Abstract :
After compactification of the ten-dimensional heterotic superstring theory L onto a six-dimensional torus gμν, the resultant four-dimensional Lagrangian density L is known to be invariant under the duality symmetry (modular transformation) Br ↔ Br−1, where Br is the radius squared of the internal space ds2 in units of the Regge slope α′, provided that the axion Bi is constant. By taking the ten-dimensional terms R4 into account and allowing gμν to be curved, shown here that L acquires a dependence upon Br and hence is no longer invariant under the duality transformation. In particular, the Newton gravitational constant is given by the formula GN = G0[1 + 15ζ(3)χ16λBr3]−1, where ζ(3) ≈ 1.2 is the Riemann zeta function and χ is the Euler characteristic of the internal space, whose volume form is H∫d6ug λV6α′3, V6 being the unit six-volume for the topology S6. For the only known, multiply-connected Calabi-Yau manifold that gives rise to three generations, χ = −6, and the subsequent conditions upon Br and λ ensuring that GN is positive are discussed, together with the possible effect of the (unknown) higher-order terms Rn, n ≥ 5.
