The non-invariance of the heterotic superstring four-action under the modular transformation

Abstract

Abstract After compactification of the ten-dimensional heterotic superstring theory L onto a six-dimensional torus g μν , the resultant four-dimensional Lagrangian density L is known to be invariant under the duality symmetry (modular transformation) B r ↔ B r −1 , where B r is the radius squared of the internal space ds 2 in units of the Regge slope α′, provided that the axion B i is constant. By taking the ten-dimensional terms R 4 into account and allowing g μν to be curved, shown here that L acquires a dependence upon B r and hence is no longer invariant under the duality transformation. In particular, the Newton gravitational constant is given by the formula G N = G 0 [ 1 + 15ζ(3) χ 16λB r 3 ] −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∫d 6 u g  λ V 6 α′ 3 , V 6 being the unit six-volume for the topology S 6 . For the only known, multiply-connected Calabi-Yau manifold that gives rise to three generations, χ = −6 , and the subsequent conditions upon B r and λ ensuring that G N is positive are discussed, together with the possible effect of the (unknown) higher-order terms R n , n ≥ 5.