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 = g _ij , signified by L ₃/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, κ ⁶ ρ ³ γ ²(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.