Noncommutative Geometry and Spacetime Gauge Symmetries of String Theory

Abstract

We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a universal gauge group which overlies all of the dynamical symmetries of the string spacetime. We also describe some aspects of toroidal compactifications with a light-like coordinate and show how certain generalized Kac-Moody symmetries, such as the Monster sporadic group, arise as gauge symmetries of the resulting spacetime and of superstring theories.