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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?

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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.

Related: https://math.stackexchange.com/questions/2651652/do-the-euler-lagrange-equations-hold-meaning-for-an-infinite-action

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