Dirac equations, light cone supersymmetry, and superconformal algebras

After a brief historical survey that emphasizes the role of the algebra obeyed by the Dirac operator, we examine an algebraic Dirac operator associated with Lie algebras and Lie algebra cosets. For symmetric cosets, its ``massless'' solutions display non-relativistic supersymmetry, and can be identified with the massless degrees of freedom of some supersymmetric theories: N=1 supergravity in eleven dimensions (M-theory), type IIB string theory in ten and four dimensions, and in four dimensions, N=8 supergravity, N=4 super-Yang-Mills, and the N=1 Wess-Zumino multiplet. By generalizing this Dirac operator to the affine case, we generate superconformal algebras associated with cosets ${\bf g}/\bf h$, where $\bf h$ contains the {\it space} little group. Only for eleven dimensional supergravity is $\bf h$ simple. This suggests, albeit in a non-relativistic setting, that these may be the limit of theories with underlying two-dimensional superconformal structure.