Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1155see the mode-expanded forms of the coordinates and the momenta at τ = 0X j (σ) = G jk y k + 1 π θjk (σ − π 2 )p k+ 2 √ ∑∞ [α ′ G jk cos(nσ)x n,k + 12πα ′ θjk sin(nσ) 1 ]n p n,k ,n=1P i (σ) = 1 π p i + 1π √ α ′∞ ∑n=1cos(nσ)p n,i ,where x j = G jk y k , the coordinates and the momenta for the oscillatormodes arex n,k = i √22 n (a n,k − a † n,k ) = √ i (α n,k − α −n,k ),2n√ np n,k =2 (a n,k + a † n,k ) = √ 1 (α n,k + α −n,k ).2The nonvanishing commutators are given by[x n,k , p m,l ] = iG kl δ n,m , [y k , p l ] = iG kl . (6.252)We should note that the metric appearing in eqs. (6.252) is G ij , insteadof g ij . So it can be seen that if we employ the variables with the loweredspace–time indices y k , p k , x n,k and p n,k , the metric used in the expressionof the overlaps is G ij not g ij .The continuity condition (6.241) is universal for any background, andthe mode expansion of the momenta P i (σ)’s is of the same form as in theNeumann case, thus the continuity conditions for the momenta in terms ofthe modes p n,i are identical with those in the Neumann case. Also, sincep n,i ’s mutually commute, it is natural to find a solution of the continuitycondition, assuming the following form for the overlap vertices:| ˆV N 〉 X 1···N = exp[i4πα θijN ∑r,s=1p (r)n,i Zrs nmp (s)m,j]|V N 〉 X 1···N , (6.253)where | ˆV N 〉 X 1···N and |V N〉 X 1···N are the overlaps in the background correspondingto the world sheet actions (6.242) and (6.238) respectively, theexplicit form of the latter is given in appendix A. Clearly the expression(6.253) satisfies the continuity conditions for the modes of the momenta,and the coefficients Znm rs are determined so that the continuity conditionsfor the coordinates are satisfied [Sugino (2000)].