Importance and Origin of level-matching condition

Question

I want to know the origin of level-matching condition of string theory.

And want to know why many string theory textbook mention that this condition is crucial.

Is this level-matching condition always right? Does all string theory satisfy the level-matching condition?

Answer 1

1. The level-matching condition (LMC) for closed strings arises from the reparametrization invariance $\sigma\to \sigma +\sigma_0$ under a constant shift.

2. Open strings do not have this reparametrization symmetry under a constant shift because of the two endpoints, nor do they have the LMC.

3. To more concretely see how the LMC arises in the Hamiltonian formulation of the closed Nambu-Goto string, see e.g. eq. (22) in my Phys.SE answer here.