Open Access
April 2014 Special polynomial rings, quasi modular forms and duality of topological strings
Murad Alim, Emanuel Scheidegger, Shing-Tung Yau, Jie Zhou
Adv. Theor. Math. Phys. 18(2): 401-467 (April 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 mirrors including local $\mathbb{P}^2$ and local del Pezzo geometries with $E_5$, $E_6$, $E_7$ and $E_8$ type singularities. We provide the analogous special polynomial ring for the quintic.


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Murad Alim. Emanuel Scheidegger. Shing-Tung Yau. Jie Zhou. "Special polynomial rings, quasi modular forms and duality of topological strings." Adv. Theor. Math. Phys. 18 (2) 401 - 467, April 2014.


Published: April 2014
First available in Project Euclid: 27 October 2014

zbMATH: 1314.14081
MathSciNet: MR3273318

Rights: Copyright © 2014 International Press of Boston

Vol.18 • No. 2 • April 2014
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