The couplings in the effective spacetime field theory, in string theory, may be calculated as the correlation functions using the supersymmetric vacua of the underlying world-sheet field theory. The latter are often most easily calculated in Landau-Ginzburg orbifolds, but may also be obtained. (to a varying degree of accuracy) in other limits of the general supersymmetric sigma model generalization.
Now, supersymmetric vacua form (‘chiral’) rings. When the underlying world-sheet field theory has an interpretation as a sigma model with a Calabi-Yau target space, these rings correspond to certain cohomology rings. In any case, however, these rings determine the moduli space geometry and parameter space dynamics. Furthermore, since the supersymmetric vacua are solutions of certain coupled systems of partial differential equations, the features of this system provide additional information about the moduli space geometry and parameter space dynamics. In particular, the integrability equation of the coupled system of PDEs must contain the so-called Picard-Fuchs equation as a factor.
© Tristan Hübsch, 2008