Ivancevic_Applied-Diff-Geom

1150 Applied Differential Geometry: A Modern Introductionwhere the structure of the product ∗ in the kinetic and potential termsis encoded to that of the overlap vertices 〈V 2 | and 〈V 3 | respectively (here,subscripts put to vectors in the Fock space label the strings concerning thevertices).As a preparation for giving the explicit form of the overlaps, let usconsider open strings in 26-dimensional space–time with the constant metricG ij in the Neumann boundary condition. The world sheet action is givenbyS W S = 1 ∫ ∫ π4πα ′ dτ dσG ij (∂ τ X i ∂ τ X j − ∂ σ X i ∂ σ X j ) + S gh , (6.238)0where S gh is the action of the bc−ghosts:S gh =i ∫ ∫ πdτ dσ[c + (∂ τ − ∂ σ )b + + c − (∂ τ + ∂ σ )b − ]. (6.239)2π0Under the Neumann boundary condition, the string coordinates have thestandard mode expansions:X j (τ, σ) = x j + 2α ′ τp j + i √ 2α ′ ∑ n≠0also the mode expansions of the ghosts are given by1n αj ne −inτ cos(nσ), (6.240)c ± (τ, σ) = ∑ n∈Zc n e −in(τ±σ) ≡ c(τ, σ) ± iπ b (τ, σ),b ± (τ, σ) = ∑ n∈Zb n e −in(τ±σ) ≡ π c (τ, σ) ∓ ib(τ, σ).As a result of the quantization, the modes obey the commutation relatons:[x i , p j ] = iG ij , [α i n, α j m] = nG ij δ n+m,0 , {b n , c m } = δ n+m,0 ,while the other therms vanish.The overlap |V N 〉 = |V N 〉 X |V N 〉 gh , (N = 1, 2, · · · ) is the state satisfyingthe continuity conditions for the string coordinates and the ghostsat the N−string vertex of the string field theory. The superscripts X andgh show the contribution of the sectors of the coordinates and the ghostsrespectively. The continuity conditions for the coordinates are(X (r)j (σ)−X (r−1)j (π−σ))|V N 〉 X = 0, (P (r)i (σ)+P (r−1)i (π−σ))|V N 〉 X = 0,(6.241)for 0 ≤ σ ≤ π 2and r = 1, · · · , N. Here P i(σ) is the momentum conjugateto the coordinate X j (σ) at τ = 0, and the superscript (r) labels the string