Quantum Gravity

294 STRING THEORY(and beyond) for closed strings. The critical value is, of course, obtained forR ∼ R D ∼ l s , which is just the length of the string, as expected.It is useful for the discussion below to go again to Euclidean space,Equation (9.47) then readsz =e σ2 −iσ 1 , σ 1 = σ,σ 2 =iτ.X µ (z, ¯z) = X µ R (z)+X µ L√ (¯z)√= xµ +˜x µ α′α− i2 2 (˜αµ 0 + ′αµ 0 )σ2 +2 (˜αµ 0 − αµ 0 )σ1 + ...T−→ −X µ R (z)+X µ L(¯z) . (9.60)What happens for the open string? In the limit R → 0, it has no possibilityof winding around the compactified dimension and therefore seems to live onlyin D − 1 dimensions. There are, however, closed strings present in the theoryof open strings, and the open string can in particular have vibrations into the25th dimension. Only the endpoints of the open string are constrained to lie on a(D−1)-dimensional hypersurface. For the vibrational part in the 25th dimension,one can use the expression (9.60) for the closed string. Choosing X 25 ≡ X inthis equation yieldsX = x +˜x√ √α′α+2 2 (˜α 0 − α 0 )σ 1 ′− i2 (˜α 0 + α 0 )σ 2 + ... , (9.61)so that one gets for its dualX D = x +˜x√ √α′α+2 2 (˜α 0 + α 0 )σ 1 ′− i2 (˜α 0 − α 0 )σ 2 + ... , (9.62)that is, X ↔ X D corresponds to −iσ 2 ↔ σ 1 , leading in particular toIntegration yieldsX ′ D ≡ ∂X D∂σ 1↔ i ∂X∂σ 2 ≡ iẊ .X D (π) − X D (0) =∫ πInserting Ẋ following from (9.3) yields0dσ 1 X ′ D =i∫ π0dσ 1 Ẋ.X D (π) − X D (0) = 2πα ′ p = 2πα′ nR =2πnR D ,n∈ Z ,where only vibrational modes were considered. The dual coordinates thus obeya Dirichlet-type condition (an exact Dirichlet condition would arise for n =0;