nLab string coupling constant

Idea

The coupling constant in perturbative string theory.

Properties

Behaviour under M/IIA duality

Under duality between type IIA string theory and M-theory the string coupling constant relates to the radius of the 11d circle-fiber:

Consider the M-theory scales

• $\ell_P$, the Planck length in 11-dimensions;

• $R_{10}$ the length (circumference) of the $S^1_{10}$ circle fiber for KK-compactification to 10 dimensions

and the string theory scales

• $\ell_s$, the string length scale;

• $g_{s}$, the string coupling constant of perturbative string theory.

Then under the duality between M-theory and type IIA string theory these scales are related as follows:

$$\ell_P \;=\; g_{st}^{1/3} \ell_s \,, \phantom{AAA} R_{10} \;=\; g_{st} \ell_s$$

equivalently

$$\ell_s \;=\; R_{10}/ g_{st} \,, \phantom{AAA} \ell_P \;=\; g_{st}^{-2/3} R_{10}$$

equivalently

$$g_{st} \;=\; (R_{10}/\ell_P)^{3/2} \,, \phantom{AAA} \ell_s \;=\; \ell_P (R_{10}/\ell_P)^{-1/2} \,.$$

Hence a membrane instanton, which on a 3-cycle $C_3$ gives a contribution

$$\exp\left( - \frac{ vol(C_3) }{ \ell^3_P } \right)$$

becomes

1. if the cycle wraps, $C_3 = C_2 \cup S^1_{10}$, a worldsheet instanton

   $$\exp\left( - \frac{ vol(C_3) }{ \ell_P^3 } \right) \;=\; \exp\left( - \frac{ R_{10} vol(C_2) }{ g_{st} \ell_s^3 } \right) \;=\; \exp\left( - \frac{ vol(C_2) }{ \ell_s^2 } \right)$$

2. the cycle does not wrap, a spacetime instanton contribution, specifically a D2-brane instanton?

   $$\exp\left( - \frac{ vol(C_3) }{ \ell_P^3 } \right) \;=\; \exp\left( - \frac{ vol(C_3)/\ell_s^3 }{ g_{st} } \right)$$

(This unification of the two different non-perturbative effects in perturbative string theory (worldsheet instantons and spacetime instantons), to a single type of effect (membrane instanton) in M-theory was maybe first made explicit in Becker-Becker-Strominger 95. Brief review includes Marino 15, sections 1.2 and 1.3).

Related concepts

• axio-dilaton

• Yukawa coupling

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

References

The identification of non-perturbative effects in string theory with brane contributions is due to

• Katrin Becker, Melanie Becker, Andrew Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456, 130 (1995) (hep-th/9507158)

Review includes

• Marcos Mariño, Non-perturbative effects in string theory and AdS/CFT, Spring School on Superstring Theory and Related Topics 2015 (pdf, pdf, recording)