Abstract
We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree. The denominators of these forms have a simple universal form with the property that the poles of the meromorphic form lie only at torsion points. The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter. This leads to very strong all genus results on these geometries, which are checked against results from curve counting. The structure can be viewed as an indication that an N=2 analog of the reciprocal of the Igusa cusp form exists that might govern the topological string theory on these Calabi-Yau manifolds completely.
Highlights
Introduction and summaryTopological strings on non-compact Calabi-Yau geometries are solvable and have a very interesting structure with a wealth of connections to gauge theories, integrable models, large N-dualities, Chern-Simons theories, supersymmetric localisation and matrix models
We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree
The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter
Summary
Topological strings on non-compact Calabi-Yau geometries are solvable and have a very interesting structure with a wealth of connections to gauge theories, integrable models, large N-dualities, Chern-Simons theories, supersymmetric localisation and matrix models. Knowing these actions we can restrict the possible constants in the rational ambiguity fg to roughly one fourth Using this information, the conifold gap and the regularity at the other points in the moduli space, especially the orbifold point we can solve the topological string for all classes to genus 9 on an elliptic fibration over P2, with one section. The conifold gap and the regularity at the other points in the moduli space, especially the orbifold point we can solve the topological string for all classes to genus 9 on an elliptic fibration over P2, with one section This is already the strongest available result for compact multiparameter Calabi-Yau manifolds and needs only very mild results on the vanishing of the BPS numbers. The formulas we find suggest a possible refinement, which we shortly indicate in (5.1.2)
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