# Different action principle for describing (maybe different) dynamics of string

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.

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).