# On the finiteness of worldsheet area

It is commom to define the wordlsheet of a classical open string, for example, as the $$2$$-dimensional smooth manifold with boundary as $$\mathbb{R} \times [0,\pi]$$. With the appropriate embedding $$X: \Sigma \to \mathbb{R}^D$$ into minkowski space one defines the area of the worldsheet via induced metric $$\gamma = X^*\eta$$, i.e.

$$S[X] = -T \int_\Sigma \sqrt{-\det \gamma} \ d^2\xi = -T\int_\mathbb{R}\int_{0}^{\pi} \sqrt{-\det \gamma} \ d \sigma d \tau \tag1.$$

Is there any chance that the integral $$(1)$$ be divergent because of the integral over $$\mathbb{R}$$?

I don't know if the convergence of the integral is something that is implicitly assumed or it is stablished by some result.

Should I define an open string worldsheet as $$[0,\pi] \times [0, \pi]$$ (In this case it would be a manifold with corners instead) for example, in order to guarante finite volume of worldsheet?

Strictly speaking an infinite action is not well defined. But what you can do is to take generic $$t_i$$ and $$t_f$$, compute the equation of motion and then do the limit for very large times.