a comprehensive pedagogical overview of U-duality in M-theory, exploring its role in classifying BPS states, analyzing compactifications, and connecting to Matrix theory and DLCQ frameworks.

Contribution

It introduces new U-duality invariant mass formulas and multiplets for BPS states in toroidal compactifications, and discusses the realization of U-duality in Matrix gauge theories.

Findings

Derived U-duality multiplets for BPS particles and strings.

Presented U-duality invariant mass formulas for various BPS states.

Explored the realization of U-duality as electric-magnetic dualities in Matrix theory.

Abstract

This work is intended as a pedagogical introduction to M-theory and to its maximally supersymmetric toroidal compactifications, 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 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 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 states are encountered in D<=3, whose existence is implied by 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 electric-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.