On Tue, Jul 08, 2003 at 07:49:25AM -0700, Eugene Salamin wrote:
1. Riemann surfaces are complex manifolds. Why does the string surface have the extra structure of C^1 rather than being simply R^2 ?
Don't forget, this is physics, so most spaces come with metrics. In this instance, the way these surfaces arise is really as conformal manifolds. The classical action only depends on the conformal class of the metric. (Hence the name of the related field, "Conformal Field Theory".)
2. I can understand trying out the idea that particles are little curves. But why can't they propagate in ordinary 4-dimensional spacetime; why do we need 11 dimensions?
On a quantum-mechanical level, the classical conformal symmetry is a little broken unless the space-time is the critical dimension. Exactly what the critical dimension is depends on the precise theory. (Yes, I know that's very vague.) Peace, Dylan