## 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 Links

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.

### Lecture Notes:

Lectures on Mirror Symmetry and Topological String Theory.

Murad Alim

arXiv:1207.0496, Surveys of Modern Mathematics, Volume 6, Open Problems and Surveys of Contemporary Mathematics .