THE SUPERSTRING WORLD SHEET IN CURVED SPACE–TIME

Abstract

The heterotic superstring theory is formulated on the world sheet hab(τ,σ), in a flat space–time background [Formula: see text], by combining a 26-dimensional left-moving sector, consisting only of bosonic fields [Formula: see text], with a 10-dimensional right-moving sector, consisting of bosonic fields [Formula: see text] and Majorana–Weyl fermionic fields ψA(τ - σ), the string coordinates in 10-dimensional space–time being the sum [Formula: see text] and the right-moving sector supersymmetric due to the decoupling of left- and right-moving modes for closed strings. Here, we generalize the background to a curved space–time ĝAB(XC). The equation of motion for XA(τ,σ), given by Short, is then a non-linear modification of the linear wave equation □XA = 0 which yields decoupled left- and right-moving sectors in flat space–time. The linearity of the transverse modes can be maintained, however, if the metric, after reduction to four dimensions, is allowed to depend only upon co-moving time t ∝ X0, although it can be anisotropic, gij = gij(X0). Specializing to the isotropic, Friedmann cosmological space–time ds2 = dt2 - r2(t) dx2, where the radius function of the three-space dx2, assumed flat, is r(t), with d ln r/dt = 2/3γt for a perfect-fluid source whose pressure and energy-density are related by p = (γ - 1)ρ, we find that the string coordinates Xα(τ,σ)(α = 1,2,3), multiplied by the overall prefactor r ≡ t2/3γ, are purely oscillatory only in Minkowski space or for the value γ = 2/3. This result is equivalent to requiring that the Nordström energy-density ρ N ≡ (3γ - 2)ρ vanish. The case ρ = 2/3 corresponds to a space–time generated by an ensemble of cosmic strings, including the superstrings themselves in a self-consistent solution containing no other matter, and defines the Milne universe. The string is only world-sheet-supersymmetric if ρ N = 0, that is if ρ = 0 or γ = 2/3.