Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1153Note that the indices of p l and α l n were lowered by the metric g ij not G ij .Recall the definition of the propagator〈X i (z, ¯z)X j (z ′ , ¯z ′ )〉 ≡ R(X i (z, ¯z)X j (z ′ , ¯z ′ )) − N(X i (z, ¯z)X j (z ′ , ¯z ′ )),(6.249)where R and N stand for the radial ordering and the normal orderingrespectively. We take a prescription for the normal ordering which pushesp i to the right and ˜x j to the left with respect to the zero–modes p i and ˜x j .It corresponds to considering the vacuum satisfyingp j |0〉 = α n,j |0〉 = 0 (n > 0), 〈0|α n,j = 0 (n < 0), (6.250)which is the standard prescription for calculating the propagator of themassless scalar field in 2D conformal field theory from the operator formalism.Making use of (6.249), (6.250) and techniques of the contour integration,it is easy to get the commutators[α n,i , α m,j ] = nδ n+m,0 G ij , [˜x i , p j ] = iδ i j,where the first equation holds for all integers with α 0,i ≡ √ 2α ′ p i . Theconstant D ij is written as α ′ D ij = −〈0|˜x i˜x j |0〉. Let us fix D ij as α ′ D ij =− i 2 θij , which is the convention taken in [Seiberg and Witten (1999)]. Thenthe coordinates ˜x i become noncommutative:[˜x i , ˜x j ] = iθ ij ,but the center of mass coordinates x i ≡ ˜x i + 1 2 θij p j can be seen to commuteeach other.Now we have the mode–expanded form of the string coordinates andthe commutation relations between the modes, which areX j (τ, σ) = x j + 2α ′ (G jk τ + 12πα ′ θjk (σ − π 2 ) )p k+ i √ 2α ∑ []′ 1n e−inτ G jk 1cos(nσ) − i2πα ′ θjk sin(nσ) α n,k ,n≠0[α n,i , α m,j ] = nδ n+m,0 G ij , [x i , p j ] = iδ i j,with all the other commutators vanishing.Also, due to the formula∞∑n=12n sin(n(σ + σ′ )) ={ π − σ − σ ′ , (σ + σ ′ ≠ 0, 2π)0, (σ + σ ′ = 0, 2π),