Polynomial Rings and Topological Strings

An overview is given of the construction of a differential polynomial ring of functions on the moduli space of Calabi-Yau threefolds. These rings coincide with the rings of quasi modular forms for geometries with duality groups for which these are known. They provide a generalization thereof otherwise. Higher genus topological string amplitudes can be expressed in terms of the generators of this ring giving them a global description in the moduli space. An action of a duality exchanging large volume and conifold loci in moduli space is discussed. The connection to quasi modular forms is illustrated by the local P² geometry and its mirror, the generalization is extended to several compact geometries with one-dimensional moduli spaces.