Closed string

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.

The left and right transverse oscillator sectors each have generating function ${\prod }_{m\ge 1}(1-q^m{)}^{-24}$. Its coefficient satisfies $d_N\sim 2^{-1/2}(N-1{)}^{-27/4}{e}^{4\pi \sqrt{N-1}}$. Closed-string level matching requires equal levels but leaves their polarizations independent, so $g_N=d_N^2$. With ${\alpha }^{\prime }{M}^2=4(N-1)$, the exponential is $e^{{\beta }_{H}M}$ with ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. The smooth rest-mass density is proportional to $M^{-26}{e}^{{\beta }_{H}M}$, since $dN/dM={\alpha }^{\prime }M/2$. This distinguishes degeneracy per level from density per unit mass. The polynomial prefactor matters at the boundary of canonical convergence.

A free closed-string embedding coordinate is the sum of center-of-mass zero modes and independent periodic left- and right-moving oscillator expansions.