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string coupling constant

Contents

Contents

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

and the string theory scales

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

P=g st 1/3 s,AAAR 10=g st s \ell_P \;=\; g_{st}^{1/3} \ell_s \,, \phantom{AAA} R_{10} \;=\; g_{st} \ell_s

equivalently

s=R 10/g st,AAA P=g st 2/3R 10 \ell_s \;=\; R_{10}/ g_{st} \,, \phantom{AAA} \ell_P \;=\; g_{st}^{-2/3} R_{10}

equivalently

g st=(R 10/ P) 3/2,AAA s= P(R 10/ P) 1/2. 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 3C_3 gives a contribution

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

becomes

  1. if the cycle wraps, C 3=C 2S 10 1C_3 = C_2 \cup S^1_{10}, a worldsheet instanton

    exp(vol(C 3) P 3)=exp(R 10vol(C 2)g st s 3)=exp(vol(C 2) s 2) \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(vol(C 3) P 3)=exp(vol(C 3)/ s 3g st) \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).

References

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

Review includes

Created on May 18, 2019 at 14:20:19. See the history of this page for a list of all contributions to it.