D-branes and Short Distances in String Theory

We study the behavior of D-branes at distances far shorter than the string length scale~$l_s$. We argue that short-distance phenomena are described by the IR behavior of the D-brane world-volume quantum theory. This description is valid until the brane motion becomes relativistic. At weak string coupling $\gs$ this corresponds to momenta and energies far above string scale. We use 0-brane quantum mechanics to study 0-brane collisions and find structure at length scales corresponding to the eleven-dimensional Planck length ($\lp11 \sim \gs^{1/3} l_s$) and to the radius of the eleventh dimension in M-theory ($\R11 \sim \gs l_s$). We use 0-branes to probe non-trivial geometries and topologies at sub-stringy scales. We study the 0-brane 4-brane system, calculating the 0-brane moduli space metric, and find the bound state at threshold, which has characteristic size $\lp11$. We examine the blowup of an orbifold and are able to resolve the resulting $S^2$ down to size $\lp11$. A 0-brane with momentum approaching $1/\R11$ is able to explore a larger configuration space in which the blowup is embedded. Analogous phenomena occur for small instantons. We finally turn to 1-branes and calculate the size of a bound state to be $\sim \gs^{1/2} l_s$, the 1-brane tension scale.