THE VACUUM STATE IN THE HETEROTIC SUPERSTRING THEORY

Abstract

The gravitational vacuum state of the heterotic superstring theory is derived by substituting the maximally symmetric D-space [Formula: see text], where [Formula: see text] is the cosmological constant, into the classical field equations obtained from the effective ten-Lagrangian including quartic higher-derivative terms, [Formula: see text]. If the theory is reduced to the physical dimensionality D = 4, as required by supersymmetry and phenomenology, the ground state, due to [Formula: see text] and [Formula: see text], is anti-de Sitter space with [Formula: see text], where [Formula: see text] is the inverse gauge coupling and κ2 ≡ 8πG N is the gravitational coupling, G N being the Newton constant. The term [Formula: see text], derived from the Euler-number density [Formula: see text], is a total divergence and the quadratic term [Formula: see text], derived from [Formula: see text], vanishes identically, while the quadratic anomaly [Formula: see text], which alone would give rise to a positive Λ(anom), is ignorable for the reduced [Formula: see text] heterotic string, containing n v = 488 vector fields, because Λ( anom ) ≳ -Λ unless n v ≳ 7,000. For hypothetical reduction to the higher dimensonalities D = 5, 9, 10, [Formula: see text] has the effect of augmenting the Boulware–Deser, anti-de Sitter space vacuum due to [Formula: see text], which becomes exact when D = 8, for which [Formula: see text] 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 [Formula: see text] vanish. The remaining terms [Formula: see text] and [Formula: see text] are then almost comparable in effect, bringing into question the convergence of the Lagrangian power series [Formula: see text] in the Einstein space, and consequently the validity of the results obtained. Without knowledge of the yet higher-order terms [Formula: see text], 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 [Formula: see text] found by Freedman and Das for extended supergravity). This leads to the estimate B r ~ 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.