**Polynomial Rings and Topological Strings**

In these projects, we developed a differential polynomial ring structure which is defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We showed that these rings are isomorphic to the rings of quasi-modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof.

#### Further Literature

- Special Polynomial Rings, Quasi Modular Forms and Duality of Topological Strings.

Murad Alim, Emanuel Scheidegger, Shing-Tung Yau, Jie Zhou

arXiv:1306.0002. - Topological Strings on Elliptic Fibrations.

Murad Alim, Emanuel Scheidegger

arXiv:1205.1784. - Global Properties of Topological String Amplitudes and Orbifold Invariants.

Murad Alim, Jean Dominique Länge, Peter Mayr

arXiv:809.4253. JHEP 1003 (2010) 113. - Polynomial Structure of the (Open) Topological String Partition Function.

Murad Alim, Jean Dominique Länge

arXiv:0708.2886. JHEP 0710 (2007) 045. - Talk on this topic at String Math 2013.