Now we consider the limit in which we send the string length to zero; l$_{s}$ → 0 ( α ′ → 0). All other dimensionless parameters (the string coupling constant, g$_{s}$ , and N ) we keep fixed. Doing this means the coupling of the strings to each other ( ∼ g$_{s}$α ′ $^{2}$) goes to zero, and hence the strings do not interact: S$_{int}$ is turned off, and we are left with two decoupled systems: classical ten dimensional gravity in the bulk, S$_{bulk}$ , and a four dimensional gauge theory on the surface of the branes, S$_{brane}$ . What is the exact nature of the four dimensional gauge theory? The ends of the open strings are labelled by the branes they are attached to. The strings are orientated, so there are two types of strings stretching between any two branes: one whose left end is on the first brane and whose right end is on the second brane, and vice versa. These labels are called Chan-Paton factors, and in the case of a coincidental stack of branes, as we have here, there is a clear re-labelling symmetry of the Chan-Paton factors. There are N 2 possible string relabellings, which fills the adjoint repre- sentation of the U(N) Lie group. The strings are free to move around on the surfaces of the branes, so the relabelling can be performed at any point: hence the U(N) symmetry is a local one. In other words it is a gauge theory, and the Chan-Paton labels can be interpreted as labelling the gauge charge of each end of the string. In the large N limit, the propagators of adjoint fields in a U ( N ) and SU ( N ) gauge theory differ by a vanishing term, proportional to N − 1 [22], so when N → ∞ , we can treat U ( N ) and SU ( N ) gauge theories as identical. The spectrum of strings is given by the dimensional reduction of an N = 1 35