Contents

# 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 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 $\Sigma$.

## 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.

fundamental scales (fundamental 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: