Special polynomial rings, quasi modular forms and duality of topological strings

Murad Alim Emanuel Scheidegger Shing-Tung Yau Jie Zhou

Mathematical Physics mathscidoc:1912.43537

Advances in Theoretical and Mathematical Physics, 18, (2), 401-467, 2014
We study the differential polynomial rings which are 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 show 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. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their
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@inproceedings{murad2014special,
  title={Special polynomial rings, quasi modular forms and duality of topological strings},
  author={Murad Alim, Emanuel Scheidegger, Shing-Tung Yau, and Jie Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203910537240101},
  booktitle={Advances in Theoretical and Mathematical Physics},
  volume={18},
  number={2},
  pages={401-467},
  year={2014},
}
Murad Alim, Emanuel Scheidegger, Shing-Tung Yau, and Jie Zhou. Special polynomial rings, quasi modular forms and duality of topological strings. 2014. Vol. 18. In Advances in Theoretical and Mathematical Physics. pp.401-467. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203910537240101.
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