Closed string

A closed string has a cylindrical worldsheet without boundary. Periodicity permits independent left- and right-moving oscillator sectors and, in a compact target direction, winding modes.

Table of contents
- Large-level degeneracy of a closed bosonic string Closed string
- Closed-string mode expansion Closed string

Large-level degeneracy of a closed bosonic string ( $g_{N} \sim \frac{1}{2} (N - 1)^{- 27/2} e^{8 π N - 1}$ )

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The left and right transverse oscillator sectors each have generating function

\prod_{m \geq 1} (1 - q^{m})^{- 24}

. Its coefficient satisfies

d_{N} \sim 2^{- 1/2} (N - 1)^{- 27/4} e^{4 π N - 1}

. Closed-string level matching requires equal levels but leaves their polarizations independent, so

g_{N} = d_{N}^{2}

. With

α^{'} M^{2} = 4 (N - 1)

, the exponential is

e^{β_{H} M}

with

β_{H} = 4 π α^{'}

. The smooth rest-mass density is proportional to

M^{- 26} e^{β_{H} M}

, since

d N / d M = α^{'} M /2

. This distinguishes degeneracy per level from density per unit mass. The polynomial prefactor matters at the boundary of canonical convergence.

Closed-string mode expansion

 0  0

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