THE VACUUM STATE IN THE HETEROTIC SUPERSTRING THEORY

摘要

The gravitational vacuum state of the heterotic superstring theory is derived by substituting the maximally symmetric D-space , where is the cosmological constant, into the classical field equations obtained from the effective ten-Lagrangian including quartic higher-derivative terms, . If the theory is reduced to the physical dimensionality D = 4, as required by supersymmetry and phenomenology, the ground state, due to and , is anti-de Sitter space with , where is the inverse gauge coupling and κ2 ≡ 8πGN is the gravitational coupling, GN being the Newton constant. The term , derived from the Euler-number density , is a total divergence and the quadratic term , derived from , vanishes identically, while the quadratic anomaly , which alone would give rise to a positive Λ(anom), is ignorable for the reduced heterotic string, containing nv = 488 vector fields, because Λ(anom) ? -Λ unless nv ? 7,000. For hypothetical reduction to the higher dimensonalities D = 5, 9, 10, has the effect of augmenting the Boulware–Deser, anti-de Sitter space vacuum due to , which becomes exact when D = 8, for which vanishes identically, but leads to a de Sitter space for D = 6, 7 thus justifying the Ricci-flat vacuum state for the six-dimensional internal space. For simplicity, we assume compactification onto a toroidal internal space when D ≥ 5, so that all contributions of the form vanish. The remaining terms and are then almost comparable in effect, bringing into question the convergence of the Lagrangian power series in the Einstein space, and consequently the validity of the results obtained. Without knowledge of the yet higher-order terms , n ≥ 6, however, no further analysis is possible. As they stand the results constitute a realization of the limiting-curvature hypothesis of Frolov et al., and are also discussed from the viewpoint of causality. Finally, the dimensional parameter a, introduced by Volkov and Akulov to define the non-linear global supersymmetry transformations, gives rise to a negative cosmological constant -κ2/a, which can therefore be identified with Λ (which has the functional dependence found by Freedman and Das for extended supergravity). This leads to the estimate Br ~ 2 for the dimensionless radius-squared of the internal space, implying a small radius of compactification, in agreement with previous estimates obtained via supersymmetry, and provides a realization of the super-Higgs effect.