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Lectures on Higher-Gauge Symmetries from Nambu Brackets and Covariantized M(atrix) Theory

This lecture consists of three parts. In part I, an overview is given on the so-called Matrix theory in the light-front gauge as a proposal for a concrete and non-perturbative formulation of M-theory. I emphasize motivations towards its covariant formulation. Then, in part II, I turn the subject to the so-called Nambu bracket and Nambu mechanics, which were proposed by Nambu in 1973 as a possible extension of the ordinary Hamiltonian mechanics. After reviewing briefly Nambu's original work, it will be explained why his idea may be useful in exploring higher symmetries which would be required for covariant formulations of Matrix theory. Then, using this opportunity, some comments on the nature of Nambu mechanics and its quantization are given incidentally: though they are not particularly relevant for our specialized purpose of constructing covariant Matrix theory, they may be of some interests for further developments in view of possible other applications of Nambu mechanics. The details will be relegated to forthcoming publications. In part III, I give an expository account of the basic ideas and main results from my recent attempt to construct a covariantized Matrix theory on the basis of a simple matrix version of Nambu bracket equipped with some auxiliary variables, which characterize the scale of M-theory and simultaneously play a crucial role in realizing (dynamical) supersymmetry in a covariant fashion.