RENORMALIZATION EFFECTS IN THE SUPERSTRING MODULI FROM THE HIGHER-DERIVATIVE TERMS

The dimensional reduction of the effective ten-action of the heterotic superstring theory to the physical four-action S[g_ij] results in the appearance of three moduli B_(a), whose real parts , set equal for simplicity, define the radius and shape of the compact internal six-space , in addition to the dilaton , the ten-interval being . These scalar fields are massless at tree level and can be put into canonical form with coefficients for the positive kinetic-energy terms, when higher-derivative terms are ignored, as found by Witten, so that . Previously, we have shown that σ_A and σ_B acquire a potential from the higher-derivative terms and , which becomes large close to the Planck era. Here, we discuss the renormalization of the kinetic-energy terms due to which, after diagonalization, results in a mixing of ∇σ_B with ∇σ_A, while the remaining coefficient of (∇σ_B)² vanishes at t_c ≈ t_P/12 in a radiation-dominated Universe, corresponding to a temperature T_c ≈ 5 × 10¹⁷GeV, where the four-theory is still classical. At earlier times, the energy is unbounded from below, signalling that the four-theory has become unphysical, and that the string must still be in its uncompactified form with one dilaton ϕ, whose canonical kinetic energy is positive in the Einstein metric. This mechanism depends upon the equation of state of the source for the Friedmann expansion assumed, and is only effective for values of the adiabatic index in the range 1.14 < γ < 2.63, which thus includes radiation (γ = 4/3) and the Zel'dovich equation of state (γ = 2).