 Closed-string level matching

Closed-string level matching is the constraint that the left-moving and right-moving worldsheet sectors combine into an allowed periodic physical state. In an uncompactified free closed bosonic string, their zero-mode momenta agree and the condition reduces to equality of string oscillator levels, $N=\tilde{N}$.

More generally, the constraint is equality of the full left and right zero-mode Virasoro generators, including momentum and winding contributions. On a compact circle, these contributions can compensate unequal oscillator levels, as in compact-circle closed-string level matching. Thus the two chiral polarization choices remain independent subject to the total constraint, while imposing equal oscillator levels in every winding sector would remove valid states.