Explicit formulae for one, two, three and fourloop string amplitudes
Abstract
In the critical dimension d = 26 the ploop scattering amplitude for tachyons with momenta k_{α} in the bosonic string theory can be expressed through a finitefold integral over the space M_{p} of moduli of Riemann surfaces of genus p: ∫Dg_{ab}Dx^{μ} exp(1/2M^{2}∫d^{2}ξ√gg^{ab}∂_{a}x^{μ}∂_{b}x^{μ})_{α}^{Π}∫d^{2}ξ_{α}√g(ξ_{α}) exp[ik_{α}x(ξ_{α})] ~ _{Mp}∫ dν_{p}(det N_{0}/det' ∆_{0})^{13} det' ∆_{1}(_{α}^{Π}∫d^{2}ξ_{α} _{α,β}^{Π} exp[(1/2M^{2}) k_{α}k_{β}G(ξ_{α}, ξ_{β})]). In the r.h.s. of this formula the twodimensional metric is supposed to have the conformal form g_{ab} = ϱ(ξ) δ_{ab} ∆_{j} = ϱ^{j1}∂ϱ^{j}∂ stands for the Laplace operator ∂^{+}∂ acting on jdifferentials f(ξ, ξ) dξ^{j} N_{j} are scalar product matrices of zero modes of the operators i.e. of holomorphic jdifferentials: det N_{j}  det_{(mn)}∫ƒ;_{m}(ξ) dξ^{j} ƒ;_{n}(ξ) dξ^{j}/[ϱ(ξ, ϱ) dξ dξ] j1 = det_{(mn)}∫ϱ^{1j}ƒ;_{m}ƒ;_{n}d^{2}ξ. The functional determinant det' ∆_{0} comes from integration over the fields x^{μ} (twodimensional scalars), while det'. ∆_{1} is the FaddeevPopov determinant arising due to fixation of the conformal gauge for g_{ab} (the parameters of general coordinate transformations are vectors, i.e. j = 1 differentials). Through G_{R} the regularized Green function on the Riemann surface is denoted.
 Publication:

Physics Letters B
 Pub Date:
 January 1987
 DOI:
 10.1016/03702693(87)905636
 Bibcode:
 1987PhLB..184..171M