Special polynomial rings, quasi modular forms and duality of topological strings
The authors define differential polynomial rings using B-model special geometry structure on Calabi-Yau threefolds. Higher genus topological string amplitudes are polynomials in the generators of these rings, and yield generating functions of Gromov-Witten invariants at particular loci. Special choice of generators and of a coordinate \(\tau\) leads to a grading on these rings that has nice properties. Namely, the derivative with respect to \(\tau\) increases the degree by \(2\) and functions corresponding to the amplitudes with \(n\) insertions have degree \(-n\). In a number of non-compact examples, namely \(\mathbb{P}^2\) and local del Pezzo surfaces with singularities of exceptional types, the rings are identified with the rings of the quasi modular forms, and the degree corresponds to the modular weight. The action of the Fricke involution, which exchanges expansions of the quasi modular forms at two different cusps, is shown to correspond to the duality on amplitudes, which exchanges the large complex structure limit with the conifold loci. This is a form of electric-magnetic duality. The construction is then formally extended to the case of the quintic, where it provides a generalization of the ring of quasi modular forms, but without a geometric or physical interpretation of the duality at this time.
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