nLab string length scale

1. Idea

In perturbative string theory, the string length scale $\ell_s$ is a unit of length which sets the scale for the extension of strings described by sigma-model worldsheet field theories. Specifically, the Nambu-Goto action for the string is nothing but the relativistic volume-functional, and the string length determines in which units the volume is measured.

The square of the string length is known as the Regge slope (as in: slope of Regge trajectories) and traditionally denoted by $\alpha^\prime$:

$$\alpha^\prime \;=\; \ell_s^2 \,.$$

The inverse of the string length squared/Regge slope is called the string tension, traditionally denoted

$$T_s \;=\; \frac{1}{2\pi \alpha^\prime} = \frac{1}{2 \pi \ell_s^2} \,.$$

This way the Nambu-Goto action for the string with proper units attached is

$$L_{NG} \;=\; T vol_{\Sigma} \,,$$

where $vol_{\Sigma}$ is the (induced) volume form on the worldsheet $\Sigma$.

2. Properties

Relation to Planck length and string coupling

For discussion of relation to Planck length and string coupling constant see at non-perturbative effect the section Worldsheet and brane instantons

Vanishing tension limit

In the limit $T_s \to 0$, $\ell_s \to \infty$ of vanishing string tension, string field theory is supposed to become Vasiliev’s higher spin gauge theory. See there for more.

3. Related concepts

• Regge slope

fundamental scales (fundamental/natural physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

  • Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

  • Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

  • Compton wavelength$\,$ $\lambda_m = \hbar / m c$

  • Schwarzschild radius$\,$ $2 m G / c^2$

  • depending also on a given charge $e$

    • Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$

• GUT scale

• string scale

  • string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

  • string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

  • string coupling constant$\,$ $g_s = e^\lambda$

4. References

For instance:

• David Tong, around p. 16 in Lectures on String Theory (arXiv:0908.0333)