One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action. However, once you see how to make this 'generalization', it becomes obvious how to write down the action, not just for a string, but for manifolds of higher dimension as well. In fact, Becker-Becker-Schwarz (the main source I happen to be learning from) actually do this. But (as far as I have read), they merely write down the action and do nothing further with it.
My question is: what happens when we proceed along the same lines as string theory, but when replacing a string with a 2-manifold, the simplest example of which would be the 2-sphere, a "shell/membrane"? Assuming 3 spatial dimensions, this is the highest dimensional manifold we can consider (because we don't allow for non-compact manifolds). Furthermore, there is only one compact manifold of dimension 1; however, there are infinitely many compact manifolds of dimension 2, which could potentially make the theory much richer (and probably much more difficult). For example, in string theory, we are stuck with $S^1$ (if you insist upon having no boundaries), but if you allow for 2 dimensions, we could consider the sphere, the torus, etc.
Because it seems that this path is not ever presented (in fact, I've never even heard of it), I would presume that something goes wrong. So then, what exactly does go wrong?
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