Actually, this is something I wonder about. Technically, what string theory requires is that the central charge be zero. Right?In physics, string theory has 11 dimensions. In mathematics, there is no limit - for example, Hilbert space is infinite dimensional.
I think that question may be linked to the other question as to whether there is an infinite regression into smaller and smaller particles. As I recall there is a theorem which states that there is a limit to discreteness of particles. I think it has something to do with entropy, event horizons, and number of microstates. Maybe someone here can give more info on that. But anyway, if there were an infinite number of dimensions, then wouldn't that mean there were an infinite number of degrees of freedom, and you'd have the same issues as with an infinite number of infinitesimal particles. Right?Is it possible?