In the lectures of string theory by David Tong, he gave a simple idea to come up with action of closed string, it's the area of worldsheet since its independent of our coordinate choice (reparameterization invariant). This leads to the Nambu-Goto action.

But the idea to use an invariant, lead me to think if it's possible to come up with a different invariant with the worldsheet, since ultimately worldsheet is a manifold embedded in Minkowski space. So I want to ask (people who know topology, manifolds) whether there exists different invariant (like Betti number, Euler characteristic) which could give me some sort of equation of motion of a string.


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The worldsheet theory of a string is a two dimensional quantum theory of gravity and, as such, is expected that the entire quantum dynamics can be described as a sum over worldsheet geometries and topologies.

From the geometric side the only contributions come from the unique metric invariant of the worldsheet, namely, its area. That gives you the Nambu-Goto or Polyakov part of the string action.

Note: Naively one might think that the Ricci scalar is another scalar that can be constructed from the worldsheet metric. And that's correct, but the only consistent way of adding it to the string action is as $$\int_{Worldsheet}d^{2}\sigma (-g)^{\frac{1}{2}}R \ ,$$ a topological invariant ($4\pi$ times the Euler characteristic of the worldsheet).

From the topological side. Fortunately, not so many topological invariant exists for two dimensional surfaces. An exhaustive classification of two dimensional Riemann surfaces exist and the result is that surfaces are classified by its orientability and its genus. That's it.

No other geometrical or topological quantities are relevant for the classical string dynamics. Other subtle and important topological and algebro-geometric aspects of Riemann surfaces (such as the number of Beltrami differentials on a given surface, the existence of a virtual fundamental classes or metrics over Teichmuller spaces etc.) are relevant but they become important only after quantization.


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