Contents

# Contents

## 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

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).

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

Review includes