...)% latex2html id marker 4951 \setcounter{footnote}{1}\fnsymbol{footnote}
Actually, W is a section of a line bundle [33]. 
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..., % latex2html id marker 4975 \setcounter{footnote}{2}\fnsymbol{footnote}
From this section on, we will depart from the conventions of the previous sections about the definitions of $S,T,U$, the only change is to make $M\rightarrow iM$ for $M=S,T,U$ then the axionic component is now the imaginary component of the complex field and the compactification size is the real component of $T$. The reason for this change is to have consistency with the standard supergravity conventions. In particular, the $SL(2,{\bf Z})_T$ duality is now the transfoamtion $T\rightarrow (aT-ib)/(icT+d)$. 
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... by% latex2html id marker 5096 \setcounter{footnote}{3}\fnsymbol{footnote}
The physical Yukawa couplings depend also on the Kähler potential, see for instance [45]. 
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Both of the last two statements have been recently modified at the non-perturbative level, by the studies of strong-weak coupling duality symmetries (for a recent review see [104] ). In particular there is some evidence for the existence of an 11D theory from which all the different strings could emerge. There is also evidence for the appearance of nonperturbative gauge groups that can raise the rank beyond $r=22$ [55]. 
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...$M_p$% latex2html id marker 5172 \setcounter{footnote}{5}\fnsymbol{footnote}
This result will be drastically changed in the brane-world scenario later where the string scale can be as small as 1 TeV 
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This last claim depends on some CFT assumptions [58] or that the breaking of supersymetry comes from an $F$-term [57]. There could be counterexamples evading the assumptions. 
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... scale.% latex2html id marker 5199 \setcounter{footnote}{7}\fnsymbol{footnote}
For type I strings the gravitational and gauge couplings are independent, so we have the freedom to adjust the unification scale with experiment as in usual GUTs 
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... have% latex2html id marker 5231 \setcounter{footnote}{8}\fnsymbol{footnote}
The classical field $U$ defined here has no relation with the moduli fields $U$ of the previous sections. 
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... 3)% latex2html id marker 5276 \setcounter{footnote}{9}\fnsymbol{footnote}
Notice that the scalar potential blows up at large radius which is weak sigma-model coupling. This is anti-intuitive, since we would have expected the potential to vanish at weak coupling. A way to understand this is to realize that $1/g_4^2=R^6/g_{10}^2$ that means that for large $R$ and fixed $g_4$, the original 10D string coupling becomes large, so the potential is blowing-up at strong string coupling from the 10D point of view (we thank J. Polchinski and S.-J. Rey for explaining this point). 
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