string length scale in nLab

Contents

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 $\Sigma$ .

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.

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 $\,$ $g_s = e^\lambda$

For instance: