Type-IIB-string–M-theory duality and longitudinal membranes inM(atrix) theory

In this paper we study duality properties of the $M$(atrix) theory compactified on a circle. We present evidence for the equivalence of this theory to the strong coupling limit of type-IIB string theory compactified on a circle. In the $M$(atrix) theory context, our evidence for this duality consists of showing the appearance (upon compactification) of a topological term recently discovered in the $D$-string action, identifying the BPS states of type-IIB strings in the spectrum and finding the remnant symmetry of $\mathrm{SL}\mathrm{}(2,Z)\mathrm{}$ and the associated τ moduli. By this type-IIB-string–$M$-theory duality, a number of insights are gained into the physics of longitudinal membranes in the infinite momentum frame.