On the application of the field-redefinition theorem to the heterotic superstring theory

Abstract

The ten-dimensional effective action \(\hat S\) which defines the heterotic superstring theory at low energy is constructed by hypothesis in such a way that the resulting classical equation of motion for the space-time metric \(\hat g_{AB}\) simultaneously implies the vanishing of the beta-function for the N = 1 supersymmetric non-linear sigma-model on the world sheet. At four-loop order it was found by Grisaru and Zanon (see also Freeman et al.) that the effective Lagrangian \(\hat L\) so constructed differs in the numerical coefficient of the term \(\hat L_3 = \hat R\hat R_{ABCD} \left( {\hat R_{EF}^{A D} \hat R^{CEFB} + \tfrac{1} {4}\hat R_{ EF}^{AB} \hat R^{CDEF} } \right)\) from that obtained directly from the four-point gravitational scattering amplitude. The two expressions can be related via a metric field redefinition \(\hat g_{AB} \to \hat g'_{AB} = \left( {1 + \delta '\hat L_3 /\hat R} \right)\hat g_{AB}\), activation of which, however, results in the appearance of ghosts at higher gravitational order \(\mathcal{R}^n\), n > 4, as shown by Lawrence. Here, we prove, after reduction of \(\hat S\) to the physical dimensionality D = 4, that the corresponding field redefinition yields the identity g′ij = gij, signified by L3/R = 0, in a Friedmann space-time generated by a perfect-fluid source characterized by adiabatic index γ ≡ 1 + p/ρ, where p is the pressure and ρ is the energy density, if, and only if, κ6ρ3γ2(γ − 1) = 0. That is, the theory remains free of ghosts in Minkowski space ρ = 0, in a maximally symmetric space-time γ = 0, or in a dust Universe γ = 1. Further aspects of ghost freedom and dimensional reduction, especially to D = 4, are discussed.