= Closed string
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= Closed bosonic string
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A closed string is a loop with no endpoints. Free propagation sweeps out a cylindrical <worldsheet>, with a periodic spatial coordinate; <interactions> and loop amplitudes involve other <worldsheet> topologies rather than only a cylinder.
The free <closed-string mode expansion> has independent left-moving and right-moving <string oscillators>. Their combination determines the physical state: in the massless <bosonic> sector, a tensor product of vector polarizations contains the <graviton>, <Kalb–Ramond field> and <dilaton>. <Closed-string level matching> relates the two sectors. If a target direction is compact, a string may also wrap that direction, giving <momentum and winding modes> alongside quantized <momentum>. Periodicity therefore constrains both the spectrum and the allowed global configurations.
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