= 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, $N=\widetilde N$.
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.
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